Projective modules over overrings of polynomial rings and a question of Quillen

TL;DR

This paper proves that projective modules of rank at least t over certain overrings of polynomial rings are free, extending known results to a broader class of rings under specific regularity and field conditions.

Contribution

The paper generalizes the freeness of projective modules over overrings of polynomial rings, addressing a question posed by Quillen and extending Popescu's Laurent polynomial case.

Findings

Projective modules of rank ≥ t are free over the specified rings.

The result applies to rings with regular parameters linearly independent modulo b1^2.

Extends known cases to more general overrings involving inverses of polynomials.

Abstract

Let $(R,\mm,K)$ be a regular local ring containing a field $k$ such that either char $k=0$ or char $k=p$ and tr-deg $K/\BF_p\geq 1$. Let $g_1,\ldots,g_t$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let $A=R_{g_1\cdots g_t} [Y_1,\ldots,Y_m,f_1(l_1)^{-1},\ldots, f_n(l_n)^{-1}]$, where $f_i(T)\in k[T]$ and $l_i=a_{i1}Y_1+\ldots+a_{im}Y_m$ with $(a_{i1},\ldots,a_{im})\in k^m-(0)$. Then every projective $A$-module of rank $\geq t$ is free. Laurent polynomial case $f_i(l_i)=Y_i$ of this result is due to Popescu.