Definition 27.15.4. Let $X$ be a scheme. Let $x \in X$. The number of branches of $X$ at $x$ is the number of branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition 15.95.6. The number of geometric branches of $X$ at $x$ is the number of geometric branches of the local ring $\mathcal{O}_{X, x}$ as defined in More on Algebra, Definition 15.95.6.

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