Lemma 75.21.5. Let S be a scheme. Let f : X \to Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let E be an object of D(\mathcal{O}_ X). Let \mathcal{G}^\bullet be a complex of quasi-coherent \mathcal{O}_ X-modules. If
E is perfect, \mathcal{G}^\bullet is a bounded above, and \mathcal{G}^ n is flat over Y, or
E is pseudo-coherent, \mathcal{G}^\bullet is bounded, and \mathcal{G}^ n is flat over Y,
then formation of
Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet )
commutes with arbitrary base change (see proof for precise statement).
Proof.
The statement means the following. Let g : Y' \to Y be a morphism of algebraic spaces and consider the base change diagram
\xymatrix{ X' \ar[r]_ h \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }
in other words X' = Y' \times _ Y X. The lemma asserts that
Lg^*Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet ) \longrightarrow R(f')_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (L(g')^*E, (g')^*\mathcal{G}^\bullet )
is an isomorphism. Observe that on the right hand side we do not use the derived pullback on \mathcal{G}^\bullet . To prove this, we apply Lemmas 75.21.2 and 75.21.3 to see that it suffices to prove the canonical map
L(g')^*\mathcal{G}^\bullet \to (g')^*\mathcal{G}^\bullet
satisfies the equivalent conditions of Lemma 75.21.1. This was shown in the proof of Lemma 75.21.4.
\square
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