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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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