The Stacks project

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

  1. $K$ is pseudo-coherent,

  2. $K$ is pseudo-coherent relative to $R$,

  3. $K$ is pseudo-coherent relative to $A$,

  4. $\varphi _*K$ is pseudo-coherent,

  5. $\varphi _*K$ 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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