Small almost free modules with prescribed topological endomorphism rings

Small Almost free Modules

with Prescribed Topological Endomorphism Rings (*).

A. L. S. CORNER(**) - RU¨_DIGER GO¨_BEL(***)

ABSTRACT- We will realize certain topological rings as endomorphism rings of]1- free abelian groups of cardinality]1where the isomorphism is also a homeo- morphism relating the topology on the given ring to the finite topology on the endomorphism ring. This way we also find]1-free abelian groups of cardinali- ty]1such that any non-trivial summand is a proper direct sum of an infinite number of summands. This answers a problem raised by the authors in [7, p.

447, (I)], (saying that]1vat (R) in the cotorsion-free case). The fact that the size of the continuum could and in particular universes of set theory will be much larger then ]1 causes difficulties in constructing pathological abelian groupsGof size]1in «ordinary» set theory using just ZFC: There are less possibilities to prevent potential, unwanted endomorphisms ofG not to be- come members of EndG.Thus additional combinatorial arguments are need- ed. They come from [15], were improved for this paper, and are now ready for applications for other algebraic aspects. The results are formulated for mod- ules over a large class of commutative rings.

1. Introduction.

First we would like to discuss the algebraic setting of this paper. Let Rbe a commutative ring (with 1) of cardinalGl, and letSbe a countable multiplicatively closed set of non-zero-divisors in R such thatR is S-re-

(*) AMS-subject classification 20K25, 20K30, 03E50, 16S60.

(**) Indirizzo dell’A.: Department of Mathematics, University of Exeter, North Park Road, Exeter, EX4 4QE, England. E-mail: TonyHmaths.ex.ac.uk

(***) Indirizzo dell’A.: Fachbereich 6, Mathematik und Informatik, Universi- tät Essen, 45117 Essen, Germany. E-mail: R.GoebelHUni-Essen.De

ducedin the sense that_s

1

q041 , and q_n114s_nq_n² (nEv).

(1.1)

For a freeR-moduleF, the submodulesq_nF(nEv) constitute a basis of neighborhoods of 0 in the (Hausdorff) S-topology onF; the correspond- ingS-completion F×containsFas anS-puresubmodule, in the sense that s F×

OF4sF for all sS.

The endomorphism algebra End_RM of an R-module M carries the natural finite topology fin, which is generated by the annihilators Ann_EndME of all finite subsets E of M. (The finite subsets may be re- stricted to singletons, in which case the annihilators constitute a preba- sis for fin at 0 .) It follows that (EndRM, fin) is a Hausdorff, complete topological R-algebra, see [13, p. 221, Theorem 107.1] This suggest im- mediately to consider R-algebras A with 1 and endowed with a similar topology.

Let A be complete and Hausdorff in a topology admitting a basis of neighborhoods of 0 consisting of a set9 of right ideals N such that the quotientsA/N are free asR-modules. We impose the cardinality restric- tions that

N9N Gl and sup

N9NA/NNEl, (1.2)

and define

k4

g

9NA/NN

h

(1.3)

where the plus sign denotes the cardinal successor. Thenk is a regular cardinal, and

]1GkGlG2^]0. (1.4)

The cardinal l can be arbitrary, if set theory permits! Our favored example however is l4]1. The other cardinal restrictions come from our aim to produce R-modules of size close to ]1 with the prescribed topological endomorphism ring. For larger cardinals, in particular for any cardinallof the forml4l^]0this was established in [7, p. 465, Corol- lary 6.4]. Thus we can view our restriction to]1 as a particular way to avoid the cardinal restriction l4l^]0, here for l4]1.

An obvious first attempt to find the cardinal ]1vat (A), the spec- trum of cardinalsl, for which a (suitable) algebra can be expressed as A4EndRM with NMN 4l (see [7, p. 447]), is replacing the arguments using the Black Box by a different combinatorial method, because the latter requires thatl4l^]0. The Black Box prediction principle was used in [7] and relatives are discussed in Göbel [14].

