Series Logo Volume 00, Number 00, Xxxx 19xx

ON CARDINALITIES IN QUOTIENTS OF INVERSE LIMITS OF GROUPS

SAHARON SHELAH AND RAMI GROSSBERG

Abstract. Let λ be ℵ_{0} or a strong limit of cofinality ℵ_{0}. Suppose that ⟨G_{m}, π_{m,n} : m ≤ n < ω⟩ and ⟨H_{m}, π_{m,n}^{t} : m ≤ n < ω⟩ are projective systems of groups of cardinality less than λ and suppose that for every n < ω there is a homorphism σ : H_{n} → G_{n} such that all the diagrams commute.

If for every μ < λ there exists ⟨f_{i} ∈ G_{ω} : i < μ⟩ such that

i ≠ j ⟹ f_{i}f_{j}^{−1} ∉ σ_{ω}(H_{ω}) then there exists ⟨f_{i} ∈ G_{ω} : i < 2^{λ}⟩ such that i ≠ j ⟹ f_{i}f_{j}^{−1} ∉ σ_{ω}(H_{ω}).

1. Introduction

The main result of this paper was motivated by our interest in the structure of the group Ext_{p}(G, Z) for G abelian torsion free. For basic results about the structure of Ext(G, Z) the reader is refered to sections 47 and 52 of Laszlo Fuchs book [Fu], however all we need is Definition 1.21 below. Since Shelah’s proof of the independence of Whitehead’s problem of ZF C (see [Sh 44]) much was done since that paper, for a summary see the introduction to [GrSh] and Chapter XII of Eklof & Mekler’s book is dedicated ([EK]) to the structure of Ext.

In [GrSh] we have dealt with the cardinality of Ext_{p}(G, Z). The main Theorem of [GrSh] states that for a strong limit λ of cofinality ℵ_{0} for every torsion free G of cardinality λ either

|Ext_{p}(G, Z)| < λ or |Ext_{p}(G, Z)| = 2^{λ}.
\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

0000-0000/00 $1.00 + $.25 per page

Received by the editors October 13, 1997.

Partially supported by the United States-Israel Binational Science Foundation and by the National Science Fundation, NSF-DMS97-04477.

In section 2 of [GrSh] we indicated that the proof of the main theorem can be adapted to give a result concerning cardinalities of inverse systems of abelian groups subject to certain conditions (See Theorem 1.1 below). We did not include a proof there. Recently we were asked to supply a complete proof to that theorem. Charles Megibben in a widely circulated preprint [Me] (which to our knowledge did not appear yet in print) even claimed that he proved a result that contradicts Theorem 1.1.

The aim of this paper is to present a complete proof of Theorem 1.1 below.

Notice that we do not make any assumptions on the groups, in particular the groups need not be commutative and can be even locally finite. See more on the subject in [Sh 664].

Theorem 1.1. [The Main Theorem] Suppose λ is ℵ_{0} or it is strong limit cardinal of cofinality ℵ_{0}.

(1) Let ⟨G_{m}, π_{m,n} : m ≤ n < ω⟩ be an inverse system of groups of cardinality less than λ whose inverse limit is G_{ω} with π_{n,ω} such that |G_{n}| < λ. ( π_{m,n} is a homomorphism from G_{m} to G_{n}, α ≤ β ≤ γ ≤ ω ⇒ π_{α,β} ∘ π_{β,γ} = π_{α,γ} and π_{α,α} is the identity).

(2) Let I be a finite index set. Suppose that for every t ∈ I, ⟨H_{m}^{t}, π_{m,n}^{t} : m ≤ n < ω⟩ is an inverse system of groups of cardinality less than λ and H_{ω}^{t} with π_{n,ω}^{t} be the corresponding inverse limit.

(3) Let for every t ∈ I, σ_{n}^{t} : H_{n}^{t} → G_{n} be a homomorphism such that all diagrams commute (i.e. π_{m,n} ∘ σ_{n}^{t} = σ_{m}^{t} ∘ π_{m,n}^{t} for m ≤ n < ω), and let σ_{ω}^{t} be the induced homomorphism from H_{ω}^{t} into G_{ω}.

Assume that for every μ < λ there is a sequence ⟨f_{i} ∈ G_{ω} : i < μ⟩ such that for i ≠ j and t ∈ I ⇒ f_{i}f_{j}^{−1} ∉ Rang(σ_{ω}^{t}). Then there is ⟨f_{i} ∈ G_{ω} : i < 2^{λ}⟩ such that i ≠ j and t ∈ I ⇒ f_{i}f_{j}^{−1} ∉ Rang(σ_{ω}^{t}).

Notation 1.2. Since λ has cofinality ℵ_{0} we can fix λ_{n} < λ for n < ω such that λ = ∑_{n<ω} λ_{n}, for all n < ω, λ_{n} is regular and 2^{λ_{n}} < λ_{n+1} < λ and |G_{n}| + ∑_{t∈I} |H_{n}^{t}| ≤ λ_{n}.

Denote by e_{G_{α}}, e_{H_{α}^{t}} the unit elements. Without loss of generality the groups are pairwise disjoint.

Definition 1.3. (1) For α ≤ ω let H_{α} = ∏_{t∈I} H_{α}^{t} and H_{<α} = ∏_{β<α} H_{β}, H_{≤α} = ∏_{β≤α} H_{β}.

(2) For ḡ ∈ H_{α} let lev(ḡ) = α, for g ∈ H_{α}^{t} let lev(g) = α (without loss of generality this is well defined).

