The groups \(N^ rK_ 0(\mathbb{Z}\pi )\) are finitely generated \(\mathbb{Z} [\mathbb{N}^ r]\)-modules if \(\pi\) is a finite group
For any ring \(R\), any integer \(i \leq 1\) and any integer \(r \geq 0\), the \(K\)-theory nil group, \(N^r K_i (R)\) is either zero or non-finitely generated as a \(\mathbb{Z}\)-module. In this paper the authors observe that \(N^r K_i(R)\) admits a \(\mathbb{Z} [\mathbb{N}^r]\)-module structure where \(\mathbb{N}\) is the multiplicative monoid of positive integers. They then prove that, for any finite group \(\pi\), the nil group \(N^r K_i (\mathbb{Z} [\pi])\) is finitely generated as a \(\mathbb{Z} [\mathbb{N}^r]\)- module. Moreover, they exhibit a power of the order of \(\pi\) which annihilates \(N^r K_i(\mathbb{Z} [\pi])\). The methods of the paper are strictly algebraic.
scientific article; zbMATH DE number 784411