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.
$\square$

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