Definition 38.16.1. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module.

Let $s \in S$. We say $\mathcal{F}$ is

*pure along $X_ s$*if there is no impurity $(g : T \to S, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $s$ with $(T, t) \to (S, s)$ an elementary étale neighbourhood.We say $\mathcal{F}$ is

*universally pure along $X_ s$*if there does not exist any impurity of $\mathcal{F}$ above $s$.We say that $X$ is

*pure along $X_ s$*if $\mathcal{O}_ X$ is pure along $X_ s$.We say $\mathcal{F}$ is

*universally $S$-pure*, or*universally pure relative to $S$*if $\mathcal{F}$ is universally pure along $X_ s$ for every $s \in S$.We say $\mathcal{F}$ is

*$S$-pure*, or*pure relative to $S$*if $\mathcal{F}$ is pure along $X_ s$ for every $s \in S$.We say that $X$ is

*$S$-pure*or*pure relative to $S$*if $\mathcal{O}_ X$ is pure relative to $S$.

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