Morphisms of presentation-limit pre-adic spaces #
A continuous map of Huber rings carrying A⁺ into B⁺ induces a morphism
Spa(B, B⁺) ⟶ Spa(A, A⁺). This file packages the existing morphism of presentation-limit
structure presheaves as a morphism of pre-adic spaces. The main point is compatibility with the
canonical stalk valuations, proved on rational germs using base change for completed rational
localizations.
Like presentationLimitPresheafComap, the construction assumes that the structure presheaf of
Spa(B, B⁺) is a sheaf, and nothing more: φ need not be adic. Every point of Spa(B, B⁺) over a
rational open R(T/s) of Spa(A, A⁺) has a rational neighbourhood R(q) inside the preimage of
R(T/s), and the stalk valuations are compared through the geometric base change
A⟨T/s⟩ → B⟨q⟩.
The comparison of stalk valuations only uses that the map of sections is the geometric base change
A⟨T/s⟩ → B⟨q⟩ on rational opens. For mutually inverse continuous ring homomorphisms the
transport presentationLimitPresheafIsoPushforward of the structure presheaf has this property
without any sheaf condition, so an isomorphism of Huber pairs induces an isomorphism of pre-adic
spaces. This applies, for instance, to the Frobenius automorphism of Spa(A_inf, A_inf), where
A_inf is not a Tate ring, so that Tate acyclicity does not make its structure presheaf a sheaf.
Main definitions #
TauCeti.ValuationSpectrum.presentationLimitPresheafedSpaceComap: the underlying morphism of presheafed spaces.TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComap: the induced morphism of pre-adic spaces.TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComapIso: the isomorphism of pre-adic spaces induced by mutually inverse continuous ring homomorphisms.
Main results #
TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComap_id,TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComap_comp: the construction is contravariantly functorial.presentationLimitRationalGerm_comp_stalkMap_presentationLimitPreAdicSpaceComap: on the germs of a rational open, the stalk maps are the base change of completed rational localisations.TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComapIso_hom_toHom: the isomorphism is the map of adic spectra on points and the transport of the structure presheaf on sections.TauCeti.ValuationSpectrum.presentationLimitPreAdicSpaceComapIso_inv_base: on points, its inverse is the map of adic spectra induced byψ.
References #
- T. Wedhorn, Adic Spaces, arXiv:1910.05934v1, §8.1.
Pulling a rational-localization point back along a morphism g : A⟨p⟩ ⟶ B⟨q⟩ of completed
rational localizations over φ gives the rational-localization point attached to the pulled-back
valuation. This applies to the base changes Presentation.mapHom and
Presentation.mapHomOfSubset.
The morphism of presentation-limit presheafed spaces induced contravariantly by a continuous ring homomorphism preserving the chosen rings of integral elements.
Instances For
The continuous map underlying presentationLimitPresheafedSpaceComap is the induced map
spaComapTopHom of adic spectra.
The map of structure presheaves underlying presentationLimitPresheafedSpaceComap is
presentationLimitPresheafComap.
Compatibility with the stalk valuations #
The stalk valuations are compared on rational germs, and the comparison only uses one property of
the map of sections: restricted from a rational open R(p) of Spa(A, A⁺) to a rational open
R(q) of Spa(B, B⁺) inside its preimage, it is the geometric base change A⟨p⟩ → B⟨q⟩. The
argument is therefore carried out for an arbitrary morphism of presheaves
c : 𝒪_{Spa A} ⟶ f_* 𝒪_{Spa B} with this property (IsBaseChangeOnRational). It applies to
presentationLimitPresheafComap, and to the transport presentationLimitPresheafIsoPushforward
along an isomorphism, which needs no sheaf condition.
Every point x of Spa(B, B⁺) over a rational open R(p) has a rational neighbourhood R(q)
inside the preimage of R(p), so no condition on φ beyond continuity is needed.
The underlying morphism of presheafed spaces preserves the canonical stalk valuations.
The morphism of presentation-limit pre-adic spaces induced contravariantly by a continuous ring homomorphism preserving the chosen rings of integral elements.
Instances For
The morphism of presheafed spaces underlying presentationLimitPreAdicSpaceComap is
presentationLimitPresheafedSpaceComap.
On points, the morphism Spa(φ) of adic spectra induced by φ is pullback of valuations
along φ.
The stalk maps on rational germs. Let R(p) be a rational open of Spa(A, A⁺) and R(q) a
rational open of Spa(B, B⁺) inside its preimage. At a point x of R(q), the stalk map of the
morphism induced by φ carries the germ of an element of A⟨p⟩ to the germ of its image under the
geometric base change A⟨p⟩ → B⟨q⟩.
Functoriality #
presentationLimitPreAdicSpaceComap depends on (φ, hφ, hplus) only through φ.
The identity of A induces the identity morphism of Spa(A, A⁺).
The morphisms are contravariantly functorial: for continuous homomorphisms
φ : A → B and ψ : B → C of Huber rings, the morphism Spa(C, C⁺) ⟶ Spa(A, A⁺) induced by
ψ ∘ φ is the morphism induced by ψ followed by the morphism induced by φ.
Isomorphisms #
Mutually inverse continuous ring homomorphisms φ : A → B and ψ : B → A carrying the plus rings
into each other induce an isomorphism Spa(B, B⁺) ≅ Spa(A, A⁺) of pre-adic spaces. Its map of
sections is the transport presentationLimitPresheafIsoPushforward, so unlike
presentationLimitPreAdicSpaceComap it needs no sheaf condition on either side.
An isomorphism of Huber pairs induces an isomorphism of pre-adic spaces. Let φ : A → B
and ψ : B → A be mutually inverse continuous ring homomorphisms carrying A⁺ into B⁺ and B⁺
into A⁺, where the plus rings consist of power-bounded elements and contain the rings of
definition of the pairs of definition P and P'. Then Spa(B, B⁺) and Spa(A, A⁺) are
isomorphic in 𝒱^pre: on points the isomorphism is the map v ↦ v ∘ φ of adic spectra
(presentationLimitPreAdicSpaceComapIso_hom_base), on sections it is the transport
presentationLimitPresheafIsoPushforward (presentationLimitPreAdicSpaceComapIso_hom_toHom), and
it matches the stalk valuations. No sheaf condition is assumed. The two pairs of definition are
arbitrary; for φ = ψ = id this compares the pre-adic spaces of two pairs of definition of one
Huber ring.
Instances For
On points, presentationLimitPreAdicSpaceComapIso is the map of adic spectra induced by
φ.
On points, the inverse of presentationLimitPreAdicSpaceComapIso is the map of adic spectra
induced by ψ.
The underlying morphism of presheafed spaces of presentationLimitPreAdicSpaceComapIso is the
map of adic spectra induced by φ on points, with the transport
presentationLimitPresheafIsoPushforward of the structure presheaf along φ and ψ on
sections.