Tilting Correspondences of Perfectoid Rings

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.