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Let $X, Y$ be Noetherian perfect derived stacks over $S$ a regular perfect derived stack. Consider the exterior tensor functor $$\text{DCoh}(X) \otimes_{\text{DCoh}(S)} \text{DCoh}(Y) \rightarrow \text{DCoh}(X \times_S Y)$$ This map is, in general, fully faithful. When is it also essentially surjective (and thus an equivalence)? There are some affirmative answers: in the appendix to Preygel's paper, it's shown that this is an equivalence when:

  • $X$ and $Y$ are almost finitely-presented Deligne-Mumford stacks with finite diagonal, and $S = \text{Spec}(k)$ a perfect field
  • $Y \rightarrow S$ is a smooth relative scheme, and $X$ is an excellent derived stack over $S$
  • this is not in the paper, but it is an easy case: when $X$ and $Y$ are both smooth over $S$, then then $\text{DCoh} = \text{Perf}$ and the result follows from work by Ben-Zvi/Francis/Nadler (for example, if $X = Y = BG$ and $G$ is affine, this says that every $G \times G$-representation is a sum of exterior tensors of $G$-representations).

I'm particularly interested in the case when we don't require either $X$ or $Y$ to be Deligne-Mumford over $S$, but more generally, allow them to be Artin stacks, and also when $X = Y$. Even more concretely, say $X = Y = Z/G$ where $Z$ is not smooth, and $S = \text{Spec}(k)$ is a characteristic zero algebraically closed. Are there known counterexamples?

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  • $\begingroup$ I guess that in your third $\bullet$ you refer to Theorem 1.2 in arxiv.org/abs/0805.0157. $\endgroup$
    – HeinrichD
    Commented Oct 10, 2016 at 20:46
  • $\begingroup$ @HeinrichD Yes, that's the one. The referenced theorems in the Preygel paper are B.3.2. and B.4.1 $\endgroup$
    – math no more
    Commented Oct 10, 2016 at 20:47
  • $\begingroup$ This sounds strange --- I would say that one should allow taking direct summands if one wants to obtain any object on the product in terms of decomposable objects. $\endgroup$
    – Sasha
    Commented Oct 10, 2016 at 20:56
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    $\begingroup$ I think you'll find answers to your questions here: math.harvard.edu/~gaitsgde/GL/Finiteness.pdf for example Cor. 4.2.3, Prop. 3.5.1 etc $\endgroup$
    – Daniel Pomerleano
    Commented Oct 10, 2016 at 21:39
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    $\begingroup$ If you're interested in functor categories rather than tensor products, you'll find more in arxiv.org/abs/1312.7164 (here it makes a big difference here if you're asking about small categories DCoh or big categories IndCoh -- the former are not dualizable so the tensor product and functor category calculations are distinct). $\endgroup$
    – David Ben-Zvi
    Commented Oct 11, 2016 at 4:39

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