The Stacks project

Lemma 64.7.1. 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 35.12.1. The following are equivalent

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

  2. 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 35.12.1). The last claim is clear from (1) and Descent, Definition 35.12.1. $\square$


Comments (2)

Comment #500 by Kestutis Cesnavicius on

Proof: 2) => 1) is clear. For the converse, take a surjective \'{e}tale with a scheme that has and let be an \'{e}tale -scheme. Then is an \'{e}tale surjection of schemes, so inherits from , which in turn inherits from .

The last claim is clear from 1) and Definition 34.11.1.


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