However, it is important to note that there are basic difficulties for lEl^]0to become a member of vat (A), as shown recently in Göbel, She- lah [16, pp. 240-243, Section 3, The non existence of realization theo- rems]: For particular classes of modules, like for abelian p-groups, or separable torsion-free abelian groups, we have ]1vat (R) for any ring R. This motivated our aim to consider the class of]1-free modules of car- dinality ]1, which is still very close to separable, torsion-free abelian groups. Recall that modules are]1-free, if all countably generated sub- modules are free.

In [15] we derived a new combinatorial method, which allows to pre- dict endomorphisms of a module (under construction) of cardinality ]1

by a list of candidates which has only ]1 members. We may assume (w.l.o.g) that for each unwanted member of this list a new element is added to the module under construction which prevents that member to become an endomorphism of the final module. Thus the length of this list of prediction should not pass beyond ]1.

Also note that by an easy counting argument our prediction of endo- morphisms must be restricted to their action on only finite subsets or singletons; otherwise the list would have length at least 2^]0 which often is larger then]1. Therefore it is not clear at the beginning if the given predictions, which provides only very little information about the actual endomorphism, are good enough to lead to our task, prescribingAas en- domorphism ring and to find out that ]1vat (A).

In [15] it was shown that any countable R-algebra with free R-module structure is the endomorphism algebra of a suitableR-module of size ]1 which is ]1-free. Thus the existence of various pathological modules of this kind, in particular the existence of indecomposable modules of cardinality ]1 which are ]1-free follows. Here we want to strengthen this result and realize any suitable topological ring as such an endomorphism ring algebraically and topologically. On the endomorphism ring EndM of an R-module M we take the finite topology fin introduced above. Moreover we want to simplify arguments

from [15], which will also shorten the proof in the discrete case.

Thus our main result reads as follows.

THEOREM 1.1. Let A be a topological R-algebra over a ring R as de- scribed above, then there exists an]1-free R-module M of cardinalityl withEndM`A where the isomorphism is an algebraic and topological isomorphism.

If the topology on A is discrete we obtain the main result in Göbel, Shelah [15] as a special case. Also other earlier results like the existence of an]1–free groupM of cardinality]1with Hom (M,Z)40 follow, see Eda [11] and Shelah [19], [10, 12].

The main difficulty here is to work exclusively in ZFC. The first example of an]1-free module which is not free is the Baer-Specker mod- ule R^v, which is the cartesian product of countably many copies of the ringR, known for sixty years; cf. Baer [1] or [13, p. 94]. Assuming CH, this module is an example of an R-module of cardinality ]142^]0. How- ever, it is surely (assuming thatRis slender) a finite but not an infinite direct sum of summandsc0 . Under the same set-theoretic assumption of the continuum hypothesis it can be shown thatAabove can be realized as the endomorphism ring of an ]1-free R-module M of cardinality ]1, see Shelah [20] for the discrete case with EndM4Z and Dugas, Göbel [8] for the discrete case and A4EndM with extensions to larger cardinals. Using Shelah’s Black Box, the existence of]¹-free modulesM with NMN 4l^]0 and EndM`A also topologically follows from Corner, Göbel [7].

Endomorphism ring results as discussed have well-known applica- tions using the appropriate also well-known R-algebras A.

IfGis any abelian semigroup, then we use the R-algebraA_G, implicit- ly discussed in Corner, Göbel [7], and constructed for particular G8s in [5] with special idempotents, with free R-module structure and NA_GN 4 4max]NGN,]0(. IfNGN Gl, we may apply the discrete case of the main theorem and find a family of]1-freeR-modulesM_a(aG) of cardinality ]1 such that for a,bG,

M_a5M_b`M<sub>a1b</sub> and M<sub>a</sub>`M_b if and only if a4b.

