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arXiv · 2305.07916

A calculation of the perfectoidization of semiperfectoid rings

Ryo Ishizuka
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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.

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Ryo Ishizuka. 2023-05-13. A calculation of the perfectoidization of semiperfectoid rings. https://doi.org/10.1017/nmj.2024.2

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