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In the context of commutative rings, a ring is completely determined by its category of modules. That is, two commutative rings $R$ and $S$ are isomorphic if and only if the category of $R$-modules is equivalent to the category of $S$-modules. In particular, we have the following result about affine schemes:

If $X=(X,\mathcal O_X)$ is a scheme, let $QCoh(X)$ denote the category of quasi-coherent $\mathcal O_X$-modules on $X$. Then, two affine schemes $X$ and $Y$ are isomorphic if and only if $QCoh(X)$ is equivalent to $QCoh(Y)$.

(This follows from the fact that if $X=Spec(R)$ for a commutative ring $R$, then $QCoh(X)$ is equivalent to the category of $R$-modules.) My question is the following:

For a general scheme $X$, to what extent does $QCoh(X)$ determine $X$?

Added: As t.b. noted below in the comments, the Gabriel-Rosenberg reconstruction theorem answers the question, at least in the quasi-compact, quasi-connected case, which is the first case proven by Gabriel. But the nLab page is not very clear about the further generalizations. In particular, I would like to know in how much generality it holds, and the uses of the quasi-compactness hypothesis.

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There's the Gabriel-Rosenberg reconstruction theorem although it is not quite clear to me to what extent this answers your question. –  t.b. Sep 8 '12 at 17:10
    
@t.b. Thank you for the link. It says that is $X$ is quasi-connected and quasi-compact, then $X$ can be reconstructed as the geometric center of the category. Is it still true in general? –  M Turgeon Sep 9 '12 at 18:00
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This is more a question for MathOverflow. –  user18119 Sep 11 '12 at 12:13
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@Matt: The bounty is mine, not M Turgeon's. I do think the reconstruction theorem covers the interesting cases, if indeed it holds for quasi-separated and quasi-compact schemes, but I must say that I find the presentation on the nlab page rather confusing as to what is actually true. I was hoping for clarifications on that. I'm fully aware that it might just be my ignorance... –  t.b. Sep 11 '12 at 13:30
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I see what you mean. I left a comment at the nForum: nforum.mathforge.org/discussion/1564/… –  Matt Sep 11 '12 at 14:31
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up vote 6 down vote accepted

If $X$ and $Y$ are quasi-separated schemes such that $\mathsf{Qcoh}(X)$ and $\mathsf{Qcoh}(Y)$ are equivalent, then $X$ and $Y$ are isomorphic. This is (claimed to be) proven in the paper:

A. Rosenberg, Spectra of 'spaces' represented by abelian categories, MPI Preprints Series, 2004 (115).

A few years ago I've studied this paper in detail and have come to conclusion that it is has several serious errors. But Gabber has told me how to correct the proof. See http://arxiv.org/abs/1310.5978 for a write-up.

I am pretty sure that the general case (without quasi-separated hypothesis) is open. Even the most simple part of the proof, namely that the canonical homomorphism $\Gamma(X,\mathcal{O}_X) \to Z(\mathsf{Qcoh}(X))$ is an isomorphism, seems to be open for general schemes. But, to be honest, who cares about schemes which are not quasi-separated ? ;)

See here for what happens when the monoidal structure is preserved.

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Thank you for the answer –  M Turgeon Feb 5 '13 at 20:14
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I've added my arXiv paper for Gabber's proof of Rosenberg reconstruction. –  Martin Brandenburg May 18 at 9:00
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