Observe that this induces all kinds of counterexamples to Kaplan- sky’s test problems for suitableG8s. If we consider the algebraAin Cor- ner [3] (see also [13, Vol. II, p. 145, Theorem 91.5]), then it is easy to see

that A_R is free and NAN 4]0. The particular idempotents in A and our main theorem provide the existence of an]1-free super-decomposableR- module of cardinality ]1. Recall that a group is super-decomposable if any non-trivial summand decomposes into a proper direct sum. Super- decomposable abelian groups of cardinality 2^]0 were also constructed (without considering endomorphism rings) in Birtz [2] and the existence of non-free but ]n-free groups of cardinality ]n (nv) (without any more specific algebraic properties) follows from Griffith [17].

Our main theorem allows us to strengthen these results and to estab- lish the existence ofk-super-decomposable groups. We say that M isk- super-decomposableif for any cardinalrEkany non-trivial summand of Mis a direct sum ofr non-trivial summands. The existence of]1-freek- super-decomposable modules G of cardinality k (e.g. for k4]1) follows by realizing the algebra given in Corner [6] as a topological endomor- phism algebra. In casek4]¹we therefore findR-modules of size]¹such that every non-trivial summand is a direct sum of aninfinitenumber of non-trivial summands, which is a property stronger then super-decom- posable and thus sometimes called very decomposable.

It is fairly straight to replace the moduleMin the theorem by a fami- ly of modules M_X with X’l such that for all X,Y’l

Hom_R(M_X,M_Y)`^._/

´ A 0

if X’Y’l otherwise . (1.5)

2. The construction of modules.

Once and for all fix an uncountable cardinallG2^]0and adopt the al- gebraic assumptions explained in the introduction.

WriteT4^vD2 for the binary tree. We will say that a subtreeT8 of

vF2 isperfectif for anynvthere is at most one finite branchfT8of length n of the form f4vOw for branches v,wT8.

Define the map

s:^v2K^v2 (vKv^s) where

(2.1) v^s(m)4

. /

´

v(n) 0

if 2ⁿ¹¹GmE2ⁿ¹² and m42ⁿ¹¹1

!

i40 n

2ⁿ²ⁱv(i) otherwise .

Clearly v^s( 0 )4v^s( 1 )40 . Consider v,w^v2 and suppose that v^s(m)4w^s(m)41. ThusmF2 andv(n)4w(n)41, where 2ⁿ¹¹GmE2ⁿ¹², and alsom42ⁿ¹¹1

!

i40 n

2ⁿ²ⁱv(i)42ⁿ¹¹1

!

i40 n

2ⁿ²ⁱw(i). From

!

i40 n

2ⁿ²ⁱv(i)4 4

!

i40 n

2ⁿ²ⁱw(i) andv(i),w(i)]0 , 1(follows thatvN³n4wN³nandvN³n1 114wN³n11 from the above. It follows that the map s is injective. If

T^s4 ]v^sN³m:v^v2 ,mv(, then we want to show the following

PROPOSITION 2.1. If s is the map given by (2.1) then (a) s:^v2K^v2 defines an injective tree embedding.

(b) T^s is a perfect subtree of T with Br (T^s)4Ims.

PROOF. Clearly T^s is a subtree of T with Br (T^s)4Ims, so it re- mains to show that it is perfect. Suppose

m4br (v^s,w^s)4br (v8^s,w8^s)

for pairsv,wandv8,w8from^v2 to branch at the same levelm. We may assume that v^s(m)4v8^s(m)41 [thus w^s(m)4w8^s(m)40]. From the preceding argument follows vN³n114v8 N³n11 where 2ⁿ¹¹GmE E2ⁿ¹². So alsov^sN³2ⁿ¹¹4v8^sN³2ⁿ¹¹and the two branchesv^sandv8^smust coincide up to the branch point m, i.e.

v^sN³m114v8^sN³m11 .

