Corollary

Tag 0CPK, permanent. Cite it as [Bourbaki, Tag 0CPK].

Commutative Algebra, chapter VIII, § 5. Anneaux locaux réguliers

A local noetherian ring A is regular if and only if its completion $\hat{A}$ is so.

In fact, $\hat{A}$ is flat over A, and one has $m_{\hat{A}} = \hat{A}.m_A$ (III, § 3, No. 4, Theorem 3 and § 2, No. 12, Cor. 2 to Prop. 16).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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