Namespace AlgebraicGeometry 3,322 theorems
— 692 · AdmissibleAlgebra 6 · AffineLimit 3 · ChowDatum 1 · ChowDatumProj 1 · DescentAction 3 · DescentCharacter 13 · Etale 6 · FGSubalgebra 2 · Flat 6 · FormallyUnramified 3 · FramedPolarisedAbelianScheme 49 · GeometricallyConnected 3 · GeometricallyIntegral 2 · GeometricallyIrreducible 2 · GeometricallyReduced 1 · GradedOAlgebra 26 · GrpObj 1 · HilbertFunctor 28 · IdealSheafData 4 · IsAffineHom 2 · IsAffineOpen 9 · IsClosedImmersion 21 · IsFinite 6 · IsIntegral 1 · IsOpenImmersion 3 · IsProper 2 · IsPullback 3 · IsSeparated 6 · IsZariskiLocalAtTarget 1 · LocallyOfFinitePresentation 2 · LocallyOfFiniteType 2 · LocallyQuasiFinite 6 · OModulePresheaf 277 · Polarisation 242 · PolarisedAbelianScheme 129 · Proj 4 · ProjSpace 57 · RelEffCartierDiv 65 · RelPicard 352 · RelTangentPoints 4 · RiemannForm 63 · Scheme 936 · SchemeHomOver 8 · SmallExtension 66 · Smooth 37 · SmoothOfRelativeDimension 18 · SmoothProperCurve 55 · Spec 3 · SplitTorus 13 · SubalgebraStages 1 · SymmRoot 2 · ThetaLevel 12 · TowerQuotientDatum 13 · TwoGluedCurves 25 · TwoGluedProjectiveLines 19 · UniversallyInjective 1 · ValuativeCommSq 1 · tilde 3
directly in AlgebraicGeometry 692
- Finite commutative group scheme over Spec R comes from a Hopf algebra
AlgebraicGeometry.exists_hopfAlgebra_of_grpObj_over_spec0 below · cited by 4 · depth 9 - Unique closed-immersion section is equivariant for intertwined automorphisms
AlgebraicGeometry.comp_hom_eq_hom_comp_of_unique_isClosedImmersion_section0 below · cited by 1 · depth 13 - Closed subsets determined by their κ-rational points
AlgebraicGeometry.eq_of_isClosed_of_forall_rationalPoint_mem_iff2 below · cited by 4 · depth 13 - Finite flat commutative group schemes over an affine base
AlgebraicGeometry.exists_hopfAlgebra_flat_of_grpObj_over_spec0 below · cited by 2 · depth 13 - Fibrewise criterion for smoothness of relative dimension one
AlgebraicGeometry.exists_mem_and_smoothOfRelativeDimension_one_of_smoothOfRelativeDimension_pullback_snd0 below · cited by 14 · depth 13 - Finite surjections onto normal schemes are locally free in codimension ≤ 1
AlgebraicGeometry.exists_opens_flat_morphismRestrict_and_finrank_eq_and_mem_of_ringKrullDim_le_one_of_isFinite1 below · cited by 5 · depth 13 - Relative Jacobian from finite-map chart data over a DVR
AlgebraicGeometry.exists_relJacobian_of_smoothOfRelativeDimension_one_of_finiteMapData689 below · cited by 4 · depth 13 - Maximal smooth open of a morphism, stable under automorphisms over S
AlgebraicGeometry.exists_smooth_maximal_and_image_eq_of_iso_over0 below · cited by 1 · depth 13 - Vanishing of H¹_{fppf}(μₚ) for odd primes
AlgebraicGeometry.fppf_natCard_H1_muP_eq_one_of_odd_of_pic_trivial0 below · cited by 2 · depth 13 - Smooth with geometrically connected fibres implies geometrically integral
AlgebraicGeometry.geometricallyIntegral_of_smooth_of_geometricallyConnected7 below · cited by 7 · depth 13 - Smooth over normal is normal, on affine sections
AlgebraicGeometry.isIntegrallyClosed_sections_of_smooth_of_forall_isIntegrallyClosed_sections4 below · cited by 5 · depth 13 - Finite type over locally Noetherian base gives finite presentation
AlgebraicGeometry.locallyOfFinitePresentation_of_comp_eq_of_isLocallyNoetherian0 below · cited by 6 · depth 13 - Fibral criterion for local quasi-finiteness of an S-endomorphism
AlgebraicGeometry.locallyQuasiFinite_of_forall_locallyQuasiFinite_schemeFibreEndo0 below · cited by 7 · depth 13 - Small- and big-fppf H⁰ and H¹ over Specℤ agree
AlgebraicGeometry.natCard_fppfCohomology_eq_natCard_fppfH_of_iso_restriction1 below · cited by 3 · depth 13 - Triviality of H¹_{fppf}(G_m) over Specℤ
AlgebraicGeometry.natCard_fppfH1_Gm_specZ_eq_one14 below · cited by 3 · depth 13 - Constant relative dimension of a smooth morphism on an irreducible source
AlgebraicGeometry.smoothOfRelativeDimension_of_irreducibleSpace0 below · cited by 20 · depth 13 - Vanishing of H¹_{fppf} of the constant sheaf ℤ/p
AlgebraicGeometry.subsingleton_fppfH1_constantZMod_specZ_of_prime19 below · cited by 4 · depth 13 - Universally closed R-schemes factor through the open part with nonempty special fibre
AlgebraicGeometry.existsUnique_comp_eq_of_universallyClosed_of_closedPoint_notMem_range0 below · cited by 1 · depth 14 - Global sections commute with affine base change (affine X)
AlgebraicGeometry.exists_algEquiv_globalSections_pullback_spec_tensorProduct0 below · cited by 7 · depth 14 - Extending a morphism to a proper scheme over a regular curve
AlgebraicGeometry.exists_comp_eq_of_isOpenImmersion_of_isProper_of_isDiscreteValuationRing_stalk0 below · cited by 3 · depth 14 - A constant ℤ/q layer receiving an H¹_{fppf}-injective map
AlgebraicGeometry.exists_hom_restriction_constantZMod_fppfCohomologyMap_injective_of_sectionsEquiv_of_ne_two20 below · cited by 2 · depth 14 - Kernel on fppf H¹ of a μ_q-comparison map over Spec ℤ
AlgebraicGeometry.exists_hom_restriction_muP_fppfCohomologyMap_ker_natCard_eq_pow_of_sectionsEquiv_of_ne_two51 below · cited by 2 · depth 14 - Affine opens through finite sets descend to open subschemes
AlgebraicGeometry.exists_isAffineOpen_opens_le_preimage_forall_mem_of_forall_finset1 below · cited by 4 · depth 14 - Finite part over a henselian local ring: open, closed, empty complementary special fibre
AlgebraicGeometry.exists_isFinite_isOpenImmersion_isClosed_cover_isEmpty_pullback_of_locallyQuasiFinite_of_henselianLocalRing4 below · cited by 2 · depth 14 - Closed points are k-rational over an algebraically closed field
AlgebraicGeometry.exists_over_hom_base_closedPoint_eq_of_isClosed_singleton0 below · cited by 21 · depth 14 - Rational enumeration of the crossing points of two closed subschemes
AlgebraicGeometry.exists_rationalPoint_enumeration_of_natCard_pullback_eq0 below · cited by 12 · depth 14 - Splitting of fppf extensions of underlineℤ with Amitsur-trivial kernel
AlgebraicGeometry.exists_section_of_fppfAmitsurTrivial5 below · cited by 2 · depth 14 - Two-component degenerate fibre persists under algebraically closed base extension
AlgebraicGeometry.exists_twoGluedSmoothCurveDegeneration_of_factor_of_isAlgClosed4 below · cited by 1 · depth 14 - Finite morphisms with pointwise free direct image are flat
AlgebraicGeometry.flat_and_locallyOfFinitePresentation_of_isFinite_of_forall_free_localizedModule0 below · cited by 5 · depth 14 - Every Ext¹ class of fppf sheaves comes from an extension
AlgebraicGeometry.fppf_extClass_surjective0 below · cited by 2 · depth 14 - Integral finite-type schemes over an algebraically closed field are geometrically integral
AlgebraicGeometry.geometricallyIntegral_of_isAlgClosed0 below · cited by 45 · depth 14 - Quasi-finite separated schemes over a one-dimensional base are affine
AlgebraicGeometry.isAffine_of_locallyQuasiFinite_of_isSeparated_of_ringKrullDim_le_one1 below · cited by 2 · depth 14 - Sections over an affine open stay a domain after base change to a field
AlgebraicGeometry.isDomain_tensorProduct_sections_of_geometricallyIntegral0 below · cited by 7 · depth 14 - Connected locally Noetherian schemes with domain stalks are integral
AlgebraicGeometry.isIntegral_of_isLocallyNoetherian_of_connectedSpace_of_forall_isDomain_stalk0 below · cited by 6 · depth 14 - Nonempty connected scheme smooth over a field is integral
AlgebraicGeometry.isIntegral_of_smooth_of_preconnectedSpace7 below · cited by 35 · depth 14 - Integrality descends from K̄ to K
AlgebraicGeometry.isIntegral_pullback_of_isIntegral_pullback_algebraicClosure0 below · cited by 5 · depth 14 - Stalks over the generic point equal stalks of the generic fibre
AlgebraicGeometry.isIso_stalkMap_pullback_fst_and_ringKrullDim_stalk_le_of_isFractionRing0 below · cited by 5 · depth 14 - Points and reducedness under algebraically closed base change
AlgebraicGeometry.isReduced_and_natCard_pullback_eq_of_finite_of_isAlgClosed3 below · cited by 2 · depth 14 - A scheme smooth over a field is reduced
AlgebraicGeometry.isReduced_of_smooth_of_field0 below · cited by 21 · depth 14 - Reducedness of field fibres descends from a perfect subfield
AlgebraicGeometry.isReduced_pullback_of_isReduced_pullback_of_perfectField2 below · cited by 3 · depth 14 - Reduced geometric special fibre gives reduced varpi-quotients on affine charts
AlgebraicGeometry.isReduced_sections_quotient_of_isReduced_pullback0 below · cited by 4 · depth 14 - Fppf points sheaf with a single geometric point vanishes
AlgebraicGeometry.isZero_of_sectionsEquiv_algHom_of_subsingleton11 below · cited by 2 · depth 14 - Schemes locally of finite type over a field are Jacobson
AlgebraicGeometry.jacobsonSpace_of_locallyOfFiniteType0 below · cited by 8 · depth 14 - Kernel of the stalk map of a scheme fibre immersion
AlgebraicGeometry.ker_fiberIota_stalkMap_eq_maximalIdeal_map0 below · cited by 10 · depth 14 - Off-crossing points of a reduced fibre lie in the smooth locus
AlgebraicGeometry.mem_smoothLocus_of_not_mem_range_of_isClosedImmersion4 below · cited by 6 · depth 14 - Proper affine morphisms have module-finite global sections
AlgebraicGeometry.moduleFinite_globalSections_of_isProper_of_isAffineHom0 below · cited by 1 · depth 14 - Equal cardinality of small- and big-fppf H¹ over Specℤ
AlgebraicGeometry.natCard_fppfCohomology_one_eq_natCard_fppfH_one_of_iso_restriction0 below · cited by 4 · depth 14 - Vanishing of H¹_{fppf}(Specℤ,G_m)
AlgebraicGeometry.subsingleton_fppfH1_Gm_specZ13 below · cited by 2 · depth 14 - Stalks of a smooth relative-dimension-one integral scheme are valuation rings
AlgebraicGeometry.valuationRing_stalk_of_smoothOfRelativeDimension_one0 below · cited by 8 · depth 14 - Proper integral schemes over ̄ k have constant global functions
AlgebraicGeometry.bijective_algebraMap_appTop_of_isProper_of_isIntegral0 below · cited by 6 · depth 15 - Sections through an étale point over a henselian local base
AlgebraicGeometry.bijective_comp_sectionsThrough_of_etale_restrict_of_isIso_residueFieldMap2 below · cited by 2 · depth 15 - Maps out of a flat reduced scheme determined by geometric generic points
AlgebraicGeometry.eq_of_forall_specMap_comp_eq_of_flat_of_isReduced_of_isSeparated0 below · cited by 8 · depth 15 - Dual-number criterion for étaleness over an algebraically closed field
AlgebraicGeometry.etale_of_forall_dualNumber_eq_comp0 below · cited by 2 · depth 15 - Function field of a base change: K(X') = Frac(K' ⊗_K K(X))
AlgebraicGeometry.exists_algHom_tensorProduct_functionField_pullback_injective_isFractionRing0 below · cited by 5 · depth 15 - Étale near a point from flat unramified stalk map
AlgebraicGeometry.exists_etale_nhd_of_flat_stalkMap_of_map_maximalIdeal_eq_of_isIso_residueFieldMap0 below · cited by 7 · depth 15 - Finite sets in a two-chart scheme lie in affine opens
AlgebraicGeometry.exists_isAffineOpen_forall_mem_of_finset_of_twoCharts2 below · cited by 3 · depth 15 - Finite sets of points in a scheme affine over Proj
AlgebraicGeometry.exists_isAffineOpen_forall_mem_of_isAffineHom_proj1 below · cited by 4 · depth 15 - Affine open shrinking around finitely many points
AlgebraicGeometry.exists_isAffineOpen_le_inf_forall_mem_of_finset0 below · cited by 2 · depth 15 - Finite part of a quasi-finite scheme over a henselian local ring
AlgebraicGeometry.exists_isFinite_isOpenImmersion_disjoint_cover_of_locallyQuasiFinite_of_henselianLocalRing3 below · cited by 4 · depth 15 - Pointwise fibre criterion for relative dimension one smoothness
AlgebraicGeometry.exists_mem_and_smoothOfRelativeDimension_one_of_smoothOfRelativeDimension_opensRestrict_pullback_snd3 below · cited by 9 · depth 15 - Representability descends along a finite étale faithfully flat extension
AlgebraicGeometry.exists_representableBy_of_representableBy_restrict_finiteEtale8 below · cited by 1 · depth 15 - R-points via a universally closed open part of X
AlgebraicGeometry.exists_section_comp_eq_iff_factors_of_universallyClosed_of_valuationRing0 below · cited by 9 · depth 15 - Morphisms from mathbf G_m to a proper k-scheme extend to A¹
AlgebraicGeometry.exists_toLaurent_comp_eq_of_isProper0 below · cited by 2 · depth 15 - Morphisms agreeing on κ-points into a separated scheme
AlgebraicGeometry.ext_of_forall_comp_eq_of_isAlgClosed0 below · cited by 8 · depth 15 - Maps agreeing on geometric generic points are equal
AlgebraicGeometry.ext_of_forall_geometricPoint_comp_eq_of_flat0 below · cited by 9 · depth 15 - Bounding Ω-points of a finite part by the special fibre's rank
AlgebraicGeometry.finite_and_natCard_le_finrank_tensorProduct_sections_of_isFinite1 below · cited by 4 · depth 15 - Finiteness and bound for sections over a henselian valuation ring
AlgebraicGeometry.finite_and_natCard_sections_le_of_finrank_specialFibre_le9 below · cited by 3 · depth 15 - Global sections of the special fibre compute the finite part
AlgebraicGeometry.finrank_sections_eq_finrank_tensorProduct_of_isPullback_residue_of_isFinite1 below · cited by 3 · depth 15 - Finite surjection onto a normal curve-like base is flat of constant generic rank
AlgebraicGeometry.flat_and_finrank_eq_of_isFinite_of_surjective_of_ringKrullDim_le_one2 below · cited by 4 · depth 15 - Zariski-local triviality gives flatness and surjectivity
AlgebraicGeometry.flat_and_surjective_of_forall_exists_iso_morphismRestrict_eq_pullback_fst0 below · cited by 1 · depth 15 - Fibrewise flatness criterion for an S-endomorphism
AlgebraicGeometry.flat_of_locallyOfFinitePresentation_of_forall_flat_schemeFibreEndo5 below · cited by 3 · depth 15 - Fibrewise flatness yields flat stalk quotients
AlgebraicGeometry.flat_stalkQuotient_of_forall_flat_schemeFibreEndo1 below · cited by 1 · depth 15 - Splitting a map of fppf sheaves onto constant ℤ
AlgebraicGeometry.fppf_exists_section_of_map_eq_unit0 below · cited by 1 · depth 15 - Splitting of fppf extensions of underlineℤ by mathbb G_m
AlgebraicGeometry.fppf_extClass_Gm_eq_zero11 below · cited by 1 · depth 15 - Smooth plus universally bijective sections gives geometrically integral
AlgebraicGeometry.geometricallyIntegral_of_bijective_algebraMap_sections_of_smooth8 below · cited by 8 · depth 15 - Dense opens of one-dimensional Noetherian affine schemes are affine
AlgebraicGeometry.isAffineOpen_of_dense_of_ringKrullDim_le_one0 below · cited by 1 · depth 15 - Reducedness of fibres of a homomorphism pair with split-torus kernel
AlgebraicGeometry.isReduced_pullback_lift_of_forall_iff_exists_torus0 below · cited by 1 · depth 15 - Reducedness of X×_Y E for E étale over κ
AlgebraicGeometry.isReduced_pullback_of_etale_of_forall_isReduced_pullback0 below · cited by 1 · depth 15 - Fppf points sheaf of an odd flat ℤ/q-model over ℤ
AlgebraicGeometry.nonempty_iso_or_exists_shortExact_of_sectionsEquiv_algHom_of_ne_two18 below · cited by 1 · depth 15 - Dichotomy for an odd flat Hopf model of μ_q
AlgebraicGeometry.nonempty_iso_or_natCard_algHom_eq_one_and_exists_shortExact_of_sectionsEquiv_convPow_of_ne_two49 below · cited by 1 · depth 15 - Stalk dimension is invariant under a finite endomorphism
AlgebraicGeometry.ringKrullDim_stalk_eq_of_isFinite_endomorphism4 below · cited by 2 · depth 15 - Locally quasi-finite endomorphisms preserve local dimension
AlgebraicGeometry.ringKrullDim_stalk_eq_of_locallyQuasiFinite_endomorphism3 below · cited by 2 · depth 15 - Five morphism properties are Zariski-local on an affine base
AlgebraicGeometry.smooth_and_isSeparated_and_quasiCompact_and_surjective_and_geometricallyConnected_of_span_eq_top0 below · cited by 1 · depth 15 - Smooth, proper, geometrically connected descend along finite étale base change
AlgebraicGeometry.smooth_isProper_geometricallyConnected_of_finiteEtale_baseChange1 below · cited by 1 · depth 15 - Unramified morphisms are rigid over a local base
AlgebraicGeometry.eq_of_comp_eq_of_residue_comp_eq_of_formallyUnramified0 below · cited by 3 · depth 16 - Function field of a base change along an algebraic extension
AlgebraicGeometry.exists_algEquiv_tensorProduct_functionField_pullback_of_isAlgebraic0 below · cited by 4 · depth 16 - Point supply over p yields test curves through a closed-fibre point
AlgebraicGeometry.exists_closedFibre_testCurves_of_integralPoints_through2 below · cited by 1 · depth 16 - Common A∩ K'-point of compatible K'- and A-points
AlgebraicGeometry.exists_comp_eq_and_comp_eq_of_valuationSubring_comap0 below · cited by 2 · depth 16 - Invariant L-points descend to K-points
AlgebraicGeometry.exists_comp_eq_of_forall_algEquiv_comp_eq0 below · cited by 1 · depth 16 - Rigidity lemma over a field
AlgebraicGeometry.exists_eq_snd_comp_of_comp_eq_const_of_isProper1 below · cited by 5 · depth 16 - Finite enumeration of coincidences with a closed immersion
AlgebraicGeometry.exists_fin_eq_of_isClosedImmersion_of_finite_pullback0 below · cited by 2 · depth 16 - Hull embedding into the Kummer sheaf μ_q over Specℤ
AlgebraicGeometry.exists_hom_injective_range_iff_of_sectionsEquiv_algHom_of_bialgHom_monoidAlgebra1 below · cited by 1 · depth 16 - Finite morphisms to a Proj glue over an idempotent decomposition
AlgebraicGeometry.exists_isFinite_to_proj_of_completeOrthogonalIdempotents0 below · cited by 1 · depth 16 - Two-chart proper P¹ over an arbitrary commutative ring
AlgebraicGeometry.exists_isProper_twoAffineLineCover0 below · cited by 1 · depth 16 - Finite étale descent of a representing scheme, orbit form
AlgebraicGeometry.exists_representableBy_of_representableBy_restrict_finiteEtale_of_forall_orbit8 below · cited by 1 · depth 16 - Sections through closed-fibre points of étale morphisms over strictly henselian rings
AlgebraicGeometry.exists_section_base_closedPoint_eq_of_etale_of_henselianLocalRing1 below · cited by 4 · depth 16 - Étale morphisms over henselian local rings lift residue points to sections
AlgebraicGeometry.exists_section_comp_eq_of_etale_of_henselianLocalRing1 below · cited by 1 · depth 16 - ℤ[ℤ/n] represents the fppf sheaf μₙ
AlgebraicGeometry.exists_sectionsEquiv_algHom_muP_apply_eq_of_bialgEquiv_monoidAlgebra0 below · cited by 2 · depth 16 - Puncturing μ_q at p: cokernel with H⁰ dividing q
AlgebraicGeometry.exists_shortExact_natCard_fppfCohomology_zero_dvd_of_injective_of_range_iff3 below · cited by 1 · depth 16 - Finiteness of H¹_{fppf}(Specℤ,L) for multiplicative-type layers
AlgebraicGeometry.finite_fppfCohomology_one_of_sectionsEquiv_algHom_of_natCard_eq_prime_of_galoisCyclotomic_of_ne_two68 below · cited by 1 · depth 16 - Finiteness of H¹_{fppf}(Specℤ, L) for odd prime order
AlgebraicGeometry.finite_fppfCohomology_one_of_sectionsEquiv_algHom_of_natCard_eq_prime_of_galoisInvariant_of_ne_two42 below · cited by 1 · depth 16 - #H¹_{fppf}(μₚ)=#H⁰_{fppf}(μₚ) when H¹(G_m) is trivial
AlgebraicGeometry.fppf_natCard_H1_muP_eq_natCard_H0_muP_of_pic_trivial0 below · cited by 1 · depth 16 - Finitely many k-points forces finiteness over an algebraically closed field
AlgebraicGeometry.isFinite_of_finite_setOf_exists_section_of_isAlgClosed0 below · cited by 4 · depth 16 - Finite m-torsion when a split torus has finite index
AlgebraicGeometry.isFinite_schemeKerStr_and_finrank_le_of_isOpenImmersion_torus1 below · cited by 1 · depth 16 - Smooth geometrically connected pointed scheme over ̄ k is integral
AlgebraicGeometry.isIntegral_of_smooth_of_geometricallyConnected7 below · cited by 6 · depth 16 - Integrality and generic point of the closed fibre of G_ℤ₍ₚ₎
AlgebraicGeometry.isIntegral_pullback_and_exists_generic_closedFibre_of_isLocalization_atPrime15 below · cited by 1 · depth 16 - Off one component, the other restricts isomorphically and smoothly
AlgebraicGeometry.isIso_morphismRestrict_and_smoothOfRelativeDimension_one_of_coe_eq_compl_range_of_isClosedImmersion0 below · cited by 4 · depth 16 - Closed subgroup of finite index on points is open
AlgebraicGeometry.isOpenImmersion_of_isClosedImmersion_of_isReduced_of_finite_index0 below · cited by 2 · depth 16 - Schemes smooth over a field are reduced
AlgebraicGeometry.isReduced_of_smooth_over_field0 below · cited by 11 · depth 16 - A smooth proper curve over a field is not affine
AlgebraicGeometry.not_isAffine_of_isProper_of_smoothOfRelativeDimension_one2 below · cited by 15 · depth 16 - Naturality of the graph-product ideal sheaf under base change
AlgebraicGeometry.prodKerGraph_comap_mapOnProdOver2 below · cited by 14 · depth 16 - Descent of smoothness, separatedness, quasi-compactness and geometric connectedness
AlgebraicGeometry.smooth_isSeparated_quasiCompact_geometricallyConnected_of_finiteEtale_baseChange1 below · cited by 1 · depth 16 - Global sections remain constants after base change over a field
AlgebraicGeometry.bijective_appTop_pullback_snd_of_bijective_appTop0 below · cited by 3 · depth 17 - Complement of the two-chart locus D(f-s)∪ D(1-sg)
AlgebraicGeometry.compl_basicOpen_sub_sup_basicOpen_one_sub_mul_of_twoCharts0 below · cited by 2 · depth 17 - Density of k-rational points over an algebraically closed field
AlgebraicGeometry.dense_setOf_exists_section_of_isAlgClosed0 below · cited by 8 · depth 17 - Rank-two split Hopf algebra represents the constant sheaf ℤ/2
AlgebraicGeometry.exists_sectionsEquiv_algHom_constantZMod_of_algEquiv_pi_two0 below · cited by 2 · depth 17 - ℤ[ℤ/2] represents the fppf sheaf μ₂
AlgebraicGeometry.exists_sectionsEquiv_algHom_muP_of_bialgEquiv_monoidAlgebra_two0 below · cited by 2 · depth 17 - Generic point of a connected smooth closed fibre over a DVR
AlgebraicGeometry.exists_specializes_closedFibre_of_smooth_of_isPreconnected8 below · cited by 1 · depth 17 - Loci t ≠ s₀, t ≠ s₁ give a two-affine cover
AlgebraicGeometry.exists_twoAffineOpenCover_eq_basicOpen_sub_sup_of_twoCharts_of_isUnit_sub0 below · cited by 1 · depth 17 - Two-affine cover by U and the locus t ≠ s
AlgebraicGeometry.exists_twoAffineOpenCover_eq_left_eq_basicOpen_sub_sup_of_twoCharts0 below · cited by 1 · depth 17 - Universally bijective global sections give geometric connectedness
AlgebraicGeometry.geometricallyConnected_of_bijective_algebraMap_sections0 below · cited by 5 · depth 17 - Smooth scheme over a DVR with connected generic fibre is integral
AlgebraicGeometry.isIntegral_of_smooth_of_isPreconnected_genericFibre4 below · cited by 1 · depth 17 - Étale over reduced locally Noetherian is reduced
AlgebraicGeometry.isReduced_of_etale3 below · cited by 2 · depth 17 - Cancellation of finite presentation along a finite-type morphism
AlgebraicGeometry.locallyOfFinitePresentation_of_comp_of_locallyOfFiniteType0 below · cited by 3 · depth 17 - Counting sections via the special fibre over a henselian valuation ring
AlgebraicGeometry.natCard_sections_eq_finrank_specialFibre_of_flat_of_isReduced10 below · cited by 5 · depth 17 - c_*𝒪=𝒪 detected on affine base changes
AlgebraicGeometry.bijective_appTop_pullback_snd_of_forall_bijective_algebraMap_sections0 below · cited by 1 · depth 18 - Unique extension of morphisms to an affine scheme across codimension ≥ 2
AlgebraicGeometry.existsUnique_extension_to_affine_of_isIntegrallyClosed_stalk5 below · cited by 2 · depth 18 - A non-singleton closed subset has a closed point other than x
AlgebraicGeometry.exists_mem_isClosed_singleton_ne_of_isIrreducible0 below · cited by 1 · depth 18 - Integrality descends from a basic open of a regular section
AlgebraicGeometry.isIntegral_of_mem_nonZeroDivisors_of_isIntegral_basicOpen1 below · cited by 5 · depth 18 - Open irreducible components when all stalks are domains
AlgebraicGeometry.isOpen_irreducibleComponent_of_isDomain_stalk0 below · cited by 9 · depth 18 - Flat morphisms preserve non-zero-divisors on affine opens
AlgebraicGeometry.map_appTop_mem_nonZeroDivisors_of_flat0 below · cited by 11 · depth 18 - Over a DVR, points with stalk dimension at most one lie in V
AlgebraicGeometry.mem_of_ringKrullDim_stalk_le_one_of_isDiscreteValuationRing0 below · cited by 2 · depth 18 - Vanishing of H¹ and h⁰=r+1-g for L(rp)
AlgebraicGeometry.subsingleton_H1_and_finrank_H0_sectionsOf_tensor_invModule_pow_ker_of_isAlgEquivZero247 below · cited by 4 · depth 18 - Vanishing of the trace for a radicial degree-p cover
AlgebraicGeometry.trace_eq_zero_of_finrank_eq_char_of_forall_isClosed_eq0 below · cited by 1 · depth 18 - Lifting K-points along surjective morphisms locally of finite type
AlgebraicGeometry.exists_comp_eq_of_surjective_of_locallyOfFiniteType_of_isAlgClosed1 below · cited by 7 · depth 19 - Surjectivity of sections and injectivity on fppf H¹
AlgebraicGeometry.fppfCohomologyMap_one_injective_of_shortExact_of_subsingleton_over0 below · cited by 1 · depth 19 - Unramified morphism: fibre at a point with trivial residue extension
AlgebraicGeometry.isIso_fiberToSpecResidueField_of_preimage_eq_singleton_of_isIso_residueFieldMap0 below · cited by 2 · depth 19 - Sections of separated unramified morphisms are open and closed immersions
AlgebraicGeometry.isOpenImmersion_and_isClosedImmersion_of_comp_eq_id0 below · cited by 1 · depth 19 - Reducedness descends from a basic open locus of a regular section
AlgebraicGeometry.isReduced_of_mem_nonZeroDivisors_of_isReduced_basicOpen0 below · cited by 1 · depth 19 - Rigidity of unramified separated morphisms over a connected base
AlgebraicGeometry.eq_of_comp_eq_of_formallyUnramified_of_preconnectedSpace0 below · cited by 2 · depth 20 - Stalks over an affine chart: germs of chart functions and constants generate up to units
AlgebraicGeometry.exists_isUnit_mul_eq_mem_closure_of_stalk_pullback1 below · cited by 3 · depth 20 - Sections of flat quasi-finite schemes over henselian valuation rings
AlgebraicGeometry.exists_section_of_flat_of_locallyQuasiFinite_of_henselianLocalRing_of_isAlgClosed7 below · cited by 1 · depth 20 - Counting sections meeting a given open of the special fibre
AlgebraicGeometry.finite_and_natCard_sections_closedPoint_mem_le_of_finrank_opens_le10 below · cited by 1 · depth 20 - Sections of L(rσ) on the n-th neighbourhood of σ
AlgebraicGeometry.finrank_sections_pushforward_thickening_and_injective_unit_app_iff39 below · cited by 1 · depth 20 - Flatness descends along flat surjective finitely presented base change
AlgebraicGeometry.flat_of_isPullback_of_flat_of_surjective0 below · cited by 2 · depth 20 - Finiteness and multiplicativity of sections for fibre products over a field
AlgebraicGeometry.isFinite_pullback_and_finrank_sections_eq_mul1 below · cited by 2 · depth 20 - Finite type and quasi-compactness from a finite cover by open immersions
AlgebraicGeometry.locallyOfFiniteType_and_quasiCompact_of_finite_openCover0 below · cited by 1 · depth 20 - Local-ring points factor through a unique chart off the closed fibre
AlgebraicGeometry.existsUnique_exists_comp_eq_of_isLocalHom_of_forall_ne0 below · cited by 1 · depth 21 - Open immersion surjective on Ω-points is an isomorphism
AlgebraicGeometry.isIso_of_isOpenImmersion_of_forall_exists_comp_eq_of_isAlgClosed0 below · cited by 5 · depth 21 - Rigidity of R-morphisms from a smooth scheme to a separated scheme
AlgebraicGeometry.eq_of_forall_specMap_comp_eq_of_smooth_of_isSeparated3 below · cited by 7 · depth 22 - Flat schemes: factoring through a closed subscheme from the generic fibre
AlgebraicGeometry.existsUnique_comp_eq_of_isClosedImmersion_of_flat_of_genericFibre_comp_eq0 below · cited by 5 · depth 22 - Gluing morphisms out of Spec S along complete orthogonal idempotents
AlgebraicGeometry.existsUnique_specMap_quotient_mk_comp_eq_of_completeOrthogonalIdempotents0 below · cited by 1 · depth 22 - Twist of a base change restricts to a model
AlgebraicGeometry.exists_restrict_twist_of_isPullback_model_of_comp_eq0 below · cited by 1 · depth 22 - Invariant Ω-points factor through the fixed subfield
AlgebraicGeometry.exists_spec_map_comp_eq_of_forall_spec_map_comp_eq_of_forall_mem_iff0 below · cited by 2 · depth 22 - Finiteness of k-points of a finite morphism over a base point
AlgebraicGeometry.finite_schemeHomOver_of_isFinite0 below · cited by 5 · depth 22 - Geometrically connected fibres are connected
AlgebraicGeometry.isConnected_preimage_singleton_of_forall_connectedSpace_pullback_of_isAlgClosed0 below · cited by 7 · depth 22 - Integrality of a geometrically integral smooth proper curve
AlgebraicGeometry.isIntegral_of_smoothOfRelativeDimension_one_of_geometricallyIntegral0 below · cited by 8 · depth 22 - Properness from a proper closed fibre and a section
AlgebraicGeometry.isProper_of_isProper_pullback_snd_of_geometricallyConnected_of_isLocalRing66 below · cited by 2 · depth 22 - Rational point on a smooth proper curve over ̄ k
AlgebraicGeometry.nonempty_schemeHomOver_id_of_isAlgClosed_of_smoothOfRelativeDimension_one0 below · cited by 2 · depth 22 - Reduction to the zero section forces v(g(b)-ε(b))<1
AlgebraicGeometry.valuation_sub_counit_lt_one_of_isClosedImmersion_of_specMap_comp_eq_zeroSection0 below · cited by 2 · depth 22 - Monotone families of irreducible closed sets are eventually constant
AlgebraicGeometry.exists_forall_le_closeds_eq_of_monotone_of_isIrreducible0 below · cited by 1 · depth 23 - Residue field of a base change as a fraction field of k ⊗_R 𝒪_{X,z}
AlgebraicGeometry.exists_ideal_residueField_pullback_algEquiv_fractionRing_tensorProduct_stalk_quotient0 below · cited by 3 · depth 23 - Automorphisms fixing one of two covering closed subschemes restrict
AlgebraicGeometry.exists_iso_hom_comp_eq_comp_hom_of_comp_hom_eq_of_isClosedImmersion_of_forall_mem_range_or0 below · cited by 1 · depth 23 - Proper clopen neighbourhood of a proper closed fibre
AlgebraicGeometry.exists_opens_isClosed_isProper_of_isProper_pullback_snd_of_isAdicComplete63 below · cited by 1 · depth 23 - Flatness of the scheme-theoretic image over a Dedekind base
AlgebraicGeometry.flat_image_comp_of_isDedekindDomain0 below · cited by 1 · depth 23 - Flat over a domain with reduced generic fibre is reduced
AlgebraicGeometry.isReduced_of_flat_of_isReduced_pullback_of_isFractionRing0 below · cited by 5 · depth 23 - Node-ring recognition: mathcal O_{X,x} as a localisation of T
AlgebraicGeometry.mem_localRing_node_iff_exists_mul_eq_of_nodeChart_of_forall_not_dominates59 below · cited by 3 · depth 23 - Sections of a regular proper flat curve over a DVR land in the maximal smooth locus
AlgebraicGeometry.range_subset_of_isRegularLocalRing_of_smoothOfRelativeDimension_maximal5 below · cited by 1 · depth 23 - Weil restriction of an affine finite-type scheme along a finite free extension
AlgebraicGeometry.exists_affine_weilRestriction_forall_existsUnique2 below · cited by 1 · depth 24 - Morphism from a reduced scheme to a closed point factors through its residue field
AlgebraicGeometry.exists_comp_fromSpecResidueField_eq_of_range_subset_singleton0 below · cited by 11 · depth 24 - Clopen subsets of the closed fibre lift to the scheme
AlgebraicGeometry.exists_isClopen_preimage_eq_of_isProper_of_isAdicComplete59 below · cited by 1 · depth 24 - Descent of geometric fibre components to the base field
AlgebraicGeometry.exists_isClosedImmersion_isPullback_of_mem_irreducibleComponents_pair_of_isReduced_pullback4 below · cited by 2 · depth 24 - Chow's lemma in envelope form over a Noetherian base
AlgebraicGeometry.exists_isProper_surjective_isOpenImmersion_comp_eq_of_isSeparated2 below · cited by 2 · depth 24 - Automorphism fixing the generic point restricts to the closed subscheme
AlgebraicGeometry.exists_iso_hom_comp_eq_comp_hom_of_isClosedImmersion_of_genericPoint_eq2 below · cited by 3 · depth 24 - Function field of a base change to a valuation ring
AlgebraicGeometry.exists_ringEquiv_functionField_pullback_of_span0 below · cited by 3 · depth 24 - Primes below the node ideal are the two branch centres
AlgebraicGeometry.forall_mem_iff_not_exists_or_of_isPrime_of_ne_nodeIdeal_of_nodeChart52 below · cited by 1 · depth 24 - Generic point of an integral generic-fibre model meets every nonempty open
AlgebraicGeometry.genericPoint_mem_preimage_comp_pullback_fst_of_injective_algebraMap0 below · cited by 10 · depth 24 - Smooth proper curve: points are closed or generic, not all generic
AlgebraicGeometry.isClosed_singleton_or_isGenericPoint_and_exists_not_isGenericPoint_of_smoothOfRelativeDimension_one4 below · cited by 2 · depth 24 - Base change of a normal Stein model along A₀ → A
AlgebraicGeometry.isIntegral_pullback_and_bijOn_specialFibre_of_stein_of_smoothLocus_of_relDimOne29 below · cited by 4 · depth 24 - Integrality of products over a DVR with smooth factors
AlgebraicGeometry.isIntegral_pullback_and_isIntegral_pullback_fst_comp_of_smooth_of_geometricallyConnected_pullback_snd_specMap10 below · cited by 1 · depth 24 - Normality of the base change of a nodal curve
AlgebraicGeometry.isIntegrallyClosed_stalk_pullback_of_ordinaryDoublePoints_of_isDiscreteValuationRing_of_relDimOne87 below · cited by 3 · depth 24 - Openness of the locus whose whole fibre maps into an open
AlgebraicGeometry.isOpen_setOf_forall_preimage_mem_of_universallyClosed0 below · cited by 4 · depth 24 - Flat surjective base change: surjective, generalising, preserves generic points
AlgebraicGeometry.surjective_and_generalizingMap_pullback_fst_of_flat0 below · cited by 3 · depth 24 - Valuation ring pinned by its trace on K(X₀)
AlgebraicGeometry.toSubring_eq_localRing_of_forall_mem_iff_of_pullback0 below · cited by 3 · depth 24 - Base change to a rank-one valuation ring: special fibre
AlgebraicGeometry.base_genericPoint_eq_and_bijOn_specialFibre_of_iso_pullback_of_residue_surjective_of_isDiscreteValuationRing2 below · cited by 2 · depth 25 - A point of an integral separated scheme is determined by its local ring in K(X)
AlgebraicGeometry.eq_of_range_algebraMap_stalk_eq_of_isSeparated0 below · cited by 3 · depth 25 - Normal relative curve: stalks are localisations of an affine model
AlgebraicGeometry.existsUnique_localRing_eq_localization_of_normal_affineModel_of_relDimOne_of_val_of_gen9 below · cited by 3 · depth 25 - Unique extension of a K-point to a section over a valuation ring
AlgebraicGeometry.existsUnique_section_comp_eq_of_universallyClosed_of_isSeparated0 below · cited by 3 · depth 25 - Chow's lemma, envelope form, for integral schemes
AlgebraicGeometry.exists_isProper_surjective_isOpenImmersion_comp_eq_of_isIntegral1 below · cited by 1 · depth 25 - Normal proper model with prescribed geometric valuations
AlgebraicGeometry.exists_normalProperModel_of_valuationSubrings_of_form_of_isAlgebraic_relDimOne_genComplete_henselian144 below · cited by 3 · depth 25 - Stalks of a base change as localisations of mathcal O_{X,z} ⊗_A k
AlgebraicGeometry.exists_ringEquiv_stalk_pullback_localization_tensorProduct_stalk0 below · cited by 3 · depth 25 - Stalks of a base change as localisations of 𝒪_{X,z}⊗_A k
AlgebraicGeometry.exists_ringEquiv_stalk_pullback_localization_tensorProduct_stalk_of_germ_snd0 below · cited by 12 · depth 25 - Function field under base change along a field homomorphism
AlgebraicGeometry.exists_ringHom_functionField_germ_eq_and_linearIndependent_of_isPullback0 below · cited by 1 · depth 25 - Closed special points lie on positive-dimensional special components
AlgebraicGeometry.exists_specializes_of_closed_of_specialFibre_of_flat0 below · cited by 3 · depth 25 - Stalks at special generic points after base change to A
AlgebraicGeometry.exists_valuationSubring_eq_range_stalk_of_iso_pullback_of_mem_smoothLocus_of_isDiscreteValuationRing_of_specializes11 below · cited by 1 · depth 25 - Global sections of a proper scheme are a finite A-module
AlgebraicGeometry.finite_appTop_of_isProper_of_isNoetherianRing54 below · cited by 7 · depth 25 - Algebraic base change with a clopen point is finite
AlgebraicGeometry.finite_pullback_of_isClopen_singleton_of_isAlgebraic0 below · cited by 2 · depth 25 - Finiteness of n-torsion on a smooth proper group scheme
AlgebraicGeometry.finite_torsion_of_isProper_of_smooth3 below · cited by 1 · depth 25 - Rationals algebraically closed in the function field
AlgebraicGeometry.functionField_mem_range_algebraMap_rat_of_isAlgebraic_of_isIntegral_pullback3 below · cited by 1 · depth 25 - Injective on points implies birational for smooth curves
AlgebraicGeometry.genericPoint_eq_and_isIso_stalkMap_of_injective_points_of_smoothOfRelativeDimension_one4 below · cited by 4 · depth 25 - Irreducibility over one algebraically closed extension gives geometric connectedness
AlgebraicGeometry.geometricallyConnected_of_irreducibleSpace_pullback_of_isAlgClosed3 below · cited by 2 · depth 25 - Integral pullbacks over algebraically closed fields give geometric integrality
AlgebraicGeometry.geometricallyIntegral_of_forall_isAlgClosed_isIntegral_pullback2 below · cited by 4 · depth 25 - Generic fibre of a reduced scheme over a characteristic-zero domain
AlgebraicGeometry.geometricallyReduced_pullback_snd_specMap_of_isReduced_of_charZero2 below · cited by 2 · depth 25 - Irreducibility criterion via bijectivity on k-points
AlgebraicGeometry.irreducibleSpace_of_bijective_sections_of_topologicalKrullDim_le_one3 below · cited by 2 · depth 25 - Finiteness from properness and finite fibres of k-points
AlgebraicGeometry.isFinite_of_isProper_of_finite_setOf_comp_eq1 below · cited by 2 · depth 25 - Integrality of a smooth proper curve over ℤ[1/M] and of its geometric fibres
AlgebraicGeometry.isIntegral_and_isIntegral_pullback_of_smooth_isProper_of_isIntegral_pullback78 below · cited by 3 · depth 25 - Flatness and integral generic fibre imply X integral
AlgebraicGeometry.isIntegral_of_flat_of_isIntegral_pullback_specMap_of_isFractionRing0 below · cited by 3 · depth 25 - Integrality of the base change of a normal Stein model
AlgebraicGeometry.isIntegral_of_iso_pullback_of_stein_of_isIntegrallyClosed_of_smoothLocus_of_isDiscreteValuationRing14 below · cited by 1 · depth 25 - Generic fibre of an integral scheme flat over a domain
AlgebraicGeometry.isIntegral_pullback_specMap_algebraMap_of_isFractionRing_of_flat0 below · cited by 2 · depth 25 - Stalks on the smooth locus over a normal base are normal
AlgebraicGeometry.isIntegrallyClosed_stalk_of_mem_smoothLocus3 below · cited by 1 · depth 25 - Normality of the base-changed stalk above an ordinary double point
AlgebraicGeometry.isIntegrallyClosed_stalk_pullback_of_ringEquiv_adicCompletion_stalk_of_isDiscreteValuationRing57 below · cited by 1 · depth 25 - Equal-rank closed immersion over a base is an isomorphism
AlgebraicGeometry.isIso_of_isClosedImmersion_of_finrank_comp_eq0 below · cited by 22 · depth 25 - Birational K-morphism of smooth curves, injective on points, is an open immersion
AlgebraicGeometry.isOpenImmersion_of_isIso_stalkMap_genericPoint_of_injective_points_of_smoothOfRelativeDimension_one4 below · cited by 4 · depth 25 - Connectedness of the closed fibre of a proper morphism
AlgebraicGeometry.isPreconnected_preimage_closedPoint_of_isProper_of_bijective_appTop55 below · cited by 7 · depth 25 - Properness, flatness and finite presentation under base change
AlgebraicGeometry.isProper_and_flat_and_locallyOfFinitePresentation_of_iso_pullback_specMap0 below · cited by 3 · depth 25 - Flat over an integral base with reduced generic fibre is reduced
AlgebraicGeometry.isReduced_of_flat_of_isReduced_fiber_genericPoint1 below · cited by 1 · depth 25 - Smoothness after base change to a rank-one valuation ring
AlgebraicGeometry.mem_smoothLocus_iff_base_mem_smoothLocus_of_iso_pullback_of_isDiscreteValuationRing3 below · cited by 1 · depth 25 - Smooth locus membership via formal smoothness of a model of the stalk
AlgebraicGeometry.mem_smoothLocus_iff_formallySmooth_of_ringEquiv_stalk0 below · cited by 11 · depth 25 - A point on two components has non-domain stalk
AlgebraicGeometry.not_isDomain_stalk_of_mem_irreducibleComponents_of_ne0 below · cited by 2 · depth 25 - Minimal primes of varpi in a stalk read on an affine chart
AlgebraicGeometry.not_subsingleton_minimalPrimes_span_germ_iff_exists_two_minimalPrimes_le_of_chart0 below · cited by 3 · depth 25 - Relative dimension of a smooth irreducible K-scheme from one rational point
AlgebraicGeometry.smoothOfRelativeDimension_of_finrank_cotangentSpace_eq1 below · cited by 2 · depth 25 - Faithfully flat descent of bijectivity on global sections
AlgebraicGeometry.bijective_appTop_of_bijective_appTop_pullback_snd_of_faithfullyFlat1 below · cited by 6 · depth 26 - Proper flat schemes over a normal base: Γ(X)=A
AlgebraicGeometry.bijective_appTop_of_isProper_of_flat_of_bijective_appTop_pullback_fractionRing56 below · cited by 1 · depth 26 - Geometric fibres of a proper flat scheme with Γ=ℤ[1/M] are connected
AlgebraicGeometry.connectedSpace_pullback_of_isProper_of_flat_of_bijective_appTop57 below · cited by 1 · depth 26 - Generic-fibre image misses the support of ker f
AlgebraicGeometry.disjoint_range_support_ker_of_comp_eq_comp_spec_of_isFractionRing_of_surjective0 below · cited by 1 · depth 26 - Euler characteristic bound for a reduced curve with two branches
AlgebraicGeometry.eulerChar_sectionsOf_le_sub_genusFF_sub_natCard_not_isRegularLocalRing188 below · cited by 1 · depth 26 - Valuative criterion of properness as unique extension of sections
AlgebraicGeometry.existsUnique_hom_comp_eq_specMap_and_specMap_comp_eq_of_isProper_of_valuationSubring0 below · cited by 1 · depth 26 - Retraction and local parameter along a section of a smooth relative curve
AlgebraicGeometry.exists_algHom_retraction_stalk_and_ker_le_span_sup_sq_of_section_of_smoothOfRelativeDimension_one0 below · cited by 4 · depth 26 - Mittag-Leffler property of sections on infinitesimal neighbourhoods
AlgebraicGeometry.exists_appTop_pullback_map_eq_appTop_pullback_fst_of_isProper54 below · cited by 1 · depth 26 - Lifting k-points along an integral surjection
AlgebraicGeometry.exists_comp_eq_of_isIntegralHom_of_surjective_of_isAlgClosed0 below · cited by 3 · depth 26 - Unramified plus equal cotangent rank implies étale near a rational point
AlgebraicGeometry.exists_etale_of_formallyUnramified_stalkMap0 below · cited by 2 · depth 26 - Morphisms over an algebraic extension descend to a finite subextension
AlgebraicGeometry.exists_intermediateField_finiteDimensional_comp_pullback_map_eq_of_isAlgebraic0 below · cited by 6 · depth 26 - Descent of qcqs finite-type schemes to a finite subextension
AlgebraicGeometry.exists_intermediateField_finiteDimensional_isPullback_of_isAlgebraic3 below · cited by 1 · depth 26 - Flat descent of a characteristic-zero domain to ℤ[1/n]
AlgebraicGeometry.exists_isDedekindDomain_ringHom_flat_of_isUnit_of_charZero0 below · cited by 1 · depth 26 - Descent of homomorphisms along a faithfully flat quasi-compact isogeny
AlgebraicGeometry.exists_isMonHom_comp_eq_of_forall_comp_eq_one_of_flat_of_surjective0 below · cited by 3 · depth 26 - Local rings of an affine model arise on the proper model
AlgebraicGeometry.exists_localRing_eq_localization_of_affineModel_of_map_maximalIdeal_le_of_isIntegrallyClosed_ofPrime7 below · cited by 3 · depth 26 - Localisations of an affine model are local rings of the proper model
AlgebraicGeometry.exists_localRing_eq_localization_of_normal_affineModel_of_map_maximalIdeal_le7 below · cited by 7 · depth 26 - Valuation rings as local rings on a universally closed model
AlgebraicGeometry.exists_localRing_eq_valuationSubring_of_isIntegrallyClosed_of_universallyClosed2 below · cited by 1 · depth 26 - Splitting of a Kummer normalisation gives a k-th root
AlgebraicGeometry.exists_pow_eq_of_section_fromNormalization_kummer0 below · cited by 2 · depth 26 - Function field of the base change to ℚ̄
AlgebraicGeometry.exists_ringEquiv_functionField_pullback_of_closure_eq_top_of_linearIndependent2 below · cited by 1 · depth 26 - Stalk of an integral scheme as a subring of F
AlgebraicGeometry.exists_ringEquiv_stalk_localRing_coe_eq0 below · cited by 3 · depth 26 - Stalks under base change with unchanged residue field
AlgebraicGeometry.exists_ringEquiv_stalk_quotient_map_maximalIdeal_of_iso_pullback_of_residue_surjective1 below · cited by 1 · depth 26 - Lifting residue-field points to sections over a valuation ring
AlgebraicGeometry.exists_section_comp_eq_of_finite_flat_valuationSubring0 below · cited by 3 · depth 26 - Smooth morphisms over a henselian local base have sections through closed-fibre rational points
AlgebraicGeometry.exists_section_comp_eq_of_smooth_of_henselianLocalRing2 below · cited by 2 · depth 26 - Sections of finite étale covers lift from the closed fibre
AlgebraicGeometry.exists_section_of_isFinite_of_etale_of_isProper_of_henselianLocalRing_of_isNoetherianRing62 below · cited by 2 · depth 26 - Finiteness of the singular locus of a two-branch proper curve
AlgebraicGeometry.finite_setOf_not_isRegularLocalRing_stalk_of_isIso_stalkMap66 below · cited by 1 · depth 26 - Closed points over a closed base point stay closed
AlgebraicGeometry.isClosed_singleton_base_of_isClosed_singleton_of_locallyOfFiniteType0 below · cited by 1 · depth 26 - DVR stalks persist under base change to a DVR extension
AlgebraicGeometry.isDiscreteValuationRing_stalk_of_isPullback_of_isDiscreteValuationRing_stalk1 below · cited by 1 · depth 26 - Stalk at a smooth generic point of the special fibre
AlgebraicGeometry.isDiscreteValuationRing_stalk_of_mem_smoothLocus_of_forall_specializes_eq2 below · cited by 1 · depth 26 - Function field tensor C is a domain for integral pullback
AlgebraicGeometry.isDomain_functionField_tensorProduct_of_isIntegral_pullback1 below · cited by 1 · depth 26 - Kummer cover of a normal proper scheme: finite étale with section over the closed fibre
AlgebraicGeometry.isFinite_and_etale_and_exists_section_fromNormalization_kummer_of_henselianLocalRing7 below · cited by 2 · depth 26 - Integrality of one algebraically closed base change suffices
AlgebraicGeometry.isIntegral_and_isIntegral_pullback_of_isIntegral_pullback_of_isAlgClosed4 below · cited by 1 · depth 26 - Integrality of X×_A L from a geometrically integral generic fibre
AlgebraicGeometry.isIntegral_pullback_specMap_of_geometricallyIntegral_pullback_snd_of_iso_pullback0 below · cited by 1 · depth 26 - Flat base change for global sections, pushout form
AlgebraicGeometry.isPushout_appTop_pullback_fst_appTop_pullback_snd_of_flat0 below · cited by 11 · depth 26 - Regularity of a ≤ 1-dimensional geometric fibre descends
AlgebraicGeometry.isRegularLocalRing_stalk_quotient_span_germ_of_isRegularLocalRing_stalk_pullback_of_ringKrullDim_le_one2 below · cited by 2 · depth 26 - Smoothness at a point where varpi₀ generates the maximal ideal
AlgebraicGeometry.mem_smoothLocus_of_forall_mem_localRing_eq_algebraMap_mul_of_perfectField0 below · cited by 1 · depth 26 - Smoothness of the generic fibre over the fraction field
AlgebraicGeometry.smooth_pullback_snd_specMap_of_forall_mem_smoothLocus_of_isFractionRing0 below · cited by 1 · depth 26 - Specialisation detected by local rings, for separated integral schemes
AlgebraicGeometry.specializes_iff_localRing_le_of_isSeparated1 below · cited by 3 · depth 26 - Base change preserves fibre dimension for smooth connected-fibred morphisms
AlgebraicGeometry.topologicalKrullDim_preimage_pullback_snd_eq_of_smooth_of_isConnected4 below · cited by 21 · depth 26 - Generic point maps to generic point when image contains an open
AlgebraicGeometry.base_genericPoint_eq_genericPoint_of_subset_range0 below · cited by 4 · depth 27 - Reduced closed fibre of a proper morphism with Γ = R
AlgebraicGeometry.bijective_appTop_pullback_snd_residue_of_bijective_appTop_of_isReduced58 below · cited by 2 · depth 27 - Characteristic zero of the function field from a characteristic-zero point
AlgebraicGeometry.charZero_functionField_of_hom_spec_of_charZero0 below · cited by 1 · depth 27 - κ-points of a finite-type scheme over an algebraically closed field
AlgebraicGeometry.eq_of_base_closedPoint_eq_and_exists_base_closedPoint_eq_and_isClosed_of_isAlgClosed0 below · cited by 6 · depth 27 - A valuation ring dominates at most one point of a separated integral scheme
AlgebraicGeometry.eq_of_stalk_le_valuationSubring_of_maximalIdeal_le_of_isSeparated0 below · cited by 1 · depth 27 - Čech Euler characteristic bounded by genus minus singularities
AlgebraicGeometry.eulerChar_sectionsOf_le_one_sub_genusFF_sub_natCard_not_isRegularLocalRing96 below · cited by 1 · depth 27 - Unique infinitesimal lifting along étale morphisms over local rings
AlgebraicGeometry.existsUnique_comp_eq_of_etale_of_isLocalRing_of_surjective_of_isNilpotent0 below · cited by 1 · depth 27 - fpqc descent of morphisms to a scheme along a finite flat cover
AlgebraicGeometry.existsUnique_forall_specMap_comp_eq_of_flat_of_forall_exists_comap_eq0 below · cited by 2 · depth 27 - Divisibility by varpi from one germ on a smooth curve family
AlgebraicGeometry.exists_eq_smul_kaehlerH0_of_germ_eq_smul_of_isIntegral_fibre_of_smoothOfRelativeDimension_one6 below · cited by 1 · depth 27 - Étale neighbourhood sections of a smooth morphism through a residue point
AlgebraicGeometry.exists_etale_nhd_section_of_smooth0 below · cited by 1 · depth 27 - Absolute Frobenius endomorphism of a scheme over 𝔽ₚ
AlgebraicGeometry.exists_frobenius_over_zmodp0 below · cited by 4 · depth 27 - Extension of morphisms to a proper κ-scheme across DVR points
AlgebraicGeometry.exists_hom_comp_eq_and_comp_eq_of_isProper_of_isDiscreteValuationRing_stalk1 below · cited by 2 · depth 27 - Isomorphisms of base changes descend to a finite subextension
AlgebraicGeometry.exists_intermediateField_finiteDimensional_iso_hom_comp_pullback_map_eq_of_isAlgebraic0 below · cited by 1 · depth 27 - Quasi-compact opens of an algebraic base change descend to a finite subextension
AlgebraicGeometry.exists_intermediateField_finiteDimensional_preimage_pullback_map_eq_of_isCompact_of_isAlgebraic0 below · cited by 1 · depth 27 - Spreading out a K-morphism of models to cofinal levels
AlgebraicGeometry.exists_intermediateField_forall_exists_hom_comp_eq_of_isPullback_of_isAlgebraic1 below · cited by 2 · depth 27 - Clopen subsets of the closed fibre lift (henselian base)
AlgebraicGeometry.exists_isClopen_preimage_eq_of_isProper_of_henselianLocalRing60 below · cited by 1 · depth 27 - Flat Dedekind base with complete Witt-type local witnesses
AlgebraicGeometry.exists_isDedekindDomain_ringHom_flat_and_forall_exists_isDiscreteValuationRing_of_isUnit_of_charZero0 below · cited by 1 · depth 27 - A clopen piece of a finite flat scheme is affine and flat
AlgebraicGeometry.exists_iso_Spec_of_isClopen_of_isFinite_of_flat0 below · cited by 1 · depth 27 - Principal kernel of a retraction of a smooth relative curve stalk
AlgebraicGeometry.exists_ker_eq_span_and_maximalIdeal_eq_of_algHom_stalk_of_smoothOfRelativeDimension_one0 below · cited by 1 · depth 27 - Section of a smooth relative curve: principal, transverse section ideal
AlgebraicGeometry.exists_ker_stalkMap_eq_span_and_maximalIdeal_eq_of_section_of_smoothOfRelativeDimension_one1 below · cited by 3 · depth 27 - Relative Jacobian of a pointed smooth proper curve over a DVR
AlgebraicGeometry.exists_relJacobian_of_smoothOfRelativeDimension_one728 below · cited by 2 · depth 27 - Kummer cover of a proper scheme splits over the closed fibre
AlgebraicGeometry.exists_section_closedFibre_fromNormalization_kummer_of_henselianLocalRing5 below · cited by 1 · depth 27 - Finitely many non-regular points on a birationally dominated proper integral curve
AlgebraicGeometry.finite_setOf_not_isRegularLocalRing_stalk_of_isIso_stalkMap_of_isIntegral64 below · cited by 1 · depth 27 - Finiteness of the fibre over a closed point after base field extension
AlgebraicGeometry.finite_setOf_pullback_fst_eq_of_isClosed_singleton0 below · cited by 2 · depth 27 - Rank of a finite reduced k-scheme counts its k-points
AlgebraicGeometry.finrank_eq_natCard_sections_of_isFinite_of_isReduced_of_isAlgClosed1 below · cited by 3 · depth 27 - Flatness over a Dedekind domain via torsion-free sections
AlgebraicGeometry.flat_iff_forall_appLE_mul_eq_zero_of_isDedekindDomain0 below · cited by 2 · depth 27 - Fibrewise criterion of flatness over a general base
AlgebraicGeometry.flat_of_locallyOfFinitePresentation_of_forall_flat_pullback_snd_fibre5 below · cited by 4 · depth 27 - Naturality of an affine-pinned Frobenius over 𝔽ₚ
AlgebraicGeometry.frobenius_comp_eq_comp_frobenius_of_forall_spec0 below · cited by 2 · depth 27 - Geometric connectedness descends from the generic fibre
AlgebraicGeometry.geometricallyConnected_of_isProper_of_flat_of_geometricallyReduced_of_geometricallyConnected_pullback_snd62 below · cited by 3 · depth 27 - Generic germ of a relative trace is the function-field trace
AlgebraicGeometry.germToFunctionField_trace_eq_traceFunAlong_germToFunctionField1 below · cited by 1 · depth 27 - Clopen agreement locus of two sections of an unramified separated morphism
AlgebraicGeometry.isClopen_preimage_diagonal_of_formallyUnramified_of_isSeparated0 below · cited by 15 · depth 27 - Kummer coverings of normal integral schemes are finite étale
AlgebraicGeometry.isFinite_and_etale_fromNormalization_kummer_of_isIntegrallyClosed3 below · cited by 1 · depth 27 - Quasi-finite morphisms of smooth proper varieties are finite flat
AlgebraicGeometry.isFinite_and_flat_and_surjective_of_locallyQuasiFinite_of_smoothOfRelativeDimension16 below · cited by 5 · depth 27 - Integrality of base changes of smooth geometrically connected fibres
AlgebraicGeometry.isIntegral_pullback_of_geometricallyConnected_of_smooth_pullback_snd9 below · cited by 1 · depth 27 - Finite étale over proper base: isomorphism from closed fibre
AlgebraicGeometry.isIso_of_isIso_pullback_closedFibre_of_isFinite_of_etale_of_isProper0 below · cited by 1 · depth 27 - Flat, formally unramified, finite type over integral implies reduced
AlgebraicGeometry.isReduced_of_flat_of_formallyUnramified_of_isIntegral0 below · cited by 4 · depth 27 - Smoothness of a finite quotient of a relative curve over a Dedekind base
AlgebraicGeometry.smoothOfRelativeDimension_one_of_isQuotient_of_isDedekindDomain30 below · cited by 1 · depth 27 - Base change along a field extension: surjective, flat, quasi-compact
AlgebraicGeometry.surjective_and_flat_and_quasiCompact_of_isPullback_specMap_algebraMap_of_field0 below · cited by 2 · depth 27 - Finite surjective k-morphisms preserve fibre Krull dimension
AlgebraicGeometry.topologicalKrullDim_preimage_eq_of_isFinite_of_surjective0 below · cited by 3 · depth 27 - Frobenius pinned by B-points is finite and topologically trivial
AlgebraicGeometry.base_apply_eq_and_surjective_and_locallyQuasiFinite_and_isFinite_of_frobenius_pin0 below · cited by 1 · depth 28 - K → Γ(X,mathcal O_X) bijective for universally closed X
AlgebraicGeometry.bijective_appTop_of_universallyClosed_of_geometricallyReduced_of_geometricallyConnected0 below · cited by 9 · depth 28 - Endomorphism identities descend along a monomorphism from Spec A
AlgebraicGeometry.comp_eq_comp_of_specMap_comp_eq_comp_of_mono_of_comp_eq0 below · cited by 1 · depth 28 - Uniqueness of K-points over a point with isomorphic residue field
AlgebraicGeometry.eq_of_comp_eq_of_base_closedPoint_eq_of_isIso_residueFieldMap0 below · cited by 2 · depth 28 - Sections of an unramified morphism agreeing at one point coincide
AlgebraicGeometry.eq_of_comp_eq_of_forall_specializes_of_lift_mem_range_diagonal0 below · cited by 1 · depth 28 - S-points of a scheme from compatible points modulo Iⁿ⁺¹
AlgebraicGeometry.existsUnique_specMap_mk_pow_comp_eq_of_isAdicComplete_of_isLocalRing0 below · cited by 1 · depth 28 - Étale-local sections through a jointly surjective étale family
AlgebraicGeometry.exists_affine_etale_cover_factor_of_forall_mem_range_of_etale0 below · cited by 2 · depth 28 - Base change of an affine chart on an open subscheme
AlgebraicGeometry.exists_baseChange_chart_isPullback_of_isPullback0 below · cited by 2 · depth 28 - Divisibility of a Čech 1-form by varpi transfers between charts
AlgebraicGeometry.exists_eq_smul_chart_of_eq_smul_chart_of_mem_kaehlerH0_of_isIntegral_fibre_of_smoothOfRelativeDimension_one2 below · cited by 1 · depth 28 - Chart divisibility by varpi of differentials from germ divisibility
AlgebraicGeometry.exists_eq_smul_chart_of_mapOfRingHom_germ_eq_smul_of_isIntegral_fibre_of_smoothOfRelativeDimension_one2 below · cited by 1 · depth 28 - Section values detect the uniformising ball of a point
AlgebraicGeometry.exists_finset_forall_pointEquiv_eq_coe_mem_ball_of_differentiableOn_appLE_of_isSeparated4 below · cited by 2 · depth 28 - Non-constant rational function on an integral scheme over K
AlgebraicGeometry.exists_functionField_ne_baseToFunctionField_of_ne_genericPoint0 below · cited by 1 · depth 28 - Clopen subschemes of a constant-rank finite étale cover are representable
AlgebraicGeometry.exists_isFinite_etale_represents_clopens_of_isFinite_of_etale3 below · cited by 2 · depth 28 - Finiteness and algebraisation of compatible systems of Xₙ-schemes
AlgebraicGeometry.exists_isFinite_isPullback_of_isProper_of_forall_points_eq_of_isAdicComplete213 below · cited by 2 · depth 28 - Closed fibres agree under residually surjective local base change
AlgebraicGeometry.exists_iso_pullback_residue_of_iso_pullback0 below · cited by 2 · depth 28 - Points over a discrete valuation ring lie on two fibres
AlgebraicGeometry.exists_mapOnProdOver_apply_eq_or_of_isFractionRing_of_surjective0 below · cited by 1 · depth 28 - Completed stalk over the crossing vertex is a UV-model
AlgebraicGeometry.exists_ringEquiv_adicCompletion_stalk_uvCrossingModel_of_flat_of_map_maximalIdeal_eq_of_isIso_residueFieldMap41 below · cited by 1 · depth 28 - Function field of a base change by a field extension
AlgebraicGeometry.exists_ringEquiv_functionField_pullback_of_closure_eq_top_of_linearIndependent_of_isIntegral_pullback2 below · cited by 1 · depth 28 - Charts and universal property of the Kummer cover T^k=g
AlgebraicGeometry.exists_root_fromNormalization_kummer_existsUnique_lift4 below · cited by 1 · depth 28 - Galois descent of morphisms between base-changed k-schemes
AlgebraicGeometry.exists_unique_eq_pullback_map_of_forall_galois_twist_comp_eq4 below · cited by 1 · depth 28 - Finitely many L-points off a non-empty open of a curve
AlgebraicGeometry.finite_sections_not_le_preimage_of_smoothOfRelativeDimension_one4 below · cited by 1 · depth 28 - Rank of a finite étale morphism at a geometric point
AlgebraicGeometry.finrank_eq_natCard_of_isFinite_of_etale_of_isAlgClosed2 below · cited by 4 · depth 28 - Miracle flatness for quasi-finite endomorphisms of smooth schemes
AlgebraicGeometry.flat_of_smooth_of_preconnectedSpace_of_locallyQuasiFinite_endomorphism18 below · cited by 1 · depth 28 - Geometric connectedness from the generic fibre, R normal
AlgebraicGeometry.geometricallyConnected_of_isProper_of_flat_of_bijective_appTop_pullback_snd60 below · cited by 1 · depth 28 - Affineness of pr₁⁻¹V and generation of its sections by Γ(V) and C
AlgebraicGeometry.isAffineOpen_preimage_fst_and_exists_eq_sum_of_isAffineOpen2 below · cited by 2 · depth 28 - Generic fibre of an S-scheme over a DVR is open
AlgebraicGeometry.isOpenImmersion_mapOnProdOver_specMap_algebraMap_of_isFractionRing_of_isDiscreteValuationRing0 below · cited by 1 · depth 28 - Specℚ̄toSpec𝒪 is an open immersion
AlgebraicGeometry.isOpenImmersion_specMap_subtype_of_liesOverPrime1 below · cited by 2 · depth 28 - Germ of varpi generates a non-zero prime in each stalk on the special fibre
AlgebraicGeometry.isPrime_span_germ_and_ne_zero_of_isIntegral_fibre_of_smoothOfRelativeDimension_one0 below · cited by 3 · depth 28 - Closed subschemes of finite reduced k-schemes are reduced
AlgebraicGeometry.isReduced_of_isClosedImmersion_of_isFinite_of_isReduced0 below · cited by 1 · depth 28 - Reducedness of the closed fibre from reduced stalks modulo mathfrak m_A
AlgebraicGeometry.isReduced_pullback_residue_of_forall_isReduced_stalk_quotient0 below · cited by 1 · depth 28 - Limit criterion for local finite presentation over an affine base
AlgebraicGeometry.locallyOfFinitePresentation_of_forall_directed_colimit3 below · cited by 2 · depth 28 - Descent of mathfrak m_Amathcal O_{X,x} along a base change
AlgebraicGeometry.mem_map_maximalIdeal_of_stalkMap_mem_map_maximalIdeal_of_iso_pullback0 below · cited by 1 · depth 28 - Rational points linked by smooth proper curves
AlgebraicGeometry.mem_of_isProper_of_forall_smoothProperCurve_mem16 below · cited by 2 · depth 28 - Base-change-stable properties pass along a composite of rings
AlgebraicGeometry.pullback_snd_specMap_comp_of_isStableUnderBaseChange0 below · cited by 2 · depth 28 - Invariants of a smooth relative curve over a Dedekind base
AlgebraicGeometry.smoothOfRelativeDimension_one_SpecMap_of_isInvariant_of_isDedekindDomain29 below · cited by 1 · depth 28 - Equivariance of a pullback lift under an intertwined endomorphism
AlgebraicGeometry.specMap_comp_pullbackLift_eq_pullbackLift_comp_pullbackMap_of_comp_eq_comp0 below · cited by 2 · depth 28 - Stalks of a base change along a surjection of rings
AlgebraicGeometry.stalkMap_surjective_and_ker_stalkMap_eq_map_ker_of_isPullback_of_surjective1 below · cited by 4 · depth 28 - Valuative criterion for universal closedness via complete DVRs
AlgebraicGeometry.universallyClosed_of_forall_isDiscreteValuationRing_isAdicComplete_finite_residueField_hasLift14 below · cited by 1 · depth 28 - Spec of an injective finite ring map is an epimorphism
AlgebraicGeometry.epi_specMap_of_injective_of_finite0 below · cited by 2 · depth 29 - Base change along a field extension is faithful into separated targets
AlgebraicGeometry.eq_of_pullback_map_eq_pullback_map_of_isSeparated0 below · cited by 2 · depth 29 - Sections over a local ring agreeing at the closed point
AlgebraicGeometry.eq_of_section_of_base_closedPoint_eq_of_ker_stalkMap_le0 below · cited by 1 · depth 29 - Kummer normalisation represents k-th roots on an affine chart
AlgebraicGeometry.existsUnique_lift_fromNormalization_kummer_of_isAffineOpen3 below · cited by 1 · depth 29 - k-points of a finite κ-scheme descend uniquely to κ-points
AlgebraicGeometry.existsUnique_section_comp_eq_of_isFinite_of_isAlgClosed0 below · cited by 1 · depth 29 - Unique extension of L-points to sections over a valuation subring
AlgebraicGeometry.existsUnique_section_comp_eq_of_isFinite_valuationSubring1 below · cited by 1 · depth 29 - Sections over p₁⁻¹V as Γ(A,V)⊗_K H
AlgebraicGeometry.exists_algEquiv_sections_pullback_fst_preimage_tensor_of_isAffineOpen1 below · cited by 9 · depth 29 - Connected component of a point on a smooth k-scheme is integral, with stabiliser
AlgebraicGeometry.exists_component_isIntegral_and_stabilizer_of_smoothOfRelativeDimension6 below · cited by 2 · depth 29 - Rigidity lemma over an algebraically closed field
AlgebraicGeometry.exists_eq_pullback_snd_comp_of_isProper0 below · cited by 3 · depth 29 - Finite surjective factorisation of a clopen curve through a clopen open
AlgebraicGeometry.exists_factor_clopen_isFinite_surjective_of_smoothOfRelativeDimension_one3 below · cited by 1 · depth 29 - Finitely presented qcqs schemes descend to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_isPullback_of_locallyOfFinitePresentation10 below · cited by 4 · depth 29 - Descent of an isomorphism of base changes to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_iso_pullback_of_iso_pullback_of_locallyOfFinitePresentation4 below · cited by 4 · depth 29 - Equality of morphisms descends to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_pullback_fst_comp_eq_of_locallyOfFiniteType1 below · cited by 8 · depth 29 - Separation of ℂ-points by values of sections
AlgebraicGeometry.exists_finset_forall_norm_appLE_sub_lt_imp_false_of_ne_of_isSeparated3 below · cited by 1 · depth 29 - Open conditions on Spec descend to a finite stage
AlgebraicGeometry.exists_forall_specMap_base_mem_of_isDirectLimit0 below · cited by 2 · depth 29 - Descent of a Galois-equivariant morphism to a finite Galois level
AlgebraicGeometry.exists_intermediateField_isGalois_galois_twist_comp_pullback_map_eq2 below · cited by 1 · depth 29 - Gluing closed subschemes given on an open cover
AlgebraicGeometry.exists_isClosedImmersion_isPullback_of_forall_iff_of_openCover0 below · cited by 1 · depth 29 - Grothendieck existence for finite morphisms over a complete base
AlgebraicGeometry.exists_isFinite_of_forall_isFinite_isPullback_of_isProper_of_isAdicComplete211 below · cited by 2 · depth 29 - Function field of a finite-group quotient is Galois
AlgebraicGeometry.exists_isGalois_functionField_of_quotient_of_finite0 below · cited by 1 · depth 29 - Finite over quasi-projective is quasi-projective over a noetherian base
AlgebraicGeometry.exists_isImmersion_projSpace_comp_of_isFinite_of_isImmersion_of_isNoetherianRing9 below · cited by 1 · depth 29 - Properness gives a uniform denominator for L-valued points
AlgebraicGeometry.exists_ne_zero_forall_exists_mul_appLE_mem_range_algebraMap_of_isProper0 below · cited by 1 · depth 29 - Finite-group quotient clauses restrict to an open of the target
AlgebraicGeometry.exists_restrict_action_quotient_clauses_morphismRestrict0 below · cited by 1 · depth 29 - Restricting a finite quotient presentation to a clopen connected piece
AlgebraicGeometry.exists_restrict_action_quotient_clauses_of_isClopen_of_isConnected0 below · cited by 1 · depth 29 - Completed stalk of uv=wvarpi^E at a vertex point
AlgebraicGeometry.exists_ringEquiv_adicCompletion_stalk_crossingScheme_uvCrossingModel_of_mem_asIdeal35 below · cited by 1 · depth 29 - Isomorphism of integral schemes induces germ-compatible function field isomorphism
AlgebraicGeometry.exists_ringEquiv_functionField_germToFunctionField_eq_of_isIso0 below · cited by 1 · depth 29 - Generic-point-preserving morphism induces map of function fields
AlgebraicGeometry.exists_ringHom_functionField_germ_eq_of_base_genericPoint_eq0 below · cited by 5 · depth 29 - Smooth proper model covering the k-points of an affine curve
AlgebraicGeometry.exists_smoothProperCurve_hom_comp_eq_of_ringKrullDim_eq_one3 below · cited by 1 · depth 29 - Sequential compactness of the ℂ-points of a proper smooth curve
AlgebraicGeometry.exists_tendsto_appLE_of_isProper_of_smoothOfRelativeDimension_one_complex94 below · cited by 2 · depth 29 - Injective holomorphic chart lifts evaluation-convergent sequences of ℂ-points
AlgebraicGeometry.exists_tendsto_forall_eq_of_injOn_of_differentiableOn_appLE_of_tendsto_appLE7 below · cited by 1 · depth 29 - Galois descent of morphisms along a finite Galois extension
AlgebraicGeometry.exists_unique_eq_pullback_map_of_forall_galois_twist_comp_eq_of_finiteDimensional0 below · cited by 2 · depth 29 - Cotangent space at a rational point has dimension n
AlgebraicGeometry.finrank_cotangentSpace_eq_of_smoothOfRelativeDimension0 below · cited by 3 · depth 29 - Flatness over mathbb Z_q from non-zero-divisors at closed points
AlgebraicGeometry.flat_of_forall_isClosed_natCast_mem_nonZeroDivisors_stalk2 below · cited by 1 · depth 29 - Base change of function fields: generation and linear disjointness
AlgebraicGeometry.functionField_pullback_generates_and_linearIndependent2 below · cited by 1 · depth 29 - Zariski connectedness: Stein proper morphisms are geometrically connected
AlgebraicGeometry.geometricallyConnected_of_isProper_of_bijective_appTop59 below · cited by 1 · depth 29 - Base change of a closed immersion into a fibre product
AlgebraicGeometry.isClosedImmersion_pullbackLift_of_isClosedImmersion_pullbackLift_of_isPullback0 below · cited by 1 · depth 29 - Residue field map is an isomorphism at a κ(y)-valued point
AlgebraicGeometry.isIso_residueFieldMap_of_comp_eq_fromSpecResidueField0 below · cited by 1 · depth 29 - Stalk isomorphism off the complementary closed component
AlgebraicGeometry.isIso_stalkMap_of_isClosedImmersion_of_not_mem_range0 below · cited by 3 · depth 29 - Openness of the smooth, irreducible, g-dimensional fibre locus
AlgebraicGeometry.isOpen_setOf_smooth_irreducibleSpace_geometricFibre_of_isProper_of_flat141 below · cited by 7 · depth 29 - Regularity of a stalk pro-representing a regular local ring
AlgebraicGeometry.isRegularLocalRing_stalk_and_natCast_ne_zero_of_isProrepresentedBy2 below · cited by 2 · depth 29 - Local equation of a section generates 𝔪 at its generic point
AlgebraicGeometry.maximalIdeal_stalk_eq_span_stalkSpecializes_of_ker_stalkMap_eq_span0 below · cited by 1 · depth 29 - Finite-group quotient of a flat proper π-adic tower exists
AlgebraicGeometry.nonempty_towerQuotientDatum_of_isProper_of_flat9 below · cited by 1 · depth 29 - Local ring at a rational point of a smooth relative curve is a DVR
AlgebraicGeometry.ringKrullDim_stalk_eq_one_and_isDiscreteValuationRing_of_section_of_smoothOfRelativeDimension_one8 below · cited by 1 · depth 29 - Invariants of a smooth relative curve over a complete DVR
AlgebraicGeometry.smoothOfRelativeDimension_one_SpecMap_of_isInvariant_of_isAdicComplete_of_isAlgClosed_residueField27 below · cited by 1 · depth 29 - Smoothness of the quotient of a smooth affine curve
AlgebraicGeometry.smoothOfRelativeDimension_one_SpecMap_of_isInvariant_of_isAlgClosed6 below · cited by 2 · depth 29 - Smoothness from formal smoothness at closed points
AlgebraicGeometry.smooth_of_locallyOfFinitePresentation_of_forall_isClosed_formallySmooth_stalkMap0 below · cited by 1 · depth 29 - Germs agree after base change of a Spec-chart
AlgebraicGeometry.stalkMap_germ_app_appIso_inv_appTop_eq_germ_appTop_of_comp_spec_map_eq_isoOfEq_hom_comp_morphismRestrict_comp0 below · cited by 1 · depth 29 - Finite morphisms of smooth integral curves over a field are surjective
AlgebraicGeometry.surjective_of_isFinite_of_smoothOfRelativeDimension_one2 below · cited by 1 · depth 29 - Valuative criterion with finite-residue-field DVRs over a ℤ-finite-type base
AlgebraicGeometry.universallyClosed_of_forall_isDiscreteValuationRing_finite_residueField_hasLift11 below · cited by 2 · depth 29 - mathcal O_B → p_*mathcal O_X is an isomorphism for proper flat morphisms with geometrically reduced connected fibres
AlgebraicGeometry.bijective_app_of_isProper_of_flat_of_geometricallyReduced_of_geometricallyConnected41 below · cited by 5 · depth 30 - Rigidity over a base with a section and closed projection
AlgebraicGeometry.comp_section_comp_eq_of_isClosedMap_of_surjective_app0 below · cited by 4 · depth 30 - Finite reduced schemes over a perfect field are étale
AlgebraicGeometry.etale_of_isFinite_of_isReduced_of_perfectField0 below · cited by 1 · depth 30 - Descent of morphisms along a faithfully flat affine base change
AlgebraicGeometry.existsUnique_comp_eq_of_isPullback_of_faithfullyFlat0 below · cited by 6 · depth 30 - Completed local rings of invariant subrings of smooth relative curves
AlgebraicGeometry.exists_adicCompletion_atPrime_ringEquiv_powerSeries_of_isInvariant_of_smoothOfRelativeDimension_one11 below · cited by 1 · depth 30 - Extending a map from a nonempty open of a smooth curve to a proper scheme
AlgebraicGeometry.exists_comp_eq_of_smoothOfRelativeDimension_one_of_isProper7 below · cited by 1 · depth 30 - Constancy of B-points of an unramified k-scheme
AlgebraicGeometry.exists_eq_specMap_comp_of_formallyUnramified_of_forall_isIdempotentElem_of_isAlgClosed3 below · cited by 1 · depth 30 - Descent of a morphism of base changes to a finitely generated stage
AlgebraicGeometry.exists_fg_subalgebra_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation1 below · cited by 9 · depth 30 - Gluing two quasi-compact opens that descend to finite stages
AlgebraicGeometry.exists_fg_subalgebra_isPullback_of_iSup_eq_top_of_locallyOfFinitePresentation7 below · cited by 1 · depth 30 - Affine model over a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_isPullback_of_isAffine_of_locallyOfFinitePresentation1 below · cited by 1 · depth 30 - Noetherian descent of a smooth proper projective scheme with section
AlgebraicGeometry.exists_fg_subalgebra_isPullback_smooth_isProper_of_isClosedImmersion_proj104 below · cited by 2 · depth 30 - Points of a finite scheme factor through a finite algebra
AlgebraicGeometry.exists_finite_algebra_specMap_comp_eq_of_isFinite0 below · cited by 1 · depth 30 - Levelwise maps to the relative spectrum and pushout squares
AlgebraicGeometry.exists_hom_glued_comp_toBase_eq_and_isPushout_of_affHom_pushforwardUnit_of_coequifibered5 below · cited by 2 · depth 30 - G-stable affine neighbourhood in a separated scheme
AlgebraicGeometry.exists_isAffineOpen_mem_forall_preimage_eq_of_finite_group0 below · cited by 1 · depth 30 - Affine chart D₊(F) around a finite set in Proj
AlgebraicGeometry.exists_mem_isAffineOpen_isClosedImmersion_morphismRestrict_basicOpen_of_isImmersion_proj1 below · cited by 2 · depth 30 - Separating tensor for two distinct ℂ-points of a separated scheme
AlgebraicGeometry.exists_sum_appLE_mul_appLE_ne_zero_and_forall_eq_zero_of_ne_of_isSeparated2 below · cited by 1 · depth 30 - DVR squares with finite residue field for immediate specialisations
AlgebraicGeometry.exists_valuativeCommSq_isDiscreteValuationRing_finite_residueField_of_specializes5 below · cited by 1 · depth 30 - Finite residue field at a closed point over a finite residue field
AlgebraicGeometry.finite_residueField_stalk_of_isClosed_of_locallyOfFiniteType0 below · cited by 2 · depth 30 - Flatness over a Dedekind base tested at closed fibre points
AlgebraicGeometry.flat_of_forall_isClosed_germ_mul_eq_zero_of_isDedekindDomain1 below · cited by 1 · depth 30 - Connected proper schemes over algebraically closed fields are geometrically connected
AlgebraicGeometry.geometricallyConnected_of_isAlgClosed_of_isProper_of_connectedSpace59 below · cited by 7 · depth 30 - Proper smooth X/K is geometrically irreducible iff Γ(X,mathcal O_X)=K
AlgebraicGeometry.geometricallyIrreducible_iff_bijective_appTop_of_isProper_of_smooth16 below · cited by 6 · depth 30 - Closed immersion from injectivity on dual-number points
AlgebraicGeometry.isClosedImmersion_of_isProper_of_forall_dualNumber_comp_eq5 below · cited by 3 · depth 30 - Proper over a base and closed immersion on geometric fibres
AlgebraicGeometry.isClosedImmersion_of_isProper_of_forall_geometricFibre_isClosedImmersion4 below · cited by 1 · depth 30 - Rank of a fibre product of finite locally free morphisms
AlgebraicGeometry.isFinite_flat_and_finrank_pullback_fst_comp_eq_mul_of_finrank_eq_const1 below · cited by 1 · depth 30 - Closed immersion of equal-rank finite flat schemes is an isomorphism
AlgebraicGeometry.isIso_of_isClosedImmersion_of_finrank_eq1 below · cited by 3 · depth 30 - Local Noetherianity descends along flat surjective quasi-compact morphisms
AlgebraicGeometry.isLocallyNoetherian_of_flat_of_surjective_of_quasiCompact0 below · cited by 2 · depth 30 - Openness of the geometrically irreducible fibre locus over any base
AlgebraicGeometry.isOpen_setOf_forall_irreducibleSpace_pullback_of_isProper_of_smooth130 below · cited by 1 · depth 30 - Openness of the geometrically smooth fibre locus
AlgebraicGeometry.isOpen_setOf_forall_smooth_pullback_snd_of_universallyClosed_of_flat4 below · cited by 1 · depth 30 - Openness of the locus of geometric fibres of given dimension
