The Stacks project

Lemma 100.7.4. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$ be a point of $\mathcal{X}$. Let $\mathcal{P}$ be a property of germs of schemes which is smooth local, see Descent, Definition 35.21.1. The following are equivalent

  1. for any smooth morphism $U \to \mathcal{X}$ with $U$ a scheme and $u \in U$ with $a(u) = x$ we have $\mathcal{P}(U, u)$,

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

  3. for any smooth morphism $U \to \mathcal{X}$ with $U$ an algebraic space and $u \in |U|$ with $a(u) = x$ the algebraic space $U$ has property $\mathcal{P}$ at $u$, and

  4. for some smooth morphism $U \to \mathcal{X}$ with $U$ a an algebraic space and some $u \in |U|$ with $a(u) = x$ the algebraic space $U$ has property $\mathcal{P}$ at $u$.

If $\mathcal{X}$ is representable, then this is equivalent to $\mathcal{P}(\mathcal{X}, x)$. If $\mathcal{X}$ is an algebraic space then this is equivalent to $\mathcal{X}$ having property $\mathcal{P}$ at $x$.

Proof. Let $a : U \to \mathcal{X}$ and $u \in |U|$ as in (3). Let $b : V \to \mathcal{X}$ be another smooth morphism with $V$ an algebraic space and $v \in |V|$ with $b(v) = x$ also. Choose a scheme $U'$, an étale morphism $U' \to U$ and $u' \in U'$ mapping to $u$. Choose a scheme $V'$, an étale morphism $V' \to V$ and $v' \in V'$ mapping to $v$. By Lemma 100.4.3 there exists a point $\overline{w} \in |V' \times _\mathcal {X} U'|$ mapping to $u'$ and $v'$. Choose a scheme $W$ and a surjective étale morphism $W \to V' \times _\mathcal {X} U'$. We may choose a $w \in |W|$ mapping to $\overline{w}$ (see Properties of Spaces, Lemma 66.4.4). Then $W \to V'$ and $W \to U'$ are smooth morphisms of schemes as compositions of étale and smooth morphisms of algebraic spaces, see Morphisms of Spaces, Lemmas 67.39.6 and 67.37.2. Hence

\[ \mathcal{P}(U, u) \Leftrightarrow \mathcal{P}(U', u') \Leftrightarrow \mathcal{P}(W, w) \Leftrightarrow \mathcal{P}(V', v') \Leftrightarrow \mathcal{P}(V, v) \]

The outer two equivalences by Properties of Spaces, Definition 66.7.5 and the other two by what it means to be a smooth local property of germs of schemes. This proves (4) $\Rightarrow $ (3).

The implications (1) $\Rightarrow $ (2), (2) $\Rightarrow $ (4), and (3) $\Rightarrow $ (1) are immediate. $\square$

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