Specialization maps for Scholze’s category of diamonds

Abstract

We introduce the specialization map in Scholze’s theory of diamonds. We consider v-sheaves that “behave like formal schemes" and call them kimberlites. We attach to them: a reduced special fiber, an analytic locus, a specialization map, a Zariski site, and an étale site. When the kimberlite comes from a formal scheme, our sites recover the classical ones. We prove that unramified p-adic Beilinson–Drinfeld Grassmannians are kimberlites with finiteness and normality properties.