Lemma 21.20.2. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}). For K in D(\mathcal{O}) we have H^ p(U, K) = H^ p(\mathcal{C}/U, K|_{\mathcal{C}/U}).
Proof. Let \mathcal{I}^\bullet be a K-injective complex of \mathcal{O}-modules representing K. Then
H^ q(U, K) = H^ q(\Gamma (U, \mathcal{I}^\bullet )) = H^ q(\Gamma (\mathcal{C}/U, \mathcal{I}^\bullet |_{\mathcal{C}/U}))
by construction of cohomology. By Lemma 21.20.1 the complex \mathcal{I}^\bullet |_{\mathcal{C}/U} is a K-injective complex representing K|_{\mathcal{C}/U} and the lemma follows. \square
Comments (0)