Definition 15.81.4. Let $R \to A$ be a finite type ring map. Let $K^\bullet$ be a complex of $A$-modules. Let $M$ be an $A$-module. Let $m \in \mathbf{Z}$.

1. We say $K^\bullet$ is $m$-pseudo-coherent relative to $R$ if the equivalent conditions of Lemma 15.81.3 hold.

2. We say $K^\bullet$ is pseudo-coherent relative to $R$ if $K^\bullet$ is $m$-pseudo-coherent relative to $R$ for all $m \in \mathbf{Z}$.

3. We say $M$ is $m$-pseudo-coherent relative to $R$ if $M[0]$ is $m$-pseudo-coherent relative to $R$.

4. We say $M$ is pseudo-coherent relative to $R$ if $M[0]$ is pseudo-coherent relative to $R$.

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