Remark 63.10.9. Let f : X \to Y be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let \Lambda be a torsion coefficient ring and let K and L be objects of D(X_{\acute{e}tale}, \Lambda ). We claim there is a canonical map
functorial in K and L. Namely, choose j : X \to \overline{X} and \overline{f} : \overline{X} \to Y as in the construction of Rf_!. We first define a map
By the construction of internal hom in the derived category, this is the same thing as defining a map
See Cohomology on Sites, Section 21.35. The source of \beta ' is equal to
by Cohomology on Sites, Lemma 21.20.9. Hence we can set \beta ' = j_!\beta '' where \beta '' : R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \otimes _\Lambda ^\mathbf {L} K \to L corresponds to the identity on R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) via the universal property of internal hom mentioned above. By Cohomology on Sites, Remark 21.35.10 we have a canonical map
Since Rf_! = R\overline{f}_*j_! and Rf_* = R\overline{f}_* Rj_* (by Leray) we obtain the desired map \alpha = \gamma \circ R\overline{f}_*\beta .
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