Lemma 21.51.1. Let $\mathcal{C}$ be a site. Let $U$ be a weakly contractible object of $\mathcal{C}$. Then
the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is an exact functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$,
$H^ p(U, \mathcal{F}) = 0$ for every abelian sheaf $\mathcal{F}$ and all $p \geq 1$, and
for any sheaf of groups $\mathcal{G}$ any $\mathcal{G}$-torsor has a section over $U$.
Proof. The first statement follows immediately from the definition (see also Homology, Section 12.7). The higher derived functors vanish by Derived Categories, Lemma 13.16.9. Let $\mathcal{F}$ be a $\mathcal{G}$-torsor. Then $\mathcal{F} \to *$ is a surjective map of sheaves. Hence (3) follows from the definition as well. $\square$
Your email address will not be published.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)