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Let $f\colon X \rightarrow Y$ be a continuous map of sufficiently nice topological spaces (say, smooth manifolds). Let ${\cal F}=(\dots\rightarrow F_i \rightarrow F_{i+1}\to \dots)$ be a bounded complex of sheaves of vector spaces over a given field on $X$. It is well known that there exists a spectral sequence $E_{r}^{p,q}(\mathcal{F})$ converging to the cohomology sheaves of the push-forward $R^\bullet f_*({\cal F})$ (in the derived category) such that the first term of it is equal to $$E_1^{p,q}({\cal F})=R^qf_*(F^p).$$

Now let us have one more complex of sheaves of vector spaces ${\cal G}$ on $X$. The claim I am interested in is the following one: There exists a canonical morphism of the tensor product of spectral sequences of ${\cal F}$ and ${\cal G}$ to the spectral sequence of ${\cal F}\otimes {\cal G}$.

After I talked to few experts in homological algebra and algebraic geometry and made some google search, I got an impression that this fact is a folklore and basically well known to experts. I would need a reference to this fact.

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@Mark Grant: In fact I need to use this fact in a paper, potential readers of which will not be experts in homological algebra or algebraic geometry, but analysts and possibly differential geometers. In these communities the above fact is not a common knowledge. So some background reference would be helpful. – sva Aug 10 '11 at 9:25
It is certain true in general! As for references, the only thing I can suggest at the moment is Voisin's "Hodge theory and complex...II" pp 113-115. This is worked for $\mathbb{Z}$ coefficients, but it's better than nothing. (By the way, it is an $E_2$ spectral sequence.) – Donu Arapura Aug 10 '11 at 11:24
Also have you checked out Bredon's "Sheaf theory", section IV.6.5? – Mark Grant Aug 10 '11 at 14:19
@Mark Grant: Bredon in section IV.6.5 deals with the Leray spectral sequence. This is not what I mentioned. I need one of the so called hypercohomology spectral sequences, see wikipedia – sva Aug 11 '11 at 8:45
Is this fundamentally a different question from the convergence of a spectral sequence of algebras to a desired algebra? In re. your last remark, the hypercohomology s.s. is asserted on p.515 of "A User's Guide to S.S." By McCleary; two methods of proof are suggested --- composite functor s.s. OR a double complex of injective objects whose total complex has some desired homology. – some guy on the street Aug 11 '11 at 17:44

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