A remark on sheafy Huber rings

Let {A} be a (complete) Huber ring, i.e. a complete topological ring containing an open subring {A_{0}} which is adic with respect to a finitely generated ideal of definition {I}. A Huber ring is Tate if {A} contains a topologically nilpotent unit {\varpi}; replacing {\varpi} by some power, we may assume {\varpi\in I}, and in fact we have {\varpi^{m}A_{0}\subset I^{m}A_{0}\subset\varpi A_{0}} for some {m}, so the topology on {A_{0}} is the {\varpi}-adic topology. A Huber ring is uniform if the subring {A^{\circ}} of powerbounded elements is bounded (when {A} is Tate, this is equivalent to {A^{\circ}\subset\varpi^{-n}A_{0}} for some {\varpi} and {A_{0}}). An affinoid ring is a pair {(A,A^{+})} where {A} is a Huber ring and {A^{+}\subset A^{\circ}} is open and integrally closed in {A}.

Given an affinoid ring, Huber defines a topological space {X=\mathrm{Spa}(A,A^{+})} of (certain) continuous valuations on {A}, together with a structure presheaf {\mathcal{O}_{X}} of complete topological rings. One of the great bugbears in the theory of adic spaces is that {\mathcal{O}_{X}} need not be a sheaf. Whether or not {\mathcal{O}_{X}} is a sheaf depends only on {A}, and one says accordingly that {A} is sheafy or not. The sheafyness of {A} is known in the following situations:

  • (Huber) {A} is finitely generated as an algebra over a Noetherian {A_{0}}, or {A} is Tate and {A\left\langle X_{1},\dots,X_{n}\right\rangle } is Noetherian for all {n\geq0}.
  • (Scholze) {A} is perfectoid, i.e. {A} is a uniform Tate ring (over {\mathbf{Z}_{p}}, say), and we may choose {\varpi} such that {p\in\varpi^{p}A^{\circ}} and Frobenius induces an isomorphism {A^{\circ}/\varpi\overset{\sim}{\rightarrow}A^{\circ}/\varpi^{p}}.
  • (Buzzard-Verberkmoes) {A} is stably uniform, i.e. {A} is a uniform Tate ring and {\mathcal{O}_{X}(U)} is uniform for all rational subsets {U\subset X}.

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

Proposition. If {R} is perfectoid, then {R\left\langle X_{1},\dots,X_{n}\right\rangle } 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 {f:(A,A^{+})\rightarrow(B,B^{+})} be a bounded morphism of affinoid Tate rings such that {A\rightarrow B} splits in the category of {A}-Banach modules. Then if {B} is stably uniform, so is {A}.

We apply this as follows. Take {A=R\left\langle X_{1},\dots,X_{n}\right\rangle }, {A^{+}=R^{\circ}\left\langle X_{1},\dots,X_{n}\right\rangle } and {B=R\left\langle X_{1}^{\frac{1}{p^{\infty}}},\dots,X_{n}^{\frac{1}{p^{\infty}}}\right\rangle }, {B^{+}=R^{\circ}\left\langle X_{1}^{\frac{1}{p^{\infty}}},\dots,X_{n}^{\frac{1}{p^{\infty}}}\right\rangle }, with {f} the obvious inclusion map. Then {B=A\oplus M} is a splitting as {A}-Banach modules, where {M} is the obvious thing (series with zero coefficients on all monomials {\prod X_{i}^{a_{i}}}, {a_{i}\in\mathbf{N}}). Since {B} is perfectoid, it is stably uniform. Therefore {A} is stably uniform as well. {\square}

This is related to some natural questions about whether fiber products of perfectoid and rigid spaces exist as honest adic spaces. For example, suppose {\mathcal{X}_{K_{p}}=\mathrm{Sh}_{K_{p}K^{p}}(G,X)^{\mathrm{ad}}} is a Hodge type Shimura variety, with universal abelian variety {\mathcal{A}_{K_{p}}\rightarrow\mathcal{X}_{K_{p}}} and perfectoid Shimura variety {\mathcal{X}_{\infty}\sim\lim_{\substack{\leftarrow\\ K_{p}} }\mathcal{X}_{K_{p}}}. Does {\mathcal{A}_{\infty}=\mathcal{X}_{\infty}\times_{\mathcal{X}_{K_{p}}}\mathcal{A}_{K_{p}}} exist as an honest adic space? More optimistically, suppose

\displaystyle \begin{array}{rcl} & & Z\\ & & \downarrow\\ Y & \rightarrow & X\end{array}

is a diagram of adic spaces, where {X,Y} are normal rigid spaces, {Z} is a perfectoid space, and {Y\rightarrow X} is nice (e.g., smooth). Does {Y\times_{X}Z} exist as an honest adic space?

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2 Responses to A remark on sheafy Huber rings

  1. Kevin Buzzard says:

    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

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