AlgebraicGeometry.isOpen_setOf_forall_topologicalKrullDim_pullback_eq_of_isProper_of_smooth4 below · cited by 1 · depth 30 - Reducedness descends along flat surjective morphisms
AlgebraicGeometry.isReduced_of_flat_of_surjective0 below · cited by 1 · depth 30 - Base change of a quasi-compact finite-type immersion into P^M_R
AlgebraicGeometry.isSeparated_and_quasiCompact_and_locallyOfFinitePresentation_and_forall_finset_exists_isAffineOpen_of_isPullback_of_isImmersion3 below · cited by 1 · depth 30 - Rational points linked by open pieces of smooth proper curves
AlgebraicGeometry.mem_of_isSeparated_of_forall_smoothProperCurve_opens_mem16 below · cited by 2 · depth 30 - Existence of a quotient tower for a finite group action
AlgebraicGeometry.nonempty_towerQuotientDatum_of_isProper_of_flat_of_forall_exists_isAffineOpen7 below · cited by 1 · depth 30 - Descent of quasi-compactness and separatedness along faithfully flat base change
AlgebraicGeometry.quasiCompact_and_isSeparated_of_isPullback_of_faithfullyFlat0 below · cited by 2 · depth 30 - Smooth closed fibre from power series completions at closed points
AlgebraicGeometry.smoothOfRelativeDimension_one_pullback_snd_residueField_of_forall_adicCompletion_atPrime_ringEquiv_powerSeries11 below · cited by 1 · depth 30 - Smoothness of the kernel of one quotient over the other
AlgebraicGeometry.smooth_pullbackFst_comp_of_forall_iff_exists_torus_of_flat2 below · cited by 1 · depth 30 - Morphisms over ℂ preserve convergence of complex points
AlgebraicGeometry.tendsto_appLE_mapPt_complex0 below · cited by 1 · depth 30 - Convergence of ℂ-points passes to the fibre product
AlgebraicGeometry.tendsto_appLE_pullbackLift_complex2 below · cited by 1 · depth 30 - Stalks over the vertex of uv=wvarpi^E have dimension ≥ 2
AlgebraicGeometry.two_le_ringKrullDim_stalk_crossingScheme_of_mem_asIdeal1 below · cited by 1 · depth 30 - Universal closedness tested on finite affine spaces over the base
AlgebraicGeometry.universallyClosed_of_forall_finite_isClosedMap_pullback_snd_mvPolynomial2 below · cited by 1 · depth 30 - Degree-zero base change over a Noetherian local base
AlgebraicGeometry.bijective_appTop_of_isProper_of_flat_of_isNoetherianRing_of_isLocalRing3 below · cited by 3 · depth 31 - Degree zero base change over an arbitrary local base
AlgebraicGeometry.bijective_appTop_of_isProper_of_flat_of_locallyOfFinitePresentation_of_isLocalRing33 below · cited by 5 · depth 31 - mathcal O_B → p_*mathcal O_X bijective for proper flat p
AlgebraicGeometry.bijective_app_of_isProper_of_flat_of_forall_bijective_appTop_fiberToSpecResidueField39 below · cited by 1 · depth 31 - Field-valued points are determined after field extension
AlgebraicGeometry.eq_of_specMap_comp_eq_of_field0 below · cited by 2 · depth 31 - K-points through a closed point are base changes of k-points
AlgebraicGeometry.eq_specMap_comp_pointOfClosedPoint_of_apply_closedPoint_eq0 below · cited by 2 · depth 31 - Unique descent of a T-point along a faithfully flat algebra
AlgebraicGeometry.existsUnique_specMap_comp_eq_of_faithfullyFlat0 below · cited by 6 · depth 31 - Completed stalk of a smooth relative curve is W[[T]]
AlgebraicGeometry.exists_adicCompletion_stalk_ringEquiv_powerSeries_of_smoothOfRelativeDimension_one4 below · cited by 1 · depth 31 - Sections of finite algebras over valuation rings
AlgebraicGeometry.exists_algHom_of_finite_of_valuationRing_of_isAlgClosed0 below · cited by 1 · depth 31 - Image of the complement of a nonempty open under a closed immersion
AlgebraicGeometry.exists_closeds_lt_forall_notMem_imp_mem_of_isClosedImmersion_of_nonempty0 below · cited by 1 · depth 31 - Weil's extension theorem for morphisms to an abelian variety
AlgebraicGeometry.exists_comp_eq_of_isOpenImmersion_of_abelianSchemePropertyBundle29 below · cited by 3 · depth 31 - Affineness of a base change descends to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_isAffine_pullback_of_isAffine_pullback1 below · cited by 2 · depth 31 - Noetherian approximation for proper flat morphisms
AlgebraicGeometry.exists_fg_subalgebra_isProper_flat_isPullback_of_isProper_of_flat_of_locallyOfFinitePresentation25 below · cited by 5 · depth 31 - Descent of a smooth proper connected-fibred morphism to a finitely generated base
AlgebraicGeometry.exists_fg_subalgebra_isPullback_smooth_isProper_geometricallyConnected100 below · cited by 1 · depth 31 - Spreading out a smooth proper projective scheme with section
AlgebraicGeometry.exists_fg_subalgebra_isPullback_smooth_isProper_of_isClosedImmersion_proj_of_isPullback8 below · cited by 1 · depth 31 - Quasi-compact opens descend to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_opens_preimage_eq_of_isCompact1 below · cited by 1 · depth 31 - Levelwise morphisms into the relative spectrum of A
AlgebraicGeometry.exists_hom_glued_comp_toBase_eq_of_affHom_pushforwardUnit_of_coequifibered2 below · cited by 1 · depth 31 - Universal morphism glued from affine points of M
AlgebraicGeometry.exists_hom_pullback_comp_eq_and_forall_pullbackMap_comp_eq_of_forall_spec_point0 below · cited by 1 · depth 31 - Incidence ideal of a closed immersion and a point
AlgebraicGeometry.exists_ideal_forall_exists_comp_eq_specMap_comp_iff_le_ker_of_isClosedImmersion1 below · cited by 1 · depth 31 - A smooth integral curve over ̄ k has infinitely many k-points
AlgebraicGeometry.exists_injective_schemeHomOver_of_isIntegral_of_smoothOfRelativeDimension_one2 below · cited by 1 · depth 31 - Finite sets of points in a scheme immersed in Proj lie in one affine open
AlgebraicGeometry.exists_isAffineOpen_forall_mem_of_isImmersion_proj1 below · cited by 2 · depth 31 - Equaliser of two S-points of a separated scheme
AlgebraicGeometry.exists_isClosedImmersion_locallyOfFinitePresentation_iff_comp_eq_of_isSeparated0 below · cited by 2 · depth 31 - Relative Segre embedding for projective closed subschemes
AlgebraicGeometry.exists_isClosedImmersion_projSpace_pullback_of_isClosedImmersion0 below · cited by 6 · depth 31 - Projective ambient scheme with Hilb×(P^N)^k points
AlgebraicGeometry.exists_isClosedImmersion_proj_forall_bijective_pullback_points2 below · cited by 1 · depth 31 - Gluing two models of open pieces along a common open model
AlgebraicGeometry.exists_isPullback_glue_of_isPullback_of_isOpenImmersion0 below · cited by 1 · depth 31 - Flat affine closed subschemes with isomorphic generic fibres
AlgebraicGeometry.exists_iso_hom_comp_eq_of_isClosedImmersion_of_flat_of_iso_generic1 below · cited by 1 · depth 31 - Reduced closed subschemes determined by their k-points
AlgebraicGeometry.exists_iso_hom_comp_eq_of_isClosedImmersion_of_isReduced_of_forall_rationalPoint3 below · cited by 1 · depth 31 - Points of an unramified finite-type scheme over ̄ k are open and rational
AlgebraicGeometry.exists_opens_coe_eq_singleton_and_isIso_iota_comp_of_formallyUnramified_of_isAlgClosed1 below · cited by 1 · depth 31 - Finite étale schemes over an algebraically closed field split
AlgebraicGeometry.exists_opens_coe_eq_singleton_and_isIso_iota_comp_of_isFinite_of_etale0 below · cited by 4 · depth 31 - Image of a non-empty open under a dominant morphism contains a non-empty open
AlgebraicGeometry.exists_opens_nonempty_subset_image_of_apply_genericPoint_eq0 below · cited by 2 · depth 31 - Hom scheme representing S-morphisms, with Hilbert-polynomial pieces
AlgebraicGeometry.exists_scheme_represents_schemeHomOver_hilbertPieces_of_isProper_of_flat341 below · cited by 2 · depth 31 - Smooth proper curve model covering the k-points of Spec A
AlgebraicGeometry.exists_smoothProperCurve_opens_hom_comp_eq_of_ringKrullDim_eq_one3 below · cited by 1 · depth 31 - Two A-algebra maps from D into a domain differ by G
AlgebraicGeometry.exists_smul_algHom_eq_of_isInvariant_of_isDomain0 below · cited by 1 · depth 31 - Existence of the table scheme of endomorphism quadruples
AlgebraicGeometry.exists_tableScheme_of_represents_homScheme9 below · cited by 1 · depth 31 - Irreducibility of all geometric fibres over s equals geometric irreducibility of the fibre
AlgebraicGeometry.forall_irreducibleSpace_pullback_iff_geometricallyIrreducible_fiberToSpecResidueField2 below · cited by 3 · depth 31 - Dual-number injectivity implies formal unramifiedness
AlgebraicGeometry.formallyUnramified_of_forall_dualNumber_comp_eq3 below · cited by 1 · depth 31 - Formal unramifiedness from the geometric fibres over the base
AlgebraicGeometry.formallyUnramified_of_forall_geometricFibre_formallyUnramified1 below · cited by 1 · depth 31 - Uniqueness of infinitesimal lifts implies formal unramifiedness
AlgebraicGeometry.formallyUnramified_of_forall_specMap_comp_eq_imp_eq0 below · cited by 1 · depth 31 - Irreducibility over one algebraically closed extension gives geometric irreducibility
AlgebraicGeometry.geometricallyIrreducible_of_irreducibleSpace_pullback_of_isAlgClosed1 below · cited by 4 · depth 31 - Affineness along one step of a flat π-adic tower
AlgebraicGeometry.isAffineOpen_of_isAffineOpen_preimage_of_isPullback_of_flat2 below · cited by 3 · depth 31 - Universally closed, universally injective unramified morphisms are closed immersions
AlgebraicGeometry.isClosedImmersion_of_universallyClosed_of_universallyInjective_of_formallyUnramified0 below · cited by 1 · depth 31 - Closedness over a directed union of subalgebras
AlgebraicGeometry.isClosedMap_pullback_snd_of_directed_subalgebra1 below · cited by 1 · depth 31 - Connected fibres of dimension g from geometric fibres
AlgebraicGeometry.isConnected_preimage_and_topologicalKrullDim_eq_of_forall_geometricFibre146 below · cited by 2 · depth 31 - Nilpotent thickening of the base: homeomorphism on underlying spaces
AlgebraicGeometry.isHomeomorph_of_isPullback_of_surjective_of_isNilpotent_ker0 below · cited by 14 · depth 31 - Openness of the geometrically irreducible fibre locus
AlgebraicGeometry.isOpen_setOf_forall_irreducibleSpace_pullback_of_isProper_of_smooth_of_isNoetherianRing98 below · cited by 1 · depth 31 - Separatedness descends along finite surjective morphisms
AlgebraicGeometry.isSeparated_of_isFinite_of_surjective_of_comp_eq0 below · cited by 1 · depth 31 - Formally unramified and injective on k-points implies mono
AlgebraicGeometry.mono_of_formallyUnramified_of_forall_comp_eq0 below · cited by 2 · depth 31 - Fibrewise relative dimension for smooth morphisms
AlgebraicGeometry.smoothOfRelativeDimension_of_smooth_of_forall_fiber2 below · cited by 2 · depth 31 - Smoothness of relative dimension one from power series completions
AlgebraicGeometry.smoothOfRelativeDimension_one_of_forall_nonempty_adicCompletion_stalk_ringEquiv_powerSeries7 below · cited by 1 · depth 31 - Convergence of values of sections at ℂ-points is local
AlgebraicGeometry.tendsto_appLE_of_tendsto_appLE_of_isAffineOpen_complex0 below · cited by 1 · depth 31 - Universal injectivity from geometric fibres over an affine base
AlgebraicGeometry.universallyInjective_of_forall_geometricFibre_universallyInjective0 below · cited by 1 · depth 31 - Surjectivity of restriction along a flat π-adic thickening
AlgebraicGeometry.appLE_surjective_of_isAffineOpen_preimage_of_isPullback_of_flat0 below · cited by 1 · depth 32 - Global sections over a base detected at maximal ideals
AlgebraicGeometry.bijective_appTop_of_forall_isMaximal1 below · cited by 2 · depth 32 - Global functions on a proper flat scheme over a complete local base
AlgebraicGeometry.bijective_appTop_of_isProper_of_flat_of_isAdicComplete0 below · cited by 1 · depth 32 - Global sections of a fibre and of a field-valued base change
AlgebraicGeometry.bijective_appTop_pullback_snd_of_bijective_appTop_fiberToSpecResidueField1 below · cited by 2 · depth 32 - Bijectivity on sections over U from the restricted morphism
AlgebraicGeometry.bijective_app_of_bijective_appTop_morphismRestrict0 below · cited by 2 · depth 32 - Bijectivity of 𝒪_B → p_*𝒪_X from affine opens
AlgebraicGeometry.bijective_app_of_forall_isAffineOpen0 below · cited by 2 · depth 32 - Base change for Kähler differentials at a stalk of a fibre product
AlgebraicGeometry.bijective_kaehlerDifferential_map_comp_mapBaseChange_stalk_pullback3 below · cited by 1 · depth 32 - Zariski-local epimorphy of affine surjections under flat base change
AlgebraicGeometry.epi_morphismRestrict_of_isPullback_of_flat0 below · cited by 2 · depth 32 - Relative dimension of a smooth k-scheme is chart-independent
AlgebraicGeometry.eq_of_isStandardSmoothOfRelativeDimension_appLE_of_smoothOfRelativeDimension0 below · cited by 1 · depth 32 - Étale fibre from injectivity on dual-number points
AlgebraicGeometry.etale_pullback_snd_of_forall_dualNumber_comp_eq1 below · cited by 1 · depth 32 - Universal flat closed subscheme from affine-point data
AlgebraicGeometry.exists_closedSubscheme_pullback_flat_forall_isPullback_of_forall_spec_point5 below · cited by 1 · depth 32 - A generic point spreads to a geometric point
AlgebraicGeometry.exists_comp_eq_specMap_of_asIdeal_eq_bot_of_locallyOfFiniteType_of_isAlgClosed_of_injective2 below · cited by 1 · depth 32 - Effective faithfully flat descent with rigidified very ample invertible module
AlgebraicGeometry.exists_descent_of_faithfullyFlat_of_closedImmersionBySections_of_rigidified134 below · cited by 1 · depth 32 - Closed equaliser locus of finitely many pairs over a separated target
AlgebraicGeometry.exists_equalizerLocus_isClosedImmersion_of_isSeparated0 below · cited by 2 · depth 32 - Étale lifting along a nilpotent thickening of an affine base
AlgebraicGeometry.exists_etale_isPullback_forall_existsUnique_comp_eq_of_isNilpotent4 below · cited by 1 · depth 32 - Finitely many base-changed morphisms descend to one f.g. subalgebra
AlgebraicGeometry.exists_fg_subalgebra_forall_hom_pullback_of_hom_pullback_of_locallyOfFinitePresentation2 below · cited by 4 · depth 32 - Affine points lift through finite flat étale surjective covers
AlgebraicGeometry.exists_finite_faithfullyFlat_etale_isPullback_specMap_of_isFinite_of_flat_of_etale_of_surjective1 below · cited by 1 · depth 32 - Values of a regular function at K-points are rational in the point
AlgebraicGeometry.exists_forall_gammaSpecIso_appLE_specMap_comp_eq_div_pow0 below · cited by 1 · depth 32 - Natural operations induced on fibre powers of a Hom-scheme
AlgebraicGeometry.exists_hom_fibrePower_homScheme_of_naturalOperation1 below · cited by 1 · depth 32 - Difference of two tangent vectors at a point
AlgebraicGeometry.exists_hom_spec_dualNumber_fst_eq_and_snd_eq_sub0 below · cited by 1 · depth 32 - Finitely generated equaliser ideal of two affine points
AlgebraicGeometry.exists_ideal_eq_bot_iff_eq_and_map_and_fg_of_locallyOfFiniteType0 below · cited by 1 · depth 32 - Affine opens lift along a nilpotent thickening of schemes
AlgebraicGeometry.exists_isAffineOpen_isPullback_restrict_of_isPullback_of_isNilpotent4 below · cited by 2 · depth 32 - Finite sets on proper reduced curves lie in affine opens
AlgebraicGeometry.exists_isAffineOpen_of_finite_of_isProper_of_forall_smoothOfRelativeDimension_one14 below · cited by 1 · depth 32 - Equaliser locus of two S-morphisms is closed
AlgebraicGeometry.exists_isClosedImmersion_iff_comp_eq_of_isSeparated0 below · cited by 2 · depth 32 - Formally smooth birational map is an open immersion near a DVR point
AlgebraicGeometry.exists_isOpenImmersion_of_formallySmooth_stalk_of_isFractionRing_of_isDiscreteValuationRing2 below · cited by 2 · depth 32 - Fibrewise isomorphism locus is open, and an isomorphism there
AlgebraicGeometry.exists_isOpen_mem_iff_isIso_fibre_and_isIso_restrict_of_isProper_of_isProper_of_flat12 below · cited by 3 · depth 32 - Isomorphism locus of a fibrewise map is retrocompact open
AlgebraicGeometry.exists_isOpen_quasiCompact_inclusion_mem_iff_isIso_fibre_of_isProper_of_flat43 below · cited by 1 · depth 32 - Standard-smooth charts restrict to charts on the fibre
AlgebraicGeometry.exists_isStandardSmoothOfRelativeDimension_appLE_fiberToSpecResidueField0 below · cited by 1 · depth 32 - Infinitesimal twist of an affine chart is an automorphism
AlgebraicGeometry.exists_iso_isoSpec_inv_comp_eq_of_specMap_comp_eq0 below · cited by 3 · depth 32 - Local fraction presentation of sections over an affine chart of a base change
AlgebraicGeometry.exists_opens_restrict_mul_eq_restrict_of_mem_closure_chart_sections0 below · cited by 1 · depth 32 - Stalks of a fibre product localise the tensor product
AlgebraicGeometry.exists_ringEquiv_stalk_pullback_localization_tensorProduct_stalk_stalk1 below · cited by 2 · depth 32 - Relative Hilbert scheme with quasi-compact Hilbert-polynomial pieces
AlgebraicGeometry.exists_scheme_represents_flat_lfp_closedSubscheme_hilbertPieces_of_closedImmersionBySections305 below · cited by 1 · depth 32 - Base change of a π-adic tower with group action
AlgebraicGeometry.exists_tower_baseChange_of_isPullback0 below · cited by 1 · depth 32 - Flatness of the πⁿ⁺¹-truncation of a flat algebra
AlgebraicGeometry.flat_specMap_quotientMap_pow_of_flat0 below · cited by 2 · depth 32 - Affine neighbourhoods of finite sets descend along surjective closed immersions
AlgebraicGeometry.forall_finite_exists_isAffineOpen_of_isClosedImmersion_of_surjective0 below · cited by 1 · depth 32 - Constant rank of a finite flat morphism detected after base change along an injection
AlgebraicGeometry.forall_finrank_eq_of_isPullback_of_injective1 below · cited by 2 · depth 32 - Fibre dimension descends from the base change along an injection
AlgebraicGeometry.forall_topologicalKrullDim_preimage_eq_of_isPullback_of_injective_of_isConnected11 below · cited by 1 · depth 32 - Formal unramifiedness from étale fibres over k-points
AlgebraicGeometry.formallyUnramified_of_forall_etale_pullback_snd0 below · cited by 1 · depth 32 - Geometric connectedness from algebraically closed fibres
AlgebraicGeometry.geometricallyConnected_of_forall_connectedSpace_pullback_of_isAlgClosed0 below · cited by 1 · depth 32 - Affineness descends along a surjective square-zero thickening
AlgebraicGeometry.isAffineOpen_of_isAffineOpen_preimage_of_app_surjective_of_mul_eq_zero0 below · cited by 1 · depth 32 - Properness descends through a surjective closed immersion of the base
AlgebraicGeometry.isClosedImmersion_and_surjective_and_isProper_of_isPullback_of_surjective0 below · cited by 7 · depth 32 - Clopen piece of a finite étale cover is finite étale
AlgebraicGeometry.isFinite_and_etale_comp_of_isOpenImmersion_of_isClosed_range0 below · cited by 1 · depth 32 - Fraction field of a stalk under a locally open immersion
AlgebraicGeometry.isFractionRing_stalk_fractionRing_stalk_of_isOpenImmersion_restrict0 below · cited by 2 · depth 32 - Base change of a fibre product of schemes
AlgebraicGeometry.isIso_lift_baseChange_fst_baseChange_snd0 below · cited by 1 · depth 32 - Isomorphism criterion over a nilpotent thickening
AlgebraicGeometry.isIso_of_isPullback_of_isIso_of_isNilpotent_ker3 below · cited by 4 · depth 32 - Étale and universally injective implies open immersion
AlgebraicGeometry.isOpenImmersion_of_etale_of_universallyInjective0 below · cited by 1 · depth 32 - Spec of a ring fibre product along a nilpotent thickening is a pushout of schemes
AlgebraicGeometry.isPushout_specMap_of_isPullback_of_surjective_of_isNilpotent0 below · cited by 9 · depth 32 - Separatedness from a two-chart cover with closed intersection graph
AlgebraicGeometry.isSeparated_of_isOpenImmersion_of_isPullback_of_isClosedImmersion_lift0 below · cited by 1 · depth 32 - Diagonal of a morphism locally of finite type is locally of finite presentation
AlgebraicGeometry.locallyOfFinitePresentation_diagonal_of_locallyOfFiniteType0 below · cited by 2 · depth 32 - Universality of the fibrewise-isomorphism locus U
AlgebraicGeometry.range_subset_iff_isIso_of_isPullback_of_forall_mem_iff_isIso_fibre0 below · cited by 1 · depth 32 - Regular local rings at closed points give smoothness of relative dimension n
AlgebraicGeometry.smoothOfRelativeDimension_of_forall_isClosed_isRegularLocalRing_stalk3 below · cited by 1 · depth 32 - Smoothness of relative dimension d spreads from the generic fibre
AlgebraicGeometry.smoothOfRelativeDimension_of_smooth_of_genericFibre0 below · cited by 2 · depth 32 - Chart endomorphism fixing the reduction agrees with the inclusion mod I
AlgebraicGeometry.specMap_comp_fromSpec_eq_specMap_comp_of_morphismRestrict_comp_eq0 below · cited by 4 · depth 32 - Stalk of a smooth relative curve at a closed point over the closed point
AlgebraicGeometry.stalk_flat_and_maximalIdeal_eq_sup_span_of_smoothOfRelativeDimension_one1 below · cited by 1 · depth 32 - Sections along base change by Spec of a surjection
AlgebraicGeometry.surjective_app_and_ker_eq_map_of_isPullback_specMap_of_surjective0 below · cited by 1 · depth 32 - Base-change invariance of connected smooth fibre dimension
AlgebraicGeometry.topologicalKrullDim_preimage_eq_of_isPullback_of_smooth_of_isConnected6 below · cited by 7 · depth 32 - Morphisms from Spec B are determined by localisations at primes
AlgebraicGeometry.Spec_hom_ext_of_forall_localization_atPrime0 below · cited by 1 · depth 33 - Base change stability of T xrightarrow∼ Γ(A_T)
AlgebraicGeometry.bijective_algebraMap_sections_pullback_of_isPullback_of_forall_bijective0 below · cited by 3 · depth 33 - Joint holomorphy of sections along the fibre square of a relative chart
AlgebraicGeometry.differentiableOn_appLE_pullback_pair_of_relChart2 below · cited by 1 · depth 33 - A closed set containing a dense image is all of Spec R₀
AlgebraicGeometry.eq_univ_of_isClopen_of_range_specMap_subset_of_injective0 below · cited by 3 · depth 33 - Gluing morphisms out of Spec S along orthogonal idempotents
AlgebraicGeometry.existsUnique_specMap_algebraMap_away_comp_eq_of_sum_eq_one_of_orthogonal0 below · cited by 2 · depth 33 - Glued ideal sheaf yields the universal flat closed subscheme
AlgebraicGeometry.exists_closedSubscheme_pullback_flat_of_idealSheafData_comap_ker_eq2 below · cited by 1 · depth 33 - Effective faithfully flat descent for rigidified very ample line bundles
AlgebraicGeometry.exists_descent_of_faithfullyFlat_of_rigidified118 below · cited by 1 · depth 33 - Affine-local models for a representing point datum
AlgebraicGeometry.exists_forall_affineOpens_closedSubscheme_ker_comap_eq_of_forall_spec_point0 below · cited by 1 · depth 33 - An R-dense affine open in a smooth separated R-scheme
AlgebraicGeometry.exists_isAffineOpen_forall_mem_of_forall_specializes_of_smooth_of_isDiscreteValuationRing7 below · cited by 1 · depth 33 - Local isomorphism at a point from an isomorphism of stalks
AlgebraicGeometry.exists_isOpenImmersion_of_isIso_stalkMap_of_locallyOfFiniteType0 below · cited by 2 · depth 33 - Fibrewise isomorphism criterion over a neighbourhood, proper flat case
AlgebraicGeometry.exists_mem_and_isIso_pullbackMap_opens_of_isIso_pullbackMap_fromSpecResidueField9 below · cited by 1 · depth 33 - Dimension <n and affine over P^N implies n affine charts
AlgebraicGeometry.exists_orderedAffineCover_card_eq_of_isAffineHom_of_topologicalKrullDim_lt0 below · cited by 1 · depth 33 - Hilbert point in the disjoint union over Hilbert polynomials
AlgebraicGeometry.exists_pt_sigma_nat_and_eq_comp_sigmaInj_of_forall_represents_hilbertPolynomial_eq205 below · cited by 1 · depth 33 - Hilbert scheme of fixed Hilbert polynomial is proper
AlgebraicGeometry.exists_scheme_represents_flat_lfp_closedSubscheme_hilbertPolynomial_eq_of_closedImmersionBySections296 below · cited by 1 · depth 33 - Unique Artin-local liftings imply formally unramified
AlgebraicGeometry.formallyUnramified_of_forall_lift_unique_of_isArtinianRing0 below · cited by 3 · depth 33 - Geometric connectedness over an Artinian local base
AlgebraicGeometry.geometricallyConnected_of_isConnected_preimage_of_isArtinianRing_of_isAlgClosed61 below · cited by 1 · depth 33 - Affine charts of a finite flat surjective morphism are faithfully flat
AlgebraicGeometry.isAffineOpen_preimage_and_faithfullyFlat_of_isFinite_of_flat_of_surjective0 below · cited by 4 · depth 33 - Products of affine opens over an affine base, with generators
AlgebraicGeometry.isAffineOpen_pullback_fst_preimage_inf_snd_preimage_and_closure_eq_top2 below · cited by 1 · depth 33 - Reduced scheme covered by finitely many affine closed subschemes is affine
AlgebraicGeometry.isAffine_of_isClosedImmersion_of_isAffine_of_iUnion_range_eq_univ3 below · cited by 1 · depth 33 - Affineness descends along surjective closed immersions
AlgebraicGeometry.isAffine_of_isClosedImmersion_of_surjective3 below · cited by 4 · depth 33 - Clopen locus of fibres of given dimension for smooth proper morphisms
AlgebraicGeometry.isClopen_setOf_topologicalKrullDim_preimage_eq_of_smooth_of_isProper6 below · cited by 1 · depth 33 - Local cutting-out by ideals implies closedness
AlgebraicGeometry.isClosed_of_forall_exists_isOpenImmersion_forall_mem_iff_le0 below · cited by 1 · depth 33 - Quasi-compactness of the fibrewise isomorphism locus over an affine open
AlgebraicGeometry.isCompact_inter_setOf_isIso_fibre_of_isAffineOpen_of_isProper_of_flat42 below · cited by 1 · depth 33 - Isomorphism over an open neighbourhood gives isomorphism on residue fibre
AlgebraicGeometry.isIso_pullbackMap_fromSpecResidueField_of_isIso_pullbackMap_opens0 below · cited by 1 · depth 33 - Being an isomorphism over the base is local on the base
AlgebraicGeometry.isIso_pullbackMap_opens_of_forall_exists_le_isIso_pullbackMap_opens0 below · cited by 1 · depth 33 - Sections over p⁻¹V localise the sections over an affine open
AlgebraicGeometry.isLocalization_map_app_pullback_fst_preimage_of_isAffineOpen2 below · cited by 3 · depth 33 - Being cartesian is Zariski-local on one corner
AlgebraicGeometry.isPullback_of_iSup_eq_top0 below · cited by 1 · depth 33 - Fibre product of two base changes is the base change of the fibre product
AlgebraicGeometry.isPullback_pullbackMap_fst_comp_of_isPullback_of_isPullback0 below · cited by 5 · depth 33 - Spec R[1/fg] as fibre product of Spec R[1/f], Spec R[1/g]
AlgebraicGeometry.isPullback_specMap_awayToAwayRight_awayToAwayLeft0 below · cited by 3 · depth 33 - Reducedness, dimension ≤ 1 and infinite components from affine charts
AlgebraicGeometry.isReduced_and_isClosed_or_mem_irreducibleComponents_and_infinite_of_openCover0 below · cited by 1 · depth 33 - Representability, separatedness and Hilbert-polynomial strata of coprod_P C_P
AlgebraicGeometry.surj_inj_isSeparated_pieces_sigmaDesc_of_forall_represents_hilbertPolynomial_eq_of_nat_of_eq_comp_sigmaInj207 below · cited by 1 · depth 33 - Surjectivity from lifting of K-rational points
AlgebraicGeometry.surjective_of_forall_exists_comp_eq_of_isAlgClosed1 below · cited by 1 · depth 33 - Proper flat morphisms with geometrically reduced connected fibres: mathcal O_B ≅ p_*mathcal O_X
AlgebraicGeometry.bijective_app_of_isProper_of_flat_of_geometricallyReduced_of_geometricallyConnected_of_isLocallyNoetherian12 below · cited by 1 · depth 34 - Global sections of an affine fibre product are generated by the two factors
AlgebraicGeometry.closure_range_pullback_fst_appTop_union_range_snd_appTop_eq_top0 below · cited by 1 · depth 34 - Sections of a base-changed π-adic tower over pulled-back affine opens
AlgebraicGeometry.exists_addMonoidHom_tmul_sections_preimage_of_isPullback0 below · cited by 1 · depth 34 - Sections of an abstract base change on preimages of affine opens
AlgebraicGeometry.exists_algEquiv_sections_preimage_tensor_of_isPullback_of_isAffineOpen2 below · cited by 1 · depth 34 - Factorisation through closed subschemes detected on infinitesimal thickenings
AlgebraicGeometry.exists_comp_eq_iff_of_forall_quotient_maximalIdeal_pow_of_isProper1 below · cited by 1 · depth 34 - Local constancy of geometric fibre Hilbert polynomials
AlgebraicGeometry.exists_cover_forall_geomFibreH0Finrank_tensorPow_eq_eval_of_isClosedImmersion_of_flat_of_locallyOfFinitePresentation202 below · cited by 2 · depth 34 - Effective faithfully flat descent for relatively very ample invertible modules
AlgebraicGeometry.exists_descent_of_faithfullyFlat_of_cocycle109 below · cited by 1 · depth 34 - Descent of a morphism of proper flat schemes to a f.g. subalgebra
AlgebraicGeometry.exists_fg_subalgebra_isPullback_isPullback_comp_eq_of_isProper_of_flat_of_locallyOfFinitePresentation26 below · cited by 1 · depth 34 - Geometric fibre Hilbert function read off a defining ideal
AlgebraicGeometry.exists_forall_geomFibreH0Finrank_tensorPow_eq_hilbertFunctionOf_of_point_geomFibre_of_hom57 below · cited by 2 · depth 34 - Equality with a finite flat closed subscheme is a closed condition
AlgebraicGeometry.exists_idealSheafData_comap_eq_bot_iff_of_isClosedImmersion_of_isFinite_of_flat0 below · cited by 1 · depth 34 - Equality of two morphisms over a finitely generated ideal
AlgebraicGeometry.exists_ideal_fg_forall_pullback_fst_comp_eq_iff_map_eq_bot_of_isProper_of_flat345 below · cited by 1 · depth 34 - Finitely many discrete valuation points lie in one affine open
AlgebraicGeometry.exists_isAffineOpen_forall_mem_of_isDiscreteValuationRing_stalk1 below · cited by 1 · depth 34 - Transporting compatible line bundles across adic thickenings
AlgebraicGeometry.exists_isInvertible_adicThickening_forall_nonempty_pullback_iso_of_forall_pullback_algebraMap_quotient0 below · cited by 1 · depth 34 - Fibre dimension lower bound at a closed point
AlgebraicGeometry.exists_isIrreducible_topologicalKrullDim_le_add_of_apply_eq_of_isClosed_singleton2 below · cited by 1 · depth 34 - Normal proper model with a proper retraction onto an open
AlgebraicGeometry.exists_isProper_isIntegrallyClosed_stalk_isOpenImmersion_comp_eq_of_isSeparated8 below · cited by 1 · depth 34 - Making the boundary of a normal complete model divisorial
AlgebraicGeometry.exists_isProper_notMem_ringKrullDim_stalk_eq_one_of_not_isProper12 below · cited by 1 · depth 34 - Base change of A×_S A along S→ K, with slices
AlgebraicGeometry.exists_isPullback_fibre_prod_and_slices_of_section0 below · cited by 2 · depth 34 - Smooth non-empty K-schemes have finite separable points
AlgebraicGeometry.exists_isSeparable_specMap_comp_eq_of_smooth_of_nonempty0 below · cited by 2 · depth 34 - Isomorphism on one fibre gives closed immersion near that fibre
AlgebraicGeometry.exists_mem_and_isClosedImmersion_pullbackMap_opens_of_isIso_pullbackMap_fromSpecResidueField2 below · cited by 1 · depth 34 - Isomorphism over a neighbourhood from an isomorphic fibre
AlgebraicGeometry.exists_mem_and_isIso_pullbackMap_opens_of_isClosedImmersion_pullbackMap_opens5 below · cited by 1 · depth 34 - Chart ring of a flat varpi-adic tower with G-action
AlgebraicGeometry.exists_ringEquiv_quotient_sections_of_isPullback_of_flat0 below · cited by 1 · depth 34 - Dense test family forces equality of morphisms into a separated target
AlgebraicGeometry.ext_of_isSeparated_of_dense_iUnion_range_of_comp_eq0 below · cited by 1 · depth 34 - Product of affine opens is affine in a fibre product
AlgebraicGeometry.isAffineOpen_pullback_fst_preimage_inf_snd_preimage0 below · cited by 1 · depth 34 - A reduced scheme covered by two affine closed subschemes is affine
AlgebraicGeometry.isAffine_of_isClosedImmersion_of_isAffine_of_range_union_range1 below · cited by 1 · depth 34 - Nilpotent thickenings of affine schemes are affine
AlgebraicGeometry.isAffine_of_isClosedImmersion_of_isNilpotent_ker1 below · cited by 1 · depth 34 - Closed immersions descend along faithfully flat base change
AlgebraicGeometry.isClosedImmersion_of_isPullback_of_faithfullyFlat0 below · cited by 1 · depth 34 - Fibrewise isomorphy is insensitive to base change of Y
AlgebraicGeometry.isIso_fibre_iff_isIso_fibre_of_isPullback_of_isPullback1 below · cited by 1 · depth 34 - Cartesian squares of schemes are Zariski-local on a corner
AlgebraicGeometry.isPullback_of_openCover_of_isPullback_morphismRestrict0 below · cited by 1 · depth 34 - Separatedness of a coproduct of separated morphisms
AlgebraicGeometry.isSeparated_sigmaDesc_of_forall_isSeparated0 below · cited by 1 · depth 34 - Constructible sets contain generising points of their image's closure
AlgebraicGeometry.mem_image_of_mem_closure_image_of_forall_specializes1 below · cited by 1 · depth 34 - Surjectivity detected on closed points over a Jacobson base
AlgebraicGeometry.surjective_of_closedPoints_subset_range0 below · cited by 1 · depth 34 - mathcal O_B → p_*mathcal O_X bijective over a locally Noetherian base
AlgebraicGeometry.bijective_app_of_isProper_of_flat_of_forall_bijective_appTop_fiberToSpecResidueField_of_isLocallyNoetherian10 below · cited by 1 · depth 35 - From one-point test objects to all schemes over ̄ k
AlgebraicGeometry.eq_comp_of_forall_appTop_eq_zero_of_forall_subsingleton0 below · cited by 1 · depth 35 - Uniqueness of morphisms agreeing on all adic thickenings
AlgebraicGeometry.eq_of_forall_adicThickening_comp_eq_of_isAdicComplete_of_isClosedImmersion_proj2 below · cited by 2 · depth 35 - Algebraisation of compatible morphisms between adic thickenings
AlgebraicGeometry.existsUnique_hom_forall_adicThickening_comp_eq_of_isAdicComplete_of_isClosedImmersion_proj110 below · cited by 2 · depth 35 - Affine neighbourhood on which a section is invertible along fibres
AlgebraicGeometry.exists_affineOpens_le_preimage_le_basicOpen_of_universallyClosed0 below · cited by 1 · depth 35 - Proper birational morphisms are local isomorphisms at one-dimensional normal points
AlgebraicGeometry.exists_base_eq_isIso_stalkMap_of_isProper_of_ringKrullDim_stalk_eq_one0 below · cited by 1 · depth 35 - Generic point of G×_k w̄ and its local dimension
AlgebraicGeometry.exists_closure_eq_preimage_closure_and_ringKrullDim_stalk_eq_of_flat0 below · cited by 2 · depth 35 - Chart-level coaction induced by an action on N⁻¹U
AlgebraicGeometry.exists_coaction_affineOpens_eq_comp_appLE_of_preimage_comp_eq0 below · cited by 2 · depth 35 - Sections on U ∩ g⁻¹V as sums sum a_k · g^sharp b_k
AlgebraicGeometry.exists_eq_sum_mul_appLE_of_isSeparated_of_isAffineOpen0 below · cited by 1 · depth 35 - Unramified point of a map between smooth k-schemes is étale
AlgebraicGeometry.exists_etale_opensInclusion_comp_of_formallyUnramified_stalkMap_of_smoothOfRelativeDimension1 below · cited by 1 · depth 35 - Finite flat finitely presented schemes descend to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_isFinite_flat_isPullback_of_isFinite_of_flat_of_locallyOfFinitePresentation26 below · cited by 1 · depth 35 - Cotangent space at a rational point, read on an affine chart
AlgebraicGeometry.exists_finrank_cotangent_chart_eq_of_smoothOfRelativeDimension2 below · cited by 1 · depth 35 - Noetherian stabilisation of a tower of natural GL_d-conditions
AlgebraicGeometry.exists_forall_generalLinearGroup_eq_one_of_forall_le_of_isNoetherian0 below · cited by 1 · depth 35 - Hilbert function of a geometric fibre from a Hilbert point
AlgebraicGeometry.exists_forall_geomFibreH0Finrank_tensorPow_eq_hilbertFunctionOf_of_point_geomFibre56 below · cited by 1 · depth 35 - Affine neighbourhoods descend along finite locally free surjections
AlgebraicGeometry.exists_isAffineOpen_forall_mem_of_forall_preimage_mem_of_isFinite_of_flat_of_surjective5 below · cited by 1 · depth 35 - Finite sets in an open of an affine scheme lie in an affine open
AlgebraicGeometry.exists_isAffineOpen_subset_of_finite_of_isAffine0 below · cited by 1 · depth 35 - Closed subfunctor criterion with finitely generated chart ideals
AlgebraicGeometry.exists_isClosedImmersion_locallyOfFinitePresentation_forall_factors_iff_of_fg_idealCut3 below · cited by 2 · depth 35 - Normalisation of an integral scheme of finite type over a field
AlgebraicGeometry.exists_isFinite_isIntegrallyClosed_stalk_isIso_morphismRestrict_of_isIntegral5 below · cited by 3 · depth 35 - Codimension-one points after a birational modification of a proper target
AlgebraicGeometry.exists_isProper_isIso_morphismRestrict_ringKrullDim_stalk_eq_one_of_ringKrullDim_stalk_eq_one16 below · cited by 1 · depth 35 - Proper modification making the centre of a valuation one-dimensional
AlgebraicGeometry.exists_isProper_ringKrullDim_stalk_eq_one_of_valuationSubring_functionField6 below · cited by 2 · depth 35 - Chow's lemma, birational form, over a Noetherian base
AlgebraicGeometry.exists_isProper_surjective_isOpenImmersion_isIntegral_isIso_morphismRestrict_of_isIntegral1 below · cited by 1 · depth 35 - Cartesian square over a lifted open in a chart
AlgebraicGeometry.exists_isPullback_homOfLE_morphismRestrict_comp_openCover_lift0 below · cited by 4 · depth 35 - Closed immersion over a neighbourhood from fibrewise isomorphisms
AlgebraicGeometry.exists_mem_and_isClosedImmersion_pullbackMap_opens_of_isFinite_morphismRestrict_of_isIso_fiberToSpecResidueField0 below · cited by 1 · depth 35 - Flat closed immersion, isomorphic on one fibre, is locally an isomorphism
AlgebraicGeometry.exists_mem_and_isIso_morphismRestrict_of_isClosedImmersion_pullbackMap_opens_of_isIso_pullbackMap_fromSpecResidueField3 below · cited by 1 · depth 35 - Isomorphism near a fibre spreads to an open neighbourhood of y
AlgebraicGeometry.exists_mem_and_isIso_pullbackMap_opens_of_forall_exists_isIso_morphismRestrict0 below · cited by 1 · depth 35 - Sections of an affine pullback over an affine chart
AlgebraicGeometry.exists_ringEquiv_sections_pullback_tensor_of_isAffineHom_of_isAffineOpen0 below · cited by 3 · depth 35 - Shear isomorphism on a chart: S⊗_R S≅ S⊗_K H
AlgebraicGeometry.exists_ringEquiv_shear_of_isIso_pullback_lift0 below · cited by 2 · depth 35 - Representability of the relative morphism scheme Mor_S(X,Y)
AlgebraicGeometry.exists_scheme_represents_schemeHomOver_of_isProper_of_flat342 below · cited by 1 · depth 35 - Rank of a finite scheme over ̄ k as sum of stalk lengths
AlgebraicGeometry.finrank_eq_finsum_length_stalk_of_isFinite_of_isAlgClosed0 below · cited by 1 · depth 35 - Geometrically connected: proper with connected fibres and a section
AlgebraicGeometry.geometricallyConnected_of_isConnected_preimage_of_section_of_isArtinianRing62 below · cited by 2 · depth 35 - Square-zero thickenings of affine schemes are affine
AlgebraicGeometry.isAffine_of_isClosedImmersion_of_ker_mul_ker_eq_bot0 below · cited by 1 · depth 35 - Fibrewise isomorphism over κ(y) gives finiteness near y
AlgebraicGeometry.isIso_fiberToSpecResidueField_and_exists_isFinite_morphismRestrict_of_isIso_pullbackMap_fromSpecResidueField0 below · cited by 1 · depth 35 - Finite of rank one over a field is an isomorphism
AlgebraicGeometry.isIso_of_isFinite_of_finrank_closedPoint_eq_one0 below · cited by 1 · depth 35 - Finite morphism with a section and unique geometric test points is an isomorphism
AlgebraicGeometry.isIso_of_isFinite_of_section_of_forall_isAlgClosed_hom_eq0 below · cited by 1 · depth 35 - Isomorphy descends along flat surjective quasi-compact base change
AlgebraicGeometry.isIso_of_isIso_of_isPullback_of_flat_of_surjective0 below · cited by 1 · depth 35 - Product of graph kernels under cartesian base change
AlgebraicGeometry.prodKerGraph_comap_fst_eq_prodKerGraph_comap_of_isPullback3 below · cited by 1 · depth 35 - Codimension plus dimension of a closure on an integral k-scheme
AlgebraicGeometry.ringKrullDim_stalk_add_topologicalKrullDim_closure_of_isIntegral2 below · cited by 2 · depth 35 - Isomorphic non-empty opens force equal dimension
AlgebraicGeometry.topologicalKrullDim_eq_of_iso_opens2 below · cited by 1 · depth 35 - Dimension of an integral finite-type k-scheme from any affine chart
AlgebraicGeometry.topologicalKrullDim_eq_ringKrullDim_of_isAffineOpen_of_isIntegral1 below · cited by 5 · depth 35 - Fibre isomorphism gives bijection on sections after base change to κ(y)
AlgebraicGeometry.bijective_rTensor_residueField_appLE_of_isIso_pullbackMap_fromSpecResidueField0 below · cited by 1 · depth 36 - Morphisms agreeing on all jets at a point are equal
AlgebraicGeometry.eq_of_forall_specMap_quotient_maximalIdeal_pow_comp_eq_of_isSchemeTheoreticallyDominant1 below · cited by 1 · depth 36 - Factoring through an infinitesimal neighbourhood of a rational point
AlgebraicGeometry.exists_comp_specMap_quotient_maximalIdeal_pow_eq_of_section_comp_eq1 below · cited by 1 · depth 36 - Spreading out a factorisation to a finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_comp_eq_pullback_fst_comp_of_comp_eq_of_locallyOfFinitePresentation2 below · cited by 2 · depth 36 - Finitely many identities descend to one finitely generated subalgebra
AlgebraicGeometry.exists_fg_subalgebra_forall_pullback_fst_comp_eq_of_locallyOfFiniteType2 below · cited by 2 · depth 36 - Algebraisation of a compatible system of closed subschemes of adic thickenings
AlgebraicGeometry.exists_idealSheafData_forall_comap_adicThickening_eq_of_isAdicComplete_of_isClosedImmersion_proj102 below · cited by 1 · depth 36 - Weil restriction of a closed subscheme along a finite flat morphism
AlgebraicGeometry.exists_ideal_fg_forall_exists_comp_eq_pullback_fst_iff_map_eq_bot_of_isFinite_of_flat1 below · cited by 1 · depth 36 - Representability of a closed subfunctor cut by local ideals
AlgebraicGeometry.exists_isClosedImmersion_forall_factors_iff_of_idealCut2 below · cited by 1 · depth 36 - Graph closure: proper modification extending a map to a proper scheme
AlgebraicGeometry.exists_isProper_isOpenImmersion_range_eq_preimage_comp_eq_of_isProper0 below · cited by 1 · depth 36 - Lifting units of global sections along a surjection of local bases
AlgebraicGeometry.exists_isUnit_appTop_eq_of_bijective_of_surjective_of_ker_le_maximalIdeal0 below · cited by 2 · depth 36 - Deformations of an affine chart modulo a square-zero ideal are automorphisms
AlgebraicGeometry.exists_iso_of_specMap_quotient_comp_eq_fromSpec1 below · cited by 1 · depth 36 - Spreading out agreement of two closed subschemes of a proper scheme
AlgebraicGeometry.exists_not_mem_forall_factorsThrough_iff_of_forall_atPrime_of_isProper2 below · cited by 2 · depth 36 - Spreading out a lift over the spectrum of a local ring
AlgebraicGeometry.exists_opens_comp_eq_base_eq_of_isLocalHom0 below · cited by 1 · depth 36 - Restricting a codimension-one valuation along a dominant map
AlgebraicGeometry.exists_valuationSubring_functionField_of_ringKrullDim_stalk_eq_one9 below · cited by 1 · depth 36 - Self-fibre-product of a base change is affine and cartesian
AlgebraicGeometry.isAffineHom_and_isPullback_pullback_lift_of_isPullback0 below · cited by 3 · depth 36 - Graphs of compatible morphisms of adic thickenings are closed immersions
AlgebraicGeometry.isClosedImmersion_and_comap_ker_eq_ker_of_adicThickening_graph1 below · cited by 1 · depth 36 - Traces on adic thickenings of a closed subscheme cut out by graphs
AlgebraicGeometry.isIso_and_adicThickening_comp_eq_of_comap_eq_ker_of_comp_eq0 below · cited by 1 · depth 36 - Section that is an open immersion onto the preimage
AlgebraicGeometry.isIso_morphismRestrict_of_isOpenImmersion_of_range_eq0 below · cited by 1 · depth 36 - Adic thickenings detect isomorphisms of projective R-schemes
AlgebraicGeometry.isIso_of_forall_isIso_adicThickening_of_isAdicComplete_of_isClosedImmersion_proj0 below · cited by 2 · depth 36 - Birational form of Zariski's main theorem
AlgebraicGeometry.isOpenImmersion_of_locallyQuasiFinite_of_isIntegrallyClosed_stalk_of_denseRange2 below · cited by 2 · depth 36 - Spec turns pushouts of R-algebras into pullback squares
AlgebraicGeometry.isPullback_Spec_map_pushout_inl_right_inr_right0 below · cited by 1 · depth 36 - Maximal ideal at a rational point of a fibre product over a field
AlgebraicGeometry.maximalIdeal_stalk_pullback_le_of_sections1 below · cited by 1 · depth 36 - Restriction of a chartwise lift to the opens devices
AlgebraicGeometry.exists_comp_openInclusion_eq_openInclusion_comp_of_local_lift0 below · cited by 1 · depth 37 - Factoring a morphism over a direct limit through a finite stage
AlgebraicGeometry.exists_eq_comp_hom_pullback_specMap_of_isDirectLimit_of_locallyOfFinitePresentation0 below · cited by 1 · depth 37 - Agreement on all jets at a point implies agreement near it
AlgebraicGeometry.exists_mem_and_iota_comp_eq_of_forall_specMap_quotient_maximalIdeal_pow_comp_eq0 below · cited by 1 · depth 37 - Generic flatness for schemes
AlgebraicGeometry.exists_nonempty_flat_preimage_iota_comp_of_isReduced_of_locallyOfFiniteType1 below · cited by 1 · depth 37 - Uniqueness in EGA IV 8.8.2 for finite-type targets
AlgebraicGeometry.exists_pullback_fst_comp_eq_of_isDirectLimit_of_locallyOfFiniteType0 below · cited by 1 · depth 37 - Clopen subscheme containing a section, connected fibres
AlgebraicGeometry.isIso_of_isOpenImmersion_of_isClosedImmersion_of_section_of_isConnected_fibres0 below · cited by 1 · depth 37 - Open immersion criterion via Artin-local infinitesimal lifting
AlgebraicGeometry.isOpenImmersion_of_mono_of_forall_exists_lift_of_isArtinianRing_of_isLocallyNoetherian2 below · cited by 1 · depth 37 - Adic thickenings commute with fibre products
AlgebraicGeometry.isPullback_adicThickening_pullback0 below · cited by 1 · depth 37 - Base change of a self-product is cartesian
AlgebraicGeometry.isPullback_lift_fst_comp_fst_snd_comp_fst_prodStr0 below · cited by 2 · depth 37 - Maximal ideal of a fibre product at a pair of rational points
AlgebraicGeometry.maximalIdeal_stalk_pullback_eq_map_stalkMap_fst_sup_map_stalkMap_snd_of_section0 below · cited by 1 · depth 37 - Maximal ideal at a rational point of an affine fibre product
AlgebraicGeometry.maximalIdeal_stalk_pullback_le_of_sections_of_isAffine0 below · cited by 1 · depth 37 - Points formula for an action on an affine chart
AlgebraicGeometry.pullback_lift_comp_eq_specMap_lift_comp_comp_fromSpec_of_chart0 below · cited by 3 · depth 37 - Transcendence degree of a residue field equals dimx̄
AlgebraicGeometry.toENat_trdeg_residueField_eq_topologicalKrullDim_closure1 below · cited by 1 · depth 37 - Point derivations give dual-number tangent points
AlgebraicGeometry.exists_tangentPoints_appLE_eq_of_pointDerivations0 below · cited by 1 · depth 38 - A smooth k-scheme with a unique k-point is Spec k
AlgebraicGeometry.isIso_of_smooth_of_subsingleton_of_isAlgClosed1 below · cited by 1 · depth 38 - Infinitesimal lifting criterion for a monomorphism to be an open immersion
AlgebraicGeometry.isOpenImmersion_of_mono_of_forall_exists_lift_of_isArtinianRing_of_isLocalRing1 below · cited by 1 · depth 38 - Cartesianness of the product chart over Specπ
AlgebraicGeometry.isPullback_pullback_lift_morphismRestrict_of_isPullback0 below · cited by 1 · depth 38 - Chart automorphism from an algebra automorphism congruent to the identity
AlgebraicGeometry.exists_iso_comp_isoSpec_hom_eq_of_algEquiv_of_sub_mem0 below · cited by 1 · depth 39 - Reduced schemes of finite type in characteristic 0 are geometrically reduced
AlgebraicGeometry.geometricallyReduced_of_isReduced_of_charZero3 below · cited by 1 · depth 39 - Finitely many k-points forces finiteness over k
AlgebraicGeometry.isFinite_comp_of_isClosedImmersion_of_finite_setOf_exists_comp_eq0 below · cited by 1 · depth 39 - Surjectivity of Spec for nilpotent thickenings
AlgebraicGeometry.surjective_specMap_of_surjective_of_ker_le_nilradical0 below · cited by 1 · depth 39 - Grothendieck existence for finite morphisms over projective X
AlgebraicGeometry.exists_isFinite_of_forall_isFinite_isPullback_of_isClosedImmersion_proj_of_isAdicComplete110 below · cited by 1 · depth 40 - Flatness of a proper morphism from flat 𝔪-adic truncations
AlgebraicGeometry.flat_of_forall_flat_pullback_snd_specMap_quotient_maximalIdeal_pow_of_isProper3 below · cited by 1 · depth 40 - Flatness of R/J⊗_Rmathcal O_{Z,z} from flat base change
AlgebraicGeometry.flat_quotient_tensor_stalk_of_flat_pullback_snd_specMap_quotientMk0 below · cited by 1 · depth 41 - Algebraisation of morphisms between projective schemes over a complete ring
AlgebraicGeometry.exists_hom_comp_eq_forall_pullback_fst_comp_eq_of_forall_truncation_of_isFinite_proj_of_isAdicComplete217 below · cited by 4 · depth 42 - Fibre products of schemes finite over projective spaces
AlgebraicGeometry.exists_isFinite_projSpace_pullback_of_isFinite_projSpace1 below · cited by 2 · depth 42 - Gluing schemes along nilpotent thickenings over a ring fibre product
AlgebraicGeometry.exists_isPushout_isPullback_specMap_pullbackFst_pullbackSnd_of_surjective_of_isNilpotent9 below · cited by 2 · depth 42 - Isomorphism criterion modulo a nilpotent ideal, flat source
AlgebraicGeometry.isIso_of_isIso_of_isPullback_specMap_of_surjective_of_isNilpotent_of_flat_left0 below · cited by 3 · depth 42 - Morphisms to a separated scheme are determined by their adic truncations
AlgebraicGeometry.eq_of_forall_pullback_fst_truncation_comp_eq_of_isProper_of_isSeparated_of_isAdicComplete1 below · cited by 1 · depth 43 - Gluing schemes along two nilpotent base thickenings
AlgebraicGeometry.exists_isPullback_isPushout_flat_of_surjective_of_isNilpotent_pullbackRing7 below · cited by 1 · depth 43 - Descent of properties along a nilpotent thickening of the base
AlgebraicGeometry.isClosedImmersion_and_isProper_and_smooth_of_isPullback_specMap_of_surjective_of_isNilpotent_ker0 below · cited by 1 · depth 43 - Proper morphism iso on all I-adic truncations is iso
AlgebraicGeometry.isIso_of_isProper_of_forall_isIso_pullback_snd_truncation_of_isAdicComplete3 below · cited by 1 · depth 43 - Fibre products of flat glued schemes are push-outs
AlgebraicGeometry.isPushout_pullbackMap_of_isPushout_of_isPushout_of_flat12 below · cited by 2 · depth 43 - Isomorphism criterion for finite morphisms via I-adic truncations
AlgebraicGeometry.isIso_of_isFinite_of_forall_isIso_pullback_snd_truncation_of_isAdicComplete2 below · cited by 1 · depth 44 - Affine chart for gluing along a nilpotent thickening
AlgebraicGeometry.isPullback_isPushout_specMap_of_isPullback_pullbackRing_of_isPushout_of_surjective_of_isNilpotent2 below · cited by 1 · depth 44 - Flat schemes over a ring fibre product are pushouts
AlgebraicGeometry.isPushout_of_flat_of_isPullback_specMap_pullbackFst_pullbackSnd11 below · cited by 2 · depth 44 - Closed immersion detected by all I-adic truncations
AlgebraicGeometry.isClosedImmersion_of_isFinite_of_forall_isClosedImmersion_pullback_snd_truncation_of_isAdicComplete1 below · cited by 1 · depth 45 - Proper flat geometrically integral group schemes are commutative
AlgebraicGeometry.isCommMonObj_of_isProper_of_flat_of_geometricallyIntegral43 below · cited by 1 · depth 45
AlgebraicGeometry.AdmissibleAlgebra 6
- Invariants of an admissible π-adic algebra under a finite group
AlgebraicGeometry.AdmissibleAlgebra.fixedPoints_isAdicComplete_and_finite_and_finiteType0 below · cited by 3 · depth 31 - Invariants of a flat base change of an admissible G-algebra
AlgebraicGeometry.AdmissibleAlgebra.isAdicComplete_fixedPoints_and_exists_ringEquiv_quotient_of_ringEquiv_tensorProduct_quotient5 below · cited by 1 · depth 35 - Descent of approximate G-invariants under flat base change
AlgebraicGeometry.AdmissibleAlgebra.exists_forall_sub_tmul_mem_span_pow_of_flat4 below · cited by 1 · depth 36 - Uniform exponent for t-power torsion in H¹(G,R)
AlgebraicGeometry.AdmissibleAlgebra.exists_forall_cocycle_pow_smul_eq_coboundary0 below · cited by 1 · depth 37 - Descent of approximate invariants along powers of t
AlgebraicGeometry.AdmissibleAlgebra.exists_smul_eq_sub_of_forall_smul_sub_mem_span_pow0 below · cited by 1 · depth 37 - Bounded t-power torsion in homology survives flat base change
AlgebraicGeometry.AdmissibleAlgebra.forall_pow_smul_mem_range_rTensor_of_flat0 below · cited by 1 · depth 37
AlgebraicGeometry.AffineLimit 3
- Morphisms to a finitely presented R-scheme and f.g. subalgebras
AlgebraicGeometry.AffineLimit.homIsLFP_of_locallyOfFinitePresentation2 below · cited by 3 · depth 16 - Locally of finite type from factorisation through f.g. subalgebras
AlgebraicGeometry.AffineLimit.locallyOfFiniteType_of_forall_exists_fg_factor0 below · cited by 1 · depth 16 - From finite-type test schemes to all: open charts for locally finitely presented sheaves
AlgebraicGeometry.AffineLimit.presheafULift_isOpenImmersion_and_isLocallySurjective_of_locallyOfFiniteType0 below · cited by 3 · depth 16
AlgebraicGeometry.ChowDatum 1
- Chow's lemma for proper integral schemes over a Noetherian ring
AlgebraicGeometry.ChowDatum.nonempty0 below · cited by 2 · depth 18
AlgebraicGeometry.ChowDatumProj 1
- Chow datum in a product of projective spaces yields one in P^N
AlgebraicGeometry.ChowDatumProj.nonempty_of0 below · cited by 2 · depth 18
AlgebraicGeometry.DescentAction 3
- Effectivity of descent data along a finite étale base change
AlgebraicGeometry.DescentAction.effective_of_finiteEtale7 below · cited by 2 · depth 16 - Effectivity of a descent action with affine orbits
AlgebraicGeometry.DescentAction.effective_of_finiteEtale_of_forall_orbit7 below · cited by 1 · depth 17 - Effective descent from a flat surjective quasi-compact kernel pair
AlgebraicGeometry.DescentAction.effective_of_isPullback_of_flat_surjective0 below · cited by 2 · depth 17
AlgebraicGeometry.DescentCharacter 13
- Endomorphisms of an invertible module as unique base constants
AlgebraicGeometry.DescentCharacter.existsUnique_isBaseScalar_of_isInvertible_of_bijective1 below · cited by 3 · depth 40 - Unique descent of an isomorphism with trivial descent character
AlgebraicGeometry.DescentCharacter.existsUnique_iso_mapIso_eq_of_hasValue_one3 below · cited by 1 · depth 40 - Effectivity of descent data for invertible modules along affine faithfully flat maps
AlgebraicGeometry.DescentCharacter.exists_isInvertible_iso_comp_eq_of_cocycle28 below · cited by 1 · depth 40 - Descent character values multiply under composition
AlgebraicGeometry.DescentCharacter.hasValue_comp_of_comp_eq1 below · cited by 1 · depth 40 - Base change of descent-character values along commuting squares
AlgebraicGeometry.DescentCharacter.hasValue_map_pullback_of_comm_sq1 below · cited by 1 · depth 40 - Base-scalar automorphisms have descent value one
AlgebraicGeometry.DescentCharacter.hasValue_one_of_isBaseScalar1 below · cited by 2 · depth 40 - Pulled-back isomorphisms have descent value one
AlgebraicGeometry.DescentCharacter.hasValue_pullback_mapIso_one0 below · cited by 3 · depth 40 - Inverse identification carries the inverse descent value
AlgebraicGeometry.DescentCharacter.hasValue_symm_of_mul_eq_one0 below · cited by 2 · depth 40 - Multiplicativity of descent-character values under tensor product
AlgebraicGeometry.DescentCharacter.hasValue_tensor4 below · cited by 1 · depth 40 - Multiplicativity of the descent character under composition
AlgebraicGeometry.DescentCharacter.hasValue_trans0 below · cited by 4 · depth 40 - Transported descended isomorphism has descent-character value c
AlgebraicGeometry.DescentCharacter.hasValue_transport_of_pullback_map_comp_sectionScalar_eq_of_appTop_eq0 below · cited by 1 · depth 40 - Base-scalar endomorphisms pull back along φ
AlgebraicGeometry.DescentCharacter.isBaseScalar_pullback_map0 below · cited by 4 · depth 40 - Multiplication by a unit-and-cocycle function gives a descent datum
AlgebraicGeometry.DescentCharacter.pullback_map_sectionScalar_comp_canonical_unit_and_cocycle0 below · cited by 1 · depth 40
AlgebraicGeometry.Etale 6
- Étale over a normal affine base: stalks are normal domains
AlgebraicGeometry.Etale.isDomain_and_isIntegrallyClosed_stalk1 below · cited by 1 · depth 19 - Étaleness over Spec R is detected on localisations R_𝔭
AlgebraicGeometry.Etale.of_forall_pullback_snd_localization_atPrime2 below · cited by 3 · depth 28 - Infinitesimal criterion: unique lifting against affine square-zero extensions gives étale
AlgebraicGeometry.Etale.of_forall_existsUnique_lift0 below · cited by 1 · depth 29 - Artin-local infinitesimal criterion for étaleness over a Noetherian base
AlgebraicGeometry.Etale.of_forall_existsUnique_lift_of_isArtinianRing_of_isNoetherianRing16 below · cited by 2 · depth 32 - Unique infinitesimal lifting along étale morphisms
AlgebraicGeometry.Etale.existsUnique_comp_eq_of_isNilpotent_ker0 below · cited by 1 · depth 33 - Local lifting of étale morphisms along a closed immersion
AlgebraicGeometry.Etale.exists_opens_etale_isPullback_of_isClosedImmersion1 below · cited by 1 · depth 33
AlgebraicGeometry.FGSubalgebra 2
- Spec A as limit of Spec of f.g. subalgebras
AlgebraicGeometry.FGSubalgebra.nonempty_isLimit_specCone1 below · cited by 3 · depth 17 - An algebra is the filtered colimit of its f.g. subalgebras
AlgebraicGeometry.FGSubalgebra.nonempty_isColimit_cocone0 below · cited by 1 · depth 18
AlgebraicGeometry.Flat 6
- Domain stalks descend along flat morphisms of schemes
AlgebraicGeometry.Flat.isDomain_stalk_of_isDomain_stalk0 below · cited by 1 · depth 24 - Flatness over Spec R is detected after base change to all R_𝔭
AlgebraicGeometry.Flat.of_forall_pullback_snd_localization_atPrime0 below · cited by 1 · depth 29 - Flatness descends to a finitely generated subalgebra stage
AlgebraicGeometry.Flat.exists_fg_subalgebra_of_flat_pullback_snd5 below · cited by 1 · depth 32 - Flatness descends along flat surjective morphisms on the source
AlgebraicGeometry.Flat.of_comp_of_flat_of_surjective2 below · cited by 2 · depth 35 - Flatness descends along affine flat surjective morphisms
AlgebraicGeometry.Flat.of_comp_of_isAffineHom_of_flat_of_surjective1 below · cited by 1 · depth 36 - Flatness over an affine base from stalks at closed points
AlgebraicGeometry.Flat.of_forall_isClosed_flat_stalk0 below · cited by 1 · depth 41
AlgebraicGeometry.FormallyUnramified 3
- Local-ring points of an unramified morphism agreeing at the closed point
AlgebraicGeometry.FormallyUnramified.eq_of_comp_eq_of_isLocalRing0 below · cited by 1 · depth 23 - Formal unramifiedness over Spec R from localisations at primes
AlgebraicGeometry.FormallyUnramified.of_forall_pullback_snd_localization_atPrime0 below · cited by 4 · depth 27 - Unramified morphisms: uniqueness of maps after surjective base change
AlgebraicGeometry.FormallyUnramified.eq_of_comp_eq_of_surjective_of_locallyOfFiniteType0 below · cited by 2 · depth 30
AlgebraicGeometry.FramedPolarisedAbelianScheme 49
- Quasi-projective fine moduli scheme for framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isFineModuli_quasiProjective1,460 below · cited by 1 · depth 28 - Finite group acting freely on theta-adapted framings
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_finite_group_action_isThetaAdapted_free_transitive_of_sq_eq1,214 below · cited by 1 · depth 29 - The theta-adapted locus is closed and finitely presented in H
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isClosedImmersion_iff_isThetaAdapted1,324 below · cited by 1 · depth 29 - Quasi-projective fine moduli of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isFineModuli_quasiProjective_of_trunk1,454 below · cited by 1 · depth 29 - Base change of framed polarised abelian schemes exists
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isPullback_type01,097 below · cited by 8 · depth 29 - Closed Theta-locus in a framed fine moduli scheme
AlgebraicGeometry.FramedPolarisedAbelianScheme.IsFineModuli.exists_pt_of_isClosedImmersion_of_iff_exists_comp_eq0 below · cited by 1 · depth 30 - Freeness of the theta group action on framed objects
AlgebraicGeometry.FramedPolarisedAbelianScheme.eq_one_of_isReframe_inter_of_iso853 below · cited by 1 · depth 30 - Theta-adapted frames differ locally by the theta group
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_cover_isReframe_inter_iso_of_isThetaAdapted_of_iso1,168 below · cited by 1 · depth 30 - A functorial finitely generated ideal cutting out theta-adaptedness
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_ideal_fg_isThetaAdapted_iff_eq_bot1,322 below · cited by 1 · depth 30 - Embedded moduli of framed polarised abelian schemes over a Noetherian base
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isImmersion_proj_represents_embedded_of_isNoetherianRing1,449 below · cited by 1 · depth 30 - Theta-adapted framed base change from a Schrödinger frame
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isPullback_isThetaAdapted_of_schrodingerFrame38 below · cited by 1 · depth 30 - Reframing a framed polarised abelian scheme by a unit matrix
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isReframe10 below · cited by 6 · depth 30 - Functorial Γ-action descends to automorphisms of a fine moduli scheme
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_monoidHom_aut_forall_pt_act_eq_of_isFineModuli1 below · cited by 1 · depth 30 - Reframing commutes with base change
AlgebraicGeometry.FramedPolarisedAbelianScheme.isPullback_of_isPullback_of_isReframe11 below · cited by 2 · depth 30 - Theta-adaptedness descends along base change of framed schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.isThetaAdapted_of_isPullback75 below · cited by 3 · depth 30 - Reframing by an intertwiner preserves theta-adaptedness
AlgebraicGeometry.FramedPolarisedAbelianScheme.isThetaAdapted_of_isReframe_inter19 below · cited by 1 · depth 30 - Theta-adaptedness is invariant under isomorphism of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.isThetaAdapted_of_iso7 below · cited by 3 · depth 30 - Reframing by intertwiners is multiplicative up to framed isomorphism
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_of_isReframe_inter_mul7 below · cited by 1 · depth 30 - Reframing by the intertwiner of the identity gives a framed isomorphism
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_of_isReframe_inter_one69 below · cited by 1 · depth 30 - Framed rigidity: frame-compatible isomorphisms of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_of_iso_comp_toProj_eq_of_one_comp_toProj_eq_of_forall_comp_toProj_eq64 below · cited by 3 · depth 30 - Reframing preserves isomorphism of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_of_iso_of_isReframe7 below · cited by 2 · depth 30 - Theta-adapted frames give theta type étale-locally
AlgebraicGeometry.FramedPolarisedAbelianScheme.thetaTypeLocally_of_isThetaAdapted1,116 below · cited by 1 · depth 30 - A reframing matrix is invertible
AlgebraicGeometry.FramedPolarisedAbelianScheme.IsReframe.isUnit0 below · cited by 1 · depth 31 - Pulled-back frame is a basis over the trivial base change
AlgebraicGeometry.FramedPolarisedAbelianScheme.bijective_sum_baseScalar_smul_of_eq_pullbackLocalSection_frame0 below · cited by 4 · depth 31 - Theta-adaptedness of a framed family is cut out by a finitely generated ideal
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_ideal_fg_forall_isPullback_isThetaAdapted_iff_map_eq_bot1,312 below · cited by 1 · depth 31 - Two Schrödinger frames differ clopen-locally by an intertwiner matrix
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_idempotents_gam_units_schrodingerFrame_sigma_eq70 below · cited by 1 · depth 31 - Base change of a frame agrees up to one unit
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_isUnit_forall_app_pullbackLocalSection_frame_eq_baseScalar_smul59 below · cited by 1 · depth 31 - Hilbert point cutting out a framed polarised abelian scheme
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_point_forall_mem_iff_frame_toProj1,174 below · cited by 1 · depth 31 - Emptiness of framed polarised abelian schemes without Hilbert polynomial (N+1)t^g
AlgebraicGeometry.FramedPolarisedAbelianScheme.isEmpty_of_not_exists_hilbertPolynomial1,163 below · cited by 1 · depth 31 - Transitivity of base change for framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.isPullback_comp2 below · cited by 2 · depth 31 - Reflexivity of base change along id_S for framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.isPullback_id6 below · cited by 1 · depth 31 - Matrix relation between pulled-back frames yields a reframing
AlgebraicGeometry.FramedPolarisedAbelianScheme.isReframe_mk_of_forall_eq_sum_baseScalar_smul_pullbackLocalSection1 below · cited by 1 · depth 31 - Transported frame gives an isomorphism of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_mk_of_iso_hom_comp_toProj_eq0 below · cited by 1 · depth 31 - Rescaling a frame by a unit preserves the framed isomorphism class
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_mk_of_iso_of_forall_sigma_eq_smul3 below · cited by 1 · depth 31 - Framed base change is unique up to framed isomorphism
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_of_isPullback_of_isPullback1 below · cited by 1 · depth 31 - Base change of isomorphic framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_of_iso_of_isPullback_of_isPullback66 below · cited by 2 · depth 31 - Reflexivity of isomorphism of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_refl0 below · cited by 1 · depth 31 - Symmetry of isomorphism of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_symm1 below · cited by 2 · depth 31 - Transitivity of framed isomorphism of framed polarised abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.iso_trans1 below · cited by 1 · depth 31 - Framed polarised abelian schemes exist over the zero ring
AlgebraicGeometry.FramedPolarisedAbelianScheme.nonempty_of_subsingleton0 below · cited by 1 · depth 31 - Reframing a framed polarised abelian scheme by a permutation
AlgebraicGeometry.FramedPolarisedAbelianScheme.reframe_perm11 below · cited by 1 · depth 31 - Normal form of a theta point on a Schrödinger frame
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_completeOrthogonalIdempotents_forall_act_schrodingerFrame_eq62 below · cited by 1 · depth 32 - Fibrewise theta condition for a fixed matrix is cut out by a finitely generated ideal
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_ideal_fg_forall_isPullback_exists_translate_comp_eq_iff_map_eq_bot1,296 below · cited by 1 · depth 32 - Theta points versus translations of a framed projective embedding
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_thetaPt_act_eq_iff_exists_translate_comp_toProj_eq7 below · cited by 1 · depth 32 - Geometric fibre h⁰(L^{⊗ d})=(N+1)d^g for framed abelian schemes
AlgebraicGeometry.FramedPolarisedAbelianScheme.geomFibreH0Finrank_natRec_tensor_eq_of_realisation1,085 below · cited by 2 · depth 32 - Theta-adaptedness via translation lifts and standard dual theta points
AlgebraicGeometry.FramedPolarisedAbelianScheme.isThetaAdapted_iff_forall_exists_thetaPt_act_eq15 below · cited by 1 · depth 32 - Finitely generated ideal detecting a translating point
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_ideal_fg_forall_isPullback_exists_comp_toProj_eq_one_comp_iff_map_eq_bot11 below · cited by 1 · depth 33 - Ideal cutting out the theta equation for a fixed matrix
AlgebraicGeometry.FramedPolarisedAbelianScheme.exists_ideal_fg_forall_isPullback_translate_comp_eq_iff_map_eq_bot_of_section364 below · cited by 1 · depth 33 - Cancellation of framed base-change squares
AlgebraicGeometry.FramedPolarisedAbelianScheme.isPullback_of_isPullback_comp3 below · cited by 1 · depth 33
AlgebraicGeometry.GeometricallyConnected 3
- Geometric connectedness descends along surjections
AlgebraicGeometry.GeometricallyConnected.descendsAlong_surjective0 below · cited by 4 · depth 14 - Geometric connectedness descends along a surjective morphism
AlgebraicGeometry.GeometricallyConnected.of_comp_of_surjective0 below · cited by 3 · depth 15 - Geometric connectedness descends to a finitely generated subalgebra
AlgebraicGeometry.GeometricallyConnected.exists_fg_subalgebra_of_geometricallyConnected_pullback_snd75 below · cited by 2 · depth 29
AlgebraicGeometry.GeometricallyIntegral 2
- Integrality of the source of a flat, universally open, geometrically integral morphism
AlgebraicGeometry.GeometricallyIntegral.isIntegral_of_flat_of_universallyOpen0 below · cited by 4 · depth 16 - Geometric integrality descends along a field extension
AlgebraicGeometry.GeometricallyIntegral.of_isPullback_of_geometricallyIntegral0 below · cited by 4 · depth 24
AlgebraicGeometry.GeometricallyIrreducible 2
- Geometrically irreducible morphisms are geometrically connected
AlgebraicGeometry.GeometricallyIrreducible.geometricallyConnected0 below · cited by 10 · depth 14 - Irreducible schemes over an algebraically closed field are geometrically irreducible
AlgebraicGeometry.GeometricallyIrreducible.of_irreducibleSpace_of_isAlgClosed0 below · cited by 17 · depth 15
AlgebraicGeometry.GeometricallyReduced 1
- Reduced and locally of finite type over a perfect field is geometrically reduced
AlgebraicGeometry.GeometricallyReduced.of_isReduced_of_perfectField1 below · cited by 17 · depth 15
AlgebraicGeometry.GradedOAlgebra 26
- Base change of the canonical morphism to Proj
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.comp_map_eq_comp6 below · cited by 1 · depth 35 - Canonical morphism to Proj of a section ring is an isomorphism
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.isIso11 below · cited by 1 · depth 35 - Descent cocycle for the section ring of an invertible module
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.cocycle_trans_symm_of_cocycle45 below · cited by 1 · depth 35 - Section rings compare along the first coface of a flat base change
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.exists_algHom_bijective_lift_of_isPullback_includeLeft24 below · cited by 1 · depth 35 - Section rings along the second coface of a flat base change
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.exists_algHom_bijective_lift_of_isPullback_includeRight24 below · cited by 1 · depth 35 - Existence of the canonical morphism to Proj of a section ring
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.exists_isCanonicalToProj7 below · cited by 1 · depth 35 - Existence of the section ring of an invertible module
AlgebraicGeometry.GradedOAlgebra.exists_isSectionRing11 below · cited by 2 · depth 35 - Proj of a degreewise base change is cartesian
AlgebraicGeometry.GradedOAlgebra.isPullback_projMap_of_isBaseChange1 below · cited by 1 · depth 35 - Canonical morphism to Proj under base change
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.comp_of_commSq1 below · cited by 2 · depth 36 - Composing a canonical morphism to Proj with Proj of a graded map
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.comp_projMap0 below · cited by 2 · depth 36 - The canonical morphism to Proj has dense image
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.denseRange2 below · cited by 1 · depth 36 - Uniqueness of canonical morphisms to Proj from chart preimages
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.eq_of_preimage_basicOpen_eq0 below · cited by 1 · depth 36 - Empty chart forces nilpotence of a homogeneous section
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.exists_pow_eq_zero_of_preimage_basicOpen_eq_bot1 below · cited by 2 · depth 36 - Sections vanishing on X_σ are σ-torsion
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.exists_pow_mul_eq_zero_of_map_eq_zero2 below · cited by 2 · depth 36 - Functions on X_σ are ratios t/σ^k
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.exists_smul_map_pow_eq_map_of_section5 below · cited by 2 · depth 36 - Canonical morphism to Proj is an isomorphism over D₊(τ)
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.isIso_morphismRestrict_basicOpen_of_isAffineOpen7 below · cited by 1 · depth 36 - Base change: equal chart preimages and pulled-back frames
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.preimage_basicOpen_comp_projMap_eq4 below · cited by 1 · depth 36 - Affine charts of the canonical morphism at a presenting section
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.preimage_basicOpen_eq_preimage_of_projPresentation_of_isClosedImmersion2 below · cited by 2 · depth 36 - The canonical morphism to Proj is universally closed
AlgebraicGeometry.GradedOAlgebra.IsCanonicalToProj.universallyClosed0 below · cited by 1 · depth 36 - Base-change functoriality of graded section rings
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.exists_algHom_apply_eq_pullback_of_isPullback7 below · cited by 4 · depth 36 - Section rings under base change along S' → S' ⊗_S B
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.exists_algHom_bijective_lift_tensorProduct_of_isPullback24 below · cited by 1 · depth 36 - Dehomogenisation at a degree-one frame of a section ring
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.exists_ringHom_smul_map_pow_eq1 below · cited by 2 · depth 36 - Section rings commute with flat base change
AlgebraicGeometry.GradedOAlgebra.IsSectionRing.isBaseChange_of_apply_eq_pullback_of_flat14 below · cited by 3 · depth 36 - Composing section-ring comparison maps along d gg c
AlgebraicGeometry.GradedOAlgebra.apply_comp_eq_pullback_comp_of_apply_eq_pullback9 below · cited by 1 · depth 36 - Transport of the pullback comparison formula along c = c'
AlgebraicGeometry.GradedOAlgebra.apply_eq_pullback_congr_hom0 below · cited by 1 · depth 36 - Uniqueness of a degreewise pull-back-compatible ring map
AlgebraicGeometry.GradedOAlgebra.ringHom_apply_eq_of_apply_eq_pullback0 below · cited by 1 · depth 36
AlgebraicGeometry.GrpObj 1
- A group object on a proper integral scheme is determined by its unit
AlgebraicGeometry.GrpObj.mul_eq_of_one_eq1 below · cited by 1 · depth 38
AlgebraicGeometry.HilbertFunctor 28
- The Hilbert scheme of Pⁿ_ℤ is proper and finitely presented
AlgebraicGeometry.HilbertFunctor.exists_scheme_represents_isProper_locallyOfFinitePresentation_hilbertFunctionOf44 below · cited by 3 · depth 29 - Representability and projectivity of the Hilbert functor Hilb^P_{Pⁿ}
AlgebraicGeometry.HilbertFunctor.exists_scheme_represents_and_isClosedImmersion_toProjSpace_hilbertFunctionOf41 below · cited by 1 · depth 30 - Hilbert points of Pⁿ versus flat closed subschemes
AlgebraicGeometry.HilbertFunctor.exists_closedImmersion_flat_lfp_forall_mem_iff_of_point_hilbertFunctionOf231 below · cited by 1 · depth 31 - Descent of Hilbert-functor points from geometric fibres to the base
AlgebraicGeometry.HilbertFunctor.exists_point_I_eq_span_of_isClosedImmersion_of_flat_of_locallyOfFinitePresentation221 below · cited by 5 · depth 31 - Representability of the Hilbert functor for maximal-growth h
AlgebraicGeometry.HilbertFunctor.exists_scheme_represents_and_isClosedImmersion_toProjSpace_of_maximal_growth38 below · cited by 1 · depth 31 - Generation in degree m for points of the Hilbert functor
AlgebraicGeometry.HilbertFunctor.Point.I_eq_span_of_forall_finrank_piece_succ_le1 below · cited by 5 · depth 32 - Flatness over the base of the subscheme cut out by a Hilbert-functor point
AlgebraicGeometry.HilbertFunctor.Point.flat_comp_of_ideal_basicOpen_eq_span2 below · cited by 2 · depth 32 - Large-degree h⁰ of tensor powers equals the Hilbert function
