Proposition 47.7.8 (Matlis duality). Let $(R, \mathfrak m, \kappa )$ be a complete local Noetherian ring. Let $E$ be an injective hull of $\kappa$ over $R$. The functor $D(-) = \mathop{\mathrm{Hom}}\nolimits _ R(-, E)$ induces an anti-equivalence

$\left\{ \begin{matrix} R\text{-modules with the} \\ \text{descending chain condition} \end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} R\text{-modules with the} \\ \text{ascending chain condition} \end{matrix} \right\}$

and we have $D \circ D = \text{id}$ on either side of the equivalence.

Proof. By Lemma 47.7.5 we have $R = \mathop{\mathrm{Hom}}\nolimits _ R(E, E) = D(E)$. Of course we have $E = \mathop{\mathrm{Hom}}\nolimits _ R(R, E) = D(R)$. Since $E$ is injective the functor $D$ is exact. The result now follows immediately from the description of the categories in Lemma 47.7.7. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).