The Stacks project

Lemma 63.12.2. Let $f : X \to Y$ be a finite type separated morphism of schemes with $Y$ quasi-compact and quasi-separated. Let $K$ be an object of $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or of $D(X_{\acute{e}tale}, \Lambda )$ in case $\Lambda $ is torsion. Then there is a canonical isomorphism

\[ (Rf_!K)_{\overline{y}} \longrightarrow R\Gamma _ c(X_{\overline{y}}, K|_{X_{\overline{y}}}) \]

in $D(\Lambda )$ for any geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$.

Proof. Immediate consequence of Lemma 63.9.4 and the definitions. $\square$

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