Čech complexes of two-member covers

Let C be a thin category with finite products and a terminal object T, for instance the poset of open subsets of a topological space, and let P : Cᵒᵖ ⥤ A be a presheaf with values in an abelian category. For a family of two objects U 0 and U 1, the cohomology of the Čech complex of P vanishes in every degree ≥ 2, and the augmented Čech complex is exact precisely when the Mayer-Vietoris sequence 0 ⟶ P(T) ⟶ P(U 0) ⊞ P(U 1) ⟶ P(U 0 × U 1) ⟶ 0 is short exact. The Čech complex itself still has terms in every degree; only its exactness reduces to that of this sequence. This is the form in which exactness in every degree of the augmented Čech complex of a cover by two opens follows from statements in degrees 0 and 1: the sheaf condition for the cover, and the fact that every section over the intersection is a difference of restrictions.

More generally, let W be a family each of whose members maps to V 0 or to V 1, for instance an open cover refining a cover by two opens. If the restrictions of W to V 0, V 1 and V 0 × V 1 are acyclic, then W is acyclic exactly when the two-member family V is (quasiIso_cechAugmentation_iff_of_fin_two). This is the inductive step in the acyclicity of Laurent covers in rigid and adic geometry, where a Laurent cover generated by several elements refines the cover generated by one of them. The family W need not refine V if instead the restriction of V to every nonempty finite intersection of members of W is acyclic (quasiIso_cechAugmentation_iff_of_fin_two_of_quasiIso_prod); this is the step that passes acyclicity from Laurent covers to the covers they refine.

The proof restricts P to the members. For an object Y, the presheaf P(Y × -) has an exact augmented Čech complex for any family with a member receiving a map from Y (quasiIso_cechAugmentation_prod_of_hom): in a thin category the family U may be replaced by Y × U, which refines and is refined by the single object Y. Restriction along the projections then gives a short exact sequence of Čech complexes 0 ⟶ Č(W, P) ⟶ Č(W, P(V 0 × -)) ⊞ Č(W, P(V 1 × -)) ⟶ Č(W, P(V 0 × V 1 × -)) ⟶ 0, split in each degree because each product of members of W lies in V 0 or in V 1. When the restrictions of W are acyclic, its long exact cohomology sequence identifies the cohomology of Č(W, P) with that of the Mayer-Vietoris sequence of V. Without refinement, the same sequence is the sequence of Čech complexes of the short complex of presheaves P ⟶ P(V 0 × -) ⊞ P(V 1 × -) ⟶ P(V 0 × V 1 × -); it is short exact once this short complex is short exact at each product of members of W (shortExact_map_cechComplexFunctor), that is, once V restricted to each such product is acyclic.

Main results

• TauCeti.CategoryTheory.quasiIso_cechAugmentation_prod_of_hom: for Y ⟶ U i₀, the augmented Čech complex of P(Y × -) for the family U is exact.
• TauCeti.CategoryTheory.exactAt_add_two_cechComplexFunctor_obj: the Čech complex of any presheaf for a two-member family is exact in every degree ≥ 2.
• TauCeti.CategoryTheory.quasiIso_cechAugmentation_fin_two_iff: the augmented Čech complex of P for a two-member family is exact if and only if P satisfies the sheaf condition for the family and P(U 0) ⊞ P(U 1) ⟶ P(U 0 × U 1) is an epimorphism.
• TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two: a family refining a two-member family V, whose restrictions to V 0, V 1 and V 0 × V 1 are acyclic, is acyclic if and only if V is.
• TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two_of_quasiIso_prod: the same for a family W that need not refine V, provided that V restricted to each finite product of members of W is acyclic.

References

• T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Appendix A, Definition A.1, where a cover is called F-acyclic when its augmented Čech complex is exact, and Lemma 8.34(i), whose proof is an induction on the number of generators of a Laurent cover.

