The Stacks project

Lemma 36.6.5. With $X$, $f_1, \ldots , f_ c \in \Gamma (X, \mathcal{O}_ X)$, and $\mathcal{F}$ as in Remark 36.6.4 the complex (36.6.4.1) restricts to an acyclic complex over $X \setminus Z$.

Proof. Let $W \subset X \setminus Z$ be an open subset. Evaluating the complex of sheaves (36.6.4.1) on $W$ we obtain the complex

\[ \mathcal{F}(W) \to \bigoplus \nolimits _{i_0} \mathcal{F}(U_{i_0} \cap W) \to \bigoplus \nolimits _{i_0 < i_1} \mathcal{F}(U_{i_0i_1} \cap W) \to \ldots \]

In other words, we obtain the extended ordered Čech complex for the covering $W = \bigcup U_ i \cap W$ and the standard ordering on $\{ 1, \ldots , c\} $, see Cohomology, Section 20.23. By Cohomology, Lemma 20.23.7 this complex is homotopic to zero as soon as $W$ is contained in $V(f_ i)$ for some $1 \leq i \leq c$. This finishes the proof. $\square$

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