The Stacks project

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 $D(R)$.