Corollary 2

Tag 0542, permanent. Cite it as [Bourbaki, Tag 0542].

Commutative Algebra, chapter VII, § 4. Modules over integrally closed Noetherian domains

If a divisorial fractional ideal $a \neq 0$ of $\mathbf{A}$ admits a finite free resolution, it is principal.

In fact we apply Corollary 1 to a finite free resolution of $a$ :

0 \to L_n \to L_{n-1} \to \cdots \to L_0 \to a \to 0.

By virtue of Proposition 16 (v), $c(a) = 0$ and hence, by virtue of Proposition 16 (iv), $\operatorname{div}(a)$ is principal; as $a$ is assumed to be divisorial, it is principal (§ 1, no. 1).

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

Cited by

Corollary 3, § 4
Proposition 9, § 4

Cites

§ 1, no. 1
Proposition 16
Corollary 1