Generalized Cotorsion Locally Compact abelian Groups

Generalized Cotorsion Locally Compact Abelian Groups

N. I. KRYUCHKOV

ABSTRACT- Thispaper isconcerned with the generalization of the concept of co- torsion abelian group. A locally compact abelian groupLiscalled generalized cotorsion ifLcontainsa compact open subgroupKsuch that the character group ofKand the groupL=Kare cotorsion groups. Some properties and homological characteristics of generalized cotorsion groups are obtained. The classification problem of generalized cotorsion groups is discussed.

1. Introduction

Throughout, all groupswill be abelian groups. Thispaper isconcerned with the generalization of the well-known in abelian group theory concept of cotorsion group. The notion of cotorsion group was introduced and studied by Harrison [1]. Fuchs [2] and Nunke [3] discovered the same notion independently. A discrete groupAiscalled acotorsiongroup if and only if Ext(Q;A)ˆ0, whereQisthe additive group of rational numbers.

Cotorsion groups play an important role in the theory of Abelian groups.

The category of compact abelian groupsisdual to the category of discrete abelian groupsby Pontrjagin duality theory. Therefore any concept in the theory of discrete abelian groups has a dual concept in the theory of compact groups and they are both specifications of some concepts in the theory of locally compact abelian groups. Every statement about discrete groups has its dual assertion in the theory of compact groups. Both of these are usually special cases some statement about locally compact abelian groups. In this paper we construct a new class of locally compact abelian groups, which contains the class of cotorsion groups and its dual. The objects of this new class are called generalized cotorsion groups. The class

(*) Indirizzo dell'A.: Department of Mathematics, Ryazan State University, Svobody, 46 Ryazan, 390048 Russia.

E-mail: kryuchkov.n@gmail.com

of generalized cotorsion groups is enough large. It include the class of cotorsion groups and the class the duals of cotorsion groups. Every com- pact coreduced generalized cotorsion group is a character group some group of extensions. The class generalized cotorsion groups has a number interesting properties. Every extension of a compact coreduced group by a discrete reduced group may be connect with some generalized cotorsion group. Therefore, generalized cotorsion groups should play an important role in the theory of locally compact abelian groups.

In Section 2 the class ofdual cotorsion groupsisdefined. A compact groupKiscalled dual cotorsion group if and only if the character group of the groupKis a discrete cotorsion group. Some properties of compact dual cotorsion groups are investigated.

In Section 3 we define the concept ofgeneralized cotorsion groups. A locally compact group L iscalled generalized cotorsion if Lcontainsan open dual cotorsion subgroupK, such that the factor group L=K isa co- torsion group. We study properties of generalized cotorsion groups. The main results of this section are Theorem 3.3 and Theorem 3.7.

In Section 4 some homological properties of generalized cotorsion groups are investigated. The main statements of this section are Theorem 4.4 and Theorem 4.5.

In Section 5 the problem of a classification of generalized cotorsion groups is discussed.

Thispaper reliesheavily on the Pontrjagin Duality Theorem and re- lated structural and character theoretic facts about locally compact groups.

Material on discrete group theory, locally compact group theory, and Pontrjagin duality can be found in [4] - [7]. ByAdenote the class of dis- crete abelian groupsand denote byCthe class of compact abelian groups.

ByLdenote the class of locally compact abelian groups.

By Lb denote the character group of a locally compact group L con- sidering with its compact-open topology, so that it is a locally compact group. A compact groupKiscalledcoreducedif the groupKb isa reduced group. If G and H are locally compact groupsand f :G!Hisa con- tinuoushomomorphism we denote by bf :Hb !Gb the continuoushomo- morphism dual tof. IfLisa topological group,L0will denote the identity component ofL. A continuoushomomorphism will be calledproperif it is open onto its image. An exact sequence of continuous homomorphisms of

A1 ^f!¹ A2 ^f!² . . . ^fn!¹ An

issaid to beproper exactif and only iff_iisproper for eachi.

