Differential Brauer Monoids

TL;DR

This paper introduces the differential Brauer monoid for differential commutative rings, extending classical Brauer theory to include differential structures and establishing its relation to the standard Brauer monoid.

Contribution

It defines the differential Brauer monoid, relates it to classical Brauer monoids and constants, and characterizes its structure in the differential algebra setting.

Findings

The differential Brauer monoid is determined by the Brauer monoids of R and its constants.

It generalizes the classical Brauer monoid to differential rings.

The differential Brauer monoid forms a monoid, extending the Bauer group.

Abstract

The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings.