Arnab Kundu: Tilting correspondences of perfectoid rings

Abstract

We present an alternate proof of a vanishing result of étale cohomology of perfectoid rings due to Česnavičius. To establish that, we prove a tilting equivalence of étale cohomology of perfectoid rings taking values in commutative, finite étale group schemes. On the way, we algebraically establish an analogue of the tilting correspondences of Scholze, between the category of finite étale schemes over a perfectoid ring and that over its tilt, without using tools from almost ring theory or adic spaces.