(3) For α ≤ β ≤ ω, g ∈ H_{β}^{t} let g ↾ H_{α}^{t} = π_{α,β}^{t}(g) and we say g ↾ H_{α}^{t} is below g and g is above g ↾ H_{α}^{t} or extend g ↾ H_{α}^{t}.

(4) For α ≤ β ≤ ω, f ∈ G_{β} let f ↾ G_{α} = π_{α,β}(f).

We will now introduce the rank function used in the proof of Theorem 1.1, it is a measure for the possibility to extend functions in Lemma 1.7 we show that it is an ultrametric valuation.

Definition 1.4. (1) For g ∈ H_{n}^{t}, f ∈ G_{ω} we say that (g, f) is a nice t-

pair if σ_{n}^{t}(g) = f ↾ G_{n}.

(2) Define a ranking function rk_{t}(g, f) for any nice t-pair. First by induction on the ordinal α (we can fix f ∈ G_{ω}), we define when rk_{t}(g, f) ≥ α simultaneously for all n < ω and every g ∈ H_{n}^{t}

(a) rk_{t}(g, f) ≥ 0 iff (g, f) is a nice t-pair

(b) rk_{t}(g, f) ≥ δ for a limit ordinal δ iff for every β < δ we have rk_{t}(g, f) ≥ β

(c) rk_{t}(g, f) ≥ β +1 iff (g, f) is a nice t-pair, and letting n = lev(g) there exists g′ ∈ H_{n+1}^{t} extending g such that rk_{t}(g′, f) ≥ β

(d) rk_{t}(g, f) ≥ −1.

(3) For α an ordinal or −1 (stipulating −1 < α < ∞ for any ordinal α) we have rk_{t}(g, f) = α iff rk_{t}(g, f) ≥ α and it is false that rk_{t}(g, f) ≥ α + 1.

(4) rk_{t}(g, f) = ∞ iff for every ordinal α we have rk_{t}(g, f) ≥ α.

The following two claims give the principal properties of rk_{t}(g, f).

Claim 1.5. Let (g, f) be a nice t-pair.

(1) The following statements are equivalent:

(a) rk_{t}(g, f) = ∞

(b) there exists g′ ∈ H_{ω}^{t} extending g such that σ_{ω}^{t}(g′) = f.

(2) If rk_{t}(g, f) < ∞, then rk_{t}(g, f) < λ^{+}.

(3) If g′ is a proper extension of g and (g′, f) is also a nice t-pair then (a) rk_{t}(g′, f) ≤ rk_{t}(g, f) and

(b) if 0 ≤ rk_{t}(g, f) < ∞ then the inequality is strict.

Proof. (1) (a) ⇒ (b): Let n be such that g ∈ H_{n}^{t}. It is enough to define

g_{k} ∈ H_{k}^{t} for k < ω, k ≥ n such that

(i) g_{n} = g
(ii) g_{k} is below g_{k+1} that is π_{k,k+1}^{t}(g_{k+1}) = g_{k} and
(iii) rk_{t}(g_{k+1}, f) = ∞:

Let g′ := \underrightarrow{lim}g_{k} it is as required. The definition is by induction on k ≥ n. For k = n let g_{0} = g. For k ≥ n, suppose g_{k} is defined. By (iii) we have rk_{t}(g_{k}, f) = ∞, hence there exists g^{∗} ∈ H_{k+1}^{t} extending g_{k} such that rk_{t}(g^{∗}, f) = ∞, and let g_{k+1} := g^{∗}.

(b) ⇒ (a): Since g is below g′, it is enough to prove by induction on α that for every k ≥ n when g_{k} := g′ ↾ H_{k}^{t} we have that rk_{t}(g, f) ≥ α.

For α = 0, since σ_{ω}^{t}(g′) = f ↾ G_{n} clearly for every k we have σ_{k}^{t}(g_{k}) = f ↾ G_{k} so (g_{k}, f) is a nice t-pair.

For limit α, by the induction hypothesis for every β < α and every k we have rk_{t}(g_{k}, f) ≥ β, hence by Definition 1.4(2)(b), rk_{t}(g_{k}, f) ≥ α.

For α = β + 1, by the induction hypothesis for every k ≥ n we have rk_{t}(g_{k}, f) ≥ β. Let k_{0} ≥ n be given. Since g_{k_{0}} is below g_{k_{0}+1} and rk_{t}(g_{k_{0}+1}, f) ≥ β, Definition 1.4(2)(c) implies that rk_{t}(g_{k_{0}}, f) ≥ β + 1; i.e. for every k ≥ n we have rk_{t}(g_{k}, f) ≥ α. So we are done.

(2) Let g ∈ H_{n}^{t} and f ∈ G_{ω} be given. It is enough to prove that if rk_{t}(g, f) ≥ λ^{+} then rk_{t}(g, f) = ∞. Using part (1) it is enough to find g′ ∈ H_{ω}^{t} such that g is below g′ and f = σ_{ω}^{t}(g′).

We define by induction on k < ω, g_{k} ∈ H_{n+k}^{t} such that g_{k} is below g_{k+1} and rk_{t}(g_{k}, f) ≥ λ^{+}. For k = 0 let g_{k} = g. For k + 1, for every α < λ^{+}, as rk_{t}(g_{k}, f) > α by 1.4(2)(c) there is g_{k,α} ∈ G_{n+k+1} extending g_{k} such that rk_{t}(g_{k,α}, f) ≥ α. But the number of possible g_{k,α} is ≤ |H_{n+k+1}^{t}| ≤ 2^{λ_{n+k+1}} < λ^{+} hence there are a function g and a set S ⊆ λ^{+} of cardinality λ^{+} such that α ∈ S ⇒ g_{k,α} = g. Now take g_{k+1} = g.

