Corollary 1

Tag 0DW5, permanent. Cite it as [Bourbaki, Tag 0DW5].

If $B(C)$ and $Z(C)$ are flat, then for every left $A$ -module $N$ and every integer $n$ , there is an exact sequence

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0 \longrightarrow H_n(C) \otimes_A N \xrightarrow{\gamma_n} H_n(C \otimes_A N) \xrightarrow{\alpha_n} \operatorname{Tor}_1^A(H_{n-1}(C), N) \longrightarrow 0 .

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

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