An exact sequence for the graded Picent

TL;DR

This paper introduces an exact sequence relating the group of invertible graded bimodules over a strongly graded algebra to cohomology and other algebraic invariants, extending classical results.

Contribution

It establishes a new exact sequence for the graded Picent group, generalizing the Beattie-del Río sequence to the graded setting involving Dade's group.

Findings

Derived a graded version of the Beattie-del Río exact sequence.

Connected the graded Picent group with Dade's group and cohomology.

Extended classical algebraic invariants to the graded context.

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\'{\i}o exact sequence, involving Dade's group $G[B]$, which relates $\mathrm{Picent}^{\mathrm{gr}}(A)$, $\mathrm{Picent}(B)$, and group cohomology.