Comparing the Čech acyclicity of two families #
Let C be a category with finite products and a terminal object T, and let P be a presheaf
on C with values in an abelian category with exact products. For two families U and V of
objects, suppose that the restriction of V to every finite product of members of U is acyclic
for P, and that the restriction of U to every finite product of members of V is acyclic.
Then U is acyclic for P exactly when V is
(quasiIso_cechAugmentation_iff_of_quasiIso_prod). Here a family U is acyclic for P when the
augmented Čech complex 0 ⟶ P(T) ⟶ Č⁰(U, P) ⟶ Č¹(U, P) ⟶ ⋯ is exact, and its restriction to an
object X is the family U with the presheaf P(X × -).
For open covers of a space this is the classical comparison of the Čech cohomology of two covers
each of which is acyclic on the finite intersections of members of the other. Its main use is for
a cover U with a refinement V: then the restriction of U to every finite intersection of
members of V has a member containing everything, so it is acyclic, and acyclicity passes from
V to U as soon as V is acyclic on the finite intersections of members of U
(quasiIso_cechAugmentation_of_quasiIso_prod_of_hom). This is how acyclicity passes from
standard covers, which refine every finite cover of an affinoid by rational subsets, to all such
covers in the proof of Tate's acyclicity theorem.
The proof resolves P by the presheaves Y ↦ Čᵏ(V, P(Y × -)). This resolution is exact at T
and at the finite products of members of U, by the hypotheses on V. Each term is a product of
restrictions of P to finite products of members of V, hence acyclic for U
(quasiIso_cechAugmentation_pi), so quasiIso_cechAugmentation_of_resolution applies. This
replaces the usual double complex argument by dimension shifting along the resolution.
Main results #
TauCeti.CategoryTheory.quasiIso_cechAugmentation_limit: a limit of acyclic presheaves is acyclic, when limits of that shape are exact.TauCeti.CategoryTheory.quasiIso_cechAugmentation_pi: a product of acyclic presheaves is acyclic, when products of that shape are exact.TauCeti.CategoryTheory.quasiIso_cechAugmentation_of_retract_pi: a presheaf which is a retract of the product of its restrictions to the members of a family is acyclic for that family; this is the Čech acyclicity of fine sheaves.TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_quasiIso_prod: two families each acyclic on the finite products of members of the other are acyclic together.TauCeti.CategoryTheory.quasiIso_cechAugmentation_of_quasiIso_prod_of_hom: in a thin category, a family is acyclic if some acyclic refinement of it is acyclic on the finite products of its members.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Appendix A, for acyclic covers, and the proof of Theorem 8.28, which passes from standard covers to arbitrary rational covers.
- S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis, Springer, 1984, §8.1.4, for the comparison of the Čech cohomology of a covering and a refinement, as used in rigid geometry.
Limits of acyclic presheaves #
Limits of acyclic presheaves are acyclic. If limits of shape J are exact in A, a limit
of a J-shaped diagram of presheaves whose augmented Čech complexes for U are exact has an exact
augmented Čech complex for U.
Products of acyclic presheaves are acyclic. If products indexed by J are exact in A,
a product of presheaves whose augmented Čech complexes for U are exact has an exact augmented
Čech complex for U.
Retracts of products of restrictions are acyclic. Let C be thin. If products indexed by
ι are exact in A and P is a retract of the product ∏ᵢ P(U i × -) of its restrictions to
the members of U, then the augmented Čech complex of P for U is exact. Each P(U i × -) is
acyclic for U, since U i is itself a member of U (quasiIso_cechAugmentation_prod_of_hom).
This is the Čech form of the acyclicity of fine sheaves. For an open cover U of a space and a
sheaf of modules over a sheaf of rings with a partition of unity ψ i subordinate to U, the
restrictions P ⟶ ∏ᵢ P(U i ∩ -) and the map ∏ᵢ P(U i ∩ -) ⟶ P sending (sᵢ)ᵢ to the locally
finite sum ∑ᵢ ψ i • sᵢ, each term extended by zero off the support of ψ i, exhibit such a
retract.
Comparing two families #
The proof that acyclicity passes from V to U resolves P by the Čech complexes of the
restricted presheaves P(Y × -) for V: in degree k this is the presheaf
Y ↦ ∏_b P(Y × V (b 0) × ⋯ × V (b k)), which is acyclic for U by the hypothesis on U, and the
resolution is exact at T and on the finite products of members of U by the hypotheses on V.
quasiIso_cechAugmentation_of_resolution then applies.
Comparison of the Čech acyclicity of two families. In a category with finite products and
a terminal object T, let P be a presheaf with values in an abelian category with exact
products, and let U and V be two families. Suppose that V is acyclic for P(X × -) for
every product X of finitely many members of U, and that U is acyclic for P(X × -) for
every product X of finitely many members of V; for open covers, the restriction of each cover
to every finite intersection of members of the other is acyclic. Then U is acyclic for P if
and only if V is.
Acyclicity from an acyclic refinement. 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, and let V be a family each of whose members maps to a member of the family U; for
open covers, V refines U. If V is acyclic for P, and the restriction of V to every
finite product of members of U is acyclic, then U is acyclic for P.