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What is the relationship between the category of modules over a product $X\times Y$ and the pair of module categories over $X$ and $Y$ separately? Ideally here $X$ and $Y$ are smooth projective varieties and we are considering the DG category of complexes of $\mathcal{O}_X$-modules (with possibly some coherence condition if needed).

A more specific question which I am interested in is: say you have two sheaves of DG commutative algebras X, Y, both modules over the sheaf of DG commutative algebras Z (all sheaves over the same space), then how is the category of DG modules of the pushout of X and Y along Z related to the module categories for X,Y.

It seems that in Section 17 of this paper by Frenkel and Gaitsgory, there is a result to the effect that $DGQCoh(X\times Y)$ is a categorical tensor product of the categories $DGQCoh(X)$ and $DGCoh(Y)$, but I do not understand it, so if it does indeed answer my question, I would appreciate some remarks which may clarify exactly what is proved there.

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The following might be helpful:

Theorem 8.9 of Toën's

Theorem 1.2 (as well as Theorem 4.7 and Corollary 4.10) of Ben-Zvi, Nadler, Francis's

Remark: I don't yet fully understand their proofs, and I don't have any good intuition as to why these types of theorems are true, but the fact that such theorems are indeed true does seem to suggest that the $\infty$-categorical (or dg categorical) point of view is a "correct" point of view.

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