Saharon Shelah and Lutz Strungmann- A characterization of Ext(G,Z) assuming (V = L)

8/3/2019 Saharon Shelah and Lutz Strungmann- A characterization of Ext(G,Z) assuming (V = L)

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A characterization of Ext(G,Z) assuming (V = L)

Saharon Shelah and Lutz Strungmann

Abstract. In this paper we complete the characterization of Ext(G,Z) forany torsion-free abelian group G assuming Godels axiom of constructibilityplus there is no weakly compact cardinal. In particular, we prove in (V = L)

that, for a singular cardinal

of uncountable cofinality which is less than thefirst weakly compact cardinal and for every sequence of cardinals (p : p )satisfying p 2 , there is a torsion-free abelian group G of size such thatp equals the p-rank of Ext(G,Z) for every prime p and 2 is the torsion-freerank of Ext(G,Z).

1. Introduction

Since the first author solved the well-known Whitehead problem in 1977 (see [Sh1],[Sh2]) the structure of Ext(G,Z) for torsion-free abelian groups G has receivedmuch attention. Easy arguments show that Ext(G,Z) is always a divisible groupfor every torsion-free group G. Hence it is of the form

Ext(G,Z) =p

Z(p)(p) Q(0)

for some cardinals p, 0 (p ) which are uniquely determined. Thus, the obviousquestion that arises is which sequences (0, p : p ) can appear as the cardinalinvariants of Ext(G,Z) for some (which) torsion-free abelian group? On the onehand there are a few results about possible sequences (0, p : p ) provable inZF C. For instance, the trivial sequence consisting of zero entries only can be real-ized by any free abelian group. On the other hand, assuming G odels constructibleuniverse (V = L) plus there is no weakly compact cardinal it has been shown thatalmost all sequences (with natural restrictions) can be the cardinal invariants ofExt(G,Z) for some torsion-free abelian group G whenever the size of the groupG is not a singular cardinal of uncountable cofinality (see section 2 for detailsand [EkHu], [EkSh], [GrSh1], [GrSh2], [HiHuSh], [MeRoSh], [SaSh1], and[SaSh2] for references). However, the question of which sequences (0, p : p )can occur is independent of ZF C. It is the purpose of this paper to deal withthe remaining case, namely torsion-free abelian groups of cardinality where

2000 Mathematics Subject Classification. Primary 20K15, 20K20, 20K35, 20K40; Secondary

18E99, 20J05.Number 873 in Shelahs list of publications. The first author was supported by project No.

I-706-54.6/2001 of the German-Israeli Foundation for Scientific Research & Development.The second author was supported by a grant from the German Research Foundation DFG.

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2 SAHARON SHELAH AND LUTZ STRUNGMANN

is singular of cofinality cf() > 0. The idea is to use the construction principlefrom [MeRoSh] which holds under (V = L) and to apply the main theorem from

[MeRoSh] in our construction.

Our notation is standard and we write maps from the left. If H is a pure subgroupof the abelian group G, then we shall write H G. For further details on abeliangroups we refer to [Fu] and for set-theoretic methods we refer to [EkMe], [Je] or[Ku].

2. The structure of Ext(G,Z)

In this section we recall the basic results on the structure of Ext(G,Z) for torsion-free G. It is easy to see that Ext(G,Z) is divisible for torsion-free G, hence it is ofthe form

Ext(G,Z) = p

Z(p)(p) Q(0)

for some cardinals p, 0 (p ). Since the cardinals p (p ) and 0 completelydetermine the structure of Ext(G,Z) we introduce the following terminology. Wedenote by re0(G) the torsion-free rank 0 of Ext(G,Z) which is the dimension ofQ Ext(G,Z) and by rep(G) the p-rank p of Ext(G,Z) which is the dimensionof Ext(G,Z)[p] as a vector space over Z/pZ for any prime number p . Thereare only a few results provable in ZF C when G is uncountable, but assumingGodels universe an almost complete characterization is known (if there is no weaklycompact cardinal). The aim of this paper is to fill the remaining gap.We first justify our restriction to torsion-free G. Let A be any abelian group andt(A) its torsion subgroup. Then Hom(t(A),Z) = 0 and hence we obtain the shortexact sequence

0 Ext(A/t(A),Z) Ext(A,Z) Ext(t(A),Z) 0which must split since Ext(A/t(A),Z) is divisible. Thus

Ext(A,Z) = Ext(A/t(A),Z) Ext(t(A),Z).

