The Stacks project

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.

1. 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.

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

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

4. 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 $s \in S$.

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

6. We say that $X$ is $S$-pure or pure relative to $S$ if $\mathcal{O}_ X$ is pure relative to $S$.