Lemma 37.56.9. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then $E$ is $m$-pseudo-coherent relative to $S$ if and only if $R\pi _*E$ is $m$-pseudo-coherent relative to $S$.

Proof. Translation of the result of More on Algebra, Lemma 15.81.5 into the language of schemes. Observe that $R\pi _*$ indeed maps $D_\mathit{QCoh}(\mathcal{O}_ X)$ into $D_\mathit{QCoh}(\mathcal{O}_ Y)$ by Derived Categories of Schemes, Lemma 36.4.1. To do the translation use Lemma 37.56.6. $\square$

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