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

TL;DR

This paper investigates epsilon-strongly graded rings that are partial crossed products, exploring their structure, connections to Azumaya algebras, and conditions under which they relate to partial crossed products and graded modules.

Contribution

It establishes a link between epsilon-strongly graded rings and partial crossed products, introduces a partial representation on the Picard semigroup, and provides conditions for Azumaya algebra structures.

Findings

The isomorphism class [A_g] lies in the Picard semigroup of R.

A partial representation of G on Pic(R) induces a partial action on Z(R).

Conditions are given for A to be an Azumaya algebra over the fixed ring.

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 ${\bf pic}{R}$ 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 $\gamma$ 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: $B=\operatorname{M}_n(A)$ 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.