Remark 2

One can also use the resolutions $l(P)$ and $l(Q)$.

Analogously, if $N$ is a second left $A$-module, one defines a composition product $(\mu, \gamma) \mapsto \mu \circ \gamma$ denoted

by the commutative diagram

where $\sigma$ denotes the commutation isomorphisms (X, p. 71).

If $\mu \in \mathrm{Ext}_A^r(M, N)$ is the class of the morphism $g : L(M) \to L(N) (-r)$, and if $\gamma \in \mathrm{Tor}_m^A(P, M)$ is the class of the cycle $z = \sum z_{ij}$, where $z_{ij} \in L_i(P) \otimes_A L_j(M)$, then $\mu \circ \gamma$ is therefore the class of the cycle $\sum (-1)^{ir} (1 \otimes g)(z_{ij})$.

One can also represent $\gamma$ by a cycle $y \in P \otimes L_m(M)$, and $\mu \circ \gamma$ is the class of the cycle $(1 \otimes g)(y) \in P \otimes L_{m-r}(M)$.