Pure extensions of locally compact abelian groups

Pure Extensions of Locally Compact Abelian Groups.

PETERLOTH(*)

ABSTRACT- In this paper, we study the groupPext(C;A) for locally compact abelian (LCA) groupsAandC. Sufficient conditions are established forPext(C;A) to coincide withthe first Ulm subgroup ofExt(C;A). Some structural information on pure injectives in the category of LCA groups is obtained. LettingCdenote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groupsGin Cwhich are pure injective in the category of LCA groups. Finally we describe those groupsGinCsuchthat every pure extension ofGby a group inCsplits and obtain a corresponding dual result.

1. Introduction.

In this paper, all considered groups are Hausdorff topological abelian groups and will be written additively. LetLdenote the category of locally compact abelian (LCA) groups with continuous homomorphisms as morphisms. The Pontrjagin dual group of a groupGis denoted byGb and the annihilator of SGinGb is denoted by (G;b S). A morphism is called properif it is open onto its image, and a short exact sequence

0!A !^f B !^c C!0

inLis said to beproper exactiffandcare proper morphisms. In this case, the sequence is called an extension of A by C(inL), andAmay be iden- tified withf(A) andCwithB=f(A). Following Fulp and Griffith[FG1], we let Ext(C;A) denote the (discrete) group of extensions of A by C. Th e elements represented by pure extensions of Aby C form a subgroup of

(*) Indirizzo dell'A.: Department of Mathematics, Sacred Heart University, 5151 Park Avenue, Fairfield, Connecticut 06825, USA

E-mail: lothp@sacredheart.edu

Mathematics Subject Classification (2000); Primary 20K35, 22B05; Secondary 20K25, 20K40, 20K45

Ext(C;A) which is denoted by Pext(C;A). This leads to a functor Pext from LLinto the category of discrete abelian groups. The literature shows the importance of the notion of pure extensions (see for instance [F]). The concept of purity in the category of locally compact abelian groups has been studied by several authors (see e.g. [A], [B], [Fu1], [HH], [Kh], [L1], [L2] and [V]). The notion of topological purity is due to Vilenkin [V]: a subgroupHof a groupGis calledtopologically pureifnHˆH\nGfor all positive integersn. The annihilator of a closed pure subgroup of an LCA group is topologically pure (cf. [L2]) but need not be pure inGb(see e.g. [A]).

As is well known, Pext(C;A) coincides with Ext(C;A)¹ ˆ\¹

nˆ1

nExt(C;A);

the first Ulm subgroup of Ext(C;A), provided thatAandCare discrete abelian groups (see [F]). In the categoryL, a corresponding result need not hold: for groupsAandCinL, Ext(C;A)¹is a (possibly proper) subgroup of Pext(C;A), and it coincides withPext(C;A) if (a)AandC are compactly generated, or (b)AandChave no small subgroups (see Theorem 2.4). IfGis pure injective inL, thenGhas the formRTG⁰ whereRis a vector group,Tis a toral group andG⁰ is a densely divisible topological torsion group. However, the converse need not be true (cf. Theorem 2.7). LetC denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group. Then a group inCis pure injective inLif and only if it is injective inL(see Cor- ollary 2.8). LetGbe a group inC. Then every pure extension ofGby a group inCsplits if and only ifGhas the formRTABwhereRis a vector group,Tis a toral group,Ais a topological direct product of finite cyclic groups andBis a discrete bounded group. Dually, every pure extension of a group inCbyGsplits exactly ifGhas the formRCDwhereRis a vector group,Cis a compact torsion group andDis a discrete direct sum of cyclic groups (see Theorem 2.11).

The additive topological group of real numbers is denoted byR,Qis the group of rationals,Zis the group of integers,Tis the quotientR=Z,Z(n) is the cyclic group of ordernandZ(p¹) denotes the quasicyclic group. ByG_d we mean the groupGwiththe discrete topology,tGis the torsion part ofG and bG is the subgroup of all compact elements of G. Throughout this paper the term ``isomorphic'' is used for ``topologically isomorphic'', ``direct summand'' for ``topological direct summand'' and ``direct product'' for ``to- pological direct product''. We follow the standard notation in [F] and [HR].

