Lemma 59.54.1. Let $f: X \to Y$ be a morphism and $\mathcal{I}$ an injective object of $\textit{Ab}(X_{\acute{e}tale})$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$. Then
for any covering $\mathcal{V} = \{ V_ j\to V\} _{j \in J}$ we have $\check H^ p(\mathcal{V}, f_*\mathcal{I}) = 0$ for all $p > 0$,
$f_*\mathcal{I}$ is acyclic for the functor $\Gamma (V, -)$, and
if $g : Y \to Z$, then $f_*\mathcal{I}$ is acyclic for $g_*$.
Proof. Observe that $\check{\mathcal{C}}^\bullet (\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet (\mathcal{V} \times _ Y X, \mathcal{I})$ which has vanishing higher cohomology groups by Lemma 59.18.7. This proves (1). The second statement follows as a sheaf which has vanishing higher Čech cohomology groups for any covering has vanishing higher cohomology groups. This a wonderful exercise in using the Čech-to-cohomology spectral sequence, but see Cohomology on Sites, Lemma 21.10.9 for details and a more precise and general statement. Part (3) is a consequence of (2) and the description of $R^ pg_*$ in Lemma 59.51.6. $\square$
Using the formalism of Grothendieck spectral sequences, this gives the following.
Proposition 59.54.2 (Leray spectral sequence). Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ an étale sheaf on $X$. Then there is a spectral sequence
\[ E_2^{p, q} = H_{\acute{e}tale}^ p(Y, R^ qf_*\mathcal{F}) \Rightarrow H_{\acute{e}tale}^{p+q}(X, \mathcal{F}). \]
Proof. See Lemma 59.54.1 and see Derived Categories, Section 13.22. $\square$
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