The Stacks project

Remark 48.32.7. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$ and let $U \subset X$ be an open. Set $g = f|_ U : U \to Y$. Then there is a canonical morphism

\[ Rg_!(K|_ U) \longrightarrow Rf_!K \]

functorial in $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ which can be defined in at least 3 ways.

  1. Denote $i : U \to X$ the inclusion morphism. We have $Rg_! = Rf_! \circ Ri_!$ by Lemma 48.32.6 and we can use $Rf_!$ applied to the map $Ri_!(K|_ U) \to K$ which is a special case of Remark 48.31.3.

  2. Choose a compactification $j : X \to \overline{X}$ of $X$ over $Y$ with structure morphism $\overline{f} : \overline{X} \to Y$. Set $j' = j \circ i : U \to \overline{X}$. We can use that $Rf_! = R\overline{f}_* \circ Rj_!$ and $Rg_! = R\overline{f}_* \circ Rj'_!$ and we can use $R\overline{f}_*$ applied to the map $Rj'_!(K|_ U) \to Rj_!K$ of Remark 48.31.3.

  3. We can use

    \begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) & = \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) \\ & \to \mathop{\mathrm{Hom}}\nolimits _ U(K|_ U, f^!L|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(K|_ U, g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rg_!(K|_ U), L) \end{align*}

    functorial in $L$ and $K$. Here we have used Proposition 48.32.2 twice and the construction of upper shriek functors which shows that $g^! = i^* \circ f^!$. The functoriality in $L$ shows by Categories, Remark 4.22.7 that we obtain a canonical map $Rg_!(K|_ U) \to Rf_!K$ in $\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ which is functorial in $K$ by the functoriality of the arrow above in $K$.

Each of these three constructions gives the same arrow; we omit the details.

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