Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{ e} := Z(A)\cap A_{ e} = F$. For any such algebra we construct a \textit{ generic} $G$-graded algebra $\mathcal{ U}$ which is \textit{ Azumaya} in the following sense. $(1)$ \textit{ $($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $\mathcal{ U}$ and the ideals of the ring $R$, the $e$-center of $\mathcal{ U}$. $(2)$ \textit{ Artin-Procesi condition}: $\mathcal{ U}$ satisfies the $G$-graded identities of $A$ and no nonzero $G$-graded homomorphic image of $\mathcal{ U}$ satisfies properly more identities. $(3)$ \textit{ Generic}: If $B$ is a $G$-graded algebra over a field then it is a specialization of $\mathcal{ U}$ along an ideal $\mathfrak{ a} \in spec(Z(\mathcal{ U})_{ e})$ if and only if it is a $G$-graded form of $A$ over its $e$-center. We apply this to characterize finite dimensional $G$-graded simple algebras over $F$ that admit a $G$-graded division algebra form over their $e$-center.
Categories: math.RA