Global sections of the presentation limit are A #
Let A be a complete Hausdorff Huber ring and A⁺ a subring of power-bounded elements. This file
identifies the value of presentationLimitPresheaf on the whole adic spectrum X = Spa(A,A⁺)
with A itself, as an isomorphism of complete separated topological rings, and says which map
realises it: the canonical map from A to the sections over an open. This is Wedhorn's
𝒪_X(X) = A for a complete affinoid ring, stated for presentationLimit.
The argument #
For every presentation p the structure map A → A⟨p⟩ is a morphism of
CompleteSeparatedTopCommRingCat (Presentation.toCompletionLocObjHom), and the restriction
morphisms of refinements are compatible with these structure maps. They therefore form a cone
over the diagram of any open V, whose lift is toPresentationLimit : A ⟶ presentationLimit V.
At V = ⊤ take the trivial presentation ({1}, 1). Its rational subset R({1}/1) is the whole
spectrum (rationalSubset_singleton_one), so the projection of the limit at it is an isomorphism
(isIso_presentationLimitπ); and for complete Hausdorff A its structure map A → A⟨1/1⟩ is an
isomorphism (toCompletionLocHomeomorphDenomOne, a consequence of the universal property of
A⟨T/s⟩). Since toPresentationLimit followed by that projection is that structure map, it is
itself an isomorphism.
Main definitions #
TauCeti.Huber.PairOfDefinition.Presentation.toCompletionLocObjHom: the structure mapA → A⟨p⟩as a morphism ofCompleteSeparatedTopCommRingCat.TauCeti.ValuationSpectrum.toPresentationLimit: the canonical mapA ⟶ presentationLimit V.TauCeti.ValuationSpectrum.presentationLimitTopIso: the isomorphismpresentationLimit Aplus ⊤ ≅ A.
Main results #
TauCeti.Huber.PairOfDefinition.Presentation.toCompletionLocObjHom_comp_completionLocObjHomandTauCeti.Huber.PairOfDefinition.Presentation.toCompletionLocObjHom_comp_restrictionHom: comparison and restriction morphisms commute with the structure maps.TauCeti.ValuationSpectrum.toPresentationLimit_comp_πToPresentationandTauCeti.ValuationSpectrum.toPresentationLimit_comp_presentationLimitMap: the canonical map projects to the structure maps and commutes with restriction.TauCeti.ValuationSpectrum.isIso_toPresentationLimit_top: on the whole spectrum the canonical map is an isomorphism.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), §8.1, where
𝒪_X(X) = Afor a complete affinoid ring is the caseU = X = R({1}/1)of𝒪_X(U) = A_U.
The canonical map from A to the sections over an open #
The canonical map A ⟶ presentationLimit V: the lift of the cone formed by the structure
maps A → A⟨T/s⟩ of the presentations refining V: Wedhorn's map A → 𝒪_X(V), stated for
presentationLimit.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The canonical map projects to the structure maps: its component at a presentation
refining V is the structure map A → A⟨T/s⟩.
The canonical map projects to the structure maps: its component at a presentation
refining V is the structure map A → A⟨T/s⟩.
The canonical map commutes with restriction: restricting the image of A in the sections
over V to W ≤ V gives its image in the sections over W.
The canonical map commutes with restriction: restricting the image of A in the sections
over V to W ≤ V gives its image in the sections over W.
Global sections #
On the whole spectrum the canonical map is an isomorphism: for a complete Hausdorff Huber
ring and a subring A⁺ of power-bounded elements, A ⟶ presentationLimit Aplus ⊤ is an
isomorphism of complete separated topological rings.
The global sections are A: for a complete Hausdorff Huber ring and a subring A⁺ of
power-bounded elements, the presentation limit over the whole adic spectrum is isomorphic to A as
a complete separated topological ring. This is Wedhorn §8.1's 𝒪_X(X) = A, stated for
presentationLimit. Its inverse is the canonical map toPresentationLimit
(presentationLimitTopIso_inv).
Equations
Instances For
The inverse of presentationLimitTopIso is the canonical map from A.