A Calculation of the Perfectoidization of Semiperfectoid Rings

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 perfectoidizations and uniform completions, 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.