Bourbaki's theorem and prime ideals

Abstract

If R is a normal domain and M a finitely generated torsion free module, then by Bourbaki's theorem [2, Sect. 9, Theorem 6] there is a free submodule F of M such that M/F is isomorphic to an ideal. We shall call such an ideal a Bourbaki ideal for M. Of particular interest is the case when this ideal may be chosen to be prime; if I 'MfF, then homological properties of M will be inherited by the domain Rfl, for instance via the isomorphisms Exti(M, R) ~ Exti(J, R) ' Exti+1(Rfl, R), for i ~ 2. In this situation, one can, by taking for M direct sums of suitable syzygies of given modules of finite length, produce a domain Rfl with a prescribed sequence of finite length local cohomology modules. A detailed statement and proof, as well as further applications, may be found in the paper of Evans and Griffith [7]. In this paper, we present sufficient conditions on the ring R that all finitely generated reflexive modules give rise to prime Bourbaki ideals. We will also present a partial converse, which demonstrates that, for the most part, these conditions are actually necessary. The proof, and especially the construction of the ideal in part I, follows that of [7], although it is somewhat more general, and avoids the use of Swan's theorem (concerning projectives over Laurent polynomial rings). The use of Bertini's theorem was first suggested in a preprint of [7]; the timely appearance of a particularly apropos version of this theorem by Flenner [8] enabled us to complete the proof along the lines suggested. This part of the proof actually demonstrates more; the prime ideals which are produced in fact are analytically irreducible, and even normal under certain conditions. We may moreover produce such ideals with arbitrarily many minimal generators. Moh has given an explicit sequence of primes in k[[ X, Y, Z]] having no bound on the number of minimal generators [I2], and Bresinsky has done the same for power series in four or more variables [3]. In each of