An exact sequence for the graded Picent

To a strongly $G$-graded algebra $A$ with $1$-component $B$ we associate the group ${\mathrm{P}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}}^{\mathrm{g}\mathrm{r}}(A)$ of isomorphism classes of invertible $G$-graded $(A,A)$-bimodules over the centralizer of $B$ in $A$. Our main result is a $\mathrm{P}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}$ version of the Beattie-del Río exact sequence, involving Dade’s group $G[B]$, which relates ${\mathrm{P}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}}^{\mathrm{g}\mathrm{r}}(A)$, $\mathrm{P}\mathrm{i}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}(B)$, and group cohomology.