Etale cohomology of diamonds

TL;DR

This paper develops a six functor formalism for the étale cohomology of diamonds and v-stacks, connecting it to existing frameworks and addressing problems in p-adic geometry and the local Langlands program.

Contribution

It introduces a new formalism for étale cohomology of diamonds and v-stacks, extending existing theories and linking analytic adic spaces with diamonds.

Findings

Formalism recovers Huber's framework in noetherian cases

Connects étale sites of adic spaces and diamonds

Addresses cohomological problems in p-adic geometry

Abstract

Motivated by problems on the \'etale cohomology of Rapoport--Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a six functor formalism for the \'etale cohomology of diamonds, and more generally small v-stacks on the category of perfectoid spaces of characteristic $p$. Using a natural functor from analytic adic spaces over $\mathbb Z_p$ to diamonds which identifies \'etale sites, this induces a similar formalism in that setting, which in the noetherian setting recovers the formalism from Huber's book.