Flat quasi-coherent modules as colimits of locally free ones 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 ... 
 A: In the paper "A Lazard-like theorem for quasi-coherent sheaves" by Sergio Estrada, Pedro A. Guil Asensio and Sinem Odabasi (arXiv), the following Theorem is proven:

Let $X$ be a quasi-compact and semi-separated scheme having enough locally countably generated vector bundles (for instance if $X$ is noetherian, separated, integral and locally factorial). Let $F$ be a flat quasi-coherent sheaf on $X$. Then $F = \lim\limits_{\longrightarrow} F_i$, where $F_i$ is locally countably generated and flat with $\mathcal{V} \dim F_i \leq 1$ (where $\mathcal{V}$ is the class of all vector bundles on $X$).

This is already a strong result, going into the direction of my question.
A: This is a local property. [if $M$ comes from an algebra over $A$]. It reduces to consider the case that $X=\text{Spec}(A)$ is an affine scheme.
If the ring $A$ is a PID, I think the statement is true (although I did not give a proof indeed).
For an arbitary ring $A$, the statement may be false. Please check the remark, Page 22, in: Commutative Algebra, 2ed, by Matsumura, 1980.  
