The complexity of the embeddability relation between torsion-free abelian groups of uncountable size
aa r X i v : . [ m a t h . L O ] A ug THE COMPLEXITY OF THE EMBEDDABILITY RELATIONBETWEEN TORSION-FREE ABELIAN GROUPS OFUNCOUNTABLE SIZE
FILIPPO CALDERONI
Abstract.
We prove that for every uncountable cardinal κ such that κ <κ = κ , the quasi-order of embeddability on the κ -space of κ -sized graphs Borelreduces to the embeddability on the κ -space of κ -sized torsion-free abeliangroups. Then we use the same techniques to prove that the former Borelreduces to the embeddability on the κ -space of κ -sized R -modules, for every S -cotorsion-free ring R of cardinality less than the continuum. As a consequencewe get that all the previous are complete Σ quasi-order. Introduction
A subset of a topological space is κ -Borel if it is in the smallest κ -algebra con-taining the open sets. Given two spaces X, Y , a function f : X → Y is κ -Borel(measurable) if the preimage through f of every open subset of Y is κ -Borel. Twospaces X, Y are said κ -Borel isomorphic if there is a κ -Borel bijection X → Y whose inverse is κ -Borel too. When κ = ℵ these notions coincide with the ones ofBorel sets, Borel functions, Borel isomorphism (see the classical reference [Kec95]).More background details and examples will be given in the next section.Let κ be an infinite cardinal. A topological space X is a κ -space if it admitsa basis of size ≤ κ . We denote by κ κ the generalized Baire space ; i.e., the set offunctions from κ to itself endowed with the topology generated by the sets of thosefunctions extending a fixed function from a bounded subset of κ to κ . We assumethe hypothesis κ <κ = κ , which implies that κ κ is a κ -space. A κ -space is standardBorel if it is κ + -Borel isomorphic to a κ + -Borel subset of κ κ . If X is a standardBorel κ -space, we say that A ⊆ X is κ -analytic (or Σ ) if it is a continuous imageof a closed subset of κ κ . The set of κ -analytic subsets of X is usually denotedby Σ ( X ) . We are interested in quasi-orders (i.e., reflexive and transitive binaryrelations) defined over standard Borel κ -spaces. A quasi-order Q on X is analytic ,or Σ , if and only if Q ∈ Σ ( X × X ) . Let P, Q be Σ quasi-orders on the standard Date : August 6, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Borel reducibility; torsion-free abelian groups; generalized descriptiveset theory.The author is very grateful to Adam J. Przeździecki for explaining some crucial parts of [Prz14]and for some invaluable discussions which eventually lead to the proof of the main results of thispaper. The author would like to thank Tapani Hyttinen for some useful comments that contributedto the last section. Moreover, the author thanks Andrew Brooke-Taylor, Raphaël Carroy, SimonThomas, and Matteo Viale for comments and interesting discussions. The results presented in thisarticle will be part of the PhD thesis of the author and carried out under the supervision of LucaMotto Ros. This work was supported by the “National Group for the Algebraic and GeometricStructures and their Applications” (GNSAGA–INDAM).
Borel κ -spaces X and Y , respectively. We say that P Borel reduces to Q if there isa κ + -Borel function f : X → Y such that ∀ x x ∈ X ( x P x ⇔ f ( x ) Q f ( x )) . Theorem 1.1 (essentially Williams [Wil14]) . For every infinite cardinal κ suchthat κ <κ = κ , the embeddability quasi-order on the κ -space of κ -sized graphs Borelreduces to the embeddability on the κ -space of κ -sized groups. The above result was proved in [Wil14, Theorem 5.1] for κ = ω , and the sameproof works for κ uncountable as well. In view of a result of Louveau and Rosendal(see [LR05, Theorem 3.1]), Theorem 1.1 in case κ = ω yields that the embeddabilityrelation on countable groups is a complete Σ quasi-order (i.e., a maximum amongall Σ quasi-orders up to Borel reducibility). To prove Theorem 1.1, Williamsmaps every countable graph to a group generated by the vertices of the graph andsatisfying some small cancellation hypothesis. Such groups are not abelian and havemany torsion elements that are used to encode the edge relation of the correspondinggraphs. So one may wonder whether there exists another Borel reduction fromembeddability between countable graphs to embeddability on countable torsion-free abelian groups.At about the same time as Theorem 1.1 was proved, Przeździecki showed in[Prz14] that the category of graphs almost-fully embeds into the category of abeliangroups. I.e., there exists a functor G : G raphs → A b such that for every two graphs T, V there is a natural isomorphism Z [Hom( T, V )] ∼ = Hom( GT, GV ) , where Z [ B ] is defined as the free abelian group with basis B . A closer look into theconstruction of the functor reveals that it takes values in the subcategory of torsion-free abelian groups. Unfortunately for us, the restriction of G to the standard Borelspace of countable graphs is not a Borel reduction in the classical sense becausecountable graphs are sent to groups of size the continuum.In this paper we work in the framework of generalized descriptive set theory. Bytweaking the construction of [Prz14] we show the following. Theorem 1.2.