IfE:0!L1!^a L2!^b L3!0 is a proper short exact sequence of lo- cally compact groupsthen by Eb denote a proper short exact sequence 0!cL3!b_b Lc2!ba Lc1!0. Byf denote the forgetful functor from the cate- gory of locally compact groups to the category of discrete groups, which forgetsthe topological structure. By Ext_Zdenote the extension functor on the category of discrete abelian groups. By Ext denote the extension functor on the category of locally compact abelian groups discussed by Fulp and Griffith [8]. Ext isan additive functor fromLLtoA. IfAandB are locally compact groups, then Ext(A;B) isthe (discrete) group of all extensions ofBbyA. The functor Ext generalizesthe functor Ext_Z. IfA and Bare discrete groups, then Ext(A;B)Ext_Z(A;B). If Aisdiscrete andL islocally compact, then Ext(A;L)Ext_Z(A;f(L)). IfAandBare locally compact groups, then there is the natural isomorphism Ext(A;B) Ext(B;b A). Ext commuteswith finite direct productsin each arguments. Byb Hom(A;B) denote the (discrete) group of continuous homomorphisms from AtoB. There are long exact sequences which connect Hom to Ext.

BySdenote the character group of the additive group of rational num- bers. We denote byRthe additive group of real numberswith itsnatural topology. LetZ_pbe the additive group ofp-adic integerswith itsnatural topology. A discrete group is said to bealgebraically compactif it isa direct summand of every group containing it as a pure subgroup. A group is al- gebraically compact if and only if it isan algebraically direct summand of some discrete group which admits a compact topology [4], Theorem 38.1 d).

Hence ifKisany compact group, thenf(K) isalgebraically compact. Every algebraically compact group iscotorsion.

IfLisany locally compact group thenLRⁿM, whereMcontains an open compact subgroup ([6], Theorem 24.30).

2. Dual cotorsion groups

We shall that a compact group K iscalleddual cotorsion if the char- acter groupKb of the groupKisa cotorsion group. We claim that a compact groupGisdual cotorsion if and only if Ext(G;S)ˆ0. Indeed, thisfollows from Pontrjagin duality and the definition of cotorsion group. From ele- mentary propertiesof cotorsion groupswe conclude that a compact group Gisa dual cotorsion group if and only ifGsatisfies Ext(G;C)ˆ0 for all compact connected groups C. Thusif K isany compact group, C isits connected subgroup andK=Cisdual cotorsion, thenKC(K=C).

Denote byCT andDCthe classes of cotorsion groups and dual cotorsion groupsrespectively. From the definition of dual cotorsion group, proper- ties of cotorsion groups (see the book [4], chapter 9, § 54) by Pontrjagin duality we have next statements.

1.A2 CT ,Ab 2 DC

2.The class of dual cotorsion groups is closed under taking subgroups.

PROOF. Let Lbe any subgroup of the dual cotorsion group K. The group Lb isthe epimorphic image of the group K, henceb Lb iscotorsion ([4], § 54, A). It followsthat Lisdual cotorsion. p 3.Let L be any coreduced dual cotorsion group. The factor group L=M is a dual cotorsion group if and only if the subgroup M is coreduced.

PROOF. The groupL=Md isthe subgroup of the groupLb and the cor- responding factor group is isomorphic toM. Henceb L=Md iscotorsion if and only ifMb isreduced ([4], § 54, B). Therefore,L=Misdual cotorsion if and

only ifMiscoreduced. p

4.Let L be any coreduced dual cotorsion group and letWbe any con- tinuous endomorphism of L. Then kerW and imW are dual cotorsion groups.

PROOF. Thisstatement followsfrom [4], § 54, C). p 5.If K is the subgroup of L such that K and L=K are dual cotorsion, then L is dual cotorsion.

PROOF. The groupsKb andL=Kd are cotorsion, it follows that Lb isco- torsion ([4], § 54, D). ThusLisdual cotorsion. p 6.LetfG_ig_fi2Igbe a family of discrete groups and letG be the characterb group of the direct product Q

i2IG_i. Then the groupG is a dual cotorsionb group if and only if each groupcGiis a dual cotorsion group.