AlgebraicGeometry.HilbertFunctor.exists_forall_geomFibreH0Finrank_tensorPow_eq_hilbertFunctionOf_of_point50 below · cited by 3 · depth 32 - Gotzmann saturation for Hilbert functor points over any ring
AlgebraicGeometry.HilbertFunctor.exists_forall_mem_of_forall_X_pow_mul_mem_hilbertFunctionOf17 below · cited by 2 · depth 32 - Uniform Gotzmann regularity for points of the Hilbert functor
AlgebraicGeometry.HilbertFunctor.exists_forall_subsingleton_HSucc_twist_and_forall_H0_exists_of_point_hilbertFunctionOf169 below · cited by 1 · depth 32 - Hilbert–Serre for a closed subscheme of Pⁿ_k
AlgebraicGeometry.HilbertFunctor.exists_ideal_forall_mem_iff_app_awayToSection_eq_zero_and_polynomial4 below · cited by 4 · depth 32 - The rank-r locus of the degree-(m+1) piece is cut out by an ideal
AlgebraicGeometry.HilbertFunctor.exists_ideal_forall_projective_piece_succ_iff2 below · cited by 1 · depth 32 - Base change of points of the Hilbert functor
AlgebraicGeometry.HilbertFunctor.exists_point_I_eq_map1 below · cited by 3 · depth 32 - Gotzmann persistence over a base ring: I defines a Hilbert-functor point
AlgebraicGeometry.HilbertFunctor.exists_point_I_eq_of_projective_piece_succ17 below · cited by 1 · depth 32 - Graded quotient data defines a point of the Hilbert functor
AlgebraicGeometry.HilbertFunctor.exists_point_I_eq_span_of_forall_surjective_of_forall_projective0 below · cited by 1 · depth 32 - Hilbert-functor point from a closed subscheme with h⁰ = P
AlgebraicGeometry.HilbertFunctor.exists_point_forall_mem_iff_of_isClosedImmersion_of_forall_geomFibreH0Finrank_eq_eval76 below · cited by 1 · depth 32 - Truncating a homogeneous ideal to a Hilbert-functor point
AlgebraicGeometry.HilbertFunctor.exists_point_hilbertFunctionOf_forall_mem_iff_of_forall_finrank_piece_eq0 below · cited by 3 · depth 32 - Gotzmann regularity in uniform numerical form
AlgebraicGeometry.HilbertFunctor.exists_forall_finrank_piece_eq_eval_of_isClosedImmersion_of_forall_mem_iff_of_eventually_eq21 below · cited by 2 · depth 33 - Gotzmann saturation over a field in degrees beyond D₀
AlgebraicGeometry.HilbertFunctor.exists_forall_mem_of_forall_X_pow_mul_mem_hilbertFunctionOf_field16 below · cited by 1 · depth 33 - Base change of the graded pieces of a homogeneous ideal
AlgebraicGeometry.HilbertFunctor.exists_linearEquiv_baseChange_piece_map0 below · cited by 9 · depth 33 - Gotzmann freeness step over a local base
AlgebraicGeometry.HilbertFunctor.free_piece_of_isLocalRing_of_forall_relation_mem_span1 below · cited by 1 · depth 33 - Hilbert points versus flat closed subschemes of Pⁿ
AlgebraicGeometry.HilbertFunctor.exists_closedImmersion_flat_lfp_forall_mem_iff_ker_ideal_eq_of_point_hilbertFunctionOf231 below · cited by 1 · depth 34 - A homogeneous ideal cuts a finitely presented closed subscheme of Hilb
AlgebraicGeometry.HilbertFunctor.exists_isClosedImmersion_locallyOfFinitePresentation_forall_factors_iff_forall_mem_of_isHomogeneous_of_le_of_fg7 below · cited by 1 · depth 34 - Uniform truncation level for geometric fibres in Pⁿ
AlgebraicGeometry.HilbertFunctor.exists_uniform_cover_forall_geomFibre_ideal_eq_point_of_isClosedImmersion_of_flat_of_locallyOfFinitePresentation181 below · cited by 2 · depth 34 - Ideals cut out chartwise on Pⁿ_k are variable-saturated
AlgebraicGeometry.HilbertFunctor.mem_of_forall_exists_X_pow_mul_mem_of_forall_mem_iff_app_awayToSection_eq_zero0 below · cited by 1 · depth 34 - Locally constant Hilbert polynomial of flat finitely presented families
AlgebraicGeometry.HilbertFunctor.exists_cover_forall_finrank_piece_eq_of_isClosedImmersion_of_flat_of_locallyOfFinitePresentation160 below · cited by 1 · depth 35 - Membership in a base-changed Hilbert point via graded pieces
AlgebraicGeometry.HilbertFunctor.map_mem_iff_tmul_mkQ_piece_eq_zero0 below · cited by 1 · depth 35 - Local constancy of geometric fibre Hilbert polynomials, Noetherian base
AlgebraicGeometry.HilbertFunctor.exists_cover_forall_finrank_piece_eq_of_isClosedImmersion_of_flat_of_isNoetherianRing131 below · cited by 1 · depth 36
AlgebraicGeometry.IdealSheafData 4
- Saturated ideal sheaves give flat closed subschemes over a DVR
AlgebraicGeometry.IdealSheafData.flat_subschemeInclusion_comp_of_forall_mul_mem_of_isDiscreteValuationRing0 below · cited by 1 · depth 29 - Comap of the glued ideal sheaf along a classifying map
AlgebraicGeometry.IdealSheafData.comap_pullbackMap_eq_ker_of_forall_affineOpens_comap_eq_ker_of_forall_spec_point0 below · cited by 1 · depth 34 - Flatness and finite presentation of the glued closed subscheme
AlgebraicGeometry.IdealSheafData.flat_and_locallyOfFinitePresentation_subschemeInclusion_comp_of_forall_affineOpens_comap_eq_ker0 below · cited by 1 · depth 34 - An ideal sheaf killed by all truncations vanishes
AlgebraicGeometry.IdealSheafData.eq_bot_of_forall_le_ker_pullback_fst_truncation_of_isProper_of_isAdicComplete0 below · cited by 3 · depth 44
AlgebraicGeometry.IsAffineHom 2
- Affineness of morphisms descends along faithfully flat quasi-compact maps
AlgebraicGeometry.IsAffineHom.descendsAlong_surjective_inf_flat_inf_quasiCompact0 below · cited by 2 · depth 26 - Sections over nested affine opens of an affine morphism form a pushout
AlgebraicGeometry.IsAffineHom.isPushout_map_appLE_appLE_map_of_isAffineOpen0 below · cited by 1 · depth 34
AlgebraicGeometry.IsAffineOpen 9
- Stalk Krull dimension bounded by that of an affine chart
AlgebraicGeometry.IsAffineOpen.ringKrullDim_stalk_le0 below · cited by 13 · depth 13 - Regular stalks from a regular affine coordinate ring
AlgebraicGeometry.IsAffineOpen.isRegularLocalRing_stalk_of_isRegularRing0 below · cited by 5 · depth 18 - Sections on an affine open as intersection of local rings
AlgebraicGeometry.IsAffineOpen.range_algebraMap_functionField_eq_iInf0 below · cited by 2 · depth 18 - Holomorphic evaluation of sections along a holomorphic family of characters
AlgebraicGeometry.IsAffineOpen.isOpen_and_exists_differentiableOn_eval_appLE_of_forall_differentiableOn0 below · cited by 2 · depth 29 - K-points in an affine open versus ring maps on its sections
AlgebraicGeometry.IsAffineOpen.eq_of_appLE_eq_and_exists_appLE_eq_of_ringHom0 below · cited by 4 · depth 31 - Holomorphy of section values spreads from an affine open
AlgebraicGeometry.IsAffineOpen.isOpen_and_exists_differentiableOn_appLE_of_forall_section0 below · cited by 4 · depth 31 - Localisation at a maximal point of a reduced scheme
AlgebraicGeometry.IsAffineOpen.isLocalization_primeCompl_of_isPreimmersion_of_forall_specializes0 below · cited by 1 · depth 32 - Pinned tensor identifications are compatible with restriction
AlgebraicGeometry.IsAffineOpen.ringEquiv_tensor_map_eq_presheaf_map_of_specMap_comp_fromSpec_eq0 below · cited by 3 · depth 35 - Transport of σ along a morphism of special fibres
AlgebraicGeometry.IsAffineOpen.ringEquiv_tensor_map_eq_map_app_of_specMap_comp_fromSpec_eq0 below · cited by 1 · depth 38
AlgebraicGeometry.IsClosedImmersion 21
- Factoring through a closed immersion over a reduced scheme
AlgebraicGeometry.IsClosedImmersion.existsUnique_comp_eq_of_range_subset_of_isReduced1 below · cited by 22 · depth 13 - Sections over a reduced scheme covered by two closed subschemes
AlgebraicGeometry.IsClosedImmersion.app_injective_and_exists_of_app_pullback_eq_of_isReduced2 below · cited by 4 · depth 16 - Mayer–Vietoris for a closed cover, after base change
AlgebraicGeometry.IsClosedImmersion.app_curveChange_injective_and_exists_of_app_eq_of_isReduced0 below · cited by 1 · depth 17 - Affine Mayer–Vietoris for two closed subschemes covering a reduced scheme
AlgebraicGeometry.IsClosedImmersion.app_injective_and_exists_of_app_pullback_eq_of_isAffineOpen0 below · cited by 3 · depth 17 - A reduced closed subscheme is determined by its image
AlgebraicGeometry.IsClosedImmersion.exists_iso_hom_comp_eq_of_range_eq0 below · cited by 8 · depth 19 - Factoring through a closed immersion via generic geometric points
AlgebraicGeometry.IsClosedImmersion.existsUnique_comp_eq_of_forall_geometricPoint_exists_comp_eq_of_flat2 below · cited by 1 · depth 22 - Integral points factor through closed subschemes
AlgebraicGeometry.IsClosedImmersion.existsUnique_comp_eq_of_specMap_subtype_comp_eq1 below · cited by 1 · depth 22 - Factorisation through a closed subscheme over a reduced scheme
AlgebraicGeometry.IsClosedImmersion.existsUnique_comp_eq_of_denseRange_of_comp_eq0 below · cited by 1 · depth 23 - Residue field at the generic point of an integral closed subscheme
AlgebraicGeometry.IsClosedImmersion.exists_ringEquiv_residueField_functionField_of_isIntegral0 below · cited by 3 · depth 23 - Stalk isomorphism for closed immersions at interior points
AlgebraicGeometry.IsClosedImmersion.isIso_stalkMap_of_mem_interior_range0 below · cited by 1 · depth 27 - Factoring through a closed immersion descends along kerπ=bot
AlgebraicGeometry.IsClosedImmersion.exists_comp_eq_of_exists_comp_eq_comp_of_ker_eq_bot0 below · cited by 1 · depth 30 - Closed immersions descend to a finitely generated subalgebra
AlgebraicGeometry.IsClosedImmersion.exists_fg_subalgebra_of_isClosedImmersion_pullback_map3 below · cited by 2 · depth 30 - Field-valued points factor through a closed immersion
AlgebraicGeometry.IsClosedImmersion.exists_comp_eq_iff_apply_closedPoint_mem_range0 below · cited by 2 · depth 31 - Closed immersion after base change descends to a finitely generated subalgebra
AlgebraicGeometry.IsClosedImmersion.exists_fg_subalgebra_of_isClosedImmersion_pullback_map_of_quasiCompact4 below · cited by 5 · depth 32 - Local coordinates of a ℂ-point family push forward along a closed immersion
AlgebraicGeometry.IsClosedImmersion.exists_hasFDerivAt_appLE_comp_of_hasFDerivAt_appLE0 below · cited by 1 · depth 32 - Closed subschemes of an affine scheme are cut out by an ideal
AlgebraicGeometry.IsClosedImmersion.exists_ideal_forall_exists_comp_eq_specMap_iff_forall_map_eq_zero0 below · cited by 2 · depth 32 - Closed immersions preserve closed, quasi-compact-over-the-base finite intersections
AlgebraicGeometry.IsClosedImmersion.isClosed_iInf_preimage_and_quasiCompact0 below · cited by 1 · depth 32 - Degree of a finite closed subscheme as a sum of stalk lengths
AlgebraicGeometry.IsClosedImmersion.exists_finset_finrank_comp_eq_sum_toNat_length_stalk_quotient_ker_stalkMap1 below · cited by 1 · depth 34 - Finitely presented closed subschemes of affine schemes, functorially
AlgebraicGeometry.IsClosedImmersion.exists_ideal_fg_forall_exists_comp_eq_specMap_iff_map_eq_bot0 below · cited by 3 · depth 34 - Factoring Spec(mathcal O_{A,y}/J) through a closed immersion
AlgebraicGeometry.IsClosedImmersion.exists_comp_eq_specMap_comp_fromSpecStalk_iff_ker_stalkMap_le0 below · cited by 3 · depth 35 - Equality of chart ideals after localising at 𝔭
AlgebraicGeometry.IsClosedImmersion.map_ideal_ker_eq_of_forall_factorsThrough_iff_atPrime0 below · cited by 1 · depth 37
AlgebraicGeometry.IsFinite 6
- Stalk fibre ring of a finite morphism is zero-dimensional
AlgebraicGeometry.IsFinite.ringKrullDim_stalk_quotient_eq_zero0 below · cited by 3 · depth 15 - Stalk map of a finite morphism at an isolated fibre point
AlgebraicGeometry.IsFinite.finite_hom_stalkMap_of_forall_base_eq1 below · cited by 1 · depth 26 - Quasi-compact smooth of relative dimension zero over a field is finite
AlgebraicGeometry.IsFinite.of_smoothOfRelativeDimension_zero_of_field0 below · cited by 2 · depth 32 - Proper over a local ring with finite closed fibre is finite
AlgebraicGeometry.IsFinite.of_isProper_of_finite_preimage_closedPoint0 below · cited by 1 · depth 35 - Proper plus finite after surjective precomposition implies finite
AlgebraicGeometry.IsFinite.of_isFinite_comp_of_surjective_of_isProper0 below · cited by 1 · depth 39 - Finiteness of g descends along a surjection
AlgebraicGeometry.IsFinite.of_comp_of_surjective0 below · cited by 1 · depth 40
AlgebraicGeometry.IsIntegral 1
- Normality of affine sections from integrally closed stalks
AlgebraicGeometry.IsIntegral.isIntegrallyClosed_sections_of_forall_isIntegrallyClosed_stalk0 below · cited by 3 · depth 14
AlgebraicGeometry.IsOpenImmersion 3
- Krull dimension of stalks is an open-immersion invariant
AlgebraicGeometry.IsOpenImmersion.ringKrullDim_stalk_eq0 below · cited by 10 · depth 13 - Regularity of stalks is invariant under open immersions
AlgebraicGeometry.IsOpenImmersion.isRegularLocalRing_stalk_iff0 below · cited by 4 · depth 16 - Flat closed subscheme of an étale scheme is open
AlgebraicGeometry.IsOpenImmersion.of_isClosedImmersion_of_flat_comp_of_etale0 below · cited by 8 · depth 27
AlgebraicGeometry.IsProper 2
- Properness descends along surjective flat quasi-compact morphisms
AlgebraicGeometry.IsProper.descendsAlong_surjective_inf_flat_inf_quasiCompact2 below · cited by 4 · depth 22 - Properness descends to a finitely generated subalgebra
AlgebraicGeometry.IsProper.exists_fg_subalgebra_of_isProper_pullback_snd4 below · cited by 3 · depth 29
AlgebraicGeometry.IsPullback 3
- Surjectivity of the residue field map under base change
AlgebraicGeometry.IsPullback.surjective_residueFieldMap_of_isIso_residueFieldMap0 below · cited by 1 · depth 29 - Affine charts of a pullback along a nilpotent thickening
AlgebraicGeometry.IsPullback.exists_iso_Spec_quotient_comp_morphismRestrict_eq0 below · cited by 20 · depth 32 - Affine opens of a nil-thickening: sections surject with kernel JΓ
AlgebraicGeometry.IsPullback.surjective_app_and_ker_app_eq_map_ker_of_isAffineOpen1 below · cited by 6 · depth 37
AlgebraicGeometry.IsSeparated 6
- Separatedness gives uniqueness of valuation-ring points
AlgebraicGeometry.IsSeparated.eq_of_spec_map_subtype_comp_eq0 below · cited by 17 · depth 14 - Separatedness descends along universally closed surjections
AlgebraicGeometry.IsSeparated.of_comp_of_universallyClosed_of_surjective0 below · cited by 3 · depth 16 - Separatedness from closed immersions on an open cover
AlgebraicGeometry.IsSeparated.of_isClosedImmersion_mapDesc_openCover0 below · cited by 2 · depth 20 - Separatedness descends along faithfully flat quasi-compact morphisms
AlgebraicGeometry.IsSeparated.descendsAlong_surjective_inf_flat_inf_quasiCompact0 below · cited by 2 · depth 23 - Separatedness descends to a finitely generated subalgebra
AlgebraicGeometry.IsSeparated.exists_fg_subalgebra_of_isSeparated_pullback_snd4 below · cited by 3 · depth 29 - Intersection of affine opens for a separated morphism to an affine scheme
AlgebraicGeometry.IsSeparated.isAffineOpen_inf_and_exists_eq_sum_mul_of_isAffineOpen0 below · cited by 4 · depth 31
AlgebraicGeometry.IsZariskiLocalAtTarget 1
- Zariski-local-at-target properties pass to coproducts of morphisms
AlgebraicGeometry.IsZariskiLocalAtTarget.sigmaMap0 below · cited by 1 · depth 34
AlgebraicGeometry.LocallyOfFinitePresentation 2
- Descent of locally of finite presentation along finite flat surjections
AlgebraicGeometry.LocallyOfFinitePresentation.of_comp_of_isFinite_of_flat_of_surjective6 below · cited by 2 · depth 29 - Finite presentation descends along flat quasi-compact surjections
AlgebraicGeometry.LocallyOfFinitePresentation.of_comp_of_flat_of_surjective7 below · cited by 2 · depth 35
AlgebraicGeometry.LocallyOfFiniteType 2
- Finite type descends along finite flat surjective morphisms
AlgebraicGeometry.LocallyOfFiniteType.of_comp_of_isFinite_of_flat_of_surjective0 below · cited by 2 · depth 16 - Locally of finite type descends along fpqc morphisms
AlgebraicGeometry.LocallyOfFiniteType.descendsAlong_surjective_inf_flat_inf_quasiCompact0 below · cited by 1 · depth 23
AlgebraicGeometry.LocallyQuasiFinite 6
- Flat, locally finite type with quasi-finite generic fibre is locally quasi-finite
AlgebraicGeometry.LocallyQuasiFinite.of_flat_of_locallyQuasiFinite_genericFiber2 below · cited by 6 · depth 14 - Formally unramified plus locally of finite type gives locally quasi-finite
AlgebraicGeometry.LocallyQuasiFinite.of_formallyUnramified_of_locallyOfFiniteType0 below · cited by 2 · depth 14 - Locally quasi-finite morphisms have zero-dimensional stalk fibres
AlgebraicGeometry.LocallyQuasiFinite.ringKrullDim_stalk_quotient_eq_zero0 below · cited by 3 · depth 15 - Locally quasi-finite morphisms descend along faithfully flat quasi-compact base change
AlgebraicGeometry.LocallyQuasiFinite.descendsAlong_surjective_inf_flat_inf_quasiCompact1 below · cited by 1 · depth 19 - Generic finiteness of separated quasi-finite morphisms
AlgebraicGeometry.LocallyQuasiFinite.exists_isFinite_morphismRestrict_of_irreducibleSpace2 below · cited by 1 · depth 36 - Generic finiteness at a minimal prime over an affine base
AlgebraicGeometry.LocallyQuasiFinite.exists_not_mem_isFinite_morphismRestrict_basicOpen_of_mem_minimalPrimes1 below · cited by 1 · depth 37
AlgebraicGeometry.OModulePresheaf 277
- Čech finiteness for a two-member cover via H⁰ and H¹
AlgebraicGeometry.OModulePresheaf.cechFinite_toOrderedAffineCover_iff0 below · cited by 18 · depth 15 - Čech finiteness of the structure sheaf of a proper scheme
AlgebraicGeometry.OModulePresheaf.cechFinite_unit_of_isProper53 below · cited by 7 · depth 15 - Finiteness of Čech cohomology of locally trivial 𝒪-modules
AlgebraicGeometry.OModulePresheaf.cechFinite_ofModules_of_locallyTrivial58 below · cited by 40 · depth 16 - Finiteness of Čech cohomology for proper morphisms
AlgebraicGeometry.OModulePresheaf.cechFinite_of_isProper52 below · cited by 22 · depth 16 - Locally trivial 𝒪-modules give quasi-coherent module presheaves
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_ofModules_of_locallyTrivial2 below · cited by 49 · depth 16 - Affine Čech acyclicity for quasi-coherent module presheaves
AlgebraicGeometry.OModulePresheaf.ker_d_succ_le_range_d_of_isQuasicoherent0 below · cited by 8 · depth 16 - Isomorphic mathcal O_V-modules have isomorphic ordered Čech cohomology
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_ofModules_of_iso0 below · cited by 34 · depth 16 - Dévissage driver for Čech finiteness over a proper base
AlgebraicGeometry.OModulePresheaf.cechFinite_of_forall_integral15 below · cited by 1 · depth 17 - Descent of the Čech-finiteness hypothesis to a closed subscheme
AlgebraicGeometry.OModulePresheaf.cechFinite_preimage_of_ih2 below · cited by 1 · depth 17 - Čech finiteness is preserved by pushforward along a closed immersion
AlgebraicGeometry.OModulePresheaf.cechFinite_pushforward_iff1 below · cited by 2 · depth 17 - Finiteness of Čech cohomology of mathcal O_Z for integral proper Z
AlgebraicGeometry.OModulePresheaf.cechFinite_unit_of_isIntegral_of_ih36 below · cited by 1 · depth 17 - Locally trivial mathcal O_V-modules are coherent on affine opens
AlgebraicGeometry.OModulePresheaf.isCoherent_ofModules_of_locallyTrivial3 below · cited by 23 · depth 17 - Vanishing of check Hⁱ⁺¹ for quasi-coherent data on affine schemes
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_of_isQuasicoherent1 below · cited by 2 · depth 17 - Chow short exact sequence for the relative check H⁰ presheaf
AlgebraicGeometry.OModulePresheaf.Leray.exists_chowSES18 below · cited by 1 · depth 18 - Coherence of relative Čech cohomology for a Chow datum
AlgebraicGeometry.OModulePresheaf.Leray.isCoherent_relHPresheaf_chow13 below · cited by 2 · depth 18 - Quasi-coherence of the relative Čech cohomology presheaf
AlgebraicGeometry.OModulePresheaf.Leray.isQuasicoherent_relHPresheaf_chow0 below · cited by 2 · depth 18 - E₂ page of the Čech–Leray double complex
AlgebraicGeometry.OModulePresheaf.Leray.nonempty_E2I_equiv0 below · cited by 1 · depth 18 - Total cohomology of the Čech–Leray double complex
AlgebraicGeometry.OModulePresheaf.Leray.nonempty_HTot_equiv5 below · cited by 1 · depth 18 - Relative Čech cohomology of a Chow datum vanishes over U
AlgebraicGeometry.OModulePresheaf.Leray.supportedIn_relHPresheaf_chow2 below · cited by 1 · depth 18 - Čech finiteness transfers along an isomorphism of cochain complexes
AlgebraicGeometry.OModulePresheaf.cechFinite_iff_of_cochain_equiv0 below · cited by 1 · depth 18 - Čech finiteness of the left term of an affine exact sequence
AlgebraicGeometry.OModulePresheaf.cechFinite_of_affSES_left3 below · cited by 2 · depth 18 - Čech-finiteness of the middle term of an affine-exact sequence
AlgebraicGeometry.OModulePresheaf.cechFinite_of_affSES_mid2 below · cited by 4 · depth 18 - Čech-finiteness from Čech-finiteness of the graded pieces
AlgebraicGeometry.OModulePresheaf.cechFinite_of_forall_cechFinite_idealPowQuot6 below · cited by 1 · depth 18 - Dévissage step: Čech finiteness along an integral closed subscheme
AlgebraicGeometry.OModulePresheaf.cechFinite_pushforward_of_isIntegral_of_ih7 below · cited by 1 · depth 18 - Finiteness of Čech cohomology of 𝒪 on closed subschemes of Pⁿ_A
AlgebraicGeometry.OModulePresheaf.cechFinite_unit_of_isClosedImmersion_proj10 below · cited by 2 · depth 18 - Degree-zero Čech cocycles are families of restrictions
AlgebraicGeometry.OModulePresheaf.d_zero_ofModules_eq_zero_iff_existsUnique0 below · cited by 12 · depth 18 - Existence of one dévissage step
AlgebraicGeometry.OModulePresheaf.hasDevissageStep1 below · cited by 2 · depth 18 - Coherence of the graded pieces I^kF/I^{k+1}F
AlgebraicGeometry.OModulePresheaf.isCoherent_idealPowQuot0 below · cited by 3 · depth 18 - Quasi-coherence of the graded pieces I^kF/I^{k+1}F
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_idealPowQuot0 below · cited by 3 · depth 18 - Vanishing of Čech H¹ is independent of the ordered affine cover
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_zero_ofModules_of_subsingleton6 below · cited by 2 · depth 18 - Graded pieces of the ideal-power filtration stay supported in Y
AlgebraicGeometry.OModulePresheaf.supportedIn_idealPowQuot0 below · cited by 2 · depth 18 - Connecting maps for Čech cohomology of an affine-exact sequence
AlgebraicGeometry.OModulePresheaf.AffSES.exists_connectingHom1 below · cited by 4 · depth 19 - Edge augmentation commutes with the vertical Čech differential
AlgebraicGeometry.OModulePresheaf.Leray.dV_comp_biAug0 below · cited by 3 · depth 19 - Degree-zero relative Čech presheaf is p_*mathcal O_{V'}
AlgebraicGeometry.OModulePresheaf.Leray.exists_hom_relHPresheaf_zero0 below · cited by 2 · depth 19 - Relative Čech presheaves on an affine base open
AlgebraicGeometry.OModulePresheaf.Leray.nonempty_relHPresheaf_obj_equiv_of_isAffineOpen0 below · cited by 1 · depth 19 - Exactness of the rows of the Čech–Leray double complex
AlgebraicGeometry.OModulePresheaf.Leray.rows_exact3 below · cited by 3 · depth 19 - Čech finiteness passes to the quotient in an affine short exact sequence
AlgebraicGeometry.OModulePresheaf.cechFinite_of_affSES_right3 below · cited by 2 · depth 19 - The Čech differential of an 𝒪-module datum squares to zero
AlgebraicGeometry.OModulePresheaf.d_comp_d0 below · cited by 30 · depth 19 - Generic freeness on an integral closed subscheme of a proper scheme
AlgebraicGeometry.OModulePresheaf.exists_basicOpen_sections_free_of_isIntegral0 below · cited by 2 · depth 19 - Base change of the Čech complex of a locally trivial module
AlgebraicGeometry.OModulePresheaf.exists_cochain_baseChange_equiv_of_locallyTrivial9 below · cited by 9 · depth 19 - Vanishing-ideal powers eventually annihilate a coherent datum supported in Y
AlgebraicGeometry.OModulePresheaf.exists_idealPowSub_eq_bot0 below · cited by 2 · depth 19 - Flatness of Čech cochains of a locally trivial module
AlgebraicGeometry.OModulePresheaf.flat_cochain_ofModules_of_locallyTrivial6 below · cited by 10 · depth 19 - Coherence passes to open-by-open cokernels
AlgebraicGeometry.OModulePresheaf.isCoherent_coker0 below · cited by 2 · depth 19 - Quasi-coherence passes to cokernels of presheaf module maps
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_coker0 below · cited by 2 · depth 19 - Top Snapper coefficient equals generic rank times that of L
AlgebraicGeometry.OModulePresheaf.coeff_eq_rankAtStalk_mul_coeff_of_forall_eulerChar_twist_tensorPow_eq87 below · cited by 2 · depth 25 - Snapper polynomiality for coherent 𝒪-module presheaf data
AlgebraicGeometry.OModulePresheaf.exists_polynomial_forall_eulerChar_twist_tensorPow_eq86 below · cited by 5 · depth 25 - Čech cohomology independent of the ordered affine cover
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_ofModules_of_isQuasicoherent_of_isSeparated7 below · cited by 18 · depth 25 - Čech cohomology of γ^*N agrees with that of γ_*mathcal O_W⊗ N
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_ofModules_pullback_comap_twist_pushforwardUnit7 below · cited by 8 · depth 25 - Affine-locally bijective morphisms induce Čech cohomology isomorphisms
AlgebraicGeometry.OModulePresheaf.AffHom.nonempty_H0_equiv_and_HSucc_equiv_of_bijective0 below · cited by 11 · depth 26 - Additivity of the Čech Euler characteristic in short exact sequences
AlgebraicGeometry.OModulePresheaf.eulerChar_eq_add_of_affSES3 below · cited by 8 · depth 26 - Tensoring an affine-wise short exact sequence by a flat datum
AlgebraicGeometry.OModulePresheaf.exists_affSES_tensor_of_flat0 below · cited by 9 · depth 26 - Kleiman's twisting step for Euler characteristics
AlgebraicGeometry.OModulePresheaf.exists_eulerChar_twist_pushforwardUnit_succ_sub_eq78 below · cited by 1 · depth 26 - Snapper–Kleiman polynomiality of Čech Euler characteristics
AlgebraicGeometry.OModulePresheaf.exists_mvPolynomial_totalDegree_le_forall_eulerChar_tensor_eq92 below · cited by 1 · depth 26 - Dévissage for coherent data with support on a proper scheme
AlgebraicGeometry.OModulePresheaf.forall_coherent_of_forall_integral11 below · cited by 4 · depth 26 - Push-forward of 𝒪_Z along a closed immersion: coherence and support
AlgebraicGeometry.OModulePresheaf.isCoherent_isQuasicoherent_supportedIn_pushforwardUnit0 below · cited by 4 · depth 26 - Open-by-open tensor product preserves quasi-coherence, coherence, support
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_isCoherent_supportedIn_tensor0 below · cited by 10 · depth 26 - Adjoining a largest affine chart preserves Čech cohomology
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_ofModules_of_orderEmbedding_of_forall_lt4 below · cited by 1 · depth 26 - Čech cohomology is invariant under order-isomorphic re-indexing
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_of_orderIso2 below · cited by 1 · depth 26 - Invariance of Čech cohomology under order-reversing re-indexing
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_of_orderIso_orderDual2 below · cited by 1 · depth 26 - Artin–Rees for Čech 0-coboundaries over a proper base
AlgebraicGeometry.OModulePresheaf.exists_d_eq_d_of_forall_d_mem_pow_of_isProper53 below · cited by 3 · depth 27 - Snapper polynomiality of χ(M⊗ L^{⊗ n})
AlgebraicGeometry.OModulePresheaf.exists_polynomial_forall_eulerChar_tensor_tensorPow_eq88 below · cited by 3 · depth 27 - Dévissage along the mathcal I_Y-adic filtration
AlgebraicGeometry.OModulePresheaf.forall_of_forall_idealAnnihilates6 below · cited by 1 · depth 27 - Dévissage step along an integral closed subscheme
AlgebraicGeometry.OModulePresheaf.forall_pushforward_of_isIntegral1 below · cited by 1 · depth 27 - Coherence of the kernel presheaf over a locally Noetherian base
AlgebraicGeometry.OModulePresheaf.isCoherent_ker0 below · cited by 3 · depth 27 - Quasi-coherence passes to kernels of morphisms
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_ker0 below · cited by 4 · depth 27 - Vanishing of alternating Čech data above the number of charts
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_and_eulerChar_eq_sum_range_of_card_le0 below · cited by 4 · depth 27 - Snapper polynomial has degree < r for r sections without common zero
AlgebraicGeometry.OModulePresheaf.degree_lt_of_forall_eulerChar_twist_tensorPow_eq_of_inter_iInter_support_zeroSchemeIdeal_eq_empty91 below · cited by 1 · depth 28 - Vanishing on a finite basic-open cover forces vanishing
AlgebraicGeometry.OModulePresheaf.eq_zero_of_forall_res_basicOpen_eq_zero0 below · cited by 4 · depth 28 - Quasi-coherence of the ideal-power subdatum I^k F
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_idealPow0 below · cited by 1 · depth 28 - Snapper induction step: first difference of twisted Euler characteristics
AlgebraicGeometry.OModulePresheaf.exists_eulerChar_twist_pushforwardUnit_succ_sub_eq_of_not_subset_support_zeroSchemeIdeal84 below · cited by 1 · depth 29 - Finite algebra structure on the algebraised coherent module
AlgebraicGeometry.OModulePresheaf.exists_coequifibered_addEquiv_of_affHom_pushforwardUnit_of_isAdicComplete_of_isProper71 below · cited by 2 · depth 30 - Grothendieck existence theorem for proper morphisms
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_of_isProper_of_isAdicComplete200 below · cited by 2 · depth 30 - h⁰=χ when higher Čech cohomology vanishes
AlgebraicGeometry.OModulePresheaf.finrank_sections_eq_eulerChar_of_iso_of_forall_subsingleton_HSucc1 below · cited by 7 · depth 30 - Coherence of pushforwards along a compatible system of finite morphisms
AlgebraicGeometry.OModulePresheaf.isCoherent_pushforwardUnit_and_exists_affHom_of_forall_isFinite_isPullback0 below · cited by 4 · depth 30 - Base change of Čech cohomology of a locally trivial module
AlgebraicGeometry.OModulePresheaf.nonempty_cech_baseChange_equiv_of_locallyTrivial11 below · cited by 15 · depth 30 - Field extension invariance of the Čech Euler characteristic
AlgebraicGeometry.OModulePresheaf.eulerChar_pullback_eq_eulerChar_of_isPullback_of_field26 below · cited by 3 · depth 31 - Unique lifting of compatible maps F → mathcal Gₙ to G
AlgebraicGeometry.OModulePresheaf.existsUnique_affHom_comp_eq_of_isAdicComplete_of_isProper68 below · cited by 5 · depth 31 - Semilinear family η induces an affine morphism into the Čech pushforward
AlgebraicGeometry.OModulePresheaf.exists_affHom_cechPushforward_apply_eq_of_forall_res_eq0 below · cited by 1 · depth 31 - Čech direct image along a proper map of an algebraised system
AlgebraicGeometry.OModulePresheaf.exists_affHom_cechPushforward_comp_eq_of_forall_ker_eq_pow_smul_top_of_isProper63 below · cited by 1 · depth 31 - Pull-back as graded R-algebra map of Čech rings
AlgebraicGeometry.OModulePresheaf.exists_algHom_cls_eq_cls_unitPullback22 below · cited by 5 · depth 31 - Coimage of a morphism of I-adic systems of coherent data
AlgebraicGeometry.OModulePresheaf.exists_coimage_adicSystem_of_forall_ker_le_range_sup_pow_smul_top1 below · cited by 1 · depth 31 - Reduction of an I-adic system modulo an ideal sheaf
AlgebraicGeometry.OModulePresheaf.exists_forall_ker_eq_idealPowSub_one_of_forall_ker_eq_pow_smul_top2 below · cited by 1 · depth 31 - Cokernel and kernel of ̂ u killed by a power of J
AlgebraicGeometry.OModulePresheaf.exists_forall_smul_mem_range_of_cechPushforward_of_isIso_pullback_snd_of_isProper96 below · cited by 1 · depth 31 - Graded Čech ring of the structure sheaf on an ordered affine cover
AlgebraicGeometry.OModulePresheaf.exists_gradedMonoid_cls_cup_unit3 below · cited by 5 · depth 31 - Inverse image of an I-adic system of coherent modules
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_forall_exists_linearEquiv_tensorProduct_of_hom5 below · cited by 1 · depth 31 - Grothendieck existence for closed subschemes of P^r_A
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_of_isClosedImmersion_proj_of_isAdicComplete99 below · cited by 3 · depth 31 - Algebraisability of kernels in adic systems on proper schemes
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_of_range_eq_ker_of_isProper_of_isAdicComplete74 below · cited by 1 · depth 31 - Algebraisability of adic systems is stable under extensions
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_of_surjective_of_range_eq_ker_of_isProper_of_isAdicComplete106 below · cited by 1 · depth 31 - Kernel and cokernel systems of a morphism of I-adic systems
AlgebraicGeometry.OModulePresheaf.exists_kernel_cokernel_adicSystem_of_affHom_of_forall_ker_eq_pow_smul_top8 below · cited by 2 · depth 31 - Adic kernel system annihilated by an ideal-sheaf power
AlgebraicGeometry.OModulePresheaf.forall_smul_eq_zero_of_comp_eq_zero_of_forall_smul_mem_pow_smul_top0 below · cited by 1 · depth 31 - Annihilation of a K-system by J₂^{ t} via Artin–Rees bookkeeping
AlgebraicGeometry.OModulePresheaf.forall_smul_eq_zero_of_range_eq_idealPowSub_of_forall_ker_le_pow_smul_top0 below · cited by 1 · depth 31 - Levelwise pushout squares for an algebraised adic system
AlgebraicGeometry.OModulePresheaf.isPushout_app_of_affHom_pushforwardUnit_of_ker_eq_pow_smul_top1 below · cited by 1 · depth 31 - Čech H⁰ of an ordered affine cover is Γ(V,M)
AlgebraicGeometry.OModulePresheaf.nonempty_sections_top_equiv_H0_ofModules0 below · cited by 7 · depth 31 - Vanishing of higher Čech cohomology descends along a field extension
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_of_forall_subsingleton_HSucc_baseChange_of_field12 below · cited by 3 · depth 31 - Affine-local kernel of a morphism of quasi-coherent module data
AlgebraicGeometry.OModulePresheaf.AffHom.exists_isQuasicoherent_injective_range_eq_ker0 below · cited by 3 · depth 32 - Affine-local cokernel of a map of quasi-coherent module data
AlgebraicGeometry.OModulePresheaf.AffHom.exists_isQuasicoherent_surjective_ker_eq_range0 below · cited by 3 · depth 32 - Base change preserves Čech acyclicity of an invertible module
AlgebraicGeometry.OModulePresheaf.H0_eq_bot_and_subsingleton_HSucc_baseChange_of_isInvertible_of_flat19 below · cited by 1 · depth 32 - Čech vanishing on an ordered affine cover is base-ring independent
AlgebraicGeometry.OModulePresheaf.H0_eq_bot_and_subsingleton_HSucc_iff_of_ofModules0 below · cited by 1 · depth 32 - Čech vanishing is local on the base via bi-Čech
AlgebraicGeometry.OModulePresheaf.H0_eq_bot_and_subsingleton_HSucc_of_forall_idx_preimage_of_isAffineOpen_inf20 below · cited by 1 · depth 32 - Fibrewise acyclicity implies acyclicity over the base
AlgebraicGeometry.OModulePresheaf.H0_eq_bot_and_subsingleton_HSucc_of_forall_isMaximal_baseChange_quotient16 below · cited by 1 · depth 32 - Vanishing of checkH^* transports along a scheme isomorphism
AlgebraicGeometry.OModulePresheaf.H0_eq_bot_and_subsingleton_HSucc_of_iso_pullback_of_isIso15 below · cited by 2 · depth 32 - Affine-local inverse image of a quasi-coherent module datum
AlgebraicGeometry.OModulePresheaf.IsQuasicoherent.exists_isQuasicoherent_forall_exists_linearEquiv_tensorProduct_of_hom3 below · cited by 1 · depth 32 - Čech ranks of an invertible module are invariant under isomorphism
AlgebraicGeometry.OModulePresheaf.cechFinrank_ofModules_pullback_eq_of_isIso14 below · cited by 11 · depth 32 - Graded commutativity of Čech cup products on classes
AlgebraicGeometry.OModulePresheaf.cls_mul_comm_graded19 below · cited by 3 · depth 32 - Associativity of the Čech cup product on cochains
AlgebraicGeometry.OModulePresheaf.cup_cup0 below · cited by 1 · depth 32 - Leibniz rule for the Čech cup product
AlgebraicGeometry.OModulePresheaf.d_cup0 below · cited by 3 · depth 32 - Refinement pull-back of Čech cochains is a chain map
AlgebraicGeometry.OModulePresheaf.d_unitPullback2 below · cited by 16 · depth 32 - Vanishing of I-adically divisible Čech 0-cocycles over proper schemes
AlgebraicGeometry.OModulePresheaf.eq_zero_of_d_eq_zero_of_forall_mem_pow_smul_of_isProper1 below · cited by 1 · depth 32 - Twisting by a torsion line bundle preserves the Euler characteristic
AlgebraicGeometry.OModulePresheaf.eulerChar_ofModules_tensor_eq_of_tensorPow_iso_unit90 below · cited by 2 · depth 32 - Degree-zero theorem on formal functions: separatedness half
AlgebraicGeometry.OModulePresheaf.exists_H0_inf_pow_smul_le_pow_smul_H0_of_isProper55 below · cited by 3 · depth 32 - Refinement induces isomorphisms on Čech cohomology of mathcal O_X
AlgebraicGeometry.OModulePresheaf.exists_HSucc_equiv_unitPullback_id_of_isSeparated16 below · cited by 10 · depth 32 - Adic envelope of an Artin–Rees-stable subsystem
AlgebraicGeometry.OModulePresheaf.exists_adicSystem_range_eq_range_of_forall_pow_smul_top_inf_range_le0 below · cited by 1 · depth 32 - Algebraisation of an extension of formal coherent systems over a proper adic-complete base
AlgebraicGeometry.OModulePresheaf.exists_affHom_affHom_range_eq_ker_comp_eq_of_surjective_of_range_eq_ker_of_isProper_of_isAdicComplete104 below · cited by 1 · depth 32 - Pull-back of a morphism of quasi-coherent module data
AlgebraicGeometry.OModulePresheaf.exists_affHom_apply_eq_of_forall_exists_linearEquiv_tensorProduct2 below · cited by 1 · depth 32 - Levelwise cokernels of a map into an I-adic system
AlgebraicGeometry.OModulePresheaf.exists_cokernel_adicSystem_of_affHom_of_forall_ker_eq_pow_smul_top1 below · cited by 1 · depth 32 - Uniform Artin–Rees bound for Čech 0-coboundaries, coherent case
AlgebraicGeometry.OModulePresheaf.exists_d_eq_d_of_forall_d_mem_pow_smul_of_isProper55 below · cited by 1 · depth 32 - Globalising the exponent for Čech thread comparison over P
AlgebraicGeometry.OModulePresheaf.exists_forall_affineOpens_thread_smul_of_forall_exists_forall_le_of_isProper1 below · cited by 1 · depth 32 - Separatedness half of formal functions, relative chart version
AlgebraicGeometry.OModulePresheaf.exists_forall_eq_sum_smul_of_forall_mem_pow_smul_preimage_of_isProper56 below · cited by 1 · depth 32 - Kernels of I-adic systems of coherent module presheaves
AlgebraicGeometry.OModulePresheaf.exists_forall_range_eq_ker_of_forall_ker_eq_pow_smul_top4 below · cited by 1 · depth 32 - Thread-level bounds imply level-wise bounds for u_k
AlgebraicGeometry.OModulePresheaf.exists_forall_smul_mem_range_of_forall_thread_of_cechPushforward9 below · cited by 1 · depth 32 - Uniform Mittag-Leffler property for Čech 0-cochains along p
AlgebraicGeometry.OModulePresheaf.exists_forall_sub_mem_pow_smul_of_forall_res_sub_res_mem_pow_smul_preimage_of_isProper56 below · cited by 1 · depth 32 - Uniform annihilation of kernel and cokernel of the threaded Čech unit
AlgebraicGeometry.OModulePresheaf.exists_forall_thread_smul_eq_apply_and_smul_eq_zero_of_isIso_pullback_snd_of_isProper85 below · cited by 1 · depth 32 - Algebraisation of an adic system from a morphism of extensions
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_of_extension_of_comp_eq0 below · cited by 1 · depth 32 - Descent of a compatible adic presentation to the cokernel
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_ker_eq_pow_smul_top_of_forall_ker_eq_range_sup1 below · cited by 1 · depth 32 - An I-adic system of coherent data is a quotient of one coherent datum
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_surjective_of_forall_ker_eq_pow_smul_top_of_isClosedImmersion33 below · cited by 1 · depth 32 - R-linearity of the alternating Čech pull-back of functions
AlgebraicGeometry.OModulePresheaf.exists_linearMap_apply_eq_unitPullback0 below · cited by 3 · depth 32 - Snapper–Kleiman polynomiality of Čech Euler characteristics
AlgebraicGeometry.OModulePresheaf.exists_mvPolynomial_totalDegree_le_forall_eulerChar_tensor_eq_monoidalV293 below · cited by 2 · depth 32 - Stable kernel of a morphism of I-adic systems of coherent sheaves
AlgebraicGeometry.OModulePresheaf.exists_subsystem_ker_le_range_sup_pow_smul_top_of_affHom_of_forall_ker_eq_pow_smul_top4 below · cited by 1 · depth 32 - A finite free two-term model for H⁰ under base change
AlgebraicGeometry.OModulePresheaf.exists_twoTermComplex_kerMapBaseChange_bijective_ofModules64 below · cited by 3 · depth 32 - Re-gluing relation for an obstruction cochain, read in classes
AlgebraicGeometry.OModulePresheaf.forall_mem_range_d_iff_add_map_sub_map_eq_zero_of_exists_refinement_of_pinned17 below · cited by 1 · depth 32 - Čech vanishing on X and U∩ V descends to V
AlgebraicGeometry.OModulePresheaf.forall_subsingleton_HSucc_restrict_of_sup_eq_top23 below · cited by 1 · depth 32 - Degree-zero sheaf condition for the internal Hom
AlgebraicGeometry.OModulePresheaf.internalHom_d_zero_eq_zero_iff_existsUnique2 below · cited by 2 · depth 32 - Sections of the internal Hom over an affine open are determined at U
AlgebraicGeometry.OModulePresheaf.internalHom_ext_of_apply_self_eq1 below · cited by 1 · depth 32 - Coherence of the Čech pushforward along a proper morphism
AlgebraicGeometry.OModulePresheaf.isCoherent_cechPushforward_of_isProper53 below · cited by 1 · depth 32 - Internal Hom of coherent module data is coherent
AlgebraicGeometry.OModulePresheaf.isCoherent_internalHom_and_existsUnique_eval_eq0 below · cited by 6 · depth 32 - Quasi-coherence of the Čech direct image along a separated morphism
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_cechPushforward_of_isSeparated0 below · cited by 1 · depth 32 - Finite direct powers preserve quasi-coherence of module presheaves
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_pow0 below · cited by 1 · depth 32 - Künneth injectivity for Čech cocycle class maps
AlgebraicGeometry.OModulePresheaf.kunneth_injective_of_cls_unitPullback59 below · cited by 3 · depth 32 - Čech cohomology of the unit mathcal O_V-module versus mathcal O_V
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_ofModules_tensorUnit_unit1 below · cited by 4 · depth 32 - Cover-independence of Čech cohomology of mathcal O_V
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_unit_of_isSeparated11 below · cited by 2 · depth 32 - Unit laws for the Čech cup product
AlgebraicGeometry.OModulePresheaf.one_cup_and_cup_one0 below · cited by 1 · depth 32 - Čech H⁰ projective and commuting with base change
AlgebraicGeometry.OModulePresheaf.projective_H0_and_bijective_kerBaseChangeHom_of_isReduced_of_finrank_eq75 below · cited by 1 · depth 32 - Composing pinning relations for pull-backs of Čech cocycles
AlgebraicGeometry.OModulePresheaf.unitPullback_comp_sub_unitPullback_id_mem21 below · cited by 1 · depth 32 - Pull-back of cup products of Čech cocycles, up to coboundary
AlgebraicGeometry.OModulePresheaf.unitPullback_cup_sub_cup_unitPullback_mem_of_mem_ker19 below · cited by 2 · depth 32 - Refinement pull-back independent of index map up to coboundary
AlgebraicGeometry.OModulePresheaf.unitPullback_sub_unitPullback_mem_of_d_eq_zero18 below · cited by 5 · depth 32 - Refinement independence of differences of Čech pullbacks
AlgebraicGeometry.OModulePresheaf.unitPullback_sub_unitPullback_mem_of_mem_refinement20 below · cited by 3 · depth 32 - Transport of pinned cocycle pairs along a commuting square
AlgebraicGeometry.OModulePresheaf.unitPullback_unitPullback_sub_mem_of_comp_eq_comp21 below · cited by 1 · depth 32 - Čech acyclicity on V from acyclicity of the glued cover
AlgebraicGeometry.OModulePresheaf.H0_eq_bot_and_forall_subsingleton_HSucc_restrict_of_subsingleton_HTot_biCech1 below · cited by 1 · depth 33 - Affine base change of sections of a quasi-coherent module datum
AlgebraicGeometry.OModulePresheaf.IsQuasicoherent.exists_linearEquiv_tensorProduct_apply_one_tmul_eq_res0 below · cited by 6 · depth 33 - Refinement pull-back and edge augmentations differ by a total coboundary
AlgebraicGeometry.OModulePresheaf.Leray.exists_dTot_eq_single_biAug_unitPullback_sub_single_id5 below · cited by 1 · depth 33 - Transposing the Čech–Leray double complex of id_X
AlgebraicGeometry.OModulePresheaf.Leray.exists_levelwise_equiv_transpose_id0 below · cited by 1 · depth 33 - Künneth formula for Čech ranks on a product of proper k-schemes
AlgebraicGeometry.OModulePresheaf.cechFinrank_tensor_pullback_eq_sum_mul_of_isProper99 below · cited by 3 · depth 33 - Čech degree-zero rank one for the structure sheaf
AlgebraicGeometry.OModulePresheaf.cechFinrank_unit_zero_eq_one_of_bijective0 below · cited by 2 · depth 33 - Invariance of the Čech Euler characteristic under isomorphism
AlgebraicGeometry.OModulePresheaf.eulerChar_ofModules_eq_of_iso1 below · cited by 1 · depth 33 - Euler characteristic invariance under automorphisms over the base field
AlgebraicGeometry.OModulePresheaf.eulerChar_ofModules_pullback_eq_of_iso_over27 below · cited by 1 · depth 33 - Multiplicativity of Čech Euler characteristics over a product
AlgebraicGeometry.OModulePresheaf.eulerChar_ofModules_tensor_pullback_eq_mul_of_isProper101 below · cited by 1 · depth 33 - Artin–Rees for Čech 0-cocycles over a proper base
AlgebraicGeometry.OModulePresheaf.exists_H0_inf_pow_smul_le_pow_smul_H0_unit_of_isProper53 below · cited by 1 · depth 33 - Trace retraction for finite flat morphisms of invertible rank
AlgebraicGeometry.OModulePresheaf.exists_affHom_pushforwardUnit_unit_retraction_of_finrank_eq_of_isUnit4 below · cited by 1 · depth 33 - Algebraisation of an adic extension over a basis of affine opens
AlgebraicGeometry.OModulePresheaf.exists_basisData_range_eq_ker_comp_eq_of_surjective_of_range_eq_ker_of_isProper_of_isAdicComplete103 below · cited by 1 · depth 33 - Čech 1-cocycle on X×_k Y split by both slices
AlgebraicGeometry.OModulePresheaf.exists_d_eq_of_d_comap_slice_eq_of_bijective_algebraMap18 below · cited by 4 · depth 33 - Uniform Mittag-Leffler property for Čech 0-cochains, proper case
AlgebraicGeometry.OModulePresheaf.exists_d_eq_zero_sub_mem_pow_of_d_mem_pow_of_isProper55 below · cited by 3 · depth 33 - Degree-one Čech pull-back independent of the refining index map
AlgebraicGeometry.OModulePresheaf.exists_d_zero_eq_unitPullback_sub_unitPullback_of_d_one_eq_zero0 below · cited by 2 · depth 33 - Filtration by affine-exact sequences from a direct-sum decomposition
AlgebraicGeometry.OModulePresheaf.exists_filtration_affSES_of_forall_affineOpens_bijective_sum0 below · cited by 1 · depth 33 - Čech 0-cocycle threads over a complete base: existence, uniqueness, finiteness
AlgebraicGeometry.OModulePresheaf.exists_forall_eq_thread_and_eq_zero_of_forall_eq_zero_of_isAdicComplete_of_isProper60 below · cited by 1 · depth 33 - Serre's theorem A in Čech form over a base ring
AlgebraicGeometry.OModulePresheaf.exists_forall_exists_H0_tensor_twist_span_eq_top2 below · cited by 1 · depth 33 - Gluing over a finite basic-open cover of an affine open
AlgebraicGeometry.OModulePresheaf.exists_forall_res_basicOpen_eq0 below · cited by 1 · depth 33 - Uniform surjectivity on H⁰ after twisting an I-adic system
AlgebraicGeometry.OModulePresheaf.exists_forall_surjective_H0Map_tensorMap_twist_of_forall_ker_eq_pow_smul_top26 below · cited by 1 · depth 33 - Untwisting compatible generating sections over a projective A-scheme
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_forall_surjective_of_forall_H0Map_tensorMap_eq_of_span_eq_top7 below · cited by 1 · depth 33 - Gluing coherent module data from a basis of affine opens
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_linearEquiv_of_forall_basicOpen_of_isBasis1 below · cited by 2 · depth 33 - Finite generation up to x-saturation of family frames
AlgebraicGeometry.OModulePresheaf.exists_isFG_hom_injective_saturated_familyFramesGradedModule_of_isFinite1 below · cited by 1 · depth 33 - Base change of an adic datum to a proper frame
AlgebraicGeometry.OModulePresheaf.exists_isPullback_isProper_and_exists_forall_surjective_ker_eq_pow_smul_top_of_forall_ker_eq_pow_smul_top0 below · cited by 1 · depth 33 - Snapper polynomiality of χ(M ⊗ L^{⊗ n})
AlgebraicGeometry.OModulePresheaf.exists_polynomial_forall_eulerChar_tensor_tensorPow_eq_monoidalV289 below · cited by 4 · depth 33 - Čech unit becomes bijective after inverting functions vanishing on T'
AlgebraicGeometry.OModulePresheaf.exists_pow_smul_eq_zero_and_exists_eq_pow_smul_of_mem_vanishingIdeal_of_isIso_pullback_snd4 below · cited by 1 · depth 33 - Ring homomorphism Λ^{op} → End_κ H₁ from pinned pull-backs
AlgebraicGeometry.OModulePresheaf.exists_ringHom_mulOpposite_forall_apply_eq_of_unitPullback0 below · cited by 1 · depth 33 - Base-changed module data are quasi-coherent on basic opens
AlgebraicGeometry.OModulePresheaf.forall_basicOpen_quasicoherent_of_forall_exists_linearEquiv_tensorProduct0 below · cited by 1 · depth 33 - Flat base change of Čech torsion and solvability from W₀ to W
AlgebraicGeometry.OModulePresheaf.forall_smul_eq_zero_and_exists_eq_smul_of_le_of_flat_of_bijective6 below · cited by 1 · depth 33 - Quasi-coherence makes restriction to a basic open a localisation
AlgebraicGeometry.OModulePresheaf.isLocalizedModule_res_of_isQuasicoherent0 below · cited by 5 · depth 33 - Per-degree Künneth injectivity for Čech classes
AlgebraicGeometry.OModulePresheaf.kunneth_toModule_diag_injective_of_cls_unitPullback57 below · cited by 1 · depth 33 - Refined pull-back cochains compose along an affine morphism
AlgebraicGeometry.OModulePresheaf.map_app_unitPullback_eq_unitPullback_comp0 below · cited by 1 · depth 33 - Čech cohomology of FotimesL^{⊗ d} agrees with the twist datum
AlgebraicGeometry.OModulePresheaf.nonempty_HSucc_ofModules_tensorObj_tensorPow_linearEquiv_HSucc_tensor_twist_monoidalV213 below · cited by 1 · depth 33 - Bi-Čech total cohomology computes Čech cohomology of the product cover
AlgebraicGeometry.OModulePresheaf.nonempty_HTot_biCech_equiv_prodCover_of_isQuasicoherent10 below · cited by 4 · depth 33 - Leibniz rule for the ordered Čech cup product
AlgebraicGeometry.OModulePresheaf.od_ocup0 below · cited by 2 · depth 33 - Alternating extension commutes with the Čech differential
AlgebraicGeometry.OModulePresheaf.od_oext2 below · cited by 5 · depth 33 - Reversal commutes with the ordered Čech differential
AlgebraicGeometry.OModulePresheaf.od_orev0 below · cited by 1 · depth 33 - Ordered Čech pull-back of functions commutes with d
AlgebraicGeometry.OModulePresheaf.od_ounitPullback0 below · cited by 1 · depth 33 - Restriction to increasing chains commutes with the cup product
AlgebraicGeometry.OModulePresheaf.ores_ocup0 below · cited by 2 · depth 33 - Restriction to increasing tuples is a cochain map
AlgebraicGeometry.OModulePresheaf.ores_od0 below · cited by 2 · depth 33 - Restriction of the alternating extension is the identity
AlgebraicGeometry.OModulePresheaf.ores_oext0 below · cited by 3 · depth 33 - Refined Čech pull-back factors through ordered cochains
AlgebraicGeometry.OModulePresheaf.ores_ounitPullback_oext0 below · cited by 4 · depth 33 - Reversal anti-commutes with the ordered cup product
AlgebraicGeometry.OModulePresheaf.orev_ocup0 below · cited by 1 · depth 33 - Alternating extension is invariant under index reversal
AlgebraicGeometry.OModulePresheaf.orev_oext0 below · cited by 1 · depth 33 - Ordered Čech pull-back is multiplicative for the cup product
AlgebraicGeometry.OModulePresheaf.ounitPullback_ocup0 below · cited by 1 · depth 33 - Ordered cocycles are alternating up to a coboundary
AlgebraicGeometry.OModulePresheaf.sub_oext_ores_mem_of_od_eq_zero10 below · cited by 2 · depth 33 - Graded Čech vanishing transfers to twisted sheaf Čech vanishing
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_tensor_twist_of_subsingleton_H_shift_familyFramesGradedModule2 below · cited by 2 · depth 33 - Acyclicity of the mixed bi-Čech complex on U∩ V
AlgebraicGeometry.OModulePresheaf.subsingleton_HTot_biCech_imageFamily_of_forall_subsingleton_HSucc19 below · cited by 1 · depth 33 - Vanishing of the columns of the bi-Čech double complex
AlgebraicGeometry.OModulePresheaf.subsingleton_colH_biCech_of_forall_idx_restrict9 below · cited by 1 · depth 33 - Refined alternating pull-back composes, up to coboundaries
AlgebraicGeometry.OModulePresheaf.unitPullback_unitPullback_sub_mem_of_d_eq_zero0 below · cited by 8 · depth 33 - Injectivity of check H^*(mathcal F₁)→check H^*(mathcal F₂) when check H^{>0}(mathcal F₃) vanishes
AlgebraicGeometry.OModulePresheaf.AffSES.injective_inc_HSuccMap_of_forall_subsingleton_HSucc_of_surjective_proj_H0Map2 below · cited by 1 · depth 34 - Vanishing of Čech ranks from a Künneth absorption isomorphism
AlgebraicGeometry.OModulePresheaf.cechFinrank_eq_zero_of_iso_tensor_pullback_of_H0_eq_bot_of_subsingleton_HSucc102 below · cited by 1 · depth 34 - Degree-0 theorem on formal functions: unique limit cocycle
AlgebraicGeometry.OModulePresheaf.existsUnique_d_eq_zero_forall_sub_mem_pow_smul_of_isAdicComplete_of_isProper59 below · cited by 2 · depth 34 - Künneth comparison for the box cover, pinned on cup products
AlgebraicGeometry.OModulePresheaf.exists_HTot_biCech_equiv_prodCover_cup_pinned29 below · cited by 1 · depth 34 - Pushforwards of invertible modules along adic thickenings
AlgebraicGeometry.OModulePresheaf.exists_affHom_pushforward_ofModules_adicThickening_surjective_ker_eq_pow_smul_top7 below · cited by 1 · depth 34 - Gluing chart-wise extension data over a basis of affine opens
AlgebraicGeometry.OModulePresheaf.exists_basisData_of_chartData_of_isProper_of_isAdicComplete82 below · cited by 1 · depth 34 - Bi-Čech complex of X×_k Y as a tensor double complex
AlgebraicGeometry.OModulePresheaf.exists_biCech_preimageFamily_equiv_tensor_cochain_pinned1 below · cited by 2 · depth 34 - Chart-wise module models for a formal extension of coherent sheaves
AlgebraicGeometry.OModulePresheaf.exists_chartData_of_surjective_of_range_eq_ker_of_isProper_of_isAdicComplete77 below · cited by 1 · depth 34 - Čech 1-cocycle bounding on slabs and on a section
AlgebraicGeometry.OModulePresheaf.exists_d_eq_of_d_comap_section_eq_of_forall_preimage_chart6 below · cited by 1 · depth 34 - Constancy of the Čech Euler characteristic over a local base
AlgebraicGeometry.OModulePresheaf.exists_forall_eulerChar_baseChange_eq_of_locallyTrivial_of_isLocalRing79 below · cited by 3 · depth 34 - Serre's theorem A in the charts of an affine morphism to P^N_A
AlgebraicGeometry.OModulePresheaf.exists_forall_exists_res_eq_frameUnit_pow_smul_res_and_span_eq_top0 below · cited by 2 · depth 34 - Uniform Serre vanishing for twisted kernels of an I-adic system
AlgebraicGeometry.OModulePresheaf.exists_forall_subsingleton_HSucc_tensor_twist_of_forall_ker_eq_pow_smul_top18 below · cited by 1 · depth 34 - Extending sections from chart intersections to compatible twisted frame families
AlgebraicGeometry.OModulePresheaf.exists_framesCompat_res_eq_prod_frameUnit_pow_smul_res_of_isQuasicoherent0 below · cited by 2 · depth 34 - Lifting point-derivation-valued Čech 1-cocycles to 0-cochains
AlgebraicGeometry.OModulePresheaf.exists_pointDerivations_d_zero_eq_of_d_one_eq_zero_of_isAffine_of_basicOpen3 below · cited by 1 · depth 34 - Snapper polynomiality for coherent presheaf data twisted by a line bundle
AlgebraicGeometry.OModulePresheaf.exists_polynomial_forall_eulerChar_twist_tensorPow_eq_monoidalV287 below · cited by 1 · depth 34 - Čech families over a cover inherit a-power localisation
AlgebraicGeometry.OModulePresheaf.exists_pow_smul_eq_zero_and_exists_eq_pow_smul_of_forall_isAffineOpen_basicOpen2 below · cited by 1 · depth 34 - Čech 1-cocycles trivial on the slice bound over affine slabs
AlgebraicGeometry.OModulePresheaf.exists_res_eq_sum_of_d_comap_slice_eq_of_isAffineOpen12 below · cited by 1 · depth 34 - Affineness and localisation of the unit over basic opens off T'
AlgebraicGeometry.OModulePresheaf.isAffineOpen_basicOpen_and_exists_pow_smul_eq_zero_and_exists_eq_pow_smul_of_mem_vanishingIdeal0 below · cited by 1 · depth 34 - Quasi-coherence of the sections datum restricts to opens
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_ofModules_restrict0 below · cited by 4 · depth 34 - Exactness of the columns of the iterated Čech complex
AlgebraicGeometry.OModulePresheaf.iterCech_cols_exact_of_isQuasicoherent7 below · cited by 2 · depth 34 - Exact augmented rows of the iterated Čech complex
AlgebraicGeometry.OModulePresheaf.iterCech_rows_exact_of_isQuasicoherent3 below · cited by 2 · depth 34 - Cochain-level Künneth for bi-Čech complexes of box products
AlgebraicGeometry.OModulePresheaf.nonempty_HTot_biCech_strips_equiv_HTot_tensor_ofCech19 below · cited by 1 · depth 34 - Ordered Čech pairing is adjoint to the boundary
AlgebraicGeometry.OModulePresheaf.opair_od_eq_opair_obd0 below · cited by 1 · depth 34 - Pairing a cochain against the sorted basis element of u
AlgebraicGeometry.OModulePresheaf.opair_oesort_single0 below · cited by 1 · depth 34 - Pairing is compatible with enlarging the ambient tuple
AlgebraicGeometry.OModulePresheaf.res_opair0 below · cited by 1 · depth 34 - Transport of higher Čech vanishing along an isomorphism over Spec R
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_ofModules_of_iso_pullback_of_isIso15 below · cited by 1 · depth 34 - Pulled-back unit cochain as signed transport through θ
AlgebraicGeometry.OModulePresheaf.unitPullback_apply_eq_sign_smul_of_ringEquiv_tensor_pin0 below · cited by 2 · depth 34 - Refining the box-cover cup product, up to a coboundary
AlgebraicGeometry.OModulePresheaf.unitPullback_prodCover_cup_sub_cup_unitPullback_mem24 below · cited by 1 · depth 34 - Augmented box product and cup product agree in Hⁿ
AlgebraicGeometry.OModulePresheaf.IterCech.exists_mk_single_augTot_eq_mk_single_augCech_cup6 below · cited by 1 · depth 35 - Sections over an affine preimage are the Čech 0-cocycles
AlgebraicGeometry.OModulePresheaf.bijective_cechPushforward_of_isAffineOpen_preimage1 below · cited by 1 · depth 35 - Degree-zero Čech cocycles are global sections
AlgebraicGeometry.OModulePresheaf.d_zero_eq_zero_iff_existsUnique_of_isQuasicoherent0 below · cited by 2 · depth 35 - Multiplication by generators of I on the graded pieces
AlgebraicGeometry.OModulePresheaf.exists_affHom_apply_eq_smul_comm_iSup_range_eq_top_of_forall_ker_eq_pow_smul_top0 below · cited by 1 · depth 35 - Box zig-zag data force a coboundary on the slice
AlgebraicGeometry.OModulePresheaf.exists_app_strip_eq_sum_of_box_zigzag_of_d_comap_slice_eq0 below · cited by 1 · depth 35 - Rank-one local freeness from invertible reductions modulo Iⁿ⁺¹
AlgebraicGeometry.OModulePresheaf.exists_basicOpen_bijective_smul_res_of_affHom_pushforward_adicThickening_of_le_asIdeal3 below · cited by 1 · depth 35 - Gluing chart-wise extension data along a cocycle
AlgebraicGeometry.OModulePresheaf.exists_basisData_of_chartData_of_cocycle_of_comp_eq0 below · cited by 1 · depth 35 - Box zig-zag datum for a Čech 1-cocycle over an affine open
AlgebraicGeometry.OModulePresheaf.exists_box_zigzag_of_d_eq_zero_of_isAffineOpen3 below · cited by 1 · depth 35 - Chart-wise push-out models for agreeing local Ext classes
AlgebraicGeometry.OModulePresheaf.exists_chartModels_extPushout_of_forall_res_symm_mk_eq3 below · cited by 1 · depth 35 - Twisting chart comparison maps compatible with a Čech 1-cocycle
AlgebraicGeometry.OModulePresheaf.exists_cocycle_comp_eq_of_chartData_of_isProper_of_isAdicComplete74 below · cited by 1 · depth 35 - Correcting overlap isomorphisms of chart extensions to a cocycle
AlgebraicGeometry.OModulePresheaf.exists_cocycle_of_chartData_of_isProper_of_isAdicComplete68 below · cited by 1 · depth 35 - Kleiman's twisting step for Euler characteristics
AlgebraicGeometry.OModulePresheaf.exists_eulerChar_twist_pushforwardUnit_succ_sub_eq_monoidalV279 below · cited by 1 · depth 35 - A coherent datum computing local Ext¹ of two coherent data
AlgebraicGeometry.OModulePresheaf.exists_isCoherent_linearEquiv_extQuot_of_isCoherent5 below · cited by 1 · depth 35 - Finite generation up to saturation of the frames graded module
AlgebraicGeometry.OModulePresheaf.exists_isFG_hom_injective_saturated_familyFramesGradedModule1 below · cited by 1 · depth 35 - Formal splitting data on an ordered affine cover
AlgebraicGeometry.OModulePresheaf.exists_orderedAffineCover_formalSplittingData_of_isProper_of_isAdicComplete7 below · cited by 1 · depth 35 - Box zig-zag datum with bounding strip cochain yields slab coboundary
AlgebraicGeometry.OModulePresheaf.exists_res_eq_sum_of_box_zigzag_of_exists_strip_eq_sum2 below · cited by 1 · depth 35 - Triviality of strip 1-cocycles when Γ(X)=k
AlgebraicGeometry.OModulePresheaf.exists_strip_eq_sum_of_forall_isAffineOpen_of_slice_of_bijective6 below · cited by 1 · depth 35 - Comparison of two box zig-zag data over nested affine opens
AlgebraicGeometry.OModulePresheaf.exists_strip_res_sub_eq_sum_of_box_zigzag_of_le3 below · cited by 1 · depth 35 - Affine Künneth for invertible modules, natural in boxes
AlgebraicGeometry.OModulePresheaf.exists_tensorProduct_sections_linearEquiv_sections_box_natural_of_isInvertible16 below · cited by 1 · depth 35 - Čech H¹ of mathcal O_X vanishes on an affine scheme
AlgebraicGeometry.OModulePresheaf.mem_range_d_zero_of_d_one_eq_zero_of_isAffine_of_basicOpen2 below · cited by 1 · depth 35 - Bi-Čech columns as products of Čech cohomologies
AlgebraicGeometry.OModulePresheaf.nonempty_colH_biCech_equiv_pi_cech_restrict0 below · cited by 1 · depth 35 - Degree zero: augmented external product equals augmented cup product
AlgebraicGeometry.OModulePresheaf.IterCech.augTot_single_eq_augCech_cup_zero4 below · cited by 1 · depth 36 - Product zig-zag in the iterated Čech complex
AlgebraicGeometry.OModulePresheaf.IterCech.exists_dTot_eq_single_augTot_sub_single_augCech_cup4 below · cited by 1 · depth 36 - Čech ranks of a locally trivial module under field extension
AlgebraicGeometry.OModulePresheaf.cechFinrank_baseChange_eq_of_locallyTrivial_of_field12 below · cited by 1 · depth 36 - Discrepancy 2-cochain of the chart isomorphisms is a Čech cocycle
AlgebraicGeometry.OModulePresheaf.d_eq_zero_of_forall_eq_comp_sub_of_chartData0 below · cited by 1 · depth 36 - Morphisms of 𝒪_X-modules from data on affine opens
AlgebraicGeometry.OModulePresheaf.existsUnique_hom_app_eq_of_affHom_ofModules1 below · cited by 1 · depth 36 - Push-out chart models over affine opens of a chart
AlgebraicGeometry.OModulePresheaf.exists_chartModel_extPushout_of_surjective1 below · cited by 1 · depth 36 - Čech 2-cochain of the overlap defect of chart-wise extensions
AlgebraicGeometry.OModulePresheaf.exists_cochain_internalHom_forall_eq_comp_sub_of_chartData1 below · cited by 1 · depth 36 - Cocycle twist of chart data agreeing on overlaps
AlgebraicGeometry.OModulePresheaf.exists_cocycle_comp_eq_of_chartData_of_internalHom_cocycle1 below · cited by 1 · depth 36 - Formal Čech cochains: cocycle plus formal coboundary
AlgebraicGeometry.OModulePresheaf.exists_d_eq_zero_forall_sub_sub_d_mem_pow_smul_of_isAdicComplete_of_isProper64 below · cited by 1 · depth 36 - Čech 1-cochains measuring chart defects, given cocycle data
AlgebraicGeometry.OModulePresheaf.exists_internalHom_cochain_lam_comp_eval_comp_eq_sub_of_chartData_of_cocycle6 below · cited by 1 · depth 36 - Local constancy of fibrewise Euler characteristic on a basic open
AlgebraicGeometry.OModulePresheaf.exists_notMem_forall_eulerChar_baseChange_eq_of_locallyTrivial92 below · cited by 1 · depth 36 - Overlap isomorphisms of push-out models on a separated scheme
AlgebraicGeometry.OModulePresheaf.exists_overlapIso_extPushout_of_forall_res_symm_mk_eq0 below · cited by 1 · depth 36 - Base-change-compatible projective complex computing Čech cohomology of a locally trivial module
AlgebraicGeometry.OModulePresheaf.exists_projective_complex_forall_baseChange_quasiIso_cech_of_locallyTrivial76 below · cited by 2 · depth 36 - Upper semicontinuity of fibrewise Čech ranks, residue-field form
AlgebraicGeometry.OModulePresheaf.isClosed_setOf_le_cechFinrank_baseChange_residueField_of_locallyTrivial85 below · cited by 1 · depth 36 - Quasi-coherence of the structure-sheaf module presheaf
AlgebraicGeometry.OModulePresheaf.isQuasicoherent_unit0 below · cited by 1 · depth 36 - Chartwise obstruction 2-cochain lies in Iⁿ⁺¹C²+dC¹
AlgebraicGeometry.OModulePresheaf.mem_pow_smul_sup_range_d_of_forall_eq_comp_sub_of_chartData6 below · cited by 1 · depth 36 - Cocycles trivial modulo every Iⁿ are coboundaries
AlgebraicGeometry.OModulePresheaf.mem_range_d_of_d_eq_zero_of_forall_mem_pow_smul_sup_range_d_of_isAdicComplete_of_isProper57 below · cited by 1 · depth 36 - Transport of Čech cohomology of an invertible sheaf along a scheme isomorphism
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_ofModules_of_iso_pullback_of_isIso15 below · cited by 4 · depth 36 - Čech cohomology of a strip equals that of the slice
AlgebraicGeometry.OModulePresheaf.nonempty_cechEquiv_restrict_preimage_snd_sliceAt_fromSpec16 below · cited by 1 · depth 36 - Nakayama surjectivity over a universally closed base
AlgebraicGeometry.OModulePresheaf.AffHom.surjective_app_of_range_sup_smul_top_eq_top_of_le_jacobson0 below · cited by 1 · depth 37 - Twisting chart data by a cocycle to make comparison maps compatible
AlgebraicGeometry.OModulePresheaf.exists_cocycle_comp_eq_of_chartData_of_components0 below · cited by 1 · depth 37 - Lifting compatible Čech classes in positive degree over a complete base
AlgebraicGeometry.OModulePresheaf.exists_d_eq_zero_forall_sub_mem_pow_smul_sup_range_d_of_isAdicComplete_of_isProper61 below · cited by 1 · depth 37 - Uniform Artin–Rees for Čech cocycles in degree i+1
AlgebraicGeometry.OModulePresheaf.exists_d_succ_eq_zero_sub_mem_pow_of_d_succ_mem_pow_of_isProper57 below · cited by 2 · depth 37 - Chart-comparison defects as I-adically Cauchy Čech 1-cochains
AlgebraicGeometry.OModulePresheaf.exists_internalHom_cochain_lam_comp_eval_comp_eq_sub_of_chartData5 below · cited by 2 · depth 37 - Artin–Rees separatedness for Čech cocycles in degree i+1
AlgebraicGeometry.OModulePresheaf.exists_ker_d_inf_pow_smul_le_pow_smul_ker_sup_range_of_isProper56 below · cited by 2 · depth 37 - Pull-back of unit cochains along an affine morphism
AlgebraicGeometry.OModulePresheaf.unitPullback_comap_id_apply0 below · cited by 1 · depth 37 - Uniform Artin–Rees for Čech coboundaries in positive degree
AlgebraicGeometry.OModulePresheaf.exists_d_succ_eq_d_succ_of_forall_d_succ_mem_pow_smul_of_isProper56 below · cited by 1 · depth 38 - Artin–Rees for Čech cocycles of mathcal O_P, positive degree
AlgebraicGeometry.OModulePresheaf.exists_ker_d_inf_pow_smul_le_pow_smul_ker_sup_range_unit_of_isProper53 below · cited by 1 · depth 38 - Čech complex of coherent F is a structure-sheaf retract
AlgebraicGeometry.OModulePresheaf.exists_unit_cochain_linearMap_comp_eq_d_of_isProper1 below · cited by 2 · depth 38 - Uniform Artin–Rees for Čech coboundaries of 𝒪_P
AlgebraicGeometry.OModulePresheaf.exists_d_succ_eq_d_succ_of_forall_d_succ_mem_pow_unit_of_isProper53 below · cited by 1 · depth 39 - Čech vanishing transported to the chosen fibre product
AlgebraicGeometry.OModulePresheaf.subsingleton_HSucc_ofModules_pullback_fst_of_isPullback16 below · cited by 1 · depth 40
AlgebraicGeometry.Polarisation 242
- Very ampleness of mathcal L₀^{⊗ n} for n≥ 4
AlgebraicGeometry.Polarisation.closedImmersionBySections_of_iso_tensorPow_of_kernelTrivial_of_finrank_pos783 below · cited by 2 · depth 29 - Trivial kernel forces the stabiliser points to be {e}
AlgebraicGeometry.Polarisation.kernelPts_eq_singleton_one_of_kernelTrivial2 below · cited by 6 · depth 29 - Vanishing and h⁰(M^{⊗ n}) = n^g h⁰(M) for symmetric M
AlgebraicGeometry.Polarisation.subsingleton_HSucc_and_finrank_eq_pow_mul_of_iso_tensorPow_of_isSymmetric_of_finite_kernelPts_of_finrank_pos877 below · cited by 5 · depth 29 - Symmetry descends from a local cube root
AlgebraicGeometry.Polarisation.IsSymmetric.of_locIsoOnBase_tensor_three3 below · cited by 1 · depth 30 - Symmetry is stable under base change
AlgebraicGeometry.Polarisation.IsSymmetric.pullback_of_isPullback1 below · cited by 6 · depth 30 - Local isomorphy over the base is an equivalence relation
AlgebraicGeometry.Polarisation.LocIsoOnBase.equivalence0 below · cited by 97 · depth 30 - Base-local isomorphy of modules becomes global over a product of localisations
AlgebraicGeometry.Polarisation.LocIsoOnBase.exists_forall_nonempty_pullback_iso_of_isPullback_pi_localizationAway3 below · cited by 1 · depth 30 - Local isomorphy over the base is stable under pullback
AlgebraicGeometry.Polarisation.LocIsoOnBase.pullback_of_comp_eq0 below · cited by 67 · depth 30 - Projective embedding separates dual-number points for n≥ 4
AlgebraicGeometry.Polarisation.ProjPresentation.eq_of_comp_toProj_eq_of_isSectionBasis_of_iso_tensorPow_of_kernelTrivial_of_four_le776 below · cited by 1 · depth 30 - Base-point freeness of M^{⊗ n}, n≥ 2
AlgebraicGeometry.Polarisation.exists_isFrameOn_iSup_eq_top_of_iso_tensorPow_of_finrank_pos598 below · cited by 3 · depth 30 - Descent of a square-root decomposition to geometric fibres
AlgebraicGeometry.Polarisation.exists_isInvertible_nonempty_iso_tensor_pullback_negMor_geomFibre_of_exists_faithfullyFlat39 below · cited by 1 · depth 30 - Effectivity of mathcal L₀ from that of mathcal L₀^{⊗ 2}
AlgebraicGeometry.Polarisation.finrank_sections_pos_of_iso_tensor_self_of_kernelTrivial_of_isSymmetric876 below · cited by 1 · depth 30 - A[2] represents the stabiliser of M
AlgebraicGeometry.Polarisation.isClosedImmersion_and_isFinite_and_forall_exists_comp_eq_iff_isInStabilizer_of_kernelIsTwoTorsion711 below · cited by 3 · depth 30 - Base change preserves the Mumford-kernel-is-2-torsion condition
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_baseChange_of_isInvertible2 below · cited by 4 · depth 30 - Stabiliser points of a tensor half of L₁⊗[-1]^*L₁
AlgebraicGeometry.Polarisation.kernelPts_finite_of_nonempty_iso_tensor_pullback_negMor_of_kernelPts_finite1 below · cited by 1 · depth 30 - Trivial Mumford kernel from two-torsion kernel of a square
AlgebraicGeometry.Polarisation.kernelTrivial_of_nonempty_iso_tensor_self_of_kernelIsTwoTorsion732 below · cited by 2 · depth 30 - Over a field, local isomorphy on the base is isomorphy
AlgebraicGeometry.Polarisation.locIsoOnBase_iff_nonempty_iso_of_field0 below · cited by 22 · depth 30 - Mumford vanishing for powers of a symmetric effective bundle
AlgebraicGeometry.Polarisation.subsingleton_HSucc_of_iso_tensorPow_of_isSymmetric_of_finite_kernelPts_of_finrank_pos818 below · cited by 2 · depth 30 - Local isomorphism on the base passes to Mumford bundles
AlgebraicGeometry.Polarisation.LocIsoOnBase.mumfordBundle_of_isInvertible6 below · cited by 5 · depth 31 - Descent of local isomorphism of line bundles along an injective base change
AlgebraicGeometry.Polarisation.LocIsoOnBase.of_pullback_of_isPullback_of_injective223 below · cited by 3 · depth 31 - Local isomorphism over the base is stable under base change
AlgebraicGeometry.Polarisation.LocIsoOnBase.pullback_of_isPullback0 below · cited by 9 · depth 31 - Local isomorphy over the base is stable under tensor product
AlgebraicGeometry.Polarisation.LocIsoOnBase.tensor0 below · cited by 31 · depth 31 - Local cube structure transported along an isomorphism over the base
AlgebraicGeometry.Polarisation.LocIsoOnBase.tensor_cube_pullback_inv_of_iso2 below · cited by 2 · depth 31 - Schrödinger action of ωᵃθ_hη_χ on a frame
AlgebraicGeometry.Polarisation.SchrodingerFrame.act_ofScalar_mul_lift_mul_dualLift_sigma3 below · cited by 2 · depth 31 - Rescaling a Schrödinger frame by a base unit
AlgebraicGeometry.Polarisation.SchrodingerFrame.exists_sigma_eq_baseScalar_smul_of_isUnit0 below · cited by 1 · depth 31 - Additivity and base-scalar linearity of the theta action
AlgebraicGeometry.Polarisation.ThetaPt.act_add_and_act_baseScalar_smul0 below · cited by 8 · depth 31 - Theta points realising arbitrary additive characters of H(δ)
AlgebraicGeometry.Polarisation.ThetaPt.exists_forall_act_eq_baseScalar_addChar_smul_of_forall_addMonoidHom13 below · cited by 5 · depth 31 - Base change of theta points along a cartesian square
AlgebraicGeometry.Polarisation.ThetaPt.exists_monoidHom_pt_comp_eq_act_eq_of_isPullback1 below · cited by 13 · depth 31 - Fppf-local principal square roots over a DVR from the generic fibre
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_principalSqrt_of_exists_faithfullyFlat_principalSqrt_pullback_of_isDiscreteValuationRing1,369 below · cited by 1 · depth 31 - A local root becomes global over a field extension of k
AlgebraicGeometry.Polarisation.exists_field_isInvertible_nonempty_iso_tensor_pullback_negMor_of_exists_faithfullyFlat3 below · cited by 1 · depth 31 - Base-point freeness of M^{⊗ 2} on an abelian variety
AlgebraicGeometry.Polarisation.exists_isFrameOn_iSup_eq_top_of_iso_tensor_self_of_finrank_pos592 below · cited by 1 · depth 31 - Global frames for the cube of an effective line bundle
AlgebraicGeometry.Polarisation.exists_isFrameOn_iSup_eq_top_of_iso_tensor_tensor_of_finrank_pos595 below · cited by 1 · depth 31 - Descent of a square root of L to a geometric fibre
AlgebraicGeometry.Polarisation.exists_isInvertible_nonempty_iso_tensor_pullback_negMor_geomFibre_of_field35 below · cited by 1 · depth 31 - Surjectivity of φ_L onto Pic⁰
AlgebraicGeometry.Polarisation.exists_nonempty_tensor_iso_pullback_translate_of_inPicZero_of_kernelPts_finite674 below · cited by 5 · depth 31 - Sections of mathcal L₀^{⊗ n} separate k-points for n≥ 4
AlgebraicGeometry.Polarisation.exists_pullbackSection_eq_zero_and_ne_zero_of_ne_of_iso_tensorPow_of_kernelTrivial767 below · cited by 1 · depth 31 - Tangent vectors separated by sections of mathcal L₀^{⊗ n}, n≥4
AlgebraicGeometry.Polarisation.exists_pullbackSection_eq_zero_and_pullbackSection_ne_zero_of_iso_tensorPow_of_kernelTrivial761 below · cited by 1 · depth 31 - [-1]^*LotimesL^∨ lies in Pic⁰
AlgebraicGeometry.Polarisation.inPicZero_pullback_negMor_tensor_dual591 below · cited by 6 · depth 31 - Kernel trivial: unit section represents the stabiliser
AlgebraicGeometry.Polarisation.isClosedImmersion_one_and_forall_iff_isInStabilizer_of_kernelTrivial7 below · cited by 2 · depth 31 - Stabiliser points and triviality of the sliced Mumford bundle
AlgebraicGeometry.Polarisation.isInStabilizer_iff_locIsoOnBase_pullback_sliceAt_mumfordBundle_unit6 below · cited by 13 · depth 31 - K(L)=A[2] from a faithfully flat local square root
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_of_exists_faithfullyFlat_kernelTrivial_locIsoOnBase_tensor_pullback_negMor392 below · cited by 1 · depth 31 - Finite stabiliser and positive h⁰ for a finite-by-sections line bundle
AlgebraicGeometry.Polarisation.kernelPts_finite_and_geomFibreH0Finrank_pos_of_finiteBySections903 below · cited by 4 · depth 31 - Trivial kernel is stable under base change of the base ring
AlgebraicGeometry.Polarisation.kernelTrivial_pullback_fst_of_kernelTrivial4 below · cited by 3 · depth 31 - Theorem of the square for slices of the Mumford bundle
AlgebraicGeometry.Polarisation.locIsoOnBase_sliceAt_mumfordBundle_mul_tensor579 below · cited by 5 · depth 31 - Kernel of L^{⊗ 3} is 6-torsion when K(L)=A[2]
AlgebraicGeometry.Polarisation.memKernel_tensor_tensor_iff_nsmul_six_eq_one_of_kernelIsTwoTorsion369 below · cited by 1 · depth 31 - x ∈ K(L) iff Tₓ^*L ≅ L
AlgebraicGeometry.Polarisation.mem_kernelPts_iff_nonempty_pullback_translate_iso0 below · cited by 14 · depth 31 - Symmetrising translate: Tₓ^*L symmetric when φ_L(2x) matches [-1]^*LotimesL^∨
AlgebraicGeometry.Polarisation.nonempty_iso_and_nonempty_iso_pullback_translate_of_nsmulPt_two585 below · cited by 3 · depth 31 - Additivity of the Mumford bundle in the invertible module
