How to characterize Abelian sheaves that are quasi-coherent? Let $X$ be a scheme. Suppose you are given a sheaf of Abelian groups $\mathcal{A}$ over $X$. How can you determine if $\mathcal{A}$ is the underlying Abelian sheaf of a sheaf of $O_X$-modules? In other words, is it possible to characterize in some (interesting) way the essential image of the forgetful functor from $Mod(O_X)$ to $Ab(X)$?
I'm not sure a non-tautological answer exists...
 A: 1) There is a very simple example that shows that it is impossible to answer the question of whether $\mathcal{A}$ comes from a quasi-coherent sheaf $\mathcal{F}$ on $X$ if all one is given is the underlying topological space $|X|$ and $\mathcal{A}$ as a sheaf on $|X|$.  Namely, if $|X|$ is a point and $\mathcal{A}$ is such that $\mathcal{A}(|X|)=\mathbf{Q}$, then either outcome is possible: the answer is YES if $X=\operatorname{Spec} \mathbf{Q}$, but NO if $X=\operatorname{Spec} \mathbf{F}_p$.
2) There are some nontrivial necessary conditions that one can state in terms of the topological space and the sheaf of abelian groups alone.  For example, in order for $\mathcal{A}$ to come from a quasi-coherent sheaf, there must exist an open covering $(U_i)$ of $|X|$ such that the sheaf $\mathcal{A}|_{U_i}$ on $U_i$ is acyclic for every $i$.  
3) The condition in 2) is definitely not sufficient, even if the scheme structure on $|X|$ is not specified in advance.  For instance the constant sheaf $\mathbf{Z}/6\mathbf{Z}$ on a point is acyclic, but it cannot be a quasi-coherent sheaf for any scheme structure on the point.