For every uncountable κ such that κ <κ = κ , there is a Borelreduction from the quasi-order of embeddability on the κ -space of graphs of size κ to the embeddability on the κ -space of κ -sized torsion-free abelian groups. So, in view of the results of [MR13] and [CMMR], the embeddability relationbetween κ -sized torsion-free abelian groups is a complete Σ quasi-order.In Section 2 we introduce the main definitions of Borel reducibility in the frame-work of generalized descriptive set theory. In Section 3 we recall some theoremson the existence of R -modules with prescribed endomorphism ring. Such theoremswill be used to define the reductions we present in the ensuing sections. Section4 is dedicated to the proof of Theorem 1.2: we define a Borel reduction from theembeddability relation on κ -sized graphs to embeddability on κ -sized torsion-freeabelian groups. In Section 5 we follow the main ideas of [GP14], and we exploitthe techniques used in Section 4 to prove an analogue of Theorem 1.2 for the em-beddability relation on R -modules, for every S -cotorsion-free ring R of cardinalityless than the continuum. Theorem 1.3.
Let R be a commutative S -cotorsion-free ring of cardinality lessthan the continuum. For every uncountable κ such that κ <κ = κ , there is a Borel MBEDDABILITY ON TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE 3 reduction from the embeddability quasi-order on the κ -space of graphs of size κ toembeddability on the κ -space of κ -sized R -modules. In Section 6 we address the problem of determining the Borel complexity of ∼ = κ TFA the isomorphism on κ -sized torsion-free abelian groups. We point out that a resultof [HM17] implies the following. Theorem 1.4.
Assume that
V = L and κ is inaccessible. Then the isomorphismrelation on κ -sized torsion-free abelian groups is a complete Σ equivalence relation. Preliminaries
We consider the generalized Baire space κ κ := { x | x : κ → κ } for an uncountablecardinal κ . Unless otherwise specified, κ κ is endowed with the bounded topology τ b ,i.e., the topology generated by the basic open sets N s = { x ∈ κ κ | x ⊇ s } , where s ∈ <κ κ . Notice that since κ is uncountable, the topology τ b is strictly finerthan the product topology. We assume the hypothesis(2.1) κ <κ = κ, consequently we get that κ κ is a κ -space, as the basis { N s | s ∈ <κ κ } has size κ .The generalized Cantor space κ { x ∈ κ κ | x : κ → } is a closed subset of κ κ andtherefore it is standard Borel with the relative topology. We recall the followingproposition which gives some characterizations of κ -analytic sets. Proposition 2.1.
Let X be a standard Borel κ -space and A ⊆ X nonempty. Then,the following are equivalent: (i) A is κ -analytic; (ii) A is a continuous image of some κ + -Borel B ⊆ κ κ ; (iii) A is a κ + -Borel image of some κ + -Borel B ⊆ κ κ ; (iv) A is the projection p ( F ) = { x ∈ X | ∃ y ∈ κ κ (( x, y ) ∈ F ) } of some closedsubset F ⊆ X × κ κ . A proof of Proposition 2.1 is given in [MR13, Section 3]. It is specially worth tonote that in view of (iv) we are allowed to use a generalization of the celebratedTarski-Kuratowski algorithm (see [Kec95, Appendix C]). That is, a set A ⊆ κ κ is κ -analytic if it is defined by an expression involving only κ -analytic sets, atomicconnectives, ∃ α, ∀ α (where α varies over a set of cardinality ≤ κ ), and existentialquantification over a standard Borel κ -space.In the remainder of this paper we fix some uncountable κ and study standardBorel κ -spaces with the assumption κ <κ = κ . For ease of exposition we simply sayBorel and analytic instead of saying respectively κ + -Borel and κ -analytic, whenever κ is clear from the context.2.1. Spaces of κ -sized structures. In this subsection we recall briefly how todefine the standard Borel κ -spaces of uncountable structures of size κ . Whilein [AMR16, FHK14, MR13] the authors are concerned only with countable lan-guages, we extend the basic definitions to uncountable ones. Our approach is moti-vated by the aim to develop a unique framework to treat algebraic objects with themost diverse features. For example, following our approach it is possible to definethe standard Borel κ -space of κ -sized R -modules, for any fixed ring R with | R | < κ .Such spaces will be taken into account in Section 5. F. CALDERONI
Definition 2.2. If A is a set of size κ , then any bijection f : κ → A induces abijection from κ κ to A κ , so that the bounded topology can be copied on A κ . A basisfor such topologies is given by { N As | ∃ α < κ ( f ′′ α = dom s ) } , where N As = { x ∈ A κ | s ⊆ x } .We briefly recall some useful applications of Definition 2.2.(a) If G is a group of cardinality κ then, we define the κ -space of subgroups of G by identifying each subgroup of G with its characteristic function and setting Sub G = { H ∈ G | G ∈ H ∧ ∀ x, y ∈ G ( x, y ∈ H → xy − ∈ H ) } , which is a closed subset of G and therefore is standard Borel.(b) Fix a language consisting of finitary relation symbols L = { R i | i ∈ I } , | I | < κ ,and let n i be the arity of R i . We denote by X κL the κ -space of L -structureswith domain κ . Every A ∈ X κL is a pair ( κ, { R A i | i ∈ I } ) where each R A i isan n i -ary relation on κ , so it can be identified with an element of Q i ∈ I ( ni κ ) in the obvious way. It follows that X κL can be endowed with the product ofthe bounded topologies on its factors ( ni κ ) .For an infinite cardinal κ , we consider the infinitary logic L κ + κ . In such logicformulas are defined inductively with the usual formation rules for terms, atomicformulas, negations, disjunctions and conjunctions of size ≤ κ , and quantificationsover less than κ many variables. Definition 2.3.
