On the Homological Dimension of Certain Ideals

1. Introduction. We present two closely related results connecting homological dimension theory and the ideal theory of noetherian rings. The first, Proposition 4.1, asserts that the only ideals of finite homological dimension in a local ring whose associated prime ideals all have grade one are of the form aR:bR. The second, Proposition 4.3, asserts that if R is a noetherian integral domain, then all such ideals are even projective. The discerning reader will immediately observe that the latter proposition suffices to imply unique factorization in regular local rings. Indeed, our results are to some extent an elaboration of an unpublished proof of unique factorization in regular local rings by Kaplansky. The reader will note that the devices involved in the proof are different from those used by Auslander, Buchs-baum and Nagata in their classic proof of unique factorization as given, for example, in [4]. Of the two propositions outlined above, the assumption in the second that the ring is a domain is to be considered a defect. If we could drop this hypothesis, the first proposition would be superfluous. Indeed, we know of devices whereby it may be dropped for rings of codimension at most three. For larger codimensions, however, the question is quite open. By way of extension of the above pair of propositions, M. Auslander has shown and will publish elsewhere a proof of the fact that if R is a noetherian integrally closed domain and A is a finitely generated, reflexive (A = Horn (Horn (A, R), R)) i?-module of finite homological dimension and whose endomorphism ring is projective, then A is itself projective.