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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. $\square$

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