Lemma 47.7.6. Let $(R, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $E$ be an injective hull of $\kappa$ over $R$. Then $E$ satisfies the descending chain condition.

Proof. If $E \supset M_1 \supset M_2 \supset \ldots$ is a sequence of submodules, then

$\mathop{\mathrm{Hom}}\nolimits _ R(E, E) \to \mathop{\mathrm{Hom}}\nolimits _ R(M_1, E) \to \mathop{\mathrm{Hom}}\nolimits _ R(M_2, E) \to \ldots$

is a sequence of surjections. By Lemma 47.7.5 each of these is a module over the completion $R^\wedge = \mathop{\mathrm{Hom}}\nolimits _ R(E, E)$. Since $R^\wedge$ is Noetherian (Algebra, Lemma 10.97.6) the sequence stabilizes: $\mathop{\mathrm{Hom}}\nolimits _ R(M_ n, E) = \mathop{\mathrm{Hom}}\nolimits _ R(M_{n + 1}, E) = \ldots$. Since $E$ is injective, this can only happen if $\mathop{\mathrm{Hom}}\nolimits _ R(M_ n/M_{n + 1}, E)$ is zero. However, if $M_ n/M_{n + 1}$ is nonzero, then it contains a nonzero element annihilated by $\mathfrak m$, because $E$ is $\mathfrak m$-power torsion by Lemma 47.7.3. In this case $M_ n/M_{n + 1}$ has a nonzero map into $E$, contradicting the assumed vanishing. This finishes the proof. $\square$

There are also:

• 3 comment(s) on Section 47.7: Injective hull of the residue field

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).