Corollary

If $M$ or $N$ is flat, $\mathrm{Tor}_i^A(M, N) = 0$ for $i \geqslant 0$.

Suppose that $N$ (resp. $M$) is flat. Then $p_M \otimes 1 : L(M) \otimes_A N \to M \otimes_A N$ (resp. $1 \otimes p_N : M \otimes_A L(N) \to M \otimes_A N$) is a homology morphism (X, p. 67, Proposition 4), hence $H_i(L(M) \otimes_A N)$ (resp. $H_i(M \otimes_A L(N))$) vanishes for $i > 0$.