Lemma 47.7.1. Let R \to S be a surjective map of local rings with kernel I. Let E be the injective hull of the residue field of R over R. Then E[I] is the injective hull of the residue field of S over S.
47.7 Injective hull of the residue field
Most of our results will be for Noetherian local rings in this section.
Proof. Observe that E[I] = \mathop{\mathrm{Hom}}\nolimits _ R(S, E) as S = R/I. Hence E[I] is an injective S-module by Lemma 47.3.4. Since E is an essential extension of \kappa = R/\mathfrak m_ R it follows that E[I] is an essential extension of \kappa as well. The result follows. \square
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
Lemma 47.7.3. Let (R, \mathfrak m, \kappa ) be a Noetherian local ring. Let E be an injective hull of \kappa over R. Let E_ n be an injective hull of \kappa over R/\mathfrak m^ n. Then E = \bigcup E_ n and E_ n = E[\mathfrak m^ n].
Proof. We have E_ n = E[\mathfrak m^ n] by Lemma 47.7.1. We have E = \bigcup E_ n because \bigcup E_ n = E[\mathfrak m^\infty ] is an injective R-submodule which contains \kappa , see Lemma 47.3.9. \square
The following lemma tells us the injective hull of the residue field of a Noetherian local ring only depends on the completion.
Lemma 47.7.4. Let R \to S be a flat local homomorphism of local Noetherian rings such that R/\mathfrak m_ R \cong S/\mathfrak m_ R S. Then the injective hull of the residue field of R is the injective hull of the residue field of S.
Proof. Note that \mathfrak m_ RS = \mathfrak m_ S as the quotient by the former is a field. Set \kappa = R/\mathfrak m_ R = S/\mathfrak m_ S. Let E_ R be the injective hull of \kappa over R. Let E_ S be the injective hull of \kappa over S. Observe that E_ S is an injective R-module by Lemma 47.3.2. Choose an extension E_ R \to E_ S of the identification of residue fields. This map is an isomorphism by Lemma 47.7.3 because R \to S induces an isomorphism R/\mathfrak m_ R^ n \to S/\mathfrak m_ S^ n for all n. \square
Lemma 47.7.5. Let (R, \mathfrak m, \kappa ) be a Noetherian local ring. Let E be an injective hull of \kappa over R. Then \mathop{\mathrm{Hom}}\nolimits _ R(E, E) is canonically isomorphic to the completion of R.
Proof. Write E = \bigcup E_ n with E_ n = E[\mathfrak m^ n] as in Lemma 47.7.3. Any endomorphism of E preserves this filtration. Hence
\mathop{\mathrm{Hom}}\nolimits _ R(E, E) = \mathop{\mathrm{lim}}\nolimits \mathop{\mathrm{Hom}}\nolimits _ R(E_ n, E_ n)
The lemma follows as \mathop{\mathrm{Hom}}\nolimits _ R(E_ n, E_ n) = \mathop{\mathrm{Hom}}\nolimits _{R/\mathfrak m^ n}(E_ n, E_ n) = R/\mathfrak m^ n by Lemma 47.6.2. \square
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
Lemma 47.7.7. Let (R, \mathfrak m, \kappa ) be a Noetherian local ring. Let E be an injective hull of \kappa .
For an R-module M the following are equivalent:
M satisfies the ascending chain condition,
M is a finite R-module, and
there exist n, m and an exact sequence R^{\oplus m} \to R^{\oplus n} \to M \to 0.
For an R-module M the following are equivalent:
M satisfies the descending chain condition,
M is \mathfrak m-power torsion and \dim _\kappa (M[\mathfrak m]) < \infty , and
there exist n, m and an exact sequence 0 \to M \to E^{\oplus n} \to E^{\oplus m}.
Proof. We omit the proof of (1).
Let M be an R-module with the descending chain condition. Let x \in M. Then \mathfrak m^ n x is a descending chain of submodules, hence stabilizes. Thus \mathfrak m^ nx = \mathfrak m^{n + 1}x for some n. By Nakayama's lemma (Algebra, Lemma 10.20.1) this implies \mathfrak m^ n x = 0, i.e., x is \mathfrak m-power torsion. Since M[\mathfrak m] is a vector space over \kappa it has to be finite dimensional in order to have the descending chain condition.
Assume that M is \mathfrak m-power torsion and has a finite dimensional \mathfrak m-torsion submodule M[\mathfrak m]. By Lemma 47.7.2 we see that M is a submodule of E^{\oplus n} for some n. Consider the quotient N = E^{\oplus n}/M. By Lemma 47.7.6 the module E has the descending chain condition hence so do E^{\oplus n} and N. Therefore N satisfies (2)(a) which implies N satisfies (2)(b) by the second paragraph of the proof. Thus by Lemma 47.7.2 again we see that N is a submodule of E^{\oplus m} for some m. Thus we have a short exact sequence 0 \to M \to E^{\oplus n} \to E^{\oplus m}.
Assume we have a short exact sequence 0 \to M \to E^{\oplus n} \to E^{\oplus m}. Since E satisfies the descending chain condition by Lemma 47.7.6 so does M. \square
Proposition 47.7.8 (Matlis duality). Let (R, \mathfrak m, \kappa ) be a complete local Noetherian ring. Let E be an injective hull of \kappa over R. The functor D(-) = \mathop{\mathrm{Hom}}\nolimits _ R(-, E) induces an anti-equivalence
\left\{ \begin{matrix} R\text{-modules with the}
\\ \text{descending chain condition}
\end{matrix} \right\} \longleftrightarrow \left\{ \begin{matrix} R\text{-modules with the}
\\ \text{ascending chain condition}
\end{matrix} \right\}
and we have D \circ D = \text{id} on either side of the equivalence.
Proof. By Lemma 47.7.5 we have R = \mathop{\mathrm{Hom}}\nolimits _ R(E, E) = D(E). Of course we have E = \mathop{\mathrm{Hom}}\nolimits _ R(R, E) = D(R). Since E is injective the functor D is exact. The result now follows immediately from the description of the categories in Lemma 47.7.7. \square
Remark 47.7.9. Let (R, \mathfrak m, \kappa ) be a Noetherian local ring. Let E be an injective hull of \kappa over R. Here is an addendum to Matlis duality: If N is an \mathfrak m-power torsion module and M = \mathop{\mathrm{Hom}}\nolimits _ R(N, E) is a finite module over the completion of R, then N satisfies the descending chain condition. Namely, for any submodules N'' \subset N' \subset N with N'' \not= N', we can find an embedding \kappa \subset N''/N' and hence a nonzero map N' \to E annihilating N'' which we can extend to a map N \to E annihilating N''. Thus N \supset N' \mapsto M' = \mathop{\mathrm{Hom}}\nolimits _ R(N/N', E) \subset M is an inclusion preserving map from submodules of N to submodules of M, whence the conclusion.
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