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Topological Vector Spaces

Pierre Colmez, Wiesława Nizioł

Abstract

Motivated by applications to duality theorems for $p$-adic pro-étale cohomology of rigid analytic spaces, we study the category of Topological Vector Spaces in the setting of condensed mathematics. We prove that it contains, as full subcategories, both the category of (topologically) bounded algebraic Vector Spaces and the category of perfect complexes on the Fargues-Fontaine curve. Vector Spaces coming from $p$-adic pro-étale cohomology of smooth partially proper rigid analytic varieties are examples of sheaves belonging to the former category.

Topological Vector Spaces

Abstract

Motivated by applications to duality theorems for -adic pro-étale cohomology of rigid analytic spaces, we study the category of Topological Vector Spaces in the setting of condensed mathematics. We prove that it contains, as full subcategories, both the category of (topologically) bounded algebraic Vector Spaces and the category of perfect complexes on the Fargues-Fontaine curve. Vector Spaces coming from -adic pro-étale cohomology of smooth partially proper rigid analytic varieties are examples of sheaves belonging to the former category.

Paper Structure

This paper contains 32 sections, 27 theorems, 190 equations.

Table of Contents

  1. Introduction
  2. Naive Topological Vector Spaces
  3. Topological pro-étale site
  4. Totally disconnected spaces
  5. Topological pro-étale site
  6. Naive Topological Vector Spaces
  7. Condensed and solid modules
  8. Condensed sheaves
  9. Solid sheaves
  10. Constant sheaves and monoidal structures
  11. Condensed versus solid sheaves
  12. Étale Naive Topological Vector Spaces
  13. Topological Vector Spaces
  14. Topologically enriched presheaves
  15. Topologized mapping space
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $S\in {\rm sPerf}_C$.

Theorems & Definitions (69)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Example 2.1
  • Lemma 2.5
  • proof
  • Proposition 2.7
  • Remark 2.12
  • Lemma 2.15
  • proof
  • ...and 59 more