The Stacks project

Lemma 35.33.3. Let $\mathcal{P}$ be a property of morphisms of schemes which is ├ętale local on source-and-target. Let $Q$ be the associated property of morphisms of germs, see Lemma 35.33.2. Let $f : X \to Y$ be a morphism of schemes. The following are equivalent:

  1. $f$ has property $\mathcal{P}$, and

  2. for every $x \in X$ the morphism of germs $(X, x) \to (Y, f(x))$ has property $\mathcal{Q}$.

Proof. The implication (1) $\Rightarrow $ (2) is direct from the definitions. The implication (2) $\Rightarrow $ (1) also follows from part (3) of Definition 35.32.3. $\square$

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