The Stacks project

Lemma 18.30.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be covering of $\mathcal{C}$. If $U$ is quasi-compact, then there exist a finite subset $I' \subset I$ such that the sequence

\[ \bigoplus \nolimits _{i, i' \in I'} j_{U_ i \times _ U U_{i'}!}\mathcal{O}_{U_ i \times _ U U_{i'}} \to \bigoplus \nolimits _{i \in I'} j_{U_ i!}\mathcal{O}_{U_ i} \to j_!\mathcal{O}_ U \to 0 \]

is exact.

Proof. This lemma is immediate from Lemma 18.30.1 if $U$ satisfies condition (3) of Sites, Lemma 7.17.2. We urge the reader to skip the proof in the general case. By definition there exists a covering $\mathcal{V} = \{ V_ j \to U\} _{j \in J}$ and a morphism $\mathcal{V} \to \mathcal{U}$ of families of maps with fixed target given by $\text{id} : U \to U$, $\alpha : J \to I$, and $f_ j : V_ j \to U_{\alpha (j)}$ (see Sites, Definition 7.8.1) such that the image $I' \subset I$ of $\alpha $ is finite. By Homology, Lemma 12.5.8 it suffices to show that for any sheaf of $\mathcal{O}$-modules $\mathcal{F}$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ turns the sequence of the lemma into an exact sequence. By ( we obtain the usual sequence

\[ 0 \to \mathcal{F}(U) \to \prod \nolimits _{i \in I'} \mathcal{F}(U_ i) \to \prod \nolimits _{i, i' \in I'} \mathcal{F}(U_ i \times _ U U_{i'}) \]

This is an exact sequence by Sites, Lemma 7.8.6 applied to the family of maps $\{ U_ i \to U\} _{i \in I'}$ which is refined by the covering $\mathcal{V}$. $\square$

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