The Stacks project

Lemma 28.15.6. Let $X$ be a scheme. Let $x \in X$.

  1. The number of branches of $X$ at $x$ is $1$ if and only if $X$ is unibranch at $x$.

  2. The number of geometric branches of $X$ at $x$ is $1$ if and only if $X$ is geometrically unibranch at $x$.

Proof. This lemma follows immediately from the definitions and the corresponding result for rings, see More on Algebra, Lemma 15.106.7. $\square$

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