Shelah's Singular Compactness Theorem

Publ. Mat. 52 (2008), 3–18 SHELAH’S SINGULAR COMPACTNESS THEOREM Paul C. Eklof Abstract We present Shelah’s famous theorem in a version for modules, together with a self-contained proof and some examples. This exposition is based on lectures given at CRM in October 2006. 0. Introduction The Singular Compactness Theorem is about an abstract notion of “free”. The general form of the theorem is as follows: If λis a singular cardinal and Mis a λ-generated module such that enough < κ-generated submodules are “free” for sufficiently many regular κ < λ, then Mis “free”. Of course, for this to have any chance to be a theorem (of ZFC) there need to be assumptions on the notion of “free”. These are detailed in the next section along with a precise statement of the theorem (Theorem 1.4), including a precise definition of “enough”. Another version of the Singular Compactness Theorem —with a different definition of “enough”— is given at the start of Section 3. The proof of Theorem 1.4 is given in Sections 3 and 4; some examples and applications are given in Sections 2 and 5. First a few words about the history of the theorem. 0.1. History. In 1973 Saharon Shelah proved that the Whitehead problem for abelian groups of cardinality ℵ1is undecidable in ZFC; in particular he showed under the assumption of the Axiom of Constructibility, V=L, that all Whitehead groups of cardinality ℵ1are free (see [14]). His argument easily extended, by induction, to prove that for all n∈ω, 2000 Mathematics Subject Classification. Primary: 16D70; Secondary: 03E75, 16D40, 20K99. Key words. Singular cardinal, singular compactness, almost free module, Baer module, tilting module. 4 P. C. Eklof Whitehead groups of cardinality ℵnare free. But for Whitehead groups of cardinality ℵω, the first singular cardinal, a new ingredient was needed. In fact, that ingredient already existed for singular cardinals of cofinality ωor ω1, by theorems of Paul Hill (see [9] and [10]); these imply that if an abelian group has singular cardinality λwhere λis of cofinality ω (or ω1) and has the property that every subgroup of smaller cardinality is free, then the group itself is free. In 1974 Shelah proved a general theorem which applied not only to arbitrary singular cardinals but to a general notion of “free” defined axiomatically. The case of the ordinary notion of freeness for abelian groups, combined with the argument in his first paper on Whitehead’s problem, led immediately to the conclusion that V=Limplies that Whitehead groups of arbitrary cardinality are free. (See Corollary 5.4 below.) Shelah’s theorem was applicable to much more than abelian groups, or even modules; in fact, there was another application that Shelah had in mind when he proved his theorem in a general form: that of transversal theory; the parallels between results there and results about “almost free” abelian groups had already been noted. (See [12]; see also [5] for more on the history.) Wilfrid Hodges [11] later wrote up and generalized another proof (due also to Shelah) of the Singular Compactness Theorem, one which is more user-friendly than the original. A version of this proof, adapted to modules, is given in [7]. Recently the Singular Compactness Theorem (for Q-filtered modules, as in part III of Section 2) has proved an essential tool in the study of Baer modules and tilting modules (see the references in Section 5). So it seems useful to give a self-contained and streamlined exposition, based on the one in [7]. 