Let be a (complete) Huber ring, i.e. a complete topological ring containing an open subring which is adic with respect to a finitely generated ideal of definition . A Huber ring is *Tate* if contains a topologically nilpotent unit ; replacing by some power, we may assume , and in fact we have for some , so the topology on is the -adic topology. A Huber ring is *uniform* if the subring of powerbounded elements is bounded (when is Tate, this is equivalent to for some and ). An affinoid ring is a pair where is a Huber ring and is open and integrally closed in .

Given an affinoid ring, Huber defines a topological space of (certain) continuous valuations on , together with a structure presheaf of complete topological rings. One of the great bugbears in the theory of adic spaces is that need not be a sheaf. Whether or not is a sheaf depends only on , and one says accordingly that is *sheafy *or not. The sheafyness of is known in the following situations:

- (Huber) is finitely generated as an algebra over a Noetherian , or is Tate and is Noetherian for all .
- (Scholze) is
*perfectoid*, i.e. is a uniform Tate ring (over , say), and we may choose such that and Frobenius induces an isomorphism . - (Buzzard-Verberkmoes) is
*stably uniform*, i.e. is a uniform Tate ring and is uniform for all rational subsets .

As noted by Buzzard-Verberkmoes, perfectoid rings are stably uniform. Here’s a new observation.

**Proposition. ***If is perfectoid, then is stably uniform, and therefore sheafy.*

*Proof. *Let us recall an observation from Kedlaya-Liu (this is the second paragraph of Remark 2.8.12). Let be a bounded morphism of affinoid Tate rings such that splits in the category of -Banach modules. Then if is stably uniform, so is .

We apply this as follows. Take , and , , with the obvious inclusion map. Then is a splitting as -Banach modules, where is the obvious thing (series with zero coefficients on all monomials , ). Since is perfectoid, it is stably uniform. Therefore is stably uniform as well.

This is related to some natural questions about whether fiber products of perfectoid and rigid spaces exist as honest adic spaces. For example, suppose is a Hodge type Shimura variety, with universal abelian variety and perfectoid Shimura variety . Does exist as an honest adic space? More optimistically, suppose

is a diagram of adic spaces, where are normal rigid spaces, is a perfectoid space, and is nice (e.g., smooth). Does exist as an honest adic space?

In the version of the Buzzard-Verberkmoes paper that was available when you wrote this blog post, we had strictly speaking only announced the stably-uniform-implies-sheafy result for k-algebras, with k a field complete wrt a non-trivial non-arch valuation. Since then several people (you included) seem to have quoted the result for more general Tate rings, so we have bowed to pressure and are about to rewrite the paper so that the arguments apply in the generality that everyone’s quoting them in. By “about to rewrite” I actually mean that we’ve essentially rewritten it already, and my co-author is just looking over the current version, so we should be able to make it public in a week or so.

Kevin

Great! Thanks for the information, and sorry it took me so long to approve this comment.