Lemma 1

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

Commutative Algebra, chapter I, § 2. Flat modules

Let $M' \xrightarrow{v} M \xrightarrow{w} M'' \to 0$ be an exact sequence of left A-modules and E a right A-module. Then the sequence

E \otimes_A M' \xrightarrow{\text{1}\otimes v} E \otimes_A M \xrightarrow{u \otimes w} E \otimes_A M'' \longrightarrow 0

is an exact sequence of commutative groups.

This is the Corollary to Proposition 5 of Algebra, Chapter II, § 3, no. 6.

It follows that, for any left A-module homomorphism $u : M \to N$ ,

E \otimes_A (\operatorname{Coker} u)

is canonically identified with $\operatorname{Coker}(1_E \otimes u)$ , as Lemma 1 shows when applied to the exact sequence

M \xrightarrow{u} N \longrightarrow \operatorname{Coker} u \longrightarrow 0.

In the notation of Lemma 1, we know (loc. cit.) that if v is injective, that is if the sequence $0 \to M' \xrightarrow{v} M \xrightarrow{w} M'' \to 0$ is exact, it does not necessarily follow that $1_E \otimes v$ is injective and so $E \otimes_A M'$ cannot in general be identified with a

(*) We inform readers already familiar with Homological Algebra that they will find other characterizations of flat modules in § 4.

subgroupof $E \otimes_A M$ . Recall however (Algebra, Chapter II, § 3, no. 7, Corollary 5 to Proposition 7) the following result:

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

Lemma 1

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