Lemma 15.81.9. Let $R \to A$ be a finite type ring map. Let $m \in \mathbf{Z}$. Let $K^\bullet $ be a bounded above complex of $A$-modules such that $K^ i$ is $(m - i)$-pseudo-coherent relative to $R$ for all $i$. Then $K^\bullet $ is $m$-pseudo-coherent relative to $R$. In particular, if $K^\bullet $ is a bounded above complex of $A$-modules pseudo-coherent relative to $R$, then $K^\bullet $ is pseudo-coherent relative to $R$.

**Proof.**
Immediate from Lemma 15.64.9 and the definitions.
$\square$

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