Definition 20.45.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{E}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. Let $m \in \mathbf{Z}$.

We say $\mathcal{E}^\bullet $ is

*$m$-pseudo-coherent*if there exists an open covering $X = \bigcup U_ i$ and for each $i$ a morphism of complexes $\alpha _ i : \mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{U_ i}$ where $\mathcal{E}_ i$ is strictly perfect on $U_ i$ and $H^ j(\alpha _ i)$ is an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ is surjective.We say $\mathcal{E}^\bullet $ is

*pseudo-coherent*if it is $m$-pseudo-coherent for all $m$.We say an object $E$ of $D(\mathcal{O}_ X)$ is

*$m$-pseudo-coherent*(resp.*pseudo-coherent*) if and only if it can be represented by a $m$-pseudo-coherent (resp. pseudo-coherent) complex of $\mathcal{O}_ X$-modules.

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