Remark 20.34.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset. Given $K$ and $M$ in $D(\mathcal{O}_ X)$ there is a canonical map

in $D(\mathcal{O}_ X|_ Z)$. Here $K|_ Z = i^{-1}K$ is the restriction of $K$ to $Z$ viewed as an object of $D(\mathcal{O}_ X|_ Z)$. By adjointness of $i_*$ and $R\mathcal{H}_ Z$ of Lemma 20.34.1 to construct this map it suffices to produce a canonical map

To construct this map, we choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}_ X$-modules representing $M$ and a K-flat complex $\mathcal{K}^\bullet $ of $\mathcal{O}_ X$-modules representing $K$. Observe that $\mathcal{K}^\bullet |_ Z$ is a K-flat complex of $\mathcal{O}_ X|_ Z$-modules representing $K|_ Z$, see Lemma 20.26.8. Hence we need to produce a map of complexes

of $\mathcal{O}_ X$-modules. For this it suffices to produce maps

Looking at stalks (for example), we see that the left hand side of this formula is equal to $\mathcal{K}^ a \otimes _{\mathcal{O}_ X} i_*\mathcal{H}_ Z(\mathcal{I}^ b)$ and we can use the inclusion $\mathcal{H}_ Z(\mathcal{I}^ b) \to \mathcal{I}^ b$ to get our map.

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