The Differential Brauer Group

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.