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

Solution 1:

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.