0.2. Notation and terminology. An infinite cardinal λis singular if it is the supremum of a set Sof fewer than λcardinals each less than λ; the smallest possible cardinality of such an Sis the cofinality of λ. If it is not singular, λis called regular. Every successor cardinal is regular. For any sets Xand Y,X\Ydenotes their difference, i.e., {x∈X:x /∈Y}. A chain of sets {Xν:ν < ρ}is continuous if for each limit ordinal σ < ρ,Xσ=Sν<σ Xν. Throughout we consider left R-modules, where Ris an arbitrary ring. Given a module Mand a subset Yof M,hYidenotes the submodule of Mgenerated by Y.Mis λ-generated if it has a generating set of cardinality λ, and it is ≤λ-generated if it has a generating set of cardinality ≤λ. Shelah’s Singular Compactness Theorem 5 These notes are a revised and expanded version of lectures given at the CRM in early October 2006, as part of the Programme in “Discrete and Continuous Methods in Ring Theory”. I would like to thank the organizers for the invitation to participate, and the CRM, and especially Professor Dolors Herbera, for their support and hospitality. 1. Statement of the theorem Given a class Fof modules containing the zero module, a module M is called F-free if and only if Mbelongs to F.Since Fwill be fixed, we will usually simply say Mis “free” when we mean Mis F-free. Some examples of Fare given in the next section. The following is a precise version of enough < κ-generated submodules of Mbeing “free”. Definition 1.1. For any regular uncountable cardinal κ, define Mto be κ-F-free, or simply κ-“free” if there is a set Cof < κ-generated submodules of Msuch that: (1) every element of Cis “free”; (2) every subset of Mof cardinality < κ is contained in an element of C; and (3) Cis closed under unions of well-ordered chains of length < κ. Now we can state the general form of the Singular Compactness Theorem a little more precisely as follows: If λis a singular cardinal and Mis a λ-generated module which is κ-“free” for sufficiently many regular κ < λ, then Mis “free”. As was said in the introduction, some conditions must be imposed on the notion of “free”, that is, on the class F. So our next task is to introduce the assumptions on F; the reader may want to study these in parallel with the examples given in Section 2. The hypotheses (specifically 1.2(iii)) involve a parameter µ, an infinite cardinal which occurs in the statement of Theorem 1.4 below. They also involve two other primitive notions. One is that of a “basis” of an F-free module. More precisely, we are given for each M∈ F, a non-empty set, B(M), of sets of subsets of M(so if Y∈X∈ B(M), then Yis a subset of M). Each member Xof B(M) is called a “basis” of M. We say that a submodule Aof Mis a “free” factor of Mif A=hYi for some member Yof a “basis” Xof M. For each “free” factor Aof a 6 P. C. Eklof “free” module M, we are given a set D(A, M) of pairs of bases of Aand Mrespectively; we write X′=X↾Aif (X′,X)∈ D(A, M). 1.2. Hypotheses on F.For each M∈ F and each “free” factor A of M, there are non-empty sets B(M)⊆ P(P(M)) and D(A, M)⊆ B(A)× B(M) satisfying for some infinite cardinal µthe following conditions for all X∈ B(M): (i) ∅ ∈ X; if Y∈X, then hYiis “free”; (ii) Xis closed under unions of chains, i.e., if C⊆Xsuch that for all Y,Y′∈C,Y⊆Y′or Y′⊆Y, then SC∈X; (iii) if Y∈Xand a∈M, there exists Y′∈Xsuch that Y⊆Y′, a∈ hY′iand |Y′| ≤ |Y|+µ; (iv) if Z, Y ∈Xand Z⊆Y, then Zis a member of a “basis” of hYi, so hZiis a “free” factor of hYi; (v) if Ais a “free” factor of M, then for any “basis” X′of A, there is a “basis” Xof Msuch that X′=X↾A, i.e., (X′,X)∈ D(A, M); (vi) if {Mα:α < ρ}is a continuous chain of “free” modules and for each α+ 1 < ρ,Mαis a “free” factor of Mα+1, then Sα<ρ Mαis “free”; (vii) given a chain {Mn:n∈ω}of “free” modules s.t. for each n∈ω, Mnis a “free” factor of Mn+1, and given a “basis” Xnof each Mn such that Xn=Xn+1 ↾Mnfor all n∈ω, then Sn∈ωXnis contained in some “basis” of