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_tensor_iso_tensor4 below · cited by 6 · depth 31 - Vanishing and n^g-scaling of h⁰(M^{⊗ n})
AlgebraicGeometry.Polarisation.subsingleton_HSucc_and_finrank_eq_pow_mul_of_iso_tensorPow_of_finite_kernelPts_of_finrank_pos_u0957 below · cited by 3 · depth 31 - Vanishing of higher Čech cohomology for powers of a symmetric bundle
AlgebraicGeometry.Polarisation.subsingleton_HSucc_of_iso_tensorPow_of_isSymmetric_of_finite_kernelPts_of_exists_finrank_tensorPow_pos817 below · cited by 4 · depth 31 - Vanishing Čech cohomology of a non-trivial bundle in Pic⁰
AlgebraicGeometry.Polarisation.H0_eq_bot_and_subsingleton_HSucc_of_inPicZero_of_not_iso_unit177 below · cited by 3 · depth 32 - Pic⁰ is stable under duals
AlgebraicGeometry.Polarisation.InPicZero.dual2 below · cited by 2 · depth 32 - Pic⁰ is closed under tensor product
AlgebraicGeometry.Polarisation.InPicZero.tensor3 below · cited by 3 · depth 32 - K(L)=A[2] descends along faithfully flat base change
AlgebraicGeometry.Polarisation.KernelIsTwoTorsion.of_pullback_of_faithfullyFlat81 below · cited by 4 · depth 32 - Trivial Mumford kernel is stable under base change
AlgebraicGeometry.Polarisation.KernelTrivial.pullback_of_isPullback2 below · cited by 16 · depth 32 - Local isomorphy over the base descends along faithfully flat base change
AlgebraicGeometry.Polarisation.LocIsoOnBase.of_pullback_of_faithfullyFlat_of_isSeparated27 below · cited by 5 · depth 32 - Rosati compatibility descends along faithfully flat base change
AlgebraicGeometry.Polarisation.RosatiCompatible.of_pullback_of_faithfullyFlat84 below · cited by 2 · depth 32 - Gluing theta points along a complete orthogonal family of idempotents
AlgebraicGeometry.Polarisation.ThetaPt.exists_pt_eq_comp_act_eq_of_isIdempotentElem_of_sum_eq_one8 below · cited by 3 · depth 32 - Multiplicativity of the action of theta points on sections
AlgebraicGeometry.Polarisation.ThetaPt.mul_act1 below · cited by 14 · depth 32 - Scalar theta points act by the corresponding base scalar
AlgebraicGeometry.Polarisation.ThetaPt.ofScalar_act1 below · cited by 8 · depth 32 - The identity theta point acts trivially on sections
AlgebraicGeometry.Polarisation.ThetaPt.one_act1 below · cited by 8 · depth 32 - Faithfully flat square-root clause transports along an isomorphism
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_kernelTrivial_locIsoOnBase_pullback_inv_of_iso5 below · cited by 1 · depth 32 - Fppf-local principal square roots descend along faithfully flat base change
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_principalSqrt_of_exists_pullback_of_faithfullyFlat5 below · cited by 1 · depth 32 - Principal square roots descend to a finite extension of k
AlgebraicGeometry.Polarisation.exists_finiteDimensional_principalSqrt_of_exists_faithfullyFlat_principalSqrt_of_field1,188 below · cited by 1 · depth 32 - Faithfully flat descent of invertible modules, up to base-local isomorphism
AlgebraicGeometry.Polarisation.exists_isInvertible_locIsoOnBase_pullback_of_locIsoOnBase_of_faithfullyFlat_of_section55 below · cited by 2 · depth 32 - Common local faithfully flat cover carrying both square roots
AlgebraicGeometry.Polarisation.exists_isLocalRing_faithfullyFlat_kernelTrivial_locIsoOnBase_pair_of_isLocalRing4 below · cited by 1 · depth 32 - Descending a principal square root from the generic fibre
AlgebraicGeometry.Polarisation.exists_kernelTrivial_locIsoOnBase_of_principalSqrt_generic_of_isDiscreteValuationRing1,115 below · cited by 1 · depth 32 - Finiteness by sections of M^{⊗ 3} on an abelian variety
AlgebraicGeometry.Polarisation.finiteBySections_of_iso_tensorPow_three_of_finite_kernelPts_of_finrank_pos625 below · cited by 2 · depth 32 - Finiteness of K(N) for N ≅ M^{⊗ n}
AlgebraicGeometry.Polarisation.finite_kernelPts_of_iso_tensorPow_of_finite_kernelPts731 below · cited by 2 · depth 32 - Tₓ^*L ⊗ L^∨ lies in Pic⁰
AlgebraicGeometry.Polarisation.inPicZero_pullback_translate_tensor_dual585 below · cited by 6 · depth 32 - Stabiliser of an invertible module via the Mumford bundle slice
AlgebraicGeometry.Polarisation.isInStabilizer_iff_locIsoOnBase_pullback_sliceAt_mumfordBundle_unit_of_commRing6 below · cited by 4 · depth 32 - Points not separated by |L^{⊗ n}| stabilise L^{⊗ j}
AlgebraicGeometry.Polarisation.isInStabilizer_tensorPow_mul_inv_of_forall_pullbackSection_eq_zero_imp758 below · cited by 1 · depth 32 - Tangent vector killing all vanishing sections stabilises mathcal L₀^{⊗ j}
AlgebraicGeometry.Polarisation.isInStabilizer_tensorPow_mul_inv_of_forall_pullbackSection_eq_zero_imp_dualNumber753 below · cited by 1 · depth 32 - Invariance of K(L)=A[2] under base-local isomorphism
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_of_locIsoOnBase_of_kernelIsTwoTorsion7 below · cited by 1 · depth 32 - Transport of the clause K(L)=A[2] along an isomorphism
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_pullback_inv_of_iso_of_isInvertible4 below · cited by 1 · depth 32 - Mumford kernel of a symmetrised bundle is the 2-torsion
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_tensor_pullback_negMor_of_kernelTrivial_of_commRing372 below · cited by 2 · depth 32 - Local isomorphy on the base over a local ring gives isomorphy
AlgebraicGeometry.Polarisation.locIsoOnBase_iff_nonempty_iso_of_isLocalRing0 below · cited by 13 · depth 32 - Kernel membership is invariant under base change
AlgebraicGeometry.Polarisation.memKernel_iff_memKernel_comp_of_isPullback4 below · cited by 3 · depth 32 - Kernel of L^{⊗ 3} is the 3-preimage of K(L)
AlgebraicGeometry.Polarisation.memKernel_tensor_tensor_iff_memKernel_nsmul_three368 below · cited by 1 · depth 32 - Transport of a square root mathcal L₀ along a base-change transition map
AlgebraicGeometry.Polarisation.nonempty_iso_tensor_pullback_negMor_pullback_of_comp_eq1 below · cited by 1 · depth 32 - Left slice of the Mumford bundle at a k-point
AlgebraicGeometry.Polarisation.nonempty_pullback_leftSlice_mumfordBundle_iso_pullback_translate_tensor_dual1 below · cited by 1 · depth 32 - [-1]^*M≅ M^∨ for M in Pic⁰
AlgebraicGeometry.Polarisation.nonempty_pullback_negMor_iso_dual_of_inPicZero114 below · cited by 3 · depth 32 - [n]^*L ≅ L^{⊗ n^2} for symmetric invertible sheaves
AlgebraicGeometry.Polarisation.nonempty_pullback_schemeNsmul_iso_tensorPow_sq_of_isSymmetric_monoidalV2585 below · cited by 1 · depth 32 - Slice of the Mumford bundle at a k-point of A
AlgebraicGeometry.Polarisation.nonempty_pullback_sliceAt_mumfordBundle_iso_pullback_fst_translate_tensor_dual1 below · cited by 4 · depth 32 - Transport of M ⊗ [-1]^*M along a base-change transition map
AlgebraicGeometry.Polarisation.nonempty_pullback_tensor_pullback_negMor_iso_of_comp_eq1 below · cited by 5 · depth 32 - Theorem of the square for a commutative relative group law
AlgebraicGeometry.Polarisation.nonempty_pullback_translate_mul_tensor_iso584 below · cited by 6 · depth 32 - Čech vanishing for a very ample sheaf on an abelian variety
AlgebraicGeometry.Polarisation.subsingleton_HSucc_of_closedImmersionBySections_of_isAlgClosed1,035 below · cited by 2 · depth 32 - Čech vanishing for powers of an effective nondegenerate line bundle
AlgebraicGeometry.Polarisation.subsingleton_HSucc_of_iso_tensorPow_of_finite_kernelPts_of_finrank_pos904 below · cited by 3 · depth 32 - Symmetry of a module under inversion is local on the base
AlgebraicGeometry.Polarisation.IsSymmetric.of_forall_away1 below · cited by 1 · depth 33 - Locality on the base of the two-torsion kernel condition
AlgebraicGeometry.Polarisation.KernelIsTwoTorsion.of_forall_away_of_isInvertible4 below · cited by 2 · depth 33 - Rosati compatibility is local on the base, invertible case
AlgebraicGeometry.Polarisation.RosatiCompatible.of_forall_away_of_isInvertible4 below · cited by 1 · depth 33 - Vanishing Čech ranks for L stabilised by a subscheme
AlgebraicGeometry.Polarisation.cechFinrank_eq_zero_of_forall_comp_mem_kernelPts_of_not_nonempty_pullback_iso_unit286 below · cited by 1 · depth 33 - Cocycle identity for a rigidified descent isomorphism
AlgebraicGeometry.Polarisation.cocycle_of_rigidifiedIso16 below · cited by 1 · depth 33 - Euler characteristic of a twisted Mumford bundle
AlgebraicGeometry.Polarisation.eulerChar_mumfordBundle_tensor_pullback_snd_eq_neg_one_pow_mul_finrank_of_forall_iff_isInStabilizer782 below · cited by 1 · depth 33 - Symmetric Pic⁰-twist of a non-degenerate invertible sheaf
AlgebraicGeometry.Polarisation.exists_inPicZero_isSymmetric_tensor_of_kernelPts_finite799 below · cited by 2 · depth 33 - Torsion element of Pic⁰(A) non-trivial on a subscheme Y
AlgebraicGeometry.Polarisation.exists_inPicZero_tensorPow_iso_unit_not_nonempty_pullback_iso_unit_of_isClosedImmersion833 below · cited by 1 · depth 33 - Shear automorphism trivialises the twisted Mumford bundle
AlgebraicGeometry.Polarisation.exists_iso_mumfordBundle_tensor_pullback_snd_iso_pullback2 below · cited by 1 · depth 33 - Finite stabiliser of the zero locus of a section
AlgebraicGeometry.Polarisation.finite_setOf_forall_pullbackSection_eq_zero_iff_of_finite_kernelPts37 below · cited by 1 · depth 33 - Translation invariance of dim_kΓ under a relative group law
AlgebraicGeometry.Polarisation.finrank_sections_eq_of_iso_pullback_translate1 below · cited by 1 · depth 33 - Mumford bundle trivial iff M lies in Pic⁰
AlgebraicGeometry.Polarisation.inPicZero_iff_nonempty_mumfordBundle_iso_unit113 below · cited by 5 · depth 33 - The structure sheaf lies in Pic⁰
AlgebraicGeometry.Polarisation.inPicZero_tensorUnit0 below · cited by 3 · depth 33 - Shear map (x,y)↦(x,xy) is an automorphism of A×_S A
AlgebraicGeometry.Polarisation.isIso_lift_fst_addMor0 below · cited by 2 · depth 33 - Symmetry of L⊗[-1]^*L
AlgebraicGeometry.Polarisation.isSymmetric_tensor_pullback_negMor0 below · cited by 3 · depth 33 - Invariance of the two kernel conditions under isomorphism of modules
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_iff_and_kernelTrivial_iff_of_iso0 below · cited by 4 · depth 33 - A Pic⁰-twist does not change kernelPts
AlgebraicGeometry.Polarisation.kernelPts_tensor_eq_of_inPicZero7 below · cited by 1 · depth 33 - Kernel-triviality, symmetry and square-root clauses under base change
AlgebraicGeometry.Polarisation.kernelTrivial_isSymmetric_locIsoOnBase_pullback_baseChangeSnd_of_comp_eq9 below · cited by 1 · depth 33 - Kernel triviality depends only on the isomorphism class
AlgebraicGeometry.Polarisation.kernelTrivial_of_iso0 below · cited by 4 · depth 33 - Trivial Mumford kernel transports along an isomorphism of group laws
AlgebraicGeometry.Polarisation.kernelTrivial_pullback_inv_of_iso4 below · cited by 2 · depth 33 - Translation at the unit section recovers the point
AlgebraicGeometry.Polarisation.lift_one_comp_translate_comp_fst0 below · cited by 1 · depth 33 - Inversion pull-back of a Mumford slice is its dual, locally
AlgebraicGeometry.Polarisation.locIsoOnBase_pullback_negMor_prod_sliceAt_mumfordBundle_dual_of_commRing366 below · cited by 2 · depth 33 - Slice of the Mumford bundle at the unit is locally trivial
AlgebraicGeometry.Polarisation.locIsoOnBase_pullback_sliceAt_mumfordBundle_one_unit1 below · cited by 4 · depth 33 - Unit slice of the Mumford bundle is locally trivial over the base
AlgebraicGeometry.Polarisation.locIsoOnBase_pullback_sliceAt_mumfordBundle_one_unit_of_commRing1 below · cited by 6 · depth 33 - Mumford bundle at x⁻¹ is locally the dual
AlgebraicGeometry.Polarisation.locIsoOnBase_sliceAt_mumfordBundle_inv_dual_of_commRing368 below · cited by 2 · depth 33 - Theorem of the square for Mumford bundle slices, locally on the base
AlgebraicGeometry.Polarisation.locIsoOnBase_sliceAt_mumfordBundle_mul_tensor_of_commRing366 below · cited by 4 · depth 33 - Tensoring with a line bundle pulled back from the base is locally trivial
AlgebraicGeometry.Polarisation.locIsoOnBase_tensor_pullback_of_isInvertible0 below · cited by 1 · depth 33 - k-points of K(L) via triviality of the sliced Mumford bundle
AlgebraicGeometry.Polarisation.mem_kernelPts_iff_nonempty_pullback_sliceAt_mumfordBundle_iso_unit8 below · cited by 6 · depth 33 - Projective presentation is constant on a translated tangent vector
AlgebraicGeometry.Polarisation.mul_comp_toProj_eq_const_dualNumber_of_forall_pullbackSection_eq_zero_imp_of_ne_zero601 below · cited by 1 · depth 33 - Agreement of projective presentation at translated k-points
AlgebraicGeometry.Polarisation.mul_comp_toProj_eq_mul_comp_toProj_of_forall_pullbackSection_eq_zero_imp_of_ne_zero602 below · cited by 1 · depth 33 - Equal multiplications force equal unit, negation, kernel and symmetry
AlgebraicGeometry.Polarisation.negMor_eq_and_kernelTrivial_iff_and_isSymmetric_iff_of_forall_mul_eq0 below · cited by 2 · depth 33 - Unique rigidified isomorphism of locally base-isomorphic invertible modules
AlgebraicGeometry.Polarisation.nonempty_and_subsingleton_rigidifiedIso_of_locIsoOnBase_of_forall_bijective12 below · cited by 3 · depth 33 - Trivial Mumford bundle gives μ^*M≅ p₁^*M⊗ p₂^*M
AlgebraicGeometry.Polarisation.nonempty_pullback_addMor_iso_tensor_of_mumfordBundle_iso_unit1 below · cited by 1 · depth 33 - Slicing the Mumford bundle of [-1]^*L
AlgebraicGeometry.Polarisation.nonempty_pullback_sliceAt_mumfordBundle_pullback_negMor_iso_of_commRing2 below · cited by 2 · depth 33 - Slices of the Mumford bundle are multiplicative
AlgebraicGeometry.Polarisation.nonempty_pullback_sliceAt_mumfordBundle_tensor_iso4 below · cited by 3 · depth 33 - Slices of the Mumford bundle are multiplicative in L
AlgebraicGeometry.Polarisation.nonempty_pullback_sliceAt_mumfordBundle_tensor_iso_of_commRing4 below · cited by 3 · depth 33 - Translation invariance of vanishing of higher Čech groups
AlgebraicGeometry.Polarisation.subsingleton_HSucc_of_forall_subsingleton_HSucc_pullback_translate71 below · cited by 1 · depth 33 - Nontrivial line bundle in Pic⁰ has no sections
AlgebraicGeometry.Polarisation.subsingleton_sections_of_inPicZero_of_not_iso_unit130 below · cited by 3 · depth 33 - Base change of K(L)=A[2] along a cartesian square
AlgebraicGeometry.Polarisation.KernelIsTwoTorsion.pullback_of_isPullback_of_isInvertible4 below · cited by 5 · depth 34 - Rosati compatibility is stable under cartesian base change
AlgebraicGeometry.Polarisation.RosatiCompatible.pullback_of_isPullback5 below · cited by 6 · depth 34 - Transport of theta points along a ring map of test rings
AlgebraicGeometry.Polarisation.ThetaPt.exists_comparison_monoidHom_of_ringHom2 below · cited by 1 · depth 34 - Euler characteristic of a twisted Mumford bundle as a sum of stalk lengths
AlgebraicGeometry.Polarisation.eulerChar_mumfordBundle_tensor_pullback_snd_eq_sum_alternating_length_of_forall_mem677 below · cited by 1 · depth 34 - Kernel of 2n(1+ι(b^⋆)ι(b)) equals the stabiliser of L⊗ι(b)^*L
AlgebraicGeometry.Polarisation.exists_comp_endKerIncl_eq_iff_isInStabilizer_tensor_pullback_of_rosatiCompatible_of_smooth590 below · cited by 1 · depth 34 - Free local model for the Mumford slice, with see-saw
AlgebraicGeometry.Polarisation.exists_free_complex_cech_sliceAt_stalk_and_seesaw168 below · cited by 1 · depth 34 - Translated product section: vanishing at k- and k[ε]-points
AlgebraicGeometry.Polarisation.exists_hom_forall_pullbackSection_eq_zero_iff_and_forall_dualNumber_of_finComb_eq_one594 below · cited by 2 · depth 34 - Translated sections multiply when sum cᵢ pᵢ = 0
AlgebraicGeometry.Polarisation.exists_hom_forall_pullbackSection_eq_zero_iff_exists_pullbackSection_translate_eq_zero_of_finComb_eq_one595 below · cited by 1 · depth 34 - Translations are compatible with base change along Specφ
AlgebraicGeometry.Polarisation.exists_isIso_comp_fst_comp_fst_eq_and_translate_comp_eq_of_isPullback0 below · cited by 1 · depth 34 - Shear automorphism (a,y)↦(a+j(y),y) of A×_k Y
AlgebraicGeometry.Polarisation.exists_iso_hom_fst_eq_sliceAt_addMor0 below · cited by 1 · depth 34 - A power of mathfrak m_y kills the sliced Čech cohomology
AlgebraicGeometry.Polarisation.exists_pow_maximalIdeal_smul_cech_sliceAt_stalk_eq_bot669 below · cited by 1 · depth 34 - M ∈ Pic⁰ iff all point slices of Λ(M) are trivial
AlgebraicGeometry.Polarisation.inPicZero_iff_forall_nonempty_pullback_sliceAt_mumfordBundle_iso_unit2 below · cited by 2 · depth 34 - Pullback along a homomorphism into K(L) lies in Pic⁰
AlgebraicGeometry.Polarisation.inPicZero_pullback_of_forall_comp_mem_kernelPts1 below · cited by 1 · depth 34 - Finiteness of the k-points of the stabiliser
AlgebraicGeometry.Polarisation.kernelPts_finite_of_kernelTrivial7 below · cited by 3 · depth 34 - Triviality of K(L) is translation-invariant
AlgebraicGeometry.Polarisation.kernelTrivial_pullback_translate5 below · cited by 1 · depth 34 - Invertible modules locally isomorphic over the base and trivialised along a section
AlgebraicGeometry.Polarisation.nonempty_iso_of_locIsoOnBase_of_pullback_iso_unit_of_forall_bijective8 below · cited by 2 · depth 34 - Mumford bundle commutes with base change
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_pullback_iso_pullback_mumfordBundle_of_isPullback2 below · cited by 8 · depth 34 - Multiplicativity of the Mumford bundle in the line bundle
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_tensor_iso_tensor_mumfordBundle4 below · cited by 3 · depth 34 - Translation quotient unchanged by twisting by Pic⁰
AlgebraicGeometry.Polarisation.nonempty_phi_tensor_iso_phi_of_inPicZero4 below · cited by 1 · depth 34 - Theorem of the cube over an affine base, pull-back form
AlgebraicGeometry.Polarisation.nonempty_pullback_mul_mul_tensor_iso_tensor_pullback_one_of_commRing364 below · cited by 2 · depth 34 - See-saw splitting of μ^*L on A ×_k Y
AlgebraicGeometry.Polarisation.nonempty_pullback_sliceAt_addMor_iso_tensor_of_forall_comp_mem_kernelPts217 below · cited by 1 · depth 34 - Rescaling an isomorphism to respect trivialisations along a section
AlgebraicGeometry.Polarisation.nonempty_rigidifiedIso_of_iso1 below · cited by 1 · depth 34 - Square-root datum descends from a localisation chart to the base
AlgebraicGeometry.Polarisation.principalSqrt_baseChange_of_principalSqrt_chart4 below · cited by 1 · depth 34 - Uniqueness of rigidified isomorphisms of invertible modules
AlgebraicGeometry.Polarisation.subsingleton_rigidifiedIso1 below · cited by 1 · depth 34 - Rosati compatibility is preserved by pull-back along inversion
AlgebraicGeometry.Polarisation.RosatiCompatible.pullback_negMor4 below · cited by 2 · depth 35 - Rosati compatibility passes to tensor products of line bundles
AlgebraicGeometry.Polarisation.RosatiCompatible.tensor6 below · cited by 2 · depth 35 - Theta points over a field as Mumford's theta group
AlgebraicGeometry.Polarisation.ThetaPt.exists_bijective_thetaGroup_antiHom_of_compatible3 below · cited by 1 · depth 35 - Kernel triviality upstairs forces x=e on S'-test points
AlgebraicGeometry.Polarisation.eq_one_of_locIsoOnBase_sliceAt_mumfordBundle_of_kernelTrivial_pullback_of_isPullback2 below · cited by 1 · depth 35 - Kernel membership: t-points of A versus points of A_K
AlgebraicGeometry.Polarisation.exists_comp_fst_eq_and_memKernel_pullback_iff_memKernel_comp_fst5 below · cited by 1 · depth 35 - Faithfully flat extension splitting mathcal L₀⊗[-1]^*mathcal L₀
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_kernelTrivial_locIsoOnBase_tensor_pullback_negMor5 below · cited by 1 · depth 35 - Principal square root of a symmetrised bundle, trivial cover
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_kernelTrivial_locIsoOnBase_tensor_pullback_negMor_of_commRing5 below · cited by 1 · depth 35 - Principal square roots over a finite product base
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_principalSqrt_of_forall_pullback_of_isPullback_pi15 below · cited by 1 · depth 35 - Fppf-local square root for a kernel-trivial invertible sheaf
AlgebraicGeometry.Polarisation.exists_faithfullyFlat_principalSqrt_of_kernelTrivial_of_locIsoOnBase4 below · cited by 1 · depth 35 - Free local model of the Mumford-slice Čech complex at a stalk
AlgebraicGeometry.Polarisation.exists_free_complex_quasiIso_cech_sliceAt_stalk94 below · cited by 1 · depth 35 - Closedness of the locus of trivial slices of an invertible module
AlgebraicGeometry.Polarisation.exists_isClosed_mem_iff_nonempty_pullback_sliceAt_iso_unit109 below · cited by 1 · depth 35 - Čech cohomology on an affine strip as a sum of stalk lengths
AlgebraicGeometry.Polarisation.finite_and_finrank_cech_restrict_strip_eq_sum_toNat_length_cech_sliceAt_stalk669 below · cited by 1 · depth 35 - At stabiliser points the local Čech model has h⁰=1
AlgebraicGeometry.Polarisation.finrank_ker_baseChange_residue_eq_one_of_quasiIso_cech_sliceAt_stalk_of_forall64 below · cited by 1 · depth 35 - Cocycle lifting over R/J' versus section lifting on the closed fibre
AlgebraicGeometry.Polarisation.forall_exists_baseChange_iff_forall_exists_pullbackSection_of_quasiIso_cech_sliceAt_stalk_of_forall2 below · cited by 1 · depth 35 - See-saw criterion: lifting sections detects the stabiliser
AlgebraicGeometry.Polarisation.forall_exists_pullbackSection_eq_iff_exists_comp_eq_of_mem_range68 below · cited by 1 · depth 35 - Effectivity descends from M^{⊗ n}⊗ P to M
AlgebraicGeometry.Polarisation.geomFibreH0Finrank_pos_of_iso_tensorPow_tensor_of_inPicZero_of_kernelPts_finite962 below · cited by 3 · depth 35 - Fibrewise positivity of h⁰ spreads along an injective base change
AlgebraicGeometry.Polarisation.geomFibreH0Finrank_pos_of_pos_pullback_of_kernelIsTwoTorsion_of_isLocalHom_of_injective79 below · cited by 1 · depth 35 - Positivity of h⁰ passes to mathcal L₀⊗[-1]^*mathcal L₀
AlgebraicGeometry.Polarisation.geomFibreH0Finrank_tensor_pullback_negMor_pos80 below · cited by 2 · depth 35 - Fibrewise positivity of mathcal L₀ ⊗ [-1]^*mathcal L₀ over a general base
AlgebraicGeometry.Polarisation.geomFibreH0Finrank_tensor_pullback_negMor_pos_of_commRing84 below · cited by 1 · depth 35 - Symmetry and local square-root clauses under base change
AlgebraicGeometry.Polarisation.isSymmetric_locIsoOnBase_pullback_baseChangeSnd_of_comp_eq6 below · cited by 1 · depth 35 - Two-torsion kernel condition over a finite product base
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_of_forall_kernelIsTwoTorsion_pullback_of_isPullback_pi2 below · cited by 1 · depth 35 - Mumford kernel of g^*L equals the 2-torsion
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_of_isPullback_of_kernelTrivial_of_locIsoOnBase_of_isAlgClosed678 below · cited by 1 · depth 35 - Symmetrisation of a sheaf with trivial kernel has kernel A[2]
AlgebraicGeometry.Polarisation.kernelIsTwoTorsion_tensor_pullback_negMor_of_kernelTrivial593 below · cited by 2 · depth 35 - Trivial Mumford kernel descends from a finite product decomposition
AlgebraicGeometry.Polarisation.kernelTrivial_of_forall_kernelTrivial_pullback_of_isPullback_pi_univ2 below · cited by 1 · depth 35 - Stabiliser represented by the base gives trivial kernel
AlgebraicGeometry.Polarisation.kernelTrivial_of_isIso_of_forall_iff_isInStabilizer6 below · cited by 1 · depth 35 - Triviality of K(L) descends along a field extension
AlgebraicGeometry.Polarisation.kernelTrivial_of_kernelTrivial_pullback_of_isPullback_of_field2 below · cited by 2 · depth 35 - Locality of module isomorphy over a finite product base
AlgebraicGeometry.Polarisation.locIsoOnBase_of_forall_locIsoOnBase_pullback_of_isPullback_pi0 below · cited by 3 · depth 35 - LocIsoOnBase descends from the clopen pieces of a product base
AlgebraicGeometry.Polarisation.locIsoOnBase_of_forall_locIsoOnBase_pullback_of_isPullback_pi_univ0 below · cited by 1 · depth 35 - Membership in K(L) iff translation-invariance at k-points
AlgebraicGeometry.Polarisation.memKernel_iff_nonempty_pullback_translation_iso_of_isAlgClosed8 below · cited by 1 · depth 35 - Kernel of mathcal L₀ᵃ⊗([-1]^*mathcal L₀)ᵇ is the (a+b)-torsion
AlgebraicGeometry.Polarisation.memKernel_iff_nsmul_eq_one_of_kernelTrivial_of_iso_tpow_tensor_tpow374 below · cited by 1 · depth 35 - Kernel of mathcal L₀^{⊗ a}⊗([-1]^*mathcal L₀)^{⊗ b} and (a+b)y
AlgebraicGeometry.Polarisation.memKernel_tpow_tensor_tpow_pullback_negMor_iff_memKernel_nsmul_add372 below · cited by 2 · depth 35 - Slice Čech cohomology versus base change of an affine chart
AlgebraicGeometry.Polarisation.nonempty_cechEquiv_sliceAt_comap_baseChange_of_isAffineOpen16 below · cited by 3 · depth 35 - Mumford bundle is invariant under isomorphism of L
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_iso_of_iso0 below · cited by 2 · depth 35 - Mumford bundle commutes with pull-back along an endomorphism
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_pullback_iso_pullback_map_mumfordBundle2 below · cited by 2 · depth 35 - Biadditivity of a birigidified invertible sheaf on A× A
AlgebraicGeometry.Polarisation.nonempty_pullback_addMorProd_iso_tensor_of_birigidified574 below · cited by 5 · depth 35 - Symmetry of the Mumford bundle under the flip
AlgebraicGeometry.Polarisation.nonempty_pullback_pullbackSymmetry_mumfordBundle_iso0 below · cited by 4 · depth 35 - Mumford bundle trivial along the zero slice, and after swapping
AlgebraicGeometry.Polarisation.nonempty_sliceAt_zero_mumfordBundle_iso_unit_and_swap9 below · cited by 7 · depth 35 - Rosati compatibility descends from the factors of a finite product base
AlgebraicGeometry.Polarisation.rosatiCompatible_of_forall_rosatiCompatible_pullback_of_isPullback_pi3 below · cited by 1 · depth 35 - Rosati compatibility descends from mathcal L₁⊗[-1]^*mathcal L₁ to mathcal L₁
AlgebraicGeometry.Polarisation.rosatiCompatible_of_rosatiCompatible_tensor_pullback_negMor_of_topologicalKrullDim_eq742 below · cited by 2 · depth 35 - Off-stabiliser vanishing of localised Čech cohomology on an affine chart
AlgebraicGeometry.Polarisation.subsingleton_localizedModule_cech_comap_of_not_mem_range_of_isAffineOpen666 below · cited by 2 · depth 35 - Local-on-the-base isomorphism passes to duals of invertible modules
AlgebraicGeometry.Polarisation.LocIsoOnBase.dual_of_isInvertible6 below · cited by 2 · depth 36 - Acyclicity of Mumford bundle slices off the stabiliser
AlgebraicGeometry.Polarisation.cechH0_eq_bot_and_subsingleton_HSucc_sliceAt_mumfordBundle_of_not_exists_comp_eq655 below · cited by 1 · depth 36 - Closed locus of k-points whose slice of Λ has sections
AlgebraicGeometry.Polarisation.exists_isClosed_mem_iff_exists_unit_hom_pullback_sliceAt_ne_zero81 below · cited by 1 · depth 36 - Čech h⁰=1 for the Mumford slice at a stabiliser stalk
AlgebraicGeometry.Polarisation.finrank_H0_baseChange_residue_sliceAt_stalk_eq_one63 below · cited by 1 · depth 36 - Twisting by Pic⁰ preserves h⁰ of a nondegenerate line bundle
AlgebraicGeometry.Polarisation.geomFibreH0Finrank_tensor_eq_of_inPicZero_of_kernelPts_finite681 below · cited by 1 · depth 36 - Twisting by a line bundle from the base preserves symmetric square roots
AlgebraicGeometry.Polarisation.isSymmetric_locIsoOnBase_tensor_pullback_of_isInvertible5 below · cited by 2 · depth 36 - Kernel triviality descends from the factors of a product base
AlgebraicGeometry.Polarisation.kernelTrivial_of_forall_kernelTrivial_pullback_of_isPullback_pi2 below · cited by 1 · depth 36 - Local symmetry and trivial square ascend along base change
AlgebraicGeometry.Polarisation.locIsoOnBase_negMor_and_tensor_unit_pullback_baseChangeSnd_of_comp_eq2 below · cited by 1 · depth 36 - Inversion dualises the Mumford slice, locally on the base
AlgebraicGeometry.Polarisation.locIsoOnBase_pullback_negMor_prod_sliceAt_mumfordBundle_dual585 below · cited by 1 · depth 36 - Slice of the Mumford bundle at x⁻¹ is the dual slice
AlgebraicGeometry.Polarisation.locIsoOnBase_sliceAt_mumfordBundle_inv_dual583 below · cited by 1 · depth 36 - Inversion respects a commutative relative group law on points
AlgebraicGeometry.Polarisation.mapPt_negMor_mul0 below · cited by 1 · depth 36 - Invertible module with non-zero section and co-section on a slice
AlgebraicGeometry.Polarisation.nonempty_iso_unit_of_ne_zero_section_dual_slice26 below · cited by 1 · depth 36 - A birigidified line bundle on A× A with trivial square is trivial
AlgebraicGeometry.Polarisation.nonempty_iso_unit_of_tensor_self_iso_unit_of_birigidified714 below · cited by 2 · depth 36 - Slices of Λ([-1]^*L) under inversion
AlgebraicGeometry.Polarisation.nonempty_pullback_sliceAt_mumfordBundle_pullback_negMor_iso2 below · cited by 1 · depth 36 - Triviality of the Mumford bundle along the unit axes
AlgebraicGeometry.Polarisation.nonempty_sliceAt_one_mumfordBundle_iso_unit_and_swap_of_pullback_one_iso_unit2 below · cited by 1 · depth 36 - Descent of Rosati compatibility along a field extension
AlgebraicGeometry.Polarisation.rosatiCompatible_of_rosatiCompatible_pullback_of_isPullback_of_field85 below · cited by 1 · depth 36 - Multiplication by m commutes with inversion
AlgebraicGeometry.Polarisation.schemeNsmul_comp_negMor1 below · cited by 1 · depth 36 - Symmetric square-trivial rigidified bundles as 2-torsion characters
AlgebraicGeometry.Polarisation.exists_torsionCharacter_two_bijOn_symmetric_tensor_self_rigidifiedLineBundle988 below · cited by 1 · depth 37 - Two symmetric square roots differ by an admissible bundle
AlgebraicGeometry.Polarisation.isSymmetric_locIsoOnBase_tensor_dual_of_isSymmetric_of_locIsoOnBase11 below · cited by 1 · depth 37 - Twisting a symmetric square root by a symmetric 2-torsion bundle
AlgebraicGeometry.Polarisation.isSymmetric_locIsoOnBase_tensor_of_isSymmetric_of_locIsoOnBase_tensor_unit2 below · cited by 1 · depth 37 - Scheme-theoretic triviality of K(L) from its k-points
AlgebraicGeometry.Polarisation.kernelTrivial_of_forall_mem_kernelPts_eq_one_of_charZero653 below · cited by 1 · depth 37 - Fibre sections over a closed point versus slice sections of Λ
AlgebraicGeometry.Polarisation.le_finrank_sections_residueField_fibre_iff_exists_unit_hom_pullback_sliceAt_ne_zero63 below · cited by 1 · depth 37 - Local isomorphism on a local base gives a global isomorphism
AlgebraicGeometry.Polarisation.nonempty_iso_of_locIsoOnBase_of_isLocalRing0 below · cited by 1 · depth 37 - Bi-rigidified bundle with trivial n-th tensor power is trivial
AlgebraicGeometry.Polarisation.nonempty_iso_unit_of_tensorPow_iso_unit_of_birigidified713 below · cited by 3 · depth 37 - Rosati compatibility descends to n-th roots modulo Pic⁰
AlgebraicGeometry.Polarisation.rosatiCompatible_of_iso_tensorPow_tensor_of_inPicZero_of_rosatiCompatible727 below · cited by 1 · depth 37 - Local triviality on the base descends to a finite stage
AlgebraicGeometry.Polarisation.exists_locIsoOnBase_pullback_unit_stage_of_locIsoOnBase_pullback_unit_of_isDirectLimit16 below · cited by 1 · depth 38 - Symmetric bi-rigidified line bundles on A× A are Mumford bundles
AlgebraicGeometry.Polarisation.exists_nonempty_iso_mumfordBundle_of_symmetric_of_birigidified_of_two_ne_zero791 below · cited by 1 · depth 38 - 2-torsion characters from [2]^*-trivial rigidified line bundles
AlgebraicGeometry.Polarisation.exists_torsionCharacter_two_bijOn_pullback_schemeNsmul_two_trivial_rigidifiedLineBundle108 below · cited by 2 · depth 38 - Cube face triviality transfers along a cartesian base change
AlgebraicGeometry.Polarisation.locIsoOnBase_faces_pullback_cube_iff_of_isPullback2 below · cited by 1 · depth 38 - Symmetric and locally square-trivial iff [2]^*N is trivial
AlgebraicGeometry.Polarisation.locIsoOnBase_negMor_and_tensor_self_iff_nonempty_pullback_schemeNsmul_two_iso_unit987 below · cited by 1 · depth 38 - Pullback of the Mumford bundle along 1 × [ℓ]
AlgebraicGeometry.Polarisation.nonempty_pullback_oneProdNsmul_mumfordBundle_iso_mumfordBundle_tensorPow585 below · cited by 2 · depth 38 - Isomorphic Mumford bundles give translation-invariant difference bundle
AlgebraicGeometry.Polarisation.nonempty_pullback_translation_tensor_dual_iso_of_mumfordBundle_iso_of_isAlgClosed8 below · cited by 1 · depth 38 - Twisting by a Pic⁰ class preserves mathcal L₀⊗[-1]^*mathcal L₀
AlgebraicGeometry.Polarisation.nonempty_tensor_pullback_negMor_iso_of_inPicZero115 below · cited by 1 · depth 38 - Two effective bundles with K(mathcal L₀) trivial are never in Pic⁰
AlgebraicGeometry.Polarisation.not_forall_nonempty_pullback_translate_tensor_iso_of_kernelTrivial_of_geomFibreH0Finrank_pos738 below · cited by 1 · depth 38 - Halving a symmetric bi-rigidified square root of a Mumford bundle
AlgebraicGeometry.Polarisation.exists_nonempty_iso_mumfordBundle_of_mumfordBundle_iso_tensor_self_of_two_ne_zero789 below · cited by 1 · depth 39 - Realising every 2-torsion character by a rigidified line bundle
AlgebraicGeometry.Polarisation.exists_rigidifiedLineBundle_pullback_schemeNsmul_two_trivial_hasValue_translate91 below · cited by 1 · depth 39 - Existence of the 2-torsion descent character Φ
AlgebraicGeometry.Polarisation.exists_torsionCharacter_two_hasValue_translate_of_pullback_schemeNsmul_two_trivial68 below · cited by 1 · depth 39 - Rigidified bundles killed by [2]^*: symmetry and 2-torsion
AlgebraicGeometry.Polarisation.locIsoOnBase_negMor_and_locIsoOnBase_tensor_self_of_nonempty_pullback_schemeNsmul_two_iso_unit980 below · cited by 1 · depth 39 - Rigidified line bundles with the same descent character agree
AlgebraicGeometry.Polarisation.nonempty_iso_of_hasValue_translate_eq_of_pullback_schemeNsmul_two_trivial11 below · cited by 1 · depth 39 - Mumford bundle of a diagonal-restricted symmetric birigidified bundle
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_pullback_diagonal_iso_tensor_of_symmetric_of_birigidified576 below · cited by 1 · depth 39 - Bi-rigidified bundle on A× A is additive in the second variable
AlgebraicGeometry.Polarisation.nonempty_pullback_oneProdNsmul_iso_tensorPow_of_birigidified575 below · cited by 2 · depth 39 - Local symmetry and N^{⊗ 2}≅𝒪 force [2]^*N≅𝒪
AlgebraicGeometry.Polarisation.nonempty_pullback_schemeNsmul_two_iso_unit_of_locIsoOnBase_negMor_of_locIsoOnBase_tensor_self372 below · cited by 1 · depth 39 - Naturality of the 2-torsion descent character in the test ring
AlgebraicGeometry.Polarisation.torsionCharacter_val_pullbackAlong_eq_of_hasValue_translate56 below · cited by 1 · depth 39 - Trivial Mumford kernel is local on the base
AlgebraicGeometry.Polarisation.KernelTrivial.of_forall_away_of_isInvertible4 below · cited by 1 · depth 40 - Descent function over [2] realising a 2-torsion character
AlgebraicGeometry.Polarisation.exists_appTop_eq_one_and_mul_eq_and_appTop_eq_torsionCharacter_two_val2 below · cited by 1 · depth 40 - Spreading a base-local isomorphism from a semilocalisation to a localisation
AlgebraicGeometry.Polarisation.exists_not_mem_forall_isUnit_locIsoOnBase_pullback_of_locIsoOnBase_isLocalization_primeCompl17 below · cited by 1 · depth 40 - Descent of an invertible module along [2] from a level subgroup
AlgebraicGeometry.Polarisation.exists_pullback_schemeNsmul_two_iso_of_levelSubgroup84 below · cited by 1 · depth 40 - A level-A[2] subgroup of the theta group of mathcal L₀^{⊗ 2}
AlgebraicGeometry.Polarisation.exists_subgroup_thetaGroup_tensor_self_injOn_pt_image_eq_two_torsion30 below · cited by 1 · depth 40 - Biadditivity: Λ([2]^*M')≅Λ(M'^{⊗ 2})^{⊗ 2}
AlgebraicGeometry.Polarisation.nonempty_mumfordBundle_pullback_schemeNsmul_two_iso_tensor588 below · cited by 1 · depth 40 - Theorem of the square for [2] over an affine base
AlgebraicGeometry.Polarisation.nonempty_pullback_schemeNsmul_two_iso_tensor_pullback_negMor_of_rigidified366 below · cited by 2 · depth 40 - Two-torsion points stabilise mathcal L₀ when Λ(mathcal L₀)congPotimesP
AlgebraicGeometry.Polarisation.nonempty_pullback_translation_iso_of_mumfordBundle_iso_tensor_self582 below · cited by 1 · depth 40 - Theta-cube of a rigidified bundle: rigidification and trivial faces
AlgebraicGeometry.Polarisation.exists_rigidifiedLineBundle_iso_thetaCube_and_locIsoOnBase_faces_of_rigidified5 below · cited by 1 · depth 41 - Theta-cube triviality along (x,x,-x) and [2]^*N
AlgebraicGeometry.Polarisation.nonempty_pullback_schemeNsmul_two_iso_of_nonempty_pullback_thetaCube_iso_unit5 below · cited by 1 · depth 41
AlgebraicGeometry.PolarisedAbelianScheme 129
- Finiteness of the QM locus over a finite-type base
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_finite_represents_of_isUnit_two_of_finiteType3,156 below · cited by 1 · depth 27 - Quasi-projective fine moduli of polarised abelian schemes of theta type δ
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_isFineModuli_thetaTypeLocally_quasiProjective1,852 below · cited by 1 · depth 27 - Representability of QM structures by a separated quasi-compact morphism
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_represents_isSeparated_quasiCompact_locallyOfFinitePresentation_of_isUnit_two3,117 below · cited by 1 · depth 28 - Formal unramifiedness of a scheme representing QM structures
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.formallyUnramified_of_represents854 below · cited by 1 · depth 28 - Universal closedness of a finite-type scheme representing QM structures
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.universallyClosed_of_represents_of_finiteType1,431 below · cited by 1 · depth 28 - Cyclotomic descent of fine moduli for polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_isFineModuli_of_forall_exists_isFineModuli_of_primitiveRoot1,023 below · cited by 1 · depth 28 - Fine moduli for theta-type polarisations from framed moduli
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_isFineModuli_thetaTypeLocally_of_isFineModuli_framed_quasiProjective_of_sq_eq1,610 below · cited by 1 · depth 28 - δ-theta type, étale-locally, is stable under base change
AlgebraicGeometry.PolarisedAbelianScheme.thetaTypeLocally_of_isPullback3 below · cited by 4 · depth 28 - Descent of étale-local theta type along faithfully flat étale maps
AlgebraicGeometry.PolarisedAbelianScheme.thetaTypeLocally_of_isPullback_of_faithfullyFlat_etale3 below · cited by 3 · depth 28 - Étale-local theta type is invariant under isomorphism
AlgebraicGeometry.PolarisedAbelianScheme.thetaTypeLocally_of_iso38 below · cited by 1 · depth 28 - Base change relation: congruence in φ and reflexivity
AlgebraicGeometry.PolarisedAbelianScheme.IsPullback.congr_and_id0 below · cited by 10 · depth 29 - Cancellation of base change for polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.IsPullback.of_isPullback_comp0 below · cited by 3 · depth 29 - Transitivity of base change for polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.IsPullback.trans0 below · cited by 16 · depth 29 - Base change of a quaternionic multiplication structure exists
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_isPullback6 below · cited by 9 · depth 29 - Extension of QM structures across a discrete valuation ring
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_isPullback_algebraMap_of_isDiscreteValuationRing1,421 below · cited by 1 · depth 29 - Representability of QM structures on polarised abelian surfaces
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_represents_of_representsLatticeActions_of_isUnit_two2,930 below · cited by 1 · depth 29 - Full-level fake elliptic curves package as QM polarised schemes
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.packages_surjective_and_iso_iff_and_isPullback_of_isUnit_two2,933 below · cited by 1 · depth 29 - Point map on QM structures is isomorphism-invariant
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.ptZ_eq_of_iso_of_isPullback_of_isPullback847 below · cited by 2 · depth 29 - A QM structure makes the polarisation symmetric, rooted, of type (6,6)
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.rootedSymmetricOfType_six_six_of_isUnit_six963 below · cited by 1 · depth 29 - Finite group action on a fine moduli scheme
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_aut_comp_pt_eq_and_comp_eq_of_isFineModuli_of_galois_of_isPullback26 below · cited by 1 · depth 29 - Descent of fine moduli along a faithfully flat base extension
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_isFineModuli_of_isFineModuli_of_isPullback_of_faithfullyFlat_of_iso990 below · cited by 1 · depth 29 - Fine moduli for ThetaTypeLocally from framed fine moduli
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_isFineModuli_thetaTypeLocally_of_isFineModuli_framed_thetaCore1,363 below · cited by 1 · depth 29 - Uniqueness of base-change comparison maps for level n≥ 3
AlgebraicGeometry.PolarisedAbelianScheme.eq_of_isPullback_of_isPullback_of_three_le846 below · cited by 3 · depth 29 - Existence of base changes of polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.exists_isPullback24 below · cited by 24 · depth 29 - Isomorphism invariance from base change and étale descent
AlgebraicGeometry.PolarisedAbelianScheme.of_iso_of_forall_isPullback_of_forall_faithfullyFlat_etale32 below · cited by 3 · depth 29 - Rooted symmetric polarisations of type δ are étale-locally of theta type
AlgebraicGeometry.PolarisedAbelianScheme.thetaTypeLocally_of_rootedSymmetricOfType1,343 below · cited by 1 · depth 29 - Base changes of isomorphic polarised abelian schemes are isomorphic
AlgebraicGeometry.PolarisedAbelianScheme.Iso.baseChange0 below · cited by 7 · depth 30 - Isomorphism of polarised abelian schemes is an equivalence relation
AlgebraicGeometry.PolarisedAbelianScheme.Iso.refl_symm_trans0 below · cited by 10 · depth 30 - Base change of QM structures composes along χ∘φ
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.IsPullback.trans0 below · cited by 3 · depth 30 - Base change of a QM isomorphism along φ
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.Iso.of_isPullback_of_isPullback0 below · cited by 4 · depth 30 - Isomorphy of QM structures is an equivalence relation
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.Iso.refl_symm_trans2 below · cited by 5 · depth 30 - Gluing QM structures over a basic open cover, 2 invertible
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_forall_isPullback_of_forall_away_of_isMaximalOrder_of_isUnit_two1,528 below · cited by 1 · depth 30 - QM structures transport along isomorphisms of polarised abelian surfaces
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_iso_of_polarisedAbelianScheme_iso11 below · cited by 4 · depth 30 - Packaging fake elliptic curves as polarised abelian surfaces
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_packages_of_withFullLevel_of_isUnit_two2,923 below · cited by 1 · depth 30 - A closed subscheme of E representing QM structures
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_ptZ_of_isClosedImmersion_iff_qmConditions847 below · cited by 1 · depth 30 - Unpacking a QM structure into a fake elliptic curve
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_withFullLevel_packages2 below · cited by 1 · depth 30 - Isomorphy of QM structures is local on the base
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.iso_of_forall_away_iso859 below · cited by 1 · depth 30 - Uniqueness of base change for QM structures
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.iso_of_isPullback_of_isPullback0 below · cited by 2 · depth 30 - Packaging is compatible with base change
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.withFullLevel_isPullback_of_packages_of_isPullback0 below · cited by 1 · depth 30 - Packaging respects isomorphism of full-level fake elliptic curves
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.withFullLevel_iso_iff_iso_of_packages_of_isUnit_two1,448 below · cited by 1 · depth 30 - Finite free quotient of a framed fine moduli scheme
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_isFineModuli_of_isFineModuli_framed_of_finite_free_transitive_of_three_le63 below · cited by 1 · depth 30 - Rigidity of polarised level-n structures, polarisation matched locally
AlgebraicGeometry.PolarisedAbelianScheme.eq_of_isPullback_of_isPullback_of_three_le_of_locally846 below · cited by 9 · depth 30 - Faithfully flat descent for polarised abelian schemes, n≥ 3
AlgebraicGeometry.PolarisedAbelianScheme.exists_descent_and_iso_of_faithfullyFlat_of_three_le_type0985 below · cited by 2 · depth 30 - Theta-adapted frame for a prescribed indexing after faithfully flat base change
AlgebraicGeometry.PolarisedAbelianScheme.exists_faithfullyFlat_isThetaAdapted_isPullback_of_thetaTypeLocally12 below · cited by 1 · depth 30 - Zariski gluing and separation for polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.exists_glue_and_iso_of_iso_localizationAway_of_three_le910 below · cited by 1 · depth 30 - Quaternionic conditions cut out a closed subscheme of E
AlgebraicGeometry.PolarisedAbelianScheme.exists_isClosedImmersion_iff_trace_and_exists_level_generator_and_exists_isCanonicalPolData_of_isUnit_two2,876 below · cited by 1 · depth 30 - Base change of polarised abelian schemes with level structure
AlgebraicGeometry.PolarisedAbelianScheme.exists_isPullback_univ24 below · cited by 5 · depth 30 - Polarisation-preserving automorphisms of a polarised abelian scheme are torsion
AlgebraicGeometry.PolarisedAbelianScheme.exists_schemeHomOverNpow_eq_schemeHomOverId_of_isIso_of_pullback_pol_iso_of_small845 below · cited by 2 · depth 30 - Étale-local existence of Schrödinger frames of type δ
AlgebraicGeometry.PolarisedAbelianScheme.exists_schrodingerFrame_of_rootedSymmetricOfType1,326 below · cited by 1 · depth 30 - Cube root of a polarisation yields a principal root
AlgebraicGeometry.PolarisedAbelianScheme.hasPrincipalRoot_of_isCanonicalPolData_of_locIsoOnBase_tensor_three3 below · cited by 1 · depth 30 - Polarisation of type (6,6) from a cube of canonical polarisation data
AlgebraicGeometry.PolarisedAbelianScheme.isOfType_six_six_of_isCanonicalPolData_of_locIsoOnBase_tensor_three960 below · cited by 1 · depth 30 - Base-change and isomorphism calculus for polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.isPullback_refl_comp_cancel_iso_symm_trans0 below · cited by 1 · depth 30 - Isomorphism of polarised abelian schemes descends along base change
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_isPullback_of_isPullback_of_iso0 below · cited by 2 · depth 30 - Quasi-compactness of the QM locus over the base
AlgebraicGeometry.PolarisedAbelianScheme.quasiCompact_comp_of_isClosedImmersion_iff_qmConditions1,165 below · cited by 1 · depth 30 - Base change preserves symmetric polarisations of type δ with principal root
AlgebraicGeometry.PolarisedAbelianScheme.rootedSymmetricOfType_of_isPullback8 below · cited by 1 · depth 30 - Triviality of the polarisation at the zero section descends along base change
AlgebraicGeometry.PolarisedAbelianScheme.IsPullback.nonempty_pullback_one_pol_iso_unit_of_pullback1 below · cited by 2 · depth 31 - Quaternionic action glues along a basic-open cover
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_act_of_forall_away_of_isMaximalOrder857 below · cited by 1 · depth 31 - Uniform bound on h⁰(L ⊗ i(βⱼ)^*L)
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_forall_geomFibreH0Finrank_tensor_pullback_act_le1,164 below · cited by 1 · depth 31 - From a polarised isomorphism to a QM isomorphism, locally
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.iso_of_polarisedAbelianScheme_iso_of_forall_away_iso847 below · cited by 1 · depth 31 - Local uniqueness of canonical polarisation data for QM surfaces
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.locIsoOnBase_of_isCanonicalPolData_of_isCanonicalPolData_of_isUnit_two1,443 below · cited by 1 · depth 31 - Classifying map to the quotient M by flat descent of frames
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_pt_comp_eq_of_finite_free_transitive5 below · cited by 1 · depth 31 - Every M-point comes from an object satisfying Q
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.exists_pt_eq_of_finite_free_transitive42 below · cited by 1 · depth 31 - Classifying point composed with q depends only on the underlying object
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.framedPt_comp_eq_of_iso_of_finite_free_transitive0 below · cited by 1 · depth 31 - Objects with equal point in the quotient are isomorphic
AlgebraicGeometry.PolarisedAbelianScheme.Satisfying.iso_of_pt_eq_of_finite_free_transitive5 below · cited by 1 · depth 31 - Isomorphic polarised abelian schemes share base changes, locally on S'
AlgebraicGeometry.PolarisedAbelianScheme.exists_cover_isPullback_of_isPullback_of_iso2 below · cited by 2 · depth 31 - Heisenberg level lifts give Schrödinger frames Zariski-locally
AlgebraicGeometry.PolarisedAbelianScheme.exists_cover_schrodingerFrame_of_levelLifts1,119 below · cited by 1 · depth 31 - Effective faithfully flat descent for rigidified polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.exists_descent_and_iso_of_faithfullyFlat_of_three_le_of_rigidified984 below · cited by 1 · depth 31 - Étale absorption of Schrödinger frames along a basic-open cover
AlgebraicGeometry.PolarisedAbelianScheme.exists_faithfullyFlat_etale_schrodingerFrame_of_cover10 below · cited by 1 · depth 31 - Zariski gluing of level-n polarised abelian schemes, n ≥ 3
AlgebraicGeometry.PolarisedAbelianScheme.exists_forall_isPullback_iso_of_forall_iso_localizationAway_of_three_le909 below · cited by 1 · depth 31 - Level generation by one point cuts out a closed subscheme of E
AlgebraicGeometry.PolarisedAbelianScheme.exists_isClosedImmersion_iff_exists_level_generator_lfp4 below · cited by 1 · depth 31 - Trace and canonical-cube conditions cut out a closed subscheme
AlgebraicGeometry.PolarisedAbelianScheme.exists_isClosedImmersion_iff_trace_and_exists_isCanonicalPolData_lfp_of_isUnit_two2,871 below · cited by 1 · depth 31 - Uniqueness of the base change of a polarised abelian scheme
AlgebraicGeometry.PolarisedAbelianScheme.exists_iso_nonempty_pullback_pol_iso_of_isPullback_of_isPullback0 below · cited by 1 · depth 31 - Étale-local Heisenberg lifts for rooted symmetric polarisations of type δ
AlgebraicGeometry.PolarisedAbelianScheme.exists_levelLifts_of_rootedSymmetricOfType1,312 below · cited by 1 · depth 31 - Polarised abelian variety: automorphisms have finite order
AlgebraicGeometry.PolarisedAbelianScheme.exists_schemeHomOverNpow_eq_schemeHomOverId_of_isIso_of_pullback_pol_iso_of_isAlgClosed212 below · cited by 1 · depth 31 - Automorphisms descend along a cartesian comparison map
AlgebraicGeometry.PolarisedAbelianScheme.exists_schemeHomOver_comp_eq_comp_of_isPullback0 below · cited by 1 · depth 31 - Finiteness and bound ≤ ℓ^{2g} for ℓ-torsion over ̄ k
AlgebraicGeometry.PolarisedAbelianScheme.finite_and_ncard_le_setOf_isTorsionPoint_of_isAlgClosed696 below · cited by 1 · depth 31 - Base change preserves the existence of a principal root
AlgebraicGeometry.PolarisedAbelianScheme.hasPrincipalRoot_of_isPullback4 below · cited by 1 · depth 31 - Being of type δ is stable under base change
AlgebraicGeometry.PolarisedAbelianScheme.isOfType_of_isPullback5 below · cited by 1 · depth 31 - Symmetry of the polarisation is stable under base change
AlgebraicGeometry.PolarisedAbelianScheme.isSymmetric_of_isPullback4 below · cited by 1 · depth 31 - Rigidity of polarised abelian schemes with level n ≥ 3
AlgebraicGeometry.PolarisedAbelianScheme.iso_hom_eq_id_of_three_le847 below · cited by 1 · depth 31 - Zariski-local uniqueness of polarised abelian schemes, n≥ 3
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_forall_isPullback_iso_of_three_le857 below · cited by 3 · depth 31 - Uniqueness of the base change of a polarised abelian scheme
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_isPullback_of_isPullback0 below · cited by 7 · depth 31 - Faithfully flat descent of isomorphisms of polarised abelian schemes, n≥ 3
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_iso_of_isPullback_of_faithfullyFlat_of_three_le877 below · cited by 2 · depth 31 - Action laws of a QM structure are Zariski-local
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.act_laws_of_forall_comp_eq_of_forall_away0 below · cited by 1 · depth 32 - Gluing a Λ-action along a basic open cover
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_act_comp_eq_of_forall_away_of_isMaximalOrder855 below · cited by 1 · depth 32 - Uniform fibrewise h⁰(L ⊗ s(βⱼ)^*L) for QM structures
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.exists_forall_geomFibreH0Finrank_tensor_pullback_act_eq1,163 below · cited by 1 · depth 32 - Mumford's commutator pairing from theta points over a point group
AlgebraicGeometry.PolarisedAbelianScheme.exists_commutatorPairing_of_thetaPt62 below · cited by 2 · depth 32 - Schrödinger frames on a basic-open cover from level lifts
AlgebraicGeometry.PolarisedAbelianScheme.exists_cover_schrodingerFrame_of_levelLifts_of_isSectionBasis1,115 below · cited by 1 · depth 32 - Theta points over a kernel point, Zariski-locally on the base
AlgebraicGeometry.PolarisedAbelianScheme.exists_cover_thetaPt_pt_eq_of_memKernel7 below · cited by 1 · depth 32 - Spreading out a polarised abelian variety with an automorphism
AlgebraicGeometry.PolarisedAbelianScheme.exists_fg_subalgebra_abelianScheme_closedImmersionBySections_comp_eq_comp_of_isIso_of_pullback_pol_iso163 below · cited by 1 · depth 32 - Removing the rigidification hypothesis in Zariski gluing
AlgebraicGeometry.PolarisedAbelianScheme.exists_forall_isPullback_iso_of_forall_iso_localizationAway_of_three_le_of_forall_rigidified863 below · cited by 1 · depth 32 - Gluing rigidified polarised abelian schemes along a basic-open cover
AlgebraicGeometry.PolarisedAbelianScheme.exists_forall_isPullback_iso_of_forall_iso_localizationAway_of_three_le_of_rigidified893 below · cited by 1 · depth 32 - Descent of polarised abelian scheme structure along faithfully flat base change
AlgebraicGeometry.PolarisedAbelianScheme.exists_isPullback_of_descent_of_faithfullyFlat96 below · cited by 1 · depth 32 - Étale-local Heisenberg level lifting of theta points
AlgebraicGeometry.PolarisedAbelianScheme.exists_levelLifts_of_commutatorPairing_eq_pow1,124 below · cited by 1 · depth 32 - Level lifts over a finite product of test rings
AlgebraicGeometry.PolarisedAbelianScheme.exists_levelLifts_pi_of_forall_exists_levelLifts16 below · cited by 1 · depth 32 - Rigidity yields a nerve datum for polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.exists_nerve_of_iso_pullbacks_of_three_le_of_rigidified857 below · cited by 1 · depth 32 - Zariski-local rank-d section basis after base change
AlgebraicGeometry.PolarisedAbelianScheme.exists_not_mem_forall_exists_isSectionBasis_sections_pullback_of_algebra1,096 below · cited by 1 · depth 32 - Drinfeld's trace condition cuts out a clopen locus
AlgebraicGeometry.PolarisedAbelianScheme.exists_opens_isClosed_range_subset_iff_trace42 below · cited by 1 · depth 32 - Non-degeneracy of the commutator pairing on each idempotent piece
AlgebraicGeometry.PolarisedAbelianScheme.forall_eq_zero_of_commutatorPairing_of_rootedSymmetricOfType947 below · cited by 1 · depth 32 - Gluing isomorphisms of polarised abelian schemes over a finite product
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_forall_iso_of_isPullback_evalRingHom0 below · cited by 1 · depth 32 - Flat descent of isomorphisms of rigidified polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_iso_of_isPullback_of_faithfullyFlat_of_three_le_of_rigidified865 below · cited by 1 · depth 32 - Zariski descent of isomorphisms of polarised abelian schemes
AlgebraicGeometry.PolarisedAbelianScheme.iso_of_iso_of_isPullback_pi_localizationAway_of_three_le850 below · cited by 2 · depth 32 - Base change of a Schrödinger frame along ψ : R → R'
AlgebraicGeometry.PolarisedAbelianScheme.nonempty_schrodingerFrame_comp_of_schrodingerFrame1,110 below · cited by 1 · depth 32 - Multiplicative comparison maps preserve the zero section
AlgebraicGeometry.PolarisedAbelianScheme.one_comp_eq_specMap_comp_one_of_mul0 below · cited by 3 · depth 32 - Extensionality of theta points with equal underlying section
AlgebraicGeometry.PolarisedAbelianScheme.thetaPt_eq_of_pt_eq_of_forall_act_eq3 below · cited by 6 · depth 32 - Geometric h⁰ of L⊗βⱼ^*L equals |n|
AlgebraicGeometry.PolarisedAbelianScheme.QMStructure.geomFibreH0Finrank_tensor_pullback_act_eq_natAbs_of_isAlgClosed1,157 below · cited by 1 · depth 33 - Glued charts for polarised abelian schemes over Spec S
AlgebraicGeometry.PolarisedAbelianScheme.exists_charts_forall_locIso_of_forall_iso_localizationAway_of_three_le852 below · cited by 1 · depth 33 - Isotropic theta points lift homomorphically over an étale cover
AlgebraicGeometry.PolarisedAbelianScheme.exists_faithfullyFlat_etale_levelLift_of_forall_act_comm1,115 below · cited by 1 · depth 33 - Spreading out a polarised abelian variety: the Proj presentation stage
AlgebraicGeometry.PolarisedAbelianScheme.exists_fg_subalgebra_abelianScheme_projPresentation_pullback_isClosedImmersion_of_pullback_pol_iso161 below · cited by 1 · depth 33 - Typed theta points over a geometric point with pairing ζ^B
AlgebraicGeometry.PolarisedAbelianScheme.exists_isAlgClosed_typePoints_thetaPt_of_idempotent_ne_zero8 below · cited by 1 · depth 33 - Theta points commute up to a unit of the base
AlgebraicGeometry.PolarisedAbelianScheme.exists_units_forall_thetaPt_act_act_eq_smul_act_act56 below · cited by 2 · depth 33 - Theta points at the identity act by a unit scalar
AlgebraicGeometry.PolarisedAbelianScheme.exists_units_forall_thetaPt_act_eq_smul_of_pt_eq_one53 below · cited by 3 · depth 33 - Section bases persist under base change of a polarisation
AlgebraicGeometry.PolarisedAbelianScheme.isSectionBasis_app_pullbackLocalSection_of_isSectionBasis1,096 below · cited by 2 · depth 33 - Rigidity: overlap isomorphisms commute with base-change comparisons
AlgebraicGeometry.PolarisedAbelianScheme.iso_hom_comp_eq_of_isPullback_of_isPullback_of_three_le847 below · cited by 1 · depth 33 - Level structure clauses descend along a principal open cover
AlgebraicGeometry.PolarisedAbelianScheme.level_clauses_of_forall_isPullback_away0 below · cited by 1 · depth 33 - Rigidified invertible modules isomorphic after faithfully flat base change
AlgebraicGeometry.PolarisedAbelianScheme.nonempty_iso_of_pullback_locally_iso_of_faithfullyFlat_of_rigidified78 below · cited by 1 · depth 33 - Central theta points lie over the identity section
AlgebraicGeometry.PolarisedAbelianScheme.thetaPt_pt_eq_one_of_forall_act_comm_of_isAlgClosed945 below · cited by 1 · depth 33 - Spreading out an abelian variety with a σ-invariant invertible module
AlgebraicGeometry.PolarisedAbelianScheme.exists_fg_subalgebra_abelianScheme_pullback_iso_comp_eq_comp_of_isIso_of_pullback_pol_iso140 below · cited by 1 · depth 34 - Comparison map between base changes along composable ring maps
AlgebraicGeometry.PolarisedAbelianScheme.exists_hom_comp_eq_isPullback_of_comp0 below · cited by 1 · depth 34 - Comparison map of two base changes, polarisation locally
AlgebraicGeometry.PolarisedAbelianScheme.exists_hom_comp_eq_isPullback_of_comp_of_locally0 below · cited by 1 · depth 34 - Enlarging a geometric point to realise the principal root
AlgebraicGeometry.PolarisedAbelianScheme.exists_isAlgClosed_principalRoot_thetaPt_of_rootedSymmetricOfType18 below · cited by 1 · depth 34 - A symmetric 2-cocycle from commuting theta points
AlgebraicGeometry.PolarisedAbelianScheme.exists_symmCocycle_forall_mul_act_eq_smul_act_of_forall_act_comm62 below · cited by 1 · depth 34 - Nonvanishing of a+b in K for a principal root
AlgebraicGeometry.PolarisedAbelianScheme.natCast_add_ne_zero_of_principalRoot_of_rootedSymmetricOfType_of_isAlgClosed388 below · cited by 1 · depth 34 - Theta points commuting with K(L) lie over the identity
AlgebraicGeometry.PolarisedAbelianScheme.thetaPt_pt_eq_one_of_forall_act_comm_of_principalRoot_of_ne_zero923 below · cited by 1 · depth 34 - Faithfulness of R on global sections of the pulled-back polarisation
AlgebraicGeometry.PolarisedAbelianScheme.eq_zero_of_forall_baseScalar_smul_eq_zero54 below · cited by 2 · depth 35 - Kernel points over an algebraically closed base point do not grow
AlgebraicGeometry.PolarisedAbelianScheme.exists_eq_comp_of_memKernel_of_isOfType_of_isAlgClosed1 below · cited by 1 · depth 35 - Spreading out an abelian variety with automorphism and invertible module
AlgebraicGeometry.PolarisedAbelianScheme.exists_fg_subalgebra_abelianScheme_pullback_iso_comp_eq_comp_of_isIso137 below · cited by 1 · depth 35 - Principal-root data base-change along a point of the covering algebra
AlgebraicGeometry.PolarisedAbelianScheme.exists_principalRoot_over_of_ringHom_of_forall_exists_principalRoot6 below · cited by 1 · depth 35 - Spreading out an abelian variety with an automorphism
AlgebraicGeometry.PolarisedAbelianScheme.exists_fg_subalgebra_abelianScheme_comp_eq_comp_of_isIso111 below · cited by 1 · depth 36
AlgebraicGeometry.Proj 4
- Transition cocycle for D₊(Fᵢ) on ProjA
AlgebraicGeometry.Proj.exists_cocycle_basicOpen_eq_inf_of_mem_of_pos0 below · cited by 2 · depth 30 - D₊(fg) as a basic open inside D₊(f)
AlgebraicGeometry.Proj.basicOpen_mul_eq_basicOpen_awayToSection0 below · cited by 7 · depth 32 - Value of `Proj.fromOfGlobalSections` on fractions s/r^k
AlgebraicGeometry.Proj.fromOfGlobalSections_appLE_awayToSection_mk_mul_pow0 below · cited by 1 · depth 36 - Morphisms to Proj agreeing locally on homogeneous charts coincide
AlgebraicGeometry.Proj.hom_ext_of_forall_exists_basicOpen0 below · cited by 1 · depth 36
AlgebraicGeometry.ProjSpace 57
- Serre vanishing for twists along a finite morphism to P^N_A
AlgebraicGeometry.ProjSpace.exists_forall_subsingleton_HSucc_twist18 below · cited by 5 · depth 18 - Projective space is stable under base change
AlgebraicGeometry.ProjSpace.isPullback_map0 below · cited by 39 · depth 18 - Prescribing the ratios x_k/xᵢ on the chart D₊(xᵢ)
AlgebraicGeometry.ProjSpace.exists_algHom_away_apply_ratio_eq0 below · cited by 8 · depth 19 - Finitely generated saturated graded submodule of the twist module
AlgebraicGeometry.ProjSpace.exists_isFG_hom_injective_saturated_twistGradedModule0 below · cited by 2 · depth 19 - Twist-datum sections over U_I as graded localisations
AlgebraicGeometry.ProjSpace.exists_sec_shift_twistGradedModule_equiv0 below · cited by 1 · depth 19 - Base change of Pⁿ preserves the standard charts
AlgebraicGeometry.ProjSpace.map_preimage_basicOpen_X0 below · cited by 10 · depth 19 - Closed subschemes of Pⁿ_A determined by chartwise vanishing in high degrees
AlgebraicGeometry.ProjSpace.exists_iso_comp_eq_of_isClosedImmersion_of_forall_app_awayToSection_eq_zero_iff1 below · cited by 1 · depth 31 - Every morphism to Pⁿ_A admits a Proj presentation
AlgebraicGeometry.ProjSpace.exists_projPresentation_toProj_eq_and_locallyTrivial3 below · cited by 4 · depth 31 - Base change of ZsubseteqPⁿ composes along a scalar tower
AlgebraicGeometry.ProjSpace.isPullback_comp_and_comp_eq_map_of_isScalarTower0 below · cited by 4 · depth 31 - Chart-ideal membership on D₊(xᵢ) versus xᵢ^N F ∈ I
AlgebraicGeometry.ProjSpace.awayToSection_mk_mem_span_iff_exists_X_pow_mul_mem0 below · cited by 4 · depth 32 - Base change of the ideal sheaf of a homogeneous ideal on Pⁿ
AlgebraicGeometry.ProjSpace.eq_comap_map_of_ideal_basicOpen_eq_span1 below · cited by 3 · depth 32 - Kernel ideal sheaf of a closed immersion into Pⁿ_A
AlgebraicGeometry.ProjSpace.eq_ker_of_ideal_basicOpen_eq_span_of_isClosedImmersion3 below · cited by 5 · depth 32 - Noetherian approximation of flat closed subschemes of Pⁿ
AlgebraicGeometry.ProjSpace.exists_fg_subalgebra_isClosedImmersion_flat_isPullback_comp_map_of_isClosedImmersion_of_flat_of_locallyOfFinitePresentation28 below · cited by 2 · depth 32 - Vanishing on one chart of Pⁿ_A spreads to all charts
AlgebraicGeometry.ProjSpace.exists_forall_app_awayToSection_eq_zero_of_app_awayToSection_eq_zero0 below · cited by 1 · depth 32 - Ideals of A[x₀,…,xₙ] cut out ideal sheaves on Pⁿ_A
AlgebraicGeometry.ProjSpace.exists_idealSheafData_ideal_basicOpen_eq_span3 below · cited by 6 · depth 32 - Closed subschemes of Pⁿ_A base change to Pⁿ_B
AlgebraicGeometry.ProjSpace.exists_isClosedImmersion_isPullback_comp_eq_map1 below · cited by 3 · depth 32 - Uniqueness of base-change realisations inside Pⁿ_B
AlgebraicGeometry.ProjSpace.exists_iso_hom_comp_eq_of_isPullback_of_comp_eq_map1 below · cited by 3 · depth 32 - Base change of the Čech complex of the twist mathcal O_Z(d)
AlgebraicGeometry.ProjSpace.exists_linearEquiv_baseChange_cochain_twist_of_isPullback4 below · cited by 6 · depth 32 - Degree-d forms as global sections of φ^*𝒪(d)
AlgebraicGeometry.ProjSpace.exists_linearMap_homogeneousSubmodule_twistObj_top_val_eq0 below · cited by 10 · depth 32 - Finiteness of Čech cohomology of twists on closed subschemes of Pⁿ_A
AlgebraicGeometry.ProjSpace.finite_H0_twist_and_finite_HSucc_twist_of_isClosedImmersion57 below · cited by 1 · depth 32 - Transport of check H⁰(𝒪(d)) along a realised base change
AlgebraicGeometry.ProjSpace.finite_projective_H0_twist_of_ker_baseChange_of_isPullback6 below · cited by 1 · depth 32 - Flatness and vanishing of Čech cochains of mathcal O_Z(d)
AlgebraicGeometry.ProjSpace.flat_cochain_twist_and_subsingleton_cochain_of_flat57 below · cited by 1 · depth 32 - Base-changed twisted Čech complex at a Hilbert-functor point
AlgebraicGeometry.ProjSpace.ker_baseChange_le_range_and_finrank_ker_eq_of_point_of_isPullback6 below · cited by 1 · depth 32 - Closed subschemes of Pⁿ_A from finitely generated homogeneous ideals
AlgebraicGeometry.ProjSpace.locallyOfFinitePresentation_comp_of_fg_of_ideal_basicOpen_eq_span1 below · cited by 2 · depth 32 - Compatibility of dehomogenised forms with a base-change realisation
AlgebraicGeometry.ProjSpace.restrictFun_app_app_awayToSection_eq_app_awayToSection_map0 below · cited by 2 · depth 32 - Large twists of a closed subscheme of Pⁿ_A: 0-cocycles come from forms
AlgebraicGeometry.ProjSpace.exists_forall_H0_twist_exists_isHomogeneous_forall_val_eq_of_isClosedImmersion23 below · cited by 3 · depth 33 - Chartwise spreading of ideal-sheaf membership on Pⁿ_A
AlgebraicGeometry.ProjSpace.exists_forall_awayToSection_mk_X_pow_mul_mem_ideal_of_mem_ideal0 below · cited by 2 · depth 33 - Chart triviality of the pulled-back twist datum
AlgebraicGeometry.ProjSpace.exists_linearEquiv_twistObj_of_le_pullbackChart0 below · cited by 7 · depth 33 - Field descent of representability of twist Čech 0-cocycles
AlgebraicGeometry.ProjSpace.forall_H0_twist_exists_isHomogeneous_of_baseChange_field6 below · cited by 1 · depth 33 - Čech acyclicity and surjectivity for ideals of maximal growth
AlgebraicGeometry.ProjSpace.forall_subsingleton_HSucc_twist_and_forall_H0_exists_of_maximal_growth157 below · cited by 1 · depth 33 - Coherence and flatness of the twist datum
AlgebraicGeometry.ProjSpace.isCoherent_twist_and_flat3 below · cited by 3 · depth 33 - Quasi-coherence of the twist datum on X → P^N_A
AlgebraicGeometry.ProjSpace.isQuasicoherent_twist0 below · cited by 4 · depth 33 - Linear maps on Pⁿ commute with base change
AlgebraicGeometry.ProjSpace.linMap_map_comp_map0 below · cited by 3 · depth 33 - Base change of the chart-vanishing criterion along a field extension
AlgebraicGeometry.ProjSpace.mem_map_iff_forall_app_awayToSection_eq_zero_of_baseChange_field3 below · cited by 1 · depth 33 - Čech vanishing for twists descends from a field extension
AlgebraicGeometry.ProjSpace.subsingleton_HSucc_twist_of_subsingleton_HSucc_twist_baseChange_field5 below · cited by 1 · depth 33 - Hyperplane short exact sequence of twists, with Čech comparison
AlgebraicGeometry.ProjSpace.exists_affSES_twist_succ_of_forall_mul_eq_zero_imp3 below · cited by 1 · depth 34 - Degree-d sections of a closed subscheme of Pⁿ_A come from polynomials for d gg 0
AlgebraicGeometry.ProjSpace.exists_forall_mem_grade_exists_isHomogeneous_forall_apply_eq_of_isClosedImmersion21 below · cited by 1 · depth 34 - Finitely generated homogeneous ideal cutting out a closed subscheme of Pⁿ_A
AlgebraicGeometry.ProjSpace.exists_ideal_fg_forall_ker_ideal_basicOpen_eq_span_of_isClosedImmersion8 below · cited by 1 · depth 34 - Generic linear form for an ideal of maximal growth
AlgebraicGeometry.ProjSpace.exists_linearForm_section_maximal_growth28 below · cited by 1 · depth 34 - Čech 0-cocycles on the pulled-back standard charts glue
AlgebraicGeometry.ProjSpace.exists_twistObj_top_forall_res_eq_of_mem_H0_twist0 below · cited by 1 · depth 34 - Projective space is flat over its base
AlgebraicGeometry.ProjSpace.flat_pi1 below · cited by 1 · depth 34 - Degree-d forms surject onto Čech H⁰ under maximal growth
AlgebraicGeometry.ProjSpace.forall_H0_twist_exists_of_forall_subsingleton_HSucc_of_maximal_growth136 below · cited by 1 · depth 34 - Vanishing degree-m piece forces trivial twisted Čech groups
AlgebraicGeometry.ProjSpace.forall_subsingleton_HSucc_twist_and_forall_H0_exists_of_finrank_piece_eq_zero0 below · cited by 2 · depth 34 - Closed immersion into P^N_S detected on a principal cover
AlgebraicGeometry.ProjSpace.isClosedImmersion_of_forall_isPullback_map_of_span_eq_top1 below · cited by 1 · depth 34 - Linear maps on Pⁿ_R commute with the structure morphism
AlgebraicGeometry.ProjSpace.linMap_comp_pi0 below · cited by 2 · depth 34 - Projective space is locally of finite presentation
AlgebraicGeometry.ProjSpace.locallyOfFinitePresentation_pi0 below · cited by 1 · depth 34 - Chart-independence of homogeneous coordinates of a B-point of Pⁿ_R
AlgebraicGeometry.ProjSpace.specMap_comp_awayInclusion_eq_of_forall_apply_ratio_mul_eq0 below · cited by 2 · depth 34 - Chartwise non-zero-divisor property of a linear form on Z
AlgebraicGeometry.ProjSpace.app_awayToSection_linearForm_mul_eq_zero_imp_of_forall_mul_mem_imp1 below · cited by 1 · depth 35 - Multiplication by a weight-e cocycle twists O(d) into O(d+e)
AlgebraicGeometry.ProjSpace.exists_hom_twist_val_eq_mul_and_injective_of_cocycle0 below · cited by 1 · depth 35 - Multiplying by xⱼ^k makes a twist section polynomial
AlgebraicGeometry.ProjSpace.exists_isHomogeneous_forall_xMul_pow_apply_eq_of_isClosedImmersion0 below · cited by 1 · depth 35 - Čech cohomology of the quotient twist on a hyperplane section
AlgebraicGeometry.ProjSpace.exists_linearEquiv_H0_HSucc_coker_twist_of_ker_eq_sup0 below · cited by 1 · depth 35 - Hilbert polynomial for h⁰(mathcal O_Z(d)) above a vanishing threshold
AlgebraicGeometry.ProjSpace.exists_polynomial_natDegree_le_forall_finrank_H0_twist_eq_of_forall_subsingleton_HSucc99 below · cited by 2 · depth 35 - Chart dictionary for J+(ℓ) under maximal growth
AlgebraicGeometry.ProjSpace.mem_span_sup_linearForm_iff_forall_app_eq_zero_of_maximal_growth24 below · cited by 1 · depth 35 - Snapper polynomiality of the Čech Euler characteristic on Pⁿ_k
AlgebraicGeometry.ProjSpace.exists_polynomial_natDegree_le_forall_eulerChar_twist_stdCoverPullback_eq98 below · cited by 1 · depth 36 - Closed subschemes of Pⁿ_k have dimension at most n
AlgebraicGeometry.ProjSpace.topologicalKrullDim_le_of_isClosedImmersion0 below · cited by 1 · depth 37 - Base change Pⁿ_A → Pⁿ_R is a closed immersion
AlgebraicGeometry.ProjSpace.isClosedImmersion_map_of_surjective1 below · cited by 1 · depth 39 - Projective space over R has a closed-immersion section
AlgebraicGeometry.ProjSpace.exists_isClosedImmersion_comp_pi_eq_id0 below · cited by 1 · depth 42
AlgebraicGeometry.RelEffCartierDiv 65
- Relative effective divisors on smooth relative curves are Cartier
AlgebraicGeometry.RelEffCartierDiv.isInvertible_I5 below · cited by 65 · depth 13 - Invertibility of a relative divisor supported in a smooth open
AlgebraicGeometry.RelEffCartierDiv.isInvertible_I_of_supportedIn7 below · cited by 53 · depth 13 - Base change of 𝒪(± u) for a point in the smooth locus
AlgebraicGeometry.RelEffCartierDiv.nonempty_pullback_ofPoint_lineBundle_iso_and_idealModule_iso_of_range_subset21 below · cited by 20 · depth 13 - Base change of the degree-one divisor of a point
AlgebraicGeometry.RelEffCartierDiv.pullbackAlong_ofPoint0 below · cited by 36 · depth 13 - Splitting off a graph from a relative effective divisor
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_ker_graphOver_mul3 below · cited by 2 · depth 14 - Section ideal of a point in the smooth locus is invertible
AlgebraicGeometry.RelEffCartierDiv.isInvertible_I_ofPoint_of_range_subset7 below · cited by 1 · depth 14 - Twist of a relative effective divisor is rigidified invertible
AlgebraicGeometry.RelEffCartierDiv.isInvertible_twistModule_and_nonempty_pullback_iso14 below · cited by 3 · depth 14 - Divisor of a point lying in an open is supported there
AlgebraicGeometry.RelEffCartierDiv.supportedIn_ofPoint0 below · cited by 16 · depth 14 - Geometric connectedness of a universal relative divisor scheme
AlgebraicGeometry.RelEffCartierDiv.IsUniversal.geometricallyConnected17 below · cited by 1 · depth 15 - Properness of a universal relative effective divisor base
AlgebraicGeometry.RelEffCartierDiv.IsUniversal.isProper18 below · cited by 2 · depth 15 - Inverse image of point divisor ideals under base change
AlgebraicGeometry.RelEffCartierDiv.comap_mapOnProdOver_I_ofPoint_and_mul_prod_pow3 below · cited by 1 · depth 15 - Tensoring by 𝒪(D) raises the Čech Euler characteristic by r
AlgebraicGeometry.RelEffCartierDiv.eulerChar_tensor_lineBundle_eq100 below · cited by 8 · depth 15 - Sums of S-points are relative effective divisors of degree r
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_prodKerGraph7 below · cited by 14 · depth 15 - Degree-r divisors over an algebraically closed field are sums of r points
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_prodKerGraph_of_isAlgClosed12 below · cited by 6 · depth 15 - Zero divisor of a section has degree χ(M)-χ(𝒪)
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal_of_eulerChar_eq117 below · cited by 13 · depth 15 - Existence of a universal relative effective divisor of degree r
AlgebraicGeometry.RelEffCartierDiv.exists_isUniversal30 below · cited by 3 · depth 15 - The divisor rε+r'z₀ over R and over A
AlgebraicGeometry.RelEffCartierDiv.exists_polarisation_pair_of_block28 below · cited by 1 · depth 15 - Representability of degree-r divisors supported in a smooth open
AlgebraicGeometry.RelEffCartierDiv.exists_supportedIn_universal_of_smooth_opens31 below · cited by 2 · depth 15 - Fibrewise algebraic equivalence to zero of the twist of D
AlgebraicGeometry.RelEffCartierDiv.isAlgEquivZero_twistModule_fibre37 below · cited by 3 · depth 15 - Representing scheme of divisors supported in U is of finite type, quasi-compact, separated
AlgebraicGeometry.RelEffCartierDiv.locallyOfFiniteType_quasiCompact_isSeparated_of_universal_supportedIn18 below · cited by 2 · depth 15 - Fibrewise zero-scheme criterion for support of a relative divisor
AlgebraicGeometry.RelEffCartierDiv.supportedIn_of_lineBundle_iso_of_forall_zeroScheme_supportedIn20 below · cited by 2 · depth 15 - Sum map to a universal degree-r divisor is finite flat of rank r!
