Continuous gluing for rational covers of rational subsets

For a strongly noetherian Tate ring, compatible continuous ring homomorphisms into the sections on a rational cover of a rational open glue continuously. This is the topological part of Wedhorn's Lemma 8.34 in degree zero.

The proof first treats a cover of the whole adic spectrum of a complete Hausdorff Tate ring. A standard rational refinement reduces this to continuous gluing for a standard cover generated by the unit ideal. For a general rational open, Wedhorn's Remark 8.4 transports the cover to the adic spectrum of its complete coordinate ring. Its topological-ring isomorphisms on sections commute with restriction, so they transport continuous gluing back to the original cover.

References

• T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.4, Lemma 7.54, Lemma 8.34, and Remark 8.20.
• R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), Lemma 2.6 and Theorem 2.5.

theorem TauCeti.ValuationSpectrum.isSheafFor_ofArrows_spaRationalOpens_of_iSup_eq_topCommRingCat {A : Type v} [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) {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) (E : TopCommRingCat) : CategoryTheory.Presieve.IsSheafFor ((presentationLimitPresheaf P Aplus).comp (TopCommRingCat.isCompleteSeparated.ι.comp (CategoryTheory.coyoneda.obj (Opposite.op E)))) (CategoryTheory.Presieve.ofArrows U fun (i : ι) => CategoryTheory.homOfLE ⋯)

Wedhorn's Lemma 8.34 in degree zero, with topology: rational covers of rational subsets. Let W be a rational subset of the adic spectrum of a strongly noetherian Tate pair, and let (U i) be a family of rational subsets whose union is W. Compatible continuous ring homomorphisms from a topological commutative ring E into the rings of sections on the U i glue uniquely to a continuous ring homomorphism into the sections on W.

The original ring need not be complete or Hausdorff, and the family may be infinite or empty.