An example concerning the Bass conjecture
Let \(R\) be a commutative ring with a 1 and let \(G\) be a group. If \(P\) is a finitely generated projective \(RG\)-module then there is a map \(r_ P: G \to R\), called the rank of \(P\), which is constant on the conjugacy classes of \(G\). This map can be considered as an extension of the usual trace map from square matrices over \(RG\) to \(RG\). The precise definition of \(r_ P\) is given by \textit{H. Bass} [see Lond. Math. Soc. Lect. Note Ser. 36, 1-26 (1979; Zbl 0425.20006)]. If \(F\) is a finitely generated free \(RG\)-module then \(r_ F(x) = 0\) for all \(x \in G \setminus 1\), and if \(Q\) is a finitely generated projective \(RG\)-module then \(r_{P \oplus Q} = r_ P + r_ Q\). One form of the strong Bass conjecture is the following: Let \(P\) be a finitely generated projective \(RG\)-module, and let \(1 \neq x \in G\). If \(R\) is a subring of \(\mathbb{C}\) such that \(R \cap\mathbb{Q}=\mathbb{Z}\), then \(r_ P = 0\). The following question was posed by Schafer: If \(G\) is a finitely generated group containing a subgroup \(A \cong\mathbb{Q}^ +\) such that \(A\) is contained in only finitely many conjugacy classes of \(G\), must \(G\) be of infinite cohomological dimension? [see also \textit{J. A. Schafer}, Mich. Math. J. 38, 103-109 (1991; Zbl 0722.20002)]. If the answer to this question were `yes' then the strong Bass conjecture would be true for all groups \(G\) of finite cohomological dimension. In the paper under review the author shows that the answer to the question is `no'. He gives an example of a 2-generator subgroup \(G\) of cohomological dimension 2 which contains a subgroup \(A \cong\mathbb{Q}^ +\) such that \(A \setminus 1\) is contained in a single conjugacy class of \(G\).
finitely generated projective \(RG\)-module
scientific article; zbMATH DE number 488365