Differential graded Brauer groups over dg-rings

Abstract

Brauer groups of graded rings were defined and studied by Caenepeel and van Oystaeyen. We study the question what happens to this group if a ${\mathbb Z}$-graded ring carries a differential graded structure in addition. We define a Brauer group for differential graded algebras over differential graded graded-commutative or commutative base rings. Based on previous work we give an explicit classification of dg-fields, and compute as an example the so-defined Brauer group in each case explicitly.