The presentation limit on a rational subset of a rational localisation #
Let U = R(T/s) be a rational subset of Spa(A, A⁺), let B = A⟨T/s⟩ with structure map
ρ : A → B and plus ring A_U⁺, and let j : Spa(B, A_U⁺) → Spa(A, A⁺) be induced by ρ.
Wedhorn's Remark 8.4 identifies 𝒪_X(V) with 𝒪_U(j⁻¹(V)) for every rational V ⊆ U, compatibly
with restriction. This file proves that statement for presentationLimit, the limit indexed by
admissible presentations, when A⁺ consists of power-bounded elements.
Main definitions #
TauCeti.ValuationSpectrum.presentationLimitLocIso: the isomorphismpresentationLimit Aplus V ≅ presentationLimit A_U⁺ j⁻¹(V)for a rationalV ⊆ R(T/s), wherej⁻¹(V)isTauCeti.ValuationSpectrum.locOpensComap.
Main results #
TauCeti.ValuationSpectrum.presentationLimitMap_comp_presentationLimitLocIso_hom: these isomorphisms commute with the restriction maps.TauCeti.ValuationSpectrum.presentationLimitLocIso_hom_presentationLimitMap_applyandTauCeti.ValuationSpectrum.bijective_presentationLimitLocIso_hom: the same, and bijectivity, read on sections.TauCeti.ValuationSpectrum.comap_presentationLimitLocIso_rationalLocalizationPoint: read on rational coordinate rings, these isomorphisms match the points determined byyand byj(y). This is what makes the stalk valuations compatible with Remark 8.4.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.4 and Proposition 8.2.
A presentation of V refining (T, s) #
The presentation over A⟨T/s⟩ #
The ring isomorphism as an isomorphism of objects #
The isomorphism for a chosen presentation #
Wedhorn's Remark 8.4 #
Wedhorn's Remark 8.4, for the presentation limit. Let B = A⟨T/s⟩ with plus ring A_U⁺,
where T spans an open ideal and A⁺ consists of power-bounded elements. For a rational open
V ⊆ R(T/s) of Spa(A, A⁺), the limit over the presentations inside V is isomorphic to the
limit, over B, of the presentations inside the pullback locOpensComap P Aplus T s S hden V.
It commutes with the restriction maps: presentationLimitMap_comp_presentationLimitLocIso_hom.
Equations
- TauCeti.ValuationSpectrum.presentationLimitLocIso P Aplus T s S hden hAplus hT V hV hVW = TauCeti.ValuationSpectrum.presentationLimitLocIsoAux✝ P Aplus T s S hden hAplus ⋯
Instances For
Transport of presentationLimitLocIso along an equality of rational opens: the
isomorphisms at two equal opens V = V' agree up to the transports of the two presentation limits
along that equality.
Wedhorn's Remark 8.4 is natural in V. For rational opens V' ⊆ V ⊆ R(T/s), the
isomorphisms presentationLimitLocIso at V and at V' carry the restriction map of V' ⊆ V
over A to the restriction map of locOpensComap … V' ⊆ locOpensComap … V over A⟨T/s⟩.
Wedhorn's Remark 8.4 is natural in V, on sections. This is
presentationLimitMap_comp_presentationLimitLocIso_hom evaluated at a section x over V.
Wedhorn's Remark 8.4 is a bijection on sections: presentationLimitLocIso at a rational
open V ⊆ R(T/s), applied to sections.
The points of the rational coordinate rings #
Wedhorn's Remark 8.4 matches the points of the rational coordinate rings. Let
V ⊆ R(T/s) be a rational open of Spa(A, A⁺) presented by p, let q present its pullback
j⁻¹(V) to Spa(A⟨T/s⟩, A_U⁺), and let y ∈ j⁻¹(V). The ring map
A⟨p⟩ ≅ 𝒪_X(V) ≅ 𝒪_U(j⁻¹(V)) ≅ A⟨T/s⟩⟨q⟩ induced by presentationLimitLocIso pulls the point of
A⟨T/s⟩⟨q⟩ determined by y back to the point of A⟨p⟩ determined by j(y).