Lemma 5

Let $M \to N \to P \to Q \to R$ be an exact sequence of $A$-modules. If $\mathscr{C}$ is stable, and if $M, N, Q$ and $R$ are of type $\mathscr{C}$, the module $P$ is of type $\mathscr{C}$. Set $N' = \operatorname{Coker}(M \to N)$ and $Q' = \operatorname{Ker}(Q \to R)$. The modules $N'$ and $Q'$ are of type $\mathscr{C}$, and one has an exact sequence $0 \to N' \to P \to Q' \to 0$.