论文标题
Epsilon紧张的级环：Azumaya代数和部分交叉产品
Epsilon-strongly graded rings: Azumaya algebras and partial crossed products
论文作者
论文摘要
本文的主要目的是研究部分交叉产品的Epsilon巧妙分级环。令$ g $为一个组，$ a = \ oplus_ {g \ in g} \，a_g $ a epsilon-strongrongly ronged ring和$ {\ bf pic} {r} $ $ r：= a_1 $的picard semigroup of $ r：= a_1 $。我们证明，对于g $中的所有$ g \，同构级$ [a_g] $是$ {\ bf pic} {r} $的元素。因此，协会$ g \ mapsto [a_g] $确定$ {\ bf pic} {r} $ o $ g $的部分表示，它诱导了$ r $ $ r $的中心$ z（r）$ z（r）的部分操作$γ$ $ g $。 $ a $作为Azumaya $ r^γ$ - 代数的足够条件在$ r $是可交换的情况下提出。在以下情况下，我们研究$ b $是部分交叉产品：$ b = \ operatorName {m} _n（a）$是矩阵的矩阵环，$ a $ a $或$ b = {\ bf grm} {m} {m} {m} = \ bigoplus_ {左分级$ a $ -module $ m $带学位$ l $的内态性，或$ b = {\ bf grm} {m} {m = a \ a \ otimes_ {r} n $是左$ r $ -module $ n $的诱导模块。最后，假设$ r $是半融合的，我们证明存在epsilon额外的分级$ a $ a $等级等同于部分交叉产品的分级。
The main purpose of this paper is to investigate epsilon-strongly graded rings that are partial crossed products. Let $G$ be a group, $A=\oplus_{g\in G}\,A_g$ an epsilon-strongly graded ring and ${\bf pic}{R}$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of ${\bf pic}{R}$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on ${\bf pic}{R}$ which induces a partial action $γ$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^γ$-algebra are presented in the case that $R$ is commutative. We study when $B$ is a partial crossed product in the following cases: $B=\operatorname{M}_n(A)$ is the ring of matrices with entries in $A$, or $B={\bf grm}{M}=\bigoplus_{l \in G}{\bf Mor}_A(M,M)_l$ is the direct sum of graded endomorphisms of left graded $A$-module $M$ with degree $l$, or $B={\bf grm}{M}$ where $M=A\otimes_{R}N$ is the induced module of a left $R$-module $N$. Finally, assuming that $R$ is semiperfect, we prove that there exists an epsilon-strongly graded subring of $A$ which is graded equivalent to a partial crossed product.
