Specialization maps for Scholze's category of diamonds

TL;DR

This paper introduces the specialization map in Scholze's theory of diamonds, defining kimberlites and establishing their properties, including connections to formal schemes and applications to p-adic Beilinson--Drinfeld Grassmannians.

Contribution

It defines kimberlites as v-sheaves resembling formal schemes and develops their associated sites, extending classical concepts within Scholze's framework.

Findings

Kimberlites have a reduced special fiber and an analytic locus.

Unramified p-adic Beilinson--Drinfeld Grassmannians are kimberlites with finiteness and normality.

The developed sites recover classical structures when originating from formal schemes.

Abstract

We introduce the specialization map in Scholzes 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 etale 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.