Lemma 59.84.2. Let $\Lambda $ be a Noetherian ring, let $k$ be an algebraically closed field, let $f : X \to Y$ be a finite morphism of separated finite type schemes over $k$ of dimension $\leq 1$, and let $\mathcal{F}$ be a sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$. If $H^ q_{\acute{e}tale}(X, \mathcal{F})$ is a finite $\Lambda $-module, then so is $H^ q_{\acute{e}tale}(Y, f_*\mathcal{F})$.
Proof. Namely, we have $H^ q_{\acute{e}tale}(X, \mathcal{F}) = H^ q_{\acute{e}tale}(Y, f_*\mathcal{F})$ by the vanishing of $R^ qf_*$ for $q > 0$ (Proposition 59.55.2) and the Leray spectral sequence (Cohomology on Sites, Lemma 21.14.6). $\square$
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