  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.
