Lemma 20.32.2. Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. For $K$ in $D(\mathcal{O}_ X)$ we have $H^ p(U, K) = H^ p(U, K|_ U)$.
Proof. Let $\mathcal{I}^\bullet $ be a K-injective complex of $\mathcal{O}_ X$-modules representing $K$. Then
\[ H^ q(U, K) = H^ q(\Gamma (U, \mathcal{I}^\bullet )) = H^ q(\Gamma (U, \mathcal{I}^\bullet |_ U)) \]
by construction of cohomology. By Lemma 20.32.1 the complex $\mathcal{I}^\bullet |_ U$ is a K-injective complex representing $K|_ U$ and the lemma follows. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$
). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)