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

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

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

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

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