Lemma 36.12.6 (09VB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.12: Descent finiteness properties of complexes
Lemma 36.12.6 (cite)

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Lemma 36.12.6. Let $f : X \to Y$ be a finite morphism of schemes such that $f_*\mathcal{O}_ X$ is pseudo-coherent as an $\mathcal{O}_ Y$ -module ¹. Let $E \in D_\mathit{QCoh}(\mathcal{O}_ X)$ . Then $E$ is $m$ -pseudo-coherent if and only if $Rf_*E$ is $m$ -pseudo-coherent.

Proof. This is a translation of More on Algebra, Lemma 15.64.11 into the language of schemes. To do the translation, use Lemmas 36.3.5 and 36.10.2. $\square$

[1] This means that

$f$ is pseudo-coherent, see More on Morphisms, Lemma 37.60.8.

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