Let $X,Y \rightarrow S$ be schemes over an algebraically closed field $k$. (Actually I'm interested in the case where they are stacks, but I'll ignore that for now.) The vague form of my question is: If I know the etale cohomology $H^*(X), H^*(Y), H^*(S)$, how much do I know about $H^*(X \times_S Y)$?

A more precise question: in singular cohomology, there is an Eilenberg-Moore spectral sequence $\mathrm{Tor}_{H^*(S)}(H^*(X), H^*(Y)) \implies H^*(X \times_S Y)$, at least under some conditions. Is there any analogue of this theorem in etale cohomology?

Remark: I could try to embed my problem into topology via etale homotopy type. I would then need to know what conditions on the algebraic side are necessary to guarantee that the topological requirements are satisfied.



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