Remark

Tag 043B, permanent. Cite it as [Bourbaki, Tag 043B].

Commutative Algebra, chapter I, § 2. Flat modules

If there is no ambiguity over the modules E, nor the bilinear mapping $f$ , $F_3 : F_2$ is sometimes used to denote the module denoted by T in Proposition 12 and it is called the transporter of F, to F,. The conclusion of Proposition 12 then reads

F_{3(B)} : F_{2(B)} = (F_3 : F_2)_{(B)}.

In the particular case when the $E_i$ are equal to the ring $A, f$ is multiplication and the $F_i$ ideals $a_i$ , we obtain the transporterformula

B(a_3 : a_2) = Ba_3 : Ba_2

valid when $B$ is a flat $A$ -module and $a_2$ is a finitely generated ideal.

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

Cited by

Nothing.

Cites

Proposition 12