Proposition 7

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

Commutative Algebra, chapter I, § 2. Flat modules

Let E be a right A-module, E' a submodule of E, F a left A-module and F' a submodule of F. Suppose that E/E' or F/F' is flat. Then the canonical image of $E' \otimes F'$ in $E \otimes F$ is the intersection of the canonical images of $E' \otimes F$ and $E \otimes F'$ in $E \otimes F$ .

Suppose for example that $E/E'$ is flat and consider the diagram

\begin{array}{ccc} E' \otimes F' & \longrightarrow & E \otimes F' \\ \downarrow & & \downarrow \\ E' \otimes F & \longrightarrow & E \otimes F \end{array}

\begin{array}{ccc} & & (E/E') \otimes F' \\ E \otimes F' & \longrightarrow & (E/E') \otimes F \end{array}

where the arrows are the canonical homomorphisms. This diagram is commutative and its rows are exact (no. 1, Lemma 1). As

E/E'

is flat,

u

is injective. Then our assertion is a special case of § 1, no. 4, Proposition 1, (i).

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

Proposition 7

Cited by

Cites