The Stacks project

Lemma 18.30.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\{ U_ i \to U\} $ be a covering of $\mathcal{C}$. Then the sequence

\[ \bigoplus j_{U_ i \times _ U U_ j!}\mathcal{O}_{U_ i \times _ U U_ j} \to \bigoplus j_{U_ i!}\mathcal{O}_{U_ i} \to j_!\mathcal{O}_ U \to 0 \]

is exact.

Proof. For any $\mathcal{O}$-module $\mathcal{F}$ the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(-, \mathcal{F})$ turns our sequence into the exact sequence $0 \to \mathcal{F}(U) \to \prod \mathcal{F}(U_ i) \to \prod \mathcal{F}(U_ i \times _ U U_ j)$, see (18.19.2.1). The lemma follows from this and Homology, Lemma 12.5.8. $\square$

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