Similarlyw^sN³m114w8^sN³m11 thusv^sOw^s4v8^sOw8^sandT^smust be perfect. r

IfX’^vF2 then we call [X]4 ]br (v,w) :vcwX(’vthesupport of X. Using a sequence of almost disjoint subsets of vand Proposition 2.1 we can fix for the remaining paper a sequenceT_a(aEl) of perfect trees with Br (T_a) pair-wise disjoint and [T_a] pair-wise almost disjoint for all aEl. Moreover letV_a’Br (T_a) be a subset of infinite branches of cardi- nality l for each aEl.

Identify the set

G4T393l, (2.2)

as a subset of a direct sum

B_Q4_(t,N,

5

(2.3)

of cyclic A-modules with annihilators given by Ann_A(t,N,a)4N. (2.4)

Continuity of multiplication inAimplies that every element ofB_Qis an- nihilated by some element of 9. It is clear that as an R-module B_Q is free of rankl. Each element of the S-completionB×_Qis expressible as a countable sumg4

!

(t,N,a)

g(t,N,a)for suitableg(t,N,a)( (t,N,a)A)×; we define the support of g by

[g]4 ](t,N,a)GNg(t,N,a)c0(. (2.5)

The norm of g is the ordinal

VgV4sup]aN(t,N,a)[g] for some tT, N9(; (2.6)

and the norm of a subset ofB×_Qis taken to be the supremum of the norms of its elements.

We shall call an elementbofB_Qbasalif it has the two properties that bAis a direct summand of theA-module B_Qand Ann_A(b)9. Fix a se- quenceb_a(aEl) runningltimes through the set of all basal elements in B_Q, and set

N_a4Ann_A(b_a) (aEl).

(2.7)

Next we construct a continuous increasing sequenceG_a(aGl) of sub-A- modules of theS-completionB×_Q. For a start, we takeG040 . Assume in- ductively that for some aEl, G_a has been so constructed that

VG_aV4a. (2.8)

Set

g_a4^./

´ b_a

0

if b_aG_a otherwise (2.9)

and, for vV_a, nEv, write y_v,n4

!

iFn

q_i qn

(vN³i,N_a,a)1g_a

!

iFn

q_i qn

v(i).

(2.10)

We now define

G_a114G_a1F_a, (2.11)

where

F_a4aT3 ]N_a( 3 ]a(bA1a]yv,nNvV_a,nEvbA. (2.12)

Our choice ofy_v,nandg_aensures thatG_a11is anS-pure submodule ofB×_Q and thatVG_a11V4a11 . Continuity takes care of the definition of G_aat a limit ordinal a. We define G to be equal to G_l, so that

G4G_l4_aEl

0

!

aEl

F_a. (2.13)

We note

PROPOSITION 2.2. NGN 4land G/G_ais an]¹-free R-module for each aEl.

PROOF. We consider any non-empty finite set E’G/G_b. Choose aEl minimal with E’G_a/G_b. First note that aDb must be a successor be- causeEis a proper finite set, hence g4a21Fbexists. Also note that G_a/G_b is a quotient ofA-modules, hence an A-module. By induction it is enough to show that

(2.14) E’(U1G_b) /G_b5G_g/G_b’

*G_g11/G_b

for a free A-module (U1G_b) /G_b. First we want to find inductively anA-submoduleU’G_a. LetT^mbe the set of elements inTof lengthGmfor anymv. From (2.11) we find a finite setE8’G_g11 of representatives of the elements in E, a finite set F’V_g, and a number mEv such that

E8’U1G_g where U4aT^mN]yvmg_g:vF(bA.

Moreover we may assume that [v_m]O[w_m]4¯for allvcwF. A sup- port argument shows that the defining generators of U are A-indepen- dent modulo G_g, hence U1G_b/G_b must be A-free and G_g/G_bO(U1 1G_b) /G_b40 .

