An exact sequence for the graded Picent

Abstract

To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group

of isomorphism classes of invertible 𝐺-graded

( A , A ) (A,A)

-bimodules over the centralizer of 𝐵 in 𝐴. Our main result is a

Picent \mathrm{Picent}

version of the Beattie–del Río exact sequence, involving Dade’s group

G ⁢ [ B ] G[B]

, which relates

Picent gr ⁢ ( A ) \mathrm{Picent}^{\mathrm{gr}}(A)

,

Picent ⁢ ( B ) \mathrm{Picent}(B)

, and group cohomology.