AlgebraicGeometry.RelEffCartierDiv.IsUniversal.exists_sumMap14 below · cited by 3 · depth 16 - Restriction of 𝒪(-P) along a closed immersion through P
AlgebraicGeometry.RelEffCartierDiv.comap_curveChange_ofPoint_comp_eq_and_isIso_pullbackModuleComparison_of_isIso_morphismRestrict15 below · cited by 2 · depth 16 - Ideal sheaf of a point restricts trivially to a disjoint closed subscheme
AlgebraicGeometry.RelEffCartierDiv.comap_curveChange_ofPoint_eq_top_and_isIso_pullbackModuleComparison_of_disjoint15 below · cited by 4 · depth 16 - Finite étale block as a relative effective divisor of degree d
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_ker_lift_and_supportedIn_of_finite_etale0 below · cited by 1 · depth 16 - Addition of relative effective divisors on a smooth curve
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_mul6 below · cited by 6 · depth 16 - Sums of points in the smooth locus give relative divisors
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_prodKerGraph_and_supportedIn11 below · cited by 5 · depth 16 - Zero scheme of a nonzero section on a fibre curve
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal_of_ne_zero_of_isProper8 below · cited by 2 · depth 16 - Fibrewise criterion for a zero scheme supported in U
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal_of_supportedIn24 below · cited by 1 · depth 16 - Affine neighbourhoods of finite sets in a universal divisor scheme
AlgebraicGeometry.RelEffCartierDiv.exists_isAffineOpen_of_finset_of_universal_supportedIn26 below · cited by 1 · depth 16 - Universal splitting cover of degree r! for a relative divisor
AlgebraicGeometry.RelEffCartierDiv.exists_split10 below · cited by 4 · depth 16 - Product of relative divisors supported in a smooth open
AlgebraicGeometry.RelEffCartierDiv.exists_supportedIn_I_eq_mul_of_supportedIn10 below · cited by 11 · depth 16 - Affine charts supporting a relative divisor exist locally on T
AlgebraicGeometry.RelEffCartierDiv.exists_supportedIn_of_forall_finset0 below · cited by 1 · depth 16 - Supportedness in U is an open condition on divisors
AlgebraicGeometry.RelEffCartierDiv.isOpenImmersion_presheaf_supportedIn_incl0 below · cited by 1 · depth 16 - Representability of the chart of divisors supported in an affine open
AlgebraicGeometry.RelEffCartierDiv.isRepresentable_supportedIn24 below · cited by 2 · depth 16 - Relative degree-r Cartier divisors form a Zariski sheaf
AlgebraicGeometry.RelEffCartierDiv.isSheaf_functor2 below · cited by 1 · depth 16 - Geometric fibre of 𝒪(rε+r'W) in point-ideal form
AlgebraicGeometry.RelEffCartierDiv.nonempty_lineBundle_pullbackAlong_iso_invModule_pow_ker_mul_pow_prod_ker5 below · cited by 1 · depth 16 - Base change of 𝒪(E) for divisors supported in a smooth open
AlgebraicGeometry.RelEffCartierDiv.nonempty_pullback_lineBundle_pullbackAlong_iso_of_supportedIn19 below · cited by 4 · depth 16 - Twisted divisor module commutes with base change
AlgebraicGeometry.RelEffCartierDiv.nonempty_twistModule_pullbackAlong_iso_pullback23 below · cited by 4 · depth 16 - Affine neighbourhoods of finite point sets on universal divisor schemes
AlgebraicGeometry.RelEffCartierDiv.IsUniversal.exists_isAffineOpen_of_finset25 below · cited by 1 · depth 17 - Finite-support ideal sheaves on fibres give relative divisors
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_of_isProper_of_support_finite0 below · cited by 1 · depth 17 - Divisor of a section on a smooth geometrically irreducible fibre
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal_of_eulerChar_eq_of_smooth_fibre120 below · cited by 2 · depth 17 - Restricting a relative effective divisor supported in an open
AlgebraicGeometry.RelEffCartierDiv.exists_comap_eq_of_supportedIn0 below · cited by 1 · depth 17 - Affine representability of relative effective divisors of degree r
AlgebraicGeometry.RelEffCartierDiv.exists_isUniversal_of_isAffine22 below · cited by 1 · depth 17 - Extending relative effective divisors along an open immersion
AlgebraicGeometry.RelEffCartierDiv.exists_supportedIn_comap_eq_of_isSeparated0 below · cited by 1 · depth 17 - The rigidified bundle 𝒪(D-E_T) is invertible and trivial along ε
AlgebraicGeometry.RelEffCartierDiv.isInvertible_rigidify_lineBundle_tensor_idealModule_and_nonempty_pullback_iso_of_supportedIn15 below · cited by 1 · depth 17 - Invertibility and trivialisation of the rigidified twist 𝒪(D-rε)
AlgebraicGeometry.RelEffCartierDiv.isInvertible_twistModule_and_nonempty_pullback_iso_of_supportedIn17 below · cited by 2 · depth 17 - The empty divisor represents degree-zero relative divisors
AlgebraicGeometry.RelEffCartierDiv.isUniversal_empty0 below · cited by 2 · depth 17 - Rigidified 𝒪(D-E_T) commutes with base change
AlgebraicGeometry.RelEffCartierDiv.nonempty_rigidify_lineBundle_tensor_idealModule_pullbackAlong_iso_pullback_of_supportedIn22 below · cited by 2 · depth 17 - Descent of relative effective Cartier divisors along finite flat covers
AlgebraicGeometry.RelEffCartierDiv.existsUnique_pullbackAlong_eq_of_isPullback_of_isAffine2 below · cited by 1 · depth 18 - Same-divisor relation on the fibre power: finite flat of rank r!
AlgebraicGeometry.RelEffCartierDiv.exists_sameDivisorScheme15 below · cited by 1 · depth 18 - Base change of the twist 𝒪(D-rε) along ψ
AlgebraicGeometry.RelEffCartierDiv.nonempty_twistModule_pullbackAlong_iso_pullback_of_supportedIn26 below · cited by 2 · depth 18 - Injectivity of divisor pullback along a flat surjective base change
AlgebraicGeometry.RelEffCartierDiv.pullbackAlong_injective1 below · cited by 1 · depth 18 - A degree r relative divisor over a field has at most r support points
AlgebraicGeometry.RelEffCartierDiv.card_le_of_subset_support0 below · cited by 1 · depth 19 - Zero scheme of a section as relative effective divisor
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal22 below · cited by 1 · depth 19 - Zero scheme of a section as relative effective Cartier divisor
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_zeroSchemeIdeal_and_supportedIn_of_support_subset_of_isOpenImmersion120 below · cited by 1 · depth 19 - A relative effective divisor misses a point in each fibre
AlgebraicGeometry.RelEffCartierDiv.exists_snd_apply_eq_and_notMem_support4 below · cited by 1 · depth 19 - Point divisors restrict along a closed immersion over R
AlgebraicGeometry.RelEffCartierDiv.nonempty_pullback_curveChange_ofPoint_comp_lineBundle_iso_and_idealModule_iso_of_isInvertible16 below · cited by 4 · depth 24 - Degree equality from fibrewise algebraic triviality
AlgebraicGeometry.RelEffCartierDiv.eq_of_isAlgEquivZero_pullback_mapOnProdOver_of_nonempty_tensor_lineBundle_iso_lineBundle270 below · cited by 1 · depth 27 - Line bundles on a smooth proper curve as differences of effective divisors
AlgebraicGeometry.RelEffCartierDiv.exists_nonempty_tensor_lineBundle_iso_lineBundle118 below · cited by 1 · depth 27 - Relative effective divisors transport along the base-change isomorphism
AlgebraicGeometry.RelEffCartierDiv.forall_exists_comap_pullbackFst_eq_I_and_forall_exists_I_eq_comap_pullbackFst0 below · cited by 1 · depth 27 - Algebraic equivalence to zero forces equal divisor degrees
AlgebraicGeometry.RelEffCartierDiv.eq_of_isAlgEquivZero_of_nonempty_tensor_lineBundle_iso_lineBundle258 below · cited by 1 · depth 28 - Schematic closure over a DVR of a relative divisor on the generic fibre
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_ker_and_pullbackAlong_eq_and_saturated_of_isDiscreteValuationRing6 below · cited by 1 · depth 28 - Relative divisors on a smooth relative curve are invertible
AlgebraicGeometry.RelEffCartierDiv.isInvertible_I_of_smoothOfRelativeDimension_one7 below · cited by 1 · depth 28 - Extending a relative effective Cartier divisor over a DVR
AlgebraicGeometry.RelEffCartierDiv.exists_I_eq_of_flat_of_comap_mapOnProdOver_eq_of_isDiscreteValuationRing3 below · cited by 1 · depth 29
AlgebraicGeometry.RelPicard 352
- Independence of the Pic⁰ representing scheme from the rigidifying section
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_inverse_pair_of_sections5 below · cited by 2 · depth 12 - Pull-back along e and e⁻¹ are mutually inverse
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.pullbackHom_inv_comp_pullbackHom_hom_of_iso0 below · cited by 3 · depth 12 - Unique morphism of representing schemes induced by a transformation
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.existsUnique_hom_of_transform0 below · cited by 14 · depth 13 - Norm–pullback endomorphism of the relative Pic⁰ over a DVR
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_hom_classifies_norm_pullback_poincare_of_twoGluedCurves_of_mem_of_ringKrullDim_le_one341 below · cited by 3 · depth 13 - Pullback along a non-pointed curve morphism induces a Pic⁰-homomorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_hom_classifies_rigidify_pullback_curveChange3 below · cited by 7 · depth 13 - Curve isomorphism on Pic⁰ points: N(a)· b=g
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mul_comp_eq_of_classifies_rigidify_pullback_of_ofPoint_of_isIso21 below · cited by 3 · depth 13 - Norm description of the Poincaré bundle under arbitrary base change
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_comp_iso_rigidify_normModule_of_range_subset58 below · cited by 5 · depth 13 - Poincaré bundle pulled back along a product of points
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_mul_iso0 below · cited by 49 · depth 13 - Triviality of the Poincaré bundle at the unit point
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_one_iso0 below · cited by 17 · depth 13 - Base-change compatibility of the classifying morphism of f^*
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.pullbackHom_baseChange_fst0 below · cited by 1 · depth 13 - Base-changed Picard restriction maps commute with 1×τ
AlgebraicGeometry.RelPicard.baseChangeSnd_comp_restrictHom_eq_of_baseChangeSnd_comp0 below · cited by 2 · depth 13 - Base change compatibility of the relative group law on points
AlgebraicGeometry.RelPicard.baseChange_relativeGroupLaw_mul_compat1 below · cited by 30 · depth 13 - Abel–Jacobi morphism for a represented relative Pic⁰
AlgebraicGeometry.RelPicard.exists_abelJacobi_of_representsRelSubPic29 below · cited by 15 · depth 13 - Raynaud's dictionary for Pic⁰ of a two-component curve
AlgebraicGeometry.RelPicard.exists_gluedPic0_equiv_of_twoGluedSmoothCurves346 below · cited by 2 · depth 13 - Represented relative Pic⁰ is abelian; Abel–Jacobi dictionary
AlgebraicGeometry.RelPicard.exists_pic0_equiv_points_of_representsRelSubPic_of_abelJacobi291 below · cited by 12 · depth 13 - Relative Pic⁰ representable over a discrete valuation ring
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_finiteMapData_of_isDiscreteValuationRing684 below · cited by 5 · depth 13 - Relative Pic⁰ for curves degenerating to two glued smooth curves
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_smoothLocus_of_twoGluedSmoothCurveDegenerations618 below · cited by 3 · depth 13 - Base change of a relative Pic⁰ representation
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_baseChange0 below · cited by 25 · depth 13 - Split torus in Pic⁰ of a two-component curve
AlgebraicGeometry.RelPicard.exists_torus_characterLattice_equiv_of_twoGluedSmoothCurves32 below · cited by 1 · depth 13 - Degree-zero point twists are algebraically equivalent to zero
AlgebraicGeometry.RelPicard.isAlgEquivZero_foldr_ofPoint_of_sum_filter_eq_zero275 below · cited by 15 · depth 13 - Properness and geometric connectedness of Pic⁰ after base change to a field
AlgebraicGeometry.RelPicard.isProper_and_geometricallyConnected_baseChange_toBase_of_representsRelSubPic_of_field391 below · cited by 4 · depth 13 - Picard pullback along a curve isomorphism transports divisor classes
AlgebraicGeometry.RelPicard.pullbackHom_points_eq_pic0_congr_of_iso24 below · cited by 1 · depth 13 - Group law of the base-changed relative Pic⁰
AlgebraicGeometry.RelPicard.relativeGroupLaw_baseChange_eq2 below · cited by 13 · depth 13 - Norms preserve fibrewise algebraic triviality of line bundles
AlgebraicGeometry.RelPicard.FibrewiseAlgEquivZero.ofInvertible_normModule_curveChange63 below · cited by 10 · depth 14 - Trivialisation of an algebraically trivial bundle with a section
AlgebraicGeometry.RelPicard.IsAlgEquivZero.nonempty_iso_tensorUnit_of_ne_zero140 below · cited by 11 · depth 14 - Algebraic equivalence to zero from an 𝒪(P-ε) identification
AlgebraicGeometry.RelPicard.IsAlgEquivZero.of_iso_pointSubBasepoint20 below · cited by 1 · depth 14 - Multiplicative transformations induce homomorphisms of representing Picard schemes
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.comp_mul_eq_mul_comp_of_transform0 below · cited by 12 · depth 14 - Norm of the Poincaré bundle is fibrewise algebraically trivial
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.fibrewiseAlgEquivZero_ofInvertible_norm_pullback_poincare_of_twoGluedCurves_of_mem_of_ringKrullDim_le_one335 below · cited by 1 · depth 14 - Norm morphism of relative Pic⁰ and Abel–Jacobi classes
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.mul_comp_eq_of_classifies_rigidify_normModule_of_ofPoint75 below · cited by 4 · depth 14 - Restriction morphism classifies the re-rigidified pullback bundle
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_schemeHomOverComp_pullbackHom_iso_rigidify1 below · cited by 12 · depth 14 - Pullback of the Poincaré bundle along the group law tensors
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_pullbackAlong_mul_iso0 below · cited by 13 · depth 14 - Primitivity of the rigidified norm of the Poincaré bundle
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_pullbackAlong_mul_iso_tensor_ofInvertible_norm_pullback_poincare10 below · cited by 1 · depth 14 - Pullback of the Poincaré bundle along the unit point is trivial
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_pullbackAlong_one_iso0 below · cited by 5 · depth 14 - Norm of the pulled-back Poincaré bundle is trivial along the zero section
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_pullback_zeroSection_norm_pullback_poincare_iso_unit_of_mem_of_ringKrullDim_le_one73 below · cited by 2 · depth 14 - Base change of a represented relative Pic⁰: points, group law, Poincaré bundle
AlgebraicGeometry.RelPicard.baseChange_points_mul_poincare_compat1 below · cited by 3 · depth 14 - Chart sections after a finite étale extension of a discrete valuation ring
AlgebraicGeometry.RelPicard.exists_finite_etale_hasChartSections_of_finiteMapData133 below · cited by 2 · depth 14 - Constant dim_k check H¹(𝒪) on geometric fibres
AlgebraicGeometry.RelPicard.exists_forall_finrank_H1_unit_fibreAt_eq_of_finrank_H0_eq_one87 below · cited by 4 · depth 14 - Every point of Pic⁰(X) comes from admissible gluing data
AlgebraicGeometry.RelPicard.exists_hom_admissible_eq_of_twoGluedSmoothCurves19 below · cited by 1 · depth 14 - Admissible gluing data give points of Pic⁰
AlgebraicGeometry.RelPicard.exists_hom_admissible_of_twoGluedSmoothCurves334 below · cited by 1 · depth 14 - Finite sets of points of the relative Pic⁰ lie in affine opens
AlgebraicGeometry.RelPicard.exists_isAffineOpen_of_representsRelSubPic_algEquivZeroCut_of_finiteMapData512 below · cited by 3 · depth 14 - Pic⁰(F/k)≃ J(k) with Abel–Jacobi normalisation
AlgebraicGeometry.RelPicard.exists_pic0_equiv_points_abelJacobi_of_curveModel284 below · cited by 3 · depth 14 - Group law, Abel–Jacobi map and points of a represented relative Pic⁰
AlgebraicGeometry.RelPicard.exists_relativeGroupLaw_abelJacobi_of_representsRelSubPic291 below · cited by 1 · depth 14 - Representability of fibrewise Pic⁰ over a reduced Noetherian base
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_finiteMapData_of_isReduced487 below · cited by 4 · depth 14 - Representability of the Pic⁰ cut is Zariski-local on the base
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_forall_prime_exists_localizationAway16 below · cited by 3 · depth 14 - Representability of relative Pic⁰ under two-line degenerations
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_smoothLocus_of_twoLineDegenerations628 below · cited by 1 · depth 14 - Finite étale descent of a relative Pic⁰ representing scheme
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_of_finite_etale_descent_of_finiteMapData144 below · cited by 3 · depth 14 - Restriction morphisms on Pic⁰ for two transversally glued curves
AlgebraicGeometry.RelPicard.exists_restrictHom_pair_of_twoGluedSmoothCurves6 below · cited by 7 · depth 14 - Rigidified 𝒪_X(P)⊗𝒪_X(-Q) on a two-component curve is fibrewise algebraically trivial
AlgebraicGeometry.RelPicard.exists_rigidifiedLineBundle_ofPoint_tensor_ofPoint_fibrewiseAlgEquivZero_of_twoGluedSmoothCurves30 below · cited by 4 · depth 14 - Torus G_m^{s-1} closed-immerses as kernel of the restriction pair
AlgebraicGeometry.RelPicard.exists_torus_isClosedImmersion_ker_restrictPair_of_twoGluedSmoothCurves34 below · cited by 5 · depth 14 - Two-sided pool from a one-sided pool and a swapping automorphism
AlgebraicGeometry.RelPicard.exists_twoSidedPool_of_oneSided_of_iso0 below · cited by 4 · depth 14 - Čech h⁰(𝒪)=1 on a fibre with bijective structure map
AlgebraicGeometry.RelPicard.finrank_H0_unit_fibreAt_eq_one_of_bijective_algebraMap1 below · cited by 4 · depth 14 - Faithful flatness of restriction to two glued smooth curves
AlgebraicGeometry.RelPicard.flat_surjective_restrictPair_of_twoGluedSmoothCurves57 below · cited by 2 · depth 14 - Relative Pic⁰ over a basic open, two-component degenerations
AlgebraicGeometry.RelPicard.forall_prime_exists_representsRelSubPic_algEquivZeroCut_baseChange_away_of_smoothLocus_of_twoGluedSmoothCurveDegenerations614 below · cited by 1 · depth 14 - Injectivity of the glued Pic⁰ dictionary for two components
AlgebraicGeometry.RelPicard.gluedPic0_mk_eq_zero_of_hom_admissible_eq_one_of_twoGluedSmoothCurves12 below · cited by 1 · depth 14 - Algebraic equivalence to zero equals equality of Čech Euler characteristics
AlgebraicGeometry.RelPicard.isAlgEquivZero_iff_eulerChar_sectionsOf_eq257 below · cited by 22 · depth 14 - Properness and geometric connectedness of a representing Pic⁰
AlgebraicGeometry.RelPicard.isProper_and_geometricallyConnected_of_representsRelSubPic_algEquivZeroCut_of_finiteMapData332 below · cited by 3 · depth 14 - Change of curve and change of test scheme form a cartesian square
AlgebraicGeometry.RelPicard.isPullback_baseChangeSnd_curveChange1 below · cited by 5 · depth 14 - Triviality criterion on two transversally glued smooth proper curves
AlgebraicGeometry.RelPicard.nonempty_iso_unit_of_isAlgEquivZero_of_ne_zero_of_twoGluedSmoothCurves150 below · cited by 3 · depth 14 - Trace of the smooth locus on a two-component degenerate fibre
AlgebraicGeometry.RelPicard.preimage_smoothLocus_eq_compl_range_and_openImmersion_of_twoGluedSmoothCurves13 below · cited by 16 · depth 14 - Rigidity of Pic⁰-endomorphisms from ℚ̄-points
AlgebraicGeometry.RelPicard.schemeHomOver_ext_of_forall_algebraicClosure_point4 below · cited by 1 · depth 14 - Chart sections pin the genus of every geometric fibre
AlgebraicGeometry.RelPicard.HasChartSections.forall_geometricFibre_riemannRoch_imp_eq63 below · cited by 2 · depth 15 - Algebraic equivalence to zero ascends along field extensions
AlgebraicGeometry.RelPicard.IsAlgEquivZero.baseChange0 below · cited by 7 · depth 15 - Algebraic equivalence to zero preserves the two-chart Euler characteristic
AlgebraicGeometry.RelPicard.IsAlgEquivZero.eulerChar_sectionsOf_tensor_eq81 below · cited by 18 · depth 15 - Abel–Jacobi: 𝒪(sum Pᵢ - rε) is algebraically equivalent to zero
AlgebraicGeometry.RelPicard.IsAlgEquivZero.of_iso_pointsSubBasepoint20 below · cited by 4 · depth 15 - Determinant norm over a flat open locus preserves Pic⁰
AlgebraicGeometry.RelPicard.IsAlgEquivZero.pullback_ofInvertible_of_iso_normModule_morphismRestrict63 below · cited by 2 · depth 15 - Norm-classifying morphism of relative Pic⁰ is a homomorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.comp_mul_eq_mul_comp_of_classifies_rigidify_normModule76 below · cited by 2 · depth 15 - No p-power torsion among K-points of the representing scheme
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.eq_one_of_nsmul_pow_eq_one_of_forall_fibre_pow_torsionFree0 below · cited by 1 · depth 15 - Flatness of the universal multiplication on D×_R D
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.flat_mul_fst_snd0 below · cited by 2 · depth 15 - Fibrewise algebraic triviality of sum Pᵢ-d ε
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.fibrewiseAlgEquivZero_of_iso_pointsSubBasepointModule39 below · cited by 5 · depth 15 - Euler characteristic one for 𝒪(E-D) on a fibre
AlgebraicGeometry.RelPicard.eulerChar_fibre_lineBundle_tensor_idealModule_eq_one_of_supportedIn122 below · cited by 1 · depth 15 - Two-sided chart with vanishing H¹ and zeros inside U
AlgebraicGeometry.RelPicard.exists_chart_subsingleton_H1_and_support_subset_fibre_of_twoSidedBlocks_of_injective376 below · cited by 1 · depth 15 - Fibre of a base change is the fibre, compatibly
AlgebraicGeometry.RelPicard.exists_fibreIso_hom_comp_eq0 below · cited by 16 · depth 15 - Bundles of admissible gluing data on two glued smooth curves
AlgebraicGeometry.RelPicard.exists_gluedTwist_admissible_of_twoGluedSmoothCurves119 below · cited by 1 · depth 15 - Four structural inputs for relative Picard charts
AlgebraicGeometry.RelPicard.exists_isAffineOpen_and_isInvertible_sectionIdeal_and_isInvertible_pullbackAlong_and_sectionTwist_of_isOpenImmersion_of_supportedIn44 below · cited by 2 · depth 15 - Relative Pic⁰ is finite over a Proj
AlgebraicGeometry.RelPicard.exists_isFinite_proj_of_representsRelSubPic_algEquivZeroCut_of_finiteMapData509 below · cited by 1 · depth 15 - Local triviality of the Pic⁰ restriction pair as a torus bundle
AlgebraicGeometry.RelPicard.exists_iso_preimage_restrictPair_pullback_torus_of_section_of_twoGluedSmoothCurves0 below · cited by 1 · depth 15 - Open charts cover relative Pic⁰ over a reduced base
AlgebraicGeometry.RelPicard.exists_openCharts_relSubPicPresheaf_algEquivZeroCut_of_finiteMapData_of_isReduced461 below · cited by 1 · depth 15 - Polarised open charts of the relative Pic⁰ presheaf
AlgebraicGeometry.RelPicard.exists_openCharts_relSubPicPresheaf_algEquivZeroCut_of_polarisation_supportedIn_of_fibrewise_zeroScheme201 below · cited by 1 · depth 15 - Openness of the algebraic-equivalence locus for 𝒪(D-E_T)
AlgebraicGeometry.RelPicard.exists_opens_range_subset_iff_isAlgEquivZero_rigidify_lineBundle_baseChange_of_twoGluedSmoothCurveDegenerations379 below · cited by 1 · depth 15 - Local sections of the restriction pair for two glued smooth curves
AlgebraicGeometry.RelPicard.exists_opens_section_restrictPair_of_twoGluedSmoothCurves34 below · cited by 3 · depth 15 - Degree-g zero divisor of a section on a two-component degeneration
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_polarisedChartModule_fibre_of_support_subset_of_twoGluedSmoothCurveDegenerations286 below · cited by 1 · depth 15 - Section theorem: relative divisor attached to a fibrewise h⁰=1 bundle
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_lineBundle_iso_of_forall_fibre_of_supportedIn41 below · cited by 2 · depth 15 - Representability of Pic⁰ cut by open charts
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_openCharts_of_bijective_sections130 below · cited by 3 · depth 15 - Finite étale descent of the represented relative Pic⁰
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_of_finite_etale_descent_of_bijective_sections_of_forall_orbit50 below · cited by 2 · depth 15 - Block general position for the twist 𝒪(E_Ω)
AlgebraicGeometry.RelPicard.exists_split_injective_forall_subsingleton_H1_lineBundle_and_support_subset_of_twoSidedBlocks_of_bijective_sections366 below · cited by 2 · depth 15 - Node-unit torus as kernel of restriction on Pic⁰
AlgebraicGeometry.RelPicard.exists_torus_ker_restrictPair_of_twoGluedSmoothCurves30 below · cited by 1 · depth 15 - Fibre Čech dimensions computed on a residue-field affine chart
AlgebraicGeometry.RelPicard.exists_twoAffineOpenCover_fibre_finrank_eq_finrank_cechDiff_baseChange_residueField12 below · cited by 11 · depth 15 - Two-sided chart data: sections and chart divisors on C_A
AlgebraicGeometry.RelPicard.exists_twoSidedChartData13 below · cited by 1 · depth 15 - h⁰=1 on a fibre from vanishing H¹ and Euler characteristic one
AlgebraicGeometry.RelPicard.finrank_H0_fibre_eq_one_of_subsingleton_H1_of_supportedIn_lineBundle108 below · cited by 1 · depth 15 - Descent orbits on relative Pic⁰ lie in affine opens
AlgebraicGeometry.RelPicard.forall_exists_isAffineOpen_forall_act_mem_of_twoSidedBlocks_of_isInvertible25 below · cited by 1 · depth 15 - Relative Pic⁰ on a basic open, two-line degenerations
AlgebraicGeometry.RelPicard.forall_prime_exists_representsRelSubPic_algEquivZeroCut_baseChange_away_of_smoothLocus_of_twoLineDegenerations627 below · cited by 1 · depth 15 - Geometric connectedness of a scheme representing the Pic⁰ cut
AlgebraicGeometry.RelPicard.geometricallyConnected_of_representsRelSubPic_algEquivZeroCut6 below · cited by 5 · depth 15 - Algebraic equivalence to zero detected on two glued smooth curves
AlgebraicGeometry.RelPicard.isAlgEquivZero_of_isAlgEquivZero_pullback_curveChange_of_twoGluedSmoothCurves32 below · cited by 7 · depth 15 - Trivial on both glued components implies algebraically equivalent to zero
AlgebraicGeometry.RelPicard.isAlgEquivZero_of_pullback_curveChange_iso_unit_of_twoGluedSmoothCurves18 below · cited by 6 · depth 15 - 𝒪(P-ε) is algebraically equivalent to zero
AlgebraicGeometry.RelPicard.isAlgEquivZero_pointSubBasepoint19 below · cited by 2 · depth 15 - Surjective finite-presentation half of Pic⁰ for two-glued-curve degenerations
AlgebraicGeometry.RelPicard.isLFPSurj_relSubPicPresheaf_algEquivZeroCut_baseChange_of_twoGluedSmoothCurveDegenerations398 below · cited by 1 · depth 15 - Fibrewise h¹=0, h⁰=n gives locally free direct image
AlgebraicGeometry.RelPicard.isLocallyFreeOfRank_pushforward_of_forall_fibre_of_twoAffineOpenCover89 below · cited by 3 · depth 15 - Separatedness of a scheme representing relative Pic⁰
AlgebraicGeometry.RelPicard.isSeparated_of_representsRelSubPic_algEquivZeroCut_of_finiteMapData272 below · cited by 1 · depth 15 - Pic⁰ sheaf condition for finite flat base change, via finite-map data
AlgebraicGeometry.RelPicard.isSheafFor_relSubPicPresheaf_algEquivZeroCut_finiteEtale_of_finiteMapData132 below · cited by 1 · depth 15 - Zariski sheaf property of the fibrewise Pic⁰ presheaf
AlgebraicGeometry.RelPicard.isSheaf_relSubPicPresheaf_algEquivZeroCut_zariski_of_bijective_sections13 below · cited by 5 · depth 15 - Zariski sheaf property of the relative Pic⁰ presheaf under finite-map data
AlgebraicGeometry.RelPicard.isSheaf_relSubPicPresheaf_algEquivZeroCut_zariski_of_finiteMapData107 below · cited by 2 · depth 15 - Pic⁰-representing scheme is locally of finite type
AlgebraicGeometry.RelPicard.locallyOfFiniteType_of_representsRelSubPic_algEquivZeroCut_of_finiteMapData292 below · cited by 1 · depth 15 - Divisor of r points minus rε splits as a tensor product
AlgebraicGeometry.RelPicard.nonempty_invModule_prodKerGraph_tensor_module_pow_iso_pointsSubBasepointModule9 below · cited by 2 · depth 15 - Reading the Poincaré bundle at every degree-zero class
AlgebraicGeometry.RelPicard.nonempty_poincare_pullbackAlong_iso_foldr_ofPoint_of_additive_of_pinned32 below · cited by 3 · depth 15 - Flatness and surjectivity of [n] on a relative Pic⁰ over ℤ
AlgebraicGeometry.RelPicard.nsmul_flat_surjective_locallyQuasiFinite_of_representsRelSubPic_of_locallyQuasiFinite_primePow40 below · cited by 1 · depth 15 - Counit nonzero on every fibre when h¹=0, h⁰=1
AlgebraicGeometry.RelPicard.pullback_map_counit_app_ne_zero_of_forall_fibre_of_twoAffineOpenCover96 below · cited by 3 · depth 15 - Néron extension of an endomorphism preserves the group law
AlgebraicGeometry.RelPicard.schemeHomOverComp_relativeGroupLaw_mul_endExtensionEquiv_symm3 below · cited by 3 · depth 15 - Rigidity over ℤ_{(ℓ)}: agreement on ℚ̄-points suffices
AlgebraicGeometry.RelPicard.schemeHomOver_ext_of_forall_algebraicClosure_point_of_isReduced0 below · cited by 1 · depth 15 - Smoothness of a scheme representing relative Pic⁰
AlgebraicGeometry.RelPicard.smooth_of_representsRelSubPic_algEquivZeroCut_of_finiteMapData44 below · cited by 1 · depth 15 - Surjectivity of the degree-g Abel–Jacobi morphism
AlgebraicGeometry.RelPicard.surjective_of_poincare_pullbackAlong_iso_twistModule284 below · cited by 1 · depth 15 - Surjectivity of the structure morphism of a relative Pic⁰ representing scheme
AlgebraicGeometry.RelPicard.surjective_toBase_of_representsRelSubPic_algEquivZeroCut0 below · cited by 3 · depth 15 - Base change of the two-glued-curve degeneration condition
AlgebraicGeometry.RelPicard.twoGluedSmoothCurveDegenerations_baseChange1 below · cited by 3 · depth 15 - Transporting Pic⁰ representing schemes along a curve isomorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_inverse_pair_of_iso_of_sections5 below · cited by 6 · depth 16 - Prime-to-p torsion for K-points of the Pic⁰ representing scheme
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_nsmul_eq_one_not_dvd_of_forall_fibre_exists_pow_eq_one0 below · cited by 1 · depth 16 - Classify-compatible θ is a homomorphism of relative group laws
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.postComp_mul_of_classify_rel0 below · cited by 4 · depth 16 - Norm morphism sends dual-number points over the origin to the origin
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.schemeHomOverComp_eq_one_of_dualNumber_of_classifies_rigidify_normModule_of_finrank_eq_char53 below · cited by 1 · depth 16 - Gluing rigidified line bundles along an open cover
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_gluing_openCover_of_bijective_sections10 below · cited by 1 · depth 16 - Rigidified line bundles lift along square-zero base extensions
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_of_squareZero_of_twoAffineOpenCover43 below · cited by 2 · depth 16 - Fibrewise algebraic triviality descends along open covers of the base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.fibrewiseAlgEquivZero_of_pullbackAlong_openCover0 below · cited by 1 · depth 16 - Gluing isomorphisms of rigidified line bundles along an open cover
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_of_pullbackAlong_openCover_of_bijective_sections7 below · cited by 6 · depth 16 - Euler characteristic one for 𝒪(rε)⊗𝒪(-D) on geometric fibres
AlgebraicGeometry.RelPicard.eulerChar_fibre_sectionTwist_tensor_idealModule_eq_one235 below · cited by 1 · depth 16 - Euler characteristic one for 𝒪(rε)⊗ I_D on a fibre
AlgebraicGeometry.RelPicard.eulerChar_fibre_sectionTwist_tensor_idealModule_eq_one_of_supportedIn122 below · cited by 1 · depth 16 - Euler characteristic of L(rε-D) on a fibre component
AlgebraicGeometry.RelPicard.eulerChar_pullback_fibreModule_tensor_sectionTwist_tensor_idealModule_eq242 below · cited by 4 · depth 16 - Milne charts cover Pic⁰: some chart kills H¹
AlgebraicGeometry.RelPicard.exists_chart_subsingleton_H1_fibre308 below · cited by 1 · depth 16 - Chart divisors from a split pool of sections cover Pic⁰
AlgebraicGeometry.RelPicard.exists_chart_subsingleton_H1_fibre_of_blocks_of_injective413 below · cited by 1 · depth 16 - Fibrewise Euler characteristic is well defined and locally constant
AlgebraicGeometry.RelPicard.exists_fibre_eulerChar_eq_and_isClopen_setOf_fibre_eulerChar_eq87 below · cited by 2 · depth 16 - A tensor power of the theta bundle is finite by sections
AlgebraicGeometry.RelPicard.exists_finiteBySections_tensorPow_thetaBundle_of_isAlgClosed470 below · cited by 1 · depth 16 - Block general position for a split pool on geometric fibres
AlgebraicGeometry.RelPicard.exists_injective_forall_subsingleton_H1_of_blocks_of_pool_of_bijective_sections406 below · cited by 1 · depth 16 - Extending an algebraically trivial bundle across a glued component
AlgebraicGeometry.RelPicard.exists_isAlgEquivZero_pullback_curveChange_iso_of_isAlgEquivZero_of_twoGluedSmoothCurves31 below · cited by 1 · depth 16 - Relative openness of the Pic⁰ locus over a degeneration locus
AlgebraicGeometry.RelPicard.exists_isOpen_inter_preimage_eq_setOf_isAlgEquivZero_fibre_of_smoothLocus_of_twoGluedSmoothCurveDegenerations353 below · cited by 2 · depth 16 - A polarised open chart for the relative Pic⁰ subfunctor
AlgebraicGeometry.RelPicard.exists_openChart_openImmersion_relSubPicPresheaf_algEquivZeroCut_of_polarisation_of_fibrewise_zeroScheme166 below · cited by 1 · depth 16 - An open chart of relative Pic⁰ from one divisor
AlgebraicGeometry.RelPicard.exists_openChart_relSubPicPresheaf_algEquivZeroCut_of_relEffCartierDiv369 below · cited by 1 · depth 16 - Milne charts for relative Pic⁰ inside the smooth locus
AlgebraicGeometry.RelPicard.exists_openCharts_relSubPicPresheaf_algEquivZeroCut_of_relEffCartierDiv_supportedIn_of_fibrewise_zeroScheme167 below · cited by 2 · depth 16 - Openness of the algebraically-trivial locus for twisted divisor bundles
AlgebraicGeometry.RelPicard.exists_opens_range_subset_iff_isAlgEquivZero_twistModule_baseChange_of_twoLineDegenerations430 below · cited by 1 · depth 16 - Fibrewise zero schemes as degree-g divisors inside U
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_chartModule_fibre372 below · cited by 1 · depth 16 - Degree-g divisors from sections over non-smooth geometric fibres
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_polarisedChartModule_fibre_of_support_subset_of_twoGluedSmoothCurveDegenerations_of_not_smooth282 below · cited by 1 · depth 16 - Degree-g zero divisors of theta-chart sections over smooth fibres
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_polarisedChartModule_fibre_of_support_subset_of_twoGluedSmoothCurveDegenerations_of_smooth263 below · cited by 1 · depth 16 - Representability of the relative Pic⁰ cut from theta-chart data
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_chartData200 below · cited by 1 · depth 16 - Two-sided block general position on the geometric fibres of a degenerating curve
AlgebraicGeometry.RelPicard.exists_split_injective_forall_subsingleton_H1_and_support_subset_of_twoSidedBlocks_of_bijective_sections358 below · cited by 1 · depth 16 - Fibrewise Čech H⁰-rank and H¹-vanishing under field extension
AlgebraicGeometry.RelPicard.exists_twoAffineOpenCover_fibre_finrank_H0_eq_and_subsingleton_H1_iff13 below · cited by 9 · depth 16 - Two-chart Čech cohomology transports along any cartesian fibre presentation
AlgebraicGeometry.RelPicard.exists_twoAffineOpenCover_fibre_linearEquiv_sectionsOf_of_isPullback1 below · cited by 12 · depth 16 - Fibrewise h⁰=1 from vanishing h¹ for the twisted bundle
AlgebraicGeometry.RelPicard.finrank_H0_fibre_eq_one_of_subsingleton_H1_of_supportedIn108 below · cited by 1 · depth 16 - Descent orbits of relative Pic⁰ points lie in affine charts
AlgebraicGeometry.RelPicard.forall_exists_isAffineOpen_forall_act_mem_of_blocks_of_isInvertible17 below · cited by 1 · depth 16 - Two-chart Čech cohomology of a fibre module is base-change invariant
AlgebraicGeometry.RelPicard.forall_exists_twoAffineOpenCover_linearEquiv_sectionsOf_fibreModule1 below · cited by 4 · depth 16 - Point-independence of algebraic equivalence to zero, two-curve degenerations
AlgebraicGeometry.RelPicard.isAlgEquivZero_fibre_of_range_subset_singleton_of_twoGluedSmoothCurveDegenerations297 below · cited by 2 · depth 16 - Closedness of the trivial-fibre locus for a rigidified Pic⁰-family
AlgebraicGeometry.RelPicard.isClosed_setOf_exists_fibreModule_iso_unit271 below · cited by 1 · depth 16 - Invertibility of the section ideal for a section through a smooth open
AlgebraicGeometry.RelPicard.isInvertible_sectionIdeal_of_range_subset2 below · cited by 29 · depth 16 - Invertibility of the theta bundle of a relative curve
AlgebraicGeometry.RelPicard.isInvertible_thetaBundle48 below · cited by 2 · depth 16 - Cohomology and base change for a proper flat family
AlgebraicGeometry.RelPicard.isIso_baseChangeHom_pushforward_of_forall_fibre_of_twoAffineOpenCover94 below · cited by 1 · depth 16 - Injectivity half of local finite presentation for Pic⁰
AlgebraicGeometry.RelPicard.isLFPInj_relSubPicPresheaf_algEquivZeroCut28 below · cited by 1 · depth 16 - Injectivity at affine limits for the Pic⁰ subpresheaf
AlgebraicGeometry.RelPicard.isLFPInj_relSubPicPresheaf_algEquivZeroCut_of_twoAffineOpenCover28 below · cited by 2 · depth 16 - Classes in relative Pic⁰ descend to f.g. subalgebras
AlgebraicGeometry.RelPicard.isLFPSurj_relSubPicPresheaf_algEquivZeroCut290 below · cited by 2 · depth 16 - LFP-surjectivity of the base-changed Pic⁰ presheaf
AlgebraicGeometry.RelPicard.isLFPSurj_relSubPicPresheaf_algEquivZeroCut_baseChange_of_twoLineDegenerations440 below · cited by 1 · depth 16 - Limit surjectivity for the Pic⁰ cut, given openness and point-independence
AlgebraicGeometry.RelPicard.isLFPSurj_relSubPicPresheaf_algEquivZeroCut_of_isOpen_setOf_isAlgEquivZero42 below · cited by 3 · depth 16 - Fibrewise h¹=0, h⁰=n gives locally free pushforward
AlgebraicGeometry.RelPicard.isLocallyFreeOfRank_pushforward_of_forall_fibre_of_finiteType_of_twoAffineOpenCover81 below · cited by 1 · depth 16 - Principal glued data give node-unit modules on two glued curves
AlgebraicGeometry.RelPicard.isNodeUnitModule_foldr_ofPoint_tensor_foldr_ofPoint_of_forall_eq_ord_of_twoGluedSmoothCurves117 below · cited by 1 · depth 16 - Openness of the fibrewise algebraic-equivalence locus, two-strata form
AlgebraicGeometry.RelPicard.isOpen_setOf_isAlgEquivZero_fibre_of_twoStrata80 below · cited by 4 · depth 16 - Separatedness of a scheme representing the Pic⁰ cut
AlgebraicGeometry.RelPicard.isSeparated_of_representsRelSubPic_algEquivZeroCut_of_bijective_sections82 below · cited by 1 · depth 16 - Finite faithfully flat descent for the rigidified Pic⁰ presheaf
AlgebraicGeometry.RelPicard.isSheafFor_relSubPicPresheaf_algEquivZeroCut_finite_faithfullyFlat_of_bijective_sections38 below · cited by 2 · depth 16 - Fibres of the Abel–Jacobi family over a k-point
AlgebraicGeometry.RelPicard.nonempty_ajFamily_fibre_iso18 below · cited by 1 · depth 16 - Trivialising L⊗𝒪(rε)⊗𝒪(-D) off the two supports
AlgebraicGeometry.RelPicard.nonempty_pullback_fibreModule_tensor_sectionTwist_tensor_idealModule_iso_of_supportedIn_of_disjoint37 below · cited by 3 · depth 16 - Base change of the theta bundle along R → R'
AlgebraicGeometry.RelPicard.nonempty_pullback_fst_thetaBundle_iso_baseChange76 below · cited by 1 · depth 16 - Section twists commute with base change along ψ
AlgebraicGeometry.RelPicard.nonempty_pullback_sectionTwist_iso_of_range_subset17 below · cited by 13 · depth 16 - Base change stability of fibrewise containment in the section component
AlgebraicGeometry.RelPicard.preimage_range_subset_connectedComponentIn_fibre_baseChange1 below · cited by 1 · depth 16 - Transport of the off-component block condition under base change
AlgebraicGeometry.RelPicard.preimage_range_subset_diff_connectedComponentIn_fibre_baseChange_of_not_smooth1 below · cited by 1 · depth 16 - Graph chart divisors lie on the ε-component of non-smooth fibres
AlgebraicGeometry.RelPicard.preimage_support_prodKerGraph_subset_connectedComponentIn_of_blocks0 below · cited by 2 · depth 16 - Uniqueness of divisors in the smooth locus representing M
AlgebraicGeometry.RelPicard.relEffCartierDiv_I_eq_of_lineBundle_iso_tensor_pullback_of_supportedIn30 below · cited by 1 · depth 16 - Smoothness of a representing scheme for the Pic⁰ cut
AlgebraicGeometry.RelPicard.smooth_of_representsRelSubPic_algEquivZeroCut_of_twoAffineOpenCover44 below · cited by 1 · depth 16 - Fibrewise H¹=0 and h⁰=r+1-g for the twisted Poincaré bundle
AlgebraicGeometry.RelPicard.subsingleton_H1_and_finrank_H0_fibre_poincare_tensor_sectionTwist261 below · cited by 2 · depth 16 - Vanishing of fibre two-chart H¹ descends along extensions of the residue field
AlgebraicGeometry.RelPicard.subsingleton_H1_fibre_of_subsingleton_H1_fibre_extension14 below · cited by 2 · depth 16 - Descent of fibrewise zero-locus containment along a field extension
AlgebraicGeometry.RelPicard.support_zeroSchemeIdeal_fibre_subset_of_support_zeroSchemeIdeal_fibre_subset_extension14 below · cited by 1 · depth 16 - Algebraically trivial bundles pulled back to a rational curve model
AlgebraicGeometry.RelPicard.IsAlgEquivZero.nonempty_pullback_iso_pullback_unit_and_eulerChar_eq_one_of_curveModel_ratFunc262 below · cited by 5 · depth 17 - Algebraically trivial line bundles pull back trivially to genus-zero curves
AlgebraicGeometry.RelPicard.IsAlgEquivZero.nonempty_pullback_iso_tensorUnit_of_finrank_H1_eq_zero261 below · cited by 4 · depth 17 - Descent of rigidified line bundles along finite flat base change
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_descent_finite_faithfullyFlat_of_bijective_sections35 below · cited by 1 · depth 17 - Normalising an isomorphism to respect the rigidifications
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_iso_map_pullback_rigSection_comp_eq0 below · cited by 7 · depth 17 - Fibrewise algebraic triviality descends along finite faithfully flat base change
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.fibrewiseAlgEquivZero_of_pullback_finite_faithfullyFlat0 below · cited by 1 · depth 17 - Rigidity of isomorphisms of rigidified line bundles
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.iso_eq_of_map_pullback_rigSection_comp_eq3 below · cited by 5 · depth 17 - Descent of rigidified line bundles along finite faithfully flat base change: uniqueness
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_of_pullback_finite_faithfullyFlat_of_bijective_sections9 below · cited by 2 · depth 17 - Rigidification lifts along a square-zero thickening of the base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_unit_pullback_rigSection_of_squareZero_of_pullback_iso11 below · cited by 1 · depth 17 - Rigidity: at most one rigidification-compatible isomorphism
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.subsingleton_iso_map_pullback_rigSection_comp_eq4 below · cited by 3 · depth 17 - Isomorphic pointed curves: Abel–Jacobi maps agree up to translation
AlgebraicGeometry.RelPicard.abelJacobi_comp_eq_mul_abelJacobi_of_iso_of_classify23 below · cited by 1 · depth 17 - Chart killing Čech H¹ on a non-smooth geometric fibre
AlgebraicGeometry.RelPicard.exists_chart_subsingleton_H1_fibre_of_blocks_of_not_smooth363 below · cited by 1 · depth 17 - A chart divisor killing check H¹ on a smooth geometric fibre
AlgebraicGeometry.RelPicard.exists_forall_subsingleton_H1_sectionsOf_fibreModule_chartModule_of_smooth325 below · cited by 1 · depth 17 - Block general position: prescribed h⁰ and vanishing Čech H¹
AlgebraicGeometry.RelPicard.exists_injective_forall_finrank_H0_add_eq_and_subsingleton_H1_of_blocks_of_isAlgEquivZero_of_lt_card309 below · cited by 3 · depth 17 - Block general position on a smooth geometric fibre
AlgebraicGeometry.RelPicard.exists_injective_forall_subsingleton_H1_of_blocks_of_smooth_fibre306 below · cited by 2 · depth 17 - Block general position at a two-line degenerate geometric fibre
AlgebraicGeometry.RelPicard.exists_injective_forall_subsingleton_H1_of_blocks_of_twoLineDegeneration_of_sectionInSmoothLocus364 below · cited by 1 · depth 17 - Openness of the Pic⁰ locus along the degeneration locus
AlgebraicGeometry.RelPicard.exists_isOpen_inter_preimage_eq_setOf_isAlgEquivZero_fibre_of_smoothLocus_of_twoLineDegenerations400 below · cited by 2 · depth 17 - Open chart of the relative Pic⁰ from a universal divisor
AlgebraicGeometry.RelPicard.exists_openChart_openImmersion_relSubPicPresheaf_algEquivZeroCut_of_fibrewise_zeroScheme130 below · cited by 1 · depth 17 - Openness of the fibrewise check H¹-vanishing locus
AlgebraicGeometry.RelPicard.exists_opens_range_subset_iff_forall_subsingleton_H1_fibre91 below · cited by 1 · depth 17 - Open locus where fibrewise Čech H¹ vanishes, tested on field points
AlgebraicGeometry.RelPicard.exists_opens_range_subset_iff_forall_subsingleton_H1_fibre_of_twoAffineOpenCover90 below · cited by 2 · depth 17 - Open locus of bases whose fibre sections vanish inside U
AlgebraicGeometry.RelPicard.exists_opens_range_subset_iff_forall_support_zeroSchemeIdeal_subset_of_forall_fibre110 below · cited by 1 · depth 17 - Zero schemes on non-smooth two-line geometric fibres
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_chartModule_fibre_of_not_smooth_of_isReduced369 below · cited by 1 · depth 17 - Degree-g divisors cutting out sections on smooth geometric fibres
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_chartModule_fibre_of_smooth267 below · cited by 1 · depth 17 - Polarised chart divisor over the H¹-vanishing open locus
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_supportedIn_rigidify_iso_of_subsingleton_H1_of_support_subset36 below · cited by 1 · depth 17 - Divisor chart where fibrewise H¹ of L(rε-D_γ) vanishes
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_twistModule_iso_of_subsingleton_H1325 below · cited by 1 · depth 17 - Two-sided block general position at a two-component degenerate fibre
AlgebraicGeometry.RelPicard.exists_split_injective_forall_subsingleton_H1_and_support_subset_of_twoSidedBlocks_of_twoGluedSmoothCurveDegeneration345 below · cited by 1 · depth 17 - Nonzero theta section with trivial stabiliser on J(k)
AlgebraicGeometry.RelPicard.exists_thetaSection_ne_zero_and_stabilizer_trivial390 below · cited by 1 · depth 17 - Field-extension invariance of two-chart Čech dimensions on fibres
AlgebraicGeometry.RelPicard.exists_twoAffineOpenCover_fibre_finrank_H0_eq_and_finrank_H1_eq12 below · cited by 1 · depth 17 - Geometric fibres of smooth proper curves have h⁰(𝒪)=1
AlgebraicGeometry.RelPicard.finrank_H0_unit_fibre_eq_one180 below · cited by 4 · depth 17 - Čech h¹ bound for algebraically trivial bundles on glued curves
AlgebraicGeometry.RelPicard.finrank_H1_le_finrank_H1_unit_add_one_of_isAlgEquivZero_of_twoGluedSmoothCurves270 below · cited by 1 · depth 17 - Independence of algebraic equivalence to zero along smooth fibres
AlgebraicGeometry.RelPicard.isAlgEquivZero_fibre_of_range_subset_singleton_of_smooth269 below · cited by 2 · depth 17 - Point-independence of algebraic triviality of fibres under two-line degeneration
AlgebraicGeometry.RelPicard.isAlgEquivZero_fibre_of_range_subset_singleton_of_twoLineDegenerations335 below · cited by 2 · depth 17 - h¹-test for algebraic equivalence to zero on two glued curves
AlgebraicGeometry.RelPicard.isAlgEquivZero_of_finrank_H1_tensorPow_add_two_le_of_twoGluedSmoothCurves329 below · cited by 1 · depth 17 - Algebraic equivalence of two fibres of an invertible family
AlgebraicGeometry.RelPicard.isAlgEquivZero_tensor_of_pullback_baseChangeSnd_iso_of_tensor_iso_unit3 below · cited by 1 · depth 17 - Closedness of the trivial locus of a rigidified family
AlgebraicGeometry.RelPicard.isClosed_setOf_exists_fibreModule_iso_unit_of_flat81 below · cited by 1 · depth 17 - Closedness of the locus where fibrewise h¹ is at least n
AlgebraicGeometry.RelPicard.isClosed_setOf_forall_fibre_le_finrank_H1_of_twoAffineOpenCover97 below · cited by 1 · depth 17 - Restricted divisors rε and D keep degrees r and e
AlgebraicGeometry.RelPicard.isFinite_and_finrank_subscheme_comap_sectionIdeal_pow_and_comap_I12 below · cited by 1 · depth 17 - Genus-zero geometric fibres force JtoSpec k finite
AlgebraicGeometry.RelPicard.isFinite_toBase_of_geometricFibre_genus_eq_zero283 below · cited by 1 · depth 17 - Invertibility of restricted section and divisor ideals on a curve model
AlgebraicGeometry.RelPicard.isInvertible_comap_sectionIdeal_pow_and_comap_I_of_isOpenImmersion13 below · cited by 1 · depth 17 - Affine-limit injectivity of the rigidified relative Picard presheaf
AlgebraicGeometry.RelPicard.isLFPInj_relPicardPresheaf27 below · cited by 2 · depth 17 - Surjectivity along affine limits for the rigidified relative Picard presheaf
AlgebraicGeometry.RelPicard.isLFPSurj_relPicardPresheaf38 below · cited by 2 · depth 17 - Direct image of a fibrewise acyclic invertible module, locally free
AlgebraicGeometry.RelPicard.isLocallyFreeOfRank_pushforward_of_forall_fibre37 below · cited by 7 · depth 17 - Principal divisor on one component gives a node-unit module
AlgebraicGeometry.RelPicard.isNodeUnitModule_foldr_ofPoint_of_forall_eq_ord_of_hasValue116 below · cited by 1 · depth 17 - Openness of the algebraic-equivalence-to-zero locus on the base
AlgebraicGeometry.RelPicard.isOpen_setOf_isAlgEquivZero_fibre260 below · cited by 1 · depth 17 - Norm along a degree-p cover trivialises first-order deformations
AlgebraicGeometry.RelPicard.nonempty_normModule_curveChange_dualNumber_iso_unit_of_finrank_eq_char_of_forall_isClosed_eq50 below · cited by 1 · depth 17 - Base change of the twisting module 𝒪(rε_T) along T'→ T
AlgebraicGeometry.RelPicard.nonempty_pullback_sectionTwist_iso15 below · cited by 17 · depth 17 - Base change of the theta bundle along ψ: T'→ T
AlgebraicGeometry.RelPicard.nonempty_pullback_thetaBundle_iso72 below · cited by 2 · depth 17 - Theta bundles commute with base change along κ
AlgebraicGeometry.RelPicard.nonempty_thetaBundle_baseChange_iso_thetaBundle_toR16 below · cited by 1 · depth 17 - Theorem of the square for the theta bundle on J
AlgebraicGeometry.RelPicard.nonempty_translate_thetaBundle_tensor_iso352 below · cited by 1 · depth 17 - Factorisation through W and uniqueness of the chart divisor
AlgebraicGeometry.RelPicard.relEffCartierDiv_eq_pullbackAlong_of_rigidify_iso_of_supportedIn_of_support_subset36 below · cited by 1 · depth 17 - Uniqueness: a divisor in the chart is φ^*D
AlgebraicGeometry.RelPicard.relEffCartierDiv_eq_pullbackAlong_of_twistModule_iso153 below · cited by 1 · depth 17 - Geometric fibres of the twisted Poincaré bundle: H¹=0, h⁰=r+1-g
AlgebraicGeometry.RelPicard.subsingleton_H1_and_finrank_H0_fibre_poincare_tensor_sectionTwist_of_isAlgClosed258 below · cited by 1 · depth 17 - Fibre cohomology of a rigidified bundle twisted by rε
AlgebraicGeometry.RelPicard.subsingleton_H1_and_finrank_H0_fibre_tensor_sectionTwist_of_fibrewiseAlgEquivZero_of_isAlgClosed258 below · cited by 2 · depth 17 - Rigidified line bundles on C_A descend to a finitely generated subalgebra
AlgebraicGeometry.RelPicard.LFP.exists_fg_nonempty_iso_pullbackAlong37 below · cited by 1 · depth 18 - Rigidity of isomorphisms of rigidified line bundles
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.iso_eq_of_map_pullback_rigSection_comp_eq_of_surjective1 below · cited by 1 · depth 18 - Over a field-valued point the rigidification recovers L
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_ofInvertible_L_iso_of_field5 below · cited by 5 · depth 18 - Euler characteristic g+1 on the first line of a degenerate fibre
AlgebraicGeometry.RelPicard.eulerChar_pullback_firstLine_sectionTwist_tensor_idealModule_eq259 below · cited by 2 · depth 18 - Euler characteristic g+1 on the section component of a two-line fibre
AlgebraicGeometry.RelPicard.eulerChar_pullback_tensor_invModule_pow_ker_tensor_module_prod_ker_eq_of_twoLineDegeneration263 below · cited by 1 · depth 18 - Dual-number deformation of chartwise bases of f_*𝒪
AlgebraicGeometry.RelPicard.exists_forall_basis_pushforward_dualNumberThickening_of_forall_basis4 below · cited by 2 · depth 18 - Near-side general position at a two-component degenerate fibre
AlgebraicGeometry.RelPicard.exists_injective_forall_finrank_H0_eq_zero_and_subsingleton_H1_restrict_fst_of_nearBlocks_of_twoGluedSmoothCurveDegeneration315 below · cited by 1 · depth 18 - Far blocks giving check H¹=0 and h⁰=1 on C₂
AlgebraicGeometry.RelPicard.exists_injective_forall_subsingleton_H1_and_finrank_H0_restrict_snd_of_farBlocks_of_twoGluedSmoothCurveDegeneration318 below · cited by 1 · depth 18 - Block transversals killing H¹ of L(rp-sum vⱼ)
AlgebraicGeometry.RelPicard.exists_injective_forall_subsingleton_H1_sectionsOf_tensor_of_isAlgEquivZero_of_lt_card305 below · cited by 1 · depth 18 - Normalised frames for an invertible sheaf with trivial reduction
AlgebraicGeometry.RelPicard.exists_isFrameOn_and_map_eq_oneAddEpsMul_smul_of_nonempty_pullback_iso_unit28 below · cited by 2 · depth 18 - Norm of a 1+ε g cocycle is 1+varepsilonTr(g)
AlgebraicGeometry.RelPicard.exists_isFrameOn_normModule_and_map_eq_oneAddEpsMul_trace_smul13 below · cited by 2 · depth 18 - Theta dictionary: Pic⁰(F) and k-points of J
AlgebraicGeometry.RelPicard.exists_pic0_equiv_points_nontrivial_H0_iff_ell_pos286 below · cited by 1 · depth 18 - A theta section cutting out the theta locus on J
AlgebraicGeometry.RelPicard.exists_pullbackSection_thetaBundle_poincare_eq_zero_iff326 below · cited by 1 · depth 18 - Fibrewise h⁰=1, h¹=0 forces M≅𝒪(D)otimespr₂^*N
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_lineBundle_iso_of_forall_fibre299 below · cited by 1 · depth 18 - Existence of D=D₀+D_γ trivialising the twist of L
AlgebraicGeometry.RelPicard.exists_relEffCartierDiv_supportedIn_twistModule_iso_of_subsingleton_H1_of_zeroScheme35 below · cited by 1 · depth 18 - Existence of the Jacobian over an algebraically closed field
AlgebraicGeometry.RelPicard.exists_representsRelSubPic_algEquivZeroCut_of_isAlgClosed525 below · cited by 8 · depth 18 - Vanishing of h¹ forces h⁰=1 on fibres of the twisted bundle
AlgebraicGeometry.RelPicard.finrank_H0_fibre_eq_one_of_subsingleton_H1105 below · cited by 2 · depth 18 - Euler characteristic 1 on fibres forces h¹(𝒪)=g
AlgebraicGeometry.RelPicard.finrank_H1_unit_fibre_eq_of_eulerChar_chart241 below · cited by 1 · depth 18 - Closedness of the locus where fibrewise h⁰ is at least n
AlgebraicGeometry.RelPicard.isClosed_setOf_forall_fibre_le_finrank_H0_of_twoAffineOpenCover86 below · cited by 1 · depth 18 - Invertibility and base change of section twists of relative curves
AlgebraicGeometry.RelPicard.isInvertible_and_nonempty_pullback_iso_foldr_sectionTwist_tensor_of_range_subset21 below · cited by 3 · depth 18 - Fibrewise criterion for local freeness of π_*F over a finite-type base
AlgebraicGeometry.RelPicard.isLocallyFreeOfRank_pushforward_of_forall_fibre_of_finiteType29 below · cited by 1 · depth 18 - Openness of the locus of fibrewise vanishing H¹
AlgebraicGeometry.RelPicard.isOpen_setOf_forall_fibre_subsingleton_H190 below · cited by 1 · depth 18 - Openness of the fibrewise Čech H¹-vanishing locus
AlgebraicGeometry.RelPicard.isOpen_setOf_forall_fibre_subsingleton_H1_of_twoAffineOpenCover89 below · cited by 1 · depth 18 - Triviality of rigidified genus-zero bundles algebraically equivalent to zero
AlgebraicGeometry.RelPicard.nonempty_iso_tensorUnit_of_fibrewiseAlgEquivZero_of_genus_eq_zero281 below · cited by 1 · depth 18 - Triviality on a rational model descends from Ω to k
AlgebraicGeometry.RelPicard.nonempty_pullback_iso_pullback_unit_of_isAlgEquivZero_baseChange_of_curveModel_ratFunc308 below · cited by 1 · depth 18 - Degree-zero base change for a fibrewise acyclic invertible sheaf
AlgebraicGeometry.RelPicard.nonempty_pushforward_pullback_iso_of_forall_fibre42 below · cited by 1 · depth 18 - Theorem of the square for relative theta bundles
AlgebraicGeometry.RelPicard.nonempty_thetaBundle_tensor_pullbackAlong_tensor_iso_of_fibrewiseAlgEquivZero335 below · cited by 1 · depth 18 - Base change compatibility of the section twist 𝒪(rε)
AlgebraicGeometry.RelPicard.nonempty_transport_sectionTwist_baseChange_iso14 below · cited by 1 · depth 18 - Uniqueness of D with 𝒪(D)≅ Motimespr₂^*N
AlgebraicGeometry.RelPicard.relEffCartierDiv_I_eq_of_lineBundle_iso_tensor_pullback_of_forall_fibre70 below · cited by 2 · depth 18 - Uniqueness half of the check H¹-vanishing divisor chart
AlgebraicGeometry.RelPicard.relEffCartierDiv_eq_pullbackAlong_of_twistModule_iso_of_supportedIn_of_zeroScheme33 below · cited by 1 · depth 18 - Vanishing H¹ and h⁰=1 on a two-line fibre
AlgebraicGeometry.RelPicard.subsingleton_H1_and_finrank_H0_fibre_of_twoGluedProjectiveLines357 below · cited by 1 · depth 18 - Fibrewise H¹=0 and h⁰=r+1-g over any field
AlgebraicGeometry.RelPicard.subsingleton_H1_and_finrank_H0_fibre_tensor_sectionTwist_of_fibrewiseAlgEquivZero261 below · cited by 5 · depth 18 - Fibre H¹ vanishing from one residue-field chart
AlgebraicGeometry.RelPicard.subsingleton_H1_fibre_of_subsingleton_H1_residueField_chart23 below · cited by 4 · depth 18 - Line bundle algebraically equivalent to zero is 𝒪(sum Pᵢ-dε)
AlgebraicGeometry.RelPicard.IsAlgEquivZero.exists_iso_pointsSubBasepointModule247 below · cited by 1 · depth 19 - Triviality of line bundles on Spec C[ε] detected by reduction
AlgebraicGeometry.RelPicard.Scheme.Modules.IsInvertible.nonempty_iso_unit_of_dualNumber_of_reduction11 below · cited by 4 · depth 19 - Section ideal restricted away from the section is everything
AlgebraicGeometry.RelPicard.comap_sectionIdeal_eq_top_and_finrank_eq_zero_of_forall_notMem_support0 below · cited by 1 · depth 19 - Euler characteristic of section twists on a fibre subscheme
AlgebraicGeometry.RelPicard.eulerChar_sectionsOf_pullback_foldr_sectionTwist_tensor_eq_add_sum108 below · cited by 1 · depth 19 - Unique splitting u=σ^sharp a+ε σ^sharp b over an affine open
AlgebraicGeometry.RelPicard.existsUnique_eq_appLE_add_eps_mul_appLE_of_isAffineOpen2 below · cited by 3 · depth 19 - Fibre over a rational point identifies with C, compatibly with theta twists
AlgebraicGeometry.RelPicard.exists_iso_fibre_pullback_fibreModule_tensor_sectionTwist_iso14 below · cited by 1 · depth 19 - Pic⁰ of the function field as k-points of J
AlgebraicGeometry.RelPicard.exists_pic0_equiv_points_nontrivial_H0_pullback_iff_ell_pos284 below · cited by 1 · depth 19 - Theta section cutting out the locus h⁰(M(dε))≠ 0
AlgebraicGeometry.RelPicard.exists_pullbackSection_thetaBundle_eq_zero_iff119 below · cited by 1 · depth 19 - Section ideal on a fibre component: invertible of degree one
AlgebraicGeometry.RelPicard.isInvertible_comap_sectionIdeal_and_finrank_eq_one_of_ideal_eq_bot4 below · cited by 1 · depth 19 - Transitivity of base change: C×_R T'=(C×_R T)×_T T'
AlgebraicGeometry.RelPicard.isPullback_baseChangeSnd0 below · cited by 7 · depth 19 - Base change of I(ε_T)^r along 1×ψ
AlgebraicGeometry.RelPicard.nonempty_pullback_sectionIdeal_pow_module_iso_of_smoothLocus12 below · cited by 1 · depth 19 - Theta bundle of a translate by sum Pᵢ-dε
AlgebraicGeometry.RelPicard.nonempty_thetaBundle_tensor_pointsSubBasepoint_tensor_foldr_pullback_iso318 below · cited by 1 · depth 19 - Fibrewise non-vanishing of the counit for an invertible module
AlgebraicGeometry.RelPicard.pullback_map_counit_app_ne_zero_of_forall_fibre44 below · cited by 1 · depth 19 - Pullback of a curve twist along a constant section
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_pullback_rigSection_pullbackAlong_iso_tensorUnit1 below · cited by 2 · depth 20 - Affine base change of sections along a stage morphism
AlgebraicGeometry.RelPicard.exists_algEquiv_sections_stage_baseChange_of_isAffineOpen1 below · cited by 1 · depth 20 - Base change for the direct image of an invertible module
AlgebraicGeometry.RelPicard.isIso_baseChangeHom_pushforward_of_forall_fibre42 below · cited by 2 · depth 20 - Local freeness and trivial determinant along the n-th section thickening
AlgebraicGeometry.RelPicard.isLocallyFreeOfRank_pushforward_thickening_sectionTwist_and_nonempty_det_iso75 below · cited by 1 · depth 20 - Theta bundles depend only on the underlying module
AlgebraicGeometry.RelPicard.nonempty_thetaBundle_iso_of_iso0 below · cited by 1 · depth 20 - Theta bundle twisted by 𝒪(P-ε)
AlgebraicGeometry.RelPicard.nonempty_thetaBundle_tensor_pointSubBasepoint_tensor_pullback_iso304 below · cited by 1 · depth 20 - Fibrewise Čech conditions transfer along an isomorphism of modules
AlgebraicGeometry.RelPicard.forall_subsingleton_H1_and_finrank_H0_fibreModule_of_iso1 below · cited by 1 · depth 21 - Determinant of the direct image along a section
AlgebraicGeometry.RelPicard.nonempty_det_pushforward_iso_det_pushforward_tensor_idealOfSection_tensor_pullback85 below · cited by 1 · depth 21 - Section ideal powers commute with base change
AlgebraicGeometry.RelPicard.nonempty_pullback_sectionIdeal_pow_module_iso11 below · cited by 1 · depth 21 - Pushforward of a thickening sequence along a relative curve
AlgebraicGeometry.RelPicard.shortExact_map_pushforward_thickening74 below · cited by 1 · depth 21 - Classifying morphism of a tensor product is a product
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.classify_tensor1 below · cited by 2 · depth 22 - Endomorphism of Pic⁰ induces unique additive endomorphism of J
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.existsUnique_addMonoidHom_pts_comp_fst_eq_comp_of_mul_comp_of_baseChangeIso1 below · cited by 1 · depth 22 - Semilinear group endomorphism induces a unique additive endomorphism of J
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.existsUnique_addMonoidHom_pts_comp_fst_eq_comp_of_semilinear_mul_comp_of_baseChangeIso1 below · cited by 1 · depth 22 - Group endomorphism of Pic⁰ induces unique additive endomorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.existsUnique_addMonoidHom_pts_eq_comp_of_mul_comp1 below · cited by 2 · depth 22 - A classifying point for 𝒪(v₁-v₂) over a local base
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_schemeHomOver_poincare_pullbackAlong_iso_ofPoint_lineBundle_tensor_ofPoint_idealModule_of_isLocalRing280 below · cited by 6 · depth 22 - Degree-zero point twists give B-points of the relative Pic⁰
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_schemeHomOver_poincare_pullbackAlong_iso_rigidify_pullback_foldr_ofPoint_of_sum_eq_zero281 below · cited by 1 · depth 22 - Poincaré bundle at a field-valued point over a B-point
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_postComp_iso_pullback_of_rigidify_of_field12 below · cited by 1 · depth 22 - Restricted Poincaré bundle at a reduced point as pullback of N
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_postComp_pullbackHom_iso_pullback_of_rigidify12 below · cited by 1 · depth 22 - Picard transport along a curve automorphism is a homomorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.schemeHomOverComp_mul_eq_mul_of_forall_postComp_classify_eq6 below · cited by 2 · depth 22 - Poincaré pullbacks of point twists read divisor classes
AlgebraicGeometry.RelPicard.addEquiv_apply_eq_pic0Mk_of_nonempty_poincare_pullbackAlong_iso_foldr_ofPoint55 below · cited by 1 · depth 22 - Point sequence 0→π_*(F⊗ Iₚ)→π_*F→ p^*F→ 0
AlgebraicGeometry.RelPicard.exists_shortExact_pushforward_tensor_idealOfSection_of_forall_fibre57 below · cited by 1 · depth 22 - Abel–Jacobi normalisation descends to the base-changed Poincaré datum
AlgebraicGeometry.RelPicard.nonempty_poincare_pullbackAlong_iso_lineBundle_tensor_idealModule_of_abelJacobi_baseChange18 below · cited by 1 · depth 22 - Composing semilinear Picard transport with an automorphism transport
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.classifies_rigidify_pullback_map_comp_of_classifies_pullback_curveChange_inv_of_classifies_rigidify_pullback_map11 below · cited by 1 · depth 23 - Mod p base change of the tangent–cusp form dictionary
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_dualNumber_kernel_equiv_addMonoidHom_intLattice_baseChange_of_surjective_of_ker_eq_span53 below · cited by 1 · depth 23 - Restricting a finite flat correspondence endomorphism along ν
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_hom_classifies_normModule_pullback_and_schemeHomOverComp_eq_of_comp_eq91 below · cited by 1 · depth 23 - Semilinear transport of rigidified Pic⁰ along a base automorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_hom_classifies_rigidify_pullback_map_of_comp_eq_comp2 below · cited by 4 · depth 23 - Norm characterisation of an endomorphism survives base change
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_iso_rigidify_normModule_baseChange_of_forall58 below · cited by 1 · depth 23 - Compatibility of ν with the Picard transports of W and α
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.postComp_eq_postComp_of_rigidify_pullback_curveChange_of_transport_of_hom_comp_eq5 below · cited by 1 · depth 23 - Transport along W commutes with base-change projection on points
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.postComp_transport_comp_fst_eq_comp_transport_of_baseChangeIso10 below · cited by 2 · depth 23 - Automorphism fixing components and crossings acts trivially on gluing torus
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.postComp_transport_eq_self_of_ker_restrictPair_of_iso_comp_eq_of_crossing_of_twoGluedSmoothCurves31 below · cited by 2 · depth 23 - Norm-of-pullback endomorphism of relative Pic⁰ is a homomorphism
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.schemeHomOverComp_mul_eq_mul_and_zeroSection_comp_of_classifies_normModule_pullback76 below · cited by 1 · depth 23 - Pull-back commutes with Picard transport along compatible automorphisms
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.schemeHomOverComp_pullbackHom_eq_of_transport_of_hom_comp_eq4 below · cited by 1 · depth 23 - Chart sections for a pointed curve over a field after a finite étale extension
AlgebraicGeometry.RelPicard.exists_finite_etale_hasChartSections_of_field135 below · cited by 1 · depth 23 - Deformation class on dual-number kernel points: additive bijection, natural in A
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.deformationClass_kerPoints_bijective_additive_natural40 below · cited by 2 · depth 24 - Correspondence-induced endomorphism of the representing object of relative Pic⁰
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_hom_classifies_rigidify_normModule_pullback_curveChange66 below · cited by 1 · depth 24 - Norm homomorphism between representatives of relative Pic⁰
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.exists_normHom_abelJacobi108 below · cited by 1 · depth 24 - Base change of the dual-number kernel of Pic⁰
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.kerPoints_baseChange_surjective_and_fibre42 below · cited by 1 · depth 24 - Dual-number kernel points: closure under the group law
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.kerPoints_mul_mem_and_comp_mem0 below · cited by 3 · depth 24 - Existence of the deformation-class map into Čech H¹
AlgebraicGeometry.RelPicard.exists_isDeformationClassMap3 below · cited by 5 · depth 24 - Norm–pull-back endomorphism acts by trace on Čech H¹
AlgebraicGeometry.RelPicard.IsDeformationClassMap.exists_cechH1ToH1_germ_eq_traceAlong_of_classifies_normModule_pullback_of_mono129 below · cited by 1 · depth 25 - Injectivity of the deformation-class map
AlgebraicGeometry.RelPicard.IsDeformationClassMap.injective30 below · cited by 2 · depth 25 - Additivity of the deformation class map under tensor product
AlgebraicGeometry.RelPicard.IsDeformationClassMap.map_mul29 below · cited by 2 · depth 25 - Naturality of the deformation-class map in the coefficient algebra
AlgebraicGeometry.RelPicard.IsDeformationClassMap.natural29 below · cited by 2 · depth 25 - Surjectivity of the deformation-class map δ
AlgebraicGeometry.RelPicard.IsDeformationClassMap.surjective27 below · cited by 2 · depth 25 - Dual-number kernel points classify rigidified bundles trivial modulo ε
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.kerPointsToRigKer_bijective0 below · cited by 2 · depth 25 - Multiplicativity of the dual-number kernel-point map
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.kerPointsToRigKer_mul0 below · cited by 2 · depth 25 - Poincaré bundle at aⁿ is the n-th tensor power
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_pow_iso_tensorPow2 below · cited by 1 · depth 25 - Base-change compatibility of the deformation class map
AlgebraicGeometry.RelPicard.IsDeformationClassMap.apply_baseTransport_eq_H1baseChangeMap30 below · cited by 1 · depth 26 - Tangent action of a norm-pull-back endomorphism over a field
AlgebraicGeometry.RelPicard.IsDeformationClassMap.exists_cechH1ToH1_germ_eq_traceAlong_of_classifies_normModule_pullback_of_field74 below · cited by 1 · depth 26 - Moduli description of an endomorphism transported to the base change
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.nonempty_poincare_pullbackAlong_iso_rigidify_normModule_baseChange58 below · cited by 1 · depth 26 - Unique lift of a unit-reducing dual-number point under base change
AlgebraicGeometry.RelPicard.RigKerDualNumber.existsUnique_kerPoint_baseChange_comp_fst_eq2 below · cited by 1 · depth 26 - Dual-number deformations admit frames with transition 1+ε f
AlgebraicGeometry.RelPicard.RigKerDualNumber.exists_isFrameOn_and_map_eq_oneAddEpsMul_smul28 below · cited by 6 · depth 26 - Base transport commutes with dual-number points of Pic⁰
AlgebraicGeometry.RelPicard.RigKerDualNumber.kerPointsToRigKer_baseTransport0 below · cited by 1 · depth 26 - First-order deformations of the trivial bundle and Čech H¹
AlgebraicGeometry.RelPicard.exists_trivialModDeformations_map_H1_tensor_natural40 below · cited by 1 · depth 26 - Fibrewise algebraic equivalence to zero over a discrete valuation ring
AlgebraicGeometry.RelPicard.fibrewiseAlgEquivZero_of_isAlgEquivZero_pullback_closedFibre_of_pullbackAlong_iso_tensorPow_poincare3 below · cited by 1 · depth 26 - Euler characteristic test for Pic⁰ on two glued curves
AlgebraicGeometry.RelPicard.isAlgEquivZero_of_eulerChar_sectionsOf_pullback_eq_of_twoGluedSmoothCurves282 below · cited by 1 · depth 26 - Tensor powers preserve algebraic equivalence to zero
AlgebraicGeometry.RelPicard.IsAlgEquivZero.tensorPow0 below · cited by 1 · depth 27 - Cover independence of the répartition class of a deformation
AlgebraicGeometry.RelPicard.IsDeformationClassMap.cechH1ToH1_germ_eq_of_two_covers36 below · cited by 1 · depth 27 - Two frame cocycles on two covers are cohomologous after cross refinement
AlgebraicGeometry.RelPicard.exists_crossSections_of_isFrameOn_of_map_eq_oneAddEpsMul_smul3 below · cited by 1 · depth 27 - Naturality of 1+ε t for thickening-compatible morphisms
AlgebraicGeometry.RelPicard.map01_oneAddEpsMul0 below · cited by 1 · depth 27 - Cross sections comparing two two-chart deformation representatives
AlgebraicGeometry.RelPicard.IsDeformationClassMap.exists_crossSections31 below · cited by 1 · depth 28 - Seesaw triviality of a rigidified line bundle
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_unit_of_forall_pullbackAlong_point100 below · cited by 5 · depth 28 - Triviality locus of a rigidified line bundle is closed
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_locallyOfFinitePresentation_forall_iff_of_locallyOfFinitePresentation218 below · cited by 3 · depth 32 - See-saw locus on an affine chart over a general base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_lfp_chart_iff_affineOpens_of_locallyOfFinitePresentation216 below · cited by 1 · depth 33 - Gluing chart-local triviality loci, with local finite presentation
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_lfp_iff_of_forall_affineOpens_chart59 below · cited by 1 · depth 33 - Base-changed Pic⁰ datum carries the base-changed group law
AlgebraicGeometry.RelPicard.RepresentsRelSubPic.relativeGroupLaw_mul_eq_baseChange_mul_of_nonempty_poincare_iso_ofR1 below · cited by 1 · depth 34 - Closed triviality locus of a rigidified line bundle
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_forall_nonempty_pullbackAlong_iso_unit_iff_of_isNoetherianRing160 below · cited by 1 · depth 34 - Gluing chart-local triviality loci of a rigidified line bundle
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_iff_of_forall_affineOpens_chart58 below · cited by 2 · depth 34 - Descent of the triviality locus to an affine chart
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_lfp_chart_iff_of_descent0 below · cited by 1 · depth 34 - Noetherian descent of abelian scheme, affine chart and rigidified bundle
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_noetherian_descent_affineOpens144 below · cited by 1 · depth 34 - Triviality locus of a rigidified bundle on an affine chart
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isClosedImmersion_chart_iff_affineOpens_of_isNoetherianRing158 below · cited by 1 · depth 35 - Noetherian descent of a rigidified bundle over an affine chart
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_noetherian_descent_affineOpens_of_isPullback_of_isPullback44 below · cited by 1 · depth 35 - Theorem of the cube over an arbitrary affine base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_unit_of_locIsoOnBase_unit_faces363 below · cited by 2 · depth 35 - Descent of rigidified line bundles along affine faithfully flat base change
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_descent_of_isAffineHom_of_flat_of_surjective_of_bijective_sections35 below · cited by 1 · depth 36 - Noetherian descent for a rigidified bundle with trivial faces
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isNoetherianRing_model_of_locIsoOnBase_unit_faces219 below · cited by 1 · depth 36 - Isomorphism of rigidified line bundles descends along affine faithfully flat base change
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_of_pullback_of_isAffineHom_of_flat_of_surjective_of_bijective_sections9 below · cited by 1 · depth 36 - Triviality of a rigidified line bundle is Zariski-local on the base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_unit_iff_forall_openCover58 below · cited by 1 · depth 36 - Theorem of the cube, rigidified form, noetherian base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_iso_unit_of_locIsoOnBase_unit_faces_of_isNoetherianRing301 below · cited by 1 · depth 36 - Cube datum descends to a finitely generated subring
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_fg_subalgebra_isPullback_prodStr_model211 below · cited by 1 · depth 37 - Descending the cube faces to a noetherian base
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_isNoetherianRing_model_of_fg_subalgebra_model_of_locIsoOnBase_unit_faces35 below · cited by 1 · depth 37 - Triviality of a rigidified bundle lifts along small extensions
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_pullbackAlong_iso_unit_of_smallExtension_of_locIsoOnBase_faces135 below · cited by 1 · depth 37 - Triviality of e^*M from local triviality of the third face
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.nonempty_pullbackAlong_one_iso_unit_of_locIsoOnBase_face63 below · cited by 1 · depth 37 - Rigidified cube bundle pulls back along a cartesian base change
AlgebraicGeometry.RelPicard.RigidifiedLineBundle.exists_rigidified_cube_pullback_of_isPullback0 below · cited by 1 · depth 38
AlgebraicGeometry.RelTangentPoints 4
- Naturality of translation to the unit in the parameter
AlgebraicGeometry.RelTangentPoints.comp_translate_eq_translate_comp0 below · cited by 6 · depth 32 - Relative tangent points factor through open neighbourhoods
AlgebraicGeometry.RelTangentPoints.existsUnique_comp_openInclusion_eq0 below · cited by 5 · depth 32 - Infinitesimal rigidity: a tangent field constant at one thickened point
AlgebraicGeometry.RelTangentPoints.eq_comp_zeroSection_of_thickenedPoint_comp_eq2 below · cited by 2 · depth 34 - Constant parametrised tangent vectors when Γ(Z₀,𝒪)=k
AlgebraicGeometry.RelTangentPoints.const_bijective_of_bijective_of_compactSpace1 below · cited by 1 · depth 35
AlgebraicGeometry.RiemannForm 63
- Closedness of the stabiliser K(L) on an abelian scheme
AlgebraicGeometry.RiemannForm.exists_isClosed_mem_iff_nonempty_pullback_translation_iso119 below · cited by 3 · depth 34 - Translation by a point: two spellings agree
AlgebraicGeometry.RiemannForm.translation_eq_translate_toUnitPt_and_translate_eq_translation_ofUnitPt0 below · cited by 4 · depth 34 - Non-degeneracy of the level-n Riemann pairing
AlgebraicGeometry.RiemannForm.eq_zero_of_forall_isLevelPairingValue_one_of_forall_nonempty_iso_imp65 below · cited by 1 · depth 35 - Elements of the theta group scalar at 1 are trivial