(3) Immediate. □_{1.5} Lemma 1.6. (1) Let (g, f) be a nice t-pair. Then we have rk(g, f) ≤

rk(g^{−1}, f^{−1}).

(2) For every nice t-pair (g, f) we have rk(g, f) = rk(g^{−1}, f^{−1}).

Proof. (1) By induction on α prove that rk(g, f) ≥ α ⇒ rk(g^{−1}, f^{−1}) ≥

α (see more details in Lemma 1.7).

(2) Apply part (1) twice. □_{1.6}

In the following lemma we show that the rank is indeed ultrametric (ordinal valued). Lemma 1.7. Let n < ω be fixed, and let (g_{1}, f_{1}), (g_{2}, f_{2}) be nice t-pairs with g_{ℓ} ∈ H_{n}^{t}(ℓ = 1, 2).

(1) If (g_{1}, f_{1}) and (g_{2}, f_{2}) are t-nice pairs, then (g_{1}g_{2}, f_{1}f_{2}) is a nice pair and rk_{t}(g_{1}g_{2}, f_{1}f_{2}) ≥ Min{rk_{t}(g_{ℓ}, f_{ℓ}) : ℓ = 1, 2}.

(2) Let n, (f_{1}, g_{1}) and (f_{2}, g_{2}) be as above. If rk_{t}(g_{1}, f_{1}) ≠ rk_{t}(g_{2}, f_{2}), then rk_{t}(g_{1}g_{2}, f_{1}f_{2}) = Min{rk_{t}(g_{ℓ}, f_{ℓ}) : ℓ = 1, 2}.

Proof. (1) It is easy to show that the pair is t-nice. We show by induction on α simultaneously for all n < ω and every g_{1}, g_{2} ∈ H_{n}^{t} that Min{rk(g_{ℓ}, f_{ℓ}) : ℓ = 1, 2} ≥ α implies that rk(g_{1}g_{2}, f_{1}f_{2}) ≥ α.

When α = 0 or α is a limit ordinal this is easy. Suppose α = β + 1 and that rk(g_{ℓ}, f_{ℓ}) ≥ β + 1; by the definition of rank for ℓ = 1, 2 there exists g′_{ℓ} ∈ H_{n+1}^{t} extending g_{ℓ} such that (g′_{ℓ}, f_{ℓ}) is a nice pair and rk_{t}(g′_{ℓ}, f_{ℓ}) ≥ β. By the induction assumption rk_{t}(g′_{1}g′_{2}, f_{1}f_{2}) ≥ β. Hence g′_{1}g′_{2} is as required in the definition of rk_{t}(g_{1}g_{2}, f_{1}f_{2}) ≥ β + 1.

(2) Suppose without loss of generality that rk(g_{1}, f_{1}) < rk(g_{2}, f_{2}), let α_{1} = rk(g_{1}, f_{1}) and let α_{2} = rk_{t}(g_{2}, f_{2}). By part (1), rk_{t}(g_{1}g_{2}, f_{1}f_{2}) ≥ α_{1}, by Proposition 1.6, rk_{t}(g_{2}^{−1}, f_{2}^{−1}) = α_{2} > α_{1}. So we have

α_{1} = rk_{t}(g_{1}, f_{1}) = rk_{t}(g_{1}g_{2}g_{2}^{−1}, f_{1}f_{2}f_{2}^{−1})
≥ Min{rk_{t}(g_{1}g_{2}, f_{1}f_{2}), rk_{t}(g_{2}^{−1}, f_{2}^{−1})}
= rk_{t}(g_{1}g_{2}, f_{1}f_{2}) ≥ α_{1}.

Hence the conclusion follows. □_{1.7}

Definition 1.8. (1) Let μ < λ and let ᾱ = ⟨α_{t} : t ∈ I⟩ where α_{t} is an ordinal less or equal to λ^{+}. We say that f̄ = ⟨f_{i} : i < μ⟩ μ-exemplifies ᾱ ∈ Γ_{n} (or f̄ is a μ-witness for ᾱ ∈ Γ_{n}) iff

(a) f_{i} ∈ G_{ω} and f_{i} ↾ G_{n} = e_{G_{n}}

(b) for i ≠ j and t ∈ I we have rk_{t}(e_{H_{n}^{t}}, f_{i}f_{j}^{−1}) < α_{t} (possibly is

−1).

(2) Let

Γ_{n} = {ᾱ : ᾱ = ⟨α_{t} : t ∈ I⟩, α_{t} an ordinal ≤ λ^{+},

and for every μ < λ there is a sequence ⟨f_{i} : i < μ⟩

which μ-exemplifies ᾱ ∈ Γ_{n}}.

(3) Δ_{n} = {ᾱ ∈ Γ_{n} : for no β̄ we have β̄ ∈ Γ_{n}, β̄ ≤ ᾱ (i.e. ⋀_{t∈J_{n}} β_{t} ≤ α_{t}) and β̄ ≠ ᾱ}.

Claim 1.9. (1) Γ_{n} is not empty.

