Proposition 10

— If M is an A-module of finite length, then $D(M)$ is (in general non-naturally) isomorphic to M, and the canonical map $c_M : M \to D(D(M))$ is an isomorphism. Using VII, p. 19, Th. 2 and II, p. 203, Cor. 1, we reduce to the case where $M$ is cyclic. Thus we may suppose that $M = A/tA$ with $t \neq 0$. Note that any homomorphism $u : A/tA \to K/A$ is completely determined by the image $\xi \in K/A$ under $u$ of the class $\varepsilon$ of 1 mod $tA$, and this element must satisfy the relation $t\xi = 0$ ; conversely, for any $\xi \in K/A$ such that $t\xi = 0$, there exists a homomorphism $u : A/tA \to K/A$ such that $u(\varepsilon) = \xi$. It follows that $D(M)$ is isomorphic to $t^{-1}A/A$, and since the homothety by $t$ is a bijection on K, we also have $D(M)$ isomorphic to $A/tA$, which proves the first assertion. This proves that $M$ and $D(D(M))$ are isomorphic, so have the same length ; on the other hand $c_M$ is injective, for if $y \in A$ is such that the relation $tz \in A$ (for $z \in K$) implies $yz \in A$, then taking $z = t^{-1}$ we have $y \in tA$. It follows that the image $c_M(M)$ is necessarily equal to $D(D(M))$.