Now it is easy to show thatU isR-pure in Gwhich also implies the purity in (2.14). If hG0G_g11, then an easy support argument shows thatG_g11is pure inGthat is to say thatdhG_g11for any 0cdRand in particulardhU. We may suppose thathG_g11, and by the last con-

siderations we find a finitely generated A-submodule U8 4aT_g^m8N]yvm8g_g:vF8 (bA

for some numberm8 Fmand finite setF’F8’V_gwithhU8. We may assume that m8 is chosen such that also

[v_m8]O[w_m8]4¯ for all vcwF8.

One more support argument now shows thatUis a summand ofU8, we leave it as an exercise to write down a complement ofUinU8. IfdhU for some 0cdR, thenhUfollows fromhU8, which shows that U is pure in G. r

Finally, we define

B_a4aT3 ]N_a( 3 ]a(bA for any aEl and B4

5

aElB_a. (2.15)

It is immediate from the construction that B’G’B×, (2.16)

and thatGisS-pure inB×. It is also clear that the annihilator of every ele- ment ofGis open inA. It follows that the natural mapAKEnd_R(G) is a topological embedding. To prove that it is a topological isomorphism we therefore only have to prove that it is surjective.

In fact it will be enough to prove that each endomorphismWEnd_RG has the property that bWbA for each basalbB. For suppose that W has this property.

bW4bab (bB), (2.17)

where each a_bA. For any xB we may choose b4(t,N_a,a) G0( [x]N[xW] ) with the property that Ann_A(x)*N4Ann_A(b). Easy checks show that AnnA(b1x)4N and that the direct summand bA on the right-hand side of (2.3) may be replaced by (b1x)A. Therefore b1xis basal, and we have (b1x)a_b1x4(b1x)W4bW1xW4ba_b1xW, whence

b(a_b1x2a_b)4xW2xa_b1x. (2.18)

Our choice of b implies that the supports of the two sides here are dis- joint, so both sides vanish. Thus a_b1x2a_bAnn_A(b)’Ann_A(x), and xW4xa_b1x4xa_b4xa_x. It follows that if for the moment we convert B into a directed set (B,G) by writingyGzto mean Ann_A(y)*Ann_A(z) for

y,zB, then the net (a_x)_xB contains a cofinal Cauchy subnet, namely the subnet indexed by the basal elements. By the completeness ofA, the whole net converges to an elementaA, and we have proved thatxW4 4xa(xB). ThusWagrees with scalar multiplication bya onBand, by continuity, also on G.

PIGEON-HOLELEMMA 2.3. LetaElandbGl,and assume that we have a family of elements t_vG_b (vW) indexed by a subset W’V_a of cardinal NWN 4k and a finitely generated sub-A-module H of G_b with the property that

v,wW and br (v,w)4m ¨ t_v2t_wH1q_mG_b. (2.19)

Then there exist a subset W8’W withNW8N 4kand a finite sequence of ordinals b1EREbsEb such that

t_v

!

i41 s

F_bi for all vW8. (2.20)

PROOF. We induct onb. The result is vacuous forb40 , so we sup- pose bD0 and assume the analogous result for smaller ordinals.

Take first the case of a limit ordinalb. If, for somegEb,G_gcontains a subfamilytv(vW8) of cardinal k, then since we may suppose thatgis large enough forG_gto contain also the finitely generatedH, a simple ap- plication of the modular law implies that W8 satisfies the analogue of (2.19) withgin place ofb, and the result follows at once by our inductive hypothesis. Assume then, for a contradiction, that no such gEb exists.

