On Perfectoidizaiton of Finite Algebras over a Perfectoid Ring

Abstract

We study general properties of the perfectoidization of finite algebras over a perfectoid ring, which helps to understand some precise and explicit descriptions. For example, we prove that if $A=R[t]/(m(t))$ where $m(t)$ is monic, $R$ is perfectoid and the discriminant $d$ of $m(t)$ is a non-zero divisor of $R$ satisfying a bounded torsion condition, then $dA_{\mathrm{pfd}}\subset A$. We also prove a density criterion reducing the construction of the perfectoidization to adjoining suitable $p$-power roots modulo $p$. In the second part of the paper, we compute perfectoidizations in several families of examples, including Kummer-type extensions and split finite algebras.