The following statement is true* and not hard to prove.
Let $X$ be a quasi-compact and separated scheme. Then every quasi-coherent $\mathcal{O}_X$-module is a subquotient of a free $\mathcal{O}_X$-module.
Does it already appear in the literature somewhere? Is it well-known? It would already be helpful if some algebraic geometers could write a comment if they have already heard of it or not. My first impression was that this statement is quite surprising.
Out of curiosity, does the statement also hold if $X$ is just quasi-compact and quasi-separated?
*The statement comes out of the proof of Prop. 3.18 here (due to Gabber) and will be included in the next version if it is not well-known.
Edit (after 9 days without any comments): So I just assume that it is not well-known.
