The Stacks project

Lemma 15.108.1. Let $(A, \mathfrak m)$ be a Noetherian local ring.

  1. The map $A^ h \to A^\wedge $ defines a surjective map from minimal primes of $A^\wedge $ to minimal primes of $A^ h$.

  2. The number of branches of $A$ is at most the number of branches of $A^\wedge $.

  3. The number of geometric branches of $A$ is at most the number of geometric branches of $A^\wedge $.

Proof. By Lemma 15.45.3 the map $A^ h \to A^\wedge $ is flat and injective. Combining going down (Algebra, Lemma 10.39.19) and Algebra, Lemma 10.30.5 we see that part (1) holds. Part (2) follows from this, Definition 15.106.6, and the fact that $A^\wedge $ is henselian (Algebra, Lemma 10.153.9). By Lemma 15.45.3 we have $(A^\wedge )^{sh} = A^{sh} \otimes _{A^ h} A^\wedge $. Thus we can repeat the arguments above using the flat injective map $A^{sh} \to (A^\wedge )^{sh}$ to prove (3). $\square$

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