Corollary 2 (Chevalley)

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

Commutative Algebra, chapter IX, § 4. Fermeture intégrale d'un anneau local complet

Let A be a reduced algebra of finite type over a field, and p a prime ideal of A. The completion of the local ring $A_p$ is reduced. Since A is reduced, the local ring $A_p$ is reduced; moreover A is a Nagata ring (example 1), hence $A_p$ is a Nagata ring (Prop. 4), and Cor. 2 follows from Cor. 1, applied to the ring $A_p$ .

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

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