Lemma 58.22.3 (Locality of cohomology). Let $\mathcal{C}$ be a site, $\mathcal{F}$ an abelian sheaf on $\mathcal{C}$, $U$ an object of $\mathcal{C}$, $p > 0$ an integer and $\xi \in H^ p(U, \mathcal{F})$. Then there exists a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ of $U$ in $\mathcal{C}$ such that $\xi |_{U_ i} = 0$ for all $i \in I$.

**Proof.**
Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $. Then $\xi $ is represented by a cocycle $\tilde{\xi } \in \mathcal{I}^ p(U)$ with $d^ p(\tilde{\xi }) = 0$. By assumption, the sequence $\mathcal{I}^{p - 1} \to \mathcal{I}^ p \to \mathcal{I}^{p + 1}$ in exact in $\textit{Ab}(\mathcal{C})$, which means that there exists a covering $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ such that $\tilde{\xi }|_{U_ i} = d^{p - 1}(\xi _ i)$ for some $\xi _ i \in \mathcal{I}^{p-1}(U_ i)$. Since the cohomology class $\xi |_{U_ i}$ is represented by the cocycle $\tilde{\xi }|_{U_ i}$ which is a coboundary, it vanishes. For more details see Cohomology on Sites, Lemma 21.7.3.
$\square$

## 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Your email address will not be published.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).

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.

## Comments (0)

There are also: