On a Question of Quillen

S. M. Bhatwadekar, R. A. Rao
1983 Transactions of the American Mathematical Society  
Let R be a regular local ring, and / a regular parameter of R. Quillen asked whether every projective R /-module is free. We settle this question when R is a regular local ring of an affine algebra over a field k. Further, if R has infinite residue field, we show that projective modules over Laurent polynomial extensions of Rf are also free. Introduction. In [Q] Quillen posed the following Question. Let R be a regular local ring and f a regular parameter of R. Are all finitely generated
more » ... y generated projective R¡-modules freel An affirmative answer implies the Conjecture (Bass-Quillen). Let R be a regular local ring. Then every finitely generated projective R[T]-module is free. Lindel [L, Theorem] has proved the Bass-Quillen conjecture when R is the local ring of an affine algebra over a field k at a regular point (not necessarily closed). However, it is not clear whether a positive solution to the Bass-Quillen conjecture implies the truth of Quillen's question. Therefore the latter is, apart from its application to the Bass-Quillen conjecture, of some independent interest. In this paper we settle the Quillen question affirmatively when R is a regular local ring of an affine algebra over a field k. Curiously, in this case we are able to reduce the Quillen question to the Bass-Quillen conjecture via the following interesting result (see Theorem 2.4).
doi:10.2307/1999568 fatcat:5bvjk2dpdfa6bfxyqcvqxrzp4e