Čech complexes of flasque sheaves, and degree zero Čech cohomology #
Let F be a flasque sheaf of abelian groups on a topological space X. This file proves that
for every finite open cover U of X the augmented Čech complex
0 ⟶ F(X) ⟶ Č⁰(U, F) ⟶ Č¹(U, F) ⟶ ⋯ is exact: flasque sheaves have no higher Čech cohomology
for finite covers. Together with the acyclicity of flasque sheaves for derived-functor cohomology
(TauCeti.Topology.subsingleton_H'_succ_of_isFlasque), this is the input to the comparison of
Čech and derived-functor cohomology for covers whose finite intersections are acyclic.
The proof is by induction on the number of members of the cover, applied more generally to the
restricted presheaves V ↦ F(Y ∩ V) for opens Y contained in the union of the members. For a
cover by two opens V 0, V 1, exactness reduces to the Mayer–Vietoris sequence
0 ⟶ F(Y) ⟶ F(Y ∩ V 0) ⊕ F(Y ∩ V 1) ⟶ F(Y ∩ V 0 ∩ V 1) ⟶ 0, which is exact on the left by the
sheaf condition and on the right because F is flasque. A cover by the opens U a and U i,
i ∈ s, refines the two-member cover by U a and ⋃_{i ∈ s} U i, so it is acyclic once its
restrictions to these two opens and to their intersection are
(TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two). The restrictions to U a
and to U a ∩ ⋃_{i ∈ s} U i have a member containing everything; the restriction to
⋃_{i ∈ s} U i is compared with the smaller cover by the U i, i ∈ s, which is acyclic by
induction (TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_quasiIso_prod).
For an arbitrary sheaf, with values in any category with the required products and homology, and
any open cover of X, the sheaf condition alone identifies the degree 0 Čech cohomology with the
global sections; for a sheaf of abelian groups, hence with the degree 0 sheaf cohomology
H⁰(X, F).
Main results #
TopCat.Sheaf.isSheafFor_ofArrows_prod: for any sheafF, a family of opensV iand an openYcontained in their union, the presheafV ↦ F(Y ∩ V)satisfies the sheaf condition for the familyV i.TopCat.Sheaf.quasiIso_cechAugmentation_prod_of_isFlasque: for a flasque sheafF, a finite family of opensU iand an openYcontained in their union, the augmented Čech complex ofV ↦ F(Y ∩ V)forUis exact.TopCat.Sheaf.quasiIso_cechAugmentation_of_isFlasque: a flasque sheaf has an exact augmented Čech complex for every finite open cover.TauCeti.Topology.quasiIsoAt_cechAugmentation_zero_of_iSup_eq_top: for any sheaf and any open cover ofX, the augmentation of the Čech complex is a quasi-isomorphism in degree0.TauCeti.Topology.homologyZeroCechIso: the degree0Čech cohomology of a sheaf for an open cover ofXis isomorphic to its degree0sheaf cohomology, naturally in the sheaf (TauCeti.Topology.homologyMap_comp_homologyZeroCechIso_hom).
References #
- R. Hartshorne, Algebraic Geometry, Proposition III.4.3.
- The Stacks Project, Lemma 20.11.4 (tag 01EW).
Gluing over an open contained in a union. For a sheaf F, a family of opens V i and an
open Y contained in their union, the underlying presheaf of sets of V ↦ F(Y ∩ V) satisfies the
sheaf condition for the family V i ≤ ⊤: sections over the opens Y ∩ V i that agree on the
pairwise intersections glue uniquely to a section over Y.
Flasque sheaves are acyclic for finite covers, restricted to an open. Let F be a flasque
sheaf of abelian groups on X, let U i be a finite family of opens, and let Y be an open
contained in their union. Then the augmented Čech complex of the presheaf V ↦ F(Y ∩ V) for the
family U is exact; that is, the cover of Y by the opens Y ∩ U i is acyclic for F.
Flasque sheaves are acyclic for finite covers. For a flasque sheaf F of abelian groups
on X and a finite family of opens U i covering X, the augmented Čech complex
0 ⟶ F(X) ⟶ Č⁰(U, F) ⟶ Č¹(U, F) ⟶ ⋯ is exact: F has no Čech cohomology for U in positive
degrees, and its degree 0 Čech cohomology is F(X).
A sheaf on X satisfies the sheaf condition for a family of opens covering X, in the form
used by the augmentation of the Čech complex.
For a sheaf on X and a family of opens covering X, the augmentation of the Čech complex
is a quasi-isomorphism in degree 0: the degree 0 Čech cohomology is the object of global
sections.
The degree 0 Čech cohomology of a sheaf for a cover of X is the group of global sections,
which is also the degree 0 sheaf cohomology.
Instances For
The identification of degree 0 Čech cohomology with degree 0 sheaf cohomology is natural
in the sheaf.
The identification of degree 0 Čech cohomology with degree 0 sheaf cohomology is natural
in the sheaf.
Refinement maps commute with the degree-zero global-sections identification.
Refinement maps commute with the degree-zero global-sections identification.