The Stacks project

Remark 38.16.2. Let $f : X \to S$ be a morphism which is locally of finite type and $\mathcal{F}$ a quasi-coherent finite type $\mathcal{O}_ X$-module. In this case it is still true that (1) and (2) above are equivalent because the proof of Lemma 38.15.5 does not use that $f$ is quasi-compact. It is also clear that (3) and (4) are equivalent. However, we don't know if (1) and (3) are equivalent. In this case it may sometimes be more convenient to define purity using the equivalent conditions (3) and (4) as is done in [GruRay]. On the other hand, for many applications it seems that the correct notion is really that of being universally pure.