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 $m \in \mathbf{Z}$.

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 $(U_{ij} \to V_ i, E|_{U_{ij}})$.We say $E$ is

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

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