# Finitely generated projective modules over a simple algebraic ring extension of a polynomial ring

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. – Mohan Jun 24 '15 at 23:44
• Yes. For example, consider $k[x]\subset k[x,y]$ where $y^2=x^3+1$. Then $k[x,y]$ is regular, but not a UFD. – Mohan Jun 25 '15 at 0:02
• But I wish to consider $k[x,y] \subseteq k[x,y][w]$ – user237522 Jun 25 '15 at 0:04
• Sorry, I do not understand your example; $k[x,y]$ IS a UFD – user237522 Jun 25 '15 at 0:06
• Add a variable! For example $k[x,y]\subset k{x,y][w]$ with $w^2=x^3+1$. – Mohan Jun 25 '15 at 0:07

## 1 Answer

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.