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Let $X$ be a scheme, $K^{\bullet}$ and $P^{\bullet}$ bounded complexes of abelian sheaves on $X_{\rm ét}$.

I want to compute the hypercohomology:

$$\mathbb{H}^*(X_{\rm ét}, K^{\bullet}\otimes^L_{\mathbf{Z}}P^{\bullet})$$

Is there a spectral sequence relating this to the étale hypercohomologies of $K^{\bullet}$ and $P^{\bullet}$, respectively?

Any references?

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I think that this situation is not quite suitable for spectral sequences. Having such a spectral sequence would be related to some compatibility of (derived) global sections and tensor product. In general there is a map $R\Gamma(X,A) \otimes^L R\Gamma(X,B) \to R\Gamma(X, A \otimes^L B)$ but I don't think that it is even close to be an isomorphism in general. Think for example on representations of $\mathbb{Z}/2$ in $\mathbb{F}_2$ vector spaces (You can think of it as a sheaf of modules for a sheaf of rings over $spec(\mathbb{R})_{et}$ is you like) and the modules $K = P = \mathbb{F}_2$. Then the tensor product of the global sections in the derived sense and the sections of the tensor product are completely different.

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  • $\begingroup$ Isn't that the point of a spectral sequence? It doesn't have to collapse at the $E_2$-page, right? $\endgroup$ – Sean Tilson Mar 2 '18 at 9:02
  • $\begingroup$ Well, usually isomorphisms in derived category correspond to some identities on spectral sequences, not necessarily ones that collapse at the $E^2$-page. Think for example on the case of composition of functors. It gives you a spectral sequence $R^iF R^jG --> R^{i+j}(F\circ G)$ under some conditions, but this spectral sequence usually does not converge at the $E^2$ even though it correspond to the equivalence $R(F \circ G) = RF \circ RG$ in the category of functors between derived category derived category $\endgroup$ – S. carmeli Mar 4 '18 at 6:28
  • $\begingroup$ It looked to me like you were talking about the edge map of a Kunneth spectral sequence which is why I am/was confused. What do you mean by converge at the $E^2$-page? $\endgroup$ – Sean Tilson Mar 13 '18 at 12:57

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