Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer 1

The following might be helpful:

Theorem 8.9 of Toën's http://arxiv.org/abs/math/0408337

Theorem 1.2 (as well as Theorem 4.7 and Corollary 4.10) of Ben-Zvi, Nadler, Francis's http://arxiv.org/abs/0805.0157

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.

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.