In this section we prove a very general version of the proper base change theorem in topology. It tells us that the stalks of the higher direct images $R^ pf_*$ can be computed on the fibre.
Lemma 20.18.1. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $y \in Y$. Assume that
$f$ is closed,
$f$ is separated, and
$f^{-1}(y)$ 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 20.17.1 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 20.16.3 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 13.16.7. In other words, we have to show the map
is an isomorphism. Using Lemma 20.16.3 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. $\square$
Theorem 20.18.2 (Proper base change).reference 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. $\square$
Lemma 20.18.3 (Proper base change for sheaves of sets). Consider a cartesian square of topological spaces
Assume that $f$ is proper. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ for any sheaf of sets $\mathcal{F}$ on $X$.
Proof. We argue exactly as in the proof of Theorem 20.18.2 and we find it suffices to show $(f_*\mathcal{F})_ y = \Gamma (X_ y, \mathcal{F}|_{X_ y})$. Then we argue as in Lemma 20.18.1 to reduce this to the $p = 0$ case of Lemma 20.16.3 for sheaves of sets. The first part of the proof of Lemma 20.16.3 works for sheaves of sets and this finishes the proof. Some details omitted. $\square$
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Comment #4967 by Brian Shih on
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