(2) Δ_{n} is not empty in fact (∀ᾱ ∈ Γ_{n})(∃β̄ ∈ Δ_{n})(β̄ ≤ ᾱ).

Proof. (1) Let α_{t}^{∗} = sup{rk_{t}(g, f)+1 : g ∈ H_{n}^{t}, f ∈ G^{ω} and rk_{t}(g, f) < ∞}, by 1.5(2), this is a supremum on a set of ordinals < λ^{+} (as −1 + 1 = 0) hence is an ordinal ≤ λ^{+}. So ⟨α_{t}^{∗} : t ∈ I⟩ is as required.

(2) If not, then choose by induction on ℓ < ω a sequence β̄^{ℓ} ∈ Γ_{n} such that β̄^{0} = ᾱ, β̄^{ℓ+1} ≤ β̄^{ℓ}, β̄^{ℓ+1} ≠ β^{ℓ}. So for each t ∈ I, the sequence ⟨β_{t}^{ℓ} : ℓ < ω⟩ is a non-increasing sequence of ordinals hence is eventually constant, say for some ℓ_{t} < ω we have ℓ ∈ [ℓ_{t}, ω) ⇒ β_{t}^{ℓ} = β_{t}^{ℓ_{t}}, so as I is finite, ℓ(∗) = max{ℓ_{t} : t ∈ I} < ω, so β̄^{ℓ(∗)} = β̄^{ℓ(∗)+1}, a contradiction. □_{1.9}

Claim 1.10. (1) If μ ≤ μ′ and ⟨f_{i} : i < μ′⟩, μ′-exemplify ᾱ ∈ Γ_{n} and

h : μ → μ′ is one to one, then ⟨f_{h(i)} : i < μ⟩, μ-exemplifies ᾱ ∈ Γ_{n}.

(2) If ⟨f_{i} : i < μ⟩, μ-exemplify ᾱ ∈ Γ_{n} and f_{i} ↾ G_{n+1} = f for i < μ, then ⟨f_{i}f_{0}^{−1} : i < μ⟩, μ-exemplify ᾱ ∈ Γ_{n+1}.

(3) If ᾱ ∈ Γ_{n}, then ᾱ ∈ Γ_{n+1}.

(4) If ᾱ ∈ Δ_{n}, then some β̄ ≤ ᾱ belongs to Δ_{n+1}.

(5) For some n < ω there is ᾱ ∈ ⋂_{m≥n} Δ_{n}.

(6) In clause (b) of Definition 1.8(1) it suffices to deal with i < j.

Proof. (1) Trivial.

(2) Clearly.

Clause (a):

(f_{i} ∘ f_{0}^{−1}) ↾ G_{n+1} = σ_{n+1}^{ω}(f_{i}f_{0}^{−1}) = (σ_{n+1}^{ω}(f_{i}))(σ_{n+1}^{ω}(f_{0}))^{−1} = ff^{−1} = e_{G_{n+1}}.

Clause (b):

For i ≠ j and t ∈ I, note that

(f_{i}f_{0}^{−1})(f_{j}f_{0}^{−1}) = f_{i}f_{0}^{−1}f_{0}f_{j}^{−1} = f_{i}f_{j}^{−1}

so we can use the assumption.

(3) So let μ < λ and we should find a μ-witness for ᾱ ∈ Γ_{n+1}. We can choose μ′ such that μ×|G_{n+1}| < μ′ < λ. As ᾱ ∈ Γ_{n}, clearly there is a μ′-witness ⟨f_{i} : i < μ′⟩ for it. Now the number of possible f_{i} ↾ G_{n+1} is ≤ |G_{n+1}| (really) even ≤ |Rang(π_{n+1,ω})∩ Ker(π_{n,n+1})|) hence for some f ∈ G_{n+1} and Y ⊆ μ′ we have: |Y | ≥ μ and i ∈ Y ⇒ f_{i} ↾ G_{n+1} = f. By renaming {i : i < μ} ⊆ Y, now ⟨f_{i}f_{0}^{−1} : i < μ⟩ is a μ-witness by part (1).

(4) Follows by 1.10(2) and 1.9(2).

(5) By 1.10(3) by the well foundedness of the ordinals (as in the proof of 1.9(2),(8).

(6) Because for i < j, (f_{j}f_{i}^{−1})^{−1} = (f_{i}f_{j}^{−1}) and 1.6(2). □_{1.10}

Convention 1.11. By renaming and 1.10(4), without loss of generality ᾱ^{∗} ∈ Δ_{n} for every n.

Claim 1.12. Each α_{t}^{∗}(t ∈ I) is a non-successor ordinal (i.e. limit or zero).

Proof. Fix n < ω.

Assume s ∈ I is a counterexample. So α_{s}^{∗} = β^{∗} + 1, β^{∗} ≥ 0. Let β̄ = ⟨β_{t} : t ∈ I⟩ be defined as follows: β_{t} is α_{t} if t ≠ s and is β^{∗} if t = s. We shall prove that β̄ ∈ Γ_{n+1} thus getting a contradiction. So let μ < λ and we shall find a μ-witness for β̄ ∈ Γ_{n+1}. Let μ′ be such that μ|G_{n+1}| < μ′ < λ. As ᾱ^{∗} ∈ Γ_{n} (see 1.11) there is a μ′-witness ⟨f_{i} : i < μ′⟩ for ᾱ^{∗} ∈ Γ_{n}, as earlier without loss of generality i < μ ⇒ f_{i} ↾ G_{n+1} = f for some f. We shall prove that ⟨f_{i}f_{0}^{−1} : i < μ⟩ is a μ-witness for β̄ ∈ Γ_{n+1}. Let f′_{i} = f_{i}f_{0}^{−1} for i < μ.