PROOF. Thisstatement followsfrom [4], § 54, E). p 7.Let M be any dual cotorsion group and let K be any compact group.

Then the character group ofHom(M;K)is a dual cotorsion group.

PROOF. Hom(M;K)Hom(K;b M). The group Hom(b K;b M) iscotorsionb ([4], § 54, G), hence itscharacter group isdual cotorsion. p 8.Let K be any dual cotorsion coreduced group. Then there is a natural isomorphismExt K;Q

p Z_p

K. Thus K is isomorphic to the characterb group of the groupExt K;Q

p Z_p .

PROOF. SinceK isdual cotorsion coredused thenKb iscotorsion re- duced such that there is an isomorphism Kb Ext(Q=Z;K) ([4], § 54, I).b Now, by Ext functor properties, we obtain Ext(Q=Z;K)b Ext(K;Q=Z)d  Ext K;Q

p Zp

. p

9.A compact group K is a dual cotorsion group if and only if there is some compact group M such that K is a subgroup of M and the character group of M is an algebraically compact group.

PROOF. SinceKisdual cotorsion thenKb is cotorsion such that there exist an exact sequenceM₁!^b Kb !0 where the groupM₁isan algebrai- cally compact [4], § 54, the Proposition 54.1. We get the proper exact se- quence 0!K!b_b Mc₁. Denote byMthe groupMc₁. The character group of Misan algebraically compact andKis isomorphic to a subgroup ofM. p 10. Let K be a compact connected group. Then K is dual cotorsion if and only if K is the character group of some algebraically compact group.

PROOF. The groupKb istorsion free. HenceKbiscotorsion if and only if Kb isan algebraically compact group [4], the Corollary 54.5. p 11. Let K be a compact totally disconnected group. Then K is dual cotorsion if and only if KQ

p

Q

mp

Z_p L

M, where M is a torsion compact group.

PROOF. The groupKbis torsion, sinceKisa totally disconnected group.

It followsKb iscotorsion if and only ifKb L

p

L

mp

Z(p¹) L

L, where the group L isbounded ([4], the Corollary 54.4). ThusKQ

p

Q

mp

Zp L bL.

DenoteLbbyM. The groupMgroup is torsion compact group, such that it is

dual to a bounded discrete group. p

The list of properties of dual cotorsion groups can be extended using the Pontrjagin duality. The classification of dual cotorsion groups will be discussed in section 5.

3. The properties of the class generalized cotorsion groups

DEFINITION3.1 (Main definition). A locally compact groupLiscalled a generalized cotorsion group, ifLcontains an open dual cotorsion subgroup Ksuch that the groupL=Kisa cotorsion group; in other wordsthere isa proper short exact sequence

0!K!L!A!0

whereKisa dual cotorsion group andAisa cotorsion group.

Any cotorsion group and any dual cotorsion are examples of generalized cotorsion groups. The additive group of allp-adic number with itsnatural topology isa generalized cotorsion group.

Denote byGCT the class of generalized cotorsion groups.

LEMMA3.2. A locally compact group L is a generalized cotorsion group if and only ifL is a generalized cotorsion group; in other words the class ofb generalized cotorsion groups is closed under Pontrjagin duality.

PROOF. LetKbe an open dual cotorsion subgroup ofLsuch thatL=K is a cotorsion group. Then there exists the proper short exact sequence

E:0!K!L!A!0 Hence there exists the proper short exact sequence

Eb :0!Ab !Lb!Kb !0

whereAbisa dual cotorsion group andKbisa cotorsion group. ThereforeLbis

a generalized cotorsion group. p

THEOREM 3.3. Let L by a locally compact group and L admits no subgroups isomorphic toR. The following conditions are equivalent:

1) L is a generalized cotorsion group;

2) Ext(Q;L)ˆExt(L;S)ˆ0;

3) Ext_Z(Q;f(L))ˆExt_Z(Q;f(bL))ˆ0;

4) f(L)andf(L)b are cotorsion groups;

5) Ext(A;L)ˆExt(L;C)ˆ0for all discrete torsion free groups A and for all compact connected groups C.

