论文标题
在通用$ g $ graded azumaya代数上
On generic $G$-graded Azumaya algebras
论文作者
论文摘要
令$ f $为特征零的代数封闭字段，让$ g $为有限组。考虑有限维度和$ e $ central $ f $，即$ f $，即$ z（a）_ {e}：= z（a）\ cap a_ {e} = f $。对于任何此类代数，我们构造A \ textIt {generic} $ g $ graded代数$ \ MATHCAL {u} $，它是\ textit {azumaya}，从以下意义上讲。 $（1）$ \ textIt {$（$ $ supertions $）$}：$ g $的理想是$ \ mathcal {u} $的$ g $的理想与环$ r $的理想，$ e $ e $ center的$ \ mathcal {u} $之间有一对一的信件。 $（2）$ \ textit {artin-procesi条件}：$ \ Mathcal {u} $满足$ a $ $ a $的$ g $降级身份，而没有$ \ mathcal {u} $的非零$ g $ g $同质图像满意。 $(3)$ \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 $ - 中心。 我们将其用于表征有限的尺寸$ g $ g $的简单代数，而$ f $，该代数在其$ e $ center上接受了$ g $ a的分区代数形式。
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.