Clause (a):

f′ ↾ G_{n+1} = (f_{0}f_{0}^{−1}) ↾ G_{n+1} = e_{G_{n+1}} because f_{i} ↾ G_{n+1} = f_{0} ↾ G_{n+1}.

Clause (b):

Let i ≠ j < μ. If t ∈ I\{s} then rk_{t}(e_{G_{n+1}}, f′_{i}(f′_{j})^{−1}) = rk_{t}(e_{G_{n+1}}, f_{i}f_{j}^{−1}) ≤ rk_{t}(e_{G_{n}}, f_{i}f_{j}^{−1}) ≤ α_{t}^{∗} = β_{t}. (Why? By group theory, by 1.5(3)(α), by choice of f̄, by choice of β_{t}, respectively).

If t = s, then rk_{t}(e_{G_{n}}, f_{i}f_{j}^{−1}) < rk_{t}(e_{G_{n+1}}, f_{i}f_{j}^{−1}) by 1.5(3)(β), and proceed as above. □_{1.12}

Notation 1.13. For α ≤ ω let T_{α} := ∏_{k<α} λ_{k}, T := ∏_{n<ω} T_{n} (note: treeness used).

Claim 1.14. There are for n < ω, a sequence ⟨f_{n,i} : i < λ_{n}⟩ and an ordinal γ_{n}^{t} < α_{t}^{∗} (α_{t}^{∗} is the ordinal from 1.11) such that

(1) f_{n,i} ∈ G_{ω}, f_{n,i} ↾ G_{n+1} = e_{G_{n+1}} for all i < λ_{n};

(2) for each t ∈ I for every h ∈ H_{n}^{t} and i < j < λ_{n} we have: rk_{t}(h, f_{n,i}f_{n,j}^{−1}) ≤ γ_{n}^{t};

(3) rk_{t}(e_{H_{n}^{t}}, f_{n,i}f_{n,j}^{−1}) ≥ γ_{n−1}^{t} for i < j < λ_{n} and γ_{n−1}^{t} ≥ 0 ⇒ rk_{t}(e_{H_{n}^{t}}, f_{n,i}f_{n,j}^{−1}) > γ_{n−1}^{t}

(4) γ_{n−1}^{t} < γ_{n}^{t} if α_{t}^{∗} > 0 and γ_{n}^{t} = −1 if α_{t}^{∗} = 0.

We delay the rest of proof for a while.

Convention 1.15. Let γ_{n}^{t}, g_{n,i} (n < ω, i < λ_{n}) be as in 1.14.

Definition 1.16. We set f_{η} = g_{n−1,η(n−1)}g_{n−2,η(n−2)} … g_{0,η(0)} for η ∈ T_{n}. Then define f_{η} for η ∈ T_{ω} as follows: f_{η} is the element of G^{ω} satisfying f_{η} ↾ G_{n} = f_{η↾n}. It is well defined by:

Fact 1.17. (1) For η ∈ T_{ω} and m ≤ n < ω we have

f_{η↾n} ↾ G_{n+1} = f_{η↾m} ↾ G_{n+1}.

(2) For η ∈ T_{ω} we have f_{η} ∈ G_{ω} is well defined (as the inverse limit of ⟨f_{η↾n} ↾ G_{n} : n < ω⟩, so n < ω → f_{η} ↾ G_{n} = f_{η↾n}.

Proof. (1) As π_{n,ω} is a homomorphism it is enough to prove (f_{η↾n}(f_{η↾m})^{−1}) ↾ G_{n+1} = e_{G_{n+1}}, hence it is enough to prove n ≤ k < ω ⇒ (f_{η↾k}f_{η↾(k+1)}^{−1}) ↾ G_{n+1} = e_{G_{n+1}} which follows from k < ω ⇒ f_{η↾k}f_{η↾(k+1)}^{−1} ↾ G_{k+1} = e_{G_{k+1}}, which means f_{k,η(k)} ↾ G_{k+1} = e_{G_{k+1}} which holds by clause (a) of 1.11.

(2) Follows by part (1) and G_{ω} being an inverse limit. □_{1.17}

Proposition 1.18. Let η, ν ∈ T_{ω}. If η ≠ ν and t ∈ I, then f_{η}f_{ν}^{−1} ∉ σ_{ω}^{t}(H_{ω}^{t}).

Proof. Suppose for the sake of contradiction that for some g ∈ H_{ω}^{t} we have σ_{ω}^{t}(g) = f_{η}f_{ν}^{−1}.

Let k be minimal such that η ↾ k = ν ↾ k, η(k) ≠ ν(k), without loss of generality η(k) < ν(k). For ℓ ≥ k let ξ^{ℓ} be rk_{t}(g ↾ H_{ℓ}^{t}, f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1}). We will reach a contradiction by showing that ℓ ≥ k ⇒ 0 ≤ ξ^{ℓ} ≤ γ_{k}^{t} and ℓ > k ⇒ ξ^{ℓ+1} < ξ^{ℓ}.

Note

(∗)_{1} if ℓ ≤ α ≤ ω, then rk_{t}(g ↾ H_{ℓ}^{t}, f_{η↾α}f_{ν↾α}^{−1}) ≥ 0 as σ_{ℓ}^{t}(g ↾ H_{ℓ}^{t}) = σ^{t}(g) ↾ G_{ℓ}^{t} = (f_{η}f_{ν}^{−1}) ↾ F_{ℓ}^{t} and 1.17.

