Differential graded Brauer groups

TL;DR

This paper introduces the dg Brauer group, an extension of the classical Brauer group to differential graded algebras, and proves their isomorphism over a field.

Contribution

It defines the dg Brauer group for differential graded algebras and establishes its isomorphism with the classical Brauer group of the field.

Findings

The dg Brauer group is isomorphic to the classical Brauer group.

Differential graded algebras can be classified via this new group.

The result bridges differential graded algebra theory with classical algebraic structures.

Abstract

We consider central simple $K$-algebras which happen to bedifferential graded $K$-algebras. Two such algebras $A$ and $B$are considered equivalent if there are bounded complexes of finite dimensional$K$-vector spaces $C_A$ and $C_B$ such that the differential graded algebras $A\otimes_K {\rm End}_K^\bullet(C_A)$ and $B\otimes_K {\rm End}_K^\bullet(C_B)$ are isomorphic.Equivalence classes form an abelian group, which we call thedg Brauer group.We prove that this group is isomorphic to the ordinary Brauer group of the field $K$.