Is every finitely generated projective $\mathbf{Z}[x]$module free?

While it's certainly true (per Fernando's comment) that this is a special case of the QuillenSuslin theorem, it was certainly known long before Quillen and Suslin came along. There's a paper of Murthy from the mid1960s which shows that every projective $R[x]$module is extended whenever $R$ is a regular ring of dimension at most 2. ("Extended" here means "of the form $P[x]$ where $P$ is a projective $R$module". Since all projective ${\mathbb Z}$modules are free, extended is equivalent to free in this case.) But there's an even earlier paper of Bass which covers the case where $R$ is regular of dimension 1, which is all you need. The paper is called "Torsion Free and Projective Modules". Edited to add: And the case of a PID predates even Bass; I think it's due to Seshadri in the 1950s. 


When $R$ is a PID, then every finitely generated projective $R[x]$module is free. As Steven already said, this is due to Seshadri. Here is the reference:


