I asked this question on SE a long time ago, but never received an answer:
The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely generated free (left) modules.
The theorem is contained in many textbooks like Eisenbud (Commutative Algebra) or Rotman (Introduction to Homological Algebra). However, no applications are given there.
Question: Are there interesting applications of the Govorov-Lazard Theorem ?
N.b.: The only application I've seen so far, was in a question on SE, where someone remarked that if $A,B$ are commutative $R$-algebras and $M$ is a flat $A$-module and $N$ a flat $B$-module, then it follows from Govorov-Lazard that $M\otimes_R N$ is a flat $A\otimes_R B$-module. But, of course, this follows more easily from standard properties of the tensor product.
Best Answer
One application which I find particularly beautiful is the following:
This appeared first in
Jensen, 1966, On homological dimensions of rings with countably generated ideals
It has been used more recently by my advisor, Mark Hovey, in his paper "On Freyd's Generating Hypothesis" because the ring of stable homotopy groups of spheres is countable, and many other objects of interest have homotopy which is flat over it.
It would appear that this observation of Jensen is also related to the projectivity criterion of Raynaud and Gruson. First, observe that
Using this, one gets the projectivity criterion for countably presented flat left $R$-modules, which states that $M$ is projective iff whenever $M$ is the direct limit of finitely generated free left $R$-modules $M_n$ then the inverse system (Hom$_R(M_n,R))_n$ satisfies the Mittag-Leffler condition.
A corollary of the proof is that if $R$ is countable then a left module $M$ is countably presented iff it's countably generated