The Stacks project

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

  1. 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.

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

  3. 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.

Proof. Let $\mathcal{F}^\bullet $ be any complex representing $E$ and let $X$, $\{ U_ i \to X\} $, and $\alpha _ i : \mathcal{E}_ i \to E|_{U_ i}$ be as in (1). We will show that $\mathcal{F}^\bullet $ is $m$-pseudo-coherent as a complex, which will prove (1) and (2) in case $\mathcal{C}$ has a final object. By Lemma 21.44.8 we can after refining the covering $\{ U_ i \to X\} $ 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.

Proof of (2). By the above we see that $\mathcal{F}^\bullet |_ U$ is $m$-pseudo-coherent as a complex of $\mathcal{O}_ U$-modules for all objects $U$ of $\mathcal{C}$. It is a formal consequence of the definitions that $\mathcal{F}^\bullet $ is $m$-pseudo-coherent.

Proof of (3). Follows from the definitions and Sites, Definition 7.6.2 part (2). $\square$

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