The Stacks project

Lemma 68.24.1. Let $S$ be a scheme. Let $X$ be a decent algebraic space over $S$. Let $x \in |X|$ be a point. The following are equivalent

  1. for any elementary étale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ has a unique minimal prime,

  2. for any elementary étale neighbourhood $(U, u) \to (X, x)$ there is a unique irreducible component of $U$ through $u$,

  3. for any elementary étale neighbourhood $(U, u) \to (X, x)$ the local ring $\mathcal{O}_{U, u}$ is unibranch,

  4. the henselian local ring $\mathcal{O}_{X, x}^ h$ has a unique minimal prime.

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 $\mathcal{O}_{X, x}^ h$ 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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