Since the structure of Ext(t(A),Z) =p Hom(A,Z(p

)) is well-known in ZF Cit is reasonable to assume that A is torsion-free and, of course, non-free. UsingPontryagins theorem one proves

Lemma 2.1. Suppose G is a countable torsion-free group which is not free. Thenre0(G) = 2

0.

Proof. See [EkMe, Theorem XII 4.1].

Similarly, we have for the p-ranks of G the following result due to C.U. Jensen.

Lemma 2.2. If G is a countable torsion-free group, then for any prime p, eitherrep(G) is finite or 2

0.

Proof. See [EkMe, Theorem XII 4.7].

This clarifies the structure of Ext(G,Z) for countable torsion-free groups G sincethe existence of groups as in Lemma 2.1 and Lemma 2.2 follows from Lemma 2.8.We now turn our attention to uncountable groups and assume Godels axiom ofconstructibility. The following is due to Hiller, Huber and Shelah.


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A CHARACTERIZATION OF Ext(G, Z) ASSUMING (V = L) 3

Lemma 2.3 (V=L). Suppose G is a torsion-free non-free group and let B be asubgroup of A of minimum cardinality such that A/B is free. Thenre0(G) = 2

.

In particular, re0(G)is uncountable and r

e0(G) = 2

|G|

if G

= Hom(G,Z) = 0.

Proof. See [EkMe, Theorem XII 4.4, Corollary XII 4.5].

Note that the above lemma is not true in ZF C since for any countable divisiblegroup D it is consistent that there exists an uncountable torsion-free group G withExt(G,Z) = D, hence re0(G) = 1 is possible taking D = Q (see Shelah [Sh3]).Again we turn to p-ranks. There is a useful characterization of rep(G) using theexact sequence

0 Zp

Z Z/pZ 0.

The induced sequence

Hom(G,Z)p

Hom(G,Z/pZ) Ext(G,Z)p Ext(G,Z)

shows that the dimension of

Hom(G,Z/pZ)/ Hom(G,Z)p

as a vector space over Z/pZ is exactly rep(G).The following result due to Mekler, Roslanowski and Shelah shows that under theassumption of (V = L) almost all possibilities for rep(G) can appear if the group isof regular cardinality.

Lemma 2.4 (V=L). Let be an uncountable regular cardinal less than the firstweakly compact cardinal. Suppose that (p : p ) is a sequence of cardinals suchthat for each p, 0 p 2

. Then there is an almost-free group G of cardinality such that re0(G) = 2

and for all p, rep(G) = p.

Proof. See [MeRoSh, Main Theorem 3.9].

On the other hand, if the cardinality of G is singular, then the following holds whichwas proved by Grossberg and Shelah.

Lemma 2.5. If is a singular strong limit cardinal of cofinality , then there is notorsion-free group G of cardinality such that rep(G) = for any prime p.

Proof. See [GrSh1, Theorem 1.0].

Note that Lemma 2.4 shows that the restriction in Lemma 2.5 is the only restrictionfor singular strong limit cardinals of cofinality . Namely, if < choose a regularcardinal < and apply Lemma 2.4 to obtain a torsion-free group G with

rep(G) = and |G| = . Since Ext(,Z) is a multiplicative functor we can noweasily get a torsion-free group G from G with |G| = and rep(G) = .Also the case of weakly compact cardinality was dealt with in [ SaSh1] by Sageevand Shelah.

Lemma 2.6. If G is a torsion-free group of weakly compact cardinality andrep(G) for some prime p, then r

ep(G) = 2

.

Proof. See [SaSh1, Main Theorem].


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The above results show that under the assumption of (V = L) the structure ofExt(G,Z) for torsion-free groups G of cardinality is clarified for all cardinals

except is singular but not of cofinality . This will be the subject of the nextsection.However, in a particular case, namely when G = Hom(G,Z) = 0, then a completecharacterization of Ext(G,Z) is known in Godels universe if there is no weaklycompact cardinal. The following is due to Hiller-Huber-Shelah.

Lemma 2.7. If G is torsion-free such that Hom(G,Z) = 0, then for all primes p,rep(G) is finite or of the form 2

p for some infinite cardinalp |G|.

Proof. See [EkMe, Lemma XII 5.2].

Together with Lemma 2.3 and the next result due to Hiller, Huber and Shelah thecharacterization is complete if Hom(G,Z) = 0.

Lemma 2.8. For any cardinal 0 of the form 0 = 20 for some infinite 0 andany sequence of cardinals (p : p ) less than or equal to 0 such that each p iseither finite or of the form 2p for some infinite p there is a torsion-free group Gsuch that Hom(G,Z) = 0 and re0(G) = 0, r

ep(G) = p for all primes p .

