# Projective modules over free groups

1. Consider the ring of Laurent polynomials $R := \mathbb{Z}[s,s^{-1}]$ with integer coefficients. Are all projective $R$-modules free? (Let's say left modules by convention.)

2. More generally, let $G$ be the free group on a finite set $S$ of generators, and consider the integral group ring $R := \mathbb{Z}G$. Are all projective $R$-modules free? Note that in part (1), $S$ was the singleton $\{ s \}$.

3. Is the answer the same if we allow an infinite set $S$ of generators? Note that the group $G$ itself is infinite either way.

I feel like work of Serre, Swan, Bass, Kaplansky, Suslin, Quillen, and others may be relevant, but I couldn't find the answers.

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The answer to your questions is 'yes' and, as you feel, it is a result of Bass:

MR0178032 (31 #2290) Bass, Hyman Projective modules over free groups are free. J. Algebra 1 1964 367–373.

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Thank you very much! That answers the case of finitely generated projective modules. What about non finitely generated projective modules? It looks like Bass's other paper: MR0143789 (26 #1341) Bass, Hyman Big projective modules are free. Illinois J. Math. 7 1963 24–31 could be useful, but I haven't thought through the details. Any insight? – Martin Frankland Jul 1 '11 at 7:07
Sorry for neglecting the infinitely generated case. Unfortunately, I don't think that second paper of Bass, helps... – Fernando Muro Jul 1 '11 at 8:40