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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 (18.19.2.1) 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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