The Stacks project

Proof. Let $\mathcal{F}^\bullet $ be any complex representing $E$ and let $X = \bigcup U_ i$ and $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ be as in (1). We will show that $\mathcal{F}^\bullet $ is $m$-pseudo-coherent as a complex, which will prove (1) and (2) simultaneously. By Lemma 20.46.8 we can after refining the open covering $X = \bigcup U_ i$ represent the maps $\alpha _ i$ by maps of complexes $\alpha _ i : \mathcal{E}_ i^\bullet \to \mathcal{F}^\bullet |_{U_ i}$. By assumption $H^ j(\alpha _ i)$ are isomorphisms for $j > m$, and $H^ m(\alpha _ i)$ is surjective whence $\mathcal{F}^\bullet $ is $m$-pseudo-coherent. $\square$