Let $\ell$ be a prime, and $k$ a field of characteristic $\ne \ell$. Let $f \colon X \to Y$ be a proper map of smooth projective $k$-varieties. The Leray spectral sequence says
$$
E\_{2}^{pq} = H^{p}(\bar{Y},R^{q}f\_{\*}\mathbb{Q}\_{\ell}) \ \Longrightarrow\ H^{p+q}(\bar{X}, \mathbb{Q}\_{\ell}) .
$$
In particular this means that there is a filtration on $H^{n}(\bar{X},\mathbb{Q}\_{\ell})$ such that $E\_{\infty}^{pq}$ (with $p+q = n$) are the graded pieces.



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> **Question:** Are there conditions on $f$, $X$, and/or $Y$ that make this filtration split?
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(This means that $H^{n}(\bar{X},\mathbb{Q}\_{\ell}) \cong \bigoplus\_{p+q=n} E\_{\infty}^{pq}$; even if the direct sum decomposition is not canonical.)


Let me point out explicitly that I am looking for a decomposition that is Galois equivariant.



When does the filtration in the limit of the Leray spectral sequence split?
Credits to Francesco Polizzi for putting me on the right track, by pointing to Deligne's work.


The important references for this are


* Deligne's thesis [“Théorème de Lefschetz et critères de dégénérescence de suites spectrales”](http://publications.ias.edu/sites/default/files/Number3.pdf) Publications Mathématiques de l'IHÉS. 35 (1968) pp. 107–126.
* Deligne's paper [“Décompositions dans la catégorie dérivée”](http://publications.ias.edu/sites/default/files/68_Decompositionsdans.pdf) in: Motives. Proceedings of Symposia in Pure Mathematics. 55 t1 (AMS 1994) pp. 115–128.



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> I guess if one invokes perverse sheaves there might be more general answers. If anyone would like to give such a more general answer, I would be very grateful!
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**[Edit]** As Dan points out below, working with perverse sheaves indeed helps: then there is a decomposition if $f$ is proper. **[/Edit]**


What follows requires that $f$ is smooth (and proper, which was already an assumption in the question). Here goes my attempt:


The decomposition
=================


In Deligne's thesis [“Théorème de Lefschetz et critères de dégénérescence de suites spectrales”](http://publications.ias.edu/sites/default/files/Number3.pdf) we find the following results:


* Proposition 2.1 states that in the derived category $D^{\textrm{b}}(\bar{Y})$ we have an isomorphism
$$ \textrm{R}f\_{\*} \mathbb{Q}\_{\ell} \cong \bigoplus\_{i} \textrm{R}^{i}f\_{\*} \mathbb{Q}\_{\ell}[-i]. $$
The proposition applies, since $\mathbb{Q}\_{\ell}$ satisfies the *Lefschetz condition* by the relative hard Lefschetz theorem (this uses that $f$ is smooth and proper).
* Proposition 2.16 with $Z = \textrm{Spec}(k)$, can be applied (since $Y$ is smooth over $k$; and by the previous point). This gives
$$ \textrm{R}(\Gamma f)\_{\*} \mathbb{Q}\_{\ell} \cong \bigoplus\_{k} \textrm{R}\Gamma\_{\*} (\textrm{R}f\_{\*}^{k} \mathbb{Q}\_{\ell}[-k]). $$


Taking the $n$-th homology group, then gives
$$ \textrm{H}^{n}(\bar{X}, \mathbb{Q}\_{\ell}) = \bigoplus\_{q} \textrm{H}^{n}(\bar{Y}, \textrm{R}f\_{\*}^{q} \mathbb{Q}\_{\ell}[-q]) = \bigoplus\_{q} \textrm{H}^{n-q}(\bar{Y}, \textrm{R}f\_{\*}^{q} \mathbb{Q}\_{\ell}). $$
This last expression can of course be written in the usual way:
$$ \textrm{H}^{n}(\bar{X}, \mathbb{Q}\_{\ell}) = \bigoplus\_{p+q=n} \textrm{H}^{p}(\bar{Y}, \textrm{R}f\_{\*}^{q} \mathbb{Q}\_{\ell}). $$


The Galois equivariance
=======================


If I am not mistaken, the above decomposition is Galois equivariant. I hope some expert can point out errors if I go astray. Here it goes:


* Deligne starts with a very general Proposition (1.2) where he works with the bounded derived category $D^{\textrm{b}}(A)$ of some abelian category $A$. In particular we can work with the category $D^{\textrm{b}}(Y \times\_{k} \bar{k})$ of lisse $\mathbb{Q}\_{\ell}$-sheaves *together with the Galois action*. Here one has to identify $\sigma^{\*}\mathcal{F}$ with $\mathcal{F}$, just as when one defines the Galois action on $\ell$-adic cohomology (or when proving the functoriality of $\textrm{H}\_{\textrm{ét}}^{i}(\\_, \mathbb{Q}\_{\ell})$).
* The next Theorem we need is (1.5), and this is Galois equivariant because the hard Lefschetz theorem is Galois equivariant. (Here I bluf that this also holds for the *relative* hard Lefschetz theorem.)
* Consequently the decomposition of Proposition 2.1 is Galois equivariant.
* And finally, therefore the decomposition of Proposition 2.16 is also Galois equivariant.


