Corollary 2

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

Algebra, chapter X, § 4. Produits de torsion

Let $0 \to C' \xrightarrow{u} C \xrightarrow{v} C'' \to 0$ be an exact sequence of complexes of left $A$ -modules, and let $E$ be a complex of right $A$ -modules. If $C''$ or $E$ is flat, the sequence

0 \longrightarrow E \otimes_A C' \xrightarrow{1 \otimes u} E \otimes_A C \xrightarrow{1 \otimes v} E \otimes_A C'' \longrightarrow 0

is exact.

Indeed, $\mathrm{Tor}_1^A(E, C'') = 0$ according to X, p. 69, Corollary to prop. 5.

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

Cited by

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Cites

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