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Chatelet's Theorem in Synthetic Algebraic Geometry

Thierry Coquand, Hugo Moeneclaey

TL;DR

This work extends Châtelet's theorem to the setting of synthetic algebraic geometry over an arbitrary base ring by formulating and proving an AZ$_n$–SB$_n$ correspondence via étale-modal methods. It establishes an equivalence between Azumaya algebras of rank $(n+1)^2$ and Severi-Brauer varieties of dimension $n$ through the Severi-Brauer construction, aided by delooping techniques in $T$-sheaves and a precise identification of automorphism groups with $\mathrm{PGL}_{n+1}(R)$. The approach hinges on descent for finite free modules and the étale-local structure of projective space, culminating in a synthetic proof that every SB$_n$ is isomorphic to $\mathbb{P}^n$ when the relevant right-ideal is a free module. Overall, the paper unifies Azumaya algebra theory with Severi-Brauer geometry in a constructive, modality-driven framework, thereby generalizing Châtelet's classical results to arbitrary base rings.

Abstract

We prove a version of Chatelet's Theorem about Severi-Brauer variety having rational points in the setting of synthetic algebraic geometry. We work over an arbitrary base ring.

Chatelet's Theorem in Synthetic Algebraic Geometry

TL;DR

This work extends Châtelet's theorem to the setting of synthetic algebraic geometry over an arbitrary base ring by formulating and proving an AZ–SB correspondence via étale-modal methods. It establishes an equivalence between Azumaya algebras of rank and Severi-Brauer varieties of dimension through the Severi-Brauer construction, aided by delooping techniques in -sheaves and a precise identification of automorphism groups with . The approach hinges on descent for finite free modules and the étale-local structure of projective space, culminating in a synthetic proof that every SB is isomorphic to when the relevant right-ideal is a free module. Overall, the paper unifies Azumaya algebra theory with Severi-Brauer geometry in a constructive, modality-driven framework, thereby generalizing Châtelet's classical results to arbitrary base rings.

Abstract

We prove a version of Chatelet's Theorem about Severi-Brauer variety having rational points in the setting of synthetic algebraic geometry. We work over an arbitrary base ring.

Paper Structure

This paper contains 15 sections, 31 theorems, 72 equations.

Table of Contents

  1. Étale sheaves
  2. Affine schemes are étale sheaves
  3. Schemes are étale sheaves
  4. Descent for finite free modules
  5. Azumaya algebras and their associated Severi-Brauer variety
  6. The type $\mathrm{AZ}_n$ of Azumaya algebras
  7. Quaternion algebras are Azumaya algebras
  8. A remark on Azumaya algebras
  9. The type $\mathrm{SB}_n$ of Severi-Brauer varieties
  10. Conics are Severi-Brauer varieties
  11. The equivalence $\mathrm{AZ}_n\simeq \mathrm{SB}_n$ and Châtelet's Theorem
  12. Generalities on delooping in $T$-sheaves
  13. Both $\mathrm{Aut}(M_{n+1}(R))$ and $\mathrm{Aut}(\mathbb{P}^n)$ are $\mathop{\mathrm{PGL}}\nolimits_{n+1}(R)$
  14. The Severi-Brauer construction is an equivalence
  15. Châtelet's Theorem

Key Result

Lemma 1.1

The type $R$ is an étale sheaf.

Theorems & Definitions (77)

  • Definition 1.1
  • Remark 1.2
  • Lemma 1.1
  • proof
  • Remark 1.3
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.1
  • ...and 67 more