nLab
injective object

Context

Category theory

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Definition

There is a very general notion of injective objects in a category CC, and a sequence of refinements as CC is equipped with more structure and property, in particular for CC an abelian category or a relative thereof.

General definition

Let CC be a category and JMor(C)J \subset Mor(C) a class of morphisms in CC.

Example

Frequently JJ is the class of all monomorphisms or a related class.

This is notably the case for CC is a category of chain complexes equipped with the injective model structure on chain complexes and JJ is its class of cofibrations.

Definition

An object II in CC is JJ-injective if all diagrams of the form

X I jJ Z \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow \\ Z }

admit an extension

X I jJ Z. \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} \\ Z } \,.

If JJ is the class of all monomorphisms, we speak merely of an injective object.

Definition

Ones says that a category CC has enough injectives if every object admits a monomorphism into an injective object.

The dual notion is a projective object.

Assuming the axiom of choice, we have the following easy result.

Proposition

An arbitrary small product of injective objects is injective.

Remark

If CC has a terminal object ** then these extensions are equivalently lifts

X I jJ Z * \array{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} & \downarrow \\ Z &\to& * }

and hence the JJ-injective objects are precisely those that have the right lifting property against the class JJ.

Remark

If CC is a locally small category then II is JJ-injective precisely if the hom-functor

Hom C(,I):C opSet Hom_C(-,I) : C^{op} \to Set

takes morphisms in JJ to epimorphisms in Set.

In abelian categories

The term injective object is used most frequently in the context that CC is an abelian category.

Observation

For CC an abelian category the class JJ of monomorphisms is the same as the class of morphisms f:ABf : A \to B such that 0AfB0 \to A \stackrel{f}{\to} B is exact.

Proof

By definition of abelian category every monomorphism ABA \hookrightarrow B is a kernel, hence a pullback of the form

A 0 B C \array{ A &\to& 0 \\ \downarrow && \downarrow \\ B &\to& C }

for 00 the (algebraic) zero object. By the pasting law for pullbacks we find that also the left square in

0 A 0 0 B C \array{ 0 &\to& A &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& B &\to& C }

is a pullback, hence 0AB0 \to A \to B is exact.

Corollary

An object II of an abelian category CC is then injective if it satisfies the following equivalent conditions:

  • the hom-functor Hom C(,I):C opSetHom_C(-, I) : C^{op} \to Set is exact;

  • for all morphisms f:XYf : X \to Y such that 0XY0 \to X \to Y is exact and for all k:XIk : X \to I, there exists h:YIh : Y \to I such that hf=k h\circ f = k.

0 X f Y k h I. \array{ 0 &\to& X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{k}} & \swarrow_{\mathrlap{\exists h}} \\ && I } \,.

In chain complexes

See homotopically injective object for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.

Examples

Injective modules

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).

The following criterion says that for identifying injective modules it is sufficient to test the right lifting property which characterizes injective objects by def. 1, only on those monomorphisms which include an ideal into the base ring RR.

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.

This is due to (Baer).

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, theorem 1, 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).

Injective abelian groups

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

Proposition

If the axiom of choice holds, then an abelian group AA is an injective object in Ab precisely if it is a divisible group, in that for all integers nn \in \mathbb{N} we have nG=Gn G = G.

This follows for instance using Baer's criterion, prop. 1.

An explit proof is spelled out at Planet math – abelian group is divisible if and only if it is an injective object

Example

By prop. 3 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.

Properties

Existence of enough injectives

We discuss a list of classes of categories that have enough injective according to def. 2.

Proposition

Every topos has enough injectives.

Proof

Every power object can be shown to be injective, and every object embeds into its power object by the “singletons” map.

Proposition

Assuming some form of the axiom of choice, the category of abelian groups has enough injectives.

Full AC is much more than required, however; small violations of choice suffices.

Proposition

As soon as the category Ab of abelian groups has enough injectives, so does the abelian category RRMod of modules over some ring RR.

Proof

Observe that the forgetful functor U:RModAbGpU\colon R Mod \to AbGp has both a left adjoint R !R_! (extension of scalars from \mathbb{Z} to \mathbb{R}) and a right adjoint R *R_* (coextension of scalars). Since it has a left adjoint, it is exact, and so its right adjoint R *R_* preserves injective objects. Thus given any RR-module MM, we can embed U(M)U(M) in an injective abelian group II, and then MM embeds in R *(I)R_*(I).

Proposition

For R=kR = k a field, hence RRMod = kkVect, ever object is both injective as well as projective.

Proposition

The category of abelian sheaves Ab(Sh(C))Ab(Sh(C)) on any small site CC, hence the category of abelian groups in the sheaf topos over CC, has enough injectives.

A proof of can be found in Peter Johnstone’s book Topos Theory, p261.

Remark

This is in stark contrast to the situation for projective objects, which generally do not exist in categories of sheaves.

Corollary

The category of sheaves of modules over any sheaf of rings? on any small site also enough injectives.

Proof

Combining prop. 8 with prop. 6 (which relativizes to any topos).

This slick proof of this important fact was pointed out by Colin McLarty in an email to the categories list dated 10 Oct 2010.

Injective resolutions

Proposition

Let 𝒜\mathcal{A} be an abelian category. Then for every object X𝒜X \in \mathcal{A} there is an injective resolution, hence a chain complex

J =[J 0J n]Ch (𝒜) J^\bullet = [J^0 \to \cdots \to J^n \to \cdots] \in Ch_(\mathcal{A})

equipped with a a quasi-isomorphism of cochain complexes XJ X \stackrel{\sim}{\to} J^\bullet

X 0 0 J 0 J 1 codt J n . \array{ X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots \\ \downarrow && \downarrow && && \downarrow \\ J^0 &\to& J^1 &\to& \codt &\to& J^n &\to& \cdots } \,.

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.

Using tools from the theory of accessible categories, injective objects are discussed in

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).

Revised on February 24, 2013 19:45:46 by Todd Trimble (67.81.93.26)