Weak proregularity, derived completion, adic flatness, and prisms

JOURNAL OF ALGEBRA

卷:583

Article

Yekutieli, Amnon¹ 

[1] Ben Gurion Univ Negev, Dept Math, IL-84105 Beer Sheva, Israel

关键词: Derived completion;    Weak proregularity;    Adic flatness;    Prisms;    Perfectoid rings;   

DOI  :  10.1016/j.jalgebra.2021.04.033

来源: Elsevier

【 摘 要 】

We begin by recalling the role that weak proregularityof an ideal in a commutative ring has in derived completionand adic flatness. We also introduce the new concepts of idealisticand sequentialderived completion, and prove a few results about them, including the fact that these two concepts agree iff the ideal is weakly proregular. Next we study the local nature of weak proregularity, and its behavior w.r.t. ring quotients. These results allow us to prove our main theorem, which states that weak proregularity occurs in the context of bounded prisms. Prisms belong to the new groundbreaking theory of perfectoid rings, developed by Scholze and his collaborators. Since perfectoid ring theory makes heavy use of derived completion and adic flatness, we anticipate that our results shall help simplify and improve some of the more technical aspects of this theory. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free