The Stacks project

Lemma 37.55.13. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a perfect proper morphism of schemes. Let $E \in D(\mathcal{O}_ X)$ be perfect. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$.

Proof. We claim that Derived Categories of Schemes, Lemma 36.27.1 applies. Conditions (1) and (2) are immediate. Condition (3) is local on $X$. Thus we may assume $X$ and $S$ affine and $E$ represented by a strictly perfect complex of $\mathcal{O}_ X$-modules. Thus it suffices to show that $\mathcal{O}_ X$ has finite tor dimension as a sheaf of $f^{-1}\mathcal{O}_ S$-modules. This is equivalent to being perfect by Lemma 37.55.11. $\square$


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