## 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?