The Stacks project

Lemma 38.16.3. 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 $s \in S$. The following are equivalent

  1. $\mathcal{F}$ is universally pure along $X_ s$, and

  2. for every morphism of pointed schemes $(S', s') \to (S, s)$ the pullback $\mathcal{F}_{S'}$ is pure along $X_{s'}$.

In particular, $\mathcal{F}$ is universally pure relative to $S$ if and only if every base change $\mathcal{F}_{S'}$ of $\mathcal{F}$ is pure relative to $S'$.

Proof. This is formal. $\square$

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