lutz strungmann and saharon shelah- on the p-rank of ext(a,b) for countable abelian groups a and b

8/3/2019 Lutz Strungmann and Saharon Shelah- On the p-rank of Ext(A,B) for countable abelian groups A and B

1/5

968

re

vision:2010-10-22

modified:2010-10-24

On the p-rank of Ext(A, B) for countable abeliangroups A and B

Lutz Strungmann and Saharon Shelah

Abstract. In this note we show that the p-rank of Ext(A,B) for count-able torsion-free abelian groups A and B is either countable or the sizeof the continuum.

1. Introduction

The structure of Ext(A, B) for torsion-free abelian groups A has receivedmuch attention in the literature. In particular in the case of B = Z com-plete characterizations are available in various models of ZFC (see [EkMe,Sections on the structure of Ext], [ShSt1], and [ShSt2] for references). Itis easy to see that Ext(A, B) is always a divisible group for any torsion-freegroup A. Hence it is of the form

Ext(A, B) =p

Z(p)(p) Q(0)

for some cardinals p, 0 (p ) which are uniquely determined. Here Z(p)

is the p-Prufer group and Q is the group of rational numbers. Thus, theobvious question that arises is which sequences (0, p : p ) can appearas the cardinal invariants of Ext(A, B) for some (which) torsion-free abeliangroup A and arbitrary abelian group B? As mentioned above for B = Zthe answer is pretty much known but for general B there is little knownso far. Some results were obtained in [Fr] and [FrSt] for countable abeliangroups A and B. However, one essential question was left open, namely if

the situation is similar to the case B = Z when it comes to p-ranks. It wasconjectured that the p-rank of Ext(A, B) can only be either countable orthe size of the continuum whenever A and B are countable. Here we prove

This is number 968 in the first authors list of publication. The authors would liketo thank the German Israeli Foundation for their support project No. I-963-98.6/2007 ;the second author was also supported by project STR 627/1-6 of the German ResearchFoundation DFG2000 Mathematics Subject Classification. Primary 20K15, 20K20, 20K35, 20K40; Sec-ondary 18E99, 20J05.

1


2/5

968

re

vision:2010-10-22

modified:2010-10-24

2 LUTZ STRUNGMANN AND SAHARON SHELAH

that the conjecture is true.

Our notation is standard and we write maps from the left. If H is a puresubgroup of the abelian group G, then we will write H G. The set ofnatural primes is denoted by . For further details on abelian groups andset-thoretic methods we refer to [Fu] and [EkMe].

2. Proof of the conjecture on the p-rank of Ext

It is well-known that for torsion-free abelian groups A the group of extensionsExt(A, B) is divisible for any abelian group B. Hence it is of the form

Ext(A, B) =p

Z(p)(p) Q(0)

for some cardinals p, 0 (p ) which are uniquely determined.The invariant rp(Ext(A, B)) := p is called the p-rank of Ext(A, B) whiler0(Ext(A, B)) := 0 is called the torsion-free rank of Ext(A, B). The fol-lowing was shown in [FrSt] and gives an almost complete description of thestructure of Ext(A, B) for countable torsion-free A and B. Recall that thenucleus G0 of a torsion-free abelian group G is the largest subring R ofQsuch that G is a module over R.

Proposition 2.1. LetA and B be torsion-free groups with A countable and|B| < 20. Then either

i) r0(Ext(A, B)) = 0 and A B0 is a free B0-module orii) r

0(Ext(A, B)) = 20 .

Recall that for a torsion-free abelian group G the p-rank rp(G) is defined tobe the Z/pZ-dimension of the vectorspace G/pG.

Proposition 2.2. The following holds true:

i) If A and B are countable torsion-free abelian groups and A has finite rank thenrp(Ext(A, B)) 0. Moreover, all cardinals 0can appear as the p-rank of some Ext(A, B).

ii) IfA andB are countable torsion-free abelian groups with Hom(A, B) =0 then

rp(Ext(A, B)) =

0 if rp(A) = 0 or rp(B) = 0

rp(A) rp(B) if 0 < rp(A), rp(B) < 00 if 0 < rp(A) < 0 and rp(B) = 020 if rp(A) = 0 and 0 < rp(B) 0

.

Recall the following way of calculating the p-rank of Ext(A, B) for torsion-free groups A and B (see [EkMe, page 389]).

Lemma 2.3. For a torsion-free abelian group A and an arbitrary abeliangroup B let p be the map that sends Hom(A, B) to with :


3/5

968

re

vision:2010-10-22

modified:2010-10-24

P-RANK OF EXT(A,B) FOR COUNTABLE ABELIAN GROUPS A AND B 3

B B/pB the canonical epimorphism. Then

rp(Ext(A, B)) = dimZ/pZ(Hom(A,B/pB)/p(Hom(A, B))).