2. Pure extensions of LCA groups.

We start witha result on pure extensions involving direct sums and direct products.

THEOREM2.1. Let G be inLand supposefH_i:i2Igis a collection of groups inL. If Hiis discrete for all but finitely many i2I,then

PextM

i2I

H_i;G

Y

i2I

Pext(H_i;G):

If H_iis compact for all but finitely many i2I,then Pext

G;Y

i2I

H_i

Y

i2I

Pext(G;H_i):

In general,there is no monomorphism Pext G;Y

i2I

Hi

d

!

!Y

i2I

Pext(G;(Hi)_d):

PROOF. To prove the first assertion, letpi:Hi!L

Hibe the natural injection for each i2I. Then the map f:Ext(L

H_i;G)!Q

Ext(H_i;G) defined byE7!(Ep_i) is an isomorphism (cf. [FG1] Theorem 2.13), mapping the group Pext(L

H_i;G) intoQ

Pext(H_i;G). If the groups H_i andGare stripped of their topology, the corresponding isomorphism maps the group Pext(L

(Hi)_d;Gd) ontoQ

Pext((Hi)_d;Gd) (see [F] Theorem 53.7 and p. 231, Exercise 6). Since an extension equivalent to a pure extension is pure,f maps Pext(L

H_i;G) ontoQ

Pext(H_i;G), establishing the first statement.

The proof of the second assertion is similar. To prove the last statement, let pbe a prime andHˆQ₁

nˆ1Z(pⁿ), taken discrete. Assume Ext(Q;b H)ˆ0.

By [FG2] Corollary 2.10, the sequences

Ext(Q;b H)!Ext(Q;b H=tH)!0 and

0ˆHom((Q=Z)b;H=tH)!Ext(Z;b H=tH)!Ext(Q;b H=tH) are exact, hence [FG1] Proposition 2.17 yieldsH=tHExt(Z;b H=tH)ˆ0 which is impossible. Since Qb is torsion-free, it follows that Pext(Q;b H)ˆ

ˆExt(Q;b H)6ˆ0. On the other hand, we have Y¹

nˆ1

Pext(Q;b Z(pⁿ))ˆY¹

nˆ1

Ext(Q;b Z(pⁿ))Y¹

nˆ1

Ext(Z(pⁿ);Q)ˆ0

by [FG1] Theorem 2.12 and [F] Theorem 21.1. Note that this example shows

that Proposition 6 in [Fu1] is incorrect. p

PROPOSITION 2.2. Suppose E₀: 0!A !^f B!C!0 is a proper exact sequence inL. Leta:A!A be a proper continuous homomorph- ism andathe induced endomorphism onExt(C; A)given bya(E)ˆaE.

ThenE02Imaif and only ifImf=Imfais a direct summand ofB=Imfa.

PROOF. Ifa:A!Ais a proper morphism inL, then 0!Ima!A!Imf=Imfa!0 and

0!Kera!A!Ima!0

are proper exact sequences in L (cf. [HR] Theorem 5.27). Now [FG2]

Corollary 2.10 and the proof of [F] Theorem 53.1 show thatE₀2Imaif and only if the induced proper exact sequence

0!Imf=Imfa!B=Imfa!C!0

splits. p

If A and C are groups in L, then Ext(C;A)Ext(A;b C) (see [FG1]b Theorem 2.12). We have, however:

LEMMA2.3. Let A and C be inL. Then:

(i) In general,Pext(C;A)6Pext(A;b C).b

(ii) LetKdenote a class of LCA groups satisfying the following property: If G2K,thenGb 2Kand nG is closed in G for all positive in- tegers n. ThenPext(C;A)Pext(A;b C)b whenever A and C are inK.

