The morphism of structure presheaves induced by a homomorphism of Huber pairs #
Let φ : A → B be a continuous ring homomorphism of topological rings with pairs of definition
P and P' and subrings A⁺ ⊆ A and B⁺ ⊆ B, carrying A⁺ into B⁺ and open ideals to open
ideals, and let f : Spa(B, B⁺) → Spa(A, A⁺) be the induced map of adic spectra. Following
Wedhorn §8.1, on a rational open R(T/s) of Spa(A, A⁺) the structure presheaf has the value
A⟨T/s⟩, the preimage f⁻¹(R(T/s)) = R(φ(T)/φ(s)) is a rational open of Spa(B, B⁺) with value
B⟨φ(T)/φ(s)⟩, and the base changes A⟨T/s⟩ → B⟨φ(T)/φ(s)⟩ of φ form a morphism of presheaves on
the rational opens. To extend it to a morphism 𝒪_{Spa A} → f_* 𝒪_{Spa B} on all opens, the base
changes on the rational opens R(T/s) ⊆ U have to be assembled into a map into 𝒪_{Spa B}(f⁻¹U).
The opens f⁻¹(R(T/s)) cover f⁻¹U, so this is possible when 𝒪_{Spa B} is a sheaf, which is the
case treated here. (The image under f of a rational open of f⁻¹U need not lie in a rational open
of U, so the limit description of 𝒪_{Spa B}(f⁻¹U) alone does not provide the map.)
This is the presheaf half of Wedhorn's construction of the morphism of pre-adic spaces
Spa(φ) : Spa(B, B⁺) → Spa(A, A⁺); the compatibility with the stalk valuations is not treated
here.
Main definitions #
TauCeti.ValuationSpectrum.presentationLimitComap: the component at an openUofSpa(A, A⁺), a morphism𝒪_{Spa A}(U) ⟶ 𝒪_{Spa B}(f⁻¹U).TauCeti.ValuationSpectrum.presentationLimitPresheafComap: the morphism of presheaves𝒪_{Spa A} ⟶ f_* 𝒪_{Spa B}.
Main results #
TauCeti.ValuationSpectrum.presentationLimitComap_comp_map_comp_π: on the rational openf⁻¹(R(T/s)) = R(φ(T)/φ(s))the morphism is the base changeA⟨T/s⟩ → B⟨φ(T)/φ(s)⟩.TauCeti.ValuationSpectrum.presentationLimit_hom_ext_of_isSheaf: morphisms into𝒪_{Spa B}(f⁻¹U)are determined by their restrictions to the opensf⁻¹(R(T/s))for the rational opensR(T/s) ⊆ U.TauCeti.ValuationSpectrum.presentationLimitMap_comp_presentationLimitComap: the components are natural inU.
References #
- T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Lemma 7.46 and §8.1.
The components on the rational opens #
The index of f⁻¹U induced by an index i of U: the presentation (φ(T), φ(s)) of the
rational open f⁻¹(R(T/s)) = R(φ(T)/φ(s)), for R(T/s) the rational open presented by i. This is
PresentationIndex.map for the preimage of U itself.
Equations
- TauCeti.ValuationSpectrum.PresentationIndex.comap φ hφ hopen hplus i = TauCeti.ValuationSpectrum.PresentationIndex.map φ hφ hopen hplus ⋯ i
Instances For
The glued morphism #
The component at U of the morphism 𝒪_{Spa A} → f_* 𝒪_{Spa B} induced by φ, for
𝒪_{Spa B} a sheaf: the morphism 𝒪_{Spa A}(U) → 𝒪_{Spa B}(f⁻¹U) whose restriction to the open
f⁻¹(R(i)) = R(φ(i)), for every index i of U, is the projection to A⟨i⟩ followed by the base
change A⟨i⟩ → B⟨φ(i)⟩ of φ (presentationLimitComap_comp_map_comp_π). It is obtained by
gluing these base changes along the cover of f⁻¹U by the opens f⁻¹(R(i)).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Morphisms into 𝒪_{Spa B}(f⁻¹U) are determined by their restrictions to the opens
f⁻¹(R(i)) = R(φ(i)), i ranging over the indices of U: these opens cover f⁻¹U and
𝒪_{Spa B} is a sheaf.
On a rational open the morphism is the base change: the component at U, restricted to
the open f⁻¹(R(i)) = R(φ(i)) for an index i of U and followed by the projection at an index
m of that open, is the projection at i, the base change A⟨i⟩ → B⟨φ(i)⟩ of φ and the
comparison morphism B⟨φ(i)⟩ → B⟨m⟩.
On a rational open the morphism is the base change, for an open V ⊆ f⁻¹U and an index
m of V whose rational open lies in R(φ(i)) for an index i of U: the component at U,
restricted to V and followed by the projection at m, is the projection at i, the base change
A⟨i⟩ → B⟨φ(i)⟩ of φ and the comparison morphism B⟨φ(i)⟩ → B⟨m⟩.
On a rational open the morphism is the base change, in terms of the identification
presentationLimitRationalIso of 𝒪_{Spa B}(R(φ(i))) with B⟨φ(i)⟩: the component at U,
restricted to f⁻¹(R(i)) = R(φ(i)), is the projection at i followed by the base change
A⟨i⟩ → B⟨φ(i)⟩ of φ.
Naturality #
The components are natural in U: restricting along U' ≤ U and then applying the
component at U' is applying the component at U and restricting along f⁻¹U' ≤ f⁻¹U.
The components are natural in U: restricting along U' ≤ U and then applying the
component at U' is applying the component at U and restricting along f⁻¹U' ≤ f⁻¹U.
The morphism of presheaves #
The morphism of structure presheaves 𝒪_{Spa A} ⟶ f_* 𝒪_{Spa B} induced by φ, for
𝒪_{Spa B} a sheaf: Wedhorn §8.1's f♭, with the components presentationLimitComap.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The component of presentationLimitPresheafComap at an open is presentationLimitComap,
transported along the evaluation equations of the two presheaves.