{"title":"A CALCULATION OF THE PERFECTOIDIZATION OF SEMIPERFECTOID RINGS","authors":"RYO ISHIZUKA","doi":"10.1017/nmj.2024.2","DOIUrl":null,"url":null,"abstract":"We show that perfectoidization can be (almost) calculated by using <jats:italic>p</jats:italic>-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 <jats:italic>p</jats:italic>-root closed property of integral perfectoid rings. Through this calculation, we establish a connection between a classical closure operation “<jats:italic>p</jats:italic>-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.","PeriodicalId":49785,"journal":{"name":"Nagoya Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nagoya Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/nmj.2024.2","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 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.
我们证明，在某些情况下，包括在半完形情况下，完形化（几乎）可以用 p 根封闭来计算。为此，我们重点研究了完形化和均匀完形的普遍性，以及积分完形环的 p 根封闭性质。通过这一计算，我们建立了罗伯茨在混合特征交换代数中使用的经典闭合运算 "p 根闭合 "与巴特和肖尔茨在棱柱同调理论中引入的最新概念 "完形化 "之间的联系。
期刊介绍：
The Nagoya Mathematical Journal is published quarterly. Since its formation in 1950 by a group led by Tadashi Nakayama, the journal has endeavoured to publish original research papers of the highest quality and of general interest, covering a broad range of pure mathematics. The journal is owned by Foundation Nagoya Mathematical Journal, which uses the proceeds from the journal to support mathematics worldwide.
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