Corollary 1

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

Commutative Algebra, chapter X, § 9. Modules dualisants

Let A be an integral noetherian ring. Every dualizing A-module is torsion-free and of rank 1.

Let \Omega be a dualizing A-module; it is torsion-free by prop. 4. Let K be the field of fractions of A; the K-vector space K \otimes_A \Omega is dualizing (n° 1, prop. 2), hence of dimension 1.

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

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