Remark 48.31.3 (0G4Y)—The Stacks project

Remark 48.31.3. Let $X \supset U \supset U'$ be open subschemes of a Noetherian scheme $X$. Denote $j : U \to X$ and $j' : U' \to X$ the inclusion morphisms. We claim there is a canonical map

\[ Rj'_!(K|_{U'}) \longrightarrow Rj_!K \]

functorial for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Namely, by Lemma 48.30.1 we have for any $L$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ the map

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

functorial in $L$ and $K'$. The functoriality in $L$ shows by Categories, Remark 4.22.7 that we obtain a canonical map $Rj'_!(K|_{U'}) \to Rj_!K$ which is functorial in $K$ by the functoriality of the arrow above in $K$.

Here is an explicit construction of this arrow. Namely, suppose that $\mathcal{F}^\bullet $ is a bounded complex of coherent $\mathcal{O}_ X$-modules whose restriction to $U$ represents $K$ in the derived category. We have seen in the proof of Lemma 48.30.3 that such a complex always exists. Let $\mathcal{I}$, resp. $\mathcal{I}'$ be a quasi-coherent sheaf of ideals on $X$ with $V(\mathcal{I}) = X \setminus U$, resp. $V(\mathcal{I}') = X \setminus U'$. After replacing $\mathcal{I}$ by $\mathcal{I} + \mathcal{I}'$ we may assume $\mathcal{I}' \subset \mathcal{I}$. By construction $Rj_!K$, resp. $Rj'_!(K|_{U'})$ is represented by the inverse system $(K_ n)$, resp. $(K'_ n)$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with

\[ K_ n = \mathcal{I}^ n\mathcal{F}^\bullet \quad \text{resp.}\quad K'_ n = (\mathcal{I}')^ n\mathcal{F}^\bullet \]

Clearly the map constructed above is given by the maps $K'_ n \to K_ n$ coming from the inclusions $(\mathcal{I}')^ n \subset \mathcal{I}^ n$.

The Stacks project

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