Diamonds II

This chapter evaluates the complements on the pro-étale topology. It addresses two issues raised in the previous lecture on the pro-étale topology. The first issue concerned descent, or more specifically pro-étale descent for perfectoid spaces. The other issue was that the property of being a pro-étale morphism is not local for the pro-étale topology on the target. The chapter then looks at quasi-pro-étale morphisms, as well as G-torsors. A morphism of perfectoid spaces is quasi-pro-étale if for any strictly totally disconnected perfectoid space with a map, the pullback is pro-étale. Using this definition, one can give an equivalent characterization of diamonds.