Proposition 2

Tag 0D5N, permanent. Cite it as [Bourbaki, Tag 0D5N].

Commutative Algebra, chapter X, § 4. Anneaux réguliers

A regular noetherian local ring is factorial.

By prop. 1, every finitely generated module over a regular noetherian local ring admits a projective resolution of finite length by finitely generated projective modules, hence free modules (II, § 3, No. 2, cor. 2 of prop. 5). It then follows from VII, § 4, No. 7, cor. 3 of prop. 16 that such a ring is factorial.

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

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