Picard groups of multiplicative invariants

Abstract

Let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural GL(A)-action on A. We determine the Picard group Pic R of the algebra of invariants R = SG. As an application, we produce new polycyclic group algebras with nontrivial torsion in K0.