%0 Journal Article %T The Differential Brauer Group %A Raymond T. Hoobler %J Mathematics %D 2010 %I arXiv %X 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. %U http://arxiv.org/abs/1003.1421v1