Lemma 20.43.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $E$ be an object of $D(\mathcal{O}_ X)$.

1. If there exists an open covering $X = \bigcup U_ i$, strictly perfect complexes $\mathcal{E}_ i^\bullet$ on $U_ i$, and maps $\alpha _ i : \mathcal{E}_ i^\bullet \to E|_{U_ i}$ in $D(\mathcal{O}_{U_ i})$ with $H^ j(\alpha _ i)$ an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ surjective, then $E$ is $m$-pseudo-coherent.

2. If $E$ is $m$-pseudo-coherent, then any complex representing $E$ is $m$-pseudo-coherent.

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 20.42.8 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. $\square$

Comment #2770 by on

There is a typo in the first line of the proof. It should be $\alpha_i\colon\mathcal{E}^\bullet_i\rightarrow E|_{U_i}$ ($\bullet$ is missing on $\mathcal{E}_i$).

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