Graded equivalences and Picard groups (Q1304894)

Let \(R\) be a ring graded by the group \(G\) and \(R\)-gr the category of graded \(R\)-modules. Two subgroups of the Picard group \(\text{Pic}(R\text{-gr})\) are studied: the subgroup \(\text{gr-Pic}(R\#P_G)\) of classes of graded equivalences and the subgroup \(\text{Pic}(R\#P_G)^G\) of classes of equivalences which commute with the suspensions in \(\text{Pic}(R\text{-gr})\). It is proved that in general these two subgroups are not the same and it is shown that they are related by an exact sequence of cohomology.