Purity for Perfectoidness

Abstract

There has been a long-standing question about whether being perfectoid for an algebra is local in the analytic topology. We provide affirmative answers for the algebras (e.g., over $\overline{\mathbb{Z}_p}$) whose spectra are inverse limits of semi-stable affine schemes. In fact, we established a valuative criterion for such an algebra being perfectoid, saying that it suffices to check the perfectoidness of the stalks of the associated Riemann-Zariski space. Combining with Gabber-Ramero's computation of differentials of valuation rings, we obtain a differential criterion for perfectoidness. We also establish a purity result for perfectoidness when the limit preserves generic points of the special fibres. As an application to limits of smooth $p$-adic varieties (on the generic point), assuming either the poly-stable modification conjecture or working only with curves, we prove that stalk-wise perfectoidness implies vanishing of the higher completed étale cohomology groups of the smooth varieties, which is inspired by Scholze's vanishing for Shimura varieties. Moreover, we give an explicit description of the completed étale cohomology group in the top degree in terms of the colimit of Zariski cohomology groups of the structural sheaves.