The Stacks project

Definition 77.3.1. In Situation 77.2.1.

1. We say $\mathcal{F}$ is pure above $y$ if none of the equivalent conditions of Lemma 77.2.5 hold.

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

3. We say that $X$ is pure above $y$ if $\mathcal{O}_ X$ is pure above $y$.

4. We say $\mathcal{F}$ is universally $Y$-pure, or universally pure relative to $Y$ if $\mathcal{F}$ is universally pure above $y$ for every $y \in |Y|$.

5. We say $\mathcal{F}$ is $Y$-pure, or pure relative to $Y$ if $\mathcal{F}$ is pure above $y$ for every $y \in |Y|$.

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