Lazard proved that every flat module over a ring is a directed colimit of finite free modules (Lam, Lectures on Modules and Rings, Theorem 4.34). I wonder if there is a similar theorem about flat modules on schemes.

**Question:** Let $X$ be a scheme which satisfies some finiteness conditions (at least quasicompact and quasiseparated, perhaps even noetherian and separated). Is then every flat quasi-coherent module $M$ on $X$ a directed colimit of locally free modules of finite rank?

This would be a nice characterization of flat modules since there is no allquantor. It would be very useful in the context of Tannaka Duality. Reading the proof of Lazard's theorem "backwards", it can be shown that the question is *equivalent* to:

Does every homomorphism from a quasi-coherent module of finite presentation to a flat quasi-coherent module factor through a locally free module of finite rank?

EDIT: The dissertation of Philipp Gross deals with the question when every quasi-coherent sheaf is a quotient of a locally free one. In Remark (3.5.7) it is remarked this would be true if every flat module is the directed colimit of locally free modules. But it's already hard to prove the resolution property for nice schemes ...

localdirected colimit of locally finitely presented modules. But it is a more interesting fact that every quasi-coherent module on a qs qc scheme is a directed colimit of locally finitely presented modules. The proof in (EGA I, 6.9.12) is tricky. $\endgroup$1more comment