The topology on sections of a Laurent cover

For a strongly noetherian Tate ring, restriction from a rational open to its two-piece Laurent cover is a closed embedding. Thus the topology on sections is the equalizer topology inherited from the product of the rings of sections on the two pieces. This supplies the topological part of Laurent gluing, in addition to the algebraic exactness of the restriction maps.

For the whole spectrum of a complete Hausdorff ring, the assertion follows from the closed embedding into the completed rational localizations and the topological-ring isomorphisms identifying these localizations with sections. Rational localization then gives the assertion on any rational open, without a completeness or separatedness assumption on the original ring.

References

• T. Wedhorn, Adic Spaces (arXiv:1910.05934v1), Remark 8.4, Lemma 8.33, Lemma 8.34(i), and Remark 8.20.

theorem TauCeti.ValuationSpectrum.isClosedEmbedding_presentationLimitMap_inf_laurentCoverOpen {A : Type v} [CommRing A] [UniformSpace A] [IsTopologicalRing A] [Huber.IsTateRing A] [Huber.IsStronglyNoetherian A] (P : Huber.PairOfDefinition A) {Aplus : Subring A} (hAplus : ∀ ⦃a : A⦄, a ∈ Aplus → Huber.IsPowerBounded a) {W : TopologicalSpace.Opens ↑(spa Aplus)} (hW : W ∈ spaRationalOpens Aplus) (f : A) : Topology.IsClosedEmbedding fun (x : (presentationLimit Aplus W).obj.α) => (↑(presentationLimitMap ⋯).hom x, ↑(presentationLimitMap ⋯).hom x)

Restriction from any rational open to its two-piece Laurent cover is a closed embedding. The original strongly noetherian Tate ring need not be complete or Hausdorff. Together with Laurent gluing, this identifies sections with the topological equalizer of the overlap maps.