The well-known theorem of Quillen-Suslin says that a finitely generated projective module over $k[x_1,\ldots,x_n]$ is free, See https://mathoverflow.net/questions/19584/what-is-the-insight-of-quillens-proof-that-all-projective-modules-over-a-polyno
What can be said about a finitely generated projective module over a simple algebraic ring extension of $k[x_1,\ldots,x_n]$? Namely, let $w$ be algebraic over $k[x_1,\ldots,x_n]$. What can be said about a finitely generated projective module over $k[x_1,\ldots,x_n][w]$?
I do not know if this question is difficult or trivial (=for example, if there is a known result which says what is the connection between finitely generated projective modules over a ring $R$ and finitely generated projective modules over a simple algebraic ring extension of $R$).
In general there is little you can say. For example, they could be regular, but not a UFD, giving non-trivial projective modules of rank one.