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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.

M Turgeon
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    There's the [Gabriel-Rosenberg reconstruction theorem](http://ncatlab.org/nlab/show/Gabriel-Rosenberg+theorem) although it is not quite clear to me to what extent this answers your question. – t.b. Sep 08 '12 at 17:10
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    @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 09 '12 at 18:00
  • Sorry, I don't know. I have no idea what happens beyond the quasi-separated and quasi-compact case and I'm not competent in this business at all. It would be interesting to have an informed assessment of the status of the reconstruction theorems, as there seems to be a history of unpublished notes, bug fixes, folklore examples etc. There are other versions of reconstruction theorems that throw in the tensor product as an additional datum, see e.g. publications 26 and 30 of [Paul Balmer's publication page](http://www.math.ucla.edu/~balmer/research/publications.html). – t.b. Sep 09 '12 at 21:54
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    This is more a question for MathOverflow. –  Sep 11 '12 at 12:13
  • No offense, but I wrote this off as completely answered when t.b. mentioned the reconstruction theorem. Do you really work with non-quasi-compact schemes? – Matt Sep 11 '12 at 13:00
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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
  • Thank you @t.b. for putting a bounty on this question. I too would like a clarification about the nLab's page. I will edit the question to reflect this. – M Turgeon Sep 11 '12 at 13:42
  • Very good, thanks for the edit! – t.b. Sep 11 '12 at 13:59
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    I see what you mean. I left a comment at the nForum: http://nforum.mathforge.org/discussion/1564/spectrum-of-an-abelian-category/#Comment_34548 – Matt Sep 11 '12 at 14:31
  • @MTurgeon, in the reference given by zskoda (link given by @Matt), look at 8.3(ii). It says something like there is a counterexample when $X$ is not quasi-compact. –  Sep 11 '12 at 21:13
  • @QiL I had a look at the link, but I am not sure this answers the question. As is alluded to on the nLab page, the "counterexample" simply choses that in order to have a theorem that works for all schemes, we have to change the definition of the spectrum. And this is the point on which I would like clarification/information. – M Turgeon Sep 11 '12 at 22:37
  • Thank you @t.b. for the bounty, even though it didn't work. I think I will move this question to MO. – M Turgeon Sep 20 '12 at 02:39

1 Answers1

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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.

Martin Brandenburg
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