Lemma 20.34.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $i : Z \to X$ be the inclusion of a closed subset.

$R\mathcal{H}_ Z : D(\mathcal{O}_ X) \to D(\mathcal{O}_ X|_ Z)$ is right adjoint to $i_* : D(\mathcal{O}_ X|_ Z) \to D(\mathcal{O}_ X)$.

For $K$ in $D(\mathcal{O}_ X|_ Z)$ we have $R\mathcal{H}_ Z(i_*K) = K$.

Let $\mathcal{G}$ be a sheaf of $\mathcal{O}_ X|_ Z$-modules on $Z$. Then $\mathcal{H}^ p_ Z(i_*\mathcal{G}) = 0$ for $p > 0$.

## Comments (2)

Comment #1801 by Keenan Kidwell on

Comment #1826 by Johan on