The Stacks project

Lemma 59.83.5. 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 $\leq 1$. Let $\mathcal{F}$ be a torsion abelian sheaf on $X$. If statements (1) – (8) hold for $\mathcal{F}$, then they hold for $f_*\mathcal{F}$.

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). For (8) use that formation of $f_*$ commutes with arbitrary base change (Lemma 59.55.3). $\square$

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