The Stacks project

Lemma 66.23.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. Let $n \in \{ 1, 2, \ldots \}$ be an integer. The following are equivalent

1. for any scheme $U$ and étale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number of minimal primes of the local ring $\mathcal{O}_{U, u}$ is $\leq n$ and for at least one choice of $U, a, u$ it is $n$,

2. for any scheme $U$ and étale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number irreducible components of $U$ passing through $u$ is $\leq n$ and for at least one choice of $U, a, u$ it is $n$,

3. for any scheme $U$ and étale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number of branches of $U$ at $u$ is $\leq n$ and for at least one choice of $U, a, u$ it is $n$,

4. for any scheme $U$ and étale morphism $a : U \to X$ and $u \in U$ with $a(u) = x$ the number of geometric branches of $U$ at $u$ is $n$, and

5. the number of minimal prime ideals of $\mathcal{O}_{X, \overline{x}}$ is $n$.