Lemma 15.81.17. Let $R$ be a Noetherian ring. Let $R \to A$ be a finite type ring map. Then

1. A complex of $A$-modules $K^\bullet$ is $m$-pseudo-coherent relative to $R$ if and only if $K^\bullet \in D^{-}(A)$ and $H^ i(K^\bullet )$ is a finite $A$-module for $i \geq m$.

2. A complex of $A$-modules $K^\bullet$ is pseudo-coherent relative to $R$ if and only if $K^\bullet \in D^{-}(A)$ and $H^ i(K^\bullet )$ is a finite $A$-module for all $i$.

3. An $A$-module is pseudo-coherent relative to $R$ if and only if it is finite.

Proof. Immediate consequence of Lemma 15.64.17 and the definitions. $\square$

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