The Stacks project

Lemma 57.87.12. 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 $E \in D^+(X_{\acute{e}tale})$ have torsion cohomology sheaves. Then the base change map (57.87.5.2) $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.

Proof. This is a simple consequence of the proper base change theorem (Theorem 57.87.11) using the spectral sequences

\[ E_2^{p, q} = R^ pf_*H^ q(E) \quad \text{and}\quad {E'}_2^{p, q} = R^ pf'_*(g')^{-1}H^ q(E) \]

converging to $R^ nf_*E$ and $R^ nf'_*(g')^{-1}E$. The spectral sequences are constructed in Derived Categories, Lemma 13.21.3. Some details omitted. $\square$


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