The Laurent cover restricted to a rational subset #
Let W = R(T/s) be a rational subset of X = Spa(A, A⁺) and f ∈ A. The two pieces
W ∩ {|f| ≤ 1} and W ∩ {|f| ≥ 1} cover W. When A is a strongly noetherian Tate ring and
A⁺ consists of power-bounded elements, the augmented two-piece Čech sequence of the
presentation-limit presheaf for this cover of W is exact: a section over W is determined by its
restrictions to the two pieces, sections over the pieces that agree on their overlap come from a
section over W, and every section over the overlap is a difference of restrictions from the
pieces. For W = X and complete Hausdorff A this is Wedhorn's Lemma 8.33, proved in
TauCeti.AlgebraicGeometry.AdicSpace.Spa.StructurePresheaf.LaurentCover.Basic.
The general case is the first step of the proof of Wedhorn's Lemma 8.34(i). Let B = A⟨T/s⟩ with
plus ring A_U⁺, and let j : Spa(B, A_U⁺) → X be induced by the structure map ρ : A → B
(pullback along j is locOpensComap). Then j⁻¹(W) is all of Spa(B, A_U⁺), and j⁻¹ carries
the Laurent cover of f to the Laurent cover of ρ(f) (locOpensComap_laurentCoverOpen).
Wedhorn's Remark 8.4 (presentationLimitLocIso) identifies the presentation limits over rational
opens V ⊆ W with those over j⁻¹(V), compatibly with restriction. Since B is again a complete
Hausdorff strongly noetherian Tate ring, Lemma 8.33 for B transports to the restricted cover. In
particular A itself need not be complete.
Main results #
TauCeti.ValuationSpectrum.locOpensComap_laurentCoverOpen: the pullback of a Laurent piece offalongjis the corresponding Laurent piece ofρ(f).TauCeti.ValuationSpectrum.injective_presentationLimitMap_inf_laurentCoverOpen: restriction fromR(T/s)to the two pieces is injective.TauCeti.ValuationSpectrum.exists_presentationLimitMap_eq_of_inf_laurentCoverOpen: sections over the two pieces that agree on their overlap come from a section overR(T/s).TauCeti.ValuationSpectrum.surjective_presentationLimitMap_sub_inf_laurentCoverOpen: the difference of restrictions from the two pieces onto their overlap is surjective.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.4, Lemma 8.33 and Lemma 8.34(i).
Pulling back the Laurent cover #
The Laurent cover pulled back to a rational subset. Pulling the Laurent piece
laurentCoverOpen Aplus f b back along Spa(A⟨T/s⟩, A_U⁺) → Spa(A, A⁺) gives the corresponding
Laurent piece of the image of f in A⟨T/s⟩.
The pullback of the Laurent piece of f restricted to R(T/s) is the Laurent piece of the
image of f, since the pullback of R(T/s) is the whole localized spectrum.
Transport along Wedhorn's Remark 8.4 #
Injectivity transported along rational localization. Restriction from R(T/s) to rational
opens U i ⊆ R(T/s) is injective as soon as restriction from the pullback of R(T/s) to the
pullbacks of the U i is injective.
Gluing transported along rational localization. Sections over two rational opens
U b ⊆ R(T/s) that agree on their overlap glue over R(T/s) as soon as the analogous gluing
statement holds for the pullbacks of R(T/s), the U b, and their overlap.
The Laurent cover transported along Wedhorn's Remark 8.4 #
Laurent injectivity transported along rational localization. If a section over the adic
spectrum of A⟨T/s⟩ is determined by its restrictions to the Laurent cover of the image of f,
then a section over R(T/s) is determined by its restrictions to the two pieces
R(T/s) ⊓ laurentCoverOpen Aplus f b.
Laurent gluing transported along rational localization. If compatible sections over the
Laurent cover of the image of f in A⟨T/s⟩ glue over its adic spectrum, then sections over the
two pieces R(T/s) ⊓ laurentCoverOpen Aplus f b that agree on their overlap come from a section
over R(T/s).
Lemma 8.33 on a rational subset #
Wedhorn's Lemma 8.33 on a rational subset, injectivity. Let A be a strongly noetherian
Tate ring, A⁺ a subring of power-bounded elements, R(T/s) a rational subset of Spa(A, A⁺) and
f ∈ A. A section of presentationLimit over R(T/s) is determined by its restrictions to the
two pieces R(T/s) ⊓ laurentCoverOpen Aplus f b of the Laurent cover of f restricted to
R(T/s). For the Laurent cover of the whole adic spectrum of a complete Hausdorff A, see
injective_presentationLimitMap_laurentCoverOpen.
Wedhorn's Lemma 8.33 on a rational subset, gluing. Let A be a strongly noetherian Tate
ring, A⁺ a subring of power-bounded elements, R(T/s) a rational subset of Spa(A, A⁺) and
f ∈ A. Sections x b of presentationLimit over the two pieces
R(T/s) ⊓ laurentCoverOpen Aplus f b that agree on their overlap are the restrictions of one
section over R(T/s), which is unique by injective_presentationLimitMap_inf_laurentCoverOpen.
For the Laurent cover of the whole adic spectrum of a complete Hausdorff A, see
exists_presentationLimitMap_eq_of_laurentCoverOpen.
Wedhorn's Lemma 8.33 on a rational subset, degree-one surjectivity. Let A be a strongly
noetherian Tate ring, A⁺ a subring of power-bounded elements, R(T/s) a rational subset of
Spa(A, A⁺) and f ∈ A. Every section of presentationLimit over the overlap of the two pieces
R(T/s) ⊓ laurentCoverOpen Aplus f b is the difference of the restrictions of sections over the
pieces. Together with injective_presentationLimitMap_inf_laurentCoverOpen and
exists_presentationLimitMap_eq_of_inf_laurentCoverOpen, this is exactness of the augmented Čech
complex of the Laurent cover of f restricted to R(T/s). For the Laurent cover of the whole adic
spectrum of a complete Hausdorff A, see surjective_presentationLimitMap_sub_laurentCoverOpen.