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 $g_*$.

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