Corollary 1

Tag 0DW1, permanent. Cite it as [Bourbaki, Tag 0DW1].

Algebra, chapter X, § 4. Produits de torsion

Let $0 \to E' \to E \to E'' \to 0$ be an exact sequence of right $A$ -modules. Assume that $E''$ is flat. Then $E$ is flat if and only if $E'$ is flat.

Let $F$ be a left $A$ -module. Since $\operatorname{Tor}^A_i(E'', F)=0$ for $i=1,2$ (Th. 2, (i) $\Rightarrow$ (ii)), we have an exact sequence

0\longrightarrow \operatorname{Tor}^A_1(E',F)\longrightarrow \operatorname{Tor}^A_1(E,F)\longrightarrow 0

whence the assertion (Th. 2, (i)

\Leftrightarrow

(iii)).

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

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