Torsion free and projective modules

Abstract

Introduction. Serre has asked [5, p. 243] whether projective modules over the polynomial K[XI, * * *, X], K a field, are free, and Seshadri has proved that this is so when n =2. More precisely, he showed that every R[X]-projective module 'comes from one over R if R is either a principal ideal domain [6] or the coordinate of a nonsingular affine curve over an algebraically closed field [7]. By modifying Seshardri's argument in [6] we show here (Theorem 2.4) that R may be any Dedekind ring(2). Moreover we provide an example, due to S. Schanuel, which shows that some assumption like integral closure on R, is, in general, necessary for this conclusion. Problems of the above type can often be reduced to showing that a projective module is a direct sum of modules of rank 1, and this led us to ask for what integral domains R this is true for all (finitely generated) torsion free modules. Our main result (Theorem 1.7) gives an essentially complete solution to this problem. With mild assumptions on R, a necessary and sufficient condition is that every ideal in R can be generated by two elements. Thus these integral domains are very close to, but need not be, Dedekind rings. We assume a familiarity with the basic techniques of algebra as developed in Zariski and Samuel [9]. Homological algebra is appealed to in two discrete instances; one is a reference in the proof of Proposition 1.5, and the other is the proof of Lemma 1.9. The latter result, elementary as it is, seems to defy any proof which does not either use, or essentially reconstruct, the functor Ext'. Minor portions of this work were present in the author's doctoral dissertation. He gratefully records his indebtedness to Professor Kaplansky, who communicated a computation which suggested one of the essential ideas in the proof of Theorem 1.7. 1. Torsion free modules. Throughout this paper 'module means generated module, and ring means commutative ring. If M is anRmodule PuR(M) denotes the least number of elements required to generate M, and ,.t*(R) =sup PR (@) where 2t ranges over all finitely generated ideals of R. We call R local if it has a unique maximal ideal.