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

Abstract

Abstract Let G be a group, let A = ⊕ g ∈ G A g {A=\bigoplus_{g\in G}A_{g}} be an epsilon-strongly graded ring over G, let R := A 1 {R:=A_{1}} be the homogeneous component associated with the identity of G, and let 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} be the Picard semigroup of R. In the first part of this paper, we prove that the isomorphism class [ A g ] {[A_{g}]} is an element of 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} for all g ∈ G {g\in G} . Moreover, the association g ↦ [ A g ] {g\mapsto[A_{g}]} determines a partial representation of G on 𝙿𝚒𝚌𝚂 ⁢ ( R ) {\mathtt{PicS}(R)} which induces a partial action γ of G on the center Z ⁢ ( R ) {Z(R)} of R. Sufficient conditions for A to be an Azumaya R γ {R^{\gamma}} -algebra are presented if R is commutative. In the second part, we study when B is a partial crossed product in the following cases: B = M n ⁡ ( A ) {B=\operatorname{M}_{n}(A)} is the ring of matrices with entries in A, or B = END A ( M ) = ⊕ l ∈ G Mor A ( M , M ) l {B=\operatorname{END}_{A}(M)=\bigoplus_{l\in G}\operatorname{Mor}_{A}(M,M)_{l}} is the direct sum of graded endomorphisms of graded left A-modules M with degree l, or B = END A ⁡ ( M ) {B=\operatorname{END}_{A}(M)} where M = A ⊗ R N {M=A\otimes_{R}N} is the induced module of a left R-module N. Assuming that R is semiperfect, we prove that there exists a subring of A which is an epsilon-strongly graded ring over a subgroup of G and it is graded equivalent to a partial crossed product.