Diamonds

Abstract

This chapter investigates the notion of a diamond. The idea is that there should be a functor which “forgets the structure morphism to Zp.” The desired quotient in the example provided in the chapter exists in a category of sheaves on the site of perfectoid spaces with pro-étale covers. The chapter then defines pro-étale morphisms between perfectoid spaces. A morphism of perfectoid spaces is pro-étale if it is locally (on the source and target) affinoid pro-étale. The intuitive definition of diamonds involved the tilting functor in case of perfectoid spaces of characteristic 0. For this reason, diamonds are defined as certain pro-étale sheaves on the category of perfectoid spaces of characteristic p.