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Lectures on Analytic Geometry

Peter Scholze

Abstract

This is a slightly updated version of lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. This is intended as a stable citable version of the material. In the first half of this course, we develop the basic theory of liquid real vector spaces, which we used in another course to give a new approach to complex-analytic geometry. In the second half, we gave a tentative definition of a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces. While the precise definition of analytic spaces represents an abandoned stepping stone on our way to define analytic stacks and hence should be seen as a historical artifact, much of the surrounding discussion stays very relevant.

Lectures on Analytic Geometry

Abstract

This is a slightly updated version of lectures notes for a course on analytic geometry taught in the winter term 2019/20 at the University of Bonn. The material presented is part of joint work with Dustin Clausen. This is intended as a stable citable version of the material. In the first half of this course, we develop the basic theory of liquid real vector spaces, which we used in another course to give a new approach to complex-analytic geometry. In the second half, we gave a tentative definition of a category of analytic spaces that contains (for example) adic spaces and complex-analytic spaces. While the precise definition of analytic spaces represents an abandoned stepping stone on our way to define analytic stacks and hence should be seen as a historical artifact, much of the surrounding discussion stays very relevant.

Paper Structure

This paper contains 15 sections, 105 theorems, 389 equations, 1 figure.

Table of Contents

  1. Lecture I: Introduction
  2. Lecture II: Solid modules
  3. Lecture III: Condensed $\mathbb R$-vector spaces
  4. Lecture IV: $\mathcal{M}$-complete condensed $\mathbb R$-vector spaces
  5. Lecture V: Entropy and a real $B_{\mathrm{dR}}^+$
  6. Lecture VI: Statement of main result
  7. Lecture VII: $\mathbb Z((T))_{>r}$ is a principal ideal domain
  8. Lecture VIII: Reduction to "Banach spaces"
  9. Lecture IX: End of proof
  10. Lecture X: Some computations with liquid modules
  11. Lecture XI: Animation
  12. Animation
  13. Lecture XII: Localizations
  14. Lecture XIII: Analytic spaces
  15. Lecture XIV: Varia

Key Result

Proposition 1.2

Consider the functor $T\mapsto \underline{T}$ from topological spaces to condensed sets.

Figures (1)

  • Figure 1: The Berkovich space $M(\mathbb Z)$

Theorems & Definitions (160)

  • Definition 1.1
  • Proposition 1.2
  • Lemma 1.3
  • Proposition 2.1
  • Remark 2.2
  • Definition 2.4
  • Theorem 2.5: Condensed
  • Proposition 2.6
  • Corollary 2.7
  • Proposition 2.8
  • ...and 150 more