Then one easily produces a continuous increasing sequence of ordinals b_jGb (jGk) and branches v_jW (jEk) such that

t_vjG_bj110G_bj (jEk),

and it is clear that we must in fact havebk4b; and there will be no loss if we assume also that H’G_b0. For each jEk, the coset t_vj1G_bj is a non-zero element ofG_bj11/G_bj. But by Proposition 2.2 this quotient is]1- free and therefore S-reduced. So for some m_jEv we have tv_j1G_bj q_mjG_bj11/G_bj or, in other words,

t_vjG_bj1q_mjG_bj11. (2.21)

A pigeon-hole argument shows that there is a subset C’k of cardinal NCN 4k on which m_j is constant; say m_j4m (jC). Now, if a set of

branchesv^v2 admits a bound n₀ on the levels of its branching points, sincevOvN³n₀is visibly an injection into a finite set, it is clear that the set in question must be finite. Since the set]v_nNnC(is infinite, we may certainly choose indices j and h such that br (v_j,v_h)4nDm and jEh. Then

t_vh2t_vjH1q_nG_bh11, and, since tv_j and H both lie in G_bh, it follows that

t_vhG_bh1q_mG_bh114G_bh1q_mhG_bh11, contrary to (2.21).

We now assume b4g11 and that the lemma holds for all ordinals Gg. Using support arguments, (1.1) and (2.9)-(2.12) we will show first that

G_gOF_g4g_gA. (2.22)

Choose anyvV_gand apply (2.10) for somenvwithv(n)41 . Thus s_nq_n4q_n11/q_n and

y_v,n2s_nq_ny_v,n114y_v,n2(q_n11/q_n)y_v,n114 4(vN³n,N_g,g)1

!

iFn11

(q_i/q_n)(vN³i,N_g,g)1g_g1

!

iFn11

(q_i/q_n)v(i) 2

!

iFn11

(q_i/q_n)(vN³i,N_g,g)2

!

iFn11

(q_i/q_n)v(i)4(vN³n,N_g,g)1g_gF_g. However (vN³n,N_g,g)F_g and g_gG_g, thus

g_gA’F_gOG_g.

On the other hand hF_g can be expressed as a sum h4

!

vE

a_vy_v,m1a_gg_g1t, (a_v,a_gA)

for large enoughmv, a finite subsetE’V_gandtB_g. We may assume for vcwE that

[g_g]O[y_v,m]4[y_v,m]O[y_w,m]4¯.

If also hG_g, consider any t[y_v,m] to see that a_v40 and similarly t40 . Hence h4a_gg_g and G_gOF_g’g_gA, so (2.22) follows.

Now the proof of the lemma when b4g11 will be easy.

Let H’G_g11,t_vG_g11(vW) satisfying the hypothesis of the lem- ma. Using (2.22), we can write

t_v4t_v⁰1t_v¹ (t_v⁰G_g,t_v¹F_g) for all vW, so that

t_v2t_wH1q_mG_g11 whenever br (v,w)4m.

We can also find finitely generated sub-A-modulesH⁰,H¹of G_g andF_g respectively such that H’H⁰1H¹. Thus

(t_v⁰2t_w⁰)1(t_v¹2t_w¹)1h⁰1h¹1q_mg1q_mf40

for suitable elements hⁱHⁱ and gG_g,fF_g. From (2.22) follows (t_v⁰2t_w⁰)1h⁰1q_mg4(t_w¹2t_v¹)2h¹2q_mfg_gA.

Hence (tv⁰2tw⁰)aH⁰,g_gAb1qmG_g, and by induction tv0

!

i41 s

F_gifor all vW8, a suitable W8’WwithNW8N 4 NWNandg1EREgsEg11 . If we add gs114g, then

t_v⁰

!

i41 s11

F_gi for all vW8 completes the induction. r

3. Comparing branch point.

PROPOSITION 3.1. Let m0v, b1EREbs be a sequence of ordi- nals andbs,aEl.Assume that we have a family t_v

!

i41 s

F_bifor all vW of elements indexed by a subset W’V_aof cardinalNWN 4k,then we can find natural numbers m,j*Fm₀, a subsequence of ordinals, renamed b1EREbs and another family

t_v

!

i41 s

F_bi indexed by W, which is of the form t_v4

!

i41 n

a_iy_vim in which the ais, n and m do not depend on v. The new family is ob- tained by passing to an equipotent subset of W which we rename W and adding to members of the old family a fixed element in

!

i41 s

F_bi.The new

family has additional properties for i,jGs,vW (a) _j

0

Gs[g_bj]O[y_vim]4¯

(b) [v_im]O[v_jm]4¯ for all icj and the branches v_i (vW) are pair-wise distinct.

