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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