Lemma 59.84.5. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $X$ be a separated finite type scheme over $k$ of dimension $\leq 1$, and let $0 \to \mathcal{F}_1 \to \mathcal{F} \to \mathcal{F}_2 \to 0$ be a short exact sequence of sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$. If $H^ q_{\acute{e}tale}(X, \mathcal{F}_ i)$, $i = 1, 2$ are finite $\Lambda $-modules then $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module.
Proof. Immediate from the long exact sequence of cohomology. $\square$
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