On a question of Quillen

Abstract

Let R R be a regular local ring, and f f a regular parameter of R R . Quillen asked whether every projective R f {R_f} -module is free. We settle this question when R R is a regular local ring of an affine algebra over a field k k . Further, if R R has infinite residue field, we show that projective modules over Laurent polynomial extensions of R f {R_f} are also free.