AlgebraicGeometry.RiemannForm.thetaGroup.eq_one_of_isScalarElt_one0 below · cited by 3 · depth 35 - Theta-group elements over the origin act by unique scalars
AlgebraicGeometry.RiemannForm.thetaGroup.existsUnique_isScalarElt_and_isScalarElt_mul16 below · cited by 6 · depth 35 - Theta-group commutator as a level pairing value
AlgebraicGeometry.RiemannForm.thetaGroup.isLevelPairingValue_of_isScalarElt_commutator_of_iso_tpow_tensor_tpow_negMor622 below · cited by 1 · depth 35 - Endomorphisms of an invertible module are unique constant scalars
AlgebraicGeometry.RiemannForm.existsUnique_isConstScalar15 below · cited by 4 · depth 36 - Right kernel of the level-n pairing is A[n]∩ K(L)
AlgebraicGeometry.RiemannForm.forall_isLevelPairingValue_one_iff_nonempty_pullback_translation_iso64 below · cited by 6 · depth 36 - Pullback along [n] of T_Q^*L for n-torsion Q
AlgebraicGeometry.RiemannForm.nonempty_pullback_schemeNsmul_pullback_translation_iso592 below · cited by 2 · depth 36 - Composition of multiplication morphisms and translations
AlgebraicGeometry.RiemannForm.schemeNsmul_mul_and_translation_comp_schemeNsmul0 below · cited by 3 · depth 36 - Pullback of theta-group elements along a homomorphism of group laws
AlgebraicGeometry.RiemannForm.thetaGroup.exists_monoidHom_pullback_pt_eq_and_isScalarElt0 below · cited by 1 · depth 36 - Tensor multiplicativity of theta groups
AlgebraicGeometry.RiemannForm.thetaGroup.exists_monoidHom_tensor_pt_eq_and_isScalarElt_mul3 below · cited by 2 · depth 36 - Theta group homomorphism into tensor powers, raising scalars to the nth power
AlgebraicGeometry.RiemannForm.thetaGroup.exists_monoidHom_tpow_pt_eq_and_isScalarElt_pow5 below · cited by 1 · depth 36 - Theta groups of isomorphic modules are isomorphic
AlgebraicGeometry.RiemannForm.thetaGroup.exists_mulEquiv_pt_eq_and_isScalarElt_iff_of_iso0 below · cited by 2 · depth 36 - Commutator with the level lift computes the level pairing
AlgebraicGeometry.RiemannForm.thetaGroup.isLevelPairingValue_of_isScalarElt_commutatorElement_levelLift2 below · cited by 3 · depth 36 - Kernel of the theta group projection is central
AlgebraicGeometry.RiemannForm.thetaGroup.ker_pt_le_center_and_commutatorElement_mem_ker17 below · cited by 5 · depth 36 - Translation by an m-torsion point followed by [m]
AlgebraicGeometry.RiemannForm.translation_comp_schemeNsmul_of_nsmul_eq_zero0 below · cited by 10 · depth 36 - ⋆-adjointness of a Riemann form under Rosati compatibility
AlgebraicGeometry.RiemannForm.apply_apply_eq_of_rosatiCompatible_of_involutive611 below · cited by 2 · depth 37 - Perfect Riemann form: no ℓ-power torsion in K(L)
AlgebraicGeometry.RiemannForm.eq_zero_of_isPerfPair_of_smul_eq_zero_of_nonempty_pullback_translation_iso755 below · cited by 1 · depth 37 - Existence and uniqueness of the ℓ-adic Riemann form
AlgebraicGeometry.RiemannForm.existsUnique_isRiemannForm617 below · cited by 2 · depth 37 - Stabiliser of a module under translations is a subgroup
AlgebraicGeometry.RiemannForm.exists_addSubgroup_mem_iff_nonempty_pullback_translation_iso1 below · cited by 4 · depth 37 - Translation-invariant isomorphism along [n] descends
AlgebraicGeometry.RiemannForm.exists_iso_pullback_schemeNsmul_mapIso_eq_of_forall_transportIso_eq53 below · cited by 1 · depth 37 - Divisible Riemann form forces LcongM^{⊗ℓ}otimesN
AlgebraicGeometry.RiemannForm.exists_iso_tensorPow_tensor_of_forall_dvd_of_two_ne_zero852 below · cited by 1 · depth 37 - Constant scalar multiplications are preserved by pullback over the base
AlgebraicGeometry.RiemannForm.isConstScalar_pullback_map0 below · cited by 5 · depth 37 - Constant scalars multiply under tensor product of module maps
AlgebraicGeometry.RiemannForm.isConstScalar_tensorHom0 below · cited by 4 · depth 37 - Skew-symmetry and vanishing on the diagonal of a Riemann form
AlgebraicGeometry.RiemannForm.isRiemannForm_swap_eq_neg_and_self_eq_zero27 below · cited by 3 · depth 37 - n-torsion translation: (T_Q^*LotimesL^∨)^{⊗ n} is trivial
AlgebraicGeometry.RiemannForm.nonempty_tensorPow_pullback_translation_tensor_dual_iso_unit_monoidalV2585 below · cited by 1 · depth 37 - Rosati adjointness of ι(b^⋆) for the Riemann form
AlgebraicGeometry.RiemannForm.apply_apply_eq_of_rosatiCompatible610 below · cited by 1 · depth 38 - Riemann form vanishes iff all translates of L are isomorphic
AlgebraicGeometry.RiemannForm.eq_zero_iff_forall_nonempty_pullback_translation_iso791 below · cited by 1 · depth 38 - Nonvanishing and u^g-divisibility of the Euler characteristic
AlgebraicGeometry.RiemannForm.eulerChar_ne_zero_and_pow_dvd_eulerChar_of_isRiemannForm_smul_of_isPerfPair1,007 below · cited by 1 · depth 38 - Well-definedness of the level-n pairing value
AlgebraicGeometry.RiemannForm.existsUnique_isLevelPairingValue602 below · cited by 7 · depth 38 - Existence of the ℓ-adic Riemann form of an invertible module
AlgebraicGeometry.RiemannForm.exists_isRiemannForm615 below · cited by 1 · depth 38 - Descent of the Mumford bundle along 1×[ℓ]
AlgebraicGeometry.RiemannForm.exists_pullback_oneProdNsmul_iso_mumfordBundle_of_forall_torsion147 below · cited by 1 · depth 38 - Level-ℓ pairing is trivial when ℓ divides the Riemann form
AlgebraicGeometry.RiemannForm.isLevelPairingValue_one_of_forall_dvd697 below · cited by 1 · depth 38 - Additivity of the Riemann form in the line bundle
AlgebraicGeometry.RiemannForm.isRiemannForm_add_of_iso_tensor5 below · cited by 1 · depth 38 - The zero form is a Riemann form of mathcal O_A
AlgebraicGeometry.RiemannForm.isRiemannForm_tensorUnit_zero1 below · cited by 1 · depth 38 - Uniqueness of the ℓ-adic Riemann form of a line bundle
AlgebraicGeometry.RiemannForm.isRiemannForm_unique603 below · cited by 2 · depth 38 - Translation-invariance makes e₀ a morphism of descent data
AlgebraicGeometry.RiemannForm.map_comp_descentDataHom_eq_of_forall_transportIso_inv_comp_map_eq48 below · cited by 1 · depth 38 - Skew-symmetry of the level-n pairing values
AlgebraicGeometry.RiemannForm.mul_eq_one_of_isLevelPairingValue_of_isLevelPairingValue_swap26 below · cited by 1 · depth 38 - Flip-symmetry of a bi-rigidified invertible sheaf on A× A
AlgebraicGeometry.RiemannForm.nonempty_pullback_pullbackSymmetry_iso_of_pullback_oneProdNsmul_iso_mumfordBundle718 below · cited by 1 · depth 38 - Perfectness of the Riemann form on T_ℓ A
AlgebraicGeometry.RiemannForm.perfect_of_forall_torsion_kernelPts_eq_zero741 below · cited by 1 · depth 38 - Translations of a relative group law: T₀=1_A and T_{P+Q}=T_P T_Q
AlgebraicGeometry.RiemannForm.translation_zero_and_translation_add0 below · cited by 3 · depth 38 - Bi-rigidified descent of the Mumford bundle along 1×[ℓ]
AlgebraicGeometry.RiemannForm.exists_birigidified_pullback_oneProdNsmul_iso_mumfordBundle_of_pullback_oneProdNsmul_iso_mumfordBundle10 below · cited by 1 · depth 39 - Descent of a ℓ-torsion-invariant rigidified bundle along 1×[ℓ]
AlgebraicGeometry.RiemannForm.exists_isInvertible_pullback_oneProdNsmul_iso_of_forall_torsion_pullback_oneProdTranslation_iso134 below · cited by 1 · depth 39 - Finiteness of K(M)(k) from its ℓ-power torsion
AlgebraicGeometry.RiemannForm.finite_setOf_nonempty_pullback_translation_iso_of_finite_torsion736 below · cited by 1 · depth 39 - ℓ-power torsion suffices for translation invariance of L
AlgebraicGeometry.RiemannForm.forall_nonempty_pullback_translation_iso_of_forall_torsion735 below · cited by 1 · depth 39 - Bimultiplicativity of level-n pairing values on torsion points
AlgebraicGeometry.RiemannForm.isLevelPairingValue_add_left_and_add_right26 below · cited by 2 · depth 39 - Level compatibility of Riemann pairing values
AlgebraicGeometry.RiemannForm.isLevelPairingValue_mul_of_isLevelPairingValue_nsmul2 below · cited by 1 · depth 39 - Multiplicativity of the level pairing in the line bundle
AlgebraicGeometry.RiemannForm.isLevelPairingValue_mul_of_iso_tensor4 below · cited by 1 · depth 39 - Functoriality of the level pairing along an endomorphism
AlgebraicGeometry.RiemannForm.isLevelPairingValue_of_isLevelPairingValue_pushPt_of_iso_pullback6 below · cited by 1 · depth 39 - Invariance of the Mumford bundle under 1× T_Q
AlgebraicGeometry.RiemannForm.nonempty_pullback_oneProdTranslation_mumfordBundle_iso_of_nonempty_pullback_translation_iso1 below · cited by 1 · depth 39 - Stabiliser of a line bundle with scalar Riemann form
AlgebraicGeometry.RiemannForm.nonempty_pullback_translation_iso_iff_pow_valuation_smul_eq_zero_of_isRiemannForm_smul755 below · cited by 1 · depth 39 - Rosati compatibility sliced at a k-point
AlgebraicGeometry.RiemannForm.nonempty_pullback_translation_pushPt_tensor_dual_iso_of_rosatiCompatible3 below · cited by 1 · depth 39 - Level-n pairing values are n-th roots of unity
AlgebraicGeometry.RiemannForm.pow_eq_one_of_isLevelPairingValue613 below · cited by 1 · depth 39 - Translation fixing a morphism from a non-empty scheme is trivial
AlgebraicGeometry.RiemannForm.eq_zero_of_comp_translation_eq_of_nonempty1 below · cited by 2 · depth 40 - Cocycle of translation isomorphisms for a rigidified bundle
AlgebraicGeometry.RiemannForm.exists_cocycle_oneProdTranslation_of_rigidified_of_forall_torsion_nonempty_iso58 below · cited by 1 · depth 40 - Constant scalars are preserved by whiskering
AlgebraicGeometry.RiemannForm.isConstScalar_whiskerRight_and_whiskerLeft_monoidalV21 below · cited by 1 · depth 40 - Monoidality of the transport isomorphism along g ∘ T = g
AlgebraicGeometry.RiemannForm.transportIso_tensorObj2 below · cited by 1 · depth 40 - Translation cocycle over A[n] yields descent datum along [n]
AlgebraicGeometry.RiemannForm.exists_descentData_schemeNsmul_obj_eq_of_forall_torsion_iso_pullback_translation54 below · cited by 1 · depth 41 - Level subgroups of the theta group give translation cocycles
AlgebraicGeometry.RiemannForm.thetaGroup.exists_iso_pullback_translation_of_injOn_pt_of_range_pt_eq_torsion0 below · cited by 1 · depth 41 - Every non-zero scalar is realised in the theta group
AlgebraicGeometry.RiemannForm.thetaGroup.exists_pt_eq_one_and_isScalarElt0 below · cited by 1 · depth 41 - Lifts of 2-torsion commute in G(mathcal L₀^{⊗ 2})
AlgebraicGeometry.RiemannForm.thetaGroup.mul_comm_of_two_torsion_of_forall_two_torsion_pullback_translation_iso27 below · cited by 1 · depth 41 - Tensor-square homomorphism of theta groups doubles scalars
AlgebraicGeometry.RiemannForm.thetaGroup.exists_monoidHom_tensor_self_pt_eq_and_isScalarElt_mul_self3 below · cited by 1 · depth 42
AlgebraicGeometry.Scheme 936
- Flag of Hopf quotients yields flag of fppf subsheaves
AlgebraicGeometry.Scheme.exists_fppfSubsheafFlag_of_bialgHomFlag0 below · cited by 1 · depth 12 - Small transport of the fppf Kummer row
AlgebraicGeometry.Scheme.exists_shrink_fppfKummerRow_of_epi_zsmul3 below · cited by 2 · depth 12 - Surjection of Hopf algebras gives a monomorphism of points sheaves
AlgebraicGeometry.Scheme.mono_of_sectionsEquiv_precomp_surjective0 below · cited by 2 · depth 12 - Degree-zero fppf cohomology inherits smallness from global sections
AlgebraicGeometry.Scheme.small_fppfCohomology_zero_of_small_sections0 below · cited by 2 · depth 12 - Rank at a flat point equals the function-field degree
AlgebraicGeometry.Scheme.Hom.finrank_eq_finrank_functionField_of_flat_morphismRestrict2 below · cited by 3 · depth 13 - Rank of a finite morphism is unchanged by restriction to an open
AlgebraicGeometry.Scheme.Hom.finrank_morphismRestrict_eq_finrank0 below · cited by 9 · depth 13 - Rank of a finite flat morphism is stable under base change
AlgebraicGeometry.Scheme.Hom.finrank_pullbackMap_of_comp_eq1 below · cited by 26 · depth 13 - Section of a smooth relative curve is an effective Cartier divisor
AlgebraicGeometry.Scheme.Hom.isInvertible_ker_of_comp_eq_id0 below · cited by 107 · depth 13 - Pullback of 𝒪(± nZ) along an isomorphism
AlgebraicGeometry.Scheme.Hom.nonempty_pullback_ker_pow_invModule_iso_of_isIso13 below · cited by 47 · depth 13 - Invertibility of the dual of an invertible ideal sheaf
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.isInvertible_invModule4 below · cited by 144 · depth 13 - Invertible ideal sheaves give invertible modules
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.isInvertible_module2 below · cited by 139 · depth 13 - Invertible ideal sheaf: I ⊗ I^∨ ≅ mathcal O_X, both orders
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.nonempty_module_tensor_invModule_iso5 below · cited by 56 · depth 13 - Duals of invertible ideal sheaves: (IJ)^∨ ≅ I^∨ ⊗ J^∨
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.nonempty_mul_invModule_iso_tensor7 below · cited by 49 · depth 13 - Pullback of the dual of an invertible ideal sheaf
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.nonempty_pullback_invModule_iso10 below · cited by 60 · depth 13 - Powers of an invertible ideal sheaf are invertible
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.pow0 below · cited by 115 · depth 13 - The dual of an invertible sheaf of modules is an inverse
AlgebraicGeometry.Scheme.Modules.IsInvertible.dual2 below · cited by 89 · depth 13 - Invertible sheaves of modules admit tensor inverses
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_tensor_inverse2 below · cited by 20 · depth 13 - Invertible modules on the spectrum of a field are trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_tensorUnit_of_field0 below · cited by 32 · depth 13 - Invertible modules on the spectrum of a local ring are trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_tensorUnit_of_isLocalRing5 below · cited by 30 · depth 13 - Norm of an invertible module along a finite flat morphism
AlgebraicGeometry.Scheme.Modules.IsInvertible.normModule29 below · cited by 21 · depth 13 - Tensor product of invertible sheaves of modules is invertible
AlgebraicGeometry.Scheme.Modules.IsInvertible.tensor2 below · cited by 165 · depth 13 - Norm of invertible sheaves along a finite surjective map to a normal scheme
AlgebraicGeometry.Scheme.Modules.exists_norm_isInvertible_tensor_pullback_normModule_of_isFinite_of_isIntegrallyClosed53 below · cited by 4 · depth 13 - Point formula for the norm of a rational point's line bundle
AlgebraicGeometry.Scheme.Modules.nonempty_normModule_invModule_ker_iso63 below · cited by 7 · depth 13 - Multiplicativity of the norm of invertible modules
AlgebraicGeometry.Scheme.Modules.nonempty_normModule_tensor_iso35 below · cited by 16 · depth 13 - Norm of the unit module along a finite flat map
AlgebraicGeometry.Scheme.Modules.nonempty_normModule_unit_iso29 below · cited by 19 · depth 13 - Pull-back of an iterated twist by invertible ideal sheaves
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_foldr_twist_iso13 below · cited by 19 · depth 13 - Base change for the norm of an invertible module
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_normModule_iso55 below · cited by 26 · depth 13 - fppf cohomology of G[n] is killed by n
AlgebraicGeometry.Scheme.fppfCohomology_kernel_zsmul_eq_zero0 below · cited by 1 · depth 13 - Naturality of the fppf Kummer row under endomorphisms
AlgebraicGeometry.Scheme.fppfKummerRow_naturality0 below · cited by 1 · depth 13 - Kummer row in fppf cohomology over Specℤ
AlgebraicGeometry.Scheme.fppfKummerRow_of_epi_zsmul0 below · cited by 1 · depth 13 - Functor of points is a small fppf sheaf
AlgebraicGeometry.Scheme.isSheaf_smallFppfTopology_forget_op_comp_yoneda_obj0 below · cited by 1 · depth 13 - An integer killing all sections kills fppf cohomology
AlgebraicGeometry.Scheme.nsmul_fppfCohomology_eq_zero_of_nsmul_sections_eq_zero0 below · cited by 1 · depth 13 - Smallness of fppf H¹ over Specℤ with small sections
AlgebraicGeometry.Scheme.small_fppfCohomology_one_specInt_of_small_sections1 below · cited by 1 · depth 13 - Invertibility of the ideal of a section meeting a smooth open
AlgebraicGeometry.Scheme.Hom.isInvertible_ker_of_comp_eq_id_of_mem_opens2 below · cited by 16 · depth 14 - Invertibility of ideal sheaves is preserved by open immersions
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.comap_of_isOpenImmersion0 below · cited by 18 · depth 14 - Euler characteristic additivity along a word of Cartier twists
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_pullback_foldr_pow_invModule_tensor_pow_module_tensor_eq_add_sum107 below · cited by 2 · depth 14 - Pullback comparison is an isomorphism for invertible ideal sheaves
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.isIso_pullbackModuleComparison6 below · cited by 56 · depth 14 - Invertible ideal sheaves: 𝒪(-Z₁-Z₂)≅𝒪(-Z₁)⊗𝒪(-Z₂)
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.nonempty_mul_module_iso_tensor2 below · cited by 26 · depth 14 - Invertibility descends from an open chart containing the support
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.of_comap_of_support_subset_range0 below · cited by 9 · depth 14 - Zero scheme of the canonical section of mathcal O_X(D)
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.zeroSchemeIdeal_invModuleSection13 below · cited by 10 · depth 14 - Inverse image of ideal sheaves is multiplicative
AlgebraicGeometry.Scheme.IdealSheafData.comap_mul0 below · cited by 56 · depth 14 - Flatness of the subscheme of I· J for invertible I
AlgebraicGeometry.Scheme.IdealSheafData.flat_subschemeIota_mul_comp_of_isInvertible0 below · cited by 1 · depth 14 - Descent of invertibility of ideal sheaves along finite flat surjections
AlgebraicGeometry.Scheme.IdealSheafData.isInvertible_of_isInvertible_comap0 below · cited by 1 · depth 14 - Reducedness of the vanishing-ideal subscheme, and killing of the ideal
AlgebraicGeometry.Scheme.IdealSheafData.isReduced_subscheme_vanishingIdeal_and_le_ker0 below · cited by 11 · depth 14 - Sections of Hom(P,Q) are determined by their value on a frame
AlgebraicGeometry.Scheme.Modules.IsFrameOn.existsUnique_ihomEval_eq0 below · cited by 9 · depth 14 - Existence of a dual frame on V pairing to 1
AlgebraicGeometry.Scheme.Modules.IsFrameOn.exists_isFrameOn_dual0 below · cited by 4 · depth 14 - Local frames of N_π(L) from bases of π_*mathcal O_X
AlgebraicGeometry.Scheme.Modules.IsFrameOn.exists_isFrameOn_normModule4 below · cited by 5 · depth 14 - Two frames differ by a unit on a common open
AlgebraicGeometry.Scheme.Modules.IsFrameOn.exists_isUnit_smul_eq0 below · cited by 6 · depth 14 - Frames pull back to frames along a morphism of schemes
AlgebraicGeometry.Scheme.Modules.IsFrameOn.pullbackLocalSection6 below · cited by 34 · depth 14 - The tensor product of two frames is a frame
AlgebraicGeometry.Scheme.Modules.IsFrameOn.tensorSections0 below · cited by 24 · depth 14 - Zero scheme of a section of a line bundle is locally principal
AlgebraicGeometry.Scheme.Modules.IsInvertible.coeffIdeal_le_and_ideal_zeroSchemeIdeal_eq2 below · cited by 22 · depth 14 - Dual of a tensor product of invertible sheaves
AlgebraicGeometry.Scheme.Modules.IsInvertible.dual_tensor4 below · cited by 8 · depth 14 - Invertible 𝒪_X-modules admit local frames
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isFrameOn2 below · cited by 25 · depth 14 - Invertible module with section is the zero-scheme ideal's dual
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_iso_invModule_zeroSchemeIdeal13 below · cited by 11 · depth 14 - Semilocal triviality of line bundles along finite morphisms
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_mem_and_nonempty_pullback_preimage_iso_unit_of_isFinite8 below · cited by 1 · depth 14 - Local triviality over the target along a finite morphism
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_nonempty_pullback_preimage_iso_tensorUnit_of_isFinite14 below · cited by 6 · depth 14 - Zero-scheme ideal of c Ω on affine opens inside a frame
AlgebraicGeometry.Scheme.Modules.IsInvertible.ideal_zeroSchemeIdeal_eq_span_of_app_eq_smul6 below · cited by 9 · depth 14 - Canonical evaluation X ⊗ X^∨ → 𝒪_Y is an isomorphism
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_ev_app_tensorUnit3 below · cited by 16 · depth 14 - Trivial rigidification forces L ≅ q^*σ^*L
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_pullback_pullback_of_rigidify_iso_unit0 below · cited by 3 · depth 14 - Rigidification commutes with base change
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_pullback_rigidify_iso4 below · cited by 13 · depth 14 - Pullback of the dual of an invertible sheaf of modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.pullback_dual3 below · cited by 20 · depth 14 - Local bases of sections for a locally free sheaf
AlgebraicGeometry.Scheme.Modules.IsLocallyFreeOfRank.exists_basis0 below · cited by 8 · depth 14 - Product of unit cocycles glues to the tensor product
AlgebraicGeometry.Scheme.Modules.exists_glueOfCocycle_mul_iso_tensor5 below · cited by 1 · depth 14 - The trivial cocycle glues to mathcal O_X
AlgebraicGeometry.Scheme.Modules.exists_glueOfCocycle_trivial_iso_tensorUnit4 below · cited by 2 · depth 14 - Twisting a unit cocycle yields an isomorphic glued module
AlgebraicGeometry.Scheme.Modules.exists_glueOfCocycle_twist_iso4 below · cited by 1 · depth 14 - Norm module frame: N_π(ι')(1) is Nm(g) times a frame
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_normModule_and_app_eq_norm_smul10 below · cited by 1 · depth 14 - Frames on the norm module with norm transition functions
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_normModule_forall_map_eq_norm_smul_of_isFrameOn_preimage8 below · cited by 3 · depth 14 - Trivialisation over an open yields a frame on it
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_of_pullback_iso_unit0 below · cited by 25 · depth 14 - A framed module with cocycle u is the glue of u
AlgebraicGeometry.Scheme.Modules.exists_iso_glueOfCocycle_app_eq_glueFrame3 below · cited by 8 · depth 14 - Pullback of a glued module is the glue of the pulled-back cocycle
AlgebraicGeometry.Scheme.Modules.exists_pullback_glueOfCocycle_iso10 below · cited by 4 · depth 14 - Frames for the norm module after refining the cover
AlgebraicGeometry.Scheme.Modules.exists_refinement_isFrameOn_normModule_map_eq_normFun_smul21 below · cited by 1 · depth 14 - Ratio cocycle of a family of frames
AlgebraicGeometry.Scheme.Modules.exists_unitCocycle_map_eq_smul_of_isFrameOn0 below · cited by 3 · depth 14 - Top wedge of a local basis frames detₙ M
AlgebraicGeometry.Scheme.Modules.isFrameOn_sheafificationAdjunction_unit_iotaMulti0 below · cited by 7 · depth 14 - Determinant of a locally free sheaf of rank n is invertible
AlgebraicGeometry.Scheme.Modules.isInvertible_det_of_isLocallyFreeOfRank2 below · cited by 11 · depth 14 - Modules glued from a unit cocycle are invertible
AlgebraicGeometry.Scheme.Modules.isInvertible_glueOfCocycle4 below · cited by 4 · depth 14 - Affine base change for locally trivial modules
AlgebraicGeometry.Scheme.Modules.isIso_baseChangeHom_of_isAffineHom29 below · cited by 2 · depth 14 - Morphisms matching frames on a cover are isomorphisms
AlgebraicGeometry.Scheme.Modules.isIso_of_isFrameOn_of_iSup_eq_top0 below · cited by 17 · depth 14 - Pushforward of a base-locally trivial module along a finite flat map
AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_pushforward_of_isFinite_of_flat_of_locallyTrivialOver10 below · cited by 10 · depth 14 - Determinant commutes with pullback for locally free sheaves
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_det_iso_det_pullback7 below · cited by 2 · depth 14 - Trivialisation on an open pulls back to the preimage
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_preimage_iso_unit_of_pullback_iso_unit0 below · cited by 1 · depth 14 - Wedge of a linearly transformed family scales by the determinant
AlgebraicGeometry.Scheme.Modules.sheafificationAdjunction_unit_iotaMulti_eq_det_smul_of_eq_sum_smul1 below · cited by 4 · depth 14 - Complement of a pole: affineness and finiteness over R[f]
AlgebraicGeometry.Scheme.Opens.isAffineOpen_and_finite_aeval_of_twoChart4 below · cited by 3 · depth 14 - Two-chart criterion: V affine and Γ(C,V) finite over R[g]
AlgebraicGeometry.Scheme.Opens.isAffineOpen_and_finite_aeval_of_twoChart_right4 below · cited by 5 · depth 14 - Universal f_*mathcal O_X=𝒪 over a reduced Noetherian base
AlgebraicGeometry.Scheme.TwoAffineOpenCover.bijective_algebraMap_sections_baseChange_of_isReduced5 below · cited by 5 · depth 14 - Finiteness of Čech H⁰,H¹ of mathcal O_X for proper X
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finite_H0_H1_structureSheaf55 below · cited by 7 · depth 14 - h⁰(𝒪)=1 on field-valued fibres from universal bijectivity
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_H0_unit_fibre_eq_one_of_bijective_sections1 below · cited by 9 · depth 14 - h⁰ = 1 on field fibres of a proper flat family
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_ker_cechDiff_baseChange_eq_one_of_isProper_of_geometricallyReduced_of_connected63 below · cited by 5 · depth 14 - Norm maps along finite surjections onto normal integral schemes
AlgebraicGeometry.Scheme.exists_normSections_mul_map_eq_norm_of_isFinite_of_isIntegrallyClosed4 below · cited by 1 · depth 14 - Finiteness of fppf cohomology along a finite chain
AlgebraicGeometry.Scheme.finite_fppfCohomology_of_shortExact_chain1 below · cited by 2 · depth 14 - Amitsur 1-cocycles with values in underlineℤ/p over ℤ
AlgebraicGeometry.Scheme.fppfAmitsurTrivial_constantZModSheaf10 below · cited by 1 · depth 14 - Smooth point from a smooth one-dimensional field fibre
AlgebraicGeometry.Scheme.Hom.mem_smoothLocus_of_flat_of_smoothOfRelativeDimension_pullback_snd1 below · cited by 3 · depth 15 - Twisting by rZ changes the Čech Euler characteristic by rd
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_pullback_pow_invModule_tensor_eq_add_mul105 below · cited by 4 · depth 15 - Euler characteristic drops by rd under twisting by I^r
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_pullback_pow_module_tensor_eq_sub_mul105 below · cited by 4 · depth 15 - Invertibility of an ideal sheaf is local on an open cover
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.of_forall_comap_openCover0 below · cited by 5 · depth 15 - Gluing quasi-coherent ideal sheaves along an open cover
AlgebraicGeometry.Scheme.IdealSheafData.exists_comap_eq_of_openCover0 below · cited by 4 · depth 15 - Inverse image ideal sheaf on affine opens is the extended ideal
AlgebraicGeometry.Scheme.IdealSheafData.ideal_comap_of_le0 below · cited by 18 · depth 15 - Inverse module of a principal product of Cartier divisors is trivial
AlgebraicGeometry.Scheme.IdealSheafData.nonempty_invModule_prod_pow_iso_tensorUnit_of_prod_pow_eq_zeroSchemeIdeal15 below · cited by 6 · depth 15 - Local surjectivity of the pullback comparison for ideal sheaf modules
AlgebraicGeometry.Scheme.IdealSheafData.pullbackModuleComparison_locallySurjective1 below · cited by 1 · depth 15 - Sections of 𝒪(-Z) over an affine open are the ideal
AlgebraicGeometry.Scheme.IdealSheafData.range_moduleIota_app_and_injective0 below · cited by 11 · depth 15 - Invertible modules on finite R-schemes are finite by sections
AlgebraicGeometry.Scheme.Modules.FiniteBySections.of_isFinite3 below · cited by 5 · depth 15 - Sections of the dual agree if they agree on a frame
AlgebraicGeometry.Scheme.Modules.IsFrameOn.dual_eq_of_ihomEval_eq0 below · cited by 2 · depth 15 - Basis of π_*𝒪_X times a frame gives basis of π_*L
AlgebraicGeometry.Scheme.Modules.IsFrameOn.exists_basis_smul_pushforward0 below · cited by 1 · depth 15 - Gluing two frames: modules with equal transition function are isomorphic
AlgebraicGeometry.Scheme.Modules.IsFrameOn.nonempty_iso_of_map_eq_smul_of_map_eq_smul5 below · cited by 6 · depth 15 - Coboundary transition data trivialises a two-chart framed module
AlgebraicGeometry.Scheme.Modules.IsFrameOn.nonempty_iso_tensorUnit_of_map_eq_mul5 below · cited by 7 · depth 15 - A frame trivialises a module on an open subset
AlgebraicGeometry.Scheme.Modules.IsFrameOn.nonempty_pullback_iso_unit2 below · cited by 18 · depth 15 - Frames glue: IsFrameOn is stable under suprema of opens
AlgebraicGeometry.Scheme.Modules.IsFrameOn.of_iSup0 below · cited by 10 · depth 15 - Zero-scheme ideal of a section commutes with base change
AlgebraicGeometry.Scheme.Modules.IsInvertible.comap_zeroSchemeIdeal3 below · cited by 25 · depth 15 - Sections of an invertible module agreeing at all k-points
AlgebraicGeometry.Scheme.Modules.IsInvertible.eq_of_forall_pullbackSection_eq5 below · cited by 2 · depth 15 - Hartogs extension for maps of invertible modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.existsUnique_pullback_map_eq_of_isIntegrallyClosed_stalk16 below · cited by 3 · depth 15 - Dual of an invertible module as ideal sheaf of Z(s)
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_dual_iso_module_zeroSchemeIdeal9 below · cited by 1 · depth 15 - Invertible module sheaves are reflexive: evaluation and bidual
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_ev_app_and_isIso_curry_braiding_ev4 below · cited by 3 · depth 15 - Locally surjective morphism of invertible modules is an isomorphism
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_of_locallySurjective0 below · cited by 6 · depth 15 - Rigidification is insensitive to twists pulled back from the base
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_rigidify_pullback_tensor_iso5 below · cited by 11 · depth 15 - Riemann's inequality in two-chart Čech form
AlgebraicGeometry.Scheme.Modules.IsInvertible.nontrivial_H0_sectionsOf_of_le_eulerChar_sub100 below · cited by 4 · depth 15 - Pullback of a section vanishes iff the point lies in the zero scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.pullbackSection_eq_zero_iff_mem_support4 below · cited by 9 · depth 15 - Rigidification of an invertible module along a section
AlgebraicGeometry.Scheme.Modules.IsInvertible.rigidify4 below · cited by 6 · depth 15 - Local freeness of rank n transports along isomorphisms
AlgebraicGeometry.Scheme.Modules.IsLocallyFreeOfRank.of_iso0 below · cited by 9 · depth 15 - Local freeness of rank n is stable under pullback
AlgebraicGeometry.Scheme.Modules.IsLocallyFreeOfRank.pullback0 below · cited by 7 · depth 15 - Gluing along a unit: realising a transition function on two opens
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_of_isUnit0 below · cited by 3 · depth 15 - Gluing frame pairs to an isomorphism of modules
AlgebraicGeometry.Scheme.Modules.exists_iso_app_eq_of_isFrameOn_of_eq_smul2 below · cited by 5 · depth 15 - Base change of sections over an affine open, locally trivial case
AlgebraicGeometry.Scheme.Modules.exists_linearEquiv_sections_baseChange_of_locallyTrivial8 below · cited by 6 · depth 15 - Exterior power of a morphism scales wedges by det a
AlgebraicGeometry.Scheme.Modules.exteriorPower_map_app_unit_iotaMulti_eq_det_smul0 below · cited by 3 · depth 15 - Locally framed modules on a scheme are invertible
AlgebraicGeometry.Scheme.Modules.isInvertible_of_forall_exists_isFrameOn3 below · cited by 5 · depth 15 - Modules framed on two opens covering X are invertible
AlgebraicGeometry.Scheme.Modules.isInvertible_of_isFrameOn_of_isFrameOn_of_sup_eq_top4 below · cited by 2 · depth 15 - Base-change morphism: independence of the chosen fibre product
AlgebraicGeometry.Scheme.Modules.isIso_baseChangeHom_iff_of_isPullback1 below · cited by 3 · depth 15 - Base change for direct images is local on the base
AlgebraicGeometry.Scheme.Modules.isIso_baseChangeHom_of_forall_exists_isPullback6 below · cited by 4 · depth 15 - Affine base change for global sections of 𝒪-modules
AlgebraicGeometry.Scheme.Modules.isIso_baseChange_sections_of_isIso_fromTildeGamma1 below · cited by 4 · depth 15 - Direct image of a locally trivial module to an affine base
AlgebraicGeometry.Scheme.Modules.isIso_fromTildeGamma_pushforward_of_locallyTrivial4 below · cited by 4 · depth 15 - Locally trivial modules localise on basic opens of affines
AlgebraicGeometry.Scheme.Modules.isLocalization_basicOpen_of_locallyTrivial1 below · cited by 12 · depth 15 - Local freeness of a direct image is local on the base
AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_pushforward_of_forall_exists_isPullback3 below · cited by 3 · depth 15 - Constant fibre rank: P̃ locally free of rank n
AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_tilde0 below · cited by 4 · depth 15 - Left unitor on sections: λ_N(g⊗ n)=g· n
AlgebraicGeometry.Scheme.Modules.leftUnitor_hom_app_tensorSections2 below · cited by 5 · depth 15 - bigwedgeⁿ(mathcal O_X^{⊕ n})≅mathcal O_X for the free sheaf of rank n
AlgebraicGeometry.Scheme.Modules.nonempty_exteriorPower_free_iso_unit0 below · cited by 3 · depth 15 - Exterior powers commute with restriction to an open subscheme
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_exteriorPower_iso_exteriorPower_pullback0 below · cited by 2 · depth 15 - Rigidification is trivial when σ^*L is trivial
AlgebraicGeometry.Scheme.Modules.nonempty_rigidify_iso_of_nonempty_pullback_iso_unit0 below · cited by 5 · depth 15 - s^∨(t) equals the coefficient of s along t
AlgebraicGeometry.Scheme.Modules.ofUnitSection_sectionDual_app0 below · cited by 2 · depth 15 - Sections of a locally trivial module sheaf over an affine open are projective
AlgebraicGeometry.Scheme.Modules.projective_sections_of_locallyTrivial4 below · cited by 26 · depth 15 - Pullback of a tensor of sections is the tensor of pullbacks
AlgebraicGeometry.Scheme.Modules.pullbackTensorObjIso_hom_app_pullbackLocalSection4 below · cited by 4 · depth 15 - Pull-back of the unit section is the unit section
AlgebraicGeometry.Scheme.Modules.pullbackTensorUnitObjIso_hom_app_pullbackLocalSection_unitSection4 below · cited by 6 · depth 15 - Pullbacks of module sheaves are locally spanned by unit images
AlgebraicGeometry.Scheme.Modules.pullback_locally_mem_span_unit0 below · cited by 2 · depth 15 - Tensor of morphisms acts on tensor of sections componentwise
AlgebraicGeometry.Scheme.Modules.tensorHom_app_tensorSections2 below · cited by 9 · depth 15 - Right whiskering on sections: (φrhdM)(s⊗ t)=φ(s)⊗ t
AlgebraicGeometry.Scheme.Modules.whiskerRight_app_tensorSections3 below · cited by 3 · depth 15 - Invariance of the zero-scheme ideal under isomorphisms
AlgebraicGeometry.Scheme.Modules.zeroSchemeIdeal_comp_eq_of_isIso1 below · cited by 18 · depth 15 - Zero scheme of r as a section of the trivial bundle
AlgebraicGeometry.Scheme.Modules.zeroSchemeIdeal_eq_ofIdealTop_of_app_eq_smul7 below · cited by 5 · depth 15 - Fibres of the map to A¹_k given by a transcendental section are finite
AlgebraicGeometry.Scheme.Opens.finite_preimage_singleton_toSpecPolynomial3 below · cited by 3 · depth 15 - Base change of the two-chart Čech complex of 𝒪_X
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_baseChangeIsos_structureSheaf0 below · cited by 19 · depth 15 - Global sections as degree-zero Čech cohomology on two affine charts
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_linearEquiv_sectionsOf_H00 below · cited by 36 · depth 15 - Transport of two-chart Čech cohomology along a scheme isomorphism
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_linearEquiv_sectionsOf_of_iso0 below · cited by 122 · depth 15 - h⁰=1 on field fibres with geometrically reduced connected fibres
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_ker_cechDiff_baseChange_eq_one_of_geometricallyReduced_of_connected2 below · cited by 1 · depth 15 - Sections over an affine open of a flat R-scheme are flat
AlgebraicGeometry.Scheme.TwoAffineOpenCover.flat_sections_of_flat0 below · cited by 21 · depth 15 - Local constancy of the fibrewise Euler characteristic of an invertible module
AlgebraicGeometry.Scheme.TwoAffineOpenCover.isLocallyConstant_finrank_ker_sub_finrank_coker_cechDiff_baseChange68 below · cited by 5 · depth 15 - Two-chart Čech H¹ is independent of the chart pair
AlgebraicGeometry.Scheme.TwoAffineOpenCover.nonempty_linearEquiv_H1_sectionsOf_of_isSeparated9 below · cited by 6 · depth 15 - Chart sections of an invertible sheaf: projective, finite, rank one
AlgebraicGeometry.Scheme.TwoAffineOpenCover.sectionsOf_isInvertible_projective_finite_finrank8 below · cited by 5 · depth 15 - Čech H⁰ of a two-chart cover equals global sections
AlgebraicGeometry.Scheme.TwoAffineOpenCover.toH0_bijective0 below · cited by 3 · depth 15 - An fppf sheaf turns Spec of a finite product into a product
AlgebraicGeometry.Scheme.bijective_pi_map_of_fppf_sheaf0 below · cited by 1 · depth 15 - Regular functions glue inside the function field
AlgebraicGeometry.Scheme.existsUnique_germToFunctionField_eq_of_iSup_eq0 below · cited by 1 · depth 15 - Descent of fppf sheaf sections along faithfully flat algebras
AlgebraicGeometry.Scheme.existsUnique_section_of_map_i1_eq_map_i20 below · cited by 1 · depth 15 - Fppf covering sieves of affine schemes admit finite affine refinements
AlgebraicGeometry.Scheme.exists_ofArrows_mem_fppfPrecoverage_of_mem_fppfTopology0 below · cited by 1 · depth 15 - Weil extension across η from dense test points
AlgebraicGeometry.Scheme.exists_opens_extension_of_pointwise_extension_dense8 below · cited by 2 · depth 15 - Fppf quotient sheaf represented by the Hopf kernel
AlgebraicGeometry.Scheme.exists_sectionsEquiv_of_shortExact_of_range_eq_hopfKer_of_isHopfGalois0 below · cited by 1 · depth 15 - Finiteness of fppf cohomology in the middle of an extension
AlgebraicGeometry.Scheme.finite_fppfCohomology_of_shortExact0 below · cited by 1 · depth 15 - Big fppf sheaves restrict to sheaves on the small fppf site of Specℤ
AlgebraicGeometry.Scheme.isSheaf_smallFppfTopology_specInt_forget_comp0 below · cited by 2 · depth 15 - A finite affine fppf family yields a single fppf cover
AlgebraicGeometry.Scheme.singleton_sigma_mem_fppfPrecoverage0 below · cited by 1 · depth 15 - Lifting Spec of a domain to the scheme-theoretic image
AlgebraicGeometry.Scheme.Hom.exists_lift_schemeTheoreticImage_of_isDomain1 below · cited by 1 · depth 16 - Pointwise fibrewise criterion for smoothness of a flat map
AlgebraicGeometry.Scheme.Hom.fiberInclusion_mem_smoothLocus_of_mem_smoothLocus_fiberToSpecResidueField1 below · cited by 7 · depth 16 - Ideal sheaf of a closed point generated by a uniformiser
AlgebraicGeometry.Scheme.Hom.ker_ideal_eq_span_of_span_germ_eq_maximalIdeal_of_forall_isUnit_germ0 below · cited by 2 · depth 16 - Étale on an open: stalk map carries maximal ideal onto maximal ideal
AlgebraicGeometry.Scheme.Hom.map_stalkMap_maximalIdeal_eq_of_etale_restrict0 below · cited by 4 · depth 16 - Quasi-compact finite-type morphisms hit accumulated generic points
AlgebraicGeometry.Scheme.Hom.mem_range_of_specializes_of_mem_closure1 below · cited by 1 · depth 16 - Flat base change: smooth locus of f' lies over that of f
AlgebraicGeometry.Scheme.Hom.smoothLocus_le_preimage_of_isPullback0 below · cited by 6 · depth 16 - Twisting by an invertible ideal sheaf raises χ by r
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_tensor_invModule_eq93 below · cited by 20 · depth 16 - Tensor product of invertible ideal sheaves is the product ideal
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_tensor_iso_mul_module8 below · cited by 10 · depth 16 - Frames for mathcal I₁^∨⊗mathcal I₂ with prescribed transition function
AlgebraicGeometry.Scheme.IdealSheafData.exists_isFrameOn_invModule_tensor_module_of_ideal_eq_span9 below · cited by 2 · depth 16 - Invariance of `FiniteBySections` under isomorphism of modules
AlgebraicGeometry.Scheme.Modules.FiniteBySections.of_iso0 below · cited by 9 · depth 16 - Isomorphy of mathcal O_X-module morphisms is local on an open cover
AlgebraicGeometry.Scheme.Modules.Hom.isIso_of_isIso_app_of_iSup_eq_top0 below · cited by 6 · depth 16 - Hartogs extension for sections of an invertible module
AlgebraicGeometry.Scheme.Modules.IsInvertible.bijective_presheaf_map_inf_of_isIntegrallyClosed_stalk8 below · cited by 1 · depth 16 - Invertible sheaf on a smooth curve is L(D)
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_divisor_range_eq_lSpaceOn9 below · cited by 12 · depth 16 - Pull-back of a transpose section vanishes iff the map does
AlgebraicGeometry.Scheme.Modules.IsInvertible.pullbackSection_transposeSection_eq_zero_iff3 below · cited by 2 · depth 16 - Tensor powers of an invertible sheaf of modules are invertible
AlgebraicGeometry.Scheme.Modules.IsInvertible.tensorPow3 below · cited by 16 · depth 16 - Local freeness of rank n is Zariski-local
AlgebraicGeometry.Scheme.Modules.IsLocallyFreeOfRank.of_forall_exists_opens0 below · cited by 5 · depth 16 - Horizontal pasting of base-change morphisms of module sheaves
AlgebraicGeometry.Scheme.Modules.baseChangeHom_comp_horizontal1 below · cited by 1 · depth 16 - Unit of pullback adjunction is bijective on sections inside the image
AlgebraicGeometry.Scheme.Modules.bijective_unit_app_of_le_opensRange0 below · cited by 17 · depth 16 - Fibrewise finiteness by sections descends over a Noetherian base
AlgebraicGeometry.Scheme.Modules.exists_finiteBySections_tensorPow_of_forall_geometricFibre61 below · cited by 1 · depth 16 - Milnor patching of invertible modules along a base-changed closed cover
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_pullback_curveChange_iso_of_closedCover13 below · cited by 2 · depth 16 - Base change of a family identifies its fibres
AlgebraicGeometry.Scheme.Modules.exists_iso_pullback_of_isPullback3 below · cited by 8 · depth 16 - Frames on a finite cover yield a P^N-presentation
AlgebraicGeometry.Scheme.Modules.exists_projPresentation_of_iSup_eq_top0 below · cited by 22 · depth 16 - Structure sheaf as L_{S_U}(0) inside the function field
AlgebraicGeometry.Scheme.Modules.exists_unit_range_eq_lSpaceOn_zero7 below · cited by 5 · depth 16 - Finiteness of sections of a locally trivial module on an affine open
AlgebraicGeometry.Scheme.Modules.finite_sections_of_locallyTrivial2 below · cited by 10 · depth 16 - Sections of a locally trivial module have base-change rank one
AlgebraicGeometry.Scheme.Modules.finrank_baseChange_sections_eq_one_of_locallyTrivial7 below · cited by 5 · depth 16 - Base change of a pushforward along an open immersion
AlgebraicGeometry.Scheme.Modules.isIso_baseChangeHom_of_isOpenImmersion1 below · cited by 3 · depth 16 - Recognising ̃Γ(M)toM on basic opens
AlgebraicGeometry.Scheme.Modules.isIso_fromTildeGamma_iff_isLocalizedModule1 below · cited by 2 · depth 16 - Locally trivial modules on an affine scheme come from global sections
AlgebraicGeometry.Scheme.Modules.isIso_fromTildeGamma_of_locallyTrivial5 below · cited by 6 · depth 16 - Localisation of module sections over a finite basic-open cover
AlgebraicGeometry.Scheme.Modules.isLocalization_basicOpen_of_finite_basicOpen_cover0 below · cited by 1 · depth 16 - Rank-one local freeness equals invertibility
AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_one_iff_isInvertible0 below · cited by 18 · depth 16 - Divisor of mathcal I_P^{ n} and its dual at a point
AlgebraicGeometry.Scheme.Modules.isPrincipal_sub_single_of_presentation_ker_pow99 below · cited by 2 · depth 16 - Tensor product of invertible sheaves adds divisors up to principal
AlgebraicGeometry.Scheme.Modules.isPrincipal_sub_sub_of_presentations_tensor85 below · cited by 3 · depth 16 - First exterior power of mathcal O_X-modules is the identity
AlgebraicGeometry.Scheme.Modules.nonempty_exteriorPower_one_iso_id0 below · cited by 1 · depth 16 - Triviality of a module on a finite k-scheme times T
AlgebraicGeometry.Scheme.Modules.nonempty_iso_unit_of_forall_pullback_rigSection_iso_unit5 below · cited by 2 · depth 16 - Norm of a line bundle splits along a scheme-theoretic union
AlgebraicGeometry.Scheme.Modules.nonempty_normModule_iso_normModule_tensor_normModule_of_isClosedImmersion42 below · cited by 1 · depth 16 - Rigidification is invisible on fibres over field-valued points
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_fst_rigidify_iso_of_isInvertible7 below · cited by 12 · depth 16 - Pullback commutes with tensor powers of 𝒪-modules
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_tensorPow_iso0 below · cited by 11 · depth 16 - Tensoring by a pullback trivial on V is invisible over q⁻¹V
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_tensor_pullback_iso_of_trivial_on_open2 below · cited by 3 · depth 16 - Sheafification tensorator on sections: μ(x^#⊗ y^#)=(x⊗ y)^#
AlgebraicGeometry.Scheme.Modules.sheafify_mu_app_tensorSections2 below · cited by 3 · depth 16 - Units of the pullback adjunctions compose along g ∘ f
AlgebraicGeometry.Scheme.Modules.unit_app_comp_pullbackComp_inv0 below · cited by 13 · depth 16 - Monotonicity of the zero-scheme ideal under module maps
AlgebraicGeometry.Scheme.Modules.zeroSchemeIdeal_comp_le0 below · cited by 1 · depth 16 - Two-chart Čech cohomology under base change
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_cech_sectionsOf_baseChange_equiv_of_locallyTrivial10 below · cited by 5 · depth 16 - Affine opens of the base as base changes
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_isPullback_snd_specMap_algebraOfHom3 below · cited by 15 · depth 16 - Čech H⁰,H¹ of the unit module sheaf versus structure-sheaf data
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_sectionsOf_unit_equiv_structureSheafSections0 below · cited by 5 · depth 16 - Sections on an affine open form a finite type R-algebra
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finiteType_algebraOfHom3 below · cited by 12 · depth 16 - Two-chart Čech cohomology of mathcal O_X: h¹=g, h⁰=1
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_H1_sectionsOf_unit_eq_and_finrank_H0_eq_one119 below · cited by 14 · depth 16 - Geometric fibre of a two-chart Čech complex: h⁰=1, h¹=g
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_cechDiff_baseChange_of_isAlgClosed81 below · cited by 10 · depth 16 - Extending a generic-fibre morphism across a graph-closure point
AlgebraicGeometry.Scheme.exists_opens_extension_of_mem_image_graph3 below · cited by 2 · depth 16 - Closed subsets missing the generic point are finite
AlgebraicGeometry.Scheme.finite_of_isClosed_of_genericPoint_notMem0 below · cited by 1 · depth 16 - Amitsur triviality of G_m over ℤ
AlgebraicGeometry.Scheme.fppfAmitsurTrivial_gmAbelianSheafLifted3 below · cited by 1 · depth 16 - Local-to-global passage for pairs of sections along two morphisms
AlgebraicGeometry.Scheme.Hom.app_injective_and_exists_of_forall_isAffineOpen0 below · cited by 1 · depth 17 - Reduced source factors through the scheme-theoretic image
AlgebraicGeometry.Scheme.Hom.exists_lift_schemeTheoreticImage_of_isReduced0 below · cited by 3 · depth 17 - Degree of a composite of finite flat morphisms
AlgebraicGeometry.Scheme.Hom.finrank_comp_of_finrank_eq_const0 below · cited by 8 · depth 17 - Constant fibre rank detected on a non-empty base change
AlgebraicGeometry.Scheme.Hom.finrank_eq_of_isPullback_of_irreducibleSpace0 below · cited by 1 · depth 17 - Powers of a section's ideal are finite flat of rank r
AlgebraicGeometry.Scheme.Hom.isFinite_and_finrank_subschemeIota_ker_pow_of_comp_eq_id9 below · cited by 16 · depth 17 - Integrality of the image of a quasi-compact immersion
AlgebraicGeometry.Scheme.Hom.isIntegral_image_and_isIso_stalkMap_toImage_genericPoint0 below · cited by 2 · depth 17 - Birational morphism: stalk isomorphism at non-generic points over DVRs
AlgebraicGeometry.Scheme.Hom.isIso_stalkMap_of_isIso_stalkMap_genericPoint0 below · cited by 2 · depth 17 - Order of a generator of mathcal I_P^{ n} at centred places
AlgebraicGeometry.Scheme.Hom.ord_eq_of_ker_pow_ideal_eq_span7 below · cited by 2 · depth 17 - Evaluation sequence along a thickened invertible ideal sheaf
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.epi_unit_app_tensor_invModule_pow_and_exists_shortExact27 below · cited by 4 · depth 17 - Divisor presenting the dual of an invertible ideal sheaf
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_divisor_range_invModule_eq_lSpaceOn25 below · cited by 1 · depth 17 - Divisor presentation of an invertible ideal sheaf on a curve
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_divisor_range_module_eq_lSpaceOn15 below · cited by 1 · depth 17 - Cancellation of invertible ideal sheaves
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.mul_left_cancel0 below · cited by 4 · depth 17 - Ideal sheaves agreeing on an open cover are equal
AlgebraicGeometry.Scheme.IdealSheafData.eq_of_forall_comap_openCover_eq0 below · cited by 13 · depth 17 - Framing of a locally principal ideal sheaf by a nonzerodivisor generator
AlgebraicGeometry.Scheme.IdealSheafData.exists_isFrameOn_module_of_forall_ideal_eq_span1 below · cited by 1 · depth 17 - Finiteness by sections is local on the affine base
AlgebraicGeometry.Scheme.Modules.FiniteBySections.of_forall_mem_finset_away11 below · cited by 1 · depth 17 - Finiteness by sections passes to positive tensor powers
AlgebraicGeometry.Scheme.Modules.FiniteBySections.tensorPow3 below · cited by 2 · depth 17 - Additivity of the Čech Euler characteristic in L ⊗ L'
AlgebraicGeometry.Scheme.Modules.IsInvertible.eulerChar_sectionsOf_tensor_eq_add_sub97 below · cited by 5 · depth 17 - Ratio of two sections of an invertible module
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_ratio_section5 below · cited by 2 · depth 17 - Nonzero section of a line bundle is nonzero at the generic point
AlgebraicGeometry.Scheme.Modules.IsInvertible.genericPoint_notMem_support_zeroSchemeIdeal3 below · cited by 3 · depth 17 - Global section of an invertible module frames off its zero scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.isFrameOn_app_of_disjoint_support_zeroSchemeIdeal4 below · cited by 11 · depth 17 - Fibrewise Cartier divisors in a flat family are relative
AlgebraicGeometry.Scheme.Modules.IsInvertible.isInvertible_zeroSchemeIdeal_and_flat4 below · cited by 2 · depth 17 - Sections differing by a unit have equal zero-scheme ideal
AlgebraicGeometry.Scheme.Modules.IsInvertible.zeroSchemeIdeal_eq_of_app_eq_smul3 below · cited by 3 · depth 17 - Finiteness of a L^{⊗ 3} Proj presentation
AlgebraicGeometry.Scheme.Modules.ProjPresentation.isFinite_toProj_of_forall_pullbackSection_eq_zero_iff25 below · cited by 1 · depth 17 - Left exactness of sections over an open on 𝒪_X-modules
AlgebraicGeometry.Scheme.Modules.exists_app_eq_of_exact_of_app_eq_zero0 below · cited by 9 · depth 17 - Spreading fibrewise finiteness by sections to a basic open
AlgebraicGeometry.Scheme.Modules.exists_away_finiteBySections_tensorPow_of_forall_geometricFibre53 below · cited by 1 · depth 17 - Two 𝒪(D)-presentations of a sheaf differ by a principal divisor
AlgebraicGeometry.Scheme.Modules.exists_eq_mul_and_eq_add_ord_of_presentations75 below · cited by 3 · depth 17 - Degree-one Čech vanishing for high tensor powers of L
AlgebraicGeometry.Scheme.Modules.exists_forall_subsingleton_HSucc_tensorPow_of_isFinite_toProj34 below · cited by 1 · depth 17 - Frames for L^{⊗ 3} on a cover, from the theorem of the square
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_tensorPow_three_of_forall_nonempty_pullback_tensor_iso21 below · cited by 2 · depth 17 - Milnor patching of invertible modules along a square of closed immersions
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_pullback_iso_of_milnorSquare11 below · cited by 1 · depth 17 - Affine base change of sections along a closed immersion
AlgebraicGeometry.Scheme.Modules.exists_linearEquiv_sections_pullback_of_isClosedImmersion_of_locallyTrivial8 below · cited by 2 · depth 17 - Frame kit comparing norm modules along a two-piece closed cover
AlgebraicGeometry.Scheme.Modules.exists_normModule_frameKit_of_isClosedImmersion40 below · cited by 1 · depth 17 - Closed-subscheme exact sequence tensored by a locally free module
AlgebraicGeometry.Scheme.Modules.exists_shortExact_ker_module_tensor_of_isClosedImmersion18 below · cited by 2 · depth 17 - Base change in degree zero, two-chart Čech form
AlgebraicGeometry.Scheme.Modules.isIso_baseChangeHom_of_twoAffineOpenCover24 below · cited by 3 · depth 17 - Tilde criterion via localisation on basic opens
AlgebraicGeometry.Scheme.Modules.isIso_fromTildeGamma_of_isLocalization_basicOpen2 below · cited by 1 · depth 17 - Rank-n local freeness of π_*F from fibrewise Čech data
AlgebraicGeometry.Scheme.Modules.isLocallyFreeOfRank_pushforward_of_twoAffineOpenCover23 below · cited by 2 · depth 17 - Sheaves of modules form a stack for Zariski open covers
AlgebraicGeometry.Scheme.Modules.isStackFor_openCover2 below · cited by 2 · depth 17 - Transition sections multiply under tensor product
AlgebraicGeometry.Scheme.Modules.map_tensorSections_eq_mul_smul_of_map_eq_smul5 below · cited by 1 · depth 17 - Tensor powers multiply: L^{⊗ ab}≅(L^{⊗ a})^{⊗ b}
AlgebraicGeometry.Scheme.Modules.nonempty_tensorPow_mul_iso0 below · cited by 2 · depth 17 - Pasting law for base-change isomorphisms of module pullbacks
AlgebraicGeometry.Scheme.Modules.pullbackComp_pullbackCongr_pasteSquares_app0 below · cited by 6 · depth 17 - Sections of a locally trivial module have rank one on affine opens
AlgebraicGeometry.Scheme.Modules.rankAtStalk_sections_eq_one_of_locallyTrivial3 below · cited by 1 · depth 17 - Surjectivity on affine sections when the kernel is locally trivial
AlgebraicGeometry.Scheme.Modules.surjective_app_of_shortExact_of_locallyTrivial12 below · cited by 4 · depth 17 - Full faithfulness of restriction to an open cover for 𝒪-modules
AlgebraicGeometry.Scheme.Modules.toDescentData_map_bijective_of_openCover0 below · cited by 8 · depth 17 - Sections on U are the rational functions regular on U
AlgebraicGeometry.Scheme.Opens.range_algebraMap_functionField_eq_iInf1 below · cited by 2 · depth 17 - Cohomology and base change in degree 0 for a two-chart cover
AlgebraicGeometry.Scheme.TwoAffineOpenCover.bijective_kerBaseChangeHom_and_surjective_unit_app_of_subsingleton_H177 below · cited by 1 · depth 17 - Euler characteristic of 𝒪(rp-sum vⱼ) on a smooth proper curve
AlgebraicGeometry.Scheme.TwoAffineOpenCover.eulerChar_sectionsOf_invModule_pow_ker_tensor_module_prod_ker_eq229 below · cited by 2 · depth 17 - Chart quotients for a square-zero thickening realise the stage map
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_ideal_pullback_cover_ringEquiv_quotient_appLE_of_squareZero3 below · cited by 1 · depth 17 - Invertible module glued from rank-one chart modules over a two-affine cover
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_isInvertible_sectionsOf_equiv_of_projective8 below · cited by 3 · depth 17 - Overlap base-change isomorphisms for lifted chart modules
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_linearEquiv_overlap_quotient_baseChange_of_lift_of_compat3 below · cited by 1 · depth 17 - Two-chart Čech sections data commute with base change
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_sectionsOf_baseChange_equiv_of_locallyTrivial9 below · cited by 5 · depth 17 - Sections over the overlap as a base change (invertible case)
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_sectionsOf_overlap_linearEquiv_baseChange_of_isInvertible2 below · cited by 4 · depth 17 - Two-chart Čech cohomology of a line bundle on a finite closed subscheme
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finite_H0_and_subsingleton_H1_sectionsOf_pushforward_of_isFinite1 below · cited by 3 · depth 17 - Base field extension preserves two-chart Čech h⁰ and H¹-vanishing
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_H0_sectionsOf_baseChange_eq_and_subsingleton_H1_iff11 below · cited by 3 · depth 17 - h⁰=1 on all field fibres of a two-chart family
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_ker_cechDiff_baseChange_eq_one2 below · cited by 1 · depth 17 - Invertible modules with isomorphic two-chart sections are isomorphic
AlgebraicGeometry.Scheme.TwoAffineOpenCover.nonempty_iso_of_sectionsOf_linearEquiv_of_isInvertible11 below · cited by 5 · depth 17 - Pullback of a glued lift along B → B/I recovers M
AlgebraicGeometry.Scheme.TwoAffineOpenCover.nonempty_iso_pullback_baseChangeSnd_of_sectionsOf_lift_appLE_of_overlap_iso19 below · cited by 1 · depth 17 - Global sections of a proper integral scheme over algebraically closed k
AlgebraicGeometry.Scheme.bijective_algebraMap_sections_of_isProper_of_isIntegral60 below · cited by 4 · depth 17 - Bijectivity of restriction Γ(V)→Γ(V∩ U) (algebraic Hartogs)
AlgebraicGeometry.Scheme.bijective_presheaf_map_inf_of_isIntegrallyClosed_stalk6 below · cited by 3 · depth 17 - Closed points lift to k-points over an algebraically closed extension
AlgebraicGeometry.Scheme.exists_SpecMap_comp_eq_of_isAlgClosed_of_isClosed_singleton0 below · cited by 5 · depth 17 - Crossing chart from a crossing presentation of a stalk
AlgebraicGeometry.Scheme.exists_crossingChart_of_crossingPresentation_stalk2 below · cited by 4 · depth 17 - A common algebraically closed field for two field-valued points
AlgebraicGeometry.Scheme.exists_isAlgClosed_factor_residueField_of_range_subset_singleton0 below · cited by 2 · depth 17 - Birational integral K-schemes: open immersion near the generic point
AlgebraicGeometry.Scheme.exists_isOpenImmersion_of_functionField_iso0 below · cited by 1 · depth 17 - Open immersion from a smooth proper curve extends to an isomorphism
AlgebraicGeometry.Scheme.exists_iso_comp_eq_of_isOpenImmersion_of_isProper3 below · cited by 1 · depth 17 - Spreading out a morphism from Specmathcal O_{G,η} over a base
AlgebraicGeometry.Scheme.exists_opens_extension_of_fromSpecStalk0 below · cited by 2 · depth 17 - Quotient by a finite locally free equivalence relation
AlgebraicGeometry.Scheme.exists_quotient_of_finiteLocallyFree_equivalenceRelation5 below · cited by 5 · depth 17 - An H-fixed L-point descends to the fixed field
AlgebraicGeometry.Scheme.exists_specMap_fixedField_comp_eq_of_forall_comp_eq0 below · cited by 6 · depth 17 - Base change of a big fppf sheaf is a small-site sheaf
AlgebraicGeometry.Scheme.isSheaf_smallFppfTopology_specInt_pullback_forget_comp0 below · cited by 1 · depth 17 - Non-isolated point in the fibre at a non-quasi-finite point
AlgebraicGeometry.Scheme.Hom.exists_isClosed_irreducible_subset_fiber_of_not_quasiFiniteAt0 below · cited by 2 · depth 18 - 𝒪_X → f_*𝒪_Y is bijective on a Hartogs open
AlgebraicGeometry.Scheme.Hom.isIso_app_of_isIso_morphismRestrict_of_bijective_presheaf_map0 below · cited by 2 · depth 18 - Quasi-finiteness at a point descends along base change
AlgebraicGeometry.Scheme.Hom.quasiFiniteAt_of_isPullback_of_locallyQuasiFinite2 below · cited by 1 · depth 18 - Morphism from Spec of a local ring meets only opens containing the closed point
AlgebraicGeometry.Scheme.Hom.range_subset_of_closedPoint_mem0 below · cited by 4 · depth 18 - Additivity of finrank for products of invertible ideal sheaves
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.isFinite_and_finrank_mul_subscheme_comp_eq_add97 below · cited by 2 · depth 18 - Pull-back of the ideal sheaf of a global ideal
AlgebraicGeometry.Scheme.IdealSheafData.comap_ofIdealTop1 below · cited by 21 · depth 18 - Restriction of a closed subscheme along an open immersion containing its support
AlgebraicGeometry.Scheme.IdealSheafData.exists_iso_subscheme_comap_of_support_subset_range0 below · cited by 2 · depth 18 - Invertibility of the vanishing ideal of η̄
AlgebraicGeometry.Scheme.IdealSheafData.isInvertible_vanishingIdeal_closure_of_isRegularLocalRing6 below · cited by 5 · depth 18 - Tensoring an ideal sheaf inclusion with a locally free module preserves monomorphy
AlgebraicGeometry.Scheme.IdealSheafData.mono_whiskerLeft_moduleIota1 below · cited by 4 · depth 18 - Radical ideal sheaf with pairwise disjoint support splits as a product
AlgebraicGeometry.Scheme.IdealSheafData.prod_vanishingIdeal_eq_of_pairwise_disjoint_of_support_eq_iSup0 below · cited by 8 · depth 18 - Invertible modules are finite by sections when mathcal O_X is
AlgebraicGeometry.Scheme.Modules.FiniteBySections.of_finiteBySections_unit5 below · cited by 1 · depth 18 - Epimorphisms of sheaves of modules are the locally surjective ones
AlgebraicGeometry.Scheme.Modules.Hom.epi_iff_locallySurjective0 below · cited by 9 · depth 18 - Being an isomorphism is local for sheaves of modules
AlgebraicGeometry.Scheme.Modules.Hom.isIso_of_forall_exists_isIso_pullback_map1 below · cited by 4 · depth 18 - Monomorphisms of mathcal O_X-modules are the sectionwise injections
AlgebraicGeometry.Scheme.Modules.Hom.mono_iff_injective0 below · cited by 9 · depth 18 - Euler characteristic of the dual of an invertible sheaf
AlgebraicGeometry.Scheme.Modules.IsInvertible.eulerChar_sectionsOf_dual_eq100 below · cited by 2 · depth 18 - Descent of a line bundle along a contraction, existence form
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isInvertible_and_pullback_iso_of_isIso_app6 below · cited by 2 · depth 18 - Kernel of PicXtoPicU generated by the Cᵢ
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_iso_invModule_prod_pow_of_zeroSchemeIdeal_support_disjoint16 below · cited by 1 · depth 18 - Effective descent for invertible modules along affine flat surjections
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_iso_toDescentData_of_isAffineHom_of_flat_of_surjective27 below · cited by 6 · depth 18 - Gluing trivialisations of a line bundle over a two-component cover
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_unit_of_closedCover_of_forall_pullbackSection_eq29 below · cited by 1 · depth 18 - Triviality of invertible modules lifts along square-zero quotients
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_unit_of_pullback_squareZero10 below · cited by 2 · depth 18 - Descent of morphisms of invertible modules along affine faithfully flat maps
AlgebraicGeometry.Scheme.Modules.IsInvertible.toDescentData_map_bijective_of_isAffineHom_of_flat_of_surjective2 below · cited by 6 · depth 18 - Base change of a projective presentation along a cartesian square
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_baseChange_of_isPullback3 below · cited by 25 · depth 18 - Finiteness by sections after base change near a quasi-finite fibre
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_finiteBySections_pullback_of_quasiFiniteAt5 below · cited by 1 · depth 18 - Sections of L^{⊗ m} as the twist datum of a Proj presentation
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_linearMap_sections_tensorPow_twistObj5 below · cited by 2 · depth 18 - Transport of a P^N-presentation along a module isomorphism
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_of_iso0 below · cited by 13 · depth 18 - Locally quasi-finiteness transfers along R-linear combinations of sections
AlgebraicGeometry.Scheme.Modules.ProjPresentation.locallyQuasiFinite_of_forall_exists_eq_sum_smul1 below · cited by 2 · depth 18 - Chart of a Proj presentation = locus where σᵢ is a local frame
AlgebraicGeometry.Scheme.Modules.ProjPresentation.mem_preimage_basicOpen_iff0 below · cited by 4 · depth 18 - Vanishing dichotomy on a fibre of a projective presentation
AlgebraicGeometry.Scheme.Modules.ProjPresentation.subset_support_zeroSchemeIdeal_or_disjoint6 below · cited by 2 · depth 18 - Being a generator of a module sheaf is local
AlgebraicGeometry.Scheme.Modules.bijective_smul_of_forall_exists_bijective_smul0 below · cited by 9 · depth 18 - Local frames pull back along a morphism of schemes
AlgebraicGeometry.Scheme.Modules.bijective_smul_unit_app_of_bijective_smul0 below · cited by 7 · depth 18 - Frames spread from the K-fibre to a basic open over 𝔭
AlgebraicGeometry.Scheme.Modules.exists_basicOpen_forall_exists_frame_of_frame_pullback5 below · cited by 3 · depth 18 - Base change of global sections at a field-valued point
AlgebraicGeometry.Scheme.Modules.exists_eq_sum_smul_pullbackSection_of_subsingleton_HSucc25 below · cited by 2 · depth 18 - Sections of a fibre product of mathcal O_X-modules
AlgebraicGeometry.Scheme.Modules.exists_fibreProduct_sections_bijective0 below · cited by 2 · depth 18 - Two embeddings of a sheaf into K(X) differ by a rational function
AlgebraicGeometry.Scheme.Modules.exists_forall_eq_mul_of_presentations0 below · cited by 7 · depth 18 - A frame among the terms of a frame sum
AlgebraicGeometry.Scheme.Modules.exists_frame_of_frame_sum_smul0 below · cited by 5 · depth 18 - Gluing a morphism of 𝒪_X-modules along two open immersions
AlgebraicGeometry.Scheme.Modules.exists_hom_restrict_eq_of_isOpenImmersion2 below · cited by 1 · depth 18 - A section of L^{⊗ 3} vanishing on three translates
AlgebraicGeometry.Scheme.Modules.exists_hom_tensorPow_three_support_zeroSchemeIdeal_eq14 below · cited by 3 · depth 18 - Maps out of Γ(M)̃xrightarrow∼M are determined on global sections
AlgebraicGeometry.Scheme.Modules.hom_ext_of_isIso_fromTildeGamma0 below · cited by 1 · depth 18 - Projection formula for free modules of finite rank
AlgebraicGeometry.Scheme.Modules.isIso_projectionMorphism_of_iso_free1 below · cited by 3 · depth 18 - Restriction to an open detects isomorphy of the projection morphism
AlgebraicGeometry.Scheme.Modules.isIso_pullback_map_projectionMorphism_iff9 below · cited by 3 · depth 18 - Sections over a basic open localise, for qcqs opens
AlgebraicGeometry.Scheme.Modules.isLocalization_basicOpen_of_locallyTrivial_of_qcqs2 below · cited by 5 · depth 18 - Open immersion: π: L→ f_*M bijective on small opens gives f^*L≅ M
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_iso_of_bijective_app_of_le_opensRange0 below · cited by 1 · depth 18 - Triviality of a pullback twist missing the ideal supports
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_tensor_invModule_pow_tensor_module_iso_of_forall_notMem_support28 below · cited by 4 · depth 18 - Pushforward along e followed by f via pullback along e⁻¹
AlgebraicGeometry.Scheme.Modules.nonempty_pushforward_hom_comp_iso0 below · cited by 1 · depth 18 - Projection formula for a closed immersion and a locally free sheaf
AlgebraicGeometry.Scheme.Modules.nonempty_pushforward_unit_tensor_iso_pushforward_pullback_of_isClosedImmersion14 below · cited by 3 · depth 18 - Trivialisation of φ^*mathcal O_Y on pulled-back functions
AlgebraicGeometry.Scheme.Modules.pullbackUnitIso_hom_app_pullbackLocalSection_toUnitSection0 below · cited by 10 · depth 18 - Right unitor on sections: ρ_N(n⊗ g)=g· n
AlgebraicGeometry.Scheme.Modules.rightUnitor_hom_app_tensorSections2 below · cited by 3 · depth 18 - Effectivity of Zariski descent data for sheaves of modules
AlgebraicGeometry.Scheme.Modules.toDescentData_essSurj_of_openCover0 below · cited by 4 · depth 18 - Left whiskering on sections: (L ⊗ ψ)(s ⊗ t) = s ⊗ ψ(t)
AlgebraicGeometry.Scheme.Modules.whiskerLeft_app_tensorSections3 below · cited by 2 · depth 18 - Properness of the map to A¹ cut out by a function
AlgebraicGeometry.Scheme.Opens.isProper_toSpecPolynomial_of_maximal0 below · cited by 1 · depth 18 - Valuation-ring stalks lie in the domain of the induced rational map
AlgebraicGeometry.Scheme.PartialMap.mem_domain_toRationalMap_of_valuationRing_stalk0 below · cited by 4 · depth 18 - Points of a fibre product with trivial residue extension
AlgebraicGeometry.Scheme.Pullback.eq_of_fst_eq_of_snd_eq_of_isIso_residueFieldMap0 below · cited by 10 · depth 18 - Indeterminacy of a rational map to an affine target occurs in codimension one
AlgebraicGeometry.Scheme.RationalMap.exists_specializes_ringKrullDim_le_one_of_not_mem_domain6 below · cited by 3 · depth 18 - Basic-open refinement of a two-affine cover trivialising π_*mathcal O_Y
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_basicOpen_refinement_basis_pushforward3 below · cited by 2 · depth 18 - Base change of two-affine chart rings along a stage map
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_stage_baseChangeIsos_structureSheaf2 below · cited by 9 · depth 18 - Stagewise affine base change of chart sections
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_stage_sectionsOf_baseChange_equiv_of_locallyTrivial9 below · cited by 4 · depth 18 - Affine quotient by a finite locally free equivalence relation
AlgebraicGeometry.Scheme.exists_affine_quotient_of_finiteLocallyFree_equivalenceRelation1 below · cited by 3 · depth 18 - Invariant affine neighbourhood for a finite flat equivalence relation
AlgebraicGeometry.Scheme.exists_invariant_isAffineOpen_of_finiteLocallyFree_equivalenceRelation0 below · cited by 1 · depth 18 - Gluing quotients along an invariant open cover
AlgebraicGeometry.Scheme.exists_quotient_of_forall_exists_quotient_restrict1 below · cited by 1 · depth 18 - Translation invariance of D under z⁻¹z' for z,z'∈ Z
AlgebraicGeometry.Scheme.forall_mem_iff_of_subset_union_preimage_or_disjoint1 below · cited by 2 · depth 18 - Reduced induced subscheme on an irreducible closed set is integral
AlgebraicGeometry.Scheme.isIntegral_subscheme_vanishingIdeal0 below · cited by 10 · depth 18 - Krull's principal ideal theorem in scheme form
AlgebraicGeometry.Scheme.ringKrullDim_stalk_eq_one_of_forall_specializes_notMem_basicOpen1 below · cited by 5 · depth 18 - A section missing a closed subscheme induces the unit ideal
AlgebraicGeometry.Scheme.Hom.comap_ker_eq_top_and_finrank_eq_zero_of_closedPoint_notMem_support0 below · cited by 1 · depth 19 - Section meets an integral vertical component in one rational point
AlgebraicGeometry.Scheme.Hom.isInvertible_comap_ker_and_finrank_eq_one_of_mul_eq_span_natCast2 below · cited by 1 · depth 19 - Smooth locus grows under base change
AlgebraicGeometry.Scheme.Hom.preimage_smoothLocus_le_of_isPullback0 below · cited by 4 · depth 19 - Pull-back of an invertible ideal sheaf to an integral scheme
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.comap_of_isIntegral1 below · cited by 2 · depth 19 - Euler characteristic twisted by a sum of invertible ideal sheaves
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_pullback_finsetProd_pow_invModule_tensor_and_module_tensor107 below · cited by 3 · depth 19 - Flat surjective pullback is injective on ideal sheaves
AlgebraicGeometry.Scheme.IdealSheafData.comap_injective_of_flat_of_surjective0 below · cited by 2 · depth 19 - Comaximal factors of an invertible ideal sheaf are invertible
AlgebraicGeometry.Scheme.IdealSheafData.isInvertible_and_isInvertible_of_mul_of_sup_eq_top0 below · cited by 1 · depth 19 - Reduced induced closed subscheme structure is reduced
AlgebraicGeometry.Scheme.IdealSheafData.isReduced_subscheme_vanishingIdeal0 below · cited by 5 · depth 19 - Arithmetic-progression twists by invertible ideal sheaves are trivial
AlgebraicGeometry.Scheme.IdealSheafData.nonempty_invModule_prod_pow_tensor_module_prod_pow_iso_tensorUnit_of_arithProg21 below · cited by 1 · depth 19 - Finiteness of f from a finite presentation of 𝒪_X
AlgebraicGeometry.Scheme.Modules.FiniteBySections.isFinite_of_unit0 below · cited by 1 · depth 19 - Transition function of a globally trivial framed line bundle is a coboundary
AlgebraicGeometry.Scheme.Modules.IsFrameOn.exists_isUnit_map_eq_mul_of_iso_tensorUnit5 below · cited by 2 · depth 19 - Tensor powers of a frame are frames
AlgebraicGeometry.Scheme.Modules.IsFrameOn.tensorPowSection3 below · cited by 6 · depth 19 - Vanishing of a line bundle section on a reduced scheme covered by two closed subschemes
AlgebraicGeometry.Scheme.Modules.IsInvertible.eq_zero_of_pullback_map_eq_zero_of_isReduced3 below · cited by 1 · depth 19 - Endomorphisms of an invertible module are multiplication by a global section
AlgebraicGeometry.Scheme.Modules.IsInvertible.existsUnique_app_eq_smul0 below · cited by 3 · depth 19 - Sections vanishing on a closed subscheme lift to L(-Z)
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_comp_whiskerLeft_moduleIota_eq_of_pullbackSection_ker_eq_zero15 below · cited by 1 · depth 19 - Effective descent of invertible modules along affine flat surjections
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_iso_toDescentData_of_isAffine_of_isAffineHom_of_flat_of_surjective20 below · cited by 1 · depth 19 - Descent of an invertible module along a contraction
AlgebraicGeometry.Scheme.Modules.IsInvertible.isInvertible_pushforward_and_isIso_counit_of_isIso_app5 below · cited by 1 · depth 19 - Invertible modules on Spec R are trivial when Pic R vanishes
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_unit_of_forall_invertible_free14 below · cited by 2 · depth 19 - Vanishing of Čech H¹ for invertible modules of large degree
AlgebraicGeometry.Scheme.Modules.IsInvertible.subsingleton_H1_sectionsOf_of_le_eulerChar_sub100 below · cited by 2 · depth 19 - Additivity of zero-scheme ideals under tensor product of sections
AlgebraicGeometry.Scheme.Modules.IsInvertible.zeroSchemeIdeal_tensorHom10 below · cited by 1 · depth 19 - Restriction of π_*mathcal O_Y between affine opens is a base change
AlgebraicGeometry.Scheme.Modules.exists_baseChange_sections_linearEquiv_pushforward_tensorUnit_of_affineOpen_le0 below · cited by 1 · depth 19 - Spreading out the frame locus to a basic open
AlgebraicGeometry.Scheme.Modules.exists_basicOpen_forall_exists_frame0 below · cited by 1 · depth 19 - Degree-one affine Čech acyclicity for locally trivial modules
AlgebraicGeometry.Scheme.Modules.exists_eq_sub_of_cocycle_of_isAffineOpen5 below · cited by 1 · depth 19 - Descending a local frame along a morphism of schemes
AlgebraicGeometry.Scheme.Modules.exists_frame_of_frame_pullback3 below · cited by 3 · depth 19 - Multiplication by a global function on the unit module
AlgebraicGeometry.Scheme.Modules.exists_hom_tensorUnit_app_eq_smul0 below · cited by 3 · depth 19 - Frames of N_π(L) with transition function Nm(u)
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_normModule_map_eq_norm_smul_of_isFrameOn_preimage8 below · cited by 1 · depth 19 - Sections over a reduced scheme covered by two closed subschemes
AlgebraicGeometry.Scheme.Modules.exists_unique_section_of_pullbackSection_closedCover13 below · cited by 2 · depth 19 - Local bases on affine opens extend to all opens
AlgebraicGeometry.Scheme.Modules.forall_exists_basis_map_eq_of_forall_isAffineOpen0 below · cited by 4 · depth 19 - Naturality in F of the projection morphism
AlgebraicGeometry.Scheme.Modules.projectionMorphism_naturality0 below · cited by 1 · depth 19 - Projection morphism as the mate of ε ⊗ 1
AlgebraicGeometry.Scheme.Modules.pullback_map_projectionMorphism_comp_counit0 below · cited by 1 · depth 19 - Homogeneity of degree n of s ↦ s^{⊗ n}
AlgebraicGeometry.Scheme.Modules.tensorPowSection_smul2 below · cited by 1 · depth 19 - fpqc descent of functions: mathcal O_Y → mathcal O_{Y'} descent data
AlgebraicGeometry.Scheme.Modules.toDescentData_map_bijective_unit_of_flat_of_surjective0 below · cited by 1 · depth 19 - Compact schemes admit ordered finite affine open covers
AlgebraicGeometry.Scheme.OrderedAffineCover.nonempty_of_compactSpace0 below · cited by 9 · depth 19 - Affine acyclicity of the alternating Čech complex of 𝒪
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.ker_d_succ_le_range_d_of_isAffineOpen0 below · cited by 4 · depth 19 - Frames and transition data pull back along stage maps
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_isFrameOn_pullback_stage_of_map_eq_smul6 below · cited by 4 · depth 19 - Openness of the vanishing locus of fibrewise check H¹
AlgebraicGeometry.Scheme.TwoAffineOpenCover.isOpen_setOf_subsingleton_H1_fibre70 below · cited by 2 · depth 19 - Cover-independence of vanishing of two-chart Čech H¹
AlgebraicGeometry.Scheme.TwoAffineOpenCover.subsingleton_H1_sectionsOf_of_subsingleton_H19 below · cited by 2 · depth 19 - Branch ideals at x order-reverse specialisation of generisations
AlgebraicGeometry.Scheme.branchIdeal_le_branchIdeal_iff45 below · cited by 4 · depth 19 - Quotients by finite locally free equivalence relations base-change
AlgebraicGeometry.Scheme.quotient_baseChange_of_finiteLocallyFree_of_isPullback0 below · cited by 3 · depth 19 - Basic opens meet irreducible closed sets in proper schemes all or nothing
AlgebraicGeometry.Scheme.subset_basicOpen_or_disjoint_of_isProper_of_isIrreducible0 below · cited by 1 · depth 19 - Restricted 𝒪_X(Cᵢ) raises the Čech Euler characteristic by r
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_pullback_invModule_eq_add_of_comap101 below · cited by 1 · depth 20 - Finite of rank N over k: reduced k-rational support
AlgebraicGeometry.Scheme.IdealSheafData.isFinite_and_finrank_subschemeIota_comp_eq_of_map_germ_eq_maximalIdeal0 below · cited by 3 · depth 20 - Invertibility of the ideal generated by a global non-zero-divisor on an affine scheme
