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injective module

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Definition

For RR a ring, let RRMod be the category of RR-modules.

An injective module over RR is an injective object in RModR Mod.

This is the dual notion of a projective module.

Properties

Equivalent characterizations

Let RR be a commutative ring and C=RModC = R Mod the category of RR-modules. We discuss injective modules over RR (see there for more).

Proposition

(Baer's criterion)

If the axiom of choice holds, then a module QRModQ \in R Mod is an injective module precisely if for II any left RR-ideal regarded as an RR-module, any homomorphism g:IQg : I \to Q in CC can be extended to all of RR along the inclusion IRI \hookrightarrow R.

Sketch of proof

Let i:MNi \colon M \hookrightarrow N be a monomorphism in RModR Mod, and let f:MQf \colon M \to Q be a map. We must extend ff to a map h:NQh \colon N \to Q. Consider the poset whose elements are pairs (M,f)(M', f') where MM' is an intermediate submodule between MM and NN and f:MQf' \colon M' \to Q is an extension of ff, ordered by (M,f)(M,f)(M', f') \leq (M'', f'') if MM'' contains MM' and ff'' extends ff'. By an application of Zorn's lemma, this poset has a maximal element, say (M,f)(M', f'). Suppose MM' is not all of NN, and let xNx \in N be an element not in MM'; we show that ff' extends to a map M=x+MQM'' = \langle x \rangle + M' \to Q, a contradiction.

The set {rR:rxM}\{r \in R: r x \in M'\} is an ideal II of RR, and we have a module homomorphism g:IQg \colon I \to Q defined by g(r)=f(rx)g(r) = f'(r x). By hypothesis, we may extend gg to a module map k:RQk \colon R \to Q. Writing a general element of MM'' as rx+yr x + y where yMy \in M', it may be shown that

f(rx+y)=k(r)+g(y)f''(r x + y) = k(r) + g(y)

is well-defined and extends ff', as desired.

Corollary

Assume that the axiom of choice holds.

Let RR be a Noetherian ring, and let {Q j} jJ\{Q_j\}_{j \in J} be a collection of injective modules over RR. Then the direct sum Q= jJQ jQ = \bigoplus_{j \in J} Q_j is also injective.

Proof

By Baer’s criterion, it suffices to show that for any ideal II of RR, a module homomorphism f:IQf \colon I \to Q extends to a map RQR \to Q. Since RR is Noetherian, II is finitely generated as an RR-module, say by elements x 1,,x nx_1, \ldots, x_n. Let p j:QQ jp_j \colon Q \to Q_j be the projection, and put f j=p jff_j = p_j \circ f. Then for each x ix_i, f j(x i)f_j(x_i) is nonzero for only finitely many summands. Taking all of these summands together over all ii, we see that ff factors through

jJQ j= jJQ jQ\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q

for some finite JJJ' \subset J. But a product of injectives is injective, hence ff extends to a map R jJQ jR \to \prod_{j \in J'} Q_j, which completes the proof.

Proposition

Conversely, RR is a Noetherian ring if direct sums of injective RR-modules are injective.

This is due to Bass and Papp. See (Lam, Theorem 3.46).

Existence of enough injectives

We discuss that in the presence of the axiom of choice at least, the category RRMod has enough injectives in that every module is a submodule of an injective one. We first consider this for R=R = \mathbb{Z}. We do assume prop. 4, which may be proven using Baer's criterion.

Proposition

Assuming the axiom of choice, the category \mathbb{Z}Mod \simeq Ab has enough injectives.

Proof

By prop. 4 an abelian group is an injective \mathbb{Z}-module precisely if it is a divisible group. So we need to show that every abelian group is a subgroup of a divisible group.

To start with, notice that the group \mathbb{Q} of rational numbers is divisible and hence the canonical embedding \mathbb{Z} \hookrightarrow \mathbb{Q} shows that the additive group of integers embeds into an injective \mathbb{Z}-module.

Now by the discussion at projective module every abelian group AA receives an epimorphism ( sS)A(\oplus_{s \in S} \mathbb{Z}) \to A from a free abelian group, hence is the quotient group of a direct sum of copies of \mathbb{Z}. Accordingly it embeds into a quotient A˜\tilde A of a direct sum of copies of \mathbb{Q}.

ker = ker ( sS) ( sS) A A˜ \array{ ker &\stackrel{=}{\to}& ker \\ \downarrow && \downarrow \\ (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q}) \\ \downarrow && \downarrow \\ A &\hookrightarrow& \tilde A }

Here A˜\tilde A is divisible because the direct sum of divisible groups is again divisible, and also the quotient group of a divisible groups is again divisble. So this exhibits an embedding of any AA into a divisible abelian group, hence into an injective \mathbb{Z}-module.

