Let $R$ be a ring and $X,Y$ two $R$-schemes, which you may assume to be noetherian or anything reasonable you like. Is it possible to "construct" $\text{Qcoh}(X \times\_R Y)$ out of $\text{Qcoh}(X)$ and $\text{Qcoh}(Y)$ in the $2$-category of all cocomplete $R$-linear tensor categories?


Perhaps it is the $2$-coproduct? So the question is if for every cocomplete $R$-linear tensor category $C$ the canonical functor


$\text{Hom}(\text{Qcoh}(X \times\_R Y),C) \to \text{Hom}(\text{Qcoh}(X),C) \times \text{Hom}(\text{Qcoh}(Y),C)$


$F \mapsto (F \circ (p\_X)^\*, F \circ (p\_Y)^\*)$


is an equivalence of categories. This is satisfied if $X,Y$ are affine, but I think also when $X,Y$ are projective over $R$ (EDIT: Yes, now I've proved this in detail, should I write it up?). Actually for my purposes it would be enough to prove that the functor is conservative, i.e. reflects isomorphisms.


[Here](https://mathoverflow.net/questions/35309/categorical-kunneth-theorem-for-module-categories) was a similar question on MO, but it adresses (as with the answer by David Ben-Zvi) only the derived setting, but I want to work with the usual category of quasi-coherent modules.




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The question is answered affirmatively here: <https://arxiv.org/abs/2002.00383>



Description of quasi-coherent modules on a product
More generally, I have proven that for quasi-compact and quasi-separated schemes $\mathrm{Qcoh}(X \times\_S Y)$ is the bicategorical pushout of $\mathrm{Qcoh}(X)$ and $\mathrm{Qcoh}(Y)$ over $\mathrm{Qcoh}(S)$ in the bicategory of cocomplete linear tensor categories. The technique of the proof has many other applications as well.


[Localizations of tensor categories and fiber products of schemes](https://arxiv.org/abs/2002.00383) (arXiv:2002.00383)


