Notes on finitely generated flat modules

TL;DR

This paper investigates conditions under which finitely generated flat modules over commutative rings are projective, using topological methods to extend known results and identify new classes of rings with this property.

Contribution

It introduces topological criteria for projectivity of finitely generated flat modules, generalizing existing results in the literature.

Findings

Rings with finitely many minimal or maximal primes have all finitely generated flat modules projective.

Certain chain conditions on subsets of the prime spectrum imply projectivity of finitely generated flat modules.

Generalizes major existing results on projectivity of finitely generated flat modules.

Abstract

In this article, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either a finitely many minimal primes or a finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions then every finitely generated flat module over this ring is projective. These results generalize some major results in the literature on the projectivity of finitely generated flat modules.