Lecture 18. v-sheaves associated with perfect and formal schemes

This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also certain non-analytic objects as v-sheaves. The chapter first analyzes the behavior on topological spaces. Let X be any pre-adic space over Zp. This is not a diamond, but the chapter shows that it is a v-sheaf. It assesses some properties of this construction. The chapter then looks at applications to local models and integral models of Rapoport-Zink spaces. By passage to the maximal unramified extension and Galois descent, one can assume that k is algebraically closed.