Leray spectral sequence and pullbacks

Question

I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:

Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^{p,q}(\mathcal{F})=\mathrm{H}^q(Y,R^q\pi_*\mathcal{F})\implies \mathrm{H}^{p+q}(X,\mathcal{F})$ and $(E')_2^{p,q}(\mathcal{G})\implies \mathrm{H}^{p+q}(X',\mathcal{G})$ their associated Leray spectral sequences (in étale cohomology for some étale sheafs $\mathcal{F},\mathcal{G}$). Assume that there are $f:X'\to X$ and $g:Y'\to Y$ such that the obvious square commutes. Then there is a morphism of spectral sequences $E_r^{p,q}(\mathcal{F})\to (E')^{p,q}_r(g^{-1}\mathcal{F})$ induced by the natural pullback maps on cohomology $E_2^{p,q}(\mathcal{F})\to (E')_2^{p,q}(g^{-1}\mathcal{F})$ such that the map on the infinity-pages agrees with the canonical pullback morphism $\mathrm{H}^n(X,\mathcal{F})\to \mathrm{H}^n(X',f^{-1}\mathcal{F})$.

It would be amazing if someone knows a reliable source for this frequently used result. Thanks in advance!

Answer 1

I apologize for the self promotion, but page 570 of my article The Leray spectral sequence is motivic has a very brief discussion of the functoriality of Leray.

Added In a bit more detail, here are the key points:

• To every object $(C, F)$ in the (bounded below, biregularly filtered) filtered derived category, one has a spectral sequence with $E_1^{pq}= H^{p+q}(Gr^p_FC)$. This construction is functorial. See Deligne, Théorie de Hodge II, sect. 1; III, sect. 7.
• With your notation, the spectral sequence associated to $(\mathbb{R} \pi_* \mathcal{F},\tau)$ becomes the Leray spectral sequence after reindexing to transform $E_1$ to $E_2$ [loc. cit.]. Here $\tau$ is the canonical filtration, given by truncations.
• Let $\mathcal{F}'= f^{-1}\mathcal{F}$. Then we have a morphism in the filtered derived category $$ (\mathbb{R} \pi_* \mathcal{F},\tau)\to \mathbb{R}g_*(\mathbb{R} \pi'_* \mathcal{F}',\tau)$$ This will induce a morphism of Leray spectral sequences.