On étale maps of sous-perfectoid adic spaces

We show that a map $\mathrm{S}\mathrm{p}\mathrm{a}B\to \mathrm{S}\mathrm{p}\mathrm{a}A$ of sous-perfectoid affinoid adic spaces is étale if and only if there exists a presentation $B\cong A⟨X_1,\dots ,X_n⟩\mathrm{/}(f_1,\dots ,f_n)$ such that the determinant of the associated Jacobian matrix $\mathrm{d}\mathrm{e}\mathrm{t}(\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{X}_j}{)}_{1⩽i,j⩽n}$ 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.