Definition 15.64.1. Let $R$ be a ring. Denote $D(R)$ its derived category. Let $m \in \mathbf{Z}$.

1. An object $K^\bullet$ of $D(R)$ is $m$-pseudo-coherent if there exists a bounded complex $E^\bullet$ of finite free $R$-modules and a morphism $\alpha : E^\bullet \to K^\bullet$ such that $H^ i(\alpha )$ is an isomorphism for $i > m$ and $H^ m(\alpha )$ is surjective.

2. An object $K^\bullet$ of $D(R)$ is pseudo-coherent if it is quasi-isomorphic to a bounded above complex of finite free $R$-modules.

3. An $R$-module $M$ is called $m$-pseudo-coherent if $M[0]$ is an $m$-pseudo-coherent object of $D(R)$.

4. An $R$-module $M$ is called pseudo-coherent1 if $M[0]$ is a pseudo-coherent object of $D(R)$.

[1] This clashes with what is meant by a pseudo-coherent module in .

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