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.
Download