PROOF. 1) ) 2). Let K be an open dual cotorsion subgroup of L, such that L=K is a cotorsion group. Consider the proper short exact sequence

0!K!L!L=K !0 (3:1)

We obtain the exact sequence

Ext(Q;K)!Ext(Q;L)!Ext(Q;L=K)!0 (3:2)

SinceL=Kisa cotorsion group we obtain

Ext(Q;L=K)ExtZ(Q;L=K)ˆ0

The groupf(K) isalgebraically compact, thereforef(K) isa cotorsion group and we have

Ext(Q;K)ExtZ(Q;f(K))ˆ0 From (3.2) we get Ext(Q;L)ˆ0.

From Lemma 3.2 it followsthatLbisa generalized cotorsion group and we obtain Ext(Q;L)b ˆ0. Since Ext(Q;L)b ˆ0 and Ext(L;S)Ext(Q;L),b we see that Ext(L;S)ˆ0.

2))1). LetKbe any compact open subgroup of the groupL. Then we have proper exact sequences (3.1) and (3.2). By Ext(Q;L)ˆ0, so that Ext(Q;L=K)ˆ0. It followsthat the discrete groupL=Kiscotorsion. The proper short exact sequence (3.1) yields the exact sequence

Ext(L=K;S)!Ext(L;S)!Ext(K;S)!0 (3:3)

By Ext(L;S)ˆ0, so that Ext(K;S)ˆ0, i.e. the groupKisdual cotorsion.

From Definition 3.1 it followsthatLisa generalized cotorsion group.

2),3). SinceQisa discrete group we get Ext(Q;L)Ext_Z(Q;f(L)).

Also, Ext(L;S)Ext(bS;L)b Ext(Q;L); thusExt(L;b S)Ext_Z(Q;f(L)).b 3) , 4). By definition of cotorsion group we conclude that this equivalence istrue.

4) ,5). From elementary propertiesof cotorsion groupsand Pontr- jagin duality it followsthat thisequivalence istrue. p Corollary 3.4 and Corollary 3.5 followsimmediately from Theorem 3.3.

COROLLARY3.4. IfLis any generalized cotorsion group andLis an open subgroup of a locally compact groupMsuch thatM=Lis torsion free, thenLis a direct summand ofM.

COROLLARY3.5. IfLis any locally compact group andCis compact connected subgroup ofLsuch thatL=Cis a generalized cotorsion group, thenCis a direct summand of the groupL.

PROPOSITION3.6. 1)LetAbe any discrete group. ThenAis general- ized cotorsion if and only if A is cotorsion, i.e. CT ˆ A \ GCT. Thus a discrete groupAis generalized cotorsion if and only ifExt(Q; A)ˆ0.

2) Let K be any compact group. Then K is generalized cotorsion if and only if K is dual cotorsion, i.e.DC ˆ C \ GCT. Thus a compact group K is generalized cotorsion if and only ifExt(K;S)ˆ0.

PROOF. 1) LetAbe any discrete group such thatAisgeneralized co- torsion. From Theorem 3.3 4) it follows thatAis cotorsion. Conversely, ifA isany cotorsion group, then Ext(Q;A)ˆ0. Sincef(A) isan algebraicallyb compact group, it followsthatf(A) isa cotorsion group. Sinceb f(A) isa co-b torsion group, we have Ext(A;S)ˆExt(Q;A)b ˆExt_Z(Q;f(A))b ˆ0 andAis a generalized cotorsion group.

2) can be obtained from 1) by Pontrjagin duality. p THEOREM 3.7. The class of generalized cotorsion locally compact groups is closed under taking:

1) extensions;

2) finite direct sums;

3) topologically direct summands;

4) compact subgroups;

5) discrete epimorphic images.

