Corollary 1

Tag 0DV9, permanent. Cite it as [Bourbaki, Tag 0DV9].

Algebra, chapter X, § 4. Produits de torsion

Let $C'$ be a split complex of left $A$ -modules such that $H(C')$ is flat. For every complex $C$ of right $A$ -modules, the canonical mapping

\gamma(C,C'):H(C)\otimes_AH(C')\longrightarrow H(C\otimes_A C')

is bijective.

By X, p. 35, def. 6, there exists a homotopy equivalence $u':C'\to H(C')$ . By prop. 3, $1_C\otimes u':C\otimes_A C'\to C\otimes_AH(C')$ is a homotopy equivalence; since

H(1_C\otimes u')\circ\gamma(C,C')=\gamma(C,H(C'))\circ(1_C\otimes H(u')),

and $H(1_C\otimes u')$ and $H(u')$ are bijective, it suffices to prove that $\gamma(C,H(C'))$ is bijective, and we are reduced to the case where $C'$ is flat and has zero differential. In this case the canonical exact sequences

\mathrm{(I)}\qquad 0\longrightarrow Z(C)\xrightarrow{i}C\xrightarrow{\partial}B(C)\longrightarrow0

\mathrm{(II)}\qquad 0\longrightarrow B(C)\xrightarrow{j}Z(C)\xrightarrow{\pi}H(C)\longrightarrow0

give exact sequences:

0 \longrightarrow Z(C) \otimes_A C' \xrightarrow{i\otimes 1} C \otimes_A C' \xrightarrow{\delta\otimes 1} B(C) \otimes_A C' \longrightarrow 0

0 \longrightarrow B(C) \otimes_A C' \xrightarrow{j\otimes 1} Z(C) \otimes_A C' \xrightarrow{\pi\otimes 1} H(C) \otimes_A C' \longrightarrow 0.

Since $d=i\circ j\circ\delta$ , we have $D=d\otimes 1_{C'}=(i\otimes 1)\circ(j\otimes 1)\circ(\delta\otimes 1)$ , which shows that the canonical mappings $Z(C)\otimes_A C'\to Z(C\otimes_A C')$ and $B(C)\otimes_A C'\to B(C\otimes_A C')$ are bijective, hence also $\gamma(C,C')$ by passing to quotients.

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

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Cites

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