Definition 1

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

Let $M$ be a right A-module and $N$ a left A-module. We call the torsion product of $M$ and $N$ the graded $k$ -module

(4)

\operatorname{Tor}^A(M,N)=H(L(M)\otimes_A L(N)).

The homogeneous components of $\operatorname{Tor}^A(M,N)$ are denoted

(5)

\operatorname{Tor}^A_n(M,N)=H_n(L(M)\otimes_A L(N)).

Since $L(M)$ and $L(N)$ are zero on the right, we have

(6)

\operatorname{Tor}_n^A(M, N) = 0 \quad \text{for } n < 0 .

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

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