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
is exact.
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
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
Comments (0)