Proposition

Assuming the axiom of choice, for RR a ring, the category RRMod has enough injectives.

The proof uses the following lemma.

Write U:RModAbU\colon R Mod \to Ab for the forgetful functor that forgets the RR-module structure on a module NN and just remembers the underlying abelian group U(N)U(N).

Lemma

The functor U:RModAbU\colon R Mod \to Ab has a right adjoint

R *:AbRMod R_* : Ab \to R Mod

given by sending an abelian group AA to the abelian group

U(R *(A))Ab(U(R),A) U(R_*(A)) \coloneqq Ab(U(R),A)

equipped with the RR-module struture by which for rRr \in R an element (U(R)fA)U(R *(A))(U(R) \stackrel{f}{\to} A) \in U(R_*(A)) is sent to the element rfr f given by

rf:rf(rr). r f : r' \mapsto f(r' \cdot r) \,.

This is called the coextension of scalars along the ring homomorphism R\mathbb{Z} \to R.

The unit of the (UdashR *)(U \dash R_*) adjunction

ϵ N:NR *(U(N)) \epsilon_N : N \to R_*(U(N))

is the RR-module homomorphism

ϵ N:NHom Ab(U(R),U(N)) \epsilon_N : N \to Hom_{Ab}(U(R), U(N))

given on nNn \in N by

j(n):rrn. j(n) : r \mapsto r n \,.
Proof

of prop. 3

Let NRModN \in R Mod. We need to find a monomorphism NN˜N \to \tilde N such that N˜\tilde N is an injective RR-module.

By prop. 2 there exists a monomorphism

i:U(N)D i \colon U(N) \hookrightarrow D

of the underlying abelian group into an injective abelian group DD.

Now consider the adjunct NR *(D)N \to R_*(D) of ii, hence the composite

Nη NR *(U(N))R *(i)R *(D) N \stackrel{\eta_N}{\to} R_*(U(N)) \stackrel{R_*(i)}{\to} R_*(D)

with R *R_* and η N\eta_N from lemma 1. On the underlying abelian groups this is

U(N)U(η N)Hom Ab(U(R),U(N))Hom Ab(U(R),i)Hom Ab(U(R),U(D)). U(N) \stackrel{U(\eta_N)}{\to} Hom_{Ab}(U(R), U(N)) \stackrel{Hom_{Ab}(U(R),i)}{\to} Hom_{Ab}(U(R),U(D)) \,.

Once checks on components that this is a monomorphism. Therefore it is now sufficient to see that Hom Ab(U(R),U(D))Hom_{Ab}(U(R), U(D)) is an injective RR-module.

This follows from the existence of the adjunction isomorphism given by lemma 1

Hom Ab(U(K),U(D))Hom RMod(K,Hom Ab(U(R),U(D))) Hom_{Ab}(U(K),U(D)) \simeq Hom_{R Mod}(K, Hom_{Ab}(U(R), U(D)))

natural in KRModK \in R Mod and from the injectivity of DAbD \in Ab.

U(K) D U(L)K R *D L. \array{ U(K) &\to& D \\ \downarrow & \nearrow \\ U(L) } \;\;\;\;\; \leftrightarrow \;\;\;\;\; \array{ K &\to& R_*D \\ \downarrow & \nearrow \\ L } \,.

Examples

Injective \mathbb{Z}-modules / abelian groups

Let C=ModC = \mathbb{Z} Mod \simeq Ab be the abelian category of abelian groups.

Proposition

An abelian group AA is injective as a \mathbb{Z}-module precisely if it is a divisible group, in that for all integers nn \in \mathbb{N} we have nG=Gn G = G.

Using Baer’s criterion, prop. 1.

Example

By prop. 4 the following abelian groups are injective in Ab.

The group of rational numbers \mathbb{Q} is injective in Ab, as is the additive group of real numbers \mathbb{R} and generally that underlying any field. The additive group underlying any vector space is injective. The quotient of any injective group by any other group is injective.

Example

Not injective in Ab is for instance the cyclic group /n\mathbb{Z}/n\mathbb{Z} for n>1n \gt 1.

References

The notion of injective modules was introduced in

  • R. Baer (1940)

(The dual notion of projective modules was considered explicitly only much later.)

A general discussion can be found in

The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.

Baer’s criterion is discussed in many texts, for example

  • N. Jacobsen, Basic Algebra II, W.H. Freeman and Company, 1980.

See also

  • T.-Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics 189, Springer Verlag (1999).

Section 4.2 of

For abelian sheaves over the etale site:

Revised on November 22, 2013 04:20:29 by Urs Schreiber (82.169.114.243)