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.

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