Proof. See [HiHuSh, Theorem 3(b)].

3. The singular case

In this section we prove our main theorem which completes the characterizationof Ext(G,Z) for torsion-free groups G under the assumption of Godels axiom ofconstructibility plus there is no weakly compact cardinal. The idea of the proof isas follows: For a singular cardinal of uncountable cofinality we shall constructa torsion-free abelian group G, of size , as the union of pure subgroups G suchthat G has prescribed values for re0(G) and r

ep(G) (p ). Together with the Gs

we also build homomorphisms fp for < rep(G) such that no non trivial combi-

nationl


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rep(Gp) = 2 = + but req(Gp) = 0 for all q = p. Since Ext(,Z) commutes with

direct sums it therefore suffices to assume p for all p . Let = cf() be

the cofinality of . Chose a continuous increasing sequence : < such that lim such that cf() = 0 and +1 >

Now, let S { < : cf() = 0} be stationary. Inductively we shall construct atorsion-free group G =


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for some finite subset E p and elements z Z/pZ. Then there exists suchthat < p and f,p = f

,p G= 0 for all E. Hence

E

zf,p =

E

zf,p

G= 0.

But

f,p : < p

is a basis of Lp and thus z = 0 for all E. Therefore,the f,p s ( < p) are linearly independent.Now, assume that there exists a finite linear combination 0 =

E

zf,p which can

be factored by p (0 = z Z/pZ for all Z). Hence there is 0 = g Hom(G,Z)such that

E

zf,p = pg.

Since E is finite, there exists < such that < p for all E. Therefore,

pg G=E

zf,p G=E

zf,p

for every [, ). By the linear independence of the f,p s we may assumewithout loss of generality that pg G= 0 for all [, ) since otherwise wesufficiently enlarge such that f,p G= 0 for all E. We conclude thatpg G+1 L

+1p \{0} for all and condition (v)(d) implies that g G+1 M+1

for all S, . Let = + 1 for some S, . Then

g G=k


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for all large enough. This contradicts the fact that jk [, ) for all k < k0 and. Therefore we obtain p rep(G) for all p .

It remains to prove that rep(G) = p for all p . For this it suffices to show

that f,p : < p generate Hom(G,Z/pZ) modulo Hom(G,Z)p. Hence, let 0 =

g Hom(G,Z/pZ). We have to prove that there is a finite linear combinationE

zf,p with E p and 0 = z Z/pZ such that

g E

zf,p = ph

for some h Hom(G,Z). Let g = g G for all < . Hence, by (iii), thereexist k Kp and l L

p such that g = k + l for every < since g

Hom(G,Z/pZ). Thus

l =

Ez f,p

for some finite subset E p and 0 = z Z/pZ. Since this representationis unique we may assume by a pigeon hole argument that z = z and E = E are

independent of < . Note that f,p G = f,p if

< and < p and

otherwise 0 by (v)(e). We conclude that

h = g E

zf,p

satisfies h G= k for all < . Since Bp forms a basis of K

p for all < there

is a finite subset F Bp and 0 = w

b Z/pZ for b F such that

l =bF

wb b

for all < . Again, a pigeon hole argument allows us to assume that wb = wband F = F are independent of by uniqueness. Note that b G B

p if b B

p

and > . Putting

h =bF

wb


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By the continuity conditions (v)(a) and (v)(c) this is well-defined. Hence, alsoM Hom(G,Z) and Lp Hom(G,Z/pZ) are defined canonically. Finally, (iii)

and the definition of Lp induce K

p as

Kp = {f Hom(G,Z/pZ) : f G Kp for all < }.

The corresponding set Bp = {f Hom(G,Z/pZ) : f G Bp for all < } is a

basis for Kp and the continuity condition (iv)(c) allows to define T = ( plays the role of in [MeRoSh], so it is thesuccessor of a strong limit singular cardinal). Since we are assuming (V = L) theprediction principle from [MeRoSh] holds. In Main Theorem 3.9 from [MeRoSh]

it is proved that we can find a torsion-free group G (denoted by G in [MeRoSh,

Main Theorem 3.9]) which has prescribed values p for rep(G). The construction is

very similar to our construction, i.e. homomorphisms fp Hom(G,Z/pZ) ( < p)

are constructed (denoted by fp, in [MeRoSh, Main Theorem 3.9]) which can not

be factorized by p (see the proof of Main Theorem 3.9 in [MeRoSh, page 346]).The main tool is Theorem 3.4 from [MeRoSh] which can be seen as a Step-Lemmasince it deals with the killing of only one undesired homomorphism. However, weare at stage of our construction, hence we do not want that our homomorphismsf,p ( < p ) which we have dealt with so far have no extension to G, i.e.can not be factorized by p but we just require that there are only some extensions,namely a set of extensions of f,p which is assigned to each f

