Stably Free Modules

Abstract

In [Su. Prob. 3], Suslin had asked the following question: Let A be any affine algebra of dimension n over an algebraically closed field. What is the smallest integer m such that all stably free projective modules of rank bigger than m are free? All the examples in the literature of stably free non-free modules have rank less than or equal to (n 1)/2. The aim of this note is to construct examples of such modules of large rank. We construct rank p stably free non-free modules over (p + 2)-dimensional affine algebras over algebraically closed fields, wherep is any prime. These varieties are actually smooth and rational. Over C, these are trivial as holomorphic vector bundles. [Forp > 2, this is classical. Forp = 2, see [MS]]. So these are strictly algebraic examples. I had described this construction in [MK 1] and proved the result for p = 2. We will reproduce the construction with necessary modifications in this note. Let p be any prime number and k any field. Letf (x) be any polynomial of degree p over k. Letf (0) = a E k* and Fi (xo, xl) = F(xo,x1) = xP *f (x0 /x1 ). Also let