The Stacks project

Lemma 47.7.4. Let $R \to S$ be a flat local homomorphism of local Noetherian rings such that $R/\mathfrak m_ R \cong S/\mathfrak m_ R S$. Then the injective hull of the residue field of $R$ is the injective hull of the residue field of $S$.

Proof. Note that $\mathfrak m_ RS = \mathfrak m_ S$ as the quotient by the former is a field. Set $\kappa = R/\mathfrak m_ R = S/\mathfrak m_ S$. Let $E_ R$ be the injective hull of $\kappa $ over $R$. Let $E_ S$ be the injective hull of $\kappa $ over $S$. Observe that $E_ S$ is an injective $R$-module by Lemma 47.3.2. Choose an extension $E_ R \to E_ S$ of the identification of residue fields. This map is an isomorphism by Lemma 47.7.3 because $R \to S$ induces an isomorphism $R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n$ for all $n$. $\square$

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