The Stacks project

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.

Proof. Here as usual we set $X_{i_0 \ldots i_ p} = X_{i_0} \cap \ldots \cap X_{i_ p}$ and we denote $f_{i_0 \ldots i_ p}$ the restriction of $f$ to $X_{i_0 \ldots i_ p}$. The maps in the complex are the maps constructed in Remark 63.4.6 with sign rules as in the Čech complex. Exactness follows easily from the description of stalks in Lemma 63.4.5. Details omitted. $\square$

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