AlgebraicGeometry.Scheme.IdealSheafData.isInvertible_ofIdealTop_span_singleton0 below · cited by 3 · depth 20 - ofIdealTop is multiplicative on products of ideals
AlgebraicGeometry.Scheme.IdealSheafData.ofIdealTop_mul0 below · cited by 1 · depth 20 - Base-change morphism under vertical pasting of squares
AlgebraicGeometry.Scheme.Modules.baseChangeHom_comp_vertical0 below · cited by 1 · depth 20 - Degeneracy locus of an equal-rank map as the zero locus of a section of (det E)^∨
AlgebraicGeometry.Scheme.Modules.exists_pullbackSection_dual_det_eq_zero_iff_not_isIso13 below · cited by 1 · depth 20 - Rank-one finite projective modules give invertible sheaves on Spec R
AlgebraicGeometry.Scheme.Modules.isInvertible_tilde_of_projective_rankOne2 below · cited by 1 · depth 20 - Injectivity of the Čech augmentation on Γ(V,W)
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.aug_injective0 below · cited by 5 · depth 20 - Čech H⁰ of the structure sheaf on an ordered affine cover
AlgebraicGeometry.Scheme.OrderedAffineCoverOf.ker_d_zero_eq_range_aug0 below · cited by 5 · depth 20 - Gluing copies of a scheme along an open by a cocycle
AlgebraicGeometry.Scheme.exists_isOpenImmersion_isPullback_of_glue_cocycle0 below · cited by 1 · depth 20 - Thickening short exact sequence for an invertible ideal sheaf
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.exists_shortExact_thickening_unit25 below · cited by 2 · depth 21 - Local generating section of an invertible sheaf of modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_basis_one0 below · cited by 2 · depth 21 - Local basis remains a basis after pull-back to a field point
AlgebraicGeometry.Scheme.Modules.exists_basis_pullback_of_field_point1 below · cited by 1 · depth 21 - Determinant of an extension of an invertible sheaf
AlgebraicGeometry.Scheme.Modules.nonempty_det_succ_iso_det_tensor_of_shortExact15 below · cited by 3 · depth 21 - Pushforward of a short exact sequence with locally surjective sections
AlgebraicGeometry.Scheme.Modules.shortExact_map_pushforward_of_forall_exists_surjective_app4 below · cited by 3 · depth 21 - Centre of a valuation on a universally closed integral scheme
AlgebraicGeometry.Scheme.exists_eq_closedPoint_and_forall_mem_of_valuationSubring0 below · cited by 5 · depth 21 - Valuative extension of a K-point via a dominating local ring
AlgebraicGeometry.Scheme.exists_section_comp_eq_of_exists_specMap_comp_eq_of_isLocalHom9 below · cited by 1 · depth 21 - Germ of a chart function lies in the maximal ideal iff
AlgebraicGeometry.Scheme.germ_app_appIso_inv_mem_maximalIdeal_iff0 below · cited by 2 · depth 21 - Non-extendable K-points over a DVR have closed image
AlgebraicGeometry.Scheme.isClosed_range_of_not_exists_section_comp_eq9 below · cited by 1 · depth 21 - Epimorphy of bigwedgeⁿE'otimesE→bigwedgeⁿ⁺¹E
AlgebraicGeometry.Scheme.Modules.epi_whiskerRight_wedgeVec_of_shortExact5 below · cited by 1 · depth 22 - Exterior powers above the rank of a locally free sheaf vanish
AlgebraicGeometry.Scheme.Modules.isZero_exteriorPower_of_isLocallyFreeOfRank2 below · cited by 1 · depth 22 - Short exactness of mathcal O_X-module complexes from local data
AlgebraicGeometry.Scheme.Modules.shortExact_of_app_injective_of_locallySurjective_of_locallyExact2 below · cited by 1 · depth 22 - K-points whose closure meets the special fibre extend to sections
AlgebraicGeometry.Scheme.exists_section_comp_eq_of_exists_mem_closure_range8 below · cited by 3 · depth 22 - Regular points of dimension ≤ 1 over a perfect field are smooth
AlgebraicGeometry.Scheme.Hom.mem_smoothLocus_of_isRegularLocalRing_stalk_of_ringKrullDim_le_one_of_perfectField3 below · cited by 4 · depth 23 - Sheafification of a locally surjective map of presheaves of modules is epi
AlgebraicGeometry.Scheme.Modules.epi_sheafification_map_of_locallySurjective0 below · cited by 1 · depth 23 - Vanishing of a module sheaf is a local condition
AlgebraicGeometry.Scheme.Modules.isZero_of_forall_exists_isZero_pullback_obj0 below · cited by 1 · depth 23 - Two-chart Čech vanishing gives surjectivity on the union
AlgebraicGeometry.Scheme.Modules.surjective_app_sup_of_shortExact_of_locallyTrivial13 below · cited by 1 · depth 23 - Čech h¹ of mathcal O_C equals the genus of the function field
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_H1_sectionsOf_unit_eq_genusFF_of_curveModel211 below · cited by 2 · depth 23 - Two-chart Čech H¹ vanishes if all field fibres vanish
AlgebraicGeometry.Scheme.TwoAffineOpenCover.subsingleton_H1_sectionsOf_of_forall_field11 below · cited by 1 · depth 23 - Norm of a pullback line bundle is a pullback
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_normModule_pullback_pullback_iso_pullback34 below · cited by 1 · depth 24 - Two-chart Čech H¹ is invariant under base change along R→ R
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_linearEquiv_H1StructureSheaf_symm_eq_H1baseChangeMap_self0 below · cited by 1 · depth 24 - Two affine charts of a non-affine irreducible scheme meet
AlgebraicGeometry.Scheme.TwoAffineOpenCover.nonempty_inf_of_not_isAffine0 below · cited by 1 · depth 24 - A valuation subring of K(X) has at most one centre
AlgebraicGeometry.Scheme.eq_of_forall_mem_valuationSubring_of_isSeparated0 below · cited by 14 · depth 24 - Primes of 𝒪_{X,x} are generisations of x
AlgebraicGeometry.Scheme.exists_fromSpecStalk_specializes_and_mem_iff_residue_eq_zero_and_eq_genericPoint_iff0 below · cited by 5 · depth 24 - Primes of a stalk come from generisations
AlgebraicGeometry.Scheme.exists_specializes_isLocalization_atPrime_stalk0 below · cited by 7 · depth 24 - Stalks of the special fibre of a finite-type morphism are noetherian
AlgebraicGeometry.Scheme.isNoetherianRing_stalk_quotient_map_maximalIdeal_of_locallyOfFiniteType0 below · cited by 1 · depth 24 - Separated target: S-morphisms agreeing at the generic germ coincide
AlgebraicGeometry.Scheme.Hom.eq_of_fromSpecStalk_genericPoint_comp_eq0 below · cited by 2 · depth 25 - Norm of a pulled-back invertible module is its d-th tensor power
AlgebraicGeometry.Scheme.Modules.nonempty_normModule_pullback_iso_tensorPow32 below · cited by 1 · depth 25 - Mumford's formula for symmetric line bundles: [n]^*L≅ L^{⊗ n^2}
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_zpow_iso_tensorPow_of_symmetric573 below · cited by 1 · depth 25 - Clopen open subscheme inherits the five relative properties
AlgebraicGeometry.Scheme.Opens.morphismProperties_inclusion_comp_of_isClosed0 below · cited by 2 · depth 25 - Base change of two-chart Čech H¹ along a surjection
AlgebraicGeometry.Scheme.TwoAffineOpenCover.H1baseChangeMap_surjective_and_eq_iff_of_surjective1 below · cited by 1 · depth 25 - Integral Serre pairing commutes with pull-back along τ
AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.serrePairingInt_map1 below · cited by 2 · depth 25 - Čech H⁰(Ω¹) and H¹(𝒪): freeness and base change
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_free_finrank_kaehlerH0_eq_finrank_structureSheafH1_and_baseChange_of_smoothOfRelativeDimension_one233 below · cited by 1 · depth 25 - Base change of sectional covers and completing Laurent charts
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_isSectional_pullback_and_isCompletionAlong_of_expand_map01_eq1 below · cited by 3 · depth 25 - Base change of a Laurent chart along R → A
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_laurentChart_baseChange1 below · cited by 3 · depth 25 - Base change of a coboundary-annihilating family of Laurent charts
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_laurentChart_baseChange_residuesVanishOnCoboundaries4 below · cited by 1 · depth 25 - Laurent chart along a section with prescribed parameter t₀
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_laurentChart_isCompletionAlong_expand_eq5 below · cited by 1 · depth 25 - Integral Čech Serre pairing equals the function-field residue pairing
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_serrePairingInt_eq_serrePairing_of_isCompletionAlong25 below · cited by 2 · depth 25 - Conormal module of a section of a smooth relative curve is free of rank one
AlgebraicGeometry.Scheme.TwoAffineOpenCover.free_cotangent_sectionAlgHom0 below · cited by 2 · depth 25 - Residues at boundary sections vanish on Čech coboundaries
AlgebraicGeometry.Scheme.TwoAffineOpenCover.residuesVanishOnCoboundaries_of_isSectional_of_isCompletionAlong_of_hasParameter102 below · cited by 2 · depth 25 - Integral Serre pairing perfect from the residue-field fibre
AlgebraicGeometry.Scheme.TwoAffineOpenCover.serrePairingInt_bijective_and_flip_bijective3 below · cited by 1 · depth 25 - Perfectness of the two-chart residue pairing on a smooth proper curve
AlgebraicGeometry.Scheme.TwoAffineOpenCover.serrePairingInt_bijective_and_flip_bijective_of_isSectional_of_isCompletionAlong_of_perfectField134 below · cited by 1 · depth 25 - Finite quotients are quotients for π-nilpotent point functors
AlgebraicGeometry.Scheme.existsUnique_nilpPoints_factor_of_quotient_of_isNoetherianRing2 below · cited by 2 · depth 25 - Quotient of a scheme by a finite group acting admissibly
AlgebraicGeometry.Scheme.exists_quotient_of_finite_of_forall_exists_isAffineOpen_invariant0 below · cited by 17 · depth 25 - Invariant affine neighbourhoods for a finite group action
AlgebraicGeometry.Scheme.forall_exists_isAffineOpen_invariant_of_isSeparated_of_finset0 below · cited by 13 · depth 25 - Separatedness and quasi-compactness descend along integral surjections
AlgebraicGeometry.Scheme.isSeparated_and_quasiCompact_of_isIntegralHom_of_surjective0 below · cited by 5 · depth 25 - Finite quotients of locally finite type schemes over a noetherian base
AlgebraicGeometry.Scheme.locallyOfFiniteType_of_quotient_of_isNoetherianRing1 below · cited by 2 · depth 25 - Pull-back of 𝒪(mp Z) is trivial off the support
AlgebraicGeometry.Scheme.IdealSheafData.nonempty_pullback_obj_iso_unit_of_disjoint_range_support_of_eq_module_or_eq_invModule14 below · cited by 1 · depth 26 - Positivity of the d-th coefficient of the Hilbert polynomial of L
AlgebraicGeometry.Scheme.Modules.FiniteBySections.exists_polynomial_coeff_pos_forall_eulerChar_tensorPow_eq117 below · cited by 2 · depth 26 - Finiteness by sections is stable under tensor product
AlgebraicGeometry.Scheme.Modules.FiniteBySections.tensor2 below · cited by 1 · depth 26 - Affine base change for sections of an invertible sheaf
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_baseChange_sections_linearEquiv_pullback4 below · cited by 8 · depth 26 - Theorem of the cube in pullback form
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_mul_mul_tensor_iso572 below · cited by 3 · depth 26 - Pull-back of A^{⊗ n}⊗ B^{⊗ d} when g^*AcongL, g^*B≅𝒪
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_tensorPow_tensor_tensorPow_iso_tensorPow_of_iso_of_iso_unit1 below · cited by 1 · depth 26 - Opens with the same geometric points coincide
AlgebraicGeometry.Scheme.Opens.eq_of_forall_isAlgClosed_mem_iff_of_locallyOfFiniteType0 below · cited by 1 · depth 26 - Residues commute with pull-back along a morphism over τ
AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.residue_kaehlerMap010 below · cited by 2 · depth 26 - Base change of the two-chart Čech complex of Ω¹
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_baseChangeIsos_kaehlerSections1 below · cited by 5 · depth 26 - Laurent chart from a power-series expansion along a section
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_laurentChart_isCompletionAlong_of_powerSeries0 below · cited by 1 · depth 26 - Global Čech 1-forms are the regular differentials of k(X)
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_linearEquiv_kaehlerSectionsH0_regularDifferentials_apply_eq_kaehlerToFunctionField11 below · cited by 3 · depth 26 - Laurent chart at a rational point extends to the function field
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_ringHom_functionField_laurentSeries_of_isCompletionAlong0 below · cited by 2 · depth 26 - Finiteness of Čech H⁰ and H¹ of Ω¹ for proper morphisms
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finite_H0_H1_kaehlerSections58 below · cited by 1 · depth 26 - h⁰(Ω¹)=h¹(𝒪) for a smooth proper geometrically integral curve
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_kaehlerSections_H0_eq_finrank_structureSheafSections_H1_of_geometricallyIntegral127 below · cited by 1 · depth 26 - Invariance of the two-chart Čech Euler characteristic under field base change
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_ker_sub_finrank_coker_baseChange_eq5 below · cited by 1 · depth 26 - Flatness of Ω_{A_i/R} for a two-chart affine cover
AlgebraicGeometry.Scheme.TwoAffineOpenCover.flat_kaehlerDifferential_cover_of_smooth0 below · cited by 2 · depth 26 - Formal smoothness of the first chart ring of a smooth scheme
AlgebraicGeometry.Scheme.TwoAffineOpenCover.formallySmooth_cover_A00 below · cited by 2 · depth 26 - Freeness and base change of two-chart Čech H¹
AlgebraicGeometry.Scheme.TwoAffineOpenCover.free_H1_structureSheaf_of_isReduced_of_finrank_coker_fibre_const2 below · cited by 1 · depth 26 - Freeness and base change for Čech H⁰ of differentials
AlgebraicGeometry.Scheme.TwoAffineOpenCover.free_kaehlerH0_of_isReduced_of_finrank_ker_fibre_const3 below · cited by 1 · depth 26 - Čech global 1-forms are regular differentials
AlgebraicGeometry.Scheme.TwoAffineOpenCover.kaehlerToFunctionField_mem_regularDifferentials10 below · cited by 4 · depth 26 - Descent of π-nilpotent point families along an affine finite quotient
AlgebraicGeometry.Scheme.existsUnique_nilpPoints_factor_of_quotient_of_isAffine_of_isAffine_of_isNoetherianRing1 below · cited by 1 · depth 26 - Categorical quotients by finite groups restrict to clopen stable pieces
AlgebraicGeometry.Scheme.exists_opens_isClosed_preimage_eq_existsUnique_of_quotient0 below · cited by 1 · depth 26 - V-valued tangent vectors as maps out of the cotangent space
AlgebraicGeometry.Scheme.exists_tangentPoints_equiv_linearMap_cotangentSpace2 below · cited by 4 · depth 26 - Flatness and finite type for an affine invariant quotient
AlgebraicGeometry.Scheme.flat_and_locallyOfFiniteType_of_isAffineHom_of_invariants0 below · cited by 1 · depth 26 - Module of an invertible ideal sheaf: invertibility and function-field presentation
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.isInvertible_module_and_exists_presentation_isFrameOn3 below · cited by 1 · depth 27 - Ideal sheaf pulled back off its support becomes the unit ideal
AlgebraicGeometry.Scheme.IdealSheafData.comap_eq_top_and_nonempty_module_iso_and_nonempty_invModule_iso_of_disjoint_range_support6 below · cited by 1 · depth 27 - Gluing a Cartier datum into an invertible ideal sheaf
AlgebraicGeometry.Scheme.IdealSheafData.exists_isInvertible_ideal_eq_span_of_locallyPrincipalDatum0 below · cited by 1 · depth 27 - Meet of incomparable invertible ideal sheaves with integral subschemes
AlgebraicGeometry.Scheme.IdealSheafData.inf_eq_mul_of_isInvertible_of_isIntegral_subscheme0 below · cited by 1 · depth 27 - Base change and constant rank of finite flat closed subschemes
AlgebraicGeometry.Scheme.IdealSheafData.isFinite_and_flat_and_finrank_subscheme_comap_comp_eq_of_isPullback0 below · cited by 1 · depth 27 - Descent of ideal invertibility and a tensor identity along an isomorphism
AlgebraicGeometry.Scheme.IdealSheafData.nonempty_iso_invModule_tensor_module_of_pullback_tensor_invModule_iso_invModule_of_isIso14 below · cited by 1 · depth 27 - Sections of a tensor product of invertible sheaves on an affine open
AlgebraicGeometry.Scheme.Modules.IsInvertible.bijective_lift_tensorSectionsBilin8 below · cited by 2 · depth 27 - Invertible sheaves on integral schemes embed in K(X)
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_functionField_presentation0 below · cited by 6 · depth 27 - Invertible sheaf on an integral scheme as fractional ideal
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_idealSheafData_tensor_linearEquiv_of_presentation7 below · cited by 3 · depth 27 - Invertible modules are framed near finitely many closed points
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isFrameOn_of_finite_subset_affineOpen9 below · cited by 2 · depth 27 - Two-chart gluing data of a line bundle trivial modulo a nilpotent ideal
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_sectionsOf_equiv_lineBundle_one_add_of_pullback_quotient_isNilpotent19 below · cited by 1 · depth 27 - Global sections of powers of an ample sheaf grow like nᵈ
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_forall_mul_pow_le_cechFinrank_zero_tensorPow64 below · cited by 1 · depth 27 - Finiteness of the morphism defined by L^{⊗ 3}
AlgebraicGeometry.Scheme.Modules.ProjPresentation.isFinite_toProj_of_finite_setOf_forall_pullbackSection_eq_zero_iff24 below · cited by 1 · depth 27 - Sections over an affine open of a tensor product of quasi-coherent modules
AlgebraicGeometry.Scheme.Modules.bijective_lift_tensorSectionsBilin_of_isQuasicoherent0 below · cited by 3 · depth 27 - Invertible sheaf with non-zero section of finite stabiliser
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_hom_ne_zero_finite_setOf_stabilizer43 below · cited by 1 · depth 27 - Theorem of the square for line bundles on an abelian variety
AlgebraicGeometry.Scheme.Modules.forall_nonempty_pullback_translate_tensor_iso573 below · cited by 1 · depth 27 - Theorem of the cube over an algebraically closed field
AlgebraicGeometry.Scheme.Modules.nonempty_iso_tensorUnit_of_pullback_three_slices563 below · cited by 1 · depth 27 - Frames and transition function pull back along a `HomOver`
AlgebraicGeometry.Scheme.TwoAffineOpenCover.HomOver.exists_isFrameOn_pullback_of_map_eq_smul6 below · cited by 2 · depth 27 - Transition function of a pulled-back chart-trivial line bundle
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_sectionsOf_pullback_stageHom_equiv_lineBundle_appLE13 below · cited by 1 · depth 27 - Integrality of a base change via two base-changed chart rings
AlgebraicGeometry.Scheme.TwoAffineOpenCover.isIntegral_pullback_and_nonempty_of_isDomain_tensorProduct1 below · cited by 1 · depth 27 - Two chart-trivial line bundles are isomorphic iff their cocycles agree
AlgebraicGeometry.Scheme.TwoAffineOpenCover.nonempty_iso_iff_exists_units_of_sectionsOf_equiv_lineBundle12 below · cited by 1 · depth 27 - Morphisms glue over a cover by pairwise disjoint opens
AlgebraicGeometry.Scheme.existsUnique_forall_opensInclusion_comp_eq_of_iSup_eq_top_of_disjoint0 below · cited by 1 · depth 27 - Finite-group quotients commute with flat base change
AlgebraicGeometry.Scheme.exists_iso_quotient_pullback_of_flat5 below · cited by 2 · depth 27 - Proper closed subsets of a Noetherian integral curve are finite
AlgebraicGeometry.Scheme.finite_of_isClosed_of_ne_univ_of_forall_isClosed_singleton0 below · cited by 2 · depth 27 - Stalkwise divisibility by a non-zero-divisor gives divisibility of sections
AlgebraicGeometry.Scheme.span_singleton_le_span_singleton_of_forall_germ_eq_mul0 below · cited by 1 · depth 27 - Geometric points lift along integral surjections
AlgebraicGeometry.Scheme.Hom.exists_comp_eq_of_isIntegralHom_of_surjective0 below · cited by 1 · depth 28 - Stalk map at the generic point is an isomorphism
AlgebraicGeometry.Scheme.Hom.isIso_stalkMap_genericPoint_of_isReduced_of_forall_specializes_of_forall_exists_div1 below · cited by 1 · depth 28 - A finite scheme over a field has at most dim_k points
AlgebraicGeometry.Scheme.Hom.natCard_le_finrank_of_isFinite0 below · cited by 2 · depth 28 - Invertibility of an invertible ideal sheaf pulled back to a reduced scheme
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.comap_of_isReduced_of_forall_specializes1 below · cited by 1 · depth 28 - Euler characteristic of twists by n degree-one points
AlgebraicGeometry.Scheme.IdealSheafData.IsInvertible.eulerChar_sectionsOf_tensor_tensorPow_foldr_module_eq_sub_and_invModule_eq_add_of_finrank_eq_one108 below · cited by 1 · depth 28 - Kernel of the closure of a generic-fibre subscheme over a DVR
AlgebraicGeometry.Scheme.IdealSheafData.comap_ker_subschemeIota_comp_mapOnProdOver_eq_and_saturated_of_isDiscreteValuationRing0 below · cited by 2 · depth 28 - Local principal regular generator from invertibility after open immersion
AlgebraicGeometry.Scheme.IdealSheafData.exists_map_ideal_eq_span_singleton_of_isInvertible_comap_of_isOpenImmersion0 below · cited by 1 · depth 28 - Stalkwise principal ideal sheaves are invertible
AlgebraicGeometry.Scheme.IdealSheafData.isInvertible_of_forall_exists_map_ideal_eq_span_singleton_of_mem_nonZeroDivisors0 below · cited by 1 · depth 28 - Tensor product of locally trivial 𝒪_X-modules is locally trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.tensor_monoidalV22 below · cited by 133 · depth 28 - Base-point freeness of L ⊗ H^{⊗ n} for large n
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_forall_le_forall_exists_notMem_support_zeroSchemeIdeal_tensor_tensorPow17 below · cited by 1 · depth 28 - Sections of n base-point-free bundles avoiding a small closed set
AlgebraicGeometry.Scheme.Modules.exists_inter_iInter_support_zeroSchemeIdeal_eq_empty_of_topologicalKrullDim_lt3 below · cited by 1 · depth 28 - Affine open complements as zero loci of line bundle sections
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_pullbackSection_eq_zero_iff_notMem_of_isAffineOpen41 below · cited by 1 · depth 28 - Theorem of the cube with a curve factor
AlgebraicGeometry.Scheme.Modules.nonempty_iso_tensorUnit_of_pullback_three_slices_of_smoothOfRelativeDimension_one527 below · cited by 1 · depth 28 - Dividing a Čech 0-cocycle of Kähler differentials by varpi
AlgebraicGeometry.Scheme.TwoAffineOpenCover.exists_eq_smul_kaehlerH0_and_val_eq_of_val_eq_smul0 below · cited by 1 · depth 28 - Two-chart Čech cohomology of 𝒪_C inside the function field
AlgebraicGeometry.Scheme.TwoAffineOpenCover.finrank_structureSheafSections_eq_finrank_span_germ0 below · cited by 1 · depth 28 - Natural maps on affine points come from a unique morphism
AlgebraicGeometry.Scheme.existsUnique_hom_over_of_forall_schemeHomOver0 below · cited by 7 · depth 28 - Finiteness of the stabiliser of a closed subset in a proper group scheme
AlgebraicGeometry.Scheme.finite_setOf_forall_mem_iff_mul_mem_of_isClosed_of_compl_subset_of_isAffineOpen0 below · cited by 1 · depth 28 - Membership in the image prime versus vanishing of the pulled-back function
AlgebraicGeometry.Scheme.mem_asIdeal_base_iff_residue_germ_appTop_eq_zero0 below · cited by 2 · depth 28 - Prime-element criterion for regularity at a point of an integral scheme
AlgebraicGeometry.Scheme.mem_range_algebraMap_stalk_functionField_of_forall_specializes_isUnit_of_exists_mul_eq0 below · cited by 1 · depth 28 - Noetherian approximation of natural families of π-adic points
AlgebraicGeometry.Scheme.nilpPoints.forall_eq_of_forall_eq_of_isNoetherianRing_of_forall_isIdempotentElem1 below · cited by 6 · depth 28 - Agreement on trivial-idempotent algebras implies agreement everywhere
AlgebraicGeometry.Scheme.nilpPoints.forall_isNoetherianRing_eq_of_forall_eq_of_forall_isIdempotentElem0 below · cited by 12 · depth 28 - Orbit fibres persist under flat base change
AlgebraicGeometry.Scheme.orbit_iff_of_quotient_pullback_of_flat4 below · cited by 1 · depth 28 - Flat base change preserves the invariant-sections quotient conditions
AlgebraicGeometry.Scheme.quotientInvariants_pullback_of_flat2 below · cited by 2 · depth 28 - Range of the base-change comparison X×_S T'→ X×_S T
AlgebraicGeometry.Scheme.range_pullbackMap_id_id_eq_preimage_range0 below · cited by 2 · depth 28 - Morphisms agreeing at the k-points of a dense open
AlgebraicGeometry.Scheme.Hom.eq_of_forall_comp_eq_of_dense_of_isAlgClosed1 below · cited by 3 · depth 29 - Morphisms agreeing on a dense set of field-valued points
AlgebraicGeometry.Scheme.Hom.eq_of_forall_comp_eq_of_dense_of_isReduced0 below · cited by 2 · depth 29 - Kernel of a quasi-compact morphism from an integral scheme
AlgebraicGeometry.Scheme.Hom.ker_eq_vanishingIdeal_closure_singleton_genericPoint0 below · cited by 1 · depth 29 - Germ of the kernel ideal equals kernel of the stalk map
AlgebraicGeometry.Scheme.Hom.map_germ_ker_ideal_eq_ker_stalkMap0 below · cited by 4 · depth 29 - Surjectivity at the generic point from quotients of pulled-back sections
AlgebraicGeometry.Scheme.Hom.stalkMap_genericPoint_surjective_of_forall_exists_div0 below · cited by 1 · depth 29 - Ideal sheaf of a closed immersion Spec K → C
AlgebraicGeometry.Scheme.Hom.support_ker_eq_singleton_and_map_germ_ker_ideal_eq_maximalIdeal_of_field1 below · cited by 1 · depth 29 - Principal regular stalk ideal descends to a generator in I(U)
AlgebraicGeometry.Scheme.IdealSheafData.exists_mem_ideal_and_map_eq_span_singleton_and_mem_nonZeroDivisors_of_map_germ_eq_span_singleton1 below · cited by 1 · depth 29 - Saturation and regularity pass to the stalk
AlgebraicGeometry.Scheme.IdealSheafData.forall_germ_mul_mem_map_imp_and_germ_mem_nonZeroDivisors_of_forall_mul_mem_imp0 below · cited by 1 · depth 29 - A principal ideal sheaf on a non-zero-divisor is invertible with trivial dual
AlgebraicGeometry.Scheme.IdealSheafData.isInvertible_and_nonempty_invModule_iso_tensorUnit_ofIdealTop_span_singleton22 below · cited by 1 · depth 29 - Germ ideals: chart independence, extensionality, support dichotomy
AlgebraicGeometry.Scheme.IdealSheafData.map_germ_ideal_eq_and_ext_and_support_dichotomy0 below · cited by 4 · depth 29 - Germ at x of the vanishing ideal of ξ̄ is the branch ideal
AlgebraicGeometry.Scheme.IdealSheafData.map_germ_vanishingIdeal_closure_eq_branchIdeal0 below · cited by 1 · depth 29 - Germ ideals under stalk maps: comap and kernel sheaves
AlgebraicGeometry.Scheme.IdealSheafData.map_stalkMap_map_germ_ideal_le_and_map_germ_ker_eq_bot0 below · cited by 1 · depth 29 - Failure of stalk ideal containment at a specialisation point
AlgebraicGeometry.Scheme.IdealSheafData.not_map_germ_ideal_le_of_specializes_of_notMem_support_of_mem_support1 below · cited by 1 · depth 29 - Local seesaw: fibrewise trivial invertible sheaf is locally trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_nonempty_pullback_baseChangeSnd_iso_unit_of_forall_point91 below · cited by 1 · depth 29 - Presentation by a section basis gives a closed immersion
AlgebraicGeometry.Scheme.Modules.ProjPresentation.isClosedImmersion_toProj_of_isSectionBasis_of_closedImmersionBySections5 below · cited by 3 · depth 29 - Invariant sections over basic opens after flat base change
AlgebraicGeometry.Scheme.app_basicOpen_injective_and_range_eq_invariants_of_flat0 below · cited by 1 · depth 29 - Sections as invariants: from a basis to all opens
AlgebraicGeometry.Scheme.app_injective_and_range_eq_invariants_of_isBasis0 below · cited by 1 · depth 29 - Bijectivity of A → Γ(X_A,𝒪) under base change
AlgebraicGeometry.Scheme.bijective_algebraMap_sections_baseChange_of_bijective_of_field13 below · cited by 2 · depth 29 - Invariant presentations of finite quotients are categorical quotients
AlgebraicGeometry.Scheme.existsUnique_comp_eq_of_isAffineHom_of_invariants1 below · cited by 1 · depth 29 - Yoneda for affine-valued points: morphisms from natural families
AlgebraicGeometry.Scheme.existsUnique_hom_spec_comp_eq_of_natural0 below · cited by 8 · depth 29 - Affine base change of sections over an affine open
AlgebraicGeometry.Scheme.exists_algEquiv_tensor_sections_pullback_fst_preimage_of_isAffineOpen0 below · cited by 2 · depth 29 - Formal smoothness of the stalk map via an affine chart
AlgebraicGeometry.Scheme.exists_chart_formallySmooth_stalkMap_of_formallySmooth_localization0 below · cited by 2 · depth 29 - Gluing a cartesian family over the affine opens
AlgebraicGeometry.Scheme.exists_hom_isPullback_opensInclusion_of_forall_affineOpens_isPullback0 below · cited by 1 · depth 29 - Ω-point centred at P from an R-linear residue field embedding
AlgebraicGeometry.Scheme.exists_hom_spec_comp_eq_specMap_algebraMap_and_apply_eq_of_residueField0 below · cited by 1 · depth 29 - Descent of a G-scheme along a Galois extension 𝒪 → 𝒪'
AlgebraicGeometry.Scheme.exists_quotient_isPullback_of_galois_of_finite_action22 below · cited by 1 · depth 29 - Finitely presented points over a directed colimit descend to a finite stage
AlgebraicGeometry.Scheme.exists_specMap_comp_eq_of_directed_colimit_of_locallyOfFinitePresentation0 below · cited by 1 · depth 29 - Scheme-level Artinian lifting restricts to ring-level lifting on an affine chart
AlgebraicGeometry.Scheme.forall_exists_algHom_lift_of_forall_exists_lift_of_isOpenImmersion1 below · cited by 1 · depth 29 - Uniqueness of Spec K-points above a point with prime residue field
AlgebraicGeometry.Scheme.hom_ext_of_field_of_apply_eq_of_surjective_zmod_residueField0 below · cited by 1 · depth 29 - Stalk kernels at a crossing are the branch ideals
AlgebraicGeometry.Scheme.ker_stalkMap_eq_branchIdeal_and_branchIdeal_sup_branchIdeal_eq_maximalIdeal_of_isReduced_pullback3 below · cited by 1 · depth 29 - Unique natural extension from connected Noetherian test algebras
AlgebraicGeometry.Scheme.nilpPoints.existsUnique_forall_isNoetherianRing_extension_of_forall_isIdempotentElem1 below · cited by 1 · depth 29 - Unique morphism induced by natural maps on nilpotent points
AlgebraicGeometry.Scheme.nilpPoints.existsUnique_hom_comp_eq_of_natural0 below · cited by 1 · depth 29 - Fibres of an affine invariant morphism are the H-orbits
AlgebraicGeometry.Scheme.orbit_iff_of_isAffineHom_of_invariants0 below · cited by 1 · depth 29 - Relative spectrum of a module-finite algebra is finite
AlgebraicGeometry.Scheme.AffineZariskiSite.isFinite_toBase_relativeGluingData0 below · cited by 6 · depth 30 - Pushouts of sections detect base change into a relative spectrum
AlgebraicGeometry.Scheme.AffineZariskiSite.isPullback_toBase_relativeGluingData_of_forall_isPushout0 below · cited by 2 · depth 30 - Chart cocycle data pull back along a scheme morphism
AlgebraicGeometry.Scheme.Hom.app_cocycle_and_basicOpen_app_eq_inf_of_basicOpen_eq_inf0 below · cited by 2 · depth 30 - Trivial kernel for base change along Spec of a k₀-algebra
AlgebraicGeometry.Scheme.Hom.ker_pullback_fst_specMap_eq_bot_of_field0 below · cited by 1 · depth 30 - Being a frame is local on the open: stability under suprema
AlgebraicGeometry.Scheme.Modules.IsFrameOn.of_iSup_monoidalV20 below · cited by 2 · depth 30 - Invertibility of the dual and L ⊗ L^∨ ≅ 𝒪_X
AlgebraicGeometry.Scheme.Modules.IsInvertible.dual_monoidalV20 below · cited by 119 · depth 30 - Seesaw: local framing of a fibrewise trivially invertible sheaf
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_bijective_smul_of_le_preimage_basicOpen_of_forall_isMaximal85 below · cited by 1 · depth 30 - Fibrewise criterion: global sections of an invertible module are frames
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isFrameOn_of_forall_geometricFibre_exists_isFrameOn_of_forall_eq_sum_smul_pullbackLocalSection5 below · cited by 1 · depth 30 - Finiteness of global sections of an invertible sheaf on a proper k-scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.finite_sections_of_isProper61 below · cited by 6 · depth 30 - Pullback of the dual of an invertible sheaf
AlgebraicGeometry.Scheme.Modules.IsInvertible.pullback_dual_monoidalV21 below · cited by 77 · depth 30 - Tensor powers of an invertible module are invertible
AlgebraicGeometry.Scheme.Modules.IsInvertible.tensorPow_monoidalV23 below · cited by 22 · depth 30 - Some presenting section frames M near every point
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_mem_isFrameOn0 below · cited by 6 · depth 30 - Closed immersion from a presentation by spanning pulled-back sections
AlgebraicGeometry.Scheme.Modules.ProjPresentation.isClosedImmersion_toProj_of_closedImmersionBySections_of_forall_eq_sum_smul_pullbackLocalSection5 below · cited by 1 · depth 30 - Closed immersions transfer to presentations with spanning sections
AlgebraicGeometry.Scheme.Modules.ProjPresentation.isClosedImmersion_toProj_of_forall_exists_eq_sum_smul4 below · cited by 4 · depth 30 - Transport of sections base change to any cartesian square
AlgebraicGeometry.Scheme.Modules.exists_linearEquiv_tensorProduct_sections_pullback_of_isPullback0 below · cited by 4 · depth 30 - Complete linear system as a projective presentation
AlgebraicGeometry.Scheme.Modules.exists_projPresentation_isSectionBasis_of_finite_sections_of_forall_exists_isFrameOn1 below · cited by 3 · depth 30 - Cohomology and base change in degree zero
AlgebraicGeometry.Scheme.Modules.finite_projective_sections_and_exists_linearEquiv_tensorProduct_pullbackLocalSection_of_forall_subsingleton_HSucc86 below · cited by 5 · depth 30 - Pulled-back generators span sections after base change
AlgebraicGeometry.Scheme.Modules.forall_exists_eq_sum_smul_pullbackLocalSection_of_span_eq_top_of_linearEquiv_tensorProduct0 below · cited by 2 · depth 30 - Fibrewise h⁰ is unchanged under base change of the family
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_comp_eq_of_isPullback_of_iso0 below · cited by 10 · depth 30 - Sections over a base-changed affine overlap as a tensor product
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_ringEquiv_tensor_sections_baseChange_inter1 below · cited by 17 · depth 30 - G-stable affine open containing a finite set of points
AlgebraicGeometry.Scheme.exists_isAffineOpen_forall_mem_forall_preimage_eq_of_isSeparated_of_finset0 below · cited by 1 · depth 30 - Compatible finitely generated ideals cut out a finitely presented closed subscheme
AlgebraicGeometry.Scheme.exists_isClosedImmersion_locallyOfFinitePresentation_forall_exists_comp_eq_iff_ideal_eq_bot0 below · cited by 3 · depth 30 - Immersions into projective space descend along a finite extension
AlgebraicGeometry.Scheme.exists_isImmersion_proj_comp_of_isImmersion_proj_of_finite_free2 below · cited by 1 · depth 30 - Ample chart datum yields an immersion into projective space
AlgebraicGeometry.Scheme.exists_isImmersion_proj_of_affineCover_cocycle_basicOpen_eq_of_locallyOfFiniteType4 below · cited by 2 · depth 30 - Quasi-projectivity descends along a finite-group quotient
AlgebraicGeometry.Scheme.exists_isImmersion_proj_of_isIntegralHom_of_quotient_of_finite_of_isImmersion_proj12 below · cited by 2 · depth 30 - Schemes over an affine base are determined by affine points
AlgebraicGeometry.Scheme.exists_iso_over_of_forall_schemeHomOver_equiv0 below · cited by 1 · depth 30 - Germs prime to P become a unit ratio near ξ_P
AlgebraicGeometry.Scheme.exists_opens_fromSpecStalk_mem_and_forall_exists_unit_mul_eq_of_not_mem_prime0 below · cited by 2 · depth 30 - Galois descent: an invariant affine quotient square is cartesian
AlgebraicGeometry.Scheme.isPullback_of_quotient_of_galois_of_finite_action1 below · cited by 1 · depth 30 - Points factor through finitely generated subalgebras
AlgebraicGeometry.Scheme.nilpPoints.exists_subalgebra_fg_map_eq_of_locallyOfFiniteType0 below · cited by 4 · depth 30 - Quasi-compact schemes have finite ordered affine covers
AlgebraicGeometry.Scheme.nonempty_orderedAffineCover_of_compactSpace0 below · cited by 9 · depth 30 - Image of Spec of a local ring lies in any open containing the closed point
AlgebraicGeometry.Scheme.range_subset_of_isLocalRing_of_closedPoint_mem0 below · cited by 2 · depth 30 - Immersion criterion from affine charts with surjective sections
AlgebraicGeometry.Scheme.Hom.isImmersion_of_forall_isAffineOpen_preimage_of_forall_surjective_app0 below · cited by 1 · depth 31 - Uniform exponent for ideal sheaf data on a Noetherian scheme
AlgebraicGeometry.Scheme.IdealSheafData.exists_forall_ideal_pow_le_of_forall_le_radical0 below · cited by 1 · depth 31 - Closed immersion by sections is stable under base change
AlgebraicGeometry.Scheme.Modules.ClosedImmersionBySections.pullback_of_isPullback4 below · cited by 6 · depth 31 - Tensor with a finite-by-sections module stays finite by sections
AlgebraicGeometry.Scheme.Modules.FiniteBySections.tensor_of_projPresentation_monoidalV22 below · cited by 2 · depth 31 - Tensor product of two frames is a frame
AlgebraicGeometry.Scheme.Modules.IsFrameOn.tensorSections_monoidalV20 below · cited by 12 · depth 31 - Invertible module on an affine scheme: finite principal trivialising cover
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_finite_away_cover_trivial3 below · cited by 3 · depth 31 - Injectivity on dual-number points of a complete linear system
AlgebraicGeometry.Scheme.Modules.ProjPresentation.eq_of_comp_toProj_eq_of_isSectionBasis_of_forall_exists_pullbackSection7 below · cited by 1 · depth 31 - Projective presentations with equal XtoP^N are framed-isomorphic
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_iso_forall_app_eq_of_toProj_eq2 below · cited by 3 · depth 31 - Pullback of a P^N_R-presentation along a morphism of schemes
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_projPresentation_pullback_sigma_eq_toProj_eq1 below · cited by 17 · depth 31 - Transport of a Proj presentation along an isomorphism
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_sigma_eq_app_unit_and_toProj_eq_comp_of_iso2 below · cited by 1 · depth 31 - Re-basing a P^N presentation along R → A
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_sigma_eq_toProj_eq_comp_map_of_algebraMap3 below · cited by 4 · depth 31 - Presentations with equal map to P^N differ by a base unit
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_unit_appTop_smul_eq_of_toProj_eq_of_bijective1 below · cited by 2 · depth 31 - Proj presentations of a fixed morphism differ by a global unit
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_unit_smul_eq_of_toProj_eq0 below · cited by 4 · depth 31 - Injectivity on points transfers between Proj presentations
AlgebraicGeometry.Scheme.Modules.ProjPresentation.injective_toProj_of_forall_exists_eq_sum_smul2 below · cited by 1 · depth 31 - Uniqueness of a module presenting a map to P^N_R
AlgebraicGeometry.Scheme.Modules.ProjPresentation.nonempty_iso_of_toProj_eq2 below · cited by 4 · depth 31 - Surjectivity on stalks transfers along R-linear spans of presentations
AlgebraicGeometry.Scheme.Modules.ProjPresentation.surjectiveOnStalks_of_forall_exists_eq_sum_smul1 below · cited by 1 · depth 31 - A projective presentation is determined by its sections
AlgebraicGeometry.Scheme.Modules.ProjPresentation.toProj_eq_of_forall_sections_eq_univ1 below · cited by 8 · depth 31 - Unit-proportional sections give the same morphism to P^N
AlgebraicGeometry.Scheme.Modules.ProjPresentation.toProj_eq_of_sigma_eq_smul2 below · cited by 3 · depth 31 - Degree-zero base change over a reduced base with constant h⁰
AlgebraicGeometry.Scheme.Modules.exists_eq_sum_smul_pullbackSection_of_isReduced_of_finrank_eq76 below · cited by 1 · depth 31 - Descent of tensor base change of sections along a cartesian comparison
AlgebraicGeometry.Scheme.Modules.exists_linearEquiv_tensorProduct_sections_pullback_of_forall_isPullback_of_iso0 below · cited by 1 · depth 31 - Proj presentation of a cocycle-glued module with prescribed charts
AlgebraicGeometry.Scheme.Modules.exists_projPresentation_glueOfCocycle_preimage_basicOpen_eq_of_basicOpen_eq_inf1 below · cited by 1 · depth 31 - Projective presentation from a section basis is a closed immersion
AlgebraicGeometry.Scheme.Modules.exists_projPresentation_sigma_eq_isClosedImmersion_of_isSectionBasis10 below · cited by 1 · depth 31 - Fibrewise h⁰ is invariant under field extension
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_comp_eq76 below · cited by 9 · depth 31 - Geometric-fibre h⁰ is invariant under cartesian base change
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_eq_of_isPullback0 below · cited by 25 · depth 31 - Geometric fibre h⁰ along id_k equals dim_kΓ(A,M)
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_id_eq_finrank_sections16 below · cited by 7 · depth 31 - Closedness of {n≤ h⁰} for a proper flat family
AlgebraicGeometry.Scheme.Modules.isClosed_setOf_le_finrank_sections_pullback_residueField75 below · cited by 6 · depth 31 - Basis condition transfers along compatible P^N-presentations
AlgebraicGeometry.Scheme.Modules.isSectionBasisOn_pullback_comp_iff_of_toProj_comp_eq12 below · cited by 1 · depth 31 - Section bases descend through pullback along the identity
AlgebraicGeometry.Scheme.Modules.isSectionBasis_of_isSectionBasisOn_pullback_id0 below · cited by 1 · depth 31 - Gluing module isomorphisms over a finite product of rings
AlgebraicGeometry.Scheme.Modules.nonempty_iso_of_forall_nonempty_pullback_iso_of_isPullback_pi2 below · cited by 1 · depth 31 - Pull-back comparison isomorphism on local sections
AlgebraicGeometry.Scheme.Modules.pullbackComp_hom_app_pullbackLocalSection0 below · cited by 15 · depth 31 - Triple affine refinement on a self-product along two projections and a third map
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_refinement_pullback_fst_snd_of_isSeparated1 below · cited by 8 · depth 31 - Affine product charts in a fibre product of schemes
AlgebraicGeometry.Scheme.Pullback.isAffineOpen_and_exists_algEquiv_tensor_sections_fst_preimage_inf_snd_preimage0 below · cited by 7 · depth 31 - Trivial idempotents force connectedness of a non-empty scheme
AlgebraicGeometry.Scheme.connectedSpace_of_forall_isIdempotentElem0 below · cited by 2 · depth 31 - Idempotents in Γ(X,mathcal O_X) on a preconnected scheme
AlgebraicGeometry.Scheme.eq_zero_or_eq_one_of_isIdempotentElem_of_preconnectedSpace0 below · cited by 1 · depth 31 - A-valued points of X centred at a K-point ̄ x
AlgebraicGeometry.Scheme.existsUnique_ringHom_stalk_comp_eq_and_specMap_comp_fromSpecStalk_eq0 below · cited by 3 · depth 31 - Gluing sections of a scheme over a principal open cover of Spec S
AlgebraicGeometry.Scheme.existsUnique_section_of_forall_away0 below · cited by 2 · depth 31 - Descent of an invariant cocycle chart datum along q
AlgebraicGeometry.Scheme.exists_affineCover_cocycle_basicOpen_eq_of_quotient_of_invariant0 below · cited by 1 · depth 31 - Extension of a section to a cocycle-compatible family
AlgebraicGeometry.Scheme.exists_forall_exists_eq_and_map_eq_pow_mul_map_of_basicOpen_eq_inf0 below · cited by 1 · depth 31 - Invariant affine cover with invariant cocycle units for a finite group action
AlgebraicGeometry.Scheme.exists_invariant_affineCover_cocycle_basicOpen_eq_of_finite_of_isImmersion_proj5 below · cited by 1 · depth 31 - Invariant affine open neighbourhoods for finite group actions
AlgebraicGeometry.Scheme.exists_isAffineOpen_mem_forall_preimage_eq_of_forall_finset_of_isSeparated0 below · cited by 1 · depth 31 - Four-layer locally closed subfunctor is representable
AlgebraicGeometry.Scheme.exists_isOpenImmersion_isClosedImmersion_forall_exists_comp_eq_iff_of_ideal_of_isOpen2 below · cited by 1 · depth 31 - Free finite quotients are finite flat étale G-torsors
AlgebraicGeometry.Scheme.finite_flat_and_locally_eq_comp_of_free_of_quotient3 below · cited by 1 · depth 31 - Permanence properties along a finite flat group quotient
AlgebraicGeometry.Scheme.isSeparated_quasiCompact_locallyOfFinitePresentation_of_quotient_of_finite8 below · cited by 1 · depth 31 - Quasi-compact schemes admit finite ordered affine covers
AlgebraicGeometry.Scheme.nonempty_orderedAffineCover_of_compactSpace_univ0 below · cited by 3 · depth 31 - Functoriality of the relative spectrum in the algebra
AlgebraicGeometry.Scheme.AffineZariskiSite.exists_hom_glued_comp_toBase_eq_of_comp_eq0 below · cited by 1 · depth 32 - Pulled-back frame of top differentials freely generates on charts
AlgebraicGeometry.Scheme.Hom.bijective_smul_topFormMap_of_isFrameOn_of_isPullback21 below · cited by 1 · depth 32 - Uniqueness of pullback maps on top differentials via affine charts
AlgebraicGeometry.Scheme.Hom.eq_of_map_pullbackLocalSection_topToSections_eq5 below · cited by 2 · depth 32 - Chain rule for pull-back maps on top differentials
AlgebraicGeometry.Scheme.Hom.eq_pullbackComp_inv_app_comp_map_comp_of_map_pullbackLocalSection_topToSections_eq8 below · cited by 1 · depth 32 - Local basis of Ω¹_{X/A} for smooth f of relative dimension d
AlgebraicGeometry.Scheme.Hom.exists_basis_kaehler_of_isAffineOpen_of_smoothOfRelativeDimension2 below · cited by 4 · depth 32 - Affine morphisms are relative spectra of their direct-image algebra
AlgebraicGeometry.Scheme.Hom.exists_coequifibered_iso_glued_comp_toBase_eq_of_isAffineHom0 below · cited by 1 · depth 32 - Pullback morphism on top differentials, computed on affine charts
AlgebraicGeometry.Scheme.Hom.exists_hom_pullback_topDifferentials_map_pullbackLocalSection_topToSections_eq1 below · cited by 2 · depth 32 - Additivity of finrank over a clopen decomposition of the source
AlgebraicGeometry.Scheme.Hom.finrank_restrict_add_finrank_restrict_of_isCompl0 below · cited by 1 · depth 32 - Base change isomorphism for top relative differentials, smooth case
AlgebraicGeometry.Scheme.Hom.isIso_of_map_pullbackLocalSection_topToSections_eq_of_isPullback_of_smoothOfRelativeDimension15 below · cited by 2 · depth 32 - Smooth of relative dimension d: Ω¹ locally free, ωᵈ invertible
AlgebraicGeometry.Scheme.Hom.isLocallyFreeOfRank_kaehler_and_topDifferentials_of_smoothOfRelativeDimension9 below · cited by 1 · depth 32 - Sheafification does not change differentials on affine opens
AlgebraicGeometry.Scheme.Hom.kaehlerToSections_bijective_of_isAffineOpen1 below · cited by 6 · depth 32 - Top differentials over an affine open compute as an exterior power
AlgebraicGeometry.Scheme.Hom.topToSections_bijective_of_isAffineOpen4 below · cited by 7 · depth 32 - Segre: external tensor product of closed immersions by sections
AlgebraicGeometry.Scheme.Modules.ClosedImmersionBySections.pullback_fst_tensor_pullback_snd7 below · cited by 1 · depth 32 - Positive d-th coefficient of the Hilbert polynomial of L
AlgebraicGeometry.Scheme.Modules.FiniteBySections.exists_polynomial_coeff_pos_forall_eulerChar_tensorPow_eq_monoidalV2118 below · cited by 2 · depth 32 - Zero-scheme ideal of a section commutes with base change
AlgebraicGeometry.Scheme.Modules.IsInvertible.comap_zeroSchemeIdeal_monoidalV23 below · cited by 6 · depth 32 - Sections of an invertible module spread out over a direct limit
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_app_unit_eq_of_isDirectLimit16 below · cited by 2 · depth 32 - Gluing invertible modules over a basic open cover, up to base-local isomorphism
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_locIsoOnBase_pullback_of_forall_away_of_locIsoOnBase34 below · cited by 1 · depth 32 - Surjectivity of Pic(X)toPic(U) for locally factorial X
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isInvertible_pullback_iso_of_isOpenImmersion_of_uniqueFactorizationMonoid_stalk24 below · cited by 1 · depth 32 - Finiteness of Γ of an invertible module after base change to k'
AlgebraicGeometry.Scheme.Modules.IsInvertible.finite_sections_pullback_fst_of_abelianSchemePropertyBundle72 below · cited by 2 · depth 32 - Nonzero section of a line bundle is nonvanishing at the generic point
AlgebraicGeometry.Scheme.Modules.IsInvertible.genericPoint_notMem_support_zeroSchemeIdeal_monoidalV23 below · cited by 6 · depth 32 - Sections of invertible modules frame off the zero scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.isFrameOn_app_of_disjoint_support_zeroSchemeIdeal_monoidalV24 below · cited by 5 · depth 32 - Descent of isomorphisms of invertible modules along faithfully flat base change
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_nonempty_pullback_iso_of_faithfullyFlat_of_isLocalRing18 below · cited by 2 · depth 32 - Invertible modules on a smooth R-scheme are determined generically
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_nonempty_pullback_iso_of_isDiscreteValuationRing37 below · cited by 1 · depth 32 - Zero scheme of a tensor product of sections: Z(s⊗ s')=Z(s)+Z(s')
AlgebraicGeometry.Scheme.Modules.IsInvertible.zeroSchemeIdeal_tensorHom_monoidalV29 below · cited by 3 · depth 32 - Separating sections force injectivity on k-points
AlgebraicGeometry.Scheme.Modules.ProjPresentation.eq_of_comp_toProj_eq_of_isSectionBasis_of_forall_exists_pullbackSection_of_comp_eq_id5 below · cited by 1 · depth 32 - Vanishing of a pulled-back section in a chart of a Proj presentation
AlgebraicGeometry.Scheme.Modules.ProjPresentation.pullbackSection_eq_zero_iff_appLE_sum_mul_eq_zero4 below · cited by 4 · depth 32 - Affine sections of a tensor product of quasi-coherent modules
AlgebraicGeometry.Scheme.Modules.bijective_lift_tensorSectionsBilin_of_isQuasicoherent_monoidalV20 below · cited by 4 · depth 32 - Sections over U as global sections of ι^*M, with frames
AlgebraicGeometry.Scheme.Modules.bijective_pullbackLocalSection_opensInclusion_and_isFrameOn_iff8 below · cited by 1 · depth 32 - Gluing sections of a module sheaf from affine opens
AlgebraicGeometry.Scheme.Modules.existsUnique_forall_map_eq_of_forall_affineOpens0 below · cited by 1 · depth 32 - Gluing morphisms of sheaves of modules along a cover
AlgebraicGeometry.Scheme.Modules.existsUnique_hom_app_eq_of_iSup_eq_top0 below · cited by 7 · depth 32 - Gluing isomorphisms of mathcal O_X-modules along a basis-like cover
AlgebraicGeometry.Scheme.Modules.existsUnique_iso_forall_pullback_mapIso_eq_of_iSup_eq_top1 below · cited by 6 · depth 32 - Unique gluing of module maps along a disjoint open cover
AlgebraicGeometry.Scheme.Modules.existsUnique_map_pullback_eq_of_iSup_eq_top_of_disjoint1 below · cited by 2 · depth 32 - Descent of an isomorphism of invertible modules to a finite stage
AlgebraicGeometry.Scheme.Modules.exists_forall_nonempty_pullback_iso_of_nonempty_pullback_iso_of_isDirectLimit_of_comp_eq15 below · cited by 1 · depth 32 - Serre vanishing for twists along a finite projective presentation
AlgebraicGeometry.Scheme.Modules.exists_forall_subsingleton_HSucc_tensorObj_tensorPow_of_isFinite_toProj_monoidalV240 below · cited by 2 · depth 32 - Frames for L^{⊗ 3} on a finite open cover
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_tensorPow_three_of_forall_nonempty_pullback_tensor_iso_monoidalV220 below · cited by 2 · depth 32 - Invertible modules over a directed limit base descend to a finite stage
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_forall_nonempty_pullback_iso_of_isDirectLimit_of_comp_eq25 below · cited by 2 · depth 32 - Invertible modules on a limit descend to a finite stage
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_nonempty_pullback_iso_of_isInvertible_of_isDirectLimit24 below · cited by 5 · depth 32 - Isomorphisms of invertible modules spread out from S_𝔭 to Sᵣ
AlgebraicGeometry.Scheme.Modules.exists_nonempty_iso_pullback_away_of_nonempty_iso_pullback_atPrime14 below · cited by 4 · depth 32 - Degree-zero base change via a finite free two-term complex
AlgebraicGeometry.Scheme.Modules.exists_twoTermComplex_sectionsEquiv_forall_baseChange74 below · cited by 3 · depth 32 - Global sections of pr₁^*mathcal O_X versus mathcal O_{X_A}
AlgebraicGeometry.Scheme.Modules.finrank_sections_pullback_obj_unit_eq1 below · cited by 1 · depth 32 - Theorem of the square for translations of an invertible sheaf
AlgebraicGeometry.Scheme.Modules.forall_nonempty_pullback_translate_tensor_iso_monoidalV2577 below · cited by 5 · depth 32 - Geometric-fibre h⁰ is invariant under local isomorphism on the base
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_eq_of_locIsoOnBase6 below · cited by 4 · depth 32 - Positivity of geometric-fibre h⁰ descends along faithfully flat base change
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_pos_of_forall_pos_pullback_of_faithfullyFlat77 below · cited by 1 · depth 32 - Invariance of geometric fibre h⁰ under an isomorphism over the base
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_pullback_inv_of_iso1 below · cited by 2 · depth 32 - Matching local frames forces a module morphism to be an isomorphism
AlgebraicGeometry.Scheme.Modules.isIso_of_isFrameOn_of_iSup_eq_top_monoidalV20 below · cited by 9 · depth 32 - Triviality over a finite product of rings, factorwise
AlgebraicGeometry.Scheme.Modules.nonempty_iso_unit_of_forall_pullback_piEvalRingHom2 below · cited by 1 · depth 32 - Pull-back of an invertible module through a k-point is trivial
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_comp_point_iso_unit0 below · cited by 1 · depth 32 - Theorem of the cube, pullback form
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_mul_mul_tensor_iso_monoidalV2576 below · cited by 6 · depth 32 - Pullback of the structure sheaf along a morphism of schemes
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_obj_unit_iso_unit0 below · cited by 2 · depth 32 - Pullback of 𝒪-modules commutes with tensor powers
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_tensorPow_iso_monoidalV20 below · cited by 7 · depth 32 - Zero-scheme ideal of a section is isomorphism-invariant
AlgebraicGeometry.Scheme.Modules.zeroSchemeIdeal_comp_eq_of_isIso_monoidalV21 below · cited by 3 · depth 32 - Common affine refinement of finitely many covers along morphisms
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_forall_le_preimage_of_compactSpace0 below · cited by 11 · depth 32 - Gluing an ordered affine cover along point-fixing overlap automorphisms
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_glued_of_overlap_isos_of_forall_base_eq0 below · cited by 1 · depth 32 - Pinned maps on relative charts commute with restriction
AlgebraicGeometry.Scheme.TwoAffineOpenCover.map_restrictAlgHom_eq_presheaf_map_of_tmul_eq0 below · cited by 4 · depth 32 - Sections over an open as rational functions regular there
AlgebraicGeometry.Scheme.existsUnique_section_algebraMap_germ_eq_of_forall_mem_range0 below · cited by 1 · depth 32 - Characteristic-zero generisation through a non-nilpotent integer
AlgebraicGeometry.Scheme.exists_charZero_point_specializes_of_natCast_mem_maximalIdeal_of_not_isNilpotent0 below · cited by 1 · depth 32 - Representability of n-tuples by the n-fold fibre power
AlgebraicGeometry.Scheme.exists_fibrePower_represents_tuples0 below · cited by 1 · depth 32 - Glueing a family of schemes along a common open subscheme
AlgebraicGeometry.Scheme.exists_glue_forall_isOpenImmersion_of_forall_isOpenImmersion0 below · cited by 1 · depth 32 - Open covers of a limit base change descend to a finite stage
AlgebraicGeometry.Scheme.exists_iSup_preimage_eq_top_of_isDirectLimit3 below · cited by 3 · depth 32 - Norm of a cocycle of units on the invariant cores of charts
AlgebraicGeometry.Scheme.exists_invariant_cocycle_basicOpen_eq_iInf_preimage_of_finite0 below · cited by 1 · depth 32 - Affine open neighbourhood avoiding a finite set of closed points
AlgebraicGeometry.Scheme.exists_isAffineOpen_mem_disjoint_of_finite_of_isClosed0 below · cited by 1 · depth 32 - Closed subscheme cut out by a base-change-compatible family of ideals
AlgebraicGeometry.Scheme.exists_isClosedImmersion_forall_exists_comp_eq_iff_ideal_eq_bot0 below · cited by 2 · depth 32 - Closed subfunctors of representable functors are represented by closed subschemes
AlgebraicGeometry.Scheme.exists_isClosedImmersion_represents_of_forall_exists_ideal0 below · cited by 3 · depth 32 - Fibre of A×_kSpecB over a k-rational maximal ideal
AlgebraicGeometry.Scheme.exists_iso_pullback_snd_specMap_quotient_comp_fst_fst_eq_id0 below · cited by 2 · depth 32 - Opens of X_A descend to a finite stage of a directed union
AlgebraicGeometry.Scheme.exists_mem_preimage_le_of_directed_subalgebra0 below · cited by 1 · depth 32 - Local rigidity of T-points of a free finite quotient
AlgebraicGeometry.Scheme.exists_open_eq_comp_aut_of_comp_eq_of_free_of_quotient1 below · cited by 1 · depth 32 - Natural finite labellings of points come from open decompositions
AlgebraicGeometry.Scheme.exists_opens_disjoint_forall_mem_iff_of_natural_nontrivial_of_connected0 below · cited by 1 · depth 32 - Open subfunctor criterion: a compatible family of opens comes from an open of E
AlgebraicGeometry.Scheme.exists_opens_forall_range_subset_iff_eq_univ_of_forall_isOpen0 below · cited by 1 · depth 32 - Free finite quotient maps are finite flat étale
AlgebraicGeometry.Scheme.isFinite_flat_etale_of_free_of_quotient1 below · cited by 1 · depth 32 - Stalks of a scheme with regular local stalks are regular rings
AlgebraicGeometry.Scheme.isRegularRing_stalk_of_forall_isRegularLocalRing_stalk0 below · cited by 1 · depth 32 - Yoneda on Noetherian test algebras over a Noetherian base
AlgebraicGeometry.Scheme.nilpPoints.existsUnique_hom_comp_eq_and_forall_apply_eq_comp_of_natural2 below · cited by 3 · depth 32 - Zariski gluing of sections over a finite basic-open cover
AlgebraicGeometry.Scheme.section_ext_and_exists_section_of_isLocalizationAway_of_span_eq_top0 below · cited by 6 · depth 32 - Maps from X×_k k[ε] determined by thickened points of a dense open
AlgebraicGeometry.Scheme.Hom.eq_of_forall_section_comp_eq_of_dense_of_dualNumber0 below · cited by 1 · depth 33 - Restricting a basis of Kähler differentials to a smaller affine open
AlgebraicGeometry.Scheme.Hom.exists_basis_kaehlerDifferential_map_of_isAffineOpen_le0 below · cited by 1 · depth 33 - Base change over affine bases: affine charts and sections as pushout
AlgebraicGeometry.Scheme.Hom.isAffineOpen_preimage_and_isPushout_of_isPullback0 below · cited by 10 · depth 33 - Top wedge of a Kähler basis frames detᵈ
AlgebraicGeometry.Scheme.Hom.isFrameOn_topToSections_iotaMulti_of_forall_exists_basis4 below · cited by 1 · depth 33 - Quasi-coherent ideal sheaves glue along an open cover
AlgebraicGeometry.Scheme.IdealSheafData.exists_forall_comap_iota_eq_of_iSup_eq_top0 below · cited by 1 · depth 33 - Smoothness of a reduced irreducible component, tested on an affine chart
AlgebraicGeometry.Scheme.IdealSheafData.smoothOfRelativeDimension_restrict_subscheme_vanishingIdeal_of_isOpenImmersion1 below · cited by 1 · depth 33 - A frame trivialises a module on an open subscheme
AlgebraicGeometry.Scheme.Modules.IsFrameOn.nonempty_pullback_iso_unit_monoidalV22 below · cited by 2 · depth 33 - Frames pull back to frames under pullback of modules
AlgebraicGeometry.Scheme.Modules.IsFrameOn.pullbackLocalSection_monoidalV22 below · cited by 2 · depth 33 - Zero ideal sheaf of a section of an invertible module is locally principal
AlgebraicGeometry.Scheme.Modules.IsInvertible.coeffIdeal_le_and_ideal_zeroSchemeIdeal_eq_monoidalV22 below · cited by 9 · depth 33 - Gluing isomorphisms of invertible modules across a discrete valuation point
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_basicOpen_le_mem_nonempty_pullback_iso_of_isDiscreteValuationRing_stalk3 below · cited by 1 · depth 33 - Gluing rigidified invertible modules along a localisation chart cover
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_pullback_iso_of_charts_of_rigidified_of_surjective_appTop22 below · cited by 2 · depth 33 - Finite affine frame cover for an invertible module on a quasi-compact scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isAffineOpen_iSup_eq_top_isFrameOn1 below · cited by 1 · depth 33 - Invertible module framed over affine opens from a finite stage
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isAffineOpen_isFrameOn_preimage_of_isDirectLimit3 below · cited by 1 · depth 33 - Descent of invertible modules along a faithfully flat base change
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isInvertible_pullback_iso_of_cocycle_of_isPullback30 below · cited by 2 · depth 33 - Rigidifying an invertible module along a section, locally on the base
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isInvertible_rigidified_locIsoOnBase_of_section5 below · cited by 1 · depth 33 - Extending an invertible module along an open immersion
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_locallyTrivial_opensRange_nonempty_pullback_iso0 below · cited by 1 · depth 33 - Invertible sheaf with h⁰>0 has a section nonvanishing at a k-point
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_pullbackSection_ne_zero_of_finrank_pos5 below · cited by 2 · depth 33 - Flat base change for global sections of an invertible module
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_tensorProduct_linearEquiv_sections_pullback_of_flat9 below · cited by 4 · depth 33 - Invertible modules on a scheme admit tensor inverses
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_tensor_inverse_monoidalV20 below · cited by 6 · depth 33 - Faithfully flat base change reflects isomorphisms of invertible modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_of_isIso_pullback_map_of_faithfullyFlat7 below · cited by 2 · depth 33 - Faithfully flat descent of isomorphy for invertible modules over an open
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_restrict_of_isIso_restrict_pullback_of_faithfullyFlat8 below · cited by 1 · depth 33 - Section generating after inverting u trivialises P over h⁻¹D(u)
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_restrict_preimage_basicOpen_of_forall_exists_pow_smul_eq_app4 below · cited by 1 · depth 33 - Global sections of a basewise trivial invertible module are invertible
AlgebraicGeometry.Scheme.Modules.IsInvertible.moduleInvertible_sections_of_forall_exists_nonempty_pullback_preimage_iso_unit6 below · cited by 1 · depth 33 - Locally isomorphic rigidified invertible modules are isomorphic
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_locally_of_pullback_section_trivial9 below · cited by 4 · depth 33 - Invertible modules on the spectrum of a field are trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_tensorUnit_of_field_monoidalV20 below · cited by 3 · depth 33 - Field-valued points where a section of an invertible module vanishes
AlgebraicGeometry.Scheme.Modules.IsInvertible.pullbackSection_eq_zero_iff_mem_support_monoidalV24 below · cited by 7 · depth 33 - Rigidification L ⊗ q^*((σ^*L)^∨) along a section
AlgebraicGeometry.Scheme.Modules.IsInvertible.tensor_pullback_dual_pullback_and_nonempty_pullback_iso_unit_monoidalV24 below · cited by 2 · depth 33 - Invertible sheaf with finite Proj presentation: c nᵈ ≤ h⁰(L^{⊗ n})
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_forall_mul_pow_le_cechFinrank_zero_tensorPow_monoidalV264 below · cited by 1 · depth 33 - Sections of L^{⊗ m} as the twist datum φ^*𝒪(m)
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_linearMap_sections_tensorPow_twistObj_monoidalV25 below · cited by 4 · depth 33 - Finiteness of the map defined by L^{⊗ 3}
AlgebraicGeometry.Scheme.Modules.ProjPresentation.isFinite_toProj_of_finite_setOf_forall_pullbackSection_eq_zero_iff_monoidalV223 below · cited by 1 · depth 33 - Morphisms agreeing on P^N have isomorphic pullbacks of a presented module
AlgebraicGeometry.Scheme.Modules.ProjPresentation.nonempty_pullback_iso_pullback_of_comp_toProj_eq4 below · cited by 2 · depth 33 - Reframing a projective presentation by U post-composes with Φ_U
AlgebraicGeometry.Scheme.Modules.ProjPresentation.toProj_eq_comp_linMap_of_sigma_eq_sum_smul6 below · cited by 3 · depth 33 - Uniqueness of a Proj presentation given its sections
AlgebraicGeometry.Scheme.Modules.ProjPresentation.toProj_eq_of_forall_sections_eq1 below · cited by 2 · depth 33 - Unit cocycles over a direct limit descend to a finite stage
AlgebraicGeometry.Scheme.Modules.UnitCocycle.exists_comap_eq_of_isDirectLimit6 below · cited by 1 · depth 33 - Closed immersion by sections descends along faithfully flat base change
AlgebraicGeometry.Scheme.Modules.closedImmersionBySections_of_faithfullyFlat_of_isPullback30 below · cited by 1 · depth 33 - Presentation by sections of an invertible sheaf is local on the base
AlgebraicGeometry.Scheme.Modules.closedImmersionBySections_of_forall_isPullback_away17 below · cited by 1 · depth 33 - Sheaf axiom for 𝒪_X-modules on a covered open
AlgebraicGeometry.Scheme.Modules.eq_of_forall_map_homOfLE_eq_and_exists_of_compatible0 below · cited by 5 · depth 33 - Gluing module isomorphisms along disjoint covering open immersions
AlgebraicGeometry.Scheme.Modules.existsUnique_iso_forall_pullback_mapIso_eq_of_isOpenImmersion_of_forall_inf_eq_bot3 below · cited by 1 · depth 33 - Theorem of the square: zero loci of L^{⊗ 3} sections
AlgebraicGeometry.Scheme.Modules.exists_hom_tensorPow_three_support_zeroSchemeIdeal_eq_monoidalV213 below · cited by 2 · depth 33 - Modules with frames having matching transition functions are isomorphic
AlgebraicGeometry.Scheme.Modules.exists_iso_app_eq_of_iSup_eq_top_of_forall_smul_eq3 below · cited by 4 · depth 33 - Gluing a locally trivial module along a trivial chart
AlgebraicGeometry.Scheme.Modules.exists_locallyTrivial_sup_nonempty_pullback_iso_of_pullback_inf_iso_unit13 below · cited by 1 · depth 33 - Triviality of a line bundle near a factorial point
AlgebraicGeometry.Scheme.Modules.exists_mem_nonempty_pullback_inf_iso_unit_of_uniqueFactorizationMonoid_stalk20 below · cited by 1 · depth 33 - Isomorphisms of invertible modules descend to a finite stage
AlgebraicGeometry.Scheme.Modules.exists_nonempty_iso_pullback_of_nonempty_iso_pullback_of_isDirectLimit14 below · cited by 5 · depth 33 - Geometric fibre h⁰ is detected on a principal cover of the base
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_eq_of_forall_isPullback_away1 below · cited by 1 · depth 33 - Positivity of geometric-fibre h⁰ is local on the base
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_pos_of_forall_away1 below · cited by 1 · depth 33 - Affine-local bijectivity of a↦ as gives a frame on U
AlgebraicGeometry.Scheme.Modules.isFrameOn_of_forall_affineOpens_bijective_smul0 below · cited by 3 · depth 33 - Left unitor on sections: λ_N(g⊗ n)=g· n
AlgebraicGeometry.Scheme.Modules.leftUnitor_hom_app_tensorSections_monoidalV22 below · cited by 1 · depth 33 - Local isomorphy descends along a finite principal cover
AlgebraicGeometry.Scheme.Modules.locallyIso_of_locallyIso_pullback_pi_localizationAway0 below · cited by 1 · depth 33 - Theorem of the cube over an algebraically closed field
AlgebraicGeometry.Scheme.Modules.nonempty_iso_tensorUnit_of_pullback_three_slices_monoidalV2563 below · cited by 2 · depth 33 - Generic fibre over a DVR is the basic open of varpi
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_basicOpen_iso_of_nonempty_pullback_iso_of_isFractionRing0 below · cited by 1 · depth 33 - Global basis on all opens below U makes M|_U free
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_iso_free_of_forall_exists_basis0 below · cited by 1 · depth 33 - Pullback of a tensor product of sections
AlgebraicGeometry.Scheme.Modules.pullbackTensorObjIso_hom_app_pullbackLocalSection_monoidalV20 below · cited by 3 · depth 33 - Pullback of a function under the trivialisation φ^*mathcal O_Y≅mathcal O_X
AlgebraicGeometry.Scheme.Modules.pullbackUnitIso_hom_app_pullbackLocalSection_toUnitSection_monoidalV20 below · cited by 3 · depth 33 - Tensor product of morphisms on elementary tensor sections
AlgebraicGeometry.Scheme.Modules.tensorHom_app_tensorSections_monoidalV20 below · cited by 5 · depth 33 - Monotonicity of the zero-scheme ideal under a module map
AlgebraicGeometry.Scheme.Modules.zeroSchemeIdeal_comp_le_monoidalV20 below · cited by 1 · depth 33 - Finitely many section pairs agreeing over the limit agree at one stage
AlgebraicGeometry.Scheme.exists_forall_app_eq_app_of_isCompact_of_isDirectLimit_of_isPullback4 below · cited by 1 · depth 33 - Finitely many sections over quasi-compact opens descend to one stage
AlgebraicGeometry.Scheme.exists_forall_app_eq_of_isCompact_of_isDirectLimit_of_isPullback5 below · cited by 1 · depth 33 - Gluing local effective fppf quotients of a relation
AlgebraicGeometry.Scheme.exists_fppf_quotient_of_forall_exists_quotient_restrict0 below · cited by 1 · depth 33 - Gluing schemes over Spec S along principal opens
AlgebraicGeometry.Scheme.exists_glued_over_Spec_of_cocycle_basicOpen0 below · cited by 2 · depth 33 - Overlap of two away-charts is the chart over S[1/rᵢrⱼ]
AlgebraicGeometry.Scheme.exists_isPullback_fst_awayToAwayRight_and_isPullback_snd_awayToAwayLeft1 below · cited by 1 · depth 33 - Concatenating ordered affine covers of two opens covering X
AlgebraicGeometry.Scheme.exists_orderedAffineCover_orderEmbedding_of_sup_eq_top0 below · cited by 1 · depth 33 - Affine Galois data for a free finite quotient of schemes
AlgebraicGeometry.Scheme.exists_ringAut_galois_sections_of_free_of_quotient0 below · cited by 2 · depth 33 - Sections on p₁⁻¹U∩ p₂⁻¹V as a tensor product
AlgebraicGeometry.Scheme.isAffineOpen_and_exists_linearEquiv_tensor_sections_of_isPullback1 below · cited by 7 · depth 33 - Quasi-compact opens carry finite ordered affine covers
AlgebraicGeometry.Scheme.nonempty_orderedAffineCover_of_sup_eq_top0 below · cited by 1 · depth 33 - Surjectivity of global sections under localisation of the affine base
AlgebraicGeometry.Scheme.surjective_appTop_and_forall_away_of_isPullback_of_forall_surjective_appTop_away0 below · cited by 1 · depth 33 - Sections on U×_k Y when Γ(Y,𝒪_Y)=k
AlgebraicGeometry.Scheme.Hom.bijective_app_of_isPullback_of_bijective_of_isAffineOpen0 below · cited by 1 · depth 34 - Non-isolated points of fibres of finite-type morphisms
AlgebraicGeometry.Scheme.Hom.exists_isClosed_irreducible_subset_fiber_of_not_quasiFiniteAt_monoidalV20 below · cited by 1 · depth 34 - Chart independence of reading a top form at an F-point
AlgebraicGeometry.Scheme.Hom.topFormMap_eq_topFormMap_of_specMap_comp_fromSpec_eq7 below · cited by 1 · depth 34 - Equality of ideal sheaves via vanishing comaps
AlgebraicGeometry.Scheme.IdealSheafData.eq_iff_comap_subschemeInclusion_eq_bot0 below · cited by 1 · depth 34 - Flat closed subschemes over a domain agreeing generically are equal
AlgebraicGeometry.Scheme.IdealSheafData.eq_of_flat_of_comap_pullback_fst_eq1 below · cited by 1 · depth 34 - Nilpotent sub-ideal sheaves exhaust an ideal sheaf inside the nilradical
AlgebraicGeometry.Scheme.IdealSheafData.exists_le_isNilpotent_mem_ideal_of_le_nilradical0 below · cited by 1 · depth 34 - Closed immersion by sections tensored with a presented module
AlgebraicGeometry.Scheme.Modules.ClosedImmersionBySections.tensor_of_projPresentation_monoidalV23 below · cited by 1 · depth 34 - Finite-by-sections is preserved by pull-back along a finite morphism
AlgebraicGeometry.Scheme.Modules.FiniteBySections.pullback_of_isFinite1 below · cited by 2 · depth 34 - Tensor powers of a frame section are frames
AlgebraicGeometry.Scheme.Modules.IsFrameOn.tensorPowSection_monoidalV23 below · cited by 2 · depth 34 - Global function acting as identity on an invertible module is 1
AlgebraicGeometry.Scheme.Modules.IsInvertible.eq_one_of_forall_smul_eq0 below · cited by 5 · depth 34 - Unique ε-rigidified isomorphism of trivialised invertible modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.existsUnique_iso_pullback_map_eq_trivialization_of_surjective_appTop5 below · cited by 1 · depth 34 - Faithfully flat descent of morphisms of invertible modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.existsUnique_map_eq_of_isPullback_of_faithfullyFlat3 below · cited by 1 · depth 34 - Endomorphisms of an invertible module are scalar multiplication
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_app_eq_smul0 below · cited by 7 · depth 34 - Invertible module from invertible reductions on adic thickenings
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_pullback_iso_of_affHom_pushforward_adicThickening_surjective_ker_eq_pow_smul_top24 below · cited by 1 · depth 34 - See-saw theorem over a general base ring
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isClosedImmersion_forall_iff_locallyIsoOver_of_flat_of_isProper96 below · cited by 2 · depth 34 - Cocycle transition isomorphisms of rigidified invertible modules on charts
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_pullback_iso_cocycle_of_charts_of_rigidified_of_surjective_appTop18 below · cited by 1 · depth 34 - Affine base change of sections of an invertible module
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_tensorProduct_linearEquiv_sections_pullback_preimage_of_isAffineOpen6 below · cited by 2 · depth 34 - Vanishing of a morphism of invertible modules detected on a non-empty open
AlgebraicGeometry.Scheme.Modules.IsInvertible.hom_eq_zero_of_pullback_map_eq_zero_of_isIntegral0 below · cited by 2 · depth 34 - Fibrewise isomorphic invertible modules are isomorphic over a complete local base
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_forall_nonempty_pullback_thickening_iso_of_isProper88 below · cited by 2 · depth 34 - Line bundles on open subschemes of Spec of a UFD are trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_unit_of_isOpenImmersion_of_uniqueFactorizationMonoid8 below · cited by 1 · depth 34 - Automorphisms fixing the components of Z(s) preserve M
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_pullback_iso_of_forall_maximal_isIrreducible_image_eq22 below · cited by 1 · depth 34 - Invertibility is local on a jointly surjective open cover
AlgebraicGeometry.Scheme.Modules.IsInvertible.of_forall_pullback_iso_of_isOpenImmersion0 below · cited by 4 · depth 34 - Morphisms fixing k-point vanishing preserve the zero-scheme support
AlgebraicGeometry.Scheme.Modules.IsInvertible.preimage_support_zeroSchemeIdeal_eq_of_forall_pullbackSection_eq_zero_iff_comp5 below · cited by 1 · depth 34 - Normalised isomorphism of rigidified line bundles satisfies the cocycle condition
AlgebraicGeometry.Scheme.Modules.IsInvertible.pullback_map_cocycle_of_pullback_section_map_eq5 below · cited by 1 · depth 34 - Constancy of φ∘ P on a dual-number point
AlgebraicGeometry.Scheme.Modules.ProjPresentation.comp_toProj_eq_const_of_forall_pullbackSection_eq_zero_imp6 below · cited by 1 · depth 34 - Two k-points with the same vanishing sections have equal images in P^N
AlgebraicGeometry.Scheme.Modules.ProjPresentation.comp_toProj_eq_of_forall_pullbackSection_eq_zero_imp6 below · cited by 1 · depth 34 - Reframing a projective presentation by an invertible matrix
AlgebraicGeometry.Scheme.Modules.ProjPresentation.exists_sigma_eq_sum_smul_toProj_eq_comp_linMap3 below · cited by 1 · depth 34 - Vanishing dichotomy on a fibre of a projective presentation
AlgebraicGeometry.Scheme.Modules.ProjPresentation.subset_support_zeroSchemeIdeal_or_disjoint_monoidalV26 below · cited by 1 · depth 34 - Gluing module isomorphisms along a pairwise disjoint open cover
AlgebraicGeometry.Scheme.Modules.existsUnique_iso_forall_pullback_mapIso_eq_of_forall_inf_eq_bot2 below · cited by 1 · depth 34 - Extending an isomorphism across a DVR point with uniformiser t
AlgebraicGeometry.Scheme.Modules.exists_basicOpen_le_mem_nonempty_pullback_iso_of_nonempty_iso_unit_of_isDiscreteValuationRing_stalk2 below · cited by 1 · depth 34 - Cocycle isomorphism over a kernel pair gives a descent datum
AlgebraicGeometry.Scheme.Modules.exists_descentData_obj_eq_of_cocycle_of_isPullback0 below · cited by 2 · depth 34 - Gluing sheaves of modules along a jointly surjective family of open immersions
AlgebraicGeometry.Scheme.Modules.exists_forall_pullback_iso_of_cocycle1 below · cited by 1 · depth 34 - Descent of a natural bilinear pairing to L ⊗ M → P
AlgebraicGeometry.Scheme.Modules.exists_hom_tensor_app_tensorSections_eq_of_bilinear0 below · cited by 1 · depth 34 - Rigidified descent data for line bundles satisfy the cocycle condition
AlgebraicGeometry.Scheme.Modules.exists_iso_pullback_cocycle_of_rigidified7 below · cited by 1 · depth 34 - Lifting automorphisms of a module along a section
AlgebraicGeometry.Scheme.Modules.exists_iso_pullback_map_hom_eq_of_pullback_section_trivial0 below · cited by 3 · depth 34 - Projective presentations glued along a principal cover of the base
AlgebraicGeometry.Scheme.Modules.exists_projPresentation_forall_isPullback_toProj_of_forall_away10 below · cited by 1 · depth 34 - Descending a projective presentation by sections along faithfully flat base change
AlgebraicGeometry.Scheme.Modules.exists_projPresentation_isPullback_toProj_of_faithfullyFlat22 below · cited by 1 · depth 34 - Theorem of the cube with a curve factor
AlgebraicGeometry.Scheme.Modules.nonempty_iso_tensorUnit_of_pullback_three_slices_of_smoothOfRelativeDimension_one_monoidalV2527 below · cited by 1 · depth 34 - Unit trivialisations of pullbacks are compatible with composition