PROOF. (i) The finite torsion part of a group inLneed not be a direct summand (see for instance [Kh]), so there is a finite groupFand a torsion- free groupCinLsuchthat Pext(C;F)ˆExt(C;F)6ˆ0. On the other hand, Pext(F;b C)b Pext(F;(C)b _d)ˆ0 by [F] Theorem 30.2.

(ii) Let A and C be in K and consider the isomorphism Ext(C;A) !Ext(A;b C) given byb E:0!A!B!C!07!Eb :0! Cb !Bb!Ab !0:The annihilator of a closed pure subgroup ofBis topo- logically pure inBb (cf. [L2] Proposition 2.1) and for all positive integersn, nAandnCbare closed subgroups ofAandC, respectively. Therefore,b Eis

pure if and only ifEb is pure. p

Recall that a topological group is said to have no small subgroupsif there is a neighborhood of 0 which contains no nontrivial subgroups.

Moskowitz [M] proved that the LCA groups with no small subgroups have the formRⁿT^mDwherenandmare nonnegative integers andDis a discrete group, and that their Pontrjagin duals are precisely the compactly generated LCA groups.

THEOREM2.4. For groups A and C inL,we have:

(i) Pext(C;A)Ext(C;A)¹.

(ii) Pext(C;A)6ˆExt(C;A)¹in general.

(iii) Suppose (a) A and C are compactly generated,or (b) A and C have no small subgroups. ThenPext(C;A)ˆExt(C;A)¹.

PROOF. (i) Leta:A!Abe the multiplication by a positive integern and letE:0!A !^f X!C!02nExt(C;A). Since Ext is an additive functor, there exists an extension 0!A!B!C!0 suchthat

0 ! A ! B ! C ! 0

a# # k

0 ! A !^f X ! C ! 0

is a pushout diagram in L. An easy calculation shows that nX\f(A)ˆ

ˆnf(A), hence Ext(C;A)¹is a subset of Pext(C;A).

(ii) Let Pext(C;F) be as in the proof of Lemma 2.3. Then Pext(C;F)6ˆ0 but Ext(C;F)¹ˆ0.

(iii) Suppose first thatAandCare compactly generated. Ifa:A!Ais the multiplication by a positive integern, thena(A)ˆnAis a group inL.

SinceAiss-compact,ais a proper morphism by [HR] Theorem 5.29. Let E:0!A !^f B!C!02Ext(C;A). By Proposition 2.2, E2Ima ˆ

ˆnExt(C;A) if and only iff(A)=nf(A) is a direct summand ofB=nf(A).

Now assume that Eis a pure extension. Then f(A)=nf(A) is pure in the groupB=nf(A) which is compactly generated (cf. [M] Theorem 2.6). Since the compact groupf(A)=nf(A) is topologically pure, it is a direct summand ofB=nf(A) (see [L1] Theorem 3.1). Consequently,Eis an element of the first Ulm subgroup of Ext(C;A) and by (i) the assertion follows. To prove the second part of (iii), assume thatAandChave no small subgroups. By what we have just shown and Lemma 2.3, we have Pext(C;A)

Pext(A;b C)b ˆExt(A;b C)b ¹Ext(C;A)¹: p By the structure theorem for locally compact abelian groups, any group GinLcan be written asGˆVGe whereVis a maximal vector subgroup

of GandGe contains a compact open subgroup. The groups V and Ge are uniquely determined up to isomorphism (see [HR] Theorem 24.30 and [AA]

Corollary 1).

LEMMA2.5. A group G inLis torsion-free if and only if every compact open subgroup ofG is torsion-free.e

PROOF. Only sufficiency needs to be shown. Suppose every compact open subgroup ofGe is torsion-free and assume thatGis not torsion-free.

ThenGe contains a nonzero elementxof finite order. IfK is any compact open subgroup ofG, thene K‡ hxiis compact (see [HR] Theorem 4.4) and open inGebut not torsion-free, a contradiction. p Dually, we obtain the following fact which extends [A] (4.33). Recall that a group is said to be densely divisible if it possesses a dense divisible subgroup.

