Epsilon-strongly graded rings: Azumaya algebras and partial crossed products

Abstract

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 $\mathbf{pic}R$ the Picard semigroup of $R:=A_1$. We prove that the isomorphism class $[A_g]$ is an element of $\mathbf{pic}R$, for all $g\in G$. Thus, the association $g\mapsto [A_g]$ determines a partial representation of $G$ on $\mathbf{pic}R$ which induces a partial action $\gamma$ of $G$ on the center $Z(R)$ of $R$. Sufficient conditions for $A$ to be an Azumaya $R^{\gamma }$-algebra are presented in the case that $R$ is commutative. We study when $B$ is a partial crossed product in the following cases: $B=M_n(A)$ is the ring of matrices with entries in $A$, or $B=\mathbf{grm}M={⨁}_{l\in G}{\mathbf{Mor}}_A(M,M{)}_l$ is the direct sum of graded endomorphisms of left graded $A$-module $M$ with degree $l$, or $B=\mathbf{grm}M$ 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.

24 pages