(c) v_iV_bj for some j4j(i)Gs which is independent of v. (d) Anny_vi4N_i9 for all vW.

(e) The v_iN³j* are pair-wise distinct and independent of v. (f ) a_i1N_iA/N_i0q_j*21(A/N_i)

(g) [T_bi]O[T_bj]’j* for all icj and [T_bi]O[T_a]’j* for all bica.

PROOF. We will apply several pigeon-hole arguments and after pass- ing to a subset ofW, we will silently name itW. As a first application we may assume that W in the hypothesis of the proposition satisfies

br (v,w)Dm₀ for all distinct pairs v,wW. (3.1)

By (2.10) any sum

!

l

!

a_liy_vlimi1b_i1a_ig_bi (3.2)

with b_iB_bi, a_i,a_liA, t,m_iv depending on v. However NT_biN, NAN,NvN Ek4 NWN and another pigeon-hole argument shows that these elements no longer depend onv. Similarly, using (2.10), we may as- sume thatmi4min (3.2) does not depend oniGsand that the supports satisfy (a) and the first part of (b) of the proposition. We may assume thata_liy_vlic0 for all pairsli. Since Annv_li4N_bi9as in (2.10) we have a_liN_bi and can choose m₁Dm₀ such that

a_li1N_biA/N_bi0q_m1(A/N_bi) for all pairs li. Thus (f) will follow if we choose j*Dm₁.

We have

t_v4

!

i41

s

g

^!

h

(3.3)

For any fixedv we can choosej_vDm₁such that all thev_liN³j_vare dis- tinct and by a pigeon-hole argument j_v4j* does not depend on v and surelyj*Dm1, hence (f ) holds. Also (e) of the proposition is shown. En- largingj* further if needed, condition (g) is obvious. Now we identify all pairs li with natural numbers Gn4st. Subtracting the constant ele- ment

!

i41 s

(b_i1aig_bi) we get a family of new elementstv4

!

j41 n

ajyv_jm. Apply theD-Lemma (see Jech [18, p. 225]) to the finite sets]v_j:jGn((vW).

We can pass to a subsetWagain such that]v_j: jGn(O]w_j:jGn( 4D for all distinct v,wW. Again subtract the constant element d4 4

!

xD

axyv_xm from each tv. Hence all new vis are distinct and (b) of the proposition follows, which provides finally the new family in the proposition. r

PROPOSITION 3.2. Let tv4

!

i41 n

aiyv_im

!

i41 s

F_bi (vW’V_a) be the family of elements given by Proposition 3.1 and let zG0s_zG for s_z S,s_zNq_m0 having the property

t_v2t_w6q_bzq_b11G for all v,wW with b4br (v,w).

(3.4)

Then there is a map p:]1 ,R,n( K ]1 ,R,s( such that for all iGn

(a) v_i,w_iV_p(i)

(b) br (v_i,w_i)Fbr (v,w)Fj* , in particular all br (v_i,w_i) are distinct.

(c) There is some jGs with br (v_j,w_j)4br (v,w) and p(j)4a, g_ac0 .

PROOF. By Proposition 3.1 (c) for eachiGnthere is a uniquep(i)G Gssuch thatv_i,w_iV_bp(i)come from the freeT_bp(i), hence pand (a) are obvious.

