Remark 63.9.5 (0H6X)—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

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
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 Étale 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).

Comments (1)

Comment #8768 by Cop 223 on January 10, 2024 at 15:27

Here is a manifestly canonical construction of the map $f_! \to f_\ast$ . Consider the relative diagonal $\Delta : X \to X \times_Y X$ along with the two projections $\pi_i : X \times_Y X \to X$ . Since $f$ is separated, $\Delta_! \simeq \Delta_\ast$ . Combining this with the tautological transformation $f_! \circ \pi_{2,\ast} \to f_\ast \circ \pi_{1,!}$ gives the desired morphism $f_! \simeq f_! \circ \pi_{2,\ast} \circ \Delta_\ast \to f_\ast \circ \pi_{1,!} \circ \Delta_\ast \simeq f_\ast \circ \pi_{1,!} \circ \Delta_! \simeq f_\ast.$

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2 comment(s) on Section 63.9: Derived lower shriek via compactifications

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