The Stacks project

Lemma 66.22.7. Let $\mathcal{P}$ be a property of morphisms of schemes which is étale local on the source-and-target. Consider the property $\mathcal{Q}$ of morphisms of germs associated to $\mathcal{P}$ in Descent, Lemma 35.33.2. Then

  1. $\mathcal{Q}$ is étale local on the source-and-target.

  2. given a morphism of algebraic spaces $f : X \to Y$ and $x \in |X|$ the following are equivalent

    1. $f$ has $\mathcal{Q}$ at $x$, and

    2. there is an open neighbourhood $X' \subset X$ of $x$ such that $X' \to Y$ has $\mathcal{P}$.

  3. given a morphism of algebraic spaces $f : X \to Y$ the following are equivalent:

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

    2. for every $x \in |X|$ the morphism $f$ has $\mathcal{Q}$ at $x$.

Proof. See Descent, Lemma 35.33.2 for (1). The implication (1)(a) $\Rightarrow $ (2)(b) follows on letting $X' = a(U) \subset X$ given a diagram as in Lemma 66.22.5. The implication (2)(b) $\Rightarrow $ (1)(a) is clear. The equivalence of (3)(a) and (3)(b) follows from the corresponding result for morphisms of schemes, see Descent, Lemma 35.33.3. $\square$

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