The Differential Brauer Group

TL;DR

This paper explores the structure of differential Azumaya algebras over rings with derivation, establishing their local triviality, automorphism groups, and the relationship between differential and classical Brauer groups.

Contribution

It introduces the $ ext{δ}$-flat topology, computes automorphism groups, and proves the differential Brauer group coincides with the classical Brauer group in affine cases.

Findings

Differential Azumaya algebras are locally trivial in the $ ext{δ}$-flat topology.

The differential automorphism group of $M_n(A)$ is explicitly calculated.

The differential Brauer group equals the classical Brauer group in affine settings.

Abstract

Let $A$ be a ring equipped with a derivation $\delta $. We study differential Azumaya $A$ algebras, that is, Azumaya $A$ algebras equipped with a derivation that extends $\delta $. We calculate the differential automorphism group of the trivial differential algebra, $M_{n}(A)$ with coordinatewise differentiation. We introduce the $\delta$-flat Grothendieck topology to show that any differential Azumaya $A$ algebra is locally isomorphic to a trivial one and then construct, as in the non-differential setting, the embedding of the differential Brauer group into $H^{2}(A_{\delta-pl},G_{m,\delta}) $. We conclude by showing that the differential Brauer group coincides with the usual Brauer group in the affine setting.