Localizations associated to semidirect products

Topology and its Applications 108 (2000) 169–177

Localizations associated to semidirect products

José Luis Rodrígueza,∗, Dirk Scevenelsb

a Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spainb Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium

Received 15 October 1998

Abstract

For homotopical localization with respect to any continuous map, there are results describing therelations among the localization functors associated to the maps of a given fibration. Here we treatan analogous question in a group-theoretical context: we study localization functors associated toa short exact sequence of groups. We further specialize to a split short exact sequence of groups.In particular, we describe explicitly the localization functors associated to a semidirect product offinitely generated Abelian groups. 2000 Elsevier Science B.V. All rights reserved.

Keywords:Localization functors; Short exact sequence of groups; Semidirect product;Epireflection; Nullification; Radicals

0. Introduction

Localization with respect to a group homomorphism has proved to be a useful tool inunderstanding homotopical localization with respect to any continuous map, as developedby Bousfield, Dror Farjoun and others [2,4,11]. This algebraic technique was introduced byCasacuberta in [5,6], and led to some further interesting results in, e.g., [1,7,8,10,12]. Thus,to any given group homomorphismf , there is associated a localization functorLf on thecategory of groups, renderingf invertible in a universal way. The collection of all grouphomomorphismsg yielding a localization functorLg which is naturally equivalent toLf ,is called the localization class〈f 〉 of f . In the special case wheref is of the formA→ 1,the associated localization functor is also denoted byPA, and the class〈f 〉 is denoted by〈A〉, which is called a nullification class. As we recall in Section 1, the collectionLocsofall localization classes forms a small-complete lattice for an obvious partial order relation.

∗ Corresponding author. Author was partially supported by DGICYT grant PB94-0725.E-mail addresses:[email protected] (J.L. Rodríguez), [email protected] (D. Scevenels).

0166-8641/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0166-8641(99)00134-0

170 J.L. Rodríguez, D. Scevenels / Topology and its Applications 108 (2000) 169–177

In homotopy theory there are results describing the relations between the localizationfunctors associated to the spaces and maps of a given fibration (cf. [11]). Here we studyan analogous question in the group-theoretical context. More precisely, given a short exactsequence of groups

1→Hφ−→N

π−→K→ 1, (0.1)

we relate the classes〈H 〉, 〈N〉, 〈K〉, 〈φ〉 and〈π〉 under the partial order relation onLocs(cf. Proposition 2). In the case where the short exact sequence (0.1) splits, we show inTheorem 5 that〈φ〉 = 〈K〉 ∗ 〈ε〉, whereε :H � H/[H,K] is the natural projection, and〈K〉 ∗ 〈ε〉 denotes the least upper bound of〈K〉 and〈ε〉 in the latticeLocs. Moreover, therelations between the localization classes associated to a split short exact sequence (0.1)give rise to the following diagrams of localization functors:

Id PH/[H,K] Lπ PH

PK PH/[H,K]×K PN PH×Kand

Id Lε P[H,K]

PK Lφ P[H,K]×Kwhere Id denotes the identity functor.

Finally, in Section 3 we illustrate the above results with explicit calculations in the caseof a semidirect product of finitely generated Abelian groups. We show in Theorem 9that the localizations associated to a nilpotent semidirect product of these groups areparticularly easy to describe. On the other hand, Proposition 12 shows that even for asemidirect product of cyclic groups for which the corresponding action is perfect, thesituation can be more complicated.

1. Preliminaries

Recall from [6] the definition of localization with respect to a given group homomor-phismf :A→B. A groupG is calledf -local if the induced map of sets

f ∗ : Hom(B,G)→Hom(A,G)

is a bijection. For every groupG there is a homomorphismlG :G→LfG which is initialamong all homomorphisms fromG into f -local groups.Lf is called the localizationfunctor with respect tof . A homomorphismφ is called anf -equivalenceif Lf φ is anisomorphism. A groupG is f -acyclicif LfG= 1.

For a given homomorphismf , the collection of all homomorphismsg such thatLg isnaturally equivalent toLf , is called thelocalization classof f , and it is denoted by〈f 〉.

