# Quasi-coherent sheaves, schemes, and the Gabriel-Rosenberg theorem

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.

• 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? Sep 9 '12 at 18:00
• This is more a question for MathOverflow.
– user18119
Sep 11 '12 at 12:13
• @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
• I see what you mean. I left a comment at the nForum: nforum.mathforge.org/discussion/1564/…
– Matt
Sep 11 '12 at 14:31

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:
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 ? ;)