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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