For ℓ = k, we show that ξ^{k} ≤ γ_{k}^{t}. Let i = η[k], j = ν[k]. By the choice of k, i ≠ j. In this case f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1} = f_{k,η(k)}f_{k,ν(k)}^{−1} by the minimality of k and, of course, f_{k,η(k)}f_{k,ν(k)}^{−1} = f_{k,i}f_{k,j}^{−1}, hence ξ^{k} = rk_{t}(g ↾ H_{k}^{h}, f_{k,i}f_{k,j}^{−1}) ≤ γ_{k} by clause (b) of 1.14. Note: if α_{t}^{∗} = 0, then γ_{m}^{t} = −1 for m < ω hence ξ^{k} = −1, but (f_{η}f_{ν}^{−1}) ↾ G_{k} = (f_{η↾(k+1)}f_{ν↾(k+1)}^{−1}) ↾ G_{k} immediate contradiction. So assume α_{t}^{∗} ≥ 0 hence 0 ≤ γ_{m}^{t} < γ_{m+1}^{t}.

Now we proceed inductively. We assume that ξ^{ℓ} ≤ ξ^{k} and show that ξ^{ℓ+1} < ξ^{ℓ}. Let i = η[ℓ + 1], j = ν[ℓ + 1], and let ζ = rk_{t}(g ↾ H_{ℓ+1}^{t}, f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1}). Observe:

ON CARDINALITIES IN QUOTIENTS OF INVERSE LIMITS OF GROUPS 9

(∗)_{2} ζ < rk_{t}(g ↾ H_{ℓ}^{t}, f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1}) = ξ^{ℓ} [why? by 1.5(3) and (∗)_{1} above.]

So (∗)_{3} ξ^{ℓ+1} = rk_{t}(g ↾ H_{ℓ+1}^{t}, f_{η↾(ℓ+2)}f_{ν↾(ℓ+2)}^{−1})

= rk_{t}(g ↾ H_{ℓ+1}^{t}, f_{ℓ+1,η(ℓ+1)}(f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1})f_{ℓ+1,ν(ℓ+1)})

= rk_{t}(e_{H_{ℓ+1}^{t}}(g ↾ H_{ℓ+1}^{t})e_{H_{ℓ+1}^{t}}, f_{ℓ+1,η(ℓ+1)}(f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1})f_{ℓ+1,ν(ℓ+1)}). Now:

(∗)_{4} rk_{t}(e_{H_{ℓ+1}^{t}}, f_{ℓ+1,η(ℓ+1)}) > γ_{ℓ}^{t} (why? by clause (c) of 1.14) (∗)_{5} rk_{t}(g ↾ H_{ℓ+1}^{t}, f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1}) = ξ^{ℓ} ≤ ξ^{k} ≤ γ_{k}^{t} ≤ γ_{ℓ}^{t} (why? the equality by the definition of ξ^{ℓ}, the first inequality by the induction hypothesis and the second inequality was proved above (for ℓ = k), the last inequality by 1.14 clause (d) (∗)_{6} rk_{t}(e_{H_{ℓ+1}^{t}}, g_{ℓ+1,ν(ℓ+1)}) > γ_{ℓ}^{t} (why? by clause (c) of 1.14).

Hence by 1.5(3) (∗)_{7} rk_{t}(e_{H_{ℓ+1}^{t}}(g ↾ H_{ℓ+1}^{t})e_{H_{ℓ+1}^{t}}, f_{ℓ+1,η(ℓ)}(f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1})f_{ℓ+1,ν(ℓ+1)}) = rk(g ↾ G_{ℓ+1}, f_{η↾(ℓ+1)}f_{ν↾(ℓ+1)}^{−1}).

Together we get the induction demand for ℓ + 1. □_{1.18} Before proving 1.14 and finishing we prove

Claim 1.19. Assume −1 ≤ β_{t} < α_{t}^{∗} for t ∈ I and n < ω and μ < λ. Then we can find ⟨f_{i} : i < μ⟩ such that

(1) f_{i} ∈ G_{ω} and f_{i} ↾ G_{n+1} = e_{G_{n+1}}

(2) t ∈ I and i ≠ j ⇒ rk_{t}(e_{H_{n}^{t}}, f_{i}f_{j}^{−1}) ∈ [β^{t}, α_{t}^{∗})

(3) t ∈ I and i < μ = rk_{t}(e_{H_{n}^{t}}, f_{i}) ∈ [β_{t}, α_{t}^{∗}).

Proof. For each s ∈ I we define β̄^{s} = ⟨β_{t}^{s} : t ∈ I⟩ by:

β_{t}^{s} = \begin{cases}α_{t} & if t ≠ s \\ β_{t} & if t = s \end{cases}

So β̄^{s} ≤ ᾱ^{∗}, β̄^{s} ≠ ᾱ^{∗}, so as ᾱ^{∗} ∈ ×_{n<ω}Δ_{m} necessarily β̄^{s} ∉ Γ_{n}, hence for some μ^{s} < λ there is no μ^{s}-witness for β̄^{s} and n (check the definition of Γ_{n}).

Let μ_{1} < λ be > μ + max{μ^{s} : s ∈ I}.

