## 65.7 Properties of Spaces defined by properties of schemes

Any étale local property of schemes gives rise to a corresponding property of algebraic spaces via the following lemma.

Lemma 65.7.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{P}$ be a property of schemes which is local in the étale topology, see Descent, Definition 35.15.1. The following are equivalent

for some scheme $U$ and surjective étale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$, and

for every scheme $U$ and every étale morphism $U \to X$ the scheme $U$ has property $\mathcal{P}$.

If $X$ is representable this is equivalent to $\mathcal{P}(X)$.

**Proof.**
The implication (2) $\Rightarrow $ (1) is immediate. For the converse, choose a surjective étale morphism $U \to X$ with $U$ a scheme that has $\mathcal{P}$ and let $V$ be an étale $X$-scheme. Then $U \times _ X V \rightarrow V$ is an étale surjection of schemes, so $V$ inherits $\mathcal{P}$ from $U \times _ X V$, which in turn inherits $\mathcal{P}$ from $U$ (see discussion following Descent, Definition 35.15.1). The last claim is clear from (1) and Descent, Definition 35.15.1.
$\square$

Definition 65.7.2. Let $\mathcal{P}$ be a property of schemes which is local in the étale topology. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ *has property $\mathcal{P}$* if any of the equivalent conditions of Lemma 65.7.1 hold.

Any étale local property of germs of schemes gives rise to a corresponding property of algebraic spaces. Here is the obligatory lemma.

Lemma 65.7.4. Let $\mathcal{P}$ be a property of germs of schemes which is étale local, see Descent, Definition 35.21.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. Consider étale morphisms $a : U \to X$ where $U$ is a scheme. The following are equivalent

for any $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$, and

for some $U \to X$ as above and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$.

If $X$ is representable, then this is equivalent to $\mathcal{P}(X, x)$.

**Proof.**
Omitted.
$\square$

Definition 65.7.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $\mathcal{P}$ be a property of germs of schemes which is étale local. We say $X$ *has property $\mathcal{P}$ at $x$* if any of the equivalent conditions of Lemma 65.7.4 hold.

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