Sn∈ωMn. Proposition 1.3. If Fsatisfies 1.2 for µand Mis a λ-generated F-free module where λis an uncountable cardinal, then for any regular cardinal κsuch that µ < κ ≤λ,Mis κ-F-free. Proof: Let Xbe a “basis” of M. Let C={hYi:Y∈Xand |Y|< κ}. One can check easily that Definition 1.1 is satisfied for this C. This proposition shows that the hypothesis in the following theorem is necessary for Mto be free. The theorem says that the condition is sufficient when λis singular; it will follow immediately from the two theorems (3.1 and 4.1) proved in Sections 3 and 4. 1.4. The Singular Compactness Theorem. Suppose that Fsatisfies 1.2 for µ. Let λbe a singular cardinal > µ and let Mbe a λ-generated module such that Mis κ-F-free for all regular cardinals κsuch that µ < κ < λ. Then Mis F-free. Shelah’s Singular Compactness Theorem 7 2. Some examples We give three different types of examples; there is a non-empty intersection between the different classes of examples. I. The usual notion of free. Fis the class of free modules, that is, modules which have a linearly independent generating set, or, equivalently, are isomorphic to a direct sum of copies of R. Here µ=ℵ0. If M∈ F, we let B(M) consist of all Xsuch that there is a basis B of M(in the usual sense) such that Xis the set of all subsets of B. If Ais an F-free factor of M, let (X′,X)∈ D(A, M) if and only if X′={Z∈X:Z⊆A}. It is easy to verify the conditions in 1.2. II. Modules defined by direct sum decompositions. For a given set Lof ≤µ-generated modules, let Fconsist of all modules which are isomorphic to a direct sum of the form (†)M i∈I Li where each Li∈ L, and Iis an arbitrary (possibly empty) set. (When Iis empty, we obtain the zero module.) In particular, when Lis the set of countably-generated projective modules (and µ=ℵ0), Fis the class of all projective modules, by a theorem of Kaplansky. For each L∈ L, fix a generating set SLof cardinality ≤µfor L. For M∈ F, let B(M) consist of all sets Xsuch that there is an isomorphism ϕof Mwith a direct sum of the form (†) and (∗)X=(Y:∃J⊆Is.t. Y=ϕ−1"[ i∈J SLi#). (Here we abuse notation by identifying SLiwith its image under the canonical embedding of Lias the ith summand of M.) Note that if Yis as in (∗), then ϕinduces an isomorphism of hYiwith Li∈JLi. If Ais an F-free factor of M, let D(A, M) consist of all pairs (X′,X)∈ B(A)× B(M) such that X′={Z∈X:Z⊆A}. Then one can check that 1.2 is satisfied for the parameter µ. Indeed, 1.2(i), (ii) and (iii) are clear; regarding 1.2(iv), note that if Yis as above, Z∈X, and Z⊆Y, then Z=ϕ−1[Si∈J′SLi] for some J′⊆J. So Zis a member of the “basis” of hYidetermined by the isomorphism ϕof hYi with Li∈JLi. 8 P. C. Eklof Regarding 1.2(v), if A=hYiis a “free” factor of M, then there is an isomorphism θof Mwith A⊕M i∈(I\J) Li which is the identity on A; if X′is a “basis” of A, we can define a “basis” Xof Mto consist of all Wsuch that W=Z∪θ−1"[ i∈K SLi# for some Z∈X′and some subset K(possibly empty) of I\J. The last two parts of 1.2 are also easy to check. III. Q-filtered modules. For a set of modules Qsuch that 0 ∈ Q, define Mto be Q-filtered if Mis the union of a continuous chain {Mα: α < σ}s.t. M0= 0 and Mα+1/Mα∈ Q for all α+ 1 < σ. For a fixed set Q, let Fconsist of all Q-filtered modules. A continuous chain {Mα:α < σ}with M0= 0 which demonstrates that Mis “free”, i.e., Q-filtered, will be called a “free” chain for M. We claim that if Qconsists of ≤µ-presented modules, then Fsatisfies 1.2 for µ. (We follow the proof in [7].) First we need to define the auxiliary notions B(M) and D(A, M). For M∈ F, let B(M) consist of all sets Xsuch that there is a “free” chain {Mν:ν < σ}for Msuch that Y∈Xif and only if Y⊆Mand for all ν+1 < σ,hYi∩(Mν+1 \Mν)6=∅implies Mν+1 ⊆Mν+hYi. If Ais an F-free factor of M, let D(A, M) consist of all pairs (X′,X) such that there is a “free” chain {Mν:ν < σ}for Mwith A=Mν0for some ν0,Xis the “basis” for Mdetermined by this chain, and X′={Z∈X:Z⊆A}. To prove 1.2(i), assume Y∈Xand let A=hYi. Suppose that Xis determined by the “free” chain {Mν:ν < σ}; for all ν < σ, let Aν= A∩Mν. Since Y∈X, for all ν < σ, either Aν+1/Aν= 0 or Aν+1/Aν is isomorphic to Mν+1/Mν; in either case, the quotient belongs to Q, so {Aν:ν < σ}is a “free” chain witnessing that A∈ F. Notice also that the “basis” of Adetermined by this chain is {Z∈X:Z⊆A}, so 1.2(iv) follows. Condition 1.2(ii) is obvious. For 1.2(iii) we use the assumption that the members of Qare ≤µ-presented. Suppose that Xis determined by the “free” chain {Mν:ν < σ}, as in the definition of “basis”. Let Mσ= M. We prove by induction on ν≤σthat for any Y∈Xand any subset S of Mνof cardinality ≤µ, there is an element Y′of Xsuch that Y⊆Y′, |Y′| ≤ |Y|+µand S⊆ hY′i, and such that Y′=Yif S⊆ hYi. If ν= 0, Shelah’s Singular Compactness Theorem 9 there is nothing to prove. If νis a limit ordinal, define by induction on β < ν a chain of sets Yβin Xof cardinality ≤ |Y|+µsuch that Y⊆Y0 and hYβicontains S∩Mβ+1; since Xis closed under unions of chains, we can do this by the inductive hypothesis, and Y′=Sβ<ν Yβwill be the desired set. If ν=β+ 1, suppose first that hYi ∩ (Mβ+1 \Mβ)6=∅; then Mβ+1 ⊆Mβ+hYiby the definition of a “basis”. For each a∈S (⊆Mβ+1) such that a /∈ hYi, choose ya∈ hYisuch that a−ya∈Mβ. By induction there exists Y′∈Xsuch that Y⊆Y′,|Y′| ≤ |Y|+µand {a−ya:a∈S} ⊆ hY′i; hence S⊆ hY′i. If hYi ∩ (Mβ+1 \Mβ) = ∅ and S*hYi, it suffices to show that there exists ˜ Y⊇Yin Xsuch that |˜ Y| ≤ |Y|+µand D˜ YE∩(Mβ+1 \Mβ)6=∅, for then we are reduced to the first case. Since Mβ+1/Mβis isomorphic to a member of Q, there exists a generating set, G, of Mβ+1 over Mβof cardinality ≤µsuch that hGi∩Mβ is ≤µ-generated. By induction we can choose Y′′ in Xcontaining Ysuch that |Y′′| ≤ |Y|+µand hGi ∩ Mβ⊆ hY′′i. If hY′′i ∩ (Mβ+1 \Mβ)6=∅, let ˜ Y=Y′′. Otherwise, let ˜ Y=Y′′ ∪G; in this case we must show that ˜ Y∈X, in other words, for all ν < σ,D˜ YE∩(Mν+1 \Mν)6=∅implies Mν+1 ⊆Mν+D˜ YE. For ν=βthe conclusion follows by construction. In general suppose that y+g∈Mν+1 \Mνwhere y∈ hY′′iand g∈ hGi. If ν < β, then y+g∈Mβso ybelongs to Mβ+1 (since g∈Mβ+1) and hence y∈ hY′′i ∩ Mβ+1 ⊆Mβ; but then g∈Mβ∩ hGi ⊆ hY′′i; thus y+gshows that hY′′i∩(Mν+1 \Mν)6=∅and we are done since Y′′ ∈X. The final case is when ν > β; then y∈Mν+1 \Mνsince g∈Mβ+1 ⊆Mν and therefore Mν+1 ⊆Mν+hY′′isince Y′′ ∈X. This completes the proof of 1.2(iii). As for 1.2(v), suppose that A=hYiwhere Ybelongs to the “basis” determined by the “free” chain {Mν:ν < σ}. Suppose that X′is a “basis” for Adetermined by a “free” chain {A′ ρ:ρ < τ}for A. We will define by induction an extension {A′ ρ:ρ < τ′}of the chain {A′ ρ:ρ < τ} which will be a “free” chain for M. The extension will be defined to have the property that for all ρ≥τand ν < σ, (†)A′ ρ∩(Mν+1 \Mν)6=∅implies Mν+1 ⊆Mν+A′ ρ. Let A′ τ=A. If A′ ρhas been defined for all ρ≤βfor some β≥τ, choose γminimal such that Mγ+1 *A′ β. (If there is none, then A′ β=M and we stop the construction.) Then, by continuity, Mγ⊆A′ β. Set A′ β+1 =A′ β+Mγ+1. It follows from the choice of A′ β+1 and the inductive property (†) for ρ=βand ν=γthat the natural map Mγ+1/Mγ→ A′ β+1/A′ βis an isomorphism, so A′ β+1/A′ β∈ Q. (Note that the map 10 P. C. Eklof is one-one because otherwise (†) implies Mγ+1 ⊆Mγ+A′ β⊆A′ β, a contradiction.) One can then check that (†) holds for ρ=β+ 1 and all ν. Finally, if Xis the basis determined by the chain {A′ ρ:ρ < τ′}, then (X ′,X)∈ D(A, M). The proof of 1.2(v) shows that whenever Ais a “free” factor