Theorem 20.18.2 (Proper base change). Consider a cartesian square of topological spaces
Assume that $f$ is proper. Let $E$ be an object of $D^+(X)$. Then the base change map
of Lemma 20.17.1 is an isomorphism in $D^+(Y')$.
[Expose V bis, 4.1.1, SGA4]
Theorem 20.18.2 (Proper base change). Consider a cartesian square of topological spaces Assume that $f$ is proper. Let $E$ be an object of $D^+(X)$. Then the base change map of Lemma 20.17.1 is an isomorphism in $D^+(Y')$.
Proof.
Let $y' \in Y'$ be a point with image $y \in Y$. It suffices to show that the base change map induces an isomorphism on stalks at $y'$. As $f$ is proper it follows that $f'$ is proper, the fibres of $f$ and $f'$ are quasi-compact and $f$ and $f'$ are closed, see Topology, Theorem 5.17.5 and Lemma 5.4.4. Thus we can apply Lemma 20.18.1 twice to see that
and
The induced map of fibres $(f')^{-1}(y') \to f^{-1}(y)$ is a homeomorphism of topological spaces and the pull back of $E|_{f^{-1}(y)}$ is $(g')^{-1}E|_{(f')^{-1}(y')}$. The desired result follows.
$\square$
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