Let A be a ring. I shall write p(A) for the category of finitely generated projective right A-modules and K0 (A) for its Grothendieck group. When A is an algebra over some commutative ring k let pk(A) denote the category of right A-modules M such that M ϵ P(k), the category of ‘k representations’ of A, and let Rk(A) denote its Grothendieck group.

I shall be mainly concerned with P(A) in the case when A = kG, the group algebra of a group G, and particularly the case when k = Z. This is a subject that barely exists except for some very special classes of groups G, notably finite groups and abelian groups. The following questions indicate the level of our ignorance.

1. Let G be a torsion free group.

1. (i) Is every P ϵ P(ZG) free?

No in general but there is essentially only one example known [D], Dunwoody's trefoil module. G = 〈x, y ∣x2 = y3 〉 is the trefoil group, P is a relation module arising from a presentation of G, and

P ⊕ ZG ≅ ZG ⊕ ZG.

1. (ii) Is K0(ZG) ≅ Z?

No counterexamples are known.

I mention in passing the following classical problem, which turns out to be related to the above questions in certain cases.

1. (iii) Is ZG without non trivial 0-divisors.

1. (iv) […]