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On the étale site of a scheme $X$, there is a spectral sequence associated to the data of an étale hypercover $K$ of $X$ and an abelian étale sheaf $\mathcal F$ on $X$: $$E_2^{p,q}=\check H^p(K,\underline H^q(\mathcal F))\Rightarrow H^{p+q}(X,\mathcal F).$$ (See https://stacks.math.columbia.edu/tag/01GY.) Here $\underline H^q(\mathcal F)$ is the étale presheaf with value $\underline H^q(\mathcal F)(U)=H^q(U,\mathcal F)$ on $U\to X$ étale.

Does the same spectral sequence exist for $\ell$-adic cohomology ($\mathbf{Z}_\ell$ or $\mathbf{Q}_\ell$ sheaves), with $\underline H^q(\mathcal F)$ a presheaf of $\mathbf{Z}_\ell$-modules or $\mathbf{Q}_\ell$-vector spaces? (With sufficient finiteness hypotheses on $X$ such as being of finite type over a field.) If so, is there a reference? (The closest I have been able to find to something mentioning this are the notes of Brian Conrad titled ‘Cohomological Descent.’)

Thank you.

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    $\begingroup$ If you use the pro-étale site of Bhatt and Scholze, the same tag applies. (Small warning: $\mathbf Z_\ell$ and $\mathbf Q_\ell$ are not constant sheaves; rather they carry a topology that affect the sections over a pro-étale cover.) $\endgroup$ Apr 29, 2020 at 4:41
  • $\begingroup$ Thank you very much. As for your warning, if I start with an $\mathbf{Q}_\ell$ sheaf $\mathcal F$ on, say, a variety over an algebraically closed field, and consider an étale hypercover of this variety, then $\underline H^q(\mathcal F)$ will be a presheaf on the pro-étale site. If I restrict this presheaf to the étale site, can its sections still be computed naïvely without mentioning the word ‘pro-étale?’ i.e. are sections of $\underline H^0(\mathbf{Q}_\ell)$ over a connected $U\to X$ étale still $\mathbf{Q}_\ell$ (even if the sections over a weakly étale $U\to X$ are something else)? $\endgroup$
    – delgato
    Apr 29, 2020 at 17:44
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    $\begingroup$ Yes: the sections over $U$ are continuous maps $U \to \mathbf Q_\ell$, so when $U$ has finitely many components (e.g. $U$ is of finite type over $X$) then this is the usual definition of a locally constant function (i.e. constant on each component). $\endgroup$ Apr 29, 2020 at 17:53

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