Lemma 15.82.8 (0E1T)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.82: Pseudo-coherent and perfect ring maps
Lemma 15.82.8 (cite)

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Lemma 15.82.8. Let $R \to B \to A$ be ring maps with $\varphi : B \to A$ surjective and $R \to B$ and $R \to A$ flat and of finite presentation. For $K \in D(A)$ denote $\varphi _*K \in D(B)$ the restriction. The following are equivalent

$K$ is pseudo-coherent,
$K$ is pseudo-coherent relative to $R$ ,
$K$ is pseudo-coherent relative to $A$ ,
$\varphi _*K$ is pseudo-coherent,
$\varphi _*K$ is pseudo-coherent relative to $R$ .

Similar holds for $m$ -pseudo-coherence.

Proof. Observe that $R \to A$ and $R \to B$ are perfect ring maps (Lemma 15.82.4) hence a fortiori pseudo-coherent ring maps. Thus (1) $\Leftrightarrow$ (2) and (4) $\Leftrightarrow$ (5) by Lemma 15.82.7.

Using that $A$ is pseudo-coherent relative to $R$ we use Lemma 15.81.15 to see that (2) $\Leftrightarrow$ (3). However, since $A \to B$ is surjective, we see directly from Definition 15.81.4 that (3) is equivalent with (4). $\square$

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