Class groups of multiplicative invariants
Let \(S=kA\) be the group algebra of the free abelian group \(A\) of finite rank \(d\) over the field \(k\), and let \(G\) be a finite subgroup of \(\text{GL}_d(\mathbb{Z})\). Thus \(G\) acts on \(A\) and by extension on \(S\) -- these are the multiplicative actions of the title, which have been studied in a series of papers by \textit{D. R. Farkas}, who asks [in Group Actions on Rings, Contemp. Math. 43, 69-80 (1985; Zbl 0573.16005)] for a characterisation of when the fixed ring \(R=S^G\) is a unique factorisation domain. The main result answers this question by describing the class group \(\text{Cl }R\) of \(R\). Let \(N\) be the subgroup of \(G\) generated by reflections, \(D\) the subgroup generated by the reflections diagonalizable over \(\mathbb{Z}\). Then \(\text{Cl }R\cong\text{Hom}(G/N,k^*)\oplus H^1(G/D,A^D)\). This result contrasts with the classical case of linear actions on polynomial \(k\)-algebras, where only the first summand appears, as shown by \textit{P. Samuel} [Lectures on unique factorisation domains (Tata Inst. Lect. Notes 30, 1964; Zbl 0184.06601)]. Notwithstanding the different conclusion, the technique of proof is similar to Samuel's.
finite subgroups of \(\text{GL}_ d(\mathbb{Z})\)
scientific article; zbMATH DE number 848972