On étale maps of sous-perfectoid adic spaces
Dmytro Rudenko
TL;DR
The work extends the classical étale criterion to affine sous-perfectoid adic spaces: a map $g:\mathrm{Spa}(B,{B}^{+})\to\mathrm{Spa}(A,{A}^{+})$ is étale precisely when $B$ admits a presentation $A\langle X_{1},\dots,X_{n} \rangle/(f_{1},\dots,f_{n})$ with $\det(\frac{\partial f_{i}}{\partial X_{j}})\in B^{\times}$. The proof builds a global closed immersion factoring $g$ and upgrades to a global complete intersection using Kedlaya–Liu pseudocoherent theory, enabling a Jacobian-criterion analogue in the sous-perfectoid setting. A noetherian approximation result for perfectoid rings, echoing Scholze’s S17, follows as a corollary, and the thesis provides an explicit local description of the sheaf of differentials for smooth maps in terms of completed differentials $\widehat{\Omega^{1}_{B/A}}$. These results deepen étale theory in the perfectoid/diamond framework and connect with foundational aspects of étale cohomology in this context.
Abstract
We show that a map $\mathrm{Spa}\,B \to \mathrm{Spa}\,A$ of sous-perfectoid affinoid adic spaces is étale if and only if there exists a presentation $B \cong A\langle X_{1},\dots, X_{n} \rangle/(f_{1},\dots,f_{n})$ such that the determinant of the associated Jacobian matrix $\mathrm{det}( \frac{\partial f_{i}}{\partial X_{j}})_{1\leqslant i, j\leqslant n}$ is a unit in $B$. This allows us to provide some technical details to an important claim from the theory of étale maps of perfectoid spaces. Namely, we show how our proposition implies a sort of noetherian approximation for perfectoid rings from "Étale cohomology of diamonds" by Peter Scholze [S17]. Apart from that, we give an explicit local description of the sheaf of differentials associated to a smooth map of sous-perfectoid adic spaces, as defined by Fargues-Scholze in [FS], in terms of the module of differentials.