Cartan's criterion: Čech acyclicity on a basis implies cohomological acyclicity

Let X be a topological space and B a set of opens of X which is closed under binary intersections, consists of quasi-compact opens, and is a basis of the topology inside each of its members. Let F be a sheaf of abelian groups on X which is Čech acyclic for the finite covers of members of B by members of B: for every finite nonempty family U i in B and every Y ∈ B contained in their union, the augmented Čech complex of V ↦ F(Y ∩ V) for U, that is, of the cover of Y by the opens Y ∩ U i, is exact (TauCeti.Topology.IsCechAcyclicOn). Then F has no cohomology in positive degrees on the members of B (TauCeti.Topology.subsingleton_H'_succ_of_isCechAcyclicOn).

This is how the vanishing of the higher cohomology of a quasi-coherent sheaf on an affine scheme is deduced from the exactness of the Čech complexes of standard covers, with B the basic opens.

The proof is by dimension shifting along the chosen embedding 0 ⟶ F ⟶ I ⟶ Q ⟶ 0 of F into an injective sheaf, which is flasque, so that its Čech complexes are exact for all finite covers (TopCat.Sheaf.quasiIso_cechAugmentation_prod_of_isFlasque).

• On every Y ∈ B, the map I(Y) ⟶ Q(Y) is onto. A section s of Q over Y lifts locally, hence, Y being quasi-compact, on a finite cover of Y by members of B. The image presheaf G of I ⟶ Q sits in a sequence 0 ⟶ F ⟶ I ⟶ G ⟶ 0 of presheaves which is exact on every open, so the Čech complexes of the cover form a short exact sequence, and G is Čech acyclic for the cover because F and I are (quasiIso_cechAugmentation_X₃_of_shortExact). In degree 0 this is the sheaf condition for G, so the local lifts of s glue to a section of G over Y, which comes from I(Y) and maps to s.
• Hence H¹(Y, F) = 0 for Y ∈ B, and the sequence 0 ⟶ F ⟶ I ⟶ Q ⟶ 0 is exact on the members of B, which are closed under finite intersections. Its Čech complexes for covers by members of B are then short exact, so Q is Čech acyclic on B as well.
• Finally Hⁿ⁺²(Y, F) ≅ Hⁿ⁺¹(Y, Q), and induction on the degree, for all sheaves at once, concludes.

Main definitions

• TauCeti.Topology.IsCechAcyclicOn B F: F is Čech acyclic for the finite covers of members of B by members of B.

Main results

• TauCeti.Topology.subsingleton_H'_succ_of_isCechAcyclicOn: Cartan's criterion, Hⁿ⁺¹(Y, F) = 0 for every member Y of B.
• TauCeti.Topology.prod_pi_mem_of_inf_mem: a set of opens closed under binary intersections contains the finite intersections Y ∩ U (a 0) ∩ ⋯ ∩ U (a n) of its members.

References

• The Stacks Project, Lemma 20.11.9 (Cartan's criterion), whose proof is followed here, and Lemma 30.2.2 (tag 01XB), where it is applied to the standard opens of an affine scheme.

def TauCeti.Topology.IsCechAcyclicOn {X : TopCat} (B : Set (TopologicalSpace.Opens ↑X)) (F : CategoryTheory.Sheaf (Opens.grothendieckTopology ↑X) AddCommGrpCat) : Prop

A sheaf F of abelian groups on X is Čech acyclic on a set B of opens if for every finite nonempty family U i of members of B and every member Y of B contained in their union, the augmented Čech complex 0 ⟶ F(Y) ⟶ Č⁰(U, F(Y ∩ -)) ⟶ Č¹(U, F(Y ∩ -)) ⟶ ⋯ of the cover of Y by the opens Y ∩ U i is exact.

theorem TauCeti.Topology.isCechAcyclicOn_iff {X : TopCat} {B : Set (TopologicalSpace.Opens ↑X)} {F : CategoryTheory.Sheaf (Opens.grothendieckTopology ↑X) AddCommGrpCat} : IsCechAcyclicOn B F ↔ ∀ ⦃ι : Type u⦄ [Finite ι] [Nonempty ι] (U : ι → TopologicalSpace.Opens ↑X), ∀ Y ∈ B, (∀ (i : ι), U i ∈ B) → Y ≤ ⨆ (i : ι), U i → QuasiIso (CategoryTheory.cechAugmentation U CategoryTheory.Limits.isTerminalTop ((CategoryTheory.Limits.prod.functor.obj Y).op.comp F.obj))

Unfolding IsCechAcyclicOn: Čech acyclicity on B is the exactness of the augmented Čech complexes of the finite covers of members of B by members of B.

theorem TauCeti.Topology.prod_pi_mem_of_inf_mem {X : TopCat} {B : Set (TopologicalSpace.Opens ↑X)} (hinf : ∀ U ∈ B, ∀ V ∈ B, U ⊓ V ∈ B) {ι : Type u} {U : ι → TopologicalSpace.Opens ↑X} (hU : ∀ (i : ι), U i ∈ B) {Y : TopologicalSpace.Opens ↑X} (hY : Y ∈ B) (n : ℕ) (a : Fin (n + 1) → ι) : (Y ⨯ ∏ᶜ fun (j : Fin (n + 1)) => U (a j)) ∈ B

If a set B of opens is closed under binary intersections, then the product of a member Y with finitely many members U (a 0), …, U (a n), which in the poset of opens is their intersection, is again a member.

theorem TauCeti.Topology.subsingleton_H'_succ_of_isCechAcyclicOn {X : TopCat} {B : Set (TopologicalSpace.Opens ↑X)} {F : CategoryTheory.Sheaf (Opens.grothendieckTopology ↑X) AddCommGrpCat} (hinf : ∀ U ∈ B, ∀ V ∈ B, U ⊓ V ∈ B) (hcpt : ∀ U ∈ B, IsCompact ↑U) (hbasis : ∀ U ∈ B, ∀ V ≤ U, ∀ x ∈ V, ∃ W ∈ B, x ∈ W ∧ W ≤ V) (hF : IsCechAcyclicOn B F) {Y : TopologicalSpace.Opens ↑X} (hY : Y ∈ B) (n : ℕ) : Subsingleton ↑(F.H' (n + 1) Y)

Cartan's criterion. Let B be a set of quasi-compact opens of X which is closed under binary intersections and contains, inside each of its members, a basis of the topology of that member. A sheaf of abelian groups which is Čech acyclic for the finite covers of members of B by members of B has no cohomology in positive degrees on the members of B.