Lemma 63.4.7. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. Let $X = \bigcup _{i \in I} X_ i$ be an open covering. Then there exists an exact complex

\[ \ldots \to \bigoplus \nolimits _{i_0, i_1, i_2} f_{i_0i_1i_2, !} \mathcal{F}|_{X_{i_0i_1i_2}} \to \bigoplus \nolimits _{i_0, i_1} f_{i_0i_1, !} \mathcal{F}|_{X_{i_0i_1}} \to \bigoplus \nolimits _{i_0} f_{i_0, !} \mathcal{F}|_{X_{i_0}} \to f_!\mathcal{F} \to 0 \]

functorial in $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$, see proof for details.

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