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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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