J.L. Rodríguez, D. Scevenels / Topology and its Applications 108 (2000) 169–177 171

There is an obvious partial order relation on the collectionLocsof all localization classes:we say that〈f 〉 6 〈g〉 if there exists a natural transformationLf → Lg of localizationfunctors. Note that this is equivalent to allf -equivalences beingg-equivalences, to everyg-local group beingf -local, or tof being ag-equivalence. With this partial order,Locsforms a small-complete lattice (cf. [12]). Recall that the least upper bound of two classes〈f 〉 and〈g〉, which is denoted by〈f 〉 ∗ 〈g〉, has the free productf ∗ g as a representative,i.e.,〈f 〉 ∗ 〈g〉 = 〈f ∗ g〉.

Recall further that a class〈f 〉 is called anepireflection classif G→ LfG is surjectivefor all groupsG. By Theorem 2.1 of [12], this is equivalent to〈f 〉 = 〈g〉 for someepimorphismg. In the special case whereg is of the formA→ 1, the functorLg is usuallydenoted byPA, and it is calledA-nullification(cf. [2,6]). The localization class〈g〉 is thencalled anullification class(which is also denoted by〈A〉), while theg-local groups areoften referred to asA-null. Observe that a groupG is f -acyclic if and only if〈G〉6 〈f 〉.For general properties of nullification and epireflection classes, in particular the relationwith radicals in group theory, we refer to [9] and [12].

2. Localizations associated to a short exact sequence

We start by a preliminary result, giving another description of the least upper bound oftwo localization classes.

Lemma 1. Letf andg be any group homomorphisms. Then

〈f 〉 ∗ 〈g〉 = 〈f ∗ g〉 = 〈f × g〉.

Proof. The first identity was explained in Section 1. Clearly,〈f ∗ g〉 6 〈f × g〉. Further,f × g = (f × id) ◦ (id×g), where(f × id) is an f -equivalence and(id×g) is a g-equivalence. This implies thatf × g is an(f ∗ g)-equivalence. 2

Given a short exact sequence of groups of the form

1→Hφ−→N

π−→K→ 1, (2.1)

we first clarify the relations among the localization functorsLφ , Lπ , PH , PN andPK .Observe that ifN = H × K, then we have a complete description of all involvedlocalization functors, since〈H ×K〉 = 〈H 〉 ∗ 〈K〉, 〈φ〉 = 〈K〉 and〈π〉 = 〈H 〉. In general,we have the following relations.

Proposition 2. Consider the short exact sequence(2.1). Then the following relations hold:

〈N〉 = 〈π〉 ∗ 〈K〉, 〈π〉6 〈H 〉 and

〈K〉6 〈φ〉6 〈H 〉 ∗ 〈K〉.


In particular, we have〈N〉6 〈H 〉 ∗ 〈K〉. In other words, there is a diagram of localizationfunctors, given by

Id Lπ PH

PK PN PH×K

PK Lφ PH×K

(2.2)

whereId denotes the identity functor.

Proof. Applying the functor Hom(−,G) to the exact sequence given in (2.1), yields anexact sequence of sets (whereπ∗ is an inclusion)

∗→Hom(K,G)π∗−→Hom(N,G)

φ∗−→Hom(H,G),

which implies directly the result.2An immediate consequence of the previous proposition is the following result (compare

with Corollary 4.8 of [2] and Theorem 3.D.1 of [11]).

Corollary 3. Consider the exact sequence(2.1)and letf be any group homomorphism.(1) If H is f -acyclic, thenπ is anf -equivalence.(2) If φ is anf -equivalence, thenK is f -acyclic.

However, as the next example shows, the converse of the statements of Corollary 3 arefalse in general.

Example 4.(1) Let f :Z ∗ Z→ Z ⊕ Z be the canonical projection. ThenLfG ∼= Gab for any

groupG, whereGab denotes its Abelianization. Consider the short exact sequence

1→ [N,N] φ−→ Nπ−→ N/[N,N] → 1. Clearlyπ is anf -equivalence, but the

commutator subgroup[N,N] is onlyf -acyclic ifN is soluble of derived length atmost two.

