Question

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.

Thank you for your help.

Answer 1

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.