Remark 20.34.9 (0G75)—The Stacks project

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

\[ K|_ Z \otimes _{\mathcal{O}_ X|_ Z}^\mathbf {L} R\mathcal{H}_ Z(M) \longrightarrow R\mathcal{H}_ Z(K \otimes _{\mathcal{O}_ X}^\mathbf {L} M) \]

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

\[ i_*\left(K|_ Z \otimes _{\mathcal{O}_ X|_ Z}^\mathbf {L} R\mathcal{H}_ Z(M)\right) \longrightarrow K \otimes _{\mathcal{O}_ X}^\mathbf {L} M \]

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

\[ i_*\text{Tot}\left( \mathcal{K}^\bullet |_ Z \otimes _{\mathcal{O}_ X|_ Z} \mathcal{H}_ Z(\mathcal{I}^\bullet )\right) \longrightarrow \text{Tot}(\mathcal{K}^\bullet \otimes _{\mathcal{O}_ X} \mathcal{I}^\bullet ) \]

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

\[ i_*(\mathcal{K}^ a|_ Z \otimes _{\mathcal{O}_ X|_ Z} \mathcal{H}_ Z(\mathcal{I}^ b)) \longrightarrow \mathcal{K}^ a \otimes _{\mathcal{O}_ X} \mathcal{I}^ b \]

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.

The Stacks project

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