of M, any “free” chain for Acan be extended to a “free” chain for M. This allows us, given {Mα:α < ρ}as in 1.2(vi), to inductively define a continuous chain {Bν:ν < τ}whose union is Sα<ρ Mαand such that for every α < ρ, some initial segment of the chain {Bν:ν < τ}is a “free” chain for Mα. Finally, for 1.2(vii), we will show that there is a chain {Bν:ν < τ}such that for all n∈ω, some initial segment {Bν:ν < αn}is a “free” chain for Mnwhich determines Xn. It is then clear that the “basis” determined by this chain contains Sn∈ωXn. Suppose that we have constructed {Bν:ν < αn}for some n∈ω; by assumption, there is a “free” chain {Kν:ν < σ}for Mn+1 which determines Xn+1 and is such that Mn=Kν0for some ν0and Xn={Z∈Xn+1 :Z⊆Mn}. Let Bαn+ℓ=Kν0+ℓ for all ℓ≥0 such that ν0+ℓ < σ. 3. Proof of Theorem 1.4, part 1 In this section we will prove the following version of a singular compactness theorem: Theorem 3.1. Suppose that Fsatisfies 1.2 for µ. Let λbe a singular cardinal > µ and let Mbe a λ-generated module such that Mis strongly κ+-F-free for all cardinals κsuch that µ < κ < λ. Then Mis F-free. This theorem involves the following notion: Definition 3.2. For a cardinal κ, define Mto be strongly κ+-F-free, or simply strongly κ+-“free” if there is a family Sof ≤κ-generated “free” submodules of Mcontaining 0 and such that for any subset X of Mof cardinality κand any N∈ S, there exists N′∈ S such that N′⊇N∪Xand Nis a “free” factor of N′. Remark. A module which is strongly-κ+-F-free is not necessarily κ+-Ffree. (See [17] for a counterexample.) The terminology originally arose in the context of abelian groups, where the implication does hold. Shelah’s Singular Compactness Theorem 11 The rest of this section is devoted to the proof of the theorem. Let τ= cf(λ); so τ < λ since λis singular. Choose and fix a continuous increasing sequence of cardinals hκν:ν < τi, each strictly less than λ, whose supremum is λand such that κ0>max{τ, µ}. Choose a generating set Gfor Mof cardinality λand a continuous increasing chain {Gν: ν < τ}of subsets of Gwhose union is Gand such that the cardinality of Gνequals κν.We will define by induction on n∈ωsimultaneously, for all ν < τ, the following: •a subset Cn νof Mof cardinality κν; •a “free” submodule Fn νof Mwhich is ≤κν-generated; •Xn ν∈ B(Fn ν); •Yn ν∈Xn ν+1 of cardinality κν. We require the following for all n∈ωand ν < τ: 3.3. Properties. (0) Gν⊆Fn ν⊆ hCn νi ⊆ Fn+1 ν; (1) Fn νis a “free” factor of Fn+1 ν, and Xn ν=Xn+1 ν↾Fn ν; (2) Cn−1 ρ⊆Cn νfor all ρ≤ν; (3) Yn ν⊆Yn+1 ν⊆Cn+1 ν⊆Yn+3 ν; If we let Cν=Sn∈ωhCn νi, (2) implies that the Cνform a chain. We require also that (4) {Cν:ν < τ}is a continuous chain. Assuming, for the moment, that we can do the inductive construction, we will finish the proof. By 3.3(0), Cν=Sn∈ωFn νand Sν<τ Cν=M. By 1.2(vi) and (vii) and 3.3(1), Cνis “free” and Sn∈ωXn νis contained in a “basis” of Cν; call this “basis” Xν. By 3.3(3), Cνis generated by Sn∈ωYn νand by 1.2(ii), Sn∈ωYn ν∈Xν+1. Therefore, Cνis a “free” factor of Cν+1. But then, by 1.2(vi) and 3.3(4), M=Sν<τ Cνis “free”. It remains to do the inductive construction. For each ν < τ, fix a set Sνof ≤κν-generated “free” submodules of Mwhich witness that Mis strongly κ+ ν-“free”, as in Definition 3.2; we will choose Fn νto be a member of Sν. At stage nwe choose Fn ν,Xn ν,Cn−1 ν, and Yn νas well as a set {un ν,α :α < κν}of generators of Fn ν. We begin with n= 0: Pick F0 ν∈ Sνso that it contains Gνand is ≤κν-generated. Pick X0 ν∈ B(F0 ν). Let C−1 ν=∅=Y0 ν. 18 P. C. Eklof Department of Mathematics University of California, Irvine (UCI) Irvine, CA 92697-3875 USA E-mail address:[email protected] Primera versi´o rebuda el 14 de febrer de 2007, darrera versi´o rebuda el 13 de juny de 2007.