The Stacks project

Lemma 15.64.10. Let $R$ be a ring. Let $m \in \mathbf{Z}$. Let $K^\bullet \in D^{-}(R)$ such that $H^ i(K^\bullet )$ is $(m - i)$-pseudo-coherent (resp. pseudo-coherent) for all $i$. Then $K^\bullet $ is $m$-pseudo-coherent (resp. pseudo-coherent).

Proof. Assume $K^\bullet $ is an object of $D^{-}(R)$ such that each $H^ i(K^\bullet )$ is $(m - i)$-pseudo-coherent. Let $n$ be the largest integer such that $H^ n(K^\bullet )$ is nonzero. We will prove the lemma by induction on $n$. If $n < m$, then $K^\bullet $ is $m$-pseudo-coherent by Lemma 15.64.7. If $n \geq m$, then we have the distinguished triangle

\[ (\tau _{\leq n - 1}K^\bullet , K^\bullet , H^ n(K^\bullet )[-n]) \]

(Derived Categories, Remark 13.12.4) Since $H^ n(K^\bullet )[-n]$ is $m$-pseudo-coherent by assumption, we can use Lemma 15.64.2 to see that it suffices to prove that $\tau _{\leq n - 1}K^\bullet $ is $m$-pseudo-coherent. By induction on $n$ we win. (The pseudo-coherent case follows from this and Lemma 15.64.5.) $\square$

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