Corollary 2

Tag 04CA, permanent. Cite it as [Bourbaki, Tag 04CA].

Commutative Algebra, chapter II, § 5. Finitely generated projective modules. Invertible fractional ideals

If the ring $B$ is semi-local, the group $\mathfrak{C}$ of classes of invertible sub-A-modules of B is canonically identified with the group $P(A)$ of classes of projective A-modules of rank 1.

In this case $P(B) = 0$ (no. 3, Proposition 5).

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

Cited by

Remark, § 5
Proposition 4, § 1

Cites

no. 3, Proposition 5