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

Let (R,m,K)(R,m,K) be a regular local ring containing a field k   such that either char k=0k=0 or char k=pk=p and tr-deg K/Fp⩾1K/Fp⩾1. Let g1,…,gtg1,…,gt be regular parameters of R   which are linearly independent modulo m2m2. Let A=Rg1⋯gt[Y1,…,Ym,f1(l1)−1,…,fn(ln)−1]A=Rg1⋯gt[Y1,…,Ym,f1(l1)−1,…,fn(ln)−1], where fi(T)∈k[T]fi(T)∈k[T] and li=ai1Y1+⋯+aimYmli=ai1Y1+⋯+aimYm with (ai1,…,aim)∈km−(0)(ai1,…,aim)∈km−(0). Then every projective A-module of rank ⩾t   is free. Laurent polynomial case fi(li)=Yifi(li)=Yi of this result is due to Popescu.