On Bass’ “Strong conjecture” about projective modules.

*(English)*Zbl 0688.16013Let \({\mathbb{Z}}[G]\) be the integral group ring of a group G. Let P be a finitely generated projective \({\mathbb{Z}}[G]\)-module. It has a rank \(r_ p\) which can be viewed as an integer valued function on conjugacy classes of G. The Bass’ “Strong Conjecture” [H. Bass, Invent. Math. 35, 155- 196 (1976; Zbl 0365.20008), p. 156] claims that \(r_ p(g)=0\) for \(g\neq 1\) in G. Bass proved this conjecture for a class of torsion free groups including those with faithful linear representations. In the present paper the author proves the conjecture for a large class of groups which satisfy the following condition called WD: Suppose H is a finitely generated subgroup of G, \(s\in H\), N a positive integer and s is conjugate in H to \(s^{p^ N}\) for all primes p; then \(s=1\). The author shows that the following groups G satisfy condition WD: (i) G is a torsion group. (ii) G is a nilpotent-by-Noetherian group. (iii) (cf. H. Bass [loc. cit.]) G is a linear group. (iv) (cf. P. A. Linnell [Proc. Lond. Math. Soc., III. Ser. 47, 83-127 (1983; Zbl 0531.20002)]) The group G does not contain any subgroup isomorphic to the additive group of rational numbers. (v) Every finitely generated subgroup of G satisfies condition WD. (vi) G is a finite extension of some WD-groups. (vii) G is a subdirect product of (maybe infinitely many) WD-groups. (viii) G is a free product of (maybe infinitely many) WD-groups.

Reviewer: T.Akasaki

##### MSC:

16S34 | Group rings |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

##### Keywords:

integral group ring; finitely generated projective \({\mathbb{Z}}[G]\)-module; rank; integer valued function on conjugacy classes; Bass’ “Strong Conjecture”; torsion free groups; condition WD; finitely generated subgroup; subdirect product; WD-groups; free product
PDF
BibTeX
XML
Cite

\textit{A. Strojnowski}, J. Pure Appl. Algebra 62, No. 2, 195--198 (1989; Zbl 0688.16013)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Bass, H., Euler characteristic and characters of discrete groups, Invent. math., 35, 155-196, (1976) · Zbl 0365.20008 |

[2] | Bass, H., Traces and Euler characteristics, (), 1-26, London Math. Soc. Lecture Note Series |

[3] | Cliff, G.H., Zero divisors and idempotents in group rings, Canad. J. math., 32, 596-602, (1980) · Zbl 0439.16011 |

[4] | Higman, G., The units of group rings, Proc. London math. soc., 46, 2, 231-248, (1940) · JFM 66.0104.04 |

[5] | Linell, P.A., Decomposition of augmentation ideals and relation modules, Proc. London math. soc., 47, 3, 83-127, (1983) · Zbl 0531.20002 |

[6] | J. Moody, Proof of Bass’ “Strong Conjecture” for residually finite groups, preprint. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.