The Stacks project

Definition 21.45.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{E}^\bullet$ be a complex of $\mathcal{O}$-modules. Let $m \in \mathbf{Z}$.

1. We say $\mathcal{E}^\bullet$ is $m$-pseudo-coherent if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\}$ and for each $i$ a morphism of complexes $\alpha _ i : \mathcal{E}_ i^\bullet \to \mathcal{E}^\bullet |_{U_ i}$ where $\mathcal{E}_ i^\bullet$ is a strictly perfect complex of $\mathcal{O}_{U_ i}$-modules and $H^ j(\alpha _ i)$ is an isomorphism for $j > m$ and $H^ m(\alpha _ i)$ is surjective.

2. We say $\mathcal{E}^\bullet$ is pseudo-coherent if it is $m$-pseudo-coherent for all $m$.

3. We say an object $E$ of $D(\mathcal{O})$ 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 $\mathcal{O}$-modules.