The Stacks project

Lemma 61.10.7. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a torsion ring. Let $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(Y_{\acute{e}tale}, \Lambda )$. Then

\[ Rf_!E \otimes _\Lambda ^\mathbf {L} K = Rf_!(E \otimes _\Lambda ^\mathbf {L} f^{-1}K) \]

in $D(Y_{\acute{e}tale}, \Lambda )$.

Proof. Choose $j : X \to \overline{X}$ and $\overline{f} : \overline{X} \to Y$ as in the construction of $Rf_!$. We have $j_!E \otimes _\Lambda ^\mathbf {L} \overline{f}^{-1}K = j_!(E \otimes _\Lambda ^\mathbf {L} f^{-1}K)$ by Cohomology on Sites, Lemma 21.20.9. Then we get the result by applying √Čtale Cohomology, Lemma 58.95.6 and using that $f^{-1} = j^{-1} \circ \overline{f}^{-1}$ and $Rf_! = R\overline{f}_*j_!$. $\square$


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