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Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ring $R$.

Is there a natural construction of such an $R$ using the properties of $S$?

Of course the proof of the theorem would give a category-theoretic construction of $R$ but in the case of schemes i'm wondering if there's a more algebro-geometric construction, maybe in terms of the affine "pieces" of $S$.

In the case of an affine scheme $Spec(A)$, then such an $R$ is obviously just $A$ (restricting to the full subcategory of finitely generated $A$-modules for coherent sheaves), but for more general schemes I find the question interesting. Possibly one would have to restrict to a particular class of well-behaved schemes which includes affine schemes.

Ideally it would be nice to be able to reconstruct $S$ from such a canonical $R$ generalizing how affine schemes are constructed from commutative rings. In this situation then the canonical $R$ for non-affine schemes would have to be non-commutative. But maybe this is not possible.

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    $\begingroup$ "In the case of an affine scheme $Spec(A)$, then $R$ is obviously just $A$" : only if you restrict to the subcategory of quasi-coherent sheaves on $Spec(A)$. $\endgroup$
    – abx
    Jul 7, 2014 at 19:16
  • $\begingroup$ @abx you're right, should I edit the question? $\endgroup$
    – Cristos A. Ruiz
    Jul 7, 2014 at 19:26
  • $\begingroup$ Yes, this is always better. $\endgroup$
    – abx
    Jul 7, 2014 at 19:35
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    $\begingroup$ If $S$ is quasicompact, you can use Zariski descent from a cover by finitely many affine opens. $\endgroup$
    – S. Carnahan ♦
    Jul 8, 2014 at 0:50
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    $\begingroup$ I don't have an answer but I'm skeptical about obtaining "canonically" such an R. But I would like to see myself disproved - since QCoh(X) recovers X, such a construction of R might open the door to other cute results. $\endgroup$
    – bananastack
    Jul 8, 2014 at 4:15

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