Affinoid adic spaces from strongly noetherian Tate pairs

The presentation-limit pre-adic space of a strongly noetherian Tate pair is an affinoid adic space. This packages the sheafiness theorem for strongly noetherian Tate pairs with the general criterion that a sheafy affinoid pre-adic space is adic.

The main source of strongly noetherian Tate rings is finite type over one: a ring B topologically of finite type over a strongly noetherian Tate ring is again a strongly noetherian Tate ring, and complete and Hausdorff. Over a complete nonarchimedean field k, the pair (B, B°) is then an affinoid k-algebra topologically of finite type in the sense of Wedhorn's Definition 7.56, and its adic spectrum is an affinoid adic space. These are the classical affinoids of rigid geometry: the quotients k⟨X₁, …, Xₙ⟩ ⧸ 𝔞 of Tate algebras, with the closed unit polydiscs as the case 𝔞 = 0. Conversely, every classical affinoid k-algebra with a complete ultrametric algebra norm is topologically of finite type over k (TauCeti.Huber.isTopologicallyFiniteType_algebraMap_iff_isAffinoidAlgebra), so its adic spectrum is an affinoid adic space as well.

Main results

• TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isStronglyNoetherian: the adic spectrum of a strongly noetherian Tate pair, with its presentation-limit structure sheaf, is an adic space.
• TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isTopologicallyFiniteType: the adic spectrum of a ring topologically of finite type over a strongly noetherian Tate ring, with any plus ring of power-bounded elements containing a ring of definition, is an adic space.
• TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isAffinoidAlgebra: the adic spectrum of a classical affinoid algebra over a complete nonarchimedean field, with its Banach topology, is an adic space.

References

• T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Definition 7.56, Remark 6.37, Definition 8.22 and Theorem 8.28(b).

theorem TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isStronglyNoetherian {A : Type u} [CommRing A] [UniformSpace A] [IsTopologicalRing A] [Huber.IsTateRing A] [Huber.IsStronglyNoetherian A] (P : Huber.PairOfDefinition A) {Aplus : Subring A} (hP : P.ringOfDefinition ≤ Aplus) (hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a) : (presentationLimitPreAdicSpace P Aplus hAplus hP).isAdic

The adic spectrum of a strongly noetherian Tate pair is an affinoid adic space. The structure presheaf is a sheaf by the strongly noetherian form of Tate acyclicity, and the presentation-limit pre-adic space is affinoid. The ring need not be complete or Hausdorff.

theorem TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isTopologicallyFiniteType {R : Type u_1} [CommRing R] [TopologicalSpace R] [IsTopologicalRing R] [Huber.IsTateRing R] [Huber.IsStronglyNoetherian R] {A : Type u} [CommRing A] [UniformSpace A] [IsUniformAddGroup A] [IsTopologicalRing A] (P : Huber.PairOfDefinition A) {Aplus : Subring A} {φ : R →+* A} (hφ : Huber.IsTopologicallyFiniteType φ) (hP : P.ringOfDefinition ≤ Aplus) (hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a) : (presentationLimitPreAdicSpace P Aplus hAplus hP).isAdic

The adic spectrum of a ring topologically of finite type over a strongly noetherian Tate ring is an affinoid adic space. Let φ : R → A be topologically of finite type, with R a strongly noetherian Tate ring and A a uniform additive group. Then A is a complete Hausdorff strongly noetherian Tate ring, so for any pair of definition P of A and any plus ring A⁺ of power-bounded elements containing its ring of definition, Spa(A, A⁺) is an affinoid adic space.

Over a complete nonarchimedean field R = k with A⁺ = A° this says that the adic spectrum of an affinoid k-algebra topologically of finite type (Wedhorn, Definition 7.56) is an affinoid adic space.

theorem TauCeti.ValuationSpectrum.isAdic_presentationLimitPreAdicSpace_of_isAffinoidAlgebra {K : Type u_1} [NontriviallyNormedField K] [IsUltrametricDist K] [CompleteSpace K] {B : Type u} [NormedCommRing B] [NormedAlgebra K B] [IsUltrametricDist B] [CompleteSpace B] (P : Huber.PairOfDefinition B) {Bplus : Subring B} (hB : IsAffinoidAlgebra K B) (hP : P.ringOfDefinition ≤ Bplus) (hBplus : ∀ ⦃b : B⦄, b ∈ Bplus → Huber.IsPowerBounded b) : (presentationLimitPreAdicSpace P Bplus hBplus hP).isAdic

The adic spectrum of a classical affinoid algebra is an affinoid adic space. Let K be a complete nonarchimedean field with a nontrivial absolute value and B a quotient of a Tate algebra K⟨X₁, …, Xₙ⟩, carrying a complete ultrametric algebra norm, for instance the quotient Gauss norm. Then for any pair of definition P of B and any plus ring B⁺ of power-bounded elements containing its ring of definition, such as B°, Spa(B, B⁺) is an affinoid adic space.