On generic \(G\)-graded Azumaya algebras (Q2118927)

Let \(F\) be an algebraically closed field of characteristic zero, and \(G\) a finite group. Given a finite dimensional \(G\)-graded central simple algebra \(A\) over \(F\), the authors construct a graded Azumaya algebra \(\mathcal{U}\) which is graded generic for \(A\). Namely, \(\mathcal{U}\), and all its nonzero graded quotients satisfy the same \(G\)-graded polynomial identities as \(A\); there is a one-to-one correspondence between graded ideals of \(\mathcal{U}\) and the ideals of its graded-center; and the graded prime quotients of \(\mathcal{U}\) are precisely the graded forms of \(A\). A substantial part of the paper is devoted to showing that a certain cyclotomic field is the unique minimal field of definition of~\(A\). This field is also the unique minimal field of definition of the T-ideal of graded identities of \(A\). This is done by recalling that a graded central simple algebra over an algebraically closed field is a matrix algebra over a twisted subgroup algebra for some subgroup \(H \leq G\), and adjusting the twisting cocycle. The algebra \(\mathcal{U}\) is then obtained as the localization, by a suitable polynomial, of the relatively free algebra of \(A\) over the minimal field of definition. This polynomial is chosen as an identity of all the graded algebras that have strictly more identities than \(A\), of which there are finitely many. As an application, it is shown that \(A\) admits a \(G\)-graded division algebra form over its \(e\)-center \(Z(A)_e\) if and only if its T-ideal of graded identities is strongly verbally prime having the same minimal field of definition as \(A\).