## 99.7 Properties of algebraic stacks defined by properties of schemes

Any smooth local property of schemes gives rise to a corresponding property of algebraic stacks via the following lemma. Note that a property of schemes which is smooth local is also étale local as any étale covering is also a smooth covering. Hence for a smooth local property $P$ of schemes we know what it means to say that an algebraic space has $P$, see Properties of Spaces, Section 65.7.

Lemma 99.7.1. Let $\mathcal{P}$ be a property of schemes which is local in the smooth topology, see Descent, Definition 35.15.1. Let $\mathcal{X}$ be an algebraic stack. The following are equivalent

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

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

3. for some algebraic space $U$ and some surjective smooth morphism $U \to \mathcal{X}$ the algebraic space $U$ has property $\mathcal{P}$, and

4. for every algebraic space $U$ and every smooth morphism $U \to \mathcal{X}$ the algebraic space $U$ has property $\mathcal{P}$.

If $\mathcal{X}$ is a scheme this is equivalent to $\mathcal{P}(U)$. If $\mathcal{X}$ is an algebraic space this is equivalent to $X$ having property $\mathcal{P}$.

Proof. Let $U \to \mathcal{X}$ surjective and smooth with $U$ an algebraic space. Let $V \to \mathcal{X}$ be a smooth morphism with $V$ an algebraic space. Choose schemes $U'$ and $V'$ and surjective étale morphisms $U' \to U$ and $V' \to V$. Finally, choose a scheme $W$ and a surjective étale morphism $W \to V' \times _\mathcal {X} U'$. 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 66.39.6 and 66.37.2. Moreover, $W \to V'$ is surjective as $U' \to \mathcal{X}$ is surjective. Hence, we have

$\mathcal{P}(U) \Leftrightarrow \mathcal{P}(U') \Rightarrow \mathcal{P}(W) \Rightarrow \mathcal{P}(V') \Leftrightarrow \mathcal{P}(V)$

where the equivalences are by definition of property $\mathcal{P}$ for algebraic spaces, and the two implications come from Descent, Definition 35.15.1. This proves (3) $\Rightarrow$ (4).

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

Definition 99.7.2. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{P}$ be a property of schemes which is local in the smooth topology. We say $\mathcal{X}$ has property $\mathcal{P}$ if any of the equivalent conditions of Lemma 99.7.1 hold.

Remark 99.7.3. Here is a list of properties which are local for the smooth topology (keep in mind that the fpqc, fppf, and syntomic topologies are stronger than the smooth topology):

1. locally Noetherian, see Descent, Lemma 35.16.1,

2. Jacobson, see Descent, Lemma 35.16.2,

3. locally Noetherian and $(S_ k)$, see Descent, Lemma 35.17.1,

4. Cohen-Macaulay, see Descent, Lemma 35.17.2,

5. reduced, see Descent, Lemma 35.18.1,

6. normal, see Descent, Lemma 35.18.2,

7. locally Noetherian and $(R_ k)$, see Descent, Lemma 35.18.3,

8. regular, see Descent, Lemma 35.18.4,

9. Nagata, see Descent, Lemma 35.18.5.

Any smooth local property of germs of schemes gives rise to a corresponding property of algebraic stacks. Note that a property of germs which is smooth local is also étale local. Hence for a smooth local property of germs of schemes $P$ we know what it means to say that an algebraic space $X$ has property $P$ at $x \in |X|$, see Properties of Spaces, Section 99.7.

Lemma 99.7.4. 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 99.4.3 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 65.4.4). 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 66.39.6 and 66.37.2. 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 65.7.5 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. $\square$

Definition 99.7.5. Let $\mathcal{P}$ be a property of germs of schemes which is smooth local. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. We say $\mathcal{X}$ has property $\mathcal{P}$ at $x$ if any of the equivalent conditions of Lemma 99.7.4 holds.

Comment #2678 by on

Properties of spaces reference to 90.7 at the end of the first paragraph should be to 57.7

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