(2) Consider the short exact sequence 1→ C3φ−→Σ3

π−→C2→ 1, whereΣ3 denotesthe permutation group of a set of three elements, andC2, C3 are cyclic groups

of order 2, respectively 3. Letf :∗pZ φp−→ ∗pZ, where the free product is takenover all primesp which are different from 3, and the homomorphismφp is justmultiplication byp. ThenLf is localization at the prime 3 (cf. [5]). We concludethat C2 is f -acyclic, but φ is not an f -equivalence, sinceLf (C3) = C3 andLf (Σ3)= 1.

We now specialize to the case of a split short exact sequence of groups:

1→Hφ−→H oK π−→K→ 1. (2.3)


We will denote the action of an elementx in K on an elementa in H by x · a =s(x)as(x)−1, where s :K → H o K is a splitting homomorphism forπ . Let [H,K]denote the commutator subgroup, i.e., the normal subgroup ofH generated by the elementsof the form (x · a)a−1 = xax−1a−1 (from now on we will omit writing the inclusions). Denote byε :H → H/[H,K] the canonical homomorphism. Observe that, for everygroupG, the set Hom(H o K,G) is in bijective correspondence with the subset ofHom(H,G)×Hom(K,G) consisting of those pairs(ψ, ξ) such that

ψ(x · a)= ξ(x)ψ(a)ξ(x)−1,

for all a ∈H andx ∈K.

Theorem 5. Consider the split short exact sequence given in(2.3). Then〈φ〉 = 〈K〉 ∗ 〈ε〉.

Proof. Suppose thatG is φ-local. By Proposition 2,G isK-null. Moreover, we claim thatany homomorphismψ :H → G is trivial on [H,K]. Indeed, since Hom(K,G) is trivialandφ∗ : Hom(H oK,G) ∼= Hom(H,G), it is clear thatψ satisfiesψ(x · a)= ψ(a) forall a ∈ H , x ∈ K. This is equivalent toψ being trivial on[H,K], which in turn impliesthatG is ε-local. Hence,G is bothK-null and ε-local. Conversely, suppose thatG isK-null and ε-local. Any homomorphismH o K → G then corresponds to(ψ, id) forsome homomorphismψ :H → G. Hence,φ∗ is clearly injective. Moreover, sinceG isε-local, any homomorphismψ :H →G is trivial on [H,K], so thatψ(x · a)= ψ(a) forall a ∈H , x ∈K. Hence,(ψ, id) defines a homomorphism fromH oK toG, and clearlyφ∗((ψ, id))=ψ . This shows thatφ∗ is surjective, thereby concluding the proof.2

With the terminology introduced in [12], Theorem 5 thus states that〈φ〉 is anepireflection class.

Theorem 6. Consider the split short exact sequence(2.3).Then a groupG is π -local ifand only if every homomorphism fromH oK toG is trivial onH . Moreover,⟨

H/[H,K]⟩6 〈π〉6 〈H 〉.Proof. The first claim is obvious, and the fact that〈π〉6 〈H 〉 was proved in Proposition 2.Suppose now thatG is π -local, so that any homomorphism(ψ, ξ) :H oK→G is trivialonH , i.e.,ψ = 1. Given any homomorphismf :H/[H,K]→G, the compositionf ◦ εis trivial on [H,K]. This means that(f ◦ ε,1) is a homomorphism fromH oK toG, sothatf ◦ ε = 1. Sinceε is an epimorphism, we infer thatf is trivial, and hence thatG isH/[H,K]-null. 2

Summarizing the previous results, for a split short exact sequence (2.3), we have thefollowing diagrams of localization functors:

Id PH/[H,K] Lπ PH

PK PH/[H,K]×K PHoK PH×K

(2.4)


and

Id Lε P[H,K]

PK Lφ P[H,K]×K

(2.5)

where Id denotes the identity functor.At the beginning of this section, we already dealt with the special case of a semidirect

product (2.3) where the action ofK onH is trivial. Another “extremal” case is when theaction is perfect, i.e.,[H,K] =H .

