Adic spaces

Abstract

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.