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 $n$.

Proof. The equivalence of (1) and (2) follows from the fact that irreducible components of $U$ passing through $u$ are in $1$-$1$ correspondence with minimal primes of the local ring of $U$ at $u$. Let $a : U \to X$ and $u \in U$ be as in (1). Then $\mathcal{O}_{X, \overline{x}}$ is the strict henselization of $\mathcal{O}_{U, u}$ by Lemma 66.22.1. Recall that the (geometric) number of branches of $U$ at $u$ is the number of minimal prime ideals of the (strict) henselization of $\mathcal{O}_{U, u}$. In particular (4) and (5) are equivalent. The equivalence of (2), (3), and (4) follows from More on Morphisms, Lemma 37.36.2. $\square$

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