Étale Cohomology of Diamonds

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 Zp 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.