Corollary

— Let M, N be two A-modules of finite length, and let $\varphi$ be an A-bilinear map from $M \times N$ into $K/A$, such that : 1) the relation $\varphi(x, y) = 0$ for all $y \in N$ implies $x = 0$ ; and 2) the relation $\varphi(x, y) = 0$ for all $x \in M$ implies $y = 0$. Then the A-linear maps $s_\varphi : M \to D(N)$ and $d_\varphi : N \to D(M)$ associated to $\varphi$ are isomorphisms. Indeed the hypotheses on $\varphi$ mean that $s_\varphi$ and $d_\varphi$ are injective and since $\mathrm{long}(D(N)) = \mathrm{long}(N)$ and $\mathrm{long}(D(M)) = \mathrm{long}(M)$ by Prop. 10, this implies that $\mathrm{long}(M) = \mathrm{long}(N)$, and consequently $s_\varphi$ and $d_\varphi$ are bijective.