AlgebraicGeometry.Scheme.Modules.pullbackComp_hom_app_comp_pullbackUnitIso_hom0 below · cited by 13 · depth 34 - Pullback of the unit section is the unit section
AlgebraicGeometry.Scheme.Modules.pullbackTensorUnitObjIso_hom_app_pullbackLocalSection_unitSection_monoidalV20 below · cited by 3 · depth 34 - Pullback pseudofunctor coherence along a commuting ladder
AlgebraicGeometry.Scheme.Modules.pullback_mapIso_pullbackComp_app_trans_eq0 below · cited by 1 · depth 34 - Rigidified isomorphism stays rigidified under compatible restriction
AlgebraicGeometry.Scheme.Modules.pullback_map_conj_eq_trivialization_of_pullback_map_eq_trivialization0 below · cited by 2 · depth 34 - Homogeneity of degree n of s ↦ s^{⊗ n}
AlgebraicGeometry.Scheme.Modules.tensorPowSection_smul_monoidalV22 below · cited by 1 · depth 34 - Gluing smooth local lifts along overlap isomorphisms
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_smooth_isPullback_of_local_lifts_of_overlap_isos0 below · cited by 1 · depth 34 - Entries of the homotopy's support come from σ
AlgebraicGeometry.Scheme.OrderedAffineCover.oSub_of_mem_support_ohom0 below · cited by 1 · depth 34 - Cone homotopy between identity and sorting on ordered Čech chains
AlgebraicGeometry.Scheme.OrderedAffineCover.obd_ohom_add_ohom_obd5 below · cited by 1 · depth 34 - Morphisms to a coproduct from a disjoint open cover
AlgebraicGeometry.Scheme.existsUnique_hom_sigma_of_disjoint_iSup_eq_top0 below · cited by 2 · depth 34 - Sections equal over the limit become equal at a finite stage
AlgebraicGeometry.Scheme.exists_app_eq_app_of_isCompact_of_isDirectLimit3 below · cited by 3 · depth 34 - Sections over a quasi-compact open descend to a finite stage
AlgebraicGeometry.Scheme.exists_app_eq_of_isCompact_of_isDirectLimit4 below · cited by 2 · depth 34 - Representability of an affine-locally charted Zariski sheaf on R-algebras
AlgebraicGeometry.Scheme.exists_represents_of_zariskiSheaf_of_openAffineCover2 below · cited by 2 · depth 34 - Scheme-level Artinian lifting descends to chart algebras
AlgebraicGeometry.Scheme.forall_exists_algHom_lift_of_forall_exists_lift_of_isOpenImmersion_of_isAlgClosed1 below · cited by 1 · depth 34 - Translation invariance of D along differences of points of Z
AlgebraicGeometry.Scheme.forall_mem_iff_of_subset_union_preimage_or_disjoint_monoidalV21 below · cited by 1 · depth 34 - Maps into a coproduct of schemes: clopen decomposition
AlgebraicGeometry.Scheme.isClopen_preimage_sigmaInj_and_existsUnique_lift_of_hom_sigma0 below · cited by 2 · depth 34 - Stalk at a generalisation as a localisation of the stalk
AlgebraicGeometry.Scheme.isLocalization_atPrime_stalk_of_specializes0 below · cited by 1 · depth 34 - Naturality of `topToSections` under restriction of opens
AlgebraicGeometry.Scheme.Hom.map_topToSections_eq_topToSections_topFormMap0 below · cited by 1 · depth 35 - An automorphism fixing the components of Z(s) fixes Z(s)
AlgebraicGeometry.Scheme.Modules.IsInvertible.comap_zeroSchemeIdeal_eq_of_forall_maximal_isIrreducible_image_eq13 below · cited by 1 · depth 35 - Compatible levelwise isomorphisms along a tower of thickenings
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_compatible_pullback_iso_of_forall_nonempty_pullback_iso3 below · cited by 1 · depth 35 - Trivialised n-th tensor power: local n-th powers up to one rational function
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_exists_isFrameOn_isUnit_germToFunctionField_eq_mul_pow_of_tensorPow_iso9 below · cited by 1 · depth 35 - Gluing a line bundle from an affine datum over adic thickenings
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_forall_pullback_iso_of_forall_bijective_smul_res_of_affHom_pushforward20 below · cited by 1 · depth 35 - Frames of an invertible module descend along a surjective morphism
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_iSup_eq_top_bijective_smul_of_span_pullback_of_surjective6 below · cited by 2 · depth 35 - See-saw theorem: trivialisation locus is a closed subscheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isClosedImmersion_forall_iff_locallyIsoOver_unit_of_flat_of_isProper90 below · cited by 1 · depth 35 - Rigidified isomorphism of invertible modules normalised along a section
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_iso_pullback_mapIso_eq_of_locally_of_rigidified10 below · cited by 1 · depth 35 - Evaluation and double dual for an invertible 𝒪_X-module
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_ihom_ev_app_monoidalV22 below · cited by 1 · depth 35 - Isomorphisms of an invertible module agreeing along a section
AlgebraicGeometry.Scheme.Modules.IsInvertible.iso_eq_of_pullback_section_map_eq_of_surjective_appTop3 below · cited by 4 · depth 35 - Compatible formal isomorphisms of invertible modules are algebraic
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_compatible_pullback_adicThickening_iso83 below · cited by 1 · depth 35 - Equal zero ideals force isomorphic invertible modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_zeroSchemeIdeal_eq11 below · cited by 1 · depth 35 - Invertible module with non-zero section and non-zero dual section is trivial
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_tensorUnit_of_section_ne_zero_of_dual_section_ne_zero1 below · cited by 1 · depth 35 - Rigidified isomorphisms of invertible modules compose
AlgebraicGeometry.Scheme.Modules.IsInvertible.trans_eq_of_pullback_mapIso_eq_of_surjective_appTop4 below · cited by 1 · depth 35 - Gluing invertible modules along a finite product base
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_forall_nonempty_pullback_iso_of_isPullback_pi2 below · cited by 2 · depth 35 - Gluing invertible modules along a finite product decomposition
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_forall_nonempty_pullback_iso_of_isPullback_pi_univ2 below · cited by 1 · depth 35 - Renormalising an isomorphism along a section of a rigidified module
AlgebraicGeometry.Scheme.Modules.exists_iso_pullback_map_eq_of_nonempty_iso1 below · cited by 1 · depth 35 - Automorphisms of the unit module are multiplication by global units
AlgebraicGeometry.Scheme.Modules.exists_units_forall_app_eq_smul_of_iso_unit2 below · cited by 1 · depth 35 - Cocycle condition over arbitrary test schemes from universal triple overlap
AlgebraicGeometry.Scheme.Modules.forall_pullback_cocycle_of_cocycle_pullback_snd_fst0 below · cited by 1 · depth 35 - Positivity of fibrewise h⁰ over a finite product base
AlgebraicGeometry.Scheme.Modules.geomFibreH0Finrank_pos_of_forall_pullback_of_isPullback_pi1 below · cited by 1 · depth 35 - Normalised isomorphisms remain normalised after pull-back
AlgebraicGeometry.Scheme.Modules.pullback_mapIso_pullback_mapIso_eq_of_pullback_mapIso_eq0 below · cited by 2 · depth 35 - Pullback of multiplication by a global function
AlgebraicGeometry.Scheme.Modules.pullback_map_app_eq_smul_of_forall_app_eq_smul0 below · cited by 4 · depth 35 - Restriction preserves normalisation: one module, two structure maps
AlgebraicGeometry.Scheme.Modules.pullback_map_conj_eq_trivialization_pair_of_pullback_map_eq0 below · cited by 1 · depth 35 - Universally closed morphisms: open sets over V(I) exhaust X
AlgebraicGeometry.Scheme.Opens.eq_top_of_forall_mem_of_le_jacobson_of_universallyClosed0 below · cited by 1 · depth 35 - Gluing chart morphisms along a nilpotent thickening
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_comp_eq_of_forall_idx_agree1 below · cited by 4 · depth 35 - Opens of a local lift above the opens of the base
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_opens_local_lifts_preimage_eq5 below · cited by 1 · depth 35 - Chart rings of the special fibre via local lifts
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_ringEquiv_tensor_sections_local_lifts0 below · cited by 1 · depth 35 - Ordered Čech boundary squares to zero
AlgebraicGeometry.Scheme.OrderedAffineCover.obd_obd0 below · cited by 1 · depth 35 - Cone on a vertex is a chain contraction in positive degrees
AlgebraicGeometry.Scheme.OrderedAffineCover.obd_ocone_add_ocone_obd0 below · cited by 1 · depth 35 - Signed sorting commutes with the ordered Čech boundary
AlgebraicGeometry.Scheme.OrderedAffineCover.obd_oesort2 below · cited by 1 · depth 35 - Strip affine covers of intersections exist
AlgebraicGeometry.Scheme.OrderedOpenFamily.exists_orderedAffineCover_inter_image_eq_inf0 below · cited by 1 · depth 35 - Open subfunctors with affine charts cover a Zariski sheaf
AlgebraicGeometry.Scheme.exists_isOpenImmersion_presheaf_overTotal_of_isOpen_of_chart0 below · cited by 1 · depth 35 - Units on W ∩ Xₜ near a DVR point are units times powers of t
AlgebraicGeometry.Scheme.exists_isUnit_mul_pow_eq_mul_pow_of_isDiscreteValuationRing_stalk0 below · cited by 1 · depth 35 - Restricting an isomorphism of open subschemes to smaller opens
AlgebraicGeometry.Scheme.exists_iso_comp_homOfLE_eq_homOfLE_comp_of_preimage_eq0 below · cited by 4 · depth 35 - Overlap charts and sections for a localisation-away cover
AlgebraicGeometry.Scheme.exists_overlaps_toSpecAway_section_of_charts_of_isPullback_of_surjective_appTop3 below · cited by 1 · depth 35 - Zariski comparison: algebra-valued sheaf extends to schemes over Spec R
AlgebraicGeometry.Scheme.exists_presheaf_over_equiv_isSheaf_overTotal_of_isLocalization_away0 below · cited by 1 · depth 35 - Irreducible closed subsets lie in or miss a basic open
AlgebraicGeometry.Scheme.subset_basicOpen_or_disjoint_of_isProper_of_isIrreducible_monoidalV20 below · cited by 1 · depth 35 - Stalk ideals at a fixed point are automorphism-invariant
AlgebraicGeometry.Scheme.IdealSheafData.map_germ_ideal_comap_eq_of_base_eq_of_ringKrullDim_le_one0 below · cited by 1 · depth 36 - Degree-one frames on a finite open cover of X
AlgebraicGeometry.Scheme.Modules.ClosedImmersionBySections.exists_iSup_eq_top_isFrameOn_of_isSectionRing0 below · cited by 1 · depth 36 - Ratio of two frames is a unit
AlgebraicGeometry.Scheme.Modules.IsFrameOn.isUnit_of_isFrameOn_smul0 below · cited by 1 · depth 36 - Sections of L⊗ M over an affine open
AlgebraicGeometry.Scheme.Modules.IsInvertible.bijective_lift_tensorSectionsBilin_monoidalV28 below · cited by 2 · depth 36 - Dual of a tensor product of invertible sheaves of modules
AlgebraicGeometry.Scheme.Modules.IsInvertible.dual_tensor_monoidalV24 below · cited by 3 · depth 36 - Affine-local base change of sections of an invertible pullback
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_baseChange_sections_linearEquiv_pullback_of_le6 below · cited by 1 · depth 36 - See-saw: trivialisation locus is Spec(A/I), universally
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_ideal_forall_iff_locallyIsoOver_unit_of_flat_of_isProper89 below · cited by 1 · depth 36 - Invertible 𝒪_X-modules admit local frames
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_isFrameOn_monoidalV22 below · cited by 2 · depth 36 - Non-zero sections multiply on an integral scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_unit_hom_tensor_ne_zero2 below · cited by 1 · depth 36 - Zero-scheme ideal of c Ω on affine opens
AlgebraicGeometry.Scheme.Modules.IsInvertible.ideal_zeroSchemeIdeal_eq_span_of_app_eq_smul_monoidalV26 below · cited by 1 · depth 36 - Generating section of an invertible sheaf on an affine open is a frame
AlgebraicGeometry.Scheme.Modules.IsInvertible.isFrameOn_of_isAffineOpen_of_span_singleton_eq_top7 below · cited by 1 · depth 36 - Surjective descent of invertibility for a section
AlgebraicGeometry.Scheme.Modules.IsInvertible.isIso_of_isIso_pullbackSection_of_surjective8 below · cited by 1 · depth 36 - Codimension-one points of the zero locus of a section
AlgebraicGeometry.Scheme.Modules.IsInvertible.maximal_isIrreducible_closure_singleton_of_mem_support_of_ringKrullDim_le_one4 below · cited by 1 · depth 36 - Unit of the adic thickening adjunction: surjectivity and kernel Iⁿ⁺¹Γ(U,N)
AlgebraicGeometry.Scheme.Modules.IsInvertible.unit_app_adicThickening_surjective_and_eq_zero_iff_mem_pow_smul_top7 below · cited by 1 · depth 36 - Zero-scheme ideals agreeing in codimension ≤ 1 coincide
AlgebraicGeometry.Scheme.Modules.IsInvertible.zeroSchemeIdeal_eq_of_forall_ringKrullDim_le_one_map_germ_ideal_eq9 below · cited by 1 · depth 36 - Invertible sheaves over an algebraic extension descend to a finite subextension
AlgebraicGeometry.Scheme.Modules.exists_intermediateField_isInvertible_nonempty_pullback_iso_of_isAlgebraic25 below · cited by 1 · depth 36 - Descent of a trivialised invertible module through a direct limit
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_pullback_map_iso_unit_nonempty_pullback_iso_of_isDirectLimit27 below · cited by 2 · depth 36 - Invertible module on a closed subscheme trivialises on a pulled-back ordered affine cover
AlgebraicGeometry.Scheme.Modules.exists_orderedAffineCover_nonempty_cechTrivialisation_comap_of_isInvertible0 below · cited by 5 · depth 36 - Non-zero unit sections under base change and pull-back along an automorphism
AlgebraicGeometry.Scheme.Modules.exists_unit_hom_pullback_ne_zero_of_isIso_and_tensor0 below · cited by 1 · depth 36 - Transitivity of the pullback–pushforward adjunction unit
AlgebraicGeometry.Scheme.Modules.pullbackPushforwardAdjunction_unit_app_eq_of_comp_eq0 below · cited by 1 · depth 36 - Triple-overlap compatibility of pullback identifications from a cocycle
AlgebraicGeometry.Scheme.Modules.pullback_iso_trans_eq_of_cocycle0 below · cited by 1 · depth 36 - Tensor powers of a section: g^{⊗ m}⊗ g^{⊗ n}↦ g^{⊗(m+n)}
AlgebraicGeometry.Scheme.Modules.tensorPowAdd_hom_app_tensorSections_tensorPowSection6 below · cited by 1 · depth 36 - Naturality of the restriction transition morphism ι'_*ι'^*→ι_*ι^*
AlgebraicGeometry.Scheme.Modules.transition_comp_pushforward_map_eq_pushforward_map_comp_transition_of_comp_eq0 below · cited by 1 · depth 36 - Multiplicativity of the section attached to an automorphism of the unit module
AlgebraicGeometry.Scheme.Modules.unitAutSection_trans_and_unitAutSection_refl0 below · cited by 8 · depth 36 - Restriction of a local smooth lift to an open subscheme
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_isPullback_opens_local_lift0 below · cited by 1 · depth 36 - Restriction of overlap isomorphisms to triple overlaps
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_overlap_isos_restrict_inter1 below · cited by 1 · depth 36 - Common affine refinement of two pulled-back ordered affine covers
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_refinement_preimage_preimage_of_isSeparated0 below · cited by 2 · depth 36 - Overlap isomorphism of local lifts respects the opens Oₐ
AlgebraicGeometry.Scheme.OrderedAffineCover.preimage_opens_local_lift_eq_of_iso_comp_eq0 below · cited by 1 · depth 36 - Compatible families of affine-open sections over Spec R come from R
AlgebraicGeometry.Scheme.exists_forall_eq_appLE_of_forall_map_eq_of_bijective3 below · cited by 1 · depth 36 - Invariant affine open through a finite set of points
AlgebraicGeometry.Scheme.exists_invariant_isAffineOpen_of_finite_of_finiteLocallyFree_equivalenceRelation0 below · cited by 1 · depth 36 - Invariant affine opens descend along a finite flat quotient
AlgebraicGeometry.Scheme.exists_isAffineOpen_preimage_eq_of_invariant3 below · cited by 1 · depth 36 - Clopen subscheme representing a subfunctor cut out by idempotents
AlgebraicGeometry.Scheme.exists_isOpenImmersion_isClosedImmersion_iff_of_isIdempotentElem_of_forall_exists_idempotent_of_finiteType0 below · cited by 1 · depth 36 - Triple chart overlaps are again localisation-away charts
AlgebraicGeometry.Scheme.exists_isPullback_fst_fst_toSpecAway_of_charts_of_isPullback1 below · cited by 1 · depth 36 - Overlaps of away-localisation charts are again away charts
AlgebraicGeometry.Scheme.exists_isPullback_fst_toSpecAway_of_charts_of_isPullback0 below · cited by 2 · depth 36 - Sections and surjectivity on global sections under base change to B[1/t₀]
AlgebraicGeometry.Scheme.exists_section_and_surjective_appTop_of_isPullback_of_isLocalization_away0 below · cited by 1 · depth 36 - Global ideal of an ideal sheaf pulled back along Specφ
AlgebraicGeometry.Scheme.IdealSheafData.map_ideal_comap_specMap_eq_map0 below · cited by 2 · depth 37 - Transition sections of a pulled-back Čech trivialisation
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.comap_transition1 below · cited by 3 · depth 37 - Rescaling a Čech trivialisation by units on the charts
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_forall_transition_eq_mul_mul2 below · cited by 1 · depth 37 - Invertible module with prescribed Čech transition cocycle
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_isInvertible_transition_eq4 below · cited by 2 · depth 37 - Gluing a Čech trivialisation with trivial transitions, charts preserved
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_iso_forall_unitAutSection_eq_one_of_forall_transition_eq_one3 below · cited by 1 · depth 37 - Pullback of a Čech trivialisation along a refinement map
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_refinement_transition_eq2 below · cited by 2 · depth 37 - Transition sections are units and satisfy the Čech cocycle identity
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.isUnit_transition_and_transition_face_mul_eq1 below · cited by 4 · depth 37 - Equal Čech transitions force isomorphic modules
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.nonempty_iso_of_transition_eq13 below · cited by 1 · depth 37 - Trivial transition functions give a global trivialisation
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.nonempty_iso_unit_of_forall_transition_eq_one3 below · cited by 2 · depth 37 - See-saw theorem, affine-local form over a Noetherian base
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_ideal_forall_locallyIsoOver_unit_iff_map_eq_bot88 below · cited by 1 · depth 37 - Invertible modules on an abelian variety descend along a field extension
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_of_nonempty_pullback_iso_of_isPullback81 below · cited by 2 · depth 37 - Pullback of tensor-power sections respects base scalars
AlgebraicGeometry.Scheme.Modules.app_pullbackTensorPowIso_tensorPowMapIso_baseScalar_smul0 below · cited by 1 · depth 37 - Tensor-power pull-back comparison of sections is compositional
AlgebraicGeometry.Scheme.Modules.app_pullbackTensorPowIso_tensorPowMapIso_comp8 below · cited by 1 · depth 37 - Pullback of global sections is multiplicative on tensor powers
AlgebraicGeometry.Scheme.Modules.app_pullbackTensorPowIso_tensorPowMapIso_tensorPowAdd_tensorSections3 below · cited by 1 · depth 37 - Unit section preserved by the degree-zero tensor-power comparison
AlgebraicGeometry.Scheme.Modules.app_pullbackTensorPowIso_tensorPowMapIso_unitSection1 below · cited by 1 · depth 37 - Associator on sections of a triple tensor product of module sheaves
AlgebraicGeometry.Scheme.Modules.associator_hom_app_tensorSections_monoidalV24 below · cited by 1 · depth 37 - Trivialisation over U yields a frame on U
AlgebraicGeometry.Scheme.Modules.exists_isFrameOn_of_pullback_iso_unit_monoidalV20 below · cited by 1 · depth 37 - Invertible modules over S_𝔭 descend to a basic open
AlgebraicGeometry.Scheme.Modules.exists_isInvertible_nonempty_iso_pullback_of_isInvertible_atPrime_of_isSeparated20 below · cited by 2 · depth 37 - [n]^*N ≅ N^{⊗ n} for translation-invariant invertible N
AlgebraicGeometry.Scheme.Modules.nonempty_pullback_schemeNsmul_iso_tensorPow_of_forall_pullback_translate_iso_monoidalV2115 below · cited by 1 · depth 37 - Monoidality of the pullback composition isomorphism on tensor products
AlgebraicGeometry.Scheme.Modules.pullbackComp_app_tensorObj1 below · cited by 9 · depth 37 - Monoidal unit comparison for f^* is the canonical one
AlgebraicGeometry.Scheme.Modules.pullbackTensorUnitObjIso_eq_pullbackUnitIso2 below · cited by 4 · depth 37 - Conjugated pullback automorphism of the unit sheaf on global sections
AlgebraicGeometry.Scheme.Modules.pullbackUnitIso_conj_app_top_one_eq_appTop0 below · cited by 8 · depth 37 - Right unitor on sections: n⊗ g↦ g n
AlgebraicGeometry.Scheme.Modules.rightUnitor_hom_app_tensorSections_monoidalV20 below · cited by 1 · depth 37 - Discrepancy section of a pulled-back Čech trivialisation restricts along h₀
AlgebraicGeometry.Scheme.Modules.unitAutSection_comap_refinement_eq_appLE2 below · cited by 1 · depth 37 - Restricting an automorphism of mathcal O_Y to an open
AlgebraicGeometry.Scheme.Modules.unitAutSection_pullbackUnitIso_conj_opensI_eq_app_one1 below · cited by 1 · depth 37 - Transfer of an affine cover along a surjective closed immersion
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_opens_preimage_eq_of_isClosedImmersion_of_surjective4 below · cited by 1 · depth 37 - Overlap transitions for arbitrary ordered pairs of indices
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_transitions_any_pair1 below · cited by 1 · depth 37 - Sections of φ over Spec p retract φ^sharp
AlgebraicGeometry.Scheme.Hom.exists_appLE_apply_eq_of_comp_eq_id0 below · cited by 4 · depth 38 - Dual of a Čech-trivialised module has inverse transitions
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_dual_forall_transition_mul_eq_one8 below · cited by 2 · depth 38 - Two Čech trivialisations differ by chartwise units
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_forall_transition_eq_transition_mul_mul2 below · cited by 1 · depth 38 - Invertible module from a unit Čech 1-cocycle
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_isInvertible_transition_eq_of_inf_cocycle2 below · cited by 1 · depth 38 - Tensoring Čech trivialisations multiplies transition sections
AlgebraicGeometry.Scheme.Modules.CechTrivialisation.exists_tensor_forall_transition_eq_mul5 below · cited by 2 · depth 38 - Tensor product preserves finiteness by sections
AlgebraicGeometry.Scheme.Modules.FiniteBySections.tensor_monoidalV22 below · cited by 2 · depth 38 - Non-zero sections compose non-trivially on an integral scheme
AlgebraicGeometry.Scheme.Modules.IsInvertible.comp_ne_zero_of_ne_zero_of_isIntegral2 below · cited by 1 · depth 38 - Triviality near a trivial fibre of a proper flat family
AlgebraicGeometry.Scheme.Modules.IsInvertible.exists_nonempty_pullback_preimage_basicOpen_iso_unit_of_forall_sections_linearEquiv86 below · cited by 1 · depth 38 - Triviality of an invertible module descends along a field map
AlgebraicGeometry.Scheme.Modules.IsInvertible.nonempty_iso_tensorUnit_of_nonempty_pullback_iso_tensorUnit77 below · cited by 1 · depth 38 - Unit cocycles over S_𝔭 descend to a basic open
AlgebraicGeometry.Scheme.Modules.UnitCocycle.exists_notMem_comap_eq_of_atPrime_of_isSeparated3 below · cited by 1 · depth 38 - Pullback compatibility of tensor-power multiplication isomorphisms
AlgebraicGeometry.Scheme.Modules.map_tensorPowAdd_hom_comp_pullbackTensorPowIso_tensorPowMapIso_hom0 below · cited by 1 · depth 38 - Compatibility of pullback tensor-power isomorphisms with composition
AlgebraicGeometry.Scheme.Modules.pullbackTensorPowIso_trans_tensorPowMapIso_comp6 below · cited by 1 · depth 38 - Sheafification tensorator on sections: μ(x^#⊗ y^#)=(x⊗ y)^#
AlgebraicGeometry.Scheme.Modules.sheafify_mu_app_tensorSections_monoidalV22 below · cited by 1 · depth 38 - Endomorphisms of the unit mathcal O_X-module are multiplications
AlgebraicGeometry.Scheme.Modules.unitHom_app_eq_mul0 below · cited by 2 · depth 38 - From strictly increasing to all ordered pairs for unit cocycles
AlgebraicGeometry.Scheme.OrderedAffineCover.exists_inf_cocycle_of_face_cocycle0 below · cited by 1 · depth 38 - Vanishing on a non-empty open forces vanishing (invertible target)
AlgebraicGeometry.Scheme.Modules.IsInvertible.eq_zero_of_forall_app_eq_zero_of_isIntegral0 below · cited by 1 · depth 39 - Lifting sections along a small thickening of a local base
AlgebraicGeometry.Scheme.Modules.exists_pullbackLocalSection_eq_of_ker_mul_maximalIdeal_eq_bot_of_forall_subsingleton_HSucc36 below · cited by 1 · depth 39 - Finite schemes over Artinian rings: Artinian local points are jointly epimorphic
AlgebraicGeometry.Scheme.Hom.eq_of_forall_spec_comp_eq_of_isFinite_of_isArtinianRing0 below · cited by 1 · depth 40 - Descent datum from a cocycle for a free split action
AlgebraicGeometry.Scheme.Modules.exists_descentData_obj_eq_of_cocycle_of_free_of_split5 below · cited by 2 · depth 40 - Surjectivity of pullback on sections transports along isomorphisms
AlgebraicGeometry.Scheme.Modules.exists_pullbackLocalSection_eq_of_iso_hom_comp_eq1 below · cited by 1 · depth 40 - Cocycle condition for a split translation action
AlgebraicGeometry.Scheme.Modules.cocycle_of_forall_mapIso_eq_of_split1 below · cited by 1 · depth 41 - Gluing a family of pullback isomorphisms over a pair equalised by q
AlgebraicGeometry.Scheme.Modules.exists_iso_pullback_forall_mapIso_eq_of_free_of_split2 below · cited by 1 · depth 41 - Maps out of M⊗ P are determined on elementary tensor sections
AlgebraicGeometry.Scheme.Modules.tensor_hom_ext_monoidalV20 below · cited by 1 · depth 41 - Ideal sheaves with equal relative rank coincide
AlgebraicGeometry.Scheme.IdealSheafData.eq_of_le_of_forall_finrank_subschemeIota_comp_eq1 below · cited by 1 · depth 42 - Ideal sheaf inclusion from germs at specialisations
AlgebraicGeometry.Scheme.IdealSheafData.le_of_forall_mem_support_exists_specializes_map_germ_le1 below · cited by 1 · depth 43 - Germ comparison for ideal sheaves pulled back along an automorphism
AlgebraicGeometry.Scheme.IdealSheafData.map_germ_comap_le_iff_map_germ_le0 below · cited by 1 · depth 43 - Germ comparison of ideal sheaves via localised chart ideals
AlgebraicGeometry.Scheme.IdealSheafData.map_germ_le_iff_map_localization_comap_ideal_top_le2 below · cited by 1 · depth 43 - Germ of a pulled-back ideal sheaf along an open immersion
AlgebraicGeometry.Scheme.IdealSheafData.map_germ_comap_ideal_eq_map_stalkMap_of_isOpenImmersion0 below · cited by 1 · depth 44
AlgebraicGeometry.SchemeHomOver 8
- Uniqueness of A-valued points of a separated R-scheme
AlgebraicGeometry.SchemeHomOver.eq_of_isSeparated_of_valuationRing_of_fst_eq0 below · cited by 13 · depth 12 - Rigidity: geometric points over ̄ K determine morphisms
AlgebraicGeometry.SchemeHomOver.ext_of_forall_algebraicClosure_point_of_isReduced_of_flat0 below · cited by 29 · depth 12 - Uniqueness of valuation-ring points of a separated scheme
AlgebraicGeometry.SchemeHomOver.ext_of_isSeparated_of_valuationRing0 below · cited by 4 · depth 13 - Closed immersion from a functorial factorisation criterion
AlgebraicGeometry.SchemeHomOver.isClosedImmersion_of_iff_exists_comp_eq_of_injective1 below · cited by 2 · depth 14 - Sections through regular points lie in the smooth locus
AlgebraicGeometry.SchemeHomOver.apply_closedPoint_mem_smoothLocus_of_isRegularLocalRing_stalk3 below · cited by 1 · depth 24 - Morphisms from a reduced finite-type κ-scheme agreeing on κ-points
AlgebraicGeometry.SchemeHomOver.ext_of_forall_point_of_isReduced_of_locallyOfFiniteType0 below · cited by 5 · depth 28 - Maps from a reduced finite-type scheme agreeing on all κ-points
AlgebraicGeometry.SchemeHomOver.ext_of_forall_point_of_isReduced_of_isAlgClosed1 below · cited by 1 · depth 29 - Affine representability gives Spec H ≅ Y over Spec R
AlgebraicGeometry.SchemeHomOver.exists_iso_spec_of_forall_equiv_algHom0 below · cited by 1 · depth 35
AlgebraicGeometry.SmallExtension 66
- Point-derivation form of tangent coordinates of a pair of lifts
AlgebraicGeometry.SmallExtension.exists_pointDerivations_isTangentCoordsOfPairAt_of_flat6 below · cited by 10 · depth 31 - Additivity of pair tangent coordinates along three lifts
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_add7 below · cited by 25 · depth 31 - Tangent fields of pairs add on the doubled thickening
AlgebraicGeometry.SmallExtension.IsTangentOfPair.exists_comp_map_fst_eq_and_isTangentOfPair_comp_map_add1 below · cited by 2 · depth 32 - Tangent coordinates determine the second member of a deformation pair
AlgebraicGeometry.SmallExtension.eq_of_isTangentCoordsOfPairAt_of_isTangentCoordsOfPairAt5 below · cited by 7 · depth 32 - Unique tangent morphism attached to two lifts, flat case
AlgebraicGeometry.SmallExtension.existsUnique_isTangentOfPair_of_flat2 below · cited by 8 · depth 32 - Existence of tangent coordinates at the unit for a pair
AlgebraicGeometry.SmallExtension.exists_isTangentCoordsOfPairAt4 below · cited by 19 · depth 32 - Point derivations at the unit are tangent coordinates of deformations
AlgebraicGeometry.SmallExtension.exists_isTangentCoordsOfPairAt_of_pointDerivations4 below · cited by 7 · depth 32 - Tangent coordinates across a commuting square of lifts
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_add_eq_add_of_specMap_comp_eq12 below · cited by 1 · depth 32 - Naturality of pair tangent coordinates under flat algebra maps
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_comp_of_flat4 below · cited by 23 · depth 32 - Tangent maps into an affine chart versus point derivations
AlgebraicGeometry.SmallExtension.mem_pointDerivations_tangentCoords_and_injective_and_surjective0 below · cited by 6 · depth 32 - Additivity of tangent coordinates along the sum map
AlgebraicGeometry.SmallExtension.tangentCoords_comp_map_add_eq_add0 below · cited by 2 · depth 32 - Tangent coordinates twisted by κ[φ_V] precompose the dual vector
AlgebraicGeometry.SmallExtension.tangentCoords_comp_map_trivSqZeroExt_map_apply0 below · cited by 1 · depth 32 - Zero section of a tangent field of a pair recovers u
AlgebraicGeometry.SmallExtension.IsTangentOfPair.zeroSection_comp_eq0 below · cited by 2 · depth 33 - Every tangent field over T' is tangent to a unique lift
AlgebraicGeometry.SmallExtension.existsUnique_comp_eq_and_isTangentOfPair_of_flat_of_comp_eq2 below · cited by 5 · depth 33 - Naturality of pair tangent coordinates under flat chart change
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAtVia_comp_of_flat4 below · cited by 5 · depth 33 - Tangent coordinates: the W=top case of the relative reading
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_iff_isTangentCoordsOfPairAtVia_top0 below · cited by 3 · depth 33 - Naturality of `IsTangentOfPair` along maps into flat algebras
AlgebraicGeometry.SmallExtension.isTangentOfPair_comp_of_isTangentOfPair_of_flat1 below · cited by 3 · depth 33 - Naturality of `tangentCoords` in the chart algebra
AlgebraicGeometry.SmallExtension.tangentCoords_map_comp0 below · cited by 2 · depth 33 - Lifts agreeing modulo I inject into tangent fields
AlgebraicGeometry.SmallExtension.exists_injective_isTangentOfPair_of_flat5 below · cited by 2 · depth 34 - Coboundary modification of overlap isomorphisms into a cocycle
AlgebraicGeometry.SmallExtension.exists_overlap_isos_cocycle_of_pointDerivations_two_coboundary26 below · cited by 1 · depth 34 - Obstruction 2-cocycle of a system of local smooth lifts
AlgebraicGeometry.SmallExtension.exists_pointDerivations_obstruction_two_cocycle_of_local_lifts35 below · cited by 2 · depth 34 - Pinned obstruction 2-cochain is a Čech cocycle
AlgebraicGeometry.SmallExtension.d_two_cochain_eq_zero_of_isTangentCoordsOfPairAtVia_pin17 below · cited by 1 · depth 35 - Tangent coordinates determine the deformation v
AlgebraicGeometry.SmallExtension.eq_of_isTangentCoordsOfPairAtVia_of_isTangentCoordsOfPairAtVia5 below · cited by 1 · depth 35 - Existence of tangent coordinates for a pair, via an open
AlgebraicGeometry.SmallExtension.exists_isTangentCoordsOfPairAtVia4 below · cited by 7 · depth 35 - Twisting an overlap isomorphism by a point derivation
AlgebraicGeometry.SmallExtension.exists_overlap_iso_isTangentCoordsOfPairAtVia_of_pointDerivations8 below · cited by 1 · depth 35 - Obstruction 2-cochain as a pinned point derivation at the unit
AlgebraicGeometry.SmallExtension.exists_pointDerivations_two_cochain_of_isTangentCoordsOfPairAtVia_pin7 below · cited by 1 · depth 35 - Additivity of via-tangent coordinates along a chain of lifts
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAtVia_add7 below · cited by 4 · depth 35 - Tangent coordinates are stable under postcomposition with ψ
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAtVia_comp_of_homOfLE_comp_eq0 below · cited by 5 · depth 35 - Tangent coordinates descend along a cartesian square over a monomorphism
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAtVia_of_isPullback_of_comp_mono1 below · cited by 5 · depth 35 - Chain rule for tangent coordinates under an endomorphism
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_comp_of_forall_apply_eq_pushPt_of_mul_maximalIdeal_eq_bot3 below · cited by 2 · depth 35 - Additivity of pair tangent coordinates along a commutative group law
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_mul_of_isCommutative11 below · cited by 1 · depth 35 - Naturality of the θ-twist on point derivations
AlgebraicGeometry.SmallExtension.pointDerivations_map_symm_map_rTensor_eq0 below · cited by 3 · depth 35 - Naturality of the obstruction 2-cocycle along a homomorphic lift
AlgebraicGeometry.SmallExtension.exists_d_eq_unitPullback_obstruction_two_cocycle_sub_of_local_lifts_hom27 below · cited by 1 · depth 36 - Lifting a module along a small extension when the Picard obstruction is a coboundary
AlgebraicGeometry.SmallExtension.exists_isInvertible_pullback_iso_of_isPicObstructionCocycle_of_forall_mem_range26 below · cited by 2 · depth 36 - Existence of a closed Picard deformation cocycle
AlgebraicGeometry.SmallExtension.exists_isPicDeformationCocycle_of_cechTrivialisation8 below · cited by 3 · depth 36 - Realising closed cocycles as Picard deformation data
AlgebraicGeometry.SmallExtension.exists_isPicDeformationCocycle_of_forall_d_eq_zero18 below · cited by 1 · depth 36 - Pullback of a Picard deformation cocycle along a refinement
AlgebraicGeometry.SmallExtension.exists_isPicDeformationCocycle_pullback_eq_unitPullback7 below · cited by 2 · depth 36 - Picard deformation cocycles add under tensor product
AlgebraicGeometry.SmallExtension.exists_isPicDeformationCocycle_tensor_add8 below · cited by 1 · depth 36 - Obstruction cocycle of the Mumford bundle is alternating
AlgebraicGeometry.SmallExtension.exists_isPicObstructionCocycle_mumfordBundle_eq_unitPullback_sub19 below · cited by 2 · depth 36 - Existence of a closed Picard obstruction cocycle
AlgebraicGeometry.SmallExtension.exists_isPicObstructionCocycle_of_cechTrivialisation6 below · cited by 2 · depth 36 - Exponentiating a point derivation into a deformation of u
AlgebraicGeometry.SmallExtension.exists_isTangentCoordsOfPairAtVia_of_pointDerivations4 below · cited by 1 · depth 36 - Push-forward of point derivations along a unit-preserving endomorphism
AlgebraicGeometry.SmallExtension.exists_pointDerivations_pushforward_natural_of_forall_apply_eq_pushPt0 below · cited by 1 · depth 36 - Rescaling the trivialisation in a Picard deformation cocycle
AlgebraicGeometry.SmallExtension.isPicDeformationCocycle_of_appTop_eq_unitAutSection3 below · cited by 2 · depth 36 - Picard deformation cocycles are invariant under isomorphism of modules
AlgebraicGeometry.SmallExtension.isPicDeformationCocycle_of_iso0 below · cited by 2 · depth 36 - The unit line bundle is a zero Picard deformation cocycle
AlgebraicGeometry.SmallExtension.isPicDeformationCocycle_unit_pullbackUnitIso_zero2 below · cited by 1 · depth 36 - Left translation invariance of pair tangent coordinates
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_mul_left0 below · cited by 1 · depth 36 - Vanishing Picard deformation class forces triviality of M
AlgebraicGeometry.SmallExtension.nonempty_iso_unit_of_isPicDeformationCocycle_of_forall_mem_range10 below · cited by 3 · depth 36 - Natural endomorphism determined by its value at M=k
AlgebraicGeometry.SmallExtension.pointDerivations_natural_endo_eq_symm_map_tmul_of_apply_eq0 below · cited by 1 · depth 36 - Two Picard deformation cocycles differ by a Čech coboundary
AlgebraicGeometry.SmallExtension.sub_mem_range_d_of_isPicDeformationCocycle_of_isPicDeformationCocycle7 below · cited by 2 · depth 36 - Fibre readings transport along a morphism of thickenings
AlgebraicGeometry.SmallExtension.IsFibreReading.appLE_of_comp_eq0 below · cited by 2 · depth 37 - Uniqueness of fibre readings over a flat base
AlgebraicGeometry.SmallExtension.IsFibreReading.eq_of_isFibreReading_of_flat0 below · cited by 4 · depth 37 - A section read as zero on the whole fibre vanishes
AlgebraicGeometry.SmallExtension.IsFibreReading.eq_zero_of_isFibreReading_zero_of_flat1 below · cited by 3 · depth 37 - Unit cocycle 1+ε realising a closed Čech 1-cocycle of readings
AlgebraicGeometry.SmallExtension.exists_isFibreReading_and_cocycle_one_add_of_forall_d_eq_zero4 below · cited by 1 · depth 37 - Naturality of Picard obstruction cocycles under refinement
AlgebraicGeometry.SmallExtension.exists_isPicObstructionCocycle_pullback_eq_unitPullback6 below · cited by 2 · depth 37 - Every fibre reading is realised on an affine open
AlgebraicGeometry.SmallExtension.exists_mem_map_range_and_isFibreReading_of_isAffineOpen1 below · cited by 3 · depth 37 - Naturality 1-cochain of tangent coordinates along a morphism
AlgebraicGeometry.SmallExtension.exists_one_cochain_isTangentCoordsOfPairAtVia_pin_of_local_lifts_hom8 below · cited by 1 · depth 37 - Calculus of fibre readings of sections of ι(V)·Γ(X,U)
AlgebraicGeometry.SmallExtension.isFibreReading_zero_add_mul_neg_restrict_and_exists_isFibreReading0 below · cited by 9 · depth 37 - Dual module carries the negated Picard obstruction cocycle
AlgebraicGeometry.SmallExtension.isPicObstructionCocycle_dual_neg9 below · cited by 1 · depth 37 - Picard obstruction cocycles are insensitive to isomorphism
AlgebraicGeometry.SmallExtension.isPicObstructionCocycle_of_iso0 below · cited by 1 · depth 37 - Picard obstruction cocycles add under tensor product
AlgebraicGeometry.SmallExtension.isPicObstructionCocycle_tensor_add6 below · cited by 1 · depth 37 - Tangent coordinates of an affine combination of two endomorphisms
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAt_add_smul_of_apply_eq_add_mul_sub5 below · cited by 1 · depth 37 - Picard obstruction cocycle well defined modulo coboundaries
AlgebraicGeometry.SmallExtension.sub_mem_range_d_of_isPicObstructionCocycle_of_isPicObstructionCocycle9 below · cited by 1 · depth 37 - Pulled-back obstruction cocycle minus obstruction cocycle is a coboundary
AlgebraicGeometry.SmallExtension.unitPullback_obstruction_two_cocycle_sub_eq_d_of_one_cochain_pin19 below · cited by 1 · depth 37 - Covariance of pair tangent coordinates along a fibre homomorphism
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAtVia_comp_of_mul_comp_eq0 below · cited by 1 · depth 38 - Alternating behaviour of the defect tangent coordinates
AlgebraicGeometry.SmallExtension.isTangentCoordsOfPairAtVia_defect_eq_sign_smul_of_pin15 below · cited by 1 · depth 38 - Lift–transition composites over T' and congruence modulo kerπ
AlgebraicGeometry.SmallExtension.naturality_pair_comp_eq_and_quotient_comp_eq_of_local_lifts_hom1 below · cited by 1 · depth 38
AlgebraicGeometry.Smooth 37
- Smooth morphisms over Henselian local rings have sections lifting residue points
AlgebraicGeometry.Smooth.exists_comp_eq_id_and_specMap_comp_eq_of_henselianLocalRing2 below · cited by 8 · depth 13 - Smoothness over a reduced locally Noetherian base preserves reducedness
AlgebraicGeometry.Smooth.isReduced_of_isReduced_of_isLocallyNoetherian1 below · cited by 47 · depth 13 - Stalks of a smooth scheme over a normal affine base
AlgebraicGeometry.Smooth.isDomain_and_isIntegrallyClosed_stalk2 below · cited by 15 · depth 14 - Smooth over a field: stalks are regular local rings
AlgebraicGeometry.Smooth.isRegularLocalRing_stalk4 below · cited by 37 · depth 14 - Stalks of a scheme smooth over a DVR are regular
AlgebraicGeometry.Smooth.isRegularLocalRing_stalk_of_isDiscreteValuationRing1 below · cited by 24 · depth 14 - Smoothness descends along surjective flat quasi-compact maps
AlgebraicGeometry.Smooth.descendsAlong_surjective_inf_flat_inf_quasiCompact0 below · cited by 5 · depth 15 - Smooth k-schemes are locally irreducible
AlgebraicGeometry.Smooth.exists_isOpen_isIrreducible_nhd8 below · cited by 1 · depth 15 - Smooth over a DVR: stalk at a generic point of the special fibre
AlgebraicGeometry.Smooth.isDiscreteValuationRing_stalk_of_forall_specializes3 below · cited by 13 · depth 17 - Stalks of a smooth scheme over a DVR are integrally closed domains
AlgebraicGeometry.Smooth.isDomain_and_isIntegrallyClosed_stalk_of_isDiscreteValuationRing2 below · cited by 17 · depth 18 - Sections are dense in the special fibre of a smooth morphism
AlgebraicGeometry.Smooth.dense_setOf_exists_section_of_henselianLocalRing_of_isAlgClosed4 below · cited by 1 · depth 21 - Smooth morphism with preconnected source has constant relative dimension
AlgebraicGeometry.Smooth.exists_smoothOfRelativeDimension_of_preconnectedSpace0 below · cited by 17 · depth 21 - Uniformiser generates mathfrak m_{T,η} at generic points of the special fibre
AlgebraicGeometry.Smooth.maximalIdeal_stalk_eq_span_of_forall_specializes4 below · cited by 3 · depth 21 - Hensel lifting of residue points of a smooth R-scheme
AlgebraicGeometry.Smooth.exists_comp_eq_specMap_and_specMap_comp_eq_and_stalkClosedPointTo_mul_of_henselianLocalRing3 below · cited by 5 · depth 23 - Relative Frobenius of a smooth scheme is finite flat surjective
AlgebraicGeometry.Smooth.isFinite_and_flat_and_surjective_of_isPullback_frobenius10 below · cited by 3 · depth 27 - Smoothness from lifting along small surjections of Artin local rings
AlgebraicGeometry.Smooth.of_forall_exists_lift_of_isArtinianRing_of_charP_of_finiteType_int20 below · cited by 1 · depth 28 - Tangent lines at positive-characteristic geometric points force relative dimension 1
AlgebraicGeometry.Smooth.smoothOfRelativeDimension_one_of_forall_charP_exists_forall_existsUnique_eq_comp_of_finiteType_int0 below · cited by 1 · depth 28 - Smoothness after base change descends to a finitely generated subalgebra
AlgebraicGeometry.Smooth.exists_fg_subalgebra_of_smooth_pullback_snd3 below · cited by 3 · depth 29 - Formally smooth affine chart along a section of a smooth morphism
AlgebraicGeometry.Smooth.exists_formallySmooth_chart_of_section0 below · cited by 1 · depth 31 - Smooth morphisms are locally of some relative dimension
AlgebraicGeometry.Smooth.exists_mem_and_smoothOfRelativeDimension_opensInclusion_comp0 below · cited by 2 · depth 31 - Uniformiser is prime in stalks of a smooth R-scheme
AlgebraicGeometry.Smooth.algebraMap_stalk_ne_zero_and_isPrime_span_of_apply_eq_closedPoint8 below · cited by 4 · depth 32 - Infinitesimal lifting of morphisms into a smooth affine scheme
AlgebraicGeometry.Smooth.exists_lift_comp_eq_of_isPullback_of_isAffine_of_isNilpotent0 below · cited by 2 · depth 32 - Formally smooth chart at a maximal special point is an open immersion
AlgebraicGeometry.Smooth.snd_apply_eq_and_exists_isOpenImmersion_homOfLE_comp_of_formallySmooth_stalk16 below · cited by 1 · depth 32 - Generic point of an irreducible smooth special fibre
AlgebraicGeometry.Smooth.exists_isDiscreteValuationRing_stalk_maximalIdeal_eq_span_germ_of_isIrreducible_fibre13 below · cited by 1 · depth 33 - Sections of a smooth morphism lift along nilpotent thickenings
AlgebraicGeometry.Smooth.exists_section_comp_eq_of_isPullback_of_isNilpotent_ker1 below · cited by 2 · depth 33 - Artinian-local lifting criterion for smoothness over a Noetherian base
AlgebraicGeometry.Smooth.of_forall_exists_lift_of_isArtinianRing_of_isNoetherianRing14 below · cited by 1 · depth 33 - Étale local sections of smooth schemes over henselian DVRs
AlgebraicGeometry.Smooth.exists_finite_etale_dvr_specMap_comp_eq_apply_closedPoint_mem_of_henselianLocalRing9 below · cited by 3 · depth 34 - Infinitesimal lifting along nilpotent ideals for smooth affine morphisms
AlgebraicGeometry.Smooth.exists_lift_comp_eq_of_isNilpotent_of_isAffine0 below · cited by 2 · depth 34 - Local smooth lifting over a nilpotent thickening of an affine base
AlgebraicGeometry.Smooth.exists_orderedAffineCover_basicOpen_forall_lift_comp_eq_of_isAffine_of_isNilpotent3 below · cited by 1 · depth 34 - Finite ordered affine cover with smooth affine liftings
AlgebraicGeometry.Smooth.exists_orderedAffineCover_forall_exists_smooth_isPullback_of_surjective_of_forall_isNilpotent2 below · cited by 1 · depth 34 - Zariski-local infinitesimal lifting along smooth morphisms
AlgebraicGeometry.Smooth.exists_span_eq_top_and_forall_exists_lift_away_of_isNilpotent1 below · cited by 6 · depth 34 - Smoothness descends along a smooth quasi-compact surjection
AlgebraicGeometry.Smooth.of_comp_of_smooth_of_surjective14 below · cited by 1 · depth 34 - Lifting separable residue points on smooth schemes over henselian DVRs
AlgebraicGeometry.Smooth.exists_finite_etale_dvr_specMap_comp_eq_of_isSeparable_of_henselianLocalRing7 below · cited by 1 · depth 35 - Local smooth lifting along a nilpotent surjection
AlgebraicGeometry.Smooth.exists_forall_isAffineOpen_exists_smooth_isPullback_of_surjective_of_forall_isNilpotent1 below · cited by 1 · depth 35 - Transition isomorphisms of smooth local lifts on overlaps
AlgebraicGeometry.Smooth.exists_overlap_isos_local_lifts6 below · cited by 1 · depth 35 - Smoothness descends along a smooth surjection on the source
AlgebraicGeometry.Smooth.of_comp_of_smooth_of_surjective_of_locallyOfFinitePresentation5 below · cited by 1 · depth 35 - Uniqueness of affine smooth lifts along a nilpotent thickening
AlgebraicGeometry.Smooth.exists_iso_hom_comp_eq_of_isPullback_of_isAffine_of_isNilpotent1 below · cited by 1 · depth 36 - Smooth affine local lift over an affine open of A₀×_T A₀
AlgebraicGeometry.Smooth.exists_affine_smooth_local_lift_opens_pullback6 below · cited by 1 · depth 37
AlgebraicGeometry.SmoothOfRelativeDimension 18
- Stalks at closed points of a smooth relative curve are DVRs
AlgebraicGeometry.SmoothOfRelativeDimension.isDiscreteValuationRing_stalk_of_isClosed1 below · cited by 16 · depth 14 - Smooth of relative dimension n over a field: dim X ≤ n
AlgebraicGeometry.SmoothOfRelativeDimension.topologicalKrullDim_le0 below · cited by 45 · depth 14 - Smoothness of relative dimension n descends along fpqc base change
AlgebraicGeometry.SmoothOfRelativeDimension.descendsAlong_surjective_inf_flat_inf_quasiCompact0 below · cited by 8 · depth 15 - Regularity at all maximal ideals implies smoothness over ̄ k
AlgebraicGeometry.SmoothOfRelativeDimension.of_forall_isRegularLocalRing_of_isAlgClosed2 below · cited by 6 · depth 15 - Stalk at a k-rational point of a smooth curve is a DVR
AlgebraicGeometry.SmoothOfRelativeDimension.isDiscreteValuationRing_stalk_of_section2 below · cited by 6 · depth 17 - Smooth of relative dimension n over a field forces dim X ≥ n
AlgebraicGeometry.SmoothOfRelativeDimension.le_topologicalKrullDim1 below · cited by 29 · depth 26 - Fibres of a smooth morphism of relative dimension n
AlgebraicGeometry.SmoothOfRelativeDimension.topologicalKrullDim_preimage_singleton_eq3 below · cited by 11 · depth 26 - Affine neighbourhood with étale coordinate on a smooth complex curve
AlgebraicGeometry.SmoothOfRelativeDimension.exists_isAffineOpen_smooth_rank_one_etaleCoordinate_of_point2 below · cited by 6 · depth 28 - Relative Frobenius of a smooth k-scheme has rank pⁿ
AlgebraicGeometry.SmoothOfRelativeDimension.finrank_eq_pow_of_isPullback_frobenius5 below · cited by 2 · depth 28 - Affine chart with étale coordinates at a rational point
AlgebraicGeometry.SmoothOfRelativeDimension.exists_isAffineOpen_smooth_rank_eq_etaleCoordinates_of_point7 below · cited by 1 · depth 31 - Smooth of relative dimension n descends along fpqc base change
AlgebraicGeometry.SmoothOfRelativeDimension.of_isPullback_of_flat_of_surjective0 below · cited by 2 · depth 31 - Relative holomorphic chart for a smooth scheme over an analytic chart
AlgebraicGeometry.SmoothOfRelativeDimension.exists_relChart_differentiableOn_appLE_of_analyticChart19 below · cited by 2 · depth 32 - Free differentials on stalks of a smooth R-scheme
AlgebraicGeometry.SmoothOfRelativeDimension.nonempty_basis_kaehlerDifferential_stalk_of_fromSpecStalk_comp_eq0 below · cited by 1 · depth 32 - Affine étale-coordinate chart at a point of a smooth scheme over a curve
AlgebraicGeometry.SmoothOfRelativeDimension.exists_isAffineOpen_le_etaleCoordinates_cons_algebraMap_of_point7 below · cited by 1 · depth 33 - Smooth descent along a surjection, with relative dimensions
AlgebraicGeometry.SmoothOfRelativeDimension.of_comp_of_surjective_of_field16 below · cited by 1 · depth 33 - Relative dimension subtracts along a smooth surjection
AlgebraicGeometry.SmoothOfRelativeDimension.of_comp_of_smoothOfRelativeDimension_of_surjective0 below · cited by 1 · depth 34 - Krull dimension of the stalk at a closed point of a smooth k-scheme
AlgebraicGeometry.SmoothOfRelativeDimension.ringKrullDim_stalk_eq_of_isClosed1 below · cited by 1 · depth 34 - Free rank-n differentials at stalks of a smooth k-scheme
AlgebraicGeometry.SmoothOfRelativeDimension.nonempty_basis_kaehlerDifferential_stalk0 below · cited by 1 · depth 36
AlgebraicGeometry.SmoothProperCurve 55
- Finite-map data of large degree invertible in R
AlgebraicGeometry.SmoothProperCurve.exists_finiteMapData_le_isUnit_of_twoAffineOpenCover317 below · cited by 9 · depth 13 - A one-function Bertini theorem for level sets on curves
AlgebraicGeometry.SmoothProperCurve.exists_polynomial_isUnit_aeval_imp_etale_levelSet2 below · cited by 7 · depth 13 - Finite-map data are stable under base change
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.exists_baseChange0 below · cited by 13 · depth 14 - Constancy of the fibre genus over a connected Noetherian base
AlgebraicGeometry.SmoothProperCurve.exists_genus_forall_geometricFibre_riemannRoch_imp_eq_of_connectedSpace128 below · cited by 3 · depth 14 - Two affine charts with affine overlap through a given point
AlgebraicGeometry.SmoothProperCurve.exists_twoAffineOpenCover_mem_of_isAlgClosed89 below · cited by 11 · depth 14 - Two-chart pole datum of large unit order, cover-input form
AlgebraicGeometry.SmoothProperCurve.exists_twoChartPoleDatum_transcendental_le_isUnit_of_twoAffineOpenCover302 below · cited by 1 · depth 14 - Level sets of a two-chart pole datum are free of rank m
AlgebraicGeometry.SmoothProperCurve.levelSet_free_of_twoChartPoleDatum15 below · cited by 1 · depth 14 - Finite point sets lie in affine opens over an affine base open
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.exists_isAffineOpen_le_preimage_of_finset3 below · cited by 2 · depth 15 - Geometric fibres of smooth proper curves are curve models
AlgebraicGeometry.SmoothProperCurve.exists_curveModel_iso_pullback_of_isAlgClosed71 below · cited by 9 · depth 15 - Riemann–Roch for geometric fibres of smooth proper curves
AlgebraicGeometry.SmoothProperCurve.exists_curveModel_riemannRoch_of_isAlgClosed88 below · cited by 13 · depth 15 - Large-degree finite étale multisections from finite-map data
AlgebraicGeometry.SmoothProperCurve.exists_finite_etale_isClosedImmersion_le_finrank_of_finiteMapData3 below · cited by 1 · depth 15 - Sections of 𝒪(mε) non-vanishing along ε
AlgebraicGeometry.SmoothProperCurve.exists_forall_le_exists_section_invModule_disjoint_of_twoAffineOpenCover287 below · cited by 1 · depth 15 - Constancy of the genus over geometric fibres of a curve
AlgebraicGeometry.SmoothProperCurve.exists_genus_forall_geometricFibre_riemannRoch_imp_eq_of_finiteMapData98 below · cited by 4 · depth 15 - Split multisection yields d fibrewise distinct sections
AlgebraicGeometry.SmoothProperCurve.exists_sections_injective_of_tensorProduct_algEquiv_pi0 below · cited by 2 · depth 15 - Two-chart pole datum from a section of 𝒪(mε)
AlgebraicGeometry.SmoothProperCurve.exists_twoChartPoleDatum_of_section_invModule39 below · cited by 2 · depth 15 - Degree m of the level sets of f at every field point
AlgebraicGeometry.SmoothProperCurve.finrank_levelSet_field_of_twoChartPoleDatum7 below · cited by 1 · depth 15 - Flatness of Γ(C,U) over R[f] for a two-chart pole datum
AlgebraicGeometry.SmoothProperCurve.flat_aeval_of_twoChartPoleDatum10 below · cited by 1 · depth 15 - Gluing a finite-map datum to a morphism C → P¹_R
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.exists_hom_proj_preimage_basicOpen_eq0 below · cited by 1 · depth 16 - Finiteness of Čech H⁰ and H¹ of a glued line bundle
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.finite_H0_H1_lineBundle3 below · cited by 2 · depth 16 - Finite-map data of arbitrarily large degree on fixed charts
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.forall_exists_le_m_of_one_le1 below · cited by 1 · depth 16 - A→Γ(C_A,𝒪) bijective, from finite-map data
AlgebraicGeometry.SmoothProperCurve.bijective_algebraMap_sections_baseChange_of_finiteMapData92 below · cited by 2 · depth 16 - Constant genus of geometric fibres via a two-chart cover
AlgebraicGeometry.SmoothProperCurve.exists_genus_forall_geometricFibre_riemannRoch_imp_eq_of_twoAffineOpenCover150 below · cited by 7 · depth 16 - Finite level sets become closed subschemes after base change
AlgebraicGeometry.SmoothProperCurve.exists_isClosedImmersion_levelSet0 below · cited by 2 · depth 16 - Base-point-free section of 𝒪(mε) on a K-fibre
AlgebraicGeometry.SmoothProperCurve.exists_section_pullback_invModule_pow_ker_notMem_support_of_twoAffineOpenCover277 below · cited by 3 · depth 16 - Split sections of C_{R'} factoring through a block
AlgebraicGeometry.SmoothProperCurve.exists_sections_injective_comp_fst_eq_comp_of_tensorProduct_algEquiv_pi0 below · cited by 2 · depth 16 - Two-chart coordinates from a section of 𝒪(mε)
AlgebraicGeometry.SmoothProperCurve.exists_twoChart_of_section_invModule20 below · cited by 1 · depth 16 - Freeness and rank m of Γ(V)/(g) for an m-th order neighbourhood
AlgebraicGeometry.SmoothProperCurve.free_and_finrank_quotient_span_of_generates_ker_pow10 below · cited by 1 · depth 16 - Transcendence of a two-chart coordinate over a base field
AlgebraicGeometry.SmoothProperCurve.injective_aeval_tensor_of_twoChartPoleDatum4 below · cited by 2 · depth 16 - Complement of a section has nonzero fibre algebra
AlgebraicGeometry.SmoothProperCurve.nontrivial_tensor_sections_of_twoChartPoleDatum3 below · cited by 1 · depth 16 - Sections of (mathcal I_ε^m)^∨ surject onto a surjective base change
AlgebraicGeometry.SmoothProperCurve.surjective_unit_app_top_invModule_pow_ker268 below · cited by 2 · depth 16 - Two-chart data force transcendence on every field fibre
AlgebraicGeometry.SmoothProperCurve.transcendental_app_of_twoChart_of_section_mem3 below · cited by 1 · depth 16 - Finite map data of exact degree m≥ 2g+1
AlgebraicGeometry.SmoothProperCurve.exists_finiteMapData_m_eq_of_forall_invertible_free320 below · cited by 1 · depth 17 - Constancy of the genus of geometric fibres (two-affine-cover edition)
AlgebraicGeometry.SmoothProperCurve.exists_genus_forall_geometricFibre_riemannRoch_imp_eq_of_twoAffineOpenCover_of_isDomain178 below · cited by 1 · depth 17 - Genus of C is the genus of all its geometric fibres
AlgebraicGeometry.SmoothProperCurve.forall_geometricFibre_riemannRoch_imp_eq_of_isAlgClosed126 below · cited by 3 · depth 17 - Fibrewise vanishing of H¹ and h⁰=m+1-g for 𝒪(mε)
AlgebraicGeometry.SmoothProperCurve.subsingleton_H1_and_finrank_H0_sectionsOf_pullback_invModule_pow_ker264 below · cited by 3 · depth 17 - Section of (mathcal I_ε^m)^∨ whose zero scheme misses ε
AlgebraicGeometry.SmoothProperCurve.exists_section_invModule_pow_ker_disjoint_of_forall_invertible_free291 below · cited by 1 · depth 18 - Two-chart pole datum of exact order m over a Noetherian base
AlgebraicGeometry.SmoothProperCurve.exists_twoChartPoleDatum_forall_finrank_of_section_invModule39 below · cited by 1 · depth 18 - Level sets of a two-chart pole datum are free of rank m
AlgebraicGeometry.SmoothProperCurve.levelSet_free_of_twoChartPoleDatum_of_forall_finrank15 below · cited by 1 · depth 18 - Finiteness of Čech H⁰ and H¹ for locally trivial modules
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.finite_H0_H1_sectionsOf6 below · cited by 4 · depth 19 - Section of 𝒪(mε) non-vanishing at a maximal ideal
AlgebraicGeometry.SmoothProperCurve.exists_section_invModule_pow_ker_notMem_support_of_isMaximal282 below · cited by 2 · depth 19 - Two charts from a non-vanishing section of 𝒪(mε)
AlgebraicGeometry.SmoothProperCurve.exists_twoChart_of_section_invModule_global20 below · cited by 1 · depth 19 - Finiteness and fibre rank m of Γ(V)/(g)
AlgebraicGeometry.SmoothProperCurve.finite_and_forall_finrank_baseChange_quotient_span_of_generates_ker_pow10 below · cited by 1 · depth 19 - Two-chart pole datum: every field-valued level set has rank m
AlgebraicGeometry.SmoothProperCurve.finrank_levelSet_field_of_twoChartPoleDatum_of_forall_finrank7 below · cited by 1 · depth 19 - Two-chart pole datum: Γ(C,U) is flat over R[X]
AlgebraicGeometry.SmoothProperCurve.flat_aeval_of_twoChartPoleDatum_global10 below · cited by 1 · depth 19 - Fibrewise transcendence of the two-chart coordinates f, g
AlgebraicGeometry.SmoothProperCurve.transcendental_app_of_twoChart_of_section_mem_global3 below · cited by 1 · depth 19 - Transcendence of the chart coordinate of a two-chart pole datum
AlgebraicGeometry.SmoothProperCurve.injective_aeval_tensor_of_twoChartPoleDatum_global4 below · cited by 2 · depth 20 - Nontriviality of K ⊗_R Γ(C,U) for a two-chart pole datum
AlgebraicGeometry.SmoothProperCurve.nontrivial_tensor_sections_of_twoChartPoleDatum_global3 below · cited by 1 · depth 20 - Perfect integral Serre pairing along prescribed boundary sections
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.exists_laurentChart_isCompletionAlong_hasParameter_serrePairingInt_bijective_of_isSectional292 below · cited by 1 · depth 24 - Smooth proper curve as curve model of its function field
AlgebraicGeometry.SmoothProperCurve.exists_curveModel_iso_germToFunctionField_eq_of_isAlgClosed76 below · cited by 6 · depth 24 - Residue sums vanish on Čech coboundaries over ℤ₍ₚ₎
AlgebraicGeometry.SmoothProperCurve.FiniteMapData.residuesVanishOnCoboundaries_of_isSectional_of_isCompletionAlong_of_hasParameter107 below · cited by 1 · depth 25 - Geometric fibre of a smooth proper curve as a model of its function field
AlgebraicGeometry.SmoothProperCurve.exists_curveModel_iso_pullback_germToFunctionField_eq_of_isAlgClosed71 below · cited by 6 · depth 25 - Base change of a proper smooth curve to a henselian local ring
AlgebraicGeometry.SmoothProperCurve.isProper_and_smooth_and_geometricallyIntegral_and_nonempty_section_pullback_of_henselianLocalRing5 below · cited by 2 · depth 25 - Finite map data of large unit degree on a pointed smooth proper curve
AlgebraicGeometry.SmoothProperCurve.exists_finiteMapData_le_isUnit342 below · cited by 1 · depth 28 - Section-adapted two-chart affine cover of a smooth proper curve
AlgebraicGeometry.SmoothProperCurve.exists_twoAffineOpenCover_of_section325 below · cited by 1 · depth 29 - Section of 𝒪(mε) nonvanishing along ε
AlgebraicGeometry.SmoothProperCurve.exists_section_invModule_pow_ker_disjoint309 below · cited by 1 · depth 30
AlgebraicGeometry.Spec 3
- Gluing a functorial family of Y-valued points over Spec S
AlgebraicGeometry.Spec.exists_forall_map_comp_eq_of_functorial_family_of_span_eq_top0 below · cited by 2 · depth 30 - Equality of two Spec A-points spreads from 𝔭 to a basic open
AlgebraicGeometry.Spec.exists_forall_away_specMap_comp_eq_of_atPrime_specMap_comp_eq0 below · cited by 1 · depth 33 - Comparing ideals in the stalk of Spec A at 𝔭
AlgebraicGeometry.Spec.map_germ_le_iff_map_algebraMap_localization_le0 below · cited by 1 · depth 44
AlgebraicGeometry.SplitTorus 13
- Lifting μ_m^t to a base change over a henselian ring
AlgebraicGeometry.SplitTorus.existsUnique_muLift_baseChange_of_torusFibre_of_henselian30 below · cited by 4 · depth 13 - Rigidity of closed split sub-tori up to GLₜ(ℤ)
AlgebraicGeometry.SplitTorus.exists_addEquiv_eq_specMap_mapDomain_comp_of_range_eq3 below · cited by 5 · depth 13 - Conjugating a toric morphism by an automorphism of A
AlgebraicGeometry.SplitTorus.exists_conj_muHom_baseChange0 below · cited by 3 · depth 13 - Twisting torus morphisms into G×_{R_0}κ by field automorphisms
AlgebraicGeometry.SplitTorus.exists_twist_torusHom_baseChange_of_ringEquiv0 below · cited by 3 · depth 13 - Unique lifting of μ_m^t from a split torus in the special fibre
AlgebraicGeometry.SplitTorus.existsUnique_muLift_of_torusFibre_of_henselian27 below · cited by 1 · depth 14 - Homomorphy on κ-points extends to all points of a split torus
AlgebraicGeometry.SplitTorus.forall_torusPt_mul_of_torusPtId_mul_of_isAlgClosed1 below · cited by 4 · depth 14 - Points of μ_m^t are m-torsion; reduction is multiplicative
AlgebraicGeometry.SplitTorus.convPow_eq_one_and_comp_mapDomain_convMul0 below · cited by 1 · depth 15 - m-torsion of a split torus is μ_m^t of degree m^t
AlgebraicGeometry.SplitTorus.isFinite_schemeKerStr_and_finrank_eq_of_iso_torusScheme0 below · cited by 3 · depth 20 - Rank m^t for the m-torsion over a split open subtorus
AlgebraicGeometry.SplitTorus.moduleFinite_and_finrank_sections_preimage_opensRange_schemeKer_eq_pow0 below · cited by 1 · depth 20 - Fppf-local m-th roots of points of the split torus
AlgebraicGeometry.SplitTorus.exists_flat_surjective_pow_eq_comp1 below · cited by 2 · depth 21 - Split torus: commutative group law, unit points, finite flat [n]
AlgebraicGeometry.SplitTorus.exists_relativeGroupLaw_isCommutative_torusPt_convMul_schemeNsmul_eq0 below · cited by 2 · depth 23 - Closed immersion of equal-rank split tori lifts κ-points
AlgebraicGeometry.SplitTorus.exists_schemeHomOverComp_eq_of_isClosedImmersion_torusStr_of_eq0 below · cited by 2 · depth 26 - Split torus of rank d is smooth of relative dimension d
AlgebraicGeometry.SplitTorus.smoothOfRelativeDimension_torusStr0 below · cited by 1 · depth 31
AlgebraicGeometry.SubalgebraStages 1
- Base change along a directed union of subalgebras is a limit
AlgebraicGeometry.SubalgebraStages.nonempty_isLimit_cone0 below · cited by 6 · depth 30
AlgebraicGeometry.SymmRoot 2
- Symmetric square root over B from a two-step base change
AlgebraicGeometry.SymmRoot.exists_isSymmetric_locIsoOnBase_of_symmRootPred_baseChange9 below · cited by 1 · depth 35 - Rigidified symmetric root over B after rebasing to R₀
AlgebraicGeometry.SymmRoot.exists_rigidified_symmRootPred_baseChange_of_isSymmetric_of_locIsoOnBase9 below · cited by 1 · depth 35
AlgebraicGeometry.ThetaLevel 12
- Existence of intertwiners for centre-fixing Heisenberg automorphisms
AlgebraicGeometry.ThetaLevel.exists_isIntertwiner_of_mem_gam3 below · cited by 2 · depth 30 - Centre-fixing automorphism trivial on Schrödinger matrices is the identity
AlgebraicGeometry.ThetaLevel.eq_one_of_mem_gam_of_forall_schrodMat_apply_eq0 below · cited by 1 · depth 31 - Schur's lemma for the Schrödinger matrices of Heis_δ
AlgebraicGeometry.ThetaLevel.exists_eq_smul_one_of_forall_mul_schrodMat_eq_schrodMat_mul0 below · cited by 4 · depth 31 - Intertwiner from an η-fixed vector generating the Schrödinger module
AlgebraicGeometry.ThetaLevel.exists_isIntertwiner_of_forall_schrod_eta_apply_eq_of_bijective0 below · cited by 1 · depth 31 - A unit coordinate for the η-averaged delta vector
AlgebraicGeometry.ThetaLevel.exists_isUnit_sum_schrod_eta_single_apply_of_mem_gam0 below · cited by 1 · depth 31 - Unit-coordinate averaged vector is η-invariant with basis of θ-translates
AlgebraicGeometry.ThetaLevel.forall_schrod_eta_apply_eq_and_bijective_of_isUnit_sum_schrod_eta_single_apply0 below · cited by 1 · depth 31 - Schrödinger matrices give a representation of Heis_δ
AlgebraicGeometry.ThetaLevel.schrodMat_one_and_schrodMat_mul0 below · cited by 5 · depth 31 - Piecewise: conjugators are units times chosen intertwiners
AlgebraicGeometry.ThetaLevel.exists_idempotents_gam_units_mul_eq_mul_inter_of_forall_mul_schrodMat_eq6 below · cited by 1 · depth 32 - Idempotent splitting making T conjugate Schrödinger matrices into Schrödinger matrices
AlgebraicGeometry.ThetaLevel.exists_completeOrthogonalIdempotents_forall_smul_mul_schrodMat_eq_smul_schrodMat_mul2 below · cited by 1 · depth 33 - Piecewise normal form for matrices normalising the Schrödinger matrices
AlgebraicGeometry.ThetaLevel.exists_completeOrthogonalIdempotents_smul_eq_smul_schrodMat_of_forall_mul_schrodMat_eq_smul2 below · cited by 1 · depth 33 - Conjugating relabelling arises from a centre-fixing Heisenberg automorphism
AlgebraicGeometry.ThetaLevel.exists_gam_forall_smul_mul_schrodMat_eq_of_forall_smul_mul_schrodMat_eq1 below · cited by 1 · depth 33 - ε-intertwiners are unit multiples of the chosen intertwiner
AlgebraicGeometry.ThetaLevel.exists_units_smul_eq_smul_map_inter_of_forall_smul_mul_schrodMat_eq2 below · cited by 1 · depth 33
AlgebraicGeometry.TowerQuotientDatum 13
- Global universal property of a tower quotient datum
AlgebraicGeometry.TowerQuotientDatum.existsUnique_forall_p_comp_eq0 below · cited by 1 · depth 31 - Flat base change of a tower quotient datum
AlgebraicGeometry.TowerQuotientDatum.exists_baseChange_of_flat_of_isPullback22 below · cited by 1 · depth 31 - Local universal property under flat base change of a tower quotient
AlgebraicGeometry.TowerQuotientDatum.univ_loc_of_isPullback_of_flat21 below · cited by 1 · depth 32 - Base-changed tower quotient: surjectivity, closedness, orbits, open descent
AlgebraicGeometry.TowerQuotientDatum.exists_preimage_eq_of_isPullback_of_forall_preimage_eq0 below · cited by 1 · depth 33 - Chart data on basic opens of the base-changed quotient tower
AlgebraicGeometry.TowerQuotientDatum.exists_ringEquiv_fixedPoints_quotient_basicOpen_of_isPullback_of_flat12 below · cited by 1 · depth 33 - Chart rings of a tower quotient datum over affine opens
AlgebraicGeometry.TowerQuotientDatum.exists_ringEquiv_fixedPoints_quotient_of_isAffineOpen0 below · cited by 1 · depth 33 - Descent of compatible chart functions along a tower quotient
AlgebraicGeometry.TowerQuotientDatum.exists_sections_eq_of_forall_specMap_comp_eq0 below · cited by 1 · depth 33 - Fibres of a tower quotient datum's projection are G-orbits
AlgebraicGeometry.TowerQuotientDatum.p_base_eq_iff_and_preimage_image_eq0 below · cited by 2 · depth 33 - Local universal property after flat base change, assuming charts
AlgebraicGeometry.TowerQuotientDatum.univ_loc_of_isPullback_of_flat_of_forall_exists_chart5 below · cited by 1 · depth 33 - Invariants of the base-changed chart ring: completeness and levels
AlgebraicGeometry.TowerQuotientDatum.isAdicComplete_fixedPoints_and_exists_ringEquiv_quotient_basicOpen9 below · cited by 1 · depth 34 - A flat model A' for the base-changed chart ring
AlgebraicGeometry.TowerQuotientDatum.exists_flat_ringEquiv_tensorProduct_quotient_and_sections_basicOpen2 below · cited by 1 · depth 35 - Base-changed quotient: sections over rₙ⁻¹(Vₙ) and D(φₙ b)
AlgebraicGeometry.TowerQuotientDatum.exists_ringEquiv_quotient_sections_preimage_and_basicOpen0 below · cited by 1 · depth 36 - Primed chart ring as a base change: R'/πⁿ⁺¹≅ (R⊗_A A')/πⁿ⁺¹
AlgebraicGeometry.TowerQuotientDatum.exists_ringEquiv_tensorProduct_quotient_of_ringEquiv_sections_basicOpen0 below · cited by 1 · depth 36
AlgebraicGeometry.TwoGluedCurves 25
- Isomorphic node-unit bundles have proportional gluing units
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.exists_eq_mul_of_iso9 below · cited by 4 · depth 14 - Node-unit line bundles are fibrewise algebraically trivial
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.fibrewiseAlgEquivZero6 below · cited by 5 · depth 14 - Uniqueness of node-unit modules with given gluing units
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.nonempty_iso0 below · cited by 6 · depth 14 - Node-unit modules pull back to the unit on each component
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.nonempty_pullback_curveChange_iso_unit8 below · cited by 5 · depth 14 - Node-unit modules are stable under base change in T
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.pullback_baseChangeSnd3 below · cited by 4 · depth 14 - Rescaling all gluing units by a global unit preserves node-unit modules
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.smul_units0 below · cited by 4 · depth 14 - Tensoring node-unit modules multiplies the gluing units
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.tensor11 below · cited by 6 · depth 14 - Invertible node-unit modules with prescribed gluing units exist
AlgebraicGeometry.TwoGluedCurves.exists_isInvertible_isNodeUnitModule1 below · cited by 6 · depth 14 - Bundles trivial on both components are node-unit modules
AlgebraicGeometry.TwoGluedCurves.exists_isNodeUnitModule_of_pullback_curveChange_iso_unit10 below · cited by 6 · depth 14 - Unit module is a node-unit module with gluing units 1
AlgebraicGeometry.TwoGluedCurves.isNodeUnitModule_one_unit0 below · cited by 10 · depth 14 - One-node open cover for a base-changed glued curve
AlgebraicGeometry.TwoGluedCurves.exists_opens_iSup_eq_top_nodeLocus_eq_bot0 below · cited by 1 · depth 15 - Frame criterion for the node-unit description of sections
AlgebraicGeometry.TwoGluedCurves.injective_and_range_eq_nodeCondition_of_forall_exists_isFrameOn1 below · cited by 3 · depth 15 - Algebraic equivalence to zero spreads over a preconnected base
AlgebraicGeometry.TwoGluedCurves.isAlgEquivZero_fibre_of_isAlgEquivZero_fibre_of_preconnectedSpace287 below · cited by 2 · depth 15 - Additivity of Euler characteristics for two glued subschemes
AlgebraicGeometry.TwoGluedCurves.eulerChar_sectionsOf_eq_add_sub_natCard_of_isInvertible90 below · cited by 5 · depth 17 - Genus of two smooth curves glued at n points
AlgebraicGeometry.TwoGluedCurves.finrank_H1_add_finrank_H1_add_eq_of_finrank_H1_unit_eq211 below · cited by 2 · depth 18 - Čech H¹ vanishing on two transversally glued curves
AlgebraicGeometry.TwoGluedCurves.subsingleton_H1_and_support_zeroSchemeIdeal_subset_of_restrict141 below · cited by 1 · depth 18 - Crossing subscheme in C₁ as product of n point ideals
AlgebraicGeometry.TwoGluedCurves.exists_finset_card_eq_and_prod_ker_eq_ker_fst_of_isReduced0 below · cited by 2 · depth 19 - Restriction of the two-sided chart bundle to the second component
AlgebraicGeometry.TwoGluedCurves.nonempty_pullback_chartModule_iso_snd15 below · cited by 1 · depth 19 - Chart module restricted along i₁ and twisted by crossings
AlgebraicGeometry.TwoGluedCurves.nonempty_pullback_chartModule_tensor_module_ker_fst_iso_of_isInvertible16 below · cited by 1 · depth 19 - Glued pair of smooth proper curves has h⁰(𝒪_X)=1
AlgebraicGeometry.TwoGluedCurves.finrank_H0_sectionsOf_unit_eq_one64 below · cited by 2 · depth 22 - Genus of two smooth curves glued at s points
AlgebraicGeometry.TwoGluedCurves.finrank_H1_sectionsOf_unit_add_one_eq_genusFF_add_genusFF_add_card_of_curveModel218 below · cited by 2 · depth 22 - Node-unit modules are preserved by node-fixing automorphisms
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.pullback_curveChange_of_iso_of_nodes_fixed0 below · cited by 1 · depth 24 - Node-unit modules are stable under isomorphism
AlgebraicGeometry.TwoGluedCurves.IsNodeUnitModule.of_iso0 below · cited by 2 · depth 26 - Mayer–Vietoris inequality for Euler characteristics of two-chart Čech cohomology
AlgebraicGeometry.TwoGluedCurves.eulerChar_sectionsOf_le_add_sub_natCard_of_isInvertible91 below · cited by 1 · depth 27 - Degrees of invertible ideal sheaves add over two glued components
AlgebraicGeometry.TwoGluedCurves.isFinite_and_finrank_subscheme_comp_eq_add_of_isInvertible_comap104 below · cited by 1 · depth 27
AlgebraicGeometry.TwoGluedProjectiveLines 19
- Algebraically trivial bundle with a section on two glued lines
AlgebraicGeometry.TwoGluedProjectiveLines.nonempty_iso_unit_of_isAlgEquivZero_of_ne_zero331 below · cited by 1 · depth 15 - Line bundles on two glued projective lines: Čech model
AlgebraicGeometry.TwoGluedProjectiveLines.exists_linearEquiv_sectionsOf_gluedLinesSections_and_eulerChar_pullback_of_isAlgClosed327 below · cited by 5 · depth 16 - Chart rings of two projective lines glued at nodes
AlgebraicGeometry.TwoGluedProjectiveLines.exists_algEquiv_cover_gluedLinesCover8 below · cited by 1 · depth 17 - Anchored chart dictionary for two transversally glued projective lines
AlgebraicGeometry.TwoGluedProjectiveLines.exists_algEquiv_cover_gluedLinesCover_eval2_apply_eq8 below · cited by 1 · depth 17 - Node-ratio invariant for two glued projective lines
AlgebraicGeometry.TwoGluedProjectiveLines.exists_nodeRatioHom72 below · cited by 1 · depth 17 - Two lines glued at s points: s = g+1
AlgebraicGeometry.TwoGluedProjectiveLines.eq_finrank_H1_add_one_of_finrank_H0_eq_one330 below · cited by 4 · depth 18 - Zeros of a section on two glued rational curves
AlgebraicGeometry.TwoGluedProjectiveLines.exists_relEffCartierDiv_I_eq_zeroSchemeIdeal_and_supportedIn_of_ne_zero_of_pos339 below · cited by 1 · depth 18 - Two-glued-lines presentation transports along an isomorphism of k-schemes
AlgebraicGeometry.TwoGluedProjectiveLines.exists_twoAffineOpenCover_presentation_comp_iso0 below · cited by 1 · depth 18 - Multidegree (s-1,0) bundles on two glued lines: h⁰=1, H¹=0
AlgebraicGeometry.TwoGluedProjectiveLines.finrank_H0_sectionsOf_eq_one_and_subsingleton_H1_of_eulerChar_pullback_eq_of_isAlgClosed331 below · cited by 3 · depth 18 - Trivial pullbacks of L force h⁰(M_±)<2
AlgebraicGeometry.TwoGluedProjectiveLines.finrank_H0_twists_lt_two_of_nonempty_pullback_iso_unit334 below · cited by 1 · depth 18 - Bundles trivial on both lines of a transversal gluing are algebraically equivalent to zero
AlgebraicGeometry.TwoGluedProjectiveLines.isAlgEquivZero_of_pullback_iso_unit18 below · cited by 3 · depth 18 - Triviality on both lines when h⁰ of both twists is <2
AlgebraicGeometry.TwoGluedProjectiveLines.nonempty_pullback_iso_unit_of_finrank_H0_twists_lt_two334 below · cited by 1 · depth 18 - Uniqueness up to isomorphism of node-unit modules
AlgebraicGeometry.TwoGluedProjectiveLines.IsNodeUnitModule.nonempty_iso0 below · cited by 1 · depth 19 - Node-unit modules are stable under base change in T
AlgebraicGeometry.TwoGluedProjectiveLines.IsNodeUnitModule.pullback_baseChangeSnd4 below · cited by 1 · depth 19 - Čech cohomology of a bundle on two glued projective lines
AlgebraicGeometry.TwoGluedProjectiveLines.exists_finrank_H0_sectionsOf_eq_finrank_H0_gluedLinesSections_and_eulerChar_pullback_of_isAlgClosed328 below · cited by 3 · depth 19 - Existence of node-unit line bundles on two glued lines
AlgebraicGeometry.TwoGluedProjectiveLines.exists_isInvertible_isNodeUnitModule2 below · cited by 2 · depth 19 - Bundle trivial on both lines is a node-unit module
AlgebraicGeometry.TwoGluedProjectiveLines.exists_isNodeUnitModule_pullback_of_pullback_iso_unit12 below · cited by 1 · depth 19 - Structure sheaf of two glued lines is a node-unit module
AlgebraicGeometry.TwoGluedProjectiveLines.isNodeUnitModule_one_unit0 below · cited by 3 · depth 19 - Matching places give equal κ-points of the ambient scheme
AlgebraicGeometry.TwoGluedProjectiveLines.pointAt_comp_eq_pointAt_comp0 below · cited by 2 · depth 20
AlgebraicGeometry.UniversallyInjective 1
- Injectivity on geometric points implies universal injectivity
AlgebraicGeometry.UniversallyInjective.of_forall_isAlgClosed_points_eq0 below · cited by 5 · depth 18
AlgebraicGeometry.ValuativeCommSq 1
- Reduction of valuative squares to complete discrete valuation rings
AlgebraicGeometry.ValuativeCommSq.exists_isAdicComplete_finite_residueField_hasLift_imp_hasLift1 below · cited by 1 · depth 29
AlgebraicGeometry.tilde 3
- Base-change isomorphism on global sections sends m to 1⊗ m
AlgebraicGeometry.tilde.pullbackSpecIso_hom_app_top_unit_toOpen0 below · cited by 3 · depth 15 - Base-change isomorphism for ·̃ respects adjunction units
AlgebraicGeometry.tilde.pullbackSpecIso_unit0 below · cited by 2 · depth 16 - Base change of M̃ composes: cocycle compatibility
AlgebraicGeometry.tilde.pullback_map_pullbackSpecIso_hom_comp_pullbackSpecIso_hom1 below · cited by 1 · depth 20