The Stacks project

Lemma 20.10.2. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ be an open covering. The functors $\mathcal{F} \mapsto \check{H}^ n(\mathcal{U}, \mathcal{F})$ form a $\delta $-functor from the abelian category of presheaves of $\mathcal{O}_ X$-modules to the category of $\mathcal{O}_ X(U)$-modules (see Homology, Definition 12.12.1).

Proof. By Lemma 20.10.1 a short exact sequence of presheaves of $\mathcal{O}_ X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is turned into a short exact sequence of complexes of $\mathcal{O}_ X(U)$-modules. Hence we can use Homology, Lemma 12.13.12 to get the boundary maps $\delta _{\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3} : \check{H}^ n(\mathcal{U}, \mathcal{F}_3) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F}_1)$ and a corresponding long exact sequence. We omit the verification that these maps are compatible with maps between short exact sequences of presheaves. $\square$


Comments (0)


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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01EK. Beware of the difference between the letter 'O' and the digit '0'.