Летняя математическая школа "Алгебра и геометрия"

Introduction ot Berkovich analytic spaces

Михаил Тёмкин (Hebrew University of Jerusalem, Израиль)

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Abstract

This is an introductory course to analytic geometry over a non-archimedean field as developed by Berkovich. We will define Berkovich spaces, study their basic properties, and describe relation with other approaches and theories, including analytifications and formal models. In order to cover the large amount of material we will concentrate on describing definitions and constructions and formulating the main results of the theory, although in some cases main ideas of the proofs will be outlined. The course can be divided to five parts as follows:

1. valuations, non-archimedean fields and Banach algebras,
2. affinoid algebras and spaces,
3. analytic spaces and their basic properties,
4. connection to other categories: analytification of algebraic varieties and generic fiber of formal schemes,
5. analytic curves and stable reduction theorem.

Prerequisites

A basic familiarity with commutative algebra and algebraic geometry, e.g. chapters II-IV of Hartshorne's, is the main prerequisite for the course. Some familiarity with field valuations and formal schemes may also be helpful, though I will mention briefly the facts we will need about them.

Sources

To some extent I will follow the lecture notes from a mini-course I gave in Paris in 2010. You may also wish to consult the literature cited there.