Lemma 20.19.1. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $y \in Y$. Assume that

1. $f$ is closed,

2. $f$ is separated, and

3. $f^{-1}(y)$ is quasi-compact.

Then for $E$ in $D^+(\mathcal{O}_ X)$ we have $(Rf_*E)_ y = R\Gamma (f^{-1}(y), E|_{f^{-1}(y)})$ in $D^+(\mathcal{O}_{Y, y})$.

Proof. The base change map of Lemma 20.18.1 gives a canonical map $(Rf_*E)_ y \to R\Gamma (f^{-1}(y), E|_{f^{-1}(y)})$. To prove this map is an isomorphism, we represent $E$ by a bounded below complex of injectives $\mathcal{I}^\bullet$. Set $Z = f^{-1}(\{ y\} )$. The assumptions of Lemma 20.17.3 are satisfied, see Topology, Lemma 5.4.2. Hence the restrictions $\mathcal{I}^ n|_ Z$ are acyclic for $\Gamma (Z, -)$. Thus $R\Gamma (Z, E|_ Z)$ is represented by the complex $\Gamma (Z, \mathcal{I}^\bullet |_ Z)$, see Derived Categories, Lemma 13.17.7. In other words, we have to show the map

$\mathop{\mathrm{colim}}\nolimits _ V \mathcal{I}^\bullet (f^{-1}(V)) \longrightarrow \Gamma (Z, \mathcal{I}^\bullet |_ Z)$

is an isomorphism. Using Lemma 20.17.3 we see that it suffices to show that the collection of open neighbourhoods $f^{-1}(V)$ of $Z = f^{-1}(\{ y\} )$ is cofinal in the system of all open neighbourhoods. If $f^{-1}(\{ y\} ) \subset U$ is an open neighbourhood, then as $f$ is closed the set $V = Y \setminus f(X \setminus U)$ is an open neighbourhood of $y$ with $f^{-1}(V) \subset U$. This proves the lemma. $\square$

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