Proposition 4

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

Algebra, chapter X, § 4. Produits de torsion

Let $u:C\to C'$ be a homology isomorphism of complexes of right A-modules and $E$ a complex of left A-modules, flat and bounded on the right. Then

u\otimes 1_E:C\otimes_A E\to C'\otimes_A E

is a homology isomorphism of complexes of $k$ -modules.

Indeed, by X, p. 38, Cor., $u$ (resp. $u\otimes 1_E$ ) is a homology isomorphism if and only if $H(\operatorname{Con}(u))=0$ (resp. $H(\operatorname{Con}(u\otimes 1_E))=0$ ). The conclusion then follows from Lemmas 1 and 2.

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

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