Let χ < λ be large enough (so that it will be possible to use the finite Ramsey theorem when λ = ℵ_{0} and when λ > ℵ_{0} the Erdös Rado theorem we require that χ → (μ_{1})_{θ}^{2} where θ = 2^{∑_{t} |H_{n}^{t}|}).

Let ⟨f_{i} : i < χ⟩ be a χ-witness ᾱ ∈ Γ_{n} and even ᾱ ∈ Γ_{n+1}. For each t ∈ I, h ∈ H_{n}^{t} define the two place function F_{t,h} from [χ]^{2} to {0, 1} for i < j < χ let

F_{t,h}\{i, j\} := \begin{cases}0 & if rk_{t}(h, f_{i}f_{j}^{−1}) < β_{t} \\ 1 & Otherwise. \end{cases}

Define the two-place function F from [χ]^{2}: For i < j < χ let F {i, j} = ⟨F_{t,h}(i, j) : t ∈ I, h ∈ H_{n}^{t}⟩.

Clearly |Rang(F)| ≤ 2^{∑_{t} |H_{n}^{t}|}.

Hence an application of one of the above partition theorems provides us with a set Y ⊆ χ, |Y | = μ_{1} such that F ↾ [Y]^{2} is constant. Without loss of generality Y = μ_{1}.

For each s ∈ I, clearly ⟨f_{i}f_{0}^{−1} : i < μ^{s}⟩ is not a μ^{s}-witness for β̄^{s}, but the only thing that may go wrong is the inequality, i < j < μ^{s} ⇒ rk_{s}(e_{H_{n}^{s}}, f_{i}f_{j}^{−1}) < β_{s}, so for some i < j < μ^{s} we have that rk_{s}(e_{H_{n}^{s}}, f_{i}f_{j}^{−1}) ≥ β_{s} holds, hence

(∗) s ∈ I and i < j < μ_{1} ⇒ rk_{s}(e_{H_{n}^{s}}, f_{i}f_{j}^{−1}) ≥ β_{s}.

This means clause (b) holds and clause (a) by definition of ⟨f_{i} : i < χ⟩ is a χ-witness for ᾱ ∈ Γ_{n}. Clause (c) follows. So ⟨f_{i} : i < μ⟩ is as required. □_{1.19}

Proof. of 1.14

Stipulate γ_{−1}^{t}: if α_{t}^{∗} > 0 it is 0, otherwise is it −1. Assume n < ω and ⟨γ_{n−1}^{t} : t ∈ I⟩ is well defined, γ_{n−1}^{t} < α_{t}^{∗}. Let γ_{n}^{t,∗} be: γ_{n−1}^{t} + 1 if α_{t}^{∗} is a limit ordinal and γ_{n−1}^{t} = −1 otherwise (i.e. α_{t}^{∗} = 0, see 1.12). Note that to construct the family {f_{n,i} : i < λ_{n}} we will combine Claim 1.19 with a second application of the Erdös Rado Theorem.

Let θ = (2^{|H_{n}^{t}|×|H_{n}^{t}|}) × |I| and χ < λ be such that χ → (λ_{n} + 2)_{θ}^{3} (exists by Ramsey theorem if λ = ℵ_{0} and by Erdös Rado theorem if λ > ℵ_{0}). Apply Claim 1.19 to get a family {f_{i} : i < χ} satisfying:

(1) f_{i} ↾ G_{n+1} = e_{G_{n+1}},

(2) for i ≠ j and t ∈ I, we have γ_{n−1}^{t,∗} ≤ rk_{t}(e_{H_{n}^{t}}, f_{i}f_{j}^{−1}) < α_{t}^{∗}.

For t ∈ I, ḡ = ⟨g_{1}, g_{2}⟩, g_{1}, g_{2} ∈ H_{n}^{t} such that σ_{n}^{t}(g) = e_{G_{n}} define a coloring F_{t,ḡ} of [I]^{3} by two colors according to the following scheme: for ε < ζ < ξ < χ, let

F_{t,g}\{ε, ζ, ξ\} := \begin{cases}red  if rk_{t}(g_{1}, f_{i_{ε}}f_{ζ}^{−1}) ≤ rk_{t}(g_{2}, f_{ζ}f_{ξ}^{−1}); \\ green  if rk_{t}(g_{1}, f_{i_{ε}}f_{ζ}^{−1}) > rk_{t}(g_{2}, f_{ζ}f_{ξ}^{−1}) \end{cases} .

By the Ramsey theorem (if λ = ℵ_{0}) or Erdös Rado Theorem if λ > ℵ_{0} there is a set J ⊆ χ, otp(J) = λ_{n} + 2 such that each coloring is constant on [J]^{3}. Let the value of F_{t,ḡ} on [J]^{3} be denoted c_{t,ḡ}. Observe that c_{t,ḡ} is never green as this would produce a descending ω-sequence of ordinals as if ε_{ℓ} ∈ J, ε_{ℓ} < ε_{ℓ+1} for ℓ < ω, then rk_{t}(g, f_{ε_{ℓ}}f_{ε_{ℓ+1}}^{−1}) > rk_{t}(g, f_{ε_{ℓ+1}}f_{ε_{ℓ+2}}^{−1}), so ⟨rk_{t}(g, f_{ε_{2ℓ}}f_{ε_{2ℓ+1}}^{−1}) : ℓ < ω⟩ is strictly decreasing.