,p . The role of this

set is played by {h,p, : [, )} (see (v)(b)). Thus the proof of Theorem 3.4

from [MeRoSh] carries over to our situation (for the case of successor cardinal ofa strong limit singular cardinal of cofinality 0). Since the role of in [MeRoSh,Theorem 3.4] is played by in our setting only cases B and C in the proof ofTheorem 3.4 remain. As usual one guesses the undesirable factorizations and kills

them without effecting the work towards lifting that has been done already. Theonly difference is that we do not require that there is no lifting but we allow onlythe assigned ones. Now the adjusted version of Theorem 3.4 from [MeRoSh] isused in the Main Theorem 3.9 from [MeRoSh] in the case of being a successorof a strong limit cardinal. The resulting group serves as our G.

Remark 3.2. We would like to remark that the only reason for the choice of as successor of a strong limit singular cardinal of cofinality 0 (if is a successorordinal) is that this is the easiest situation in the proof of [MeRoSh, Main Theorem3.9]. However, the strategy described in Case D of the above proof of Theorem 3.1


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(i.e. not killing all extensions of a homomorphism but allowing some of them tosurvive) works for every regular uncountable cardinal which is not weakly compact,

e.g. 1. For instance it follows easily for 1 from [EkMe, Theorem XII 4.10] using[EkMe, Lemma XII 4.8 and Lemma XII 4.9].

References

[EkHu] P.C. Eklof and M. Huber, On the rank of Ext, Math. Zeit. 174 (1980), 159185.[EkMe] P.C. Eklof and A. Mekler, Almost Free Modules, Set- Theoretic Methods (revised

edition), Amsterdam, New York, North-Holland, Math. Library.[EkSh] P.C. Eklof and S. Shelah, The structure of Ext(A,Z) and GCH: possible co-Moore

spaces, Math. Zeit. 239 (2002), 143157.[Fu] L. Fuchs, Infinite Abelian Groups, Vol. I and II, Academic Press (1970 and 1973).[GrSh1] R. Grossberg and S. Shelah, On the structure of Extp(G,Z), J. Algebra 121 (1989),

117128.[GrSh2] R. Grossberg and S. Shelah, On cardinalities in quotients of inverse limits of groups,

Math Japonica 47 (1998), 189-197.[HiHuSh] H. Hiller, M. Huber and S. Shelah, The structure of Ext(A,Z) andV = L, Math.

Zeit. 162 (1978), 3950.[Je] T. Jech, Set Theory, Academic Press, New York (1973).[Ku] K. Kunen, Set Theory - An Introduction to Independent Proofs, Studies in Logic and the

Foundations of Mathematics, North Holland, 102 (1980).[MeRoSh] A. Mekler, A. Roslanowski and S. Shelah, On the p-rank of Ext, Israel J. Math.

112 (1999), 327356.[SaSh1] G. Sageev and S. Shelah, Weak compactness and the structure of Ext(G,Z), Abelian

group theory (Oberwolfach, 1981), ed. R. Gobel and A.E. Walker, Lecture Notes in Mathe-matics 874 Springer Verlag (1981), 8792.

[SaSh2] G. Sageev and S. Shelah, On the structure of Ext(A,Z) in ZFC+, J. of SymbolicLogic 50 (1985), 302315.

[Sh1] S. Shelah, Whitehead groups may not be free even assuming CH, I, Israel J. Math. 28(1977), 193203.

[Sh2] S. Shelah, Whitehead groups may not be free even assuming CH, II, Israel J. Math. 35(1980), 257285.

[Sh3] S. Shelah, The consistency of Ext(G,Z) = Q, Israel J. Math. 39 (1981), 7482.

Department of Mathematics, The Hebrew University of Jerusalem, Israel, and Rutgers

University, New Brunswick, NJ U.S.A.

E-mail address: [email protected]

Department of Mathematics, University of Duisburg-Essen, 45117 Essen, Germany


Current address: Department of Mathematics, University of Hawaii, 2565 McCarthyMall, Honolulu, HI 96822-2273, USA


Saharon Shelah and Lutz Strungmann- A characterization of Ext(G,Z) assuming (V = L)

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