Lemma 63.10.7. 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 Étale Cohomology, Lemma 59.96.6 and using that f^{-1} = j^{-1} \circ \overline{f}^{-1} and Rf_! = R\overline{f}_*j_!.
\square
Comments (0)