Proposition 7

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

Algebra, chapter X, § 4. Produits de torsion

Let I and J be two sets, $(M_\alpha)_{\alpha \in I}$ a family of right A-modules, $(N_\beta)_{\beta \in J}$ a family of left A-modules. The homomorphism

\bigoplus_{\alpha \in I, \beta \in J} \mathrm{Tor}^A(M_\alpha, N_\beta) \to \mathrm{Tor}^A\left( \bigoplus_{\alpha \in I} M_\alpha, \bigoplus_{\beta \in J} N_\beta \right)

deduced from the canonical homomorphisms $M_\alpha \to \bigoplus M_\alpha$ and $N_\beta \to \bigoplus N_\beta$ is bijective.

It suffices to prove that for every right module $M$ (resp. every left module $N$ ), the canonical homomorphism

\bigoplus_{\beta \in J} \mathrm{Tor}^A(M, N_\beta) \to \mathrm{Tor}^A(M, \bigoplus_{\beta \in J} N_\beta)

(resp. $\bigoplus_{\alpha \in I} \mathrm{Tor}^A(M_\alpha, N) \to \mathrm{Tor}^A(\bigoplus_{\alpha \in I} M_\alpha, N)$ ) is bijective. Now this follows from the preceding, from Proposition 1 of X, p. 28, and from the canonical isomorphisms :

\bigoplus_{\beta} (L(M) \otimes_A N_\beta) \to L(M) \otimes_A (\bigoplus_{\beta} N_\beta),

\bigoplus_{\alpha} (M_\alpha \otimes_A L(N)) \to (\bigoplus_{\alpha} M_\alpha) \otimes_A L(N).

An analogous reasoning gives :

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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