The Stacks project

Lemma 15.64.9. Let $R$ be a ring. Let $m \in \mathbf{Z}$. Let $K^\bullet $ be a bounded above complex of $R$-modules such that $K^ i$ is $(m - i)$-pseudo-coherent for all $i$. Then $K^\bullet $ is $m$-pseudo-coherent. In particular, if $K^\bullet $ is a bounded above complex of pseudo-coherent $R$-modules, then $K^\bullet $ is pseudo-coherent.

Proof. We may replace $K^\bullet $ by $\sigma _{\geq m - 1}K^\bullet $ (for example) and hence assume that $K^\bullet $ is bounded. Then the complex $K^\bullet $ is $m$-pseudo-coherent as each $K^ i[-i]$ is $m$-pseudo-coherent by induction on the length of the complex: use Lemma 15.64.2 and the stupid truncations. For the final statement, it suffices to prove that $K^\bullet $ is $m$-pseudo-coherent for all $m \in \mathbf{Z}$, see Lemma 15.64.5. This follows from the first part. $\square$

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