The Čech complexes of Laurent covers of a rational subset #
Let A be a strongly noetherian Tate ring, A⁺ a subring of power-bounded elements and W a
rational subset of X = Spa(A, A⁺). This file proves that the augmented Čech complex of the
structure presheaf of X, as a presheaf of abelian groups, is exact in every degree for every
Laurent cover of W: Laurent covers are acyclic. This is Wedhorn's Lemma 8.34(i).
The covers of W are taken in the meet-semilattice Set.Iic W of opens contained in W, whose
greatest element is W, and the structure presheaf is restricted to it
(presentationLimitAddCommGrpPresheaf). The Laurent cover of W generated by f : Fin n → A is
the family laurentFamily Aplus W f indexed by signs σ : Fin n → Bool, whose member at σ is
W ∩ {|f i| ≤ 1 for σ i = true, |f i| ≥ 1 for σ i = false}.
For one element f, this is the two-piece cover W ∩ {|f| ≤ 1}, W ∩ {|f| ≥ 1}. For a
two-member cover, exactness of the augmented Čech complex in every degree follows from statements
in degrees 0 and 1
(TauCeti.CategoryTheory.quasiIso_cechAugmentation_fin_two_iff_isSheafFor_and_surjective), and
those are Lemma 8.33 on a rational subset: injective_presentationLimitMap_inf_laurentCoverOpen,
exists_presentationLimitMap_eq_of_inf_laurentCoverOpen and
surjective_presentationLimitMap_sub_inf_laurentCoverOpen.
The general case is Wedhorn's induction on the number of generators. The Laurent cover generated
by f refines the two-piece cover of f 0, and its restrictions to the two pieces and to their
overlap are, up to refinement both ways, the Laurent covers of these rational subsets generated
by the remaining elements. By TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two,
it is therefore acyclic once those restrictions are. The restriction of a cover of W to an open
Y ≤ W is acyclic exactly when its intersection with Y, as a cover of Y, is
(quasiIso_cechAugmentation_prod_iff).
Acyclicity also descends along Laurent covers to covers they need not refine. Let U be a family
of rational subsets and R ⊆ A a finite set, and suppose that the cover Y ∩ U of Y is acyclic
for every rational subset Y ⊆ W on which every element of R has a fixed sign. Then the cover
W ∩ U of W is acyclic (quasiIso_cechAugmentation_inf_of_laurentSieve). The induction on R
compares W ∩ U with the two-piece cover of W generated by one element of R, which is acyclic
after restriction to every finite intersection of members of W ∩ U, these being rational
(TauCeti.CategoryTheory.quasiIso_cechAugmentation_iff_of_fin_two_of_quasiIso_prod).
Main definitions #
TauCeti.ValuationSpectrum.laurentFamily: the Laurent cover of an openWgenerated byf : Fin n → A, as a family of opens contained inW.
Main results #
TauCeti.ValuationSpectrum.quasiIso_cechAugmentation_inf_laurentCoverOpen: over a strongly noetherian Tate ring, the augmented Čech complex of the structure presheaf for the two-piece Laurent cover of a rational subset is exact.TauCeti.ValuationSpectrum.quasiIso_cechAugmentation_laurentFamily: over a strongly noetherian Tate ring, the augmented Čech complex of the structure presheaf for every Laurent cover of a rational subset is exact.TauCeti.ValuationSpectrum.quasiIso_cechAugmentation_inf_of_laurentSieve: over a strongly noetherian Tate ring, a cover of a rational subset by rational subsets is acyclic if its restrictions to the rational subsets on which a finite set of elements has fixed signs are.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Definition A.1, Lemma 8.33 and Lemma 8.34(i).
- S. Bosch, U. Güntzer, R. Remmert, Non-Archimedean Analysis, §8.2.2, Lemma 2, the rigid-analytic original of the induction.
Lemma 8.33 in all degrees on a rational subset #
Wedhorn's Lemma 8.33 on a rational subset, in every degree. Let A be a strongly noetherian
Tate ring, A⁺ a subring of power-bounded elements, W a rational subset of Spa(A, A⁺) and
f ∈ A. The augmented Čech complex of the structure presheaf, as a presheaf of abelian groups on
the opens contained in W, for the Laurent cover W ⊓ laurentCoverOpen Aplus f true,
W ⊓ laurentCoverOpen Aplus f false of W (that is, W ∩ {|f| ≤ 1} and W ∩ {|f| ≥ 1}) is
exact: the cover is acyclic for the structure presheaf. A itself need not be complete.
Laurent covers generated by several elements #
The Laurent cover of an open W generated by f : Fin n → A, as a family of opens
contained in W indexed by signs σ : Fin n → Bool: its member at σ is the intersection of W
with the pieces laurentCoverOpen Aplus (f i) (σ i), that is, with {|f i| ≤ 1} when σ i = true
and with {|f i| ≥ 1} when σ i = false. This is Wedhorn's Laurent covering
(X(f₁^{ε₁}, …, fₙ^{εₙ}))_ε of Lemma 8.34, intersected with W.
Instances For
The member of the Laurent cover at σ is W ∩ ⋂ᵢ laurentCoverOpen Aplus (f i) (σ i).
Wedhorn's Lemma 8.34(i): Laurent covers are acyclic. Let A be a strongly noetherian
Tate ring, A⁺ a subring of power-bounded elements, W a rational subset of Spa(A, A⁺) and
f : Fin n → A. The augmented Čech complex of the structure presheaf, as a presheaf of abelian
groups on the opens contained in W, for the Laurent cover of W generated by f is exact in
every degree. A itself need not be complete.
Descent of acyclicity along Laurent covers. Let A be a strongly noetherian Tate ring,
A⁺ a subring of power-bounded elements, W a rational subset of Spa(A, A⁺), U a family of
rational subsets and R ⊆ A a finite set. Suppose that the cover Y ∩ U of Y is acyclic for the
structure presheaf for every rational subset Y ⊆ W lying in the Laurent sieve generated by R,
that is, on which every element of R has a fixed sign. Then the cover W ∩ U of W is acyclic:
its augmented Čech complex, as a presheaf of abelian groups on the opens contained in W, is exact
in every degree. A itself need not be complete.
The family U need not be refined by the Laurent cover generated by R. This is how Wedhorn's
Lemma 8.34 passes from Laurent covers to the covers they refine.