20.18 Proper base change in topology

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

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 20.17.1 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

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

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). Consider a cartesian square of topological spaces

$\xymatrix{ X' = Y' \times _ Y X \ar[d]_{f'} \ar[r]_-{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

Assume that $f$ is proper and separated. Let $E$ be an object of $D^+(X)$. Then the base change map

$g^{-1}Rf_*E \longrightarrow Rf'_*(g')^{-1}E$

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. Moreover $f'$ is separated by Topology, Lemma 5.4.4. Thus we can apply Lemma 20.18.1 twice to see that

$(Rf'_*(g')^{-1}E)_{y'} = R\Gamma ((f')^{-1}(y'), (g')^{-1}E|_{(f')^{-1}(y')})$

and

$(Rf_*E)_ y = R\Gamma (f^{-1}(y), E|_{f^{-1}(y)})$

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

$\xymatrix{ X' \ar[d]_{f'} \ar[r]_-{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

Assume that $f$ is proper and separated. 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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