A calculation of the perfectoidization of semiperfectoid rings

Abstract

We show that perfectoidization can be (almost) calculated by using $p$-root closure in certain cases, including the semiperfectoid case. To do this, we focus on the universality of perfectoidization and uniform completion, as well as the $p$-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation ``$p$-root closure'' used by Roberts in mixed characteristic commutative algebra and a more recent concept of ``perfectoidization'' introduced by Bhatt and Scholze in their theory of prismatic cohomology.