# Adic spaces

**Time and venue**

We, 8:30-10, Fr, 8:30-9:15, in MI 02.10.011

Exercises: Fr, 9:15-10

**Content**

We consider a field k which is complete with respect to a non-archimedean valuation. These fields (such as for example the p-adic numbers **Q**_{p}) play a major role in algebraic number theory. A natural question is now to ask for an analogue over k of the classical theory of manifolds (over **R** or **C**), or more generally of zero-sets of a set of convergent power series. A naive analogue can be (and has been) studied, but the resulting theory has the major disadvantage that all occurring spaces are totally disconnected.In the 70s Tate then defined rigid-analytic spaces, which have much nicer geometric properties (for example closed unit balls are connected) and a theory of coherent sheaves on them. Since then, several improvements and generalizations of this theory have been considered (among others by Raynaud, Berkovich and Huber). They have played an important role in several major results in arithmetic geometry, such as for example the proof of the local Langlands correspondence for GL_{n} by Harris and Taylor. In the past years, Huber's theory of adic spaces (originally developed in the 90s) has received particular attention. This is mainly due to the striking successes of Scholze's work on perfectoid spaces (which is biult on adic spaces), which led to progress in many deep questions in arithmetic geometry. The present course will give an introduction into Huber's theory.

In the lecture we will discuss the following topics:

- Tate rings
- valuations and continuous valuations
- Huber rings
- affinoid adic spaces
- structure presheaves and sheaf properties
- adic spaces.

### **Prerequisites**

Basic knowledge of algebraic geometry (locally ringed spaces, schemes), non-archimedean valuations

**Literature**

B. Conrad, Number theory learning seminar 2014-2015 on perfectoid spaces, online

R. Huber, Continuous valuations, Math. Z. (1993)

R. Huber, A generalization of formal schemes and rigid-analytic varieties, Math. Z. (1994)

R. Huber, Etale cohomology of rigid-analytic varieties and adic spaces, Vieweg 1996.

T. Wedhorn, Introduction to Adic Spaces.