Proposition 6

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

The mapping $(f, g) \mapsto \mathrm{Tor}^A(f, g) :$

\mathrm{Hom}_A(M, M') \times \mathrm{Hom}_A(N, N') \to \mathrm{Hom}_k(\mathrm{Tor}^A(M, N), \mathrm{Tor}^A(M', N'))

is $k$ -bilinear.

Let $f \in \mathrm{Hom}_A(M, M')$ , $g_1, g_2 \in \mathrm{Hom}_A(N, N')$ , $\lambda_1, \lambda_2 \in k$ . Then the morphisms

\lambda_1(L(f) \otimes g_1) + \lambda_2(L(f) \otimes g_2) \quad \text{and} \quad L(f) \otimes (\lambda_1 g_1 + \lambda_2 g_2)

from $L(M) \otimes_A N$ into $L(M) \otimes_A N'$ coincide; by prop. 5 and Remark 3, one therefore has

(10)

\mathrm{Tor}^A(f, \lambda_1 g_1 + \lambda_2 g_2) = \lambda_1 \mathrm{Tor}^A(f, g_1) + \lambda_2 \mathrm{Tor}^A(f, g_2).

One reasons analogously for the mapping $f \mapsto \mathrm{Tor}^A(f, g)$ .

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

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