Stably free modules over rings of Laurent polynomials

Abstract

Let Φ be a finite group and let A be the group algebra of a free abelian group over a field k. We show that, in general, A[Φ] admits nontrivial stably free modules. By contrast, if A is the group algebra of a finitely generated free group then A[Φ] has stably free cancellation. Mathematics Subject Classification (2010). Primary 20C05;Secondary 16G99, 16K50.

Stably free modules over virtually free groups

“…In contrast Johnson [7] has shown that both Z[C p × F m ] and Z[D 2p × F m ] admit no non-free stably free modules when p is prime, C p is the cyclic group of order p and D 2p is the dihedral group of order 2p. Johnson [6] has also shown that k[G × F m ] admits no non-free stably free modules when k is any field and G is any finite group.…”

Rings of Polynomials With Artinian Coefficients

“…We showed in [5] that Ω has the SFC property provided each D i is commutative; that is, provided A is strongly Eichler. Thus from Proposition 3.1 we obtain the following.…”

“…However, in the case n = 1 one may show that L 1 (D) = D[t, t −1 ] has SFC regardless of whether the division ring D is commutative or not. Indeed, in that case, D[t, t −1 ] is projective free (see [4] or [5,Proposition 2.9]). The SFC property is now preserved under finite direct products and passage to matrix rings [6, pp.…”

“…We showed in [5] that Ω has property SF C provided each D i is commutative; that is, provided A is strongly Eichler. Thus from (3.1) we obtain: As we observed in the Introduction, Ojanguran and Sridharan proved in [8] that L n (D) fails the SF C property whenever n ≥ 2 and the division ring D is noncommutative.…”

“…However, in the case n = 1 one may show that L 1 (D) = D[t, t −1 ] has SF C regardless of whether the division ring D is commutative or not. Indeed, in that case, D[t, t −1 ] is projective free (cf [4] or [5] ) is weakly Euclidean and has property SF C. As both these properties are closed under finite direct products then A/rad(A) is weakly Euclidean and has property SF C. However, rad(A) is nilpotent so that, from (1.3) and (3.1), we conclude the following which should be well known but is difficult to locate explicitly in the literature.…”