The Stacks project

Remark 63.9.5. Let $f : X \to Y$ be a finite type separated morphism of schemes with $Y$ quasi-compact and quasi-separated. Below we will construct a map

\[ Rf_!K \longrightarrow Rf_*K \]

functorial for $K$ in $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or $D(X_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion. This transformation of functors in both cases is compatible with

  1. the isomorphism $Rg_! \circ Rf_! \to R(g \circ f)_!$ of Lemma 63.9.2 and the isomorphism $Rg_* \circ Rf_* \to R(g \circ f)_*$ of Cohomology on Sites, Lemma 21.19.2 and

  2. the isomorphism $g^{-1} \circ Rf_! \to Rf'_! \circ (g')^{-1}$ of Lemma 63.9.4 and the base change map of Cohomology on Sites, Remark 21.19.3.

Namely, choose a compactification $j : X \to \overline{X}$ over $Y$ and denote $\overline{f} : \overline{X} \to Y$ the structure morphism. Since $Rf_! = R\overline{f}_* \circ j_!$ and $Rf_* = R\overline{f}_* \circ Rj_*$ it suffices to construct a transformation of functors $j_! \to Rj_*$. For this we use the canonical transformation $j_! \to j_*$ of √Čtale Cohomology, Lemma 59.70.6. We omit the proof that the resulting transformation is independent of the choice of compactification and we omit the proof of the compatibilities (1) and (2).


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