Galois and Pro-étale Cohomology of Overconvergent de Rham Period Rings

Abstract

Motivated by the theory of p-adic differential equations and p-adic geometric representation theory, we introduce overconvergent variants of Fontaine's classical period rings. In particular, we study the positive overconvergent de Rham period ring, which is the stalk of the structure sheaf of the analytic Fargues-Fontaine curve at infinity. Our main results include the computation of the Galois cohomology of these overconvergent period rings, as well as the cohomology of the associated period sheaves and period structure sheaves.