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
is exact.
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
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
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
Comments (0)