# Commentarii Mathematici Helvetici

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**Volume 72, Issue 3, 1997, pp. 389–399**

**DOI: 10.1007/s000140050023**

Picard groups of multiplicative invariants

Martin Lorenz^{[1]}(1) Department of Mathematics, Temple University, PA 19122-6094, PHILADELPHIA, UNITED STATES

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.

*Keywords: *Picard group, ring of invariants, group action, Grothendieck group, polycyclic group algebra, Cartan map, class group, 1-cohomology group

Lorenz M. Picard groups of multiplicative invariants. *Comment. Math. Helv.* 72 (1997), 389-399. doi: 10.1007/s000140050023