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Namespace ProjSpaceCech 24 theorems

GradedModule 19 · Twist 5

ProjSpaceCech.GradedModule 19

• Saturated injective maps induce isomorphisms on Čech cohomology ProjSpaceCech.GradedModule.Hom.HMap_bijective_of_saturated 4 below · cited by 2 · depth 19
• Serre finiteness for Čech cohomology on Pⁿ_R ProjSpaceCech.GradedModule.finite_cohomology_of_isFG 9 below · cited by 1 · depth 19
• Serre vanishing for finitely generated graded modules ProjSpaceCech.GradedModule.subsingleton_cohomology_shift_of_isFG 10 below · cited by 3 · depth 19
• Čech cohomology bijective when cochain maps are bijective ProjSpaceCech.GradedModule.HMap_bijective_of_cochainMap_bijective 0 below · cited by 2 · depth 20
• Saturated injective maps induce bijections on Čech cochains ProjSpaceCech.GradedModule.Hom.cochainMap_bijective_of_saturated 2 below · cited by 1 · depth 20
• Dévissage step for finiteness of Čech cohomology ProjSpaceCech.GradedModule.Presentation.finite_H_of_ses 1 below · cited by 1 · depth 20
• Kernels of graded presentations are again finitely generated ProjSpaceCech.GradedModule.Presentation.ker_isFG 0 below · cited by 3 · depth 20
• Dévissage: vanishing of Hⁱ from a presentation ProjSpaceCech.GradedModule.Presentation.subsingleton_H_of_ses 1 below · cited by 1 · depth 20
• Finiteness of Čech cohomology of a finite sum of twists ProjSpaceCech.GradedModule.finite_cohomology_pi_FD 5 below · cited by 1 · depth 20
• Two models of Hⁱ(Pⁿ_R,𝒪(d₀)) agree ProjSpaceCech.GradedModule.nonempty_HEquiv_FD 1 below · cited by 2 · depth 20
• Čech cohomology commutes with finite products of graded modules ProjSpaceCech.GradedModule.nonempty_HEquiv_pi 0 below · cited by 2 · depth 20
• Saturated injective maps induce isomorphisms on sections ProjSpaceCech.GradedModule.Hom.secMap_bijective_of_saturated 1 below · cited by 1 · depth 21
• The alternating Čech differential of a graded module squares to zero ProjSpaceCech.GradedModule.d_sq 0 below · cited by 3 · depth 21
• Finiteness of Hⁱ(Pⁿ_R,S(d₀)̃) ProjSpaceCech.GradedModule.finite_cohomology_FD 3 below · cited by 1 · depth 21
• Injectivity passes to degree-zero localisations of graded modules ProjSpaceCech.GradedModule.Hom.secMap_injective 0 below · cited by 2 · depth 22
• Serre vanishing for a saturating finitely generated graded submodule ProjSpaceCech.GradedModule.exists_forall_subsingleton_H_shift_of_isFG_of_hom_injective_saturated 11 below · cited by 2 · depth 33
• Global sections of M̃(d) come from M_d for dgg 0 ProjSpaceCech.GradedModule.exists_forall_H_zero_shift_eq_sec_mk_of_isFG 14 below · cited by 1 · depth 35
• Dévissage for Čech H⁰ along a graded presentation ProjSpaceCech.GradedModule.Presentation.forall_H_zero_shift_eq_sec_mk_of_subsingleton_H_one 2 below · cited by 1 · depth 36
• Degree-d Čech 0-cocycles of bigoplus_k S(e_k) come from F_d ProjSpaceCech.GradedModule.exists_forall_H_zero_pi_FD_shift_eq_sec_mk 0 below · cited by 1 · depth 36

ProjSpaceCech.Twist 5

• Vanishing of Hⁱ(Pⁿ_R,𝒪(d)) for i≥ 1, d≥ -n ProjSpaceCech.Twist.subsingleton_cohomology_of_neg_le 2 below · cited by 1 · depth 20
• The twisted Čech differential squares to zero ProjSpaceCech.Twist.d_sq 0 below · cited by 1 · depth 21
• Vanishing of Hⁱ for i>n in the twist complex ProjSpaceCech.Twist.subsingleton_cohomology_of_lt 0 below · cited by 1 · depth 21
• Vanishing of Hⁱ(Pⁿ_R,𝒪(d)) for 0<i<n ProjSpaceCech.Twist.subsingleton_cohomology_succ_of_le 0 below · cited by 1 · depth 21
• Finiteness of Hⁱ(Pⁿ_R,𝒪(d)) over any commutative ring ProjSpaceCech.Twist.finite_cohomology 0 below · cited by 1 · depth 22