Abstract
Abstract To a strongly 𝐺-graded algebra 𝐴 with 1-component 𝐵, we associate the group Picent gr ( A ) \mathrm{Picent}^{\mathrm{gr}}(A) 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.
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