The Stacks project

Lemma 47.7.1. Let $R \to S$ be a surjective map of local rings with kernel $I$. Let $E$ be the injective hull of the residue field of $R$ over $R$. Then $E[I]$ is the injective hull of the residue field of $S$ over $S$.

Proof. Observe that $E[I] = \mathop{\mathrm{Hom}}\nolimits _ R(S, E)$ as $S = R/I$. Hence $E[I]$ is an injective $S$-module by Lemma 47.3.4. Since $E$ is an essential extension of $\kappa = R/\mathfrak m_ R$ it follows that $E[I]$ is an essential extension of $\kappa $ as well. The result follows. $\square$

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