On finitely generated flat modules

Abstract

Introduction. The aim of this paper is to study conditions which reflect the projectivity of a given finitely generated flat module over a commutative ring. The use of the invariant factors of a module (see below for definition) are very appropriate here: By translating the description by Bourbaki [4] of finitely generated projective modules, one can state that projectivity=flatness+finitely generated invariant factors. Since the invariant factors of a flat module are very peculiar (locally they are either (1) or (0)), the presence of almost any other condition on the module precipitates their finite generation. For instance, consider the following two statements. Let M be a finitely generated flat moGule over the commutative ring R: (i) Let S be a multiplicative set in R consisting of nonzero divisors such that MS (localization of M with respect to S) is Rs-projective; then M is projective. (ii) Let J be the Jacobson radical of R and assume that MIJM is R/J-projective; then M is projective. The first is a result by Endo [8] who uses homological algebra in the proof, while the second can be viewed as a generalization of the well-known fact that over a local ring a finitely generated flat module is free. Next we apply these ideas for a look at finitely generated flat ideals. Even though they are not always projective (see below for an example of a principal flat ideal which is not projective) it can be shown this to be the case when the flat ideal is a finite intersection of primary ideals. The criterion mentioned above says that for a finitely generated flat ideal, projectivity is the same as having finitely generated annihilator. For rings with the weakened form of coherency that finitely generated ideals have finitely generated annihilators, one can even show that any finitely generated flat submodule of a projective module is projective. When used to study the prime ideals of a ring R of weak dimension one we arrive at the fact that a finitely generated prime ideal is either maximal or generated by an idempotent. It is also shown that if every principal ideal is projective then R is semihereditary; if moreover every cyclic flat module is projective, then R is a direct sum of finitely many Priufer domains.