Proposition 4

Tag 051V, permanent. Cite it as [Bourbaki, Tag 051V].

Commutative Algebra, chapter VII, § 3. Factorial domains

Let $A$ be a Zariski ring and $\hat{A}$ its completion. If $\hat{A}$ is a factorial domain, $A$ is a factorial domain.

This follows from no. 1, Definition 1 and § 1, no. 10, Proposition 16.

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

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