Corollary 4 (« Künneth formula »)

Tag 0DW9, permanent. Cite it as [Bourbaki, Tag 0DW9].

Algebra, chapter X, § 4. Produits de torsion

Suppose C bounded on the right, C and H(C) flat; then the canonical homomorphism

\gamma(C, C') : H(C) \otimes_A H(C') \to H(C \otimes_A C')

is bijective.

By the theorem, it suffices to proving that B(C) and Z(C) are flat. Now one has exact sequences

\begin{align*} 0 &\to B_n(C) \to Z_n(C) \to H_n(C) \to 0 \\ 0 &\to Z_n(C) \to C_n \to B_{n-1}(C) \to 0, \end{align*}

whence, by X, p. 75, cor. 1, implications (B_{n-1}(C) is flat) $\Rightarrow$ (Z_n(C) is flat) $\Rightarrow$ (B_n(C) is flat) ; we conclude by remarking that B_n(C) = 0 for n small enough.

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

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