TITLE: Is every projective $\mathbf{Z}[x]$-module free?
QUESTION [14 upvotes]: Is every finitely generated projective $\mathbf{Z}[x]$-module free?

REPLY [12 votes]: 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:

Seshadri, C.S., Triviality of vector bundles over the affine space $K^2$, Proc. Nat. Acad. Sci. USA 44 (1958), 456-458.