Restriction to an object

theorem TauCeti.CategoryTheory.quasiIso_cechAugmentation_prod_of_hom {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] [CategoryTheory.Limits.HasFiniteProducts C] [Quiver.IsThin C] [CategoryTheory.Preadditive A] [CategoryTheory.Limits.HasProducts A] [CategoryTheory.Limits.HasZeroObject A] [CategoryTheory.CategoryWithHomology A] {ι : Type w} (U : ι → C) {T : C} (hT : CategoryTheory.Limits.IsTerminal T) (P : CategoryTheory.Functor Cᵒᵖ A) (Y : C) {i₀ : ι} (g : Y ⟶ U i₀) : QuasiIso (cechAugmentation U hT ((CategoryTheory.Limits.prod.functor.obj Y).op.comp P))

The Čech complex of a restricted presheaf. If some member U i₀ of the family receives a morphism from Y, then the augmented Čech complex of the presheaf P(Y × -) for U is exact. For Y = T this is quasiIso_cechAugmentation_of_hom; for an open cover of T and an open Y contained in a member, it says that the cover restricted to Y is acyclic for P.

Families refining a two-member family

Throughout, V 0 and V 1 are two objects and W is a family each of whose members maps to V 0 or to V 1, through g k : W k ⟶ V (c k); the main case is W = V.

In degree n, the terms of Čech complexes for W are products indexed by a : Fin (n + 1) → κ; cechXMap builds morphisms between them factorwise.

The splitting in degree n acts factorwise. A factor of the degree n term is indexed by a : Fin (n + 1) → κ, and the product X of the members W (a j) maps to V (c (a 0)). If c (a 0) = 0, then P(X) ⟶ P(V 0 × X) and P(V 1 × X) ⟶ P(V 0 × V 1 × X) are isomorphisms; if c (a 0) = 1, then so are P(X) ⟶ P(V 1 × X) and P(V 0 × X) ⟶ P(V 0 × V 1 × X).

The cohomology of the middle and last terms of mvShortComplex is computed from acyclicity of W for the restricted presheaves P(Y × -), Y = V 0, V 1, V 0 × V 1. When W = V, these hold by quasiIso_cechAugmentation_prod_of_hom.

A family W need not refine V for mvShortComplex to be short exact. It is the sequence of Čech complexes for W of the short complex of presheaves mvPresheafSC, whose evaluation at an object X is the Mayer-Vietoris sequence of V for P(X × -); it is therefore short exact as soon as these Mayer-Vietoris sequences are, for every product X of members of W.

Two-member covers

theorem TauCeti.CategoryTheory.exactAt_add_two_cechComplexFunctor_obj {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] [CategoryTheory.Limits.HasFiniteProducts C] [Quiver.IsThin C] [CategoryTheory.Abelian A] (P : CategoryTheory.Functor Cᵒᵖ A) [CategoryTheory.Limits.HasProducts A] (U : Fin 2 → C) (n : ℕ) : HomologicalComplex.ExactAt ((CategoryTheory.cechComplexFunctor U).obj P) (n + 2)

The Čech complex of a two-member family is exact in degrees ≥ 2. In a thin category with finite products, the Čech complex of any presheaf with values in an abelian category, for a family of two objects, is exact in every degree ≥ 2.

theorem TauCeti.CategoryTheory.quasiIso_cechAugmentation_fin_two_iff {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] [CategoryTheory.Limits.HasFiniteProducts C] [Quiver.IsThin C] [CategoryTheory.Abelian A] {T : C} (hT : CategoryTheory.Limits.IsTerminal T) (P : CategoryTheory.Functor Cᵒᵖ A) [CategoryTheory.Limits.HasProducts A] (U : Fin 2 → C) : QuasiIso (cechAugmentation U hT P) ↔ (∀ (E : Aᵒᵖ), CategoryTheory.Presieve.IsSheafFor (P.comp (CategoryTheory.coyoneda.obj E)) (CategoryTheory.Presieve.ofArrows U fun (i : Fin 2) => hT.from (U i))) ∧ CategoryTheory.Epi (CategoryTheory.Limits.biprod.desc (P.map CategoryTheory.Limits.prod.fst.op) (-P.map CategoryTheory.Limits.prod.snd.op))

Exactness of the Čech complex of a two-member cover. In a thin category with finite products and a terminal object T, let P be a presheaf with values in an abelian category and let U 0, U 1 be two objects. The augmented Čech complex 0 ⟶ P(T) ⟶ Č⁰(U, P) ⟶ Č¹(U, P) ⟶ ⋯ is exact if and only if P satisfies the sheaf condition for the family U i ⟶ T and the difference of restrictions P(U 0) ⊞ P(U 1) ⟶ P(U 0 × U 1) is an epimorphism, that is, if and only if the Mayer-Vietoris sequence 0 ⟶ P(T) ⟶ P(U 0) ⊞ P(U 1) ⟶ P(U 0 × U 1) ⟶ 0 is short exact.

theorem TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] [CategoryTheory.Limits.HasFiniteProducts C] [Quiver.IsThin C] [CategoryTheory.Abelian A] {T : C} (hT : CategoryTheory.Limits.IsTerminal T) (P : CategoryTheory.Functor Cᵒᵖ A) [CategoryTheory.Limits.HasProducts A] [CategoryTheory.Limits.HasProducts A] (V : Fin 2 → C) {κ : Type w} {W : κ → C} (hW : ∀ (k : κ), ∃ (i : Fin 2), Nonempty (W k ⟶ V i)) (h₀ : QuasiIso (cechAugmentation W hT ((CategoryTheory.Limits.prod.functor.obj (V 0)).op.comp P))) (h₁ : QuasiIso (cechAugmentation W hT ((CategoryTheory.Limits.prod.functor.obj (V 1)).op.comp P))) (h₀₁ : QuasiIso (cechAugmentation W hT ((CategoryTheory.Limits.prod.functor.obj (V 0 ⨯ V 1)).op.comp P))) : QuasiIso (cechAugmentation W hT P) ↔ QuasiIso (cechAugmentation V hT P)

Acyclicity of a family refining a two-member family. In a thin category with finite products and a terminal object T, let P be a presheaf with values in an abelian category, let V 0, V 1 be two objects and let W be a family each of whose members maps to V 0 or to V 1. Suppose that the augmented Čech complexes of the restricted presheaves P(V 0 × -), P(V 1 × -) and P(V 0 × V 1 × -) for W are exact; for open covers, this says that the restrictions of W to V 0, V 1 and V 0 ∩ V 1 are acyclic. Then the augmented Čech complex of P for W is exact if and only if the one for the two-member family V is.

theorem TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two_of_quasiIso_prod {C : Type u} [CategoryTheory.Category.{v, u} C] {A : Type u'} [CategoryTheory.Category.{v', u'} A] [CategoryTheory.Limits.HasFiniteProducts C] [Quiver.IsThin C] [CategoryTheory.Abelian A] {T : C} (hT : CategoryTheory.Limits.IsTerminal T) (P : CategoryTheory.Functor Cᵒᵖ A) [CategoryTheory.Limits.HasProducts A] [CategoryTheory.Limits.HasProducts A] (V : Fin 2 → C) {κ : Type w} (W : κ → C) [∀ (n : ℕ), CategoryTheory.HasExactLimitsOfShape (CategoryTheory.Discrete (Fin (n + 1) → κ)) A] (hV : ∀ (n : ℕ) (a : Fin (n + 1) → κ), QuasiIso (cechAugmentation V hT ((CategoryTheory.Limits.prod.functor.obj (∏ᶜ fun (j : Fin (n + 1)) => W (a j))).op.comp P))) (h₀ : QuasiIso (cechAugmentation W hT ((CategoryTheory.Limits.prod.functor.obj (V 0)).op.comp P))) (h₁ : QuasiIso (cechAugmentation W hT ((CategoryTheory.Limits.prod.functor.obj (V 1)).op.comp P))) (h₀₁ : QuasiIso (cechAugmentation W hT ((CategoryTheory.Limits.prod.functor.obj (V 0 ⨯ V 1)).op.comp P))) : QuasiIso (cechAugmentation W hT P) ↔ QuasiIso (cechAugmentation V hT P)

Acyclicity relative to a two-member family acyclic on intersections. In a thin category with finite products and a terminal object T, let P be a presheaf with values in an abelian category with exact products, let V 0, V 1 be two objects and let W be a family. Suppose that the augmented Čech complexes of the restricted presheaves P(V 0 × -), P(V 1 × -) and P(V 0 × V 1 × -) for W are exact, and that the augmented Čech complex of P(X × -) for V is exact for every product X = W a₀ × ⋯ × W aₙ of one or more members of W; for open covers, this says that the restrictions of W to V 0, V 1 and V 0 ∩ V 1 are acyclic, and that the restriction of V to every nonempty finite intersection of members of W is acyclic. Then the augmented Čech complex of P for W is exact if and only if the one for V is.

Unlike quasiIso_cechAugmentation_iff_of_fin_two, the family W need not refine V.