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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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