Lemma 59.63.4. Let $X$ be separated of finite type over an algebraically closed field $k$ of characteristic $p > 0$. Then $H_{\acute{e}tale}^ q(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$ for $q \geq dim(X) + 1$.
Proof. Let $d = \dim (X)$. By the vanishing established in Lemma 59.63.1 it suffices to show that $H_{\acute{e}tale}^{d + 1}(X, \underline{\mathbf{Z}/p\mathbf{Z}}) = 0$. By Lemma 59.63.3 we see that $H^ d(X, \mathcal{O}_ X)$ is a finite dimensional $k$-vector space. Hence the long exact cohomology sequence associated to the Artin-Schreier sequence ends with
\[ H^ d(X, \mathcal{O}_ X) \xrightarrow {F - 1} H^ d(X, \mathcal{O}_ X) \to H^{d + 1}_{\acute{e}tale}(X, \mathbf{Z}/p\mathbf{Z}) \to 0 \]
By Lemma 59.63.2 the map $F - 1$ in this sequence is surjective. This proves the lemma. $\square$
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Your email address will not be published.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)
There are also: