The Stacks project

Lemma 58.95.6. Let $f : X \to Y$ be a proper morphism of 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. There is a canonical map from left to right by Cohomology on Sites, Section 21.48. We will check the equality on stalks at $\overline{y}$. By the proper base change (in the form of Lemma 58.91.3 where $Y' = \overline{y}$) this reduces to the case where $Y$ is the spectrum of an algebraically closed field. This is shown in Lemma 58.95.5 where we use that $\text{cd}(X) < \infty $ by Lemma 58.91.2. $\square$


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