The Stacks project

Remark 47.7.9. Let $(R, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa $ over $R$. Here is an addendum to Matlis duality: If $N$ is an $\mathfrak m$-power torsion module and $M = \mathop{\mathrm{Hom}}\nolimits _ R(N, E)$ is a finite module over the completion of $R$, then $N$ satisfies the descending chain condition. Namely, for any submodules $N'' \subset N' \subset N$ with $N'' \not= N'$, we can find an embedding $\kappa \subset N''/N'$ and hence a nonzero map $N' \to E$ annihilating $N''$ which we can extend to a map $N \to E$ annihilating $N''$. Thus $N \supset N' \mapsto M' = \mathop{\mathrm{Hom}}\nolimits _ R(N/N', E) \subset M$ is an inclusion preserving map from submodules of $N$ to submodules of $M$, whence the conclusion.


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