Lemma 36.30.1. Let f : X \to S be a morphism of finite presentation. Let E \in D(\mathcal{O}_ X) be a perfect object. Let \mathcal{G}^\bullet be a bounded complex of finitely presented \mathcal{O}_ X-modules, flat over S, with support proper over S. Then
K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet )
is a perfect object of D(\mathcal{O}_ S) and its formation commutes with arbitrary base change.
Proof.
The statement on base change is Lemma 36.26.4. Thus it suffices to show that K is a perfect object. If S is Noetherian, then this follows from Lemma 36.27.2. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.
The question is local on S, hence we may assume S is affine. Say S = \mathop{\mathrm{Spec}}(R). We write R = \mathop{\mathrm{colim}}\nolimits R_ i as a filtered colimit of Noetherian rings R_ i. By Limits, Lemma 32.10.1 there exists an i and a scheme X_ i of finite presentation over R_ i whose base change to R is X. By Limits, Lemma 32.10.2 we may assume after increasing i, that there exists a bounded complex of finitely presented \mathcal{O}_{X_ i}-modules \mathcal{G}_ i^\bullet whose pullback to X is \mathcal{G}^\bullet . After increasing i we may assume \mathcal{G}_ i^ n is flat over R_ i, see Limits, Lemma 32.10.4. After increasing i we may assume the support of \mathcal{G}_ i^ n is proper over R_ i, see Limits, Lemma 32.13.5 and Cohomology of Schemes, Lemma 30.26.7. Finally, by Lemma 36.29.3 we may, after increasing i, assume there exists a perfect object E_ i of D(\mathcal{O}_{X_ i}) whose pullback to X is E. Applying Lemma 36.27.2 to X_ i \to \mathop{\mathrm{Spec}}(R_ i), E_ i, \mathcal{G}_ i^\bullet and using the base change property already shown we obtain the result.
\square
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