Corollary

Tag 042X, permanent. Cite it as [Bourbaki, Tag 042X].

Commutative Algebra, chapter I, § 2. Flat modules

Let E be a right A-module and E' a submodule of E.

(i) *Suppose that E/E' is flat. Then, for every left ideal a of A,

E'a = E' \cap Ea.

(ii) Conversely, suppose that E is flat and, for every finitely generated left ideal a of A, relation (5) holds. Then E/E' is flat.

(i) It is sufficient to apply Proposition to the case $F = A$ , $F' = a$ . (ii) To prove that $E/E'$ is flat, apply criterion (c) of Proposition 4 of no. 5; it is then necessary to establish that the sequence

0 \to E'/E'a \to E/Ea \to E/(E' + Ea) \to 0

is exact at

E'/E'a

for every finitely generated left ideal a of A. Now this is precisely what relation (5) expresses.

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

Cited by

Exercise 16, § 2
Exercise 23, § 2
Proposition 9, § 3

Cites

Proposition 4