Actions of Picard groups on graded rings
Let \(R\) be a \(G\)-strongly graded ring, \(G\) any group. Let \(A\) denote \(R_e\). In this paper, the authors describe when strongly graded rings \(R\) and \(R'\) are graded isomorphic, graded equivalent (i.e., the categories of graded modules are equivalent by a functor that commutes with suspensions), or graded \(A\)-isomorphic (i.e., the isomorphism is the identity on \(A=R_e=R_e'\)). They show that \(R\) and \(R'\) are graded equivalent iff \(R'\) is graded isomorphic to \(R^C\), \(R\) acted on by a graded Morita context \(C\). Also they show that the difference between the classes of rings graded equivalent to \(R\) and graded isomorphic to \(R\) (graded equivalent to \(R\) and graded \(A\)-isomorphic to \(R\), graded isomorphic to \(R\) and graded \(A\)-isomorphic to \(R\), respectively) can be described by quotients of \(\text{Pic}(A)\) or \(\Aut(A)\). There exist graded equivalent strongly \(G\)-graded rings \(R\) and \(R'\) with \(A=R_e=R_e'\) and \(R_g\cong R_g'\) as \(A\)-bimodules for all \(g\) but \(R\) not graded isomorphic to \(R'\). Now the authors look for \(R'\) graded equivalent to \(R\) but ``simpler'' in the sense that \(R_e'\) is better understood or \(R'\) is a crossed product. In particular, if \(R_e\) is semiperfect, \(R'\) can be chosen so that \(R_e'\) is the basic ring of \(R_e\). If \(R\) and thus \(R_e\) is semisimple, \(R'\) is a crossed product over a finite product of division rings and can be described explicitly.
scientific article; zbMATH DE number 1348173