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?