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Azumaya algebras and Barr Theorem

Thierry Coquand, Henri Lombardi, Stefan Neuwirth

TL;DR

This paper develops a constructive, topos-theoretic framework for Azumaya algebras by encoding local, henselian, and separably closed local rings as coherent theories and leveraging Barr's theorem. It shows that Azumaya algebras are locally matrix algebras in the étale topology and provides a finite-tree descent criterion, supported by lifting lemmas and descent arguments. The approach unifies two definitions of Azumaya algebras constructively and yields Skolem–Noether-type insights within a constructive setting. The work lays foundational groundwork for Azumaya theory in topos-theoretic and constructive contexts and points toward Gabber’s theorem as a key future extension with cohomological implications.

Abstract

We study etale topology and the notion of Azumaya algebra over a commutative ring constructively. As an application of the syntactic version of Barr's Theorem, we show the equivalence between two definitions of Azumaya algebra.

Azumaya algebras and Barr Theorem

TL;DR

This paper develops a constructive, topos-theoretic framework for Azumaya algebras by encoding local, henselian, and separably closed local rings as coherent theories and leveraging Barr's theorem. It shows that Azumaya algebras are locally matrix algebras in the étale topology and provides a finite-tree descent criterion, supported by lifting lemmas and descent arguments. The approach unifies two definitions of Azumaya algebras constructively and yields Skolem–Noether-type insights within a constructive setting. The work lays foundational groundwork for Azumaya theory in topos-theoretic and constructive contexts and points toward Gabber’s theorem as a key future extension with cohomological implications.

Abstract

We study etale topology and the notion of Azumaya algebra over a commutative ring constructively. As an application of the syntactic version of Barr's Theorem, we show the equivalence between two definitions of Azumaya algebra.
Paper Structure (7 sections, 19 theorems, 10 equations)

This paper contains 7 sections, 19 theorems, 10 equations.

Table of Contents

  1. Some coherent theories
  2. Theory of local rings
  3. Theory of separably closed local rings
  4. Some lemmas about Henselian rings
  5. Azumaya algebras
  6. Theory of algebraically closed local rings
  7. Conclusion

Key Result

Lemma 1.1

Let $R$ be a local ring, residually discrete with maximal ideal $\mathsf{m}$ and residue field $F = R/\mathsf{m}$. If $f$ is a monic polynomial in $R[X]$ which has a simple root in some nontrivial $F$-algebra $A$, then $f$ is unramifiable.

Theorems & Definitions (33)

  • Lemma 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Theorem 1.1
  • proof
  • Lemma 2.1: ALP
  • proof
  • Lemma 2.2
  • ...and 23 more