Remark 2

Theorem 1 provides a description of $\mathrm{Ext}_A^n(M, N)$ as a set of equivalence classes of exact sequences; it is easy to describe the group law obtained on this set by transport of structure. Indeed, let $\theta$ (resp. $\theta'$) be the class of an exact sequence $0 \to N \xrightarrow{f_{n+1}} R_n \xrightarrow{f_n} \ldots \to R_1 \to M \to 0$ (resp. $0 \to N \xrightarrow{f'_{n+1}} R'_n \xrightarrow{f'_n} \ldots \to R'_1 \to M \to 0$). Let $\Delta : M \to M \oplus M$ and $\nabla : N \oplus N \to N$ be the $A$-linear maps defined by $\Delta(x) = (x, x)$ for $x \in M$ and $\nabla(y, z) = y + z$ for $y, z \in N$. Consider the mapping

defined in the remark, p. 121. With the notations of loc. cit., one has $\nabla \circ i_N = 1_N$ and $q_M \circ \Delta = 1_M$, and thus $\theta + \theta' = \nabla \circ m(\theta, \theta') \circ \Delta$. Taking into account loc. cit. and cor. 3, p. 120, this provides an exact sequence of class $\theta+\theta'$: for example, if $n \geq 2$, one can take the sequence

where $R_n''$ is the quotient of $R_n\oplus R'_n$ by the submodule formed by the couples

and where $R''_1=R_1\times_M R'_1$.