Lemma 34.20.28. The property $\mathcal{P}(f) =$“$f$ is unramified” is fpqc local on the base. The property $\mathcal{P}(f) =$“$f$ is G-unramified” is fpqc local on the base.

Proof. A morphism is unramified (resp. G-unramified) if and only if it is locally of finite type (resp. finite presentation) and its diagonal morphism is an open immersion (see Morphisms, Lemma 28.33.13). We have seen already that being locally of finite type (resp. locally of finite presentation) and an open immersion is fpqc local on the base (Lemmas 34.20.11, 34.20.10, and 34.20.16). Hence the result follows formally. $\square$

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