The Stacks project

Lemma 68.24.3. Let $S$ be a scheme. Let $X$ be a decent 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 elementary étale neighbourhood $(U, u) \to (X, x)$ the number of minimal primes of the local ring $\mathcal{O}_{U, u}$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$,

  2. for any elementary étale neighbourhood $(U, u) \to (X, x)$ the number irreducible components of $U$ passing through $u$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$,

  3. for any elementary étale neighbourhood $(U, u) \to (X, x)$ the number of branches of $U$ at $u$ is $\leq n$ and for at least one choice of $(U, u)$ it is $n$,

  4. the number of minimal prime ideals of $\mathcal{O}_{X, x}^ h$ 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 $u$. The ring $\mathcal{O}_{X, x}$ is the henselization of $\mathcal{O}_{U, u}$, see discussion following Definition 68.11.7. In particular (3) and (4) are equivalent by More on Algebra, Lemma 15.106.3. The equivalence of (2) and (3) follows from More on Morphisms, Lemma 37.36.2. $\square$

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