Remark 62.10.9. 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

$\alpha : Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!K, Rf_!L)$

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

$\beta : Rj_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (j_!K, j_!L)$

By the construction of internal hom in the derived category, this is the same thing as defining a map

$\beta ' : Rj_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \otimes _\Lambda ^\mathbf {L} j_!K \longrightarrow j_!L$

See Cohomology on Sites, Section 21.35. The source of $\beta '$ is equal to

$j_!\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \otimes _\Lambda ^\mathbf {L} K\right)$

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

$\gamma : R\overline{f}_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (j_!K, j_!L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (R\overline{f}_*j_!K, R\overline{f}_*j_!L)$

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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