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On Bass’ “Strong conjecture” about projective modules. (English) Zbl 0688.16013
Let \({\mathbb{Z}}[G]\) 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
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[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.
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