Lemma 15.108.7 (0C2D)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.108: Branches of the completion
Lemma 15.108.7 (cite)

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Lemma 15.108.7. Let $(A, \mathfrak m)$ be a Noetherian local ring of dimension $1$ . Then the number of (geometric) branches of $A$ and $A^\wedge$ is the same.

Proof. To see this for the number of branches, combine Lemmas 15.108.1, 15.108.2, and 15.108.5 and use that the dimension of $A^\wedge$ is one, see Lemma 15.43.1. To see this is true for the number of geometric branches we use the result for branches, the fact that the dimension does not change under strict henselization (Lemma 15.45.7), and the fact that $(A^{sh})^\wedge = ((A^\wedge )^{sh})^\wedge$ by Lemma 15.45.3. $\square$

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