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

M. K. Keshari; Swapnil A. Lokhande

paper

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

arXiv:1408.2619

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 $g_{1}, \dots, g_{t}$ be regular parameters of $R$ which are linearly independent modulo $\mm^2$. Let $A = R_{g_{1} \dots g_{t}} [Y_{1}, \dots, Y_{m}, f_{1} (l_{1})^{- 1}, \dots, f_{n} (l_{n})^{- 1}]$ , where $f_{i} (T) \in k [T]$ and $l_{i} = a_{i 1} Y_{1} + \dots + a_{im} Y_{m}$ with $(a_{i 1}, \dots, a_{im}) \in k^{m} - (0)$ . 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.