Tilting Correspondences of Perfectoid Rings

TL;DR

This paper provides an alternative proof of a vanishing result for étale cohomology on perfectoid rings, establishing a tilting equivalence for finite étale group schemes without using almost ring theory or adic spaces.

Contribution

It introduces a new algebraic approach to tilting equivalences of étale cohomology for perfectoid rings, avoiding advanced tools from almost ring theory.

Findings

Proves a tilting equivalence of étale cohomology for perfectoid rings.

Provides an algebraic analogue of Scholze's tilting correspondence.

Offers an alternative proof of a vanishing result for étale cohomology.

Abstract

In this article, we present an alternate proof of a vanishing result of \'etale cohomology on perfectoid rings due to \v{C}esnavi\v{c}ius and more recently proved by a different approach by Bhatt and Scholze. To establish that, we prove a tilting equivalence of \'etale cohomology of perfectoid rings taking values in commutative, finite \'etale group schemes. On the way, we algebraically establish an analogue of the tilting correspondences of Scholze, between the category of finite \'etale schemes over a perfectoid ring and that over its tilt, without using tools from almost ring theory or adic spaces.