The Stacks project

Definition 37.59.2. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Fix $m \in \mathbf{Z}$.

1. We say $E$ is $m$-pseudo-coherent relative to $S$ if there exists an affine open covering $S = \bigcup V_ i$ and for each $i$ an affine open covering $f^{-1}(V_ i) = \bigcup U_{ij}$ such that the equivalent conditions of Lemma 37.59.1 are satisfied for each of the pairs $(U_{ij} \to V_ i, E|_{U_{ij}})$.

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

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

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