Adic spaces

TL;DR

This paper provides an introductory overview of adic spaces, explaining their construction, key classes like rigid analytic spaces and formal schemes, and illustrating concepts with fundamental examples.

Contribution

It offers a clear exposition of the theory of adic spaces, building on Huber pairs, and connects various important classes of non-archimedean geometric spaces.

Findings

Construction of adic spaces from Huber pairs

Connections between rigid analytic spaces and formal schemes

Illustrations with open/closed disc and affine line

Abstract

These lecture notes are based on the second course in a series of lectures at the Spring school "Non-archimedean geometry and Eigenvarieties" in March 2023 in Heidelberg. The objective of the first three courses was to give an introduction to the theory of adic spaces. Building up on the theory of Huber pairs presented in John Bergdall's lecture we explain the construction of adic spaces. We study some important classes of adic spaces such as rigid analytic spaces and formal schemes and show the connections between them. In the course of the lecture we will illustrate the respective concepts with the fundamental examples of the open and closed disc and the affine line.