LEMMA2.6. A group G inLis densely divisible if and only ifG=K ise divisible for every compact open subgroup K ofG.e

PROOF. Again, only sufficiency needs to be proved. Assume thatG=Ke is divisible for every compact open subgroupKofGeand letCbe a compact open subgroup of (G)be. Since (G)be(G=V)bwhere V is a maximal vector subgroup ofG, there exists a compact open subgroupX=VofG=Vsuchthat C((G=V)b;X=V)((G=V)=(X=V))b(see [HR] Theorems 23.25, 24.10 and 24.11). By our assumption, (G=V)=(X=V) is divisible. But thenCis torsion- free (cf. [HR] Theorem 24.23), so by Lemma 2.5,Gbis torsion-free. Finally, [R] Theorem 5.2 shows thatGis densely divisible. p Let Gbe inL. ThenG is called pure injective inLif for every pure extension 0!A !^f B!C!0 in L and continuous homomorphism f :A!G there is a continuous homomorhism f :B!G such that the diagram

0 ! A !^f B ! C ! 0

f # .f

G

is commutative. Following Robertson [R], we callGatopological torsion groupif (n!)x!0 for everyx2G. Note that a groupGinLis a topological torsion group if and only if bothGandGb are totally disconnected (cf. [R]

Theorem 3.15). Our next result improves [Fu1] Proposition 9.

THEOREM2.7. Consider the following conditions for a group G inL:

(i) G is pure injective inL.

(ii) Pext(X;G)ˆ0for all groups X inL.

(iii) GRⁿT^mG⁰ where n is a nonnegative integer, m is a cardinal and G⁰is a densely divisible topological torsion group which,as such,possesses no nontrivial pure compact open subgroups.

Then we have:(i),(ii))(iii)and(iii)6)(ii).

PROOF. If G is pure injective in L, then any pure extension 0!G!B!X!0 inLsplits because there is a commutative diagram

0 ! G ! B ! X ! 0

k . G;

hence (i) implies (ii). Conversely, assume (ii). If 0!A!B!X!0 is a pure extension inL andf :A!Gis a continuous homomorphism, then there is a pushout diagram

0 ! A ! B ! X ! 0

f # # k

0 ! G ! Y ! X ! 0:

The bottom row is an extension inL(cf. [FG1]) which is pure. By our as- sumption, it splits and (i) follows.

To show (ii))(iii), let us assume first that Pext(X;G)ˆ0 for all groups X2C. Then the proof of [L1] Theorem 4.3 shows thatGis isomorphic to RⁿT^mG⁰wherenis a nonnegative integer,mis a cardinal andG⁰is totally disconnected. Notice thatG⁰=bG⁰is discrete (cf. [HR] (9.26)(a)) and torsion-free. Since the sequence

0ˆHom((Q=Z)b;G⁰=bG⁰)!Ext(Z;b G⁰=bG⁰)!Ext(Q;b G⁰=bG⁰)ˆ0 is exact,G⁰=bG⁰is isomorphic to Ext(Z;b G⁰=bG⁰)ˆ0 and thereforeG⁰ˆbG⁰. It follows that the dual group ofG⁰is totally disconnected (cf. [HR] Theorem 24.17), thusG⁰is a topological torsion group. Suppose that Pext(X;G)ˆ0 for allX2Land letK be a compact open subgroup ofG⁰. ThenG⁰=Kis a divisible group (see [Fu2] Theorem 7 or the proof of [L1] Theorem 4.1), so by Lemma 2.6 G⁰ is densely divisible. Now assume that G⁰ has a pure compact open subgroupA. SinceAis algebraically compact, it is a direct summand ofG⁰. But thenAis divisible, hence connected (see [HR] Theorem 24.25) and thereforeAˆ0. Consequently, (ii) implies (iii).

Finally, (iii)6)(ii) because for instance, there is a nonsplitting extension ofZ(p¹) by a compact group (cf. [A] Example 6.4). p

Those groups in C which are pure injective in L are completely de- termined:

COROLLARY2.8. A groupGinCis pure injective inLif and only if GRⁿT^mwherenis a nonnegative integer andmis a cardinal.

