Lemma 47.7.3. Let $(R, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa $ over $R$. Let $E_ n$ be an injective hull of $\kappa $ over $R/\mathfrak m^ n$. Then $E = \bigcup E_ n$ and $E_ n = E[\mathfrak m^ n]$.
Proof. We have $E_ n = E[\mathfrak m^ n]$ by Lemma 47.7.1. We have $E = \bigcup E_ n$ because $\bigcup E_ n = E[\mathfrak m^\infty ]$ is an injective $R$-submodule which contains $\kappa $, see Lemma 47.3.9. $\square$
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