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Notes on Solid Geometry

Juan Esteban Rodríguez Camargo

Abstract

These are expanded notes of a seminar held in Columbia university during the Spring and Fall of 2024 about the theory of analytic stacks of Clausen and Scholze, with a focus in the theory of solid mathematics. The seminar is inspired from the Lecture Series of Analytic Stacks of Clausen and Scholze during the winter semester of 2023. All the theory of light condensed mathematics, analytic stacks and the proof of Serre duality must be attributed to Clausen and Scholze, any mistake or misconception is totally due to the author. The only original work in these notes is the discussion of smooth, étale and finite presentation morphisms of solid Huber rings in Section 7.

Notes on Solid Geometry

Abstract

These are expanded notes of a seminar held in Columbia university during the Spring and Fall of 2024 about the theory of analytic stacks of Clausen and Scholze, with a focus in the theory of solid mathematics. The seminar is inspired from the Lecture Series of Analytic Stacks of Clausen and Scholze during the winter semester of 2023. All the theory of light condensed mathematics, analytic stacks and the proof of Serre duality must be attributed to Clausen and Scholze, any mistake or misconception is totally due to the author. The only original work in these notes is the discussion of smooth, étale and finite presentation morphisms of solid Huber rings in Section 7.
Paper Structure (65 sections, 149 theorems, 387 equations)

This paper contains 65 sections, 149 theorems, 387 equations.

Table of Contents

  1. Introduction
  2. Light condensed sets
  3. Analytic rings
  4. Analytic stacks
  5. Examples
  6. Overview of the document
  7. Light condensed mathematics
  8. Light profinite sets
  9. Light condensed sets
  10. Light condensed abelian groups
  11. Light solid abelian groups
  12. Null-sequences and summability
  13. Solid abelian groups form an analytic ring
  14. Computing measures in solid abelian groups
  15. Flatness of $\prod_{\mathbb{N}}\mathbb{Z}$ and the structure of the category $\mathsf{Solid}$
  16. ...and 50 more sections

Key Result

Proposition 2.1.1

The following categories are equivalent. We let $\mathop{\mathrm{Prof}}\nolimits$ denote the category of profinite sets, considered as in (1) or (2) above.

Theorems & Definitions (438)

  • Definition 1.1.1
  • Definition 1.1.2
  • Definition 1.2.1
  • Proposition 2.1.1
  • proof
  • Proposition 2.1.2
  • proof
  • Definition 2.1.3
  • Proposition 2.1.4
  • proof
  • ...and 428 more