Corollary 7. Consider the split short exact sequence(2.3). If [H,K] = H , then〈φ〉 =〈H 〉 ∗ 〈K〉. Therefore, diagrams(2.4)and(2.5)collapse to

Id Lπ PH

PK PHoK Lφ ∼= PH×K(2.6)

Note that the above result is a special case of the general fact that〈[H,K]〉6 〈ε〉 if andonly if 〈[H,K]〉 = 〈ε〉.

We close this section by exhibiting an example showing that in general the arrows indiagram (2.6) are not necessarily natural equivalences.

Example 8. Consider the split short exact sequence

1→C3φ−→Σ3

π−→ C2→ 1.

The commutator subgroup is[C3,C2] = C3. It is easy to check that all localizations indiagram (2.6) are different, by evaluating them on the groupsC2, C3, C3×C2 andΣ3.

Other examples, generalizing Example 8, and illustrating that the arrows in dia-grams (2.4) and (2.5) are not necessarily natural equivalences, will be exhibited in the nextsection, where we specialize to split short exact sequences of finitely generated Abeliangroups.

3. Localizations associated to semidirect products of finitely generated Abeliangroups

From [4, Proposition 7.3] we know thatP(HoK)ab = PH/[H,K]×K and PHoK arenaturally equivalent when restricted to the category of nilpotent groups. In particular, ifH oK is itself nilpotent, we infer thatPHoK is equivalent toPH/[H,K]×K on the categoryof all groups. Here we show that, under the additional assumption thatH andK are finitelygenerated Abelian groups, these nullifications actually are equivalent toPH×K .


Theorem 9. Consider the split short exact sequence(2.3), whereH andK are finitelygenerated Abelian groups. IfH o K is nilpotent, then〈H/[H,K]〉 = 〈π〉 = 〈H 〉 and〈H oK〉 = 〈H ×K〉. Therefore, diagrams(2.4)and(2.5)collapse to

Id Lε P[H,K] PH ∼= Lπ ∼= PH/[H,K]

PK Lφ P[H,K]×K PH×K ∼= PHoK ∼= PH/[H,K]×K(3.1)

Proof. To prove the first claim, it suffices to show that〈H/[H,K]〉 = 〈H 〉. If H is a finiteAbelian group, then it follows easily from the fact thatH o K is nilpotent, that the setT (H) of prime numbersp for whichH hasp-torsion coincides with the setT (H/[H,K])of prime numbersp for which H/[H,K] hasp-torsion. Hence,〈H/[H,K]〉 = 〈H 〉 =〈∏p∈T (H) Cp〉, whereCp denotes the cyclic group of orderp. If H is not finite, then wecan choose generatorsa1, . . . , am for the torsion-free part ofH such that the torsion-freepart of [H,K] is generated byt1a1, . . . , tmam for some integerst1, . . . , tm with t1| · · · |tm.SinceH o K is nilpotent, we infer thattm = 0. Hence,〈H 〉 = 〈H/[H,K]〉 = 〈Z〉.Finally, if 〈π〉 = 〈H 〉, then the second claim follows immediately from Proposition 2 andLemma 1. 2

In the case of a semidirect product of finitely generated Abelian groups, we can even gofurther in our description of the associated localization functors. We henceforth adopt theconvention that the cyclic group of orderm is writtenZm when we want to use additivenotation, and that we writeCm when thought of this cyclic group as a multiplicative group.We allowm to be any positive integer or to be∞, with the convention thatC∞ = Z∞ = Z.Let

H = Zn1 × · · · ×Znk and K = Cm1 × · · · ×Cml (3.2)

be finitely generated Abelian groups (we do not exclude the case where some of the factorsof H and/orK are infinite cyclic groups). Choose generatorsxi for Cmi andaj for Znj .Suppose further that, for this choice of generators, the action ofxi on H is given by a(k×k)-matrixPi (representing the corresponding element of Aut(Zn1×· · ·×Znk )). Hence,the commutator subgroup[H,K] is the subgroup ofH generated by the images of(Pi−I)for all i, whereI denotes the identity matrix, i.e.,

[H,K] = grp{im(P1− I), . . . , im(Pl − I)

}.

We restate the results of Theorems 5 and 6 in this particular case.