PROOF. 1) Let 0!K!L!M!0 be a proper short exact sequence such thatKandMare generalized cotorsion groups. We will prove thatLis a generalized cotorsion group too. We have the exact sequences

Ext(Q;K)!Ext(Q;L)!Ext(Q;M)!0 and

Ext(M;S)!Ext(L;S)!Ext(K;S)!0

Since Ext(Q;K)ˆExt(Q;M)ˆExt(M;S)ˆExt(K;S)ˆ0, we get Ext(Q;L)ˆExt(L;S)ˆ0. ThusLisgeneralized cotorsion.

The properties2) and 3) follow from Theorem 3.3 since Ext commutes with finite direct products.

4) LetLbe any generalized cotorsion group and letKbe an arbitrary compact subgroup ofL. The proper short exact sequence

0!K!L!L=K !0 yieldsthe exact sequence

Ext(L=K;S)!Ext(L;S)!Ext(K;S)!0

Using Ext(L;S)ˆ0, we get Ext(K;S)ˆ0. From Proposition 3.6 it follows thatKisgeneralized cotorsion.

5) Let Lbe any generalized cotorsion group and let M be a discrete epimorphic image ofL. From the proper exact sequenceL!M!0 we have the exact sequence Ext(Q;L)!Ext(Q;M)!0. By Ext(Q;L)ˆ0, so that Ext(Q;M)ˆ0. From Proposition 3.6 it follows thatM isa gen-

eralized cotorsion group. p

4. Some homological properties of generalized cotorsion groups LetAbe any discrete group. ThenAiscotorsion if and only ifAhasthe injective property relative to all torsion splitting short exact sequences ([4], chapter 9, the Theorem 58.2). Remind that a short exact sequence E:0!A!B!C!0 of discrete groups is calledtorsion splittingif the sequenceEtissplitting, wheret:T(C)!Cisthe natural inclusion. The notion of compact splitting sequence, that we introduce in next definition, is a suitable generalization of the concept of torsion splitting sequence.

LetMbe any locally compact group. Denote byB(M) the subgroup of Msuch thatc2B(M) if and only ifcisa compact element. An elementcis called compact if the smallest closed subgroup which contains the element ciscompact ([6], chapter 2, the Definition 9.9).

DEFINITION 4.1. A proper short exact sequence of locally compact groups

E:0!K!L!M!0

is called compact splitting if the sequence Et issplitting, wheret isthe natural inclusiont:B(M)!M.

Let E:0!K!L!M!0 be any proper short exact sequence of locally compact groups. Obviously, if the groupMhasno compact elements thenEiscompact splitting.

LetAbe any discrete group. Then an elementa2Aiscompact if and only if a hasa finite order. Therefore, if A isa discrete group then T(A)ˆB(A) and we conclude that a short exact sequence E of discrete groupsiscompact splitting if and only ifEis torsion splitting.

Now we introduce a new concept. It isdual to the notion of compact splitting sequence.

DEFINITION 4.2. A proper short exact sequence of locally compact groups

E:0!K!L!M!0

iscalledtd-splitting (totally disconnected splitting) if the sequencesEis splitting, wheresisthe natural projections:K!K=K0.

Let E:0!K!L!M!0 be any proper short exact sequence of locally compact groups. Evidently, if the group K isconnected thenEis td-splitting. If the groupK istotally disconnected thenEistd-splitting if and only ifEissplitting. Hence ifEis a short exact sequence of discrete groups, thenEistd-splitting if and only if Eissplitting.

LetM be any locally compact group and letB(M) be the subgroup of compact elementsofM. Evidently, ifE:0!B(M)!^t Misa proper exact sequence, then Eb :Mb !bt M=b Mb0!0 isa proper exact sequence too, by Pontrjagin duality.

From the last remarks by Pontrjagin duality we obtain the next lemma.

