Lemma 47.7.2 (08Z3)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 47: Dualizing Complexes
Section 47.7: Injective hull of the residue field
Lemma 47.7.2 (cite)

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Lemma 47.7.2. Let $(R, \mathfrak m, \kappa )$ be a local ring. Let $E$ be the injective hull of $\kappa$ . Let $M$ be a $\mathfrak m$ -power torsion $R$ -module with $n = \dim _\kappa (M[\mathfrak m]) < \infty$ . Then $M$ is isomorphic to a submodule of $E^{\oplus n}$ .

Proof. Observe that $E^{\oplus n}$ is the injective hull of $\kappa ^{\oplus n} = M[\mathfrak m]$ . Thus there is an $R$ -module map $M \to E^{\oplus n}$ which is injective on $M[\mathfrak m]$ . Since $M$ is $\mathfrak m$ -power torsion the inclusion $M[\mathfrak m] \subset M$ is an essential extension (for example by Lemma 47.2.4) we conclude that the kernel of $M \to E^{\oplus n}$ is zero. $\square$

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