The Stacks project

Lemma 47.6.1. Let $(R, \mathfrak m, \kappa )$ be an artinian local ring. Let $E$ be an injective hull of $\kappa $. For every finite $R$-module $M$ we have

\[ \text{length}_ R(M) = \text{length}_ R(\mathop{\mathrm{Hom}}\nolimits _ R(M, E)) \]

In particular, the injective hull $E$ of $\kappa $ is a finite $R$-module.

Proof. Because $E$ is an essential extension of $\kappa $ we have $\kappa = E[\mathfrak m]$ where $E[\mathfrak m]$ is the $\mathfrak m$-torsion in $E$ (notation as in More on Algebra, Section 15.88). Hence $\mathop{\mathrm{Hom}}\nolimits _ R(\kappa , E) \cong \kappa $ and the equality of lengths holds for $M = \kappa $. We prove the displayed equality of the lemma by induction on the length of $M$. If $M$ is nonzero there exists a surjection $M \to \kappa $ with kernel $M'$. Since the functor $M \mapsto \mathop{\mathrm{Hom}}\nolimits _ R(M, E)$ is exact we obtain a short exact sequence

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _ R(\kappa , E) \to \mathop{\mathrm{Hom}}\nolimits _ R(M, E) \to \mathop{\mathrm{Hom}}\nolimits _ R(M', E) \to 0. \]

Additivity of length for this sequence and the sequence $0 \to M' \to M \to \kappa \to 0$ and the equality for $M'$ (induction hypothesis) and $\kappa $ implies the equality for $M$. The final statement of the lemma follows as $E = \mathop{\mathrm{Hom}}\nolimits _ R(R, E)$. $\square$

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