The Stacks project

Lemma 58.88.3. Let $f : X \to Y$ be a proper morphism of schemes. Let $g : Y' \to Y$ be a morphism of schemes. Set $X' = Y' \times _ Y X$ and denote $f' : X' \to Y'$ and $g' : X' \to X$ the projections. Let $n \geq 1$ be an integer. Let $E \in D(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z})$. Then the base change map (58.87.5.2) $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.

Proof. It is enough to prove this when $Y$ and $Y'$ are quasi-compact. By Morphisms, Lemma 29.28.5 we see that the dimension of the fibres of $f : X \to Y$ and $f' : X' \to Y'$ are bounded. Thus Lemma 58.88.2 implies that

\[ f_* : \textit{Mod}(X_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow \textit{Mod}(Y_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

and

\[ f'_* : \textit{Mod}(X'_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \longrightarrow \textit{Mod}(Y'_{\acute{e}tale}, \mathbf{Z}/n\mathbf{Z}) \]

have finite cohomological dimension in the sense of Derived Categories, Lemma 13.32.2. Choose a K-injective complex $\mathcal{I}^\bullet $ of $\mathbf{Z}/n\mathbf{Z}$-modules each of whose terms $\mathcal{I}^ n$ is an injective sheaf of $\mathbf{Z}/n\mathbf{Z}$-modules representing $E$. See Injectives, Theorem 19.12.6. By the usual proper base change theorem we find that $R^ qf'_*(g')^{-1}\mathcal{I}^ n = 0$ for $q > 0$, see Theorem 58.87.11. Hence we conclude by Derived Categories, Lemma 13.32.2 that we may compute $Rf'_*(g')^{-1}E$ by the complex $f'_*(g')^{-1}\mathcal{I}^\bullet $. Another application of the usual proper base change theorem shows that this is equal to $g^{-1}f_*\mathcal{I}^\bullet $ as desired. $\square$


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