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 15.83). 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 $E = \mathop{\mathrm{Hom}}\nolimits _ R(R, E)$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).