Lemma 15.42.1. Let $A$ be a Noetherian local ring. Then $\dim (A) = \dim (A^\wedge )$.

**Proof.**
By Algebra, Lemma 10.96.4 the map $A \to A^\wedge $ induces isomorphisms $A/\mathfrak m^ n = A^\wedge /(\mathfrak m^\wedge )^ n$ for $n \geq 1$. By Algebra, Lemma 10.51.12 this implies that

for all $n \geq 1$. Thus $d(A) = d(A^\wedge )$ and we conclude by Algebra, Proposition 10.59.8. An alternative proof is to use Algebra, Lemma 10.111.7. $\square$

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