LEMMA 4.3. 1) A proper short exact sequence E of locally compact groups is compact splitting if and only if the sequenceE is td-splitting.b

2) Let L be any locally compact group. Then L has the injective (projective) property relative to all compact splitting sequences if and only ifL has the projective (injective) property relative to all td-splitting se-b quences.

THEOREM 4.4. Let L be any generalized cotorsion locally compact group. Then L has the injective property relative to all compact splitting sequences and L has the projective property relative to all td-splitting se- quences.

PROOF. Let E:0!L1!^a L2!^b L3!0 be any proper short exact compact splitting sequence of locally compact groups andW:L₁!Lbe any continuous homomorphism. Suppose first, thatB(L₃)ˆ0. ThenL₃hasno

compact subgroups and we getL3RⁿL

A, whereAis a discrete torsion free group. We get Ext(L3;L)Ext(Rⁿ;L)L

Ext(A;L)ˆ0. The short exact sequenceEyieldsthe exact sequence

Hom(L2;L)!^a Hom(L1;L)!Ext(L3;L)ˆ0

It followsthat a isan epimorphism. Hence every continuoushomo- morphism fromL1 toLcan be extended to a continuoushomomorphism fromL2toL.

LetB(L3)6ˆ0 and lettbe a natural inclusiont:B(L3)!L3, thenEtis splitting.

We get the following commutative diagram:

Define the mapW₁:L₁L

B(L₃)!Lby the ruleW₁((a;b))ˆW(a) for every a2L1. Clearly the mapW₁isa continuoushomomorphism and the homo- morphismW₁extendsthe homomorphismW. LetL⁰₂be the inverse image of subgroup B(L3) relative b. By construction of induced extension we con- clude eachx2L⁰₂hasa unique inverse image relativelyc. It isto prove that there exists the continuous homomorphismW₂:L⁰₂!L, such that we get WˆW₂a.

Denote by L⁰₃ the factor group L2=L⁰₂. The group L⁰₃ hasno compact elements and we have the proper short exact sequence

0!L⁰₂!L2!L⁰₃!0

It follows that there exists a continuous homomorphismW₃:L2!Lsuch thatW₃extendsW₂, therefore we see thatWˆW₃a. It followsthat the group Lhas the injective property to all compact splitting sequences.

Since the groupLbisgeneralized cotorsion too, thenLbhasan injective property to all compact splitting sequences. From Lemma 4.3 it follows that the group L hasthe projective property to all td-split short exact

sequences. p

Every reduced discrete groupAcan be embedded in a reduced cotor- sion groupC(A) such thatC(A)=Ais torsion free and divisible. Moreover, this embedding is unique in the sense that ifAiscontained in a reduced cotorsion group B with B=A torsion free and divisible then there is an

isomorphismcbetweenC(A) andBwhich preserves the identity map onA.

The groupC(A) iscalled thecotorsion envelopeofA(see [4], chapter 9, the Theorem 58.1). There is an isomorphismC(A)Ext_Z(Q=Z;A).

Now we introduce a new concept. A coreduced compact groupDC(K) is called adual cotorsion coveringof the compact coreduced groupKif the following conditionshold:

i) DC(K) isdual cotorsion and coreduced;

ii) there isan open continuousepimorphismW:DC(K)!K;

iii) kerWS^a.

By Pontrjagin duality we conclude that DC(K) isisomorphic to the character group of group Ext K;Q

p Zp .

THEOREM4.5. Let K by any compact coreduced group and A by any discrete reduced group. If C(A)is the cotorsion envelope of A and DC(K)is the dual cotorsion covering of K, then there is a natural isomorphism Ext(A;K)Ext(C(A);DC(K)).