We now need some preparation from descriptive set-theory in order to proveour main result. Recall that a subset of a topological space is called perfectif it is closed and contains no isolated points. Moreover, a subset X of aPolish space V is analytic if there is a Polish space Y and a Borel (or closed)set B V Y such that X is the projection of B; that is,

X = {v V|(y Y)v, y B}.

Proposition 2.4. If E is an analytic equivalence relation on = {X : X } that satisfies

() if X, Y , n Y, X = Y {n} then X and Y are not E-equivalent

then there is a perfect subset T of of pairwise nonequivalent X .Proof. See [Sh, Lemma 13].

We are now in the position to prove our main result.

Theorem 2.5. Let A be a countable torsion-free abelian group and B anarbitrary countable abelian group. Then either

rp(Ext(A, B)) 0 or rp(Ext(A, B)) = 2

0.

Proof. The proof uses descriptive set theory, relies on [Sh, Lemma13] and is inspired by [HaSh, Lemma 2.2]. By the above Lemma 2.3 the

p-rank of Ext(A, B) is the dimension of the Z/pZ vector space L :=Hom(A,B/pB)/p(Hom(A, B)) where p is the natural map. We choose abasis {[] | < } ofL with Hom(A,B/pB) and assume that 0 < .Note that [] = 0 means, that : A B/pB has no lifting to an ele-ment Hom(A, B) such that = .Now let A =

i


4/5

968

re

vision:2010-10-22

modified:2010-10-24

4 LUTZ STRUNGMANN AND SAHARON SHELAH

It remains to prove that the size of {[] : 2} is 20 which then implies

that = 20 .

We now define an equivalence relation on = {X } in the following way:X X if and only if [X ] = [X ], where X

2 is the characteristicfunction of X. We claim that

() for X, X and n X such that X = X {n} we have X X.

But this is obvious since in this case X X = n and [n] = 0, so[X ] = [X ]. We claim that the above equivalence relation is analytic.It then follows from Proposition 2.4 that there is a perfect subset of ofpairwise nonequivalent X and hence there are 20 distinct equivalenceclasses for . Thus the p-rank of Ext(A, B) must be the size of the contin-

uum as claimed.In order to see that is analytic recall that both, the cantor space 2(and hence ) and the Baire space are Polish spaces with the nat-ural topologies. For to be analytic we therefore have to show that = {(X, X) : X, X and X X} is an analytic subset of the productspace . We enumerate A = {an : n } and B = {bn : n }and let C1

be the set of all such that induces a homo-morphism f Hom(A,B/pB), i.e. the map f : A B/pB sendingan onto b(n) + pB is a homomorphism. C2 is defined similarly replacingHom(A,B/pB) by Hom(A, B), i.e. C2 if the map g : A B withan b(n) is a homomorphism. Clearly C1 and C2 are closed subsets of. Put P = {(1, 2, ) : 1, 2 C1; C2 and f1 f2 =

p(g)}.

Then P is a closed subset of . Now the construction abovegives a continuous function from 2 to by sending to = ()where is induced by the homomorphism . Then X X

if and only ifthere is C2 such that ((X), (X ), ) P and thus is an analyticequivalence relation.

References

[EkMe] P.C. Eklof and A. Mekler, Almost Free Modules, Set- Theoretic Methods (re-vised edition), Amsterdam, New York, North-Holland, Math. Library (2002).

[Fr] S. Friedenberg, Extensions of Abelian Groups, Sudwestdeutscher Verlag furHochschulschrift (2010).

[FrSt] S. Friedenberg and L. Strungmann, The structure of Ext(A,B) for countable

abelian groups A andB, submitted (2010).[Fu] L. Fuchs, Infinite Abelian Groups, Vol. I and II, Academic Press (1970 and 1973).[HaSh] L. Harrington and S. Shelah, Counting equivalence classes for co--Souslin

relation, Proc. Conf. Prague, 1980, Logic Colloq. North-Holland, 1982, 147-152.[Sh] S. Shelah, Can the fundamental (homotopy) group of a space be the rationals?, Proc.

AMS 103 (1988), 627632.[ShSt1] S. Shelah and L. Strungmann, A characterization of Ext(G,Z) assuming

(V=L), Fundamenta Math. 193 (2007), 141-151.[ShSt2] S. Shelah and L. Strungmann, On the p-rank of Ext(G,Z) in certain models

of ZFC, Algebra and Logic 46 (2007), 200-215.


5/5

968

re

vision:2010-10-22

modified:2010-10-24

P-RANK OF EXT(A,B) FOR COUNTABLE ABELIAN GROUPS A AND B 5

Faculty of Mathematics, University of DuisburgEssen, 45117 Essen, Germany

E-mail address: [email protected]

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

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

E-mail address: [email protected]

lutz strungmann and saharon shelah- on the p-rank of ext(a,b) for countable abelian groups a and b

Documents