PROOF. The assertion follows immediately from [M] Theorem 3.2 and

the above theorem. p

The following lemma will be needed.

LEMMA 2.9. Every finite subset of a reduced torsion group A can be embedded in a finite pure subgroup of A.

PROOF. By [F] Theorem 8.4, it suffices to assume thatAis a reducedp- group. But then the assertion follows from [K] p. 23, Lemma 9 and an easy

induction. p

A pure extension 0!A!B!C!0 withdiscrete torsion group A and compact groupC need not split, as [A] Example 6.4 illustrates. Our next result shows that no such example can occur ifAis reduced.

PROPOSITION 2.10. Suppose A is a discrete reduced torsion group.

ThenPext(X; A)ˆ0for all compactly generated groupsX inL.

PROOF. SupposeE:0!A !^f B !^c X!0 represents an element of Pext(X;A) whereAis a discrete reduced torsion group andXis a compactly generated group inL. By [FG2] Theorem 2.1, there is a compactly gener- ated subgroupCofBsuchthatc(C)ˆX. If we setA⁰ˆf(A), thenA⁰\Cis discrete, compactly generated and torsion, hence finite, so by Lemma 2.9A⁰ has a finite pure subgroupFcontainingA⁰\C. Now setC⁰ˆC‡F. ThenF is a pure subgroup ofC⁰because it is pure inB. But thenFis topologically pure inC⁰sinceC⁰is compactly generated. By [L1] Theorem 3.1, there is a closed subgroup Y of C⁰ suchthat C⁰ˆFY. We have BˆA⁰‡Cˆ

ˆA⁰‡C⁰ˆA⁰‡Yand

A⁰\YˆC⁰\A⁰\Y ˆ(F‡C)\A⁰\Yˆ[F‡(C\A⁰)]\YˆF\Yˆ0;

thusBis an algebraic direct sum ofA⁰ andY. SinceY is compactly gen- erated, it iss-compact, so by [FG1] Corollary 3.2 we obtain BˆA⁰Y.

Consequently, the extensionEsplits. p

THEOREM2.11. Let G be a group inC. Then we have:

(i) Pext(X;G)ˆ0for all X2Cif and only if GRⁿT^mAB where n is a nonnegative integer,mis a cardinal,A is a direct product of finite cyclic groups and B is a discrete bounded group.

(ii) Pext(G;X)ˆ0 for all X2C if and only if GRⁿCD where n is a nonnegative integer,C is a compact torsion group and D is a discrete direct sum of cyclic groups.

PROOF. SupposeG2Cand Pext(X;G)ˆ0 for allX2C. By the proof of part (ii) ) (iii) of Theorem 2.7, Gis isomorphic to RⁿT^mAB whereAis a compact totally disconnected group andBis a discrete torsion group. By Lemma 2.3, we have Pext(A;b X)Pext(X;b A)ˆ0 for all discrete groupsX, henceAb is a direct sum of cyclic groups (see [F] Theorem 30.2) and it follows thatAis a direct product of finite cyclic groups. Again, we make use of [A] Example 6.4 and conclude thatBis reduced. But thenBis bounded since it is torsion and cotorsion. Conversely, supposeGhas the formRⁿT^mABas in the theorem and letXˆR^mYZwhereY is a compact group and Z is a discrete group. Then Pext(X;A)

Pext(A;b X)b Pext(A;b (X)b _d)ˆ0. By Theorem 2.1, Proposition 2.10 and [F] Theorem 27.5 we have

Pext(X;B)Pext(R^m;B)Pext(Y;B)Pext(Z;B)ˆ0 and conclude that

Pext(X;G)Pext(X;RⁿT^m)Pext(X;A)Pext(X;B)ˆ0:

Finally, the second assertion follows from Lemma 2.3 and duality. p

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Manoscritto pervenuto in redazione il 20 ottobre 2004.