First we claim that the k_i4br (v_i,w_i) (iGn) are pair-wise distinct, which is part of (b). If there areicjwithk_i4k_jthenp(i)4p(j) follows from k_i4k_jFj* and Proposition 3.1 (e) and (g). Hence v_i,w_i,v_j,w_j come from the same treeT_bp(i). Howeverv_iOw_icv_jOw_j are distinct by Proposition 3.1 (e) and of the same lengthki4kj which is impossible for the perfect tree T_bp(i). Hence it remains to show for (b) that

bGki for all iGn,

which we postpone until (c) is shown. For this we need the hypothesis (3.4) in the form

t_v2t_w6q_bzq_b11G’ q_b11

q_m G. From (2.10) follows

yv_im2yw_im6 q_ki q_mg_bp(i)4

q_ki11

q_m (y_vim112y_wim11) (3.5)

and ifb4br (v,w)Ek4min]ki,iGn(thenb11Gkfor alliGn. We derive y_vim2y_wim^qb11

q_m G, hence tv2tw4

!

iGn

ai(yv_im2yw_im) q_b11 q_m G andqbz^qb11

q_m Gfollows. Soz ^qb11

q_bq_mG4^s_s^bqb

m

G’szGfromszNqm₀andm0E EmEb, which contradicts the choice of z.

HencekGband supposekEb. We note that there is a uniquejGn with k4kj and kjEki for all icj by our first claim in the proof. From k11Gb and (3.4) follows t_v2t_wq_k11G’^qk11

q_m G, moreover

jci

!

a_i(y_vim2y_wim) q_k11 q_m G, hence

a_j(y_vjm2y_wjm) qk11

qm

G

anda_j ^qk

qm

g_bp(i)^qk11

qm

Gfrom (3.5). We geta_jg_bp(j)^qk11

qk

G4s_kq_kG. Recall kFj*21 and Anng_bp(j)4N_p(j), hencea_j1N_p(j)q_j*21(A/N_p(j)) contra- dicting Proposition 3.1 (f). Thus the first part of (c) follows, which imme- diately impliesp(j)4aby Proposition 3.1 (g) (the second part) and (b), (c) are shown. r

4. Proof of the Theorem.

The main theorem will follow from a corollary of the results in section 3.

COROLLARY 4.1. Let t_v4

!

i41 n

a_iy_vim

!

i41 s

F_bi(vWGF_a)be the fam- ily of elements given by Proposition 3.1 and let zG0szG be as in Proposition 3.2 with the property (3.4) [such that Annz*Anng_a], then zAg_a.

PROOF. If t_v2t_w4

!

i41 n

a_i(y_vim2y_wim) and b4br (v,w), then by Proposition 3.1 and Proposition 3.2 we have

tv2tw2aj(y_njm2yw_jm)qb11G

withp(j)4aas in Proposition 3.2 (c), where br (v_jm,w_jm)4br (v,w)4 4b. From (3.4) follows

q_bz2a_j(y_vjm2y_wjm)q_b11G’ q_b11 q_m G and similarly, using (3.5) we have

q_bz2a_j qb

qm

g_a qb11

qm

G thus qmz2ajq_a^qb11

q_b G’sbqbG.

Choose a sequence (v,w) of pairs v,wW with br (v,w)4bw converging to infinity. Thus qmz2ajg_ab

1

We finally prove the main theorem.

PROOF. By (2.17) it remains to show thatbWbAfor all basal 0cb B. We fix a basal elementbBand suppose thatz4bWbA. Choose anyaEl such that b4g_aG_aand Anng_a4N_a9. In particularzc0 and there ism0wwith zs_zG for somes_zSand s_zNq_m0. ConsiderF_a as in (2.12) and take its family

tv4yvW (vV_a) .

From (2.10) follows y_v2y_w6q_bg_aq_b11G for b4br (v,w), hence t_v2t_w6q_bzq_b11G for any v,w,V_a with b4br (v,w).

The t_v8s constitute a family satisfying the hypothesis of Proposition 3.1 and Proposition 3.2. Passing to the subfamily in Proposition 3.1 we can apply Proposition 3.2 and getzg_aAfrom Corollary 4.1, which is a contradiction. It follows bWbA as desired. r

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Manoscritto pervenuto in redazione il 23 settembre 2002 e in forma finale l’11 ottobre 2002.