Corollary 1

Let $0 \to P' \to P \to P'' \to 0$ be an exact sequence of right $A$-*modules*, $0 \to M' \to M \to M'' \to 0$ be an exact sequence of left $A$-*modules*. Then the composed homomorphisms of connecting homomorphisms

and are opposite.

Indeed, if $\theta$ and $\theta_1$ are the classes associated with the given exact sequences, and if $\gamma \in \mathrm{Tor}_n^A(P'', M'')$, the images of $\gamma$ are respectively $\theta \circ (\theta_1 \circ \gamma)$ and $\theta_1 \circ (\theta \circ \gamma)$, hence are opposite by prop. 6.

Let us resume the notations of X, p. 127 and consider the sequence $(\mathscr{S})$ of left $A$-modules and the connecting homomorphisms associated with the exact sequences (9)

we deduce from this by composition of the iterated connecting homomorphisms Then by prop. 7 and prop. 3 of X, p. 118 :