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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...

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Did you mean that you need a definition other than the one on Hartshorne? What do you mean by being non-tautological? – 7-adic Mar 25 '10 at 19:41
Note that all curves (over a fixed algebraically closed field say) are homemorophic in the Zariski topology. Hence you can take a quasi-coherent sheaf on one curve and then transfer it by a homeomorphism to another curve where it very rarely will come from a quasi-coherent sheaf. This makes it very unlikely that there is any kind of reasonable description of the essential image (other than the tautological one). – Torsten Ekedahl Mar 25 '10 at 20:41
Torsten- Are you sure that the isomorphism type of (say) the structure sheaf of a curve depends on the complex structure of the curve as a sheaf of a abelian groups? I'm not convinced either way. – Ben Webster Mar 25 '10 at 21:33
@Ben: Good question, it made me come up with an answer. On affine curves (and on $\mathbb P^1$) the first cohomology cohomology group of the structure sheaf is zero but on other curves it isn't (if one considers $k$-sheaves instead of sheaves of abelian groups one can even recover the genus of a smooth and proper curve). – Torsten Ekedahl Mar 29 '10 at 13:19
up vote 13 down vote accepted

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.

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These minimalist examples are real mathematical haikus: thanks, Bjorn. – Georges Elencwajg Mar 26 '10 at 10:02

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