Lemma 59.78.2. Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism of schemes.

1. If $\mathcal{F}$ is a torsion abelian sheaf on $X_{\acute{e}tale}$, then $R^ nf_*\mathcal{F}$ is a torsion abelian sheaf on $Y_{\acute{e}tale}$ for all $n$.

2. If $K$ in $D^+(X_{\acute{e}tale})$ has torsion cohomology sheaves, then $Rf_*K$ is an object of $D^+(Y_{\acute{e}tale})$ whose cohomology sheaves are torsion abelian sheaves.

Proof. Proof of (1). Recall that $R^ nf_*\mathcal{F}$ is the sheaf associated to the presheaf $V \mapsto H^ n_{\acute{e}tale}(X \times _ Y V, \mathcal{F})$ on $Y_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.7.4. If we choose $V$ affine, then $X \times _ Y V$ is quasi-compact and quasi-separated because $f$ is, hence we can apply Lemma 59.78.1 to see that $H^ n_{\acute{e}tale}(X \times _ Y V, \mathcal{F})$ is torsion.

Proof of (2). Recall that $R^ nf_*K$ is the sheaf associated to the presheaf $V \mapsto H^ n_{\acute{e}tale}(X \times _ Y V, K)$ on $Y_{\acute{e}tale}$. See Cohomology on Sites, Lemma 21.20.6. If we choose $V$ affine, then $X \times _ Y V$ is quasi-compact and quasi-separated because $f$ is, hence we can apply Lemma 59.78.1 to see that $H^ n_{\acute{e}tale}(X \times _ Y V, K)$ is torsion. $\square$

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