Lemma 21.45.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $E$ be an object of $D(\mathcal{O})$.

If $\mathcal{C}$ has a final object $X$ and if there exist a covering $\{ U_ i \to X\} $, strictly perfect complexes $\mathcal{E}_ i^\bullet $ of $\mathcal{O}_{U_ i}$-modules, 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.

If $E$ is $m$-pseudo-coherent, then any complex of $\mathcal{O}$-modules representing $E$ is $m$-pseudo-coherent.

If for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ such that $E|_{U_ i}$ is $m$-pseudo-coherent, then $E$ is $m$-pseudo-coherent.

## Comments (0)