Let ε(∗) = Min(J) and J_{0} = {ε ∈ J : otp(ε ∩ J) < λ} and α is the λ_{n}-th member of J, β the (λ_{n} + 1)-th member of J and let γ_{n}^{t} = rk_{t}(e_{H_{n}^{t}}, f_{α}f_{β}^{−1}), by clause (b) above γ_{n}^{t,∗} ≤ γ_{n}^{t} < α_{t}^{∗} so α_{t}^{∗} = 0 ⇒ γ_{n}^{t} = −1 and α_{t}^{∗} > 0 ⇒ γ_{n−1}^{t} < γ_{n}^{t}.

We claim that {f_{i}f_{ε(∗)}^{−1} : i ∈ J_{0}} (remember J_{0} ⊆ J, |J_{0}| = λ_{n}) provides a set that can play the role of {f_{n,i} : i < λ_{n}}. We note

(∗)_{1} rk_{t}(g, f_{ε}f_{ζ}^{−1}) ≤ γ_{t}^{n} for ε < ζ in J_{0} [why? clearly α < β < ε < ζ are in J hence by the choice of J we have rk_{t}(g, f_{ε}f_{ζ}^{−1}) ≤ rk_{t}(g, f_{ζ}f_{α}^{−1}) ≤ rk_{t}(g, f_{α}f_{β}^{−1}) = γ_{n}^{t}].

Now clauses (1), (4) of 1.14 holds by clause (1) above, clause (3) of 1.14 holds by (∗)_{1} and clause (4) of 1.14 holds by the choice of the γ_{t}^{∗}. We are left with clause (2). Let h ∈ H_{n}^{t}, as above clearly for Υ < ξ < ζ < ξ in J we have rk_{t}(h, f_{ε}f_{ζ}^{−1}) ≤ rk_{t}(h, f_{ζ}f_{ξ}^{−1}). Hence for Υε < ζ < ξ in J_{0} we have

γ_{t}^{n} ≥ rk_{t}(e_{H_{n}^{t}}, f_{ε}f_{ζ}^{−1})
= rk_{t}(h^{−1}, (f_{ε}f_{ξ}^{−1})(f_{ζ}f_{ξ}^{−1})^{−1})
≥ Min{rk_{t}(h, f_{ε}f_{ξ}^{−1}), rk_{t}(h^{−1}, (f_{ζ}f_{ξ}^{−1})^{−1}}
= Min{rk_{t}(h, f_{ε}f_{ξ}^{−1}), rk_{t}(h, f_{ζ}f_{ξ}^{−1})}
≥ Min{rk_{t}(h, f_{Υ}f_{ε}^{−1}), rk_{t}(h, f_{Υ}f_{ε}^{−1})}

= rk_{t}(h, f_{Υ}f_{ε}^{−1}).

So giving also clause (2) of 1.14.  □_{1.14}  □_{1.1}

Remark 1.20. The result about the cardinality of Ext_{p}(G, Z) can be derived from Theorem 1.1 using the following definition (which constructs an isomorphic group ot Ext_{p}(G, Z)).

Definition 1.21. Given an abelian group G, let G^{∗} := Hom(G, Z) and for a prime p denote by G^{p} the group Hom(G, Z/pZ). For g ∈ G^{∗} let g ↦ g/p be the natural homomorphism from G^{∗} into G^{p}. By G^{∗}/p denote the subgroup of G^{p} which is the image ofG^{∗}/p under g ↦ g/p. Finally

Ext_{p}(G, Z) := G^{p}/(G^{∗}/p).

Recall that when λ is ℵ_{0} or strong limit of cofinality ℵ_{0} then λ^{ℵ_{0}} = 2^{λ}.

The group H_{ω} corresponde to the subgroup G^{∗}/p and the σ’s are inclusions.

We have learned from Paul Eklof that Christian U. Jensen in his book [Jen] have a proof of Theorem 1.0 of [GrSh] for the case that λ = ℵ_{0}.

References

[EK] Paul C. Eklof and Alan H. Mekler. Almost Free Modules, set-theoretic methods. North-Holland Publishing .Co., Amsterdam, 1990.

[Fu] Lazslo Fuchs, Infinite Abelian Groups I, Academic Press 1970

[Gr] Rami Grossberg. A course in model theory. Book in preparation.

[GrSh] Rami Grossberg and Saharon Shelah. On the structure of Ext_{p}(G, Z). Journal of Algebra, 121:117–128, 1989.

[Je] Thomas Jech. Set Theory. Academic Press, New York, 1978.

[Jen] Christian U. Jensen. Les Foncteurs Derives de \underleftarrow{lim} et leurs Applications en Theorie des Modules. Springer-Verlag, Lecture Notes in Mathematics volume 254, 1972.

[Me] Charles Megibben. ..... A preprint. [Sh 44] Saharon Shelah. Infinite abelian groups, Whitehead problem and some construc-

tions. Israel Journal of Mathematics, 18:243–256, 1974. [Sh 664] Saharon Shelah. On strong dichotomy of cardinality, in preparation.

ON CARDINALITIES IN QUOTIENTS OF INVERSE LIMITS OF GROUPS 13

E-mail address, Saharon Shelah: shelah@sunset.ma.huji.ac.il Institute of Mathematics, The Hebrew University, Jerusalem, Is-

rael, & Rutgers University, Mathematics Department, New Brunswick, NJ 08903 USA

E-mail address, Rami Grossberg: rami@cmu.edu Department of Mathematical Sciences, Carnegie Mellon Univer-

sity, Pittsburgh, PA 15213