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.

Baer Invariants and the Birkhoff-Witt Theorem, Journal of Algebra 11, pp. 469-482 (1969).) $\endgroup$ – darij grinberg Apr 17 '13 at 1:48