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A new approach to $δ$-rings via Stone duality

Yuto Yamada

Abstract

We define Stone $δ$-rings as a new class of $δ$-rings. Via Stone duality, we shows that $δ$-rings relates light condensed mathematics, which is developed by Clausen-Scholze. Also, we examine some phenomena for this relationship, for example, we observe $δ$-rings which corresponds to metrizable compact Hausdorff spaces.

A new approach to $δ$-rings via Stone duality

Abstract

We define Stone -rings as a new class of -rings. Via Stone duality, we shows that -rings relates light condensed mathematics, which is developed by Clausen-Scholze. Also, we examine some phenomena for this relationship, for example, we observe -rings which corresponds to metrizable compact Hausdorff spaces.
Paper Structure (16 sections, 28 theorems, 26 equations)

This paper contains 16 sections, 28 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Stone Duality
  3. Stone $\mathbb{F}_p$-Algebras
  4. Perfect $\mathbb{F}_p$-Algebras
  5. $\delta$-Rings
  6. $\delta$-Rings
  7. Stone $\delta$-Rings
  8. Light Condensed Mathematics
  9. Light Profinite Sets
  10. Light Condensed Sets
  11. Comparisons of Sites
  12. Faithfully Flat Covers
  13. $p$-Completely Faithfully Flat Covers
  14. Connection between $\delta$-Rings and Light Condensed Sets
  15. Totally Disconnected Formal $\delta$-Stacks
  16. ...and 1 more sections

Key Result

Theorem 1.1

There exists an isomorphism of sites: where $\tau_{p\text{-fpqc}}$ denotes the Grothendieck topology defined by $p$-completely faithfully flat covers.

Theorems & Definitions (77)

  • Theorem 1.1: \ref{['thm:Stone duality 2']} and \ref{['thm:comparison of sites 2']}
  • Theorem 1.2: \ref{['cor:morphism of topoi']}
  • Definition 2.1: Ant23 and Gre24, Stone algebra
  • Theorem 2.2: Ant23 and Gre24, Stone dulality
  • proof
  • Definition 2.3: $p$-Boolean algebra
  • Proposition 2.4: Gre24
  • Lemma 2.5: Gre24
  • proof
  • Remark 2.6: Gre24
  • ...and 67 more