The Stacks project

Lemma 47.7.5. Let $(R, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa $ over $R$. Then $\mathop{\mathrm{Hom}}\nolimits _ R(E, E)$ is canonically isomorphic to the completion of $R$.

Proof. Write $E = \bigcup E_ n$ with $E_ n = E[\mathfrak m^ n]$ as in Lemma 47.7.3. Any endomorphism of $E$ preserves this filtration. Hence

\[ \mathop{\mathrm{Hom}}\nolimits _ R(E, E) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ R(E_ n, E_ n) \]

The lemma follows as $\mathop{\mathrm{Hom}}\nolimits _ R(E_ n, E_ n) = \mathop{\mathrm{Hom}}\nolimits _{R/\mathfrak m^ n}(E_ n, E_ n) = R/\mathfrak m^ n$ by Lemma 47.6.2. $\square$

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