An exact sequence for the graded Picent

Abstract

To a strongly $G$-graded algebra $A$ with $1$-component $B$ we associate the group ${\mathrm{Picent}}^{\mathrm{gr}}(A)$ of isomorphism classes of invertible $G$-graded $(A,A)$-bimodules over the centralizer of $B$ in $A$. Our main result is a $\mathrm{Picent}$ version of the Beattie-del R\'ıo exact sequence, involving Dade's group $G[B]$, which relates ${\mathrm{Picent}}^{\mathrm{gr}}(A)$, $\mathrm{Picent}(B)$, and group cohomology.