Cancellation over commutative rings of dimension one and two

Abstract

All rings in this paper are commutative and Noetherian, and all modules are finitely generated. We will study the question: If A @ C z B @ C, is A z B? (Here A, B and C are modules over a ring R.) In 1962 Chase [4] gave an affirmative answer for R = k[X, Y], the polynomial ring in two variables over an algebraically closed field k, provided A and B are torsionfree and char(k)trank(A). In the first section of this paper we extend Chase’s result to include all two-dimensional regular affine k-domains. Most of the paper deals with one-dimensional rings. In Section 2 we prove some general results, which we hope will eventally lead to some sort of structure theory for torsionfree modules over one-dimensional reduced rings with finite normalization. In this context we show that one can cancel from projectives (that is, the answer is “yes” when A and B are projective) if and only if Pit R = Pit z, where g is the normalization. When A and B are assumed to be merely torsionfree, the problem is more subtle, but we have some partial results. The last three sections deal with three naturally occurring examples of one-dimensional rings: coordinate rings of curves, quadratic orders and integral group rings of finite abelian groups. In each of these cases we are able to give a fairly complete answer to the cancellation problem for torsionfree modules. Many of the ideas in this paper are the direct or indirect results of innumerable conversations with Raymond Heitmann and Lawrence Levy. In particular, Levy discovered a serious error in the proof of the main theorem of an earlier version, and Heitmann showed by example that the statement itself was incorrect. I am extremely grateful to both of them for their interest and insight.