Stably uniform affinoids are sheafy

Let A be a stably uniform Tate ring: every rational localisation A⟨T/s⟩ is uniform. Let P be a pair of definition whose ring of definition lies in A⁺, and A⁺ a subring of power-bounded elements. Buzzard and Verberkmoes prove that the structure presheaf of X = Spa(A, A⁺) is a sheaf. This file proves this for the presentation-limit structure presheaf, as a sheaf of complete separated topological rings, and deduces that every stably uniform Tate ring is sheafy.

Stable uniformity passes to rational localisations, and over a stably uniform ring two-piece Laurent covers of rational subsets glue, with the topology on sections induced by restriction to the two pieces. So it satisfies TauCeti.ValuationSpectrum.ContinuousLaurentGluing (continuousLaurentGluing_isStablyUniform). Wedhorn's reduction of Lemma 8.34 from Laurent covers to arbitrary rational covers, and from rational covers to all opens, therefore applies, exactly as for strongly noetherian Tate rings. No noetherian hypothesis is used.

Main results

• TauCeti.ValuationSpectrum.isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq_of_isStablyUniform : over a stably uniform Tate ring, the presentation-limit presheaf of sets satisfies the sheaf condition for every cover of a rational subset by rational subsets.
• TauCeti.ValuationSpectrum.isSheaf_underlying_presentationLimitPresheaf_of_isStablyUniform : over a stably uniform Tate ring, the presheaf of sets underlying the presentation-limit structure presheaf is a sheaf on all opens.
• TauCeti.ValuationSpectrum.isSheaf_presentationLimitPresheaf_of_isStablyUniform : over a stably uniform Tate ring, the presentation-limit structure presheaf is a sheaf of complete separated topological rings.
• TauCeti.Huber.isSheafyForEveryPresentation_of_isStablyUniform : every ring of integral elements of a stably uniform Tate ring satisfies TauCeti.Huber.IsSheafyForEveryPresentation.
• TauCeti.Huber.isSheafyRing_of_isStablyUniform : a stably uniform Tate ring is sheafy.

References

• K. Buzzard, A. Verberkmoes, Stably uniform affinoids are sheafy, J. reine angew. Math. 740 (2018), 25–39, Theorem 7.
• T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Lemma 8.34 and Definition 8.26.

theorem TauCeti.ValuationSpectrum.isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq_of_isStablyUniform {A : Type v} [CommRing A] [UniformSpace A] [IsTopologicalRing A] [Huber.IsTateRing A] [Huber.IsStablyUniform A] (P : Huber.PairOfDefinition A) {Aplus : Subring A} (hP : P.ringOfDefinition ≤ Aplus) (hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a) {W : TopologicalSpace.Opens ↑(spa Aplus)} (hW : W ∈ spaRationalOpens Aplus) {ι : Type u_1} {U : ι → TopologicalSpace.Opens ↑(spa Aplus)} (hU : ∀ (i : ι), U i ∈ spaRationalOpens Aplus) (hcov : ⨆ (i : ι), U i = W) : CategoryTheory.Presieve.IsSheafFor ((presentationLimitPresheaf P Aplus).comp (TopCommRingCat.isCompleteSeparated.ι.comp (CategoryTheory.forget TopCommRingCat))) (CategoryTheory.Presieve.ofArrows U fun (i : ι) => CategoryTheory.homOfLE ⋯)

Gluing along rational covers of a stably uniform affinoid. Let A be a stably uniform Tate ring, P a pair of definition whose ring of definition lies in A⁺, A⁺ a subring of power-bounded elements and W a rational subset of Spa(A, A⁺). The presentation-limit presheaf, as a presheaf of sets, satisfies the sheaf condition for every family of rational subsets of W whose union is W. The ring need not be complete or Hausdorff, and the family may be infinite or empty.

theorem TauCeti.ValuationSpectrum.isSheaf_underlying_presentationLimitPresheaf_of_isStablyUniform {A : Type v} [CommRing A] [UniformSpace A] [IsTopologicalRing A] [Huber.IsTateRing A] [Huber.IsStablyUniform A] (P : Huber.PairOfDefinition A) {Aplus : Subring A} (hP : P.ringOfDefinition ≤ Aplus) (hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a) : CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↑(spa Aplus)) ((presentationLimitPresheaf P Aplus).comp (TopCommRingCat.isCompleteSeparated.ι.comp (CategoryTheory.forget TopCommRingCat)))

Buzzard–Verberkmoes for the presheaf of sets. Let A be a stably uniform Tate ring, P a pair of definition whose ring of definition lies in A⁺, and A⁺ a subring of power-bounded elements. The presheaf of sets underlying the presentation-limit structure presheaf of Spa(A, A⁺) is a sheaf on all opens. The ring need not be complete or Hausdorff.

theorem TauCeti.ValuationSpectrum.isSheaf_presentationLimitPresheaf_of_isStablyUniform {A : Type v} [CommRing A] [UniformSpace A] [IsTopologicalRing A] [Huber.IsTateRing A] [Huber.IsStablyUniform A] (P : Huber.PairOfDefinition A) {Aplus : Subring A} (hP : P.ringOfDefinition ≤ Aplus) (hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a) : CategoryTheory.Presheaf.IsSheaf (Opens.grothendieckTopology ↑(spa Aplus)) (presentationLimitPresheaf P Aplus)

Buzzard–Verberkmoes: stably uniform Tate pairs are sheafy. Let A be a stably uniform Tate ring, P a pair of definition whose ring of definition lies in A⁺, and A⁺ a subring of power-bounded elements. The presentation-limit structure presheaf of Spa(A, A⁺) is a sheaf of complete separated topological rings on all opens. The ring need not be complete or Hausdorff.

theorem TauCeti.Huber.isSheafyForEveryPresentation_of_isStablyUniform {A : Type v} [CommRing A] [UniformSpace A] [IsTopologicalRing A] [IsTateRing A] [IsStablyUniform A] {Aplus : Subring A} (hAplus : IsRingOfIntegralElements Aplus) : IsSheafyForEveryPresentation Aplus

Stably uniform Tate pairs are sheafy (Buzzard–Verberkmoes): every ring of integral elements A⁺ of a stably uniform Tate ring A satisfies TauCeti.Huber.IsSheafyForEveryPresentation, so the structure presheaf of Spa(A, A⁺) is a sheaf of complete separated topological rings. A need not be complete or Hausdorff.

theorem TauCeti.Huber.isSheafyRing_of_isStablyUniform {A : Type v} [CommRing A] [UniformSpace A] [IsUniformAddGroup A] [IsTopologicalRing A] [IsTateRing A] [IsStablyUniform A] : IsSheafyRing A

A stably uniform Tate ring is sheafy in the sense of Wedhorn's Definition 8.26 (TauCeti.Huber.IsSheafyRing). This is the ring-level form of the theorem of Buzzard and Verberkmoes. A need not be complete or Hausdorff: its completion Â is stably uniform (TauCeti.Huber.IsStablyUniform.completion), so every ring of integral elements of Â is sheafy by TauCeti.Huber.isSheafyForEveryPresentation_of_isStablyUniform.