Proposition 10. Consider the split short exact sequence(2.3), whereH andK are givenby (3.2). Then

(1) 〈φ〉 = 〈K〉 ∗ 〈ε〉, whereε :H �H/grp{im(P1− I), . . . , im(Pl − I)},(2) 〈H/grp{im(P1− I), . . . , im(Pl − I)}〉6 〈π〉6 〈H 〉,

wherePi is the matrix corresponding to the action ofxi onH .


For example, if everyPi is a diagonal matrix, i.e., if there exist integerstij such thatxi · aj = tij aj for all i, j , then 〈φ〉 = 〈K〉 ∗ 〈ε〉, whereε is the natural projection fromZn1 × · · · × Znk ontoZn1/grp{g1a1} × · · · ×Znk /grp{gkak}, and, furthermore,⟨

Zn1/grp{g1a1} × · · · ×Znk /grp{gkak}⟩6 〈π〉6 〈Zn1 × · · · ×Znk 〉,

where the integer

gj = gcd{tij − 1 | tij − 1 6= 0 andi = 1, . . . , k

}.

We now confine our attention to

1→ Zn = 〈a〉 φ−→ Zn oCmπ−→ Cm = 〈x〉→ 1, (3.3)

wheren is a positive integer (andm is a positive integer or∞).

Proposition 11. Consider the split short exact sequence(3.3). Let the correspondingaction be given byx · a = ta with 06 t < n. Suppose that the prime decomposition ofn is given byn = pr11 · · ·prkk (whereri > 1). Suppose further thatt − 1= ps11 · · ·pskk Q,whereQ is coprime top1, . . . , pk and06 si 6 ri . Then

(1) 〈π〉 = 〈π1〉 ∗ · · · ∗ 〈πk〉, where the exact sequence1→ Zprjj

→ Zprjj

o Cmπj−→

Cm→ 1 represents the induced action;(2) 〈ε〉 = 〈ε1〉 ∗ · · · ∗ 〈εk〉, whereεj :Z

prjj

� Zpsjj

is induced byε :Zn� Zn/[Zn,Cm].

Proof. Clearly,[Zn,Cm] = grp{(t−1)a} ∼= Zpr1−s11×· · ·×Z

prk−skk

. HenceZn/[Zn,Cm] ∼=Zps11×· · ·×Z

pskk

. Moreover, it is clear that a groupG isπ -local if and only ifG isπj -local

for all j . 2The above result enables us to reduce the general case of a semidirect product (3.3) to

the case wheren is a power of a prime number.

Proposition 12. Consider the split short exact sequence(3.3)wheren= pr for a primepandr > 1. Suppose that the corresponding action is given byx · a = ta, where06 t < pr .Suppose further thatt − 1= psQ, where06 s < r andQ is coprime top. Then

(1) 〈φ〉 = 〈Cm〉 ∗ 〈ε〉, whereε :Zpr � Zps .(2) If Zpr oCm is nilpotent(i.e.,s > 1), then〈π〉 = 〈Zp〉.(3) If the action is perfect(i.e.,s = 0), then〈π〉6 〈Zp〉.

To complete our description, we recall thatPZpG=G/Ip(G) for every groupG, whereIp(G) denotes thep-radical (orp-isolator) of the groupG. Furthermore, iff :Zpr � Zps(with r > s) is the canonical projection, then the effect ofLf on a groupG is to reduce theorder of any element of orderpk (for k > s) to ps . Finally,Lπ kills all the elementsg of agroupG for whichgp

r = 1 and for which there exists an elementy in G such thatym = 1(if m is finite) and[y,g] = gt−1.

Finally, to see that we may have〈π〉 < 〈Zp〉 in Proposition 12, it suffices to considera perfect action ofCq on Zp , wherep and q are distinct primes (cf. Example 8).


Indeed,〈π〉 = 〈Zp〉 would imply that〈Zp oCq 〉 = 〈Zp ×Cq 〉, which is impossible, sincePZpoCq (Zp ×Cq) 6= 1.

Acknowledgements

We would like to thank Carles Casacuberta for his constant interest and invaluableadvice. The second-named author also thanks the Centre de Recerca Matemàtica for itshospitality.

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