The Stacks project

Definition 76.45.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Fix $m \in \mathbf{Z}$.

1. We say $E$ is $m$-pseudo-coherent relative to $Y$ if the equivalent conditions of Lemma 76.45.2 are satisfied.

2. We say $E$ is pseudo-coherent relative to $Y$ if $E$ is $m$-pseudo-coherent relative to $Y$ for all $m \in \mathbf{Z}$.

3. We say $\mathcal{F}$ is $m$-pseudo-coherent relative to $Y$ if $\mathcal{F}$ viewed as an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is $m$-pseudo-coherent relative to $Y$.

4. We say $\mathcal{F}$ is pseudo-coherent relative to $Y$ if $\mathcal{F}$ viewed as an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ is pseudo-coherent relative to $Y$.