Lemma 20.45.8. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $m \in \mathbf{Z}$. Let $E$ be an object of $D(\mathcal{O}_ X)$. If $E$ is (locally) bounded above and $H^ i(E)$ is $(m - i)$-pseudo-coherent for all $i$, then $E$ is $m$-pseudo-coherent.

Proof. Omitted. Hint: use Lemma 20.45.4 and truncations as in the proof of More on Algebra, Lemma 15.64.10. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).