Lemma 66.7.1 (03E8)—The Stacks project

Lemma 66.7.1. 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. $\square$

Comments (2)

Comment #500 by Kestutis Cesnavicius on April 01, 2014 at 17:17

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

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

Comment #509 by Johan on April 02, 2014 at 19:50

Thanks! Proof added here.

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