The presentation limit on a rational open is A⟨T/s⟩ #
Wedhorn §8.1 defines 𝒪_X(V) for an open V ⊆ Spa(A,A⁺) as the limit of A⟨T/s⟩ over the
rational subsets R(T/s) ⊆ V, and states that on a rational open U = R(T/s) this limit is
A_U = A⟨T/s⟩ again. This file proves that statement for presentationLimit, the limit indexed by
admissible presentations: when A⁺ consists of power-bounded elements, the projection of
presentationLimit Aplus R(T/s) at the presentation (T, s) itself is an isomorphism, and under
these isomorphisms the restriction maps of presentationLimitPresheaf between rational opens are
the comparison maps of Wedhorn's Proposition 8.2(1).
The argument #
For a containment R(T'/s') ⊆ R(T/s) there is a unique continuous homomorphism
A⟨T/s⟩ → A⟨T'/s'⟩ compatible with the structure maps from A
(existsUnique_continuous_ringHom_of_rationalSubset_subset); homOfRationalSubsetSubset is it as
a morphism of CompleteSeparatedTopCommRingCat. Uniqueness makes these maps functorial, and
identifies every restriction map of a refinement with one of them.
If V ⊆ R(T/s) and (T, s) is an index of V, the comparison maps out of A⟨T/s⟩ form a cone
over the diagram of V, which gives an inverse to the projection at (T, s). That the projection
is also injective comes from the key identity presentationLimitπ_eq_π_comp: the projection at any
index j factors through the projection at any index i with R(j) ⊆ R(i). To prove it, pass to
the common refinement k of i and j, which presents R(i) ∩ R(j) = R(j). The restriction map
A_j → A_k is then a split monomorphism, since the comparison map back is a left inverse.
Main definitions #
TauCeti.Huber.PairOfDefinition.Presentation.toCompletionLocObjHom: the structure mapA → A⟨p⟩as a morphism ofCompleteSeparatedTopCommRingCat.TauCeti.ValuationSpectrum.homOfRationalSubsetSubset: the comparison morphismA⟨T/s⟩ ⟶ A⟨T'/s'⟩of a containmentR(T'/s') ⊆ R(T/s).TauCeti.ValuationSpectrum.presentationLimitRationalIso: the isomorphismpresentationLimit Aplus R(T/s) ≅ A⟨T/s⟩for an admissible presentation(T, s).
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.restrictionHom_eq_homOfRationalSubsetSubset: the restriction morphism of a refinement is the comparison morphism of the containment it induces.TauCeti.ValuationSpectrum.presentationLimitπ_eq_π_comp: projections of the limit factor through each other along comparison morphisms.TauCeti.ValuationSpectrum.isIso_presentationLimitπ: the projection at an index whose rational subset containsVis an isomorphism.TauCeti.ValuationSpectrum.presentationLimitRationalIso_inv_comp_map_comp_hom: between rational opens, the restriction maps ofpresentationLimitPresheafare the comparison morphisms.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), §8.1 and Proposition 8.2(1).
The structure maps as morphisms #
The structure map A → A⟨p⟩ of a presentation, as a morphism of
CompleteSeparatedTopCommRingCat out of the complete Hausdorff ring A.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The underlying morphism of Presentation.toCompletionLocObjHom is the structure map
toCompletionLoc, transported across CompleteSeparatedTopCommRingCat.of_obj and
completionLocObj_obj.
Comparison morphisms over A commute with the structure maps: a continuous ring
homomorphism A⟨p⟩ → A⟨q⟩ compatible with the structure maps from A, as a morphism, carries the
structure morphism of p to that of q.
Restriction commutes with the structure maps: the restriction morphism
A⟨p⟩ → A⟨q⟩ of a refinement p ≤ q carries the structure morphism of p to that of q.
Restriction commutes with the structure maps: the restriction morphism
A⟨p⟩ → A⟨q⟩ of a refinement p ≤ q carries the structure morphism of p to that of q.
The projections of the presentation limit #
Projections factor through comparison morphisms: if R(j) ⊆ R(i) for two indices of V,
the projection of the limit at j is the projection at i followed by the comparison morphism
A⟨i⟩ → A⟨j⟩.
The projection at an index whose rational subset contains V is an isomorphism: then
R(i) = V, and the limit over the presentations inside V is A⟨i⟩.
The presentation limit on a rational open is its coordinate ring: for an admissible
presentation p, the limit over the presentations inside R(p) is isomorphic to A⟨p⟩ by the
projection at p itself (presentationLimitRationalIso_hom). This is Wedhorn §8.1's
𝒪_X(U) = A_U, stated for presentationLimit.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The isomorphism presentationLimitRationalIso is the projection at the presentation itself.
The inverse of presentationLimitRationalIso, followed by the projection at an index j, is
the comparison morphism A⟨p⟩ → A⟨j⟩.
A comparison morphism followed by the transport along an equality of presentations is again a comparison morphism.
Between rational opens, restriction is the comparison morphism: for admissible
presentations p and q with R(q) ⊆ R(p), the restriction map of the presentation limit from
R(p) to R(q) becomes, under presentationLimitRationalIso, the comparison morphism of
Wedhorn's Proposition 8.2(1).