The Stacks project

Lemma 84.4.7. Let $S$ be a scheme. Let

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

be a cartesian square of algebraic spaces over $S$. Assume $f$ is proper. Let $E \in D^+(X_{\acute{e}tale})$ have torsion cohomology sheaves. Then the base change map $g^{-1}Rf_*E \to Rf'_*(g')^{-1}E$ is an isomorphism.

Proof. This is a simple consequence of the proper base change theorem (Theorem 84.4.6) 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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