An exact sequence for the graded Picent.

To a strongly 퐺-graded algebra 퐴 with 1-component 퐵, we associate the group Picent gr ⁢ (A) of isomorphism classes of invertible 퐺-graded (A , A) -bimodules over the centralizer of 퐵 in 퐴. Our main result is a Picent version of the Beattie–del Río exact sequence, involving Dade's group G ⁢ [ B ] , which relates Picent gr ⁢ (A) , Picent ⁢ (B) , and group cohomology. [ABSTRACT FROM AUTHOR]