PROOF. From the short exact sequence of discrete groups 0!A!C(A)!C(A)=A!0

we have the exact sequence

Ext(C(A)=A;K)!Ext(C(A);K)!Ext(A;K)!0;

SinceKisa compact group andC(A)=Ais a torsion free discrete group, then we get Ext(C(A)=A;K)ˆ0 ([9], Corollary 2.3) and we have an isomorphism

Ext(C(A);K)Ext(A;K) (4:1)

From the proper short exact sequence of compact groups 0!S^a !DC(K)!K !0

we have the exact sequence

Ext(C(A);S^a)!Ext(C(A);DC(K))!Ext(C(A);K)!0;

SinceS^aisa compact connected group andC(A) isa discrete group, we get Ext(C(A);S^a)ˆ0 ([9], Corollary 2.4). From the last exact sequence we have

Ext(C(A);DC(K))Ext(C(A);K) (4:2)

Using (4.1) and (4.2), we get Ext(C(A);DC(K))Ext(A;K). Thiscompletes

the proof of Theorem 4.5. p

Theorem 4.5 shows that generalized cotorsion groups play an excep- tional role in the structural theory of locally compact groups. Really, ifKis any compact coreduced group andAisany reduced discrete group, then there is one-to-one correspondence between the classes of extensionsKby Aand the classes of extensionsDC(K) byC(A). Every extensionDC(K) by C(A) isa generalized cotorsion group.

5. Some results about a classification of generalized cotorsion groups We recall now what isobserved in [4] that ifAisany cotorsion group thenADL

CL

F, whereDisa divisible group,Cisa reduced torsion free algebraically compact group andF isan adjusted group. A reduced cotorsion group F iscalled adjustedif F admitsno nonzero torsion free direct summand. Every divisible group and every algebraically compact group are characterized by cardinal invariants(see [4], chapter 4 and chapter 7). The correspondence W:T7!ExtZ(Q=Z;T) isa bijective map from the class of reduced torsion groups in the class of adjusted cotorsion groups (Harrison, [1]). The inverse mappingW ¹istaking the torsion part of the adjusted cotorsion group. Therefore any adjusted cotorsion groupF is characterized by its torsion part. The problem of the classification of torsion groups is reduced to the problem of classification of primary tor- sion groups. For example the groups from the class of totally projective groups are characterized by cardinal invariants. These are well-known Ulm invariants (see, [5], chapter 12, § 82). Another class of p-primary groups characterized by cardinal invariants is the class of torsion complete groups(see, [5], chapter 11, § 68).

We say that a coreduced dual cotorsion groupCiscalledcoadjustedif itscharacter group Cb isan adjusted group. From Pontrjagin duality we conclude that the correspondence c:C!C=C₀ isa bijective mapping from the class of coadjusted groups in the class of coreduced totally dis- connected groups. It follows that ifCisany coadjusted group, thenChas the same invariants as the quotient group C=C0. Let K be any dual co- torsion group. Then there is an isomorphismKEL

A_kL

C, whereEis a torsion free compact group (i.e. E isa character group of a divisible group) andA_kisa character group of a reduced torsion free algebraically compact group, andC iscoadjusted. The groupsEandA_k may be char- acterized by cardinal invariants. A coadjusted group Ccan be character- ized by cardinal invariants in some cases. For example, suppose that every topologicallyp-primary component of the groupC=C₀isdual to some to-

tally projective group. Then the topologically p-primary componentsof C=C0 has the cardinal invariants. There are co-Ulm invariants, where Kiefer [10] introduced, (see also [7], chapter 5). A lot of these cardinal invariantsfor all primepwill be the system invariants of the groupC.

LetKbe any compact group and letAbe any discrete group, and letP be the set of all prime numbers. Denote byTp(A) thep-primary component of the torsion part of the groupAandp(A)ˆ fp2PjTp(A)6ˆ0g. Letq(K) be the setfq2PjqK6ˆKg. In [9], isproved that Ext(A;K)ˆ0 if and only if p(A)\q(K)ˆ ;. The last equality is equivalent to p(A)\p(K)b ˆ ;.

Therefore ifLisany generalized cotorsion group andListhe extension of the compact groupK by the discrete group A, and p(A)\p(K)b ˆ ; then LKL

A. Thisyieldsthat the invariantsof the groupsKandAare the invariantsof the groupLin thiscase.

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Manoscritto pervenuto in redazione il 27 marzo 2011.