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

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

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

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

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

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

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

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

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