Given an infinite cardinal κ and an L κ + κ -sentence ϕ , we define the κ -space of κ -sized models of ϕ by X κϕ := {A ∈ X κL | A | = ϕ } . The following theorem is a generalization of a classical result by López-Escobarfor spaces of uncountable structures.
Theorem 2.4 ( κ <κ = κ ) . A set B ⊆ X κL is Borel and closed under isomorphismif and only if there is an L κ + κ -sentence ϕ such that B = X κϕ . To see a proof of Theorem 2.4 we refer the reader to [FHK14, Theorem 24]or [AMR16, Theorem 8.7]. A straightforward consequence of it is that the spacedefined in Definition 2.3 is standard Borel.We conclude this subsection with a list of those spaces of models of L κ + κ -sentences that we will use in the ensuing sections. We denote by X κ GRAPHS the space of κ -sized graphs . By graph we mean an undirected graph whose edge re-lation is irreflexive. We denote by X κ TFA the space of κ -sized torsion-free abeliangroups . For any fixed ring R such that | R | < κ , we denote by X κR - MOD , the space of κ -sized R -modules . Here observe that every r ∈ R is regarded as a unary functionalsymbol, interpreted as the left scalar multiplication by r . The axioms of R -modulesare the following • ϕ AB , i.e., the first order formula defining abelian groups, • ∀ x ∀ y (cid:0) r ( x + y ) = rx + ry (cid:1) , • ∀ x (cid:0) ( r + q ) x = rx + qx (cid:1) , • ∀ x (cid:0) r ( qx ) = ( rq ) x (cid:1) , • x = x . MBEDDABILITY ON TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE 5
Since r and q vary in R , which has size < κ , the formula defining the class of R -modules is a formula in the logic L κ + κ .2.2. Borel reducibility.
Let
X, Y be standard Borel κ -spaces and L a fixed lan-guage such that | L | < κ . Given A , B ∈ X κL , we say that A is embeddable into B , in symbols A ⊑ κL B , if there is x ∈ κ κ which realizes an isomorphism between A and B ↾ Im x . As pointed out in [AMR16, Section 7.2.2] one can show di-rectly that ⊑ κL is the projection on X κL × X κL of a closed subset of X κL × X κL × κ κ ,therefore the quasi-order ⊑ κL of embeddability between κ -sized L -structures is Σ .We denote by ⊑ κ Z the quasi-order of embeddability on the κ -space X κ Z , where Z ∈ { GRAPHS , TFA , R - MOD } .Let P, Q be Σ quasi-orders on the standard Borel κ -spaces X and Y , respec-tively. Recall that P Borel reduces to Q , in symbols P ≤ B Q , if and only if there isa κ + -Borel function f : X → Y such that ∀ x x ∈ X ( x P x ⇔ f ( x ) Q f ( x )) . Moreover, Q is a complete Σ quasi-order if for every Σ quasi-order P on a stan-dard Borel κ -space, P ≤ B Q . Similarly, we say that E is a complete Σ equivalencerelation if F ≤ B E , for every Σ equivalence relation F . Any quasi-order Q on X induces canonically an equivalence relation on X , which is denoted by E Q , anddefined by setting x E Q y if and only if x Q y and y Q x for all x, y ∈ X . It canbe easily verified that if Q is a complete Σ quasi-order then E Q is a complete Σ equivalence relation. The following theorem is a generalization of [LR05, Theorem3.1] to the embeddability relation on uncountable graphs. Theorem 2.5 (Mildenberger-Motto Ros [CMMR]) . If κ is uncountable such that κ <κ = κ , then the relation of embeddability ⊑ κ GRAPHS on the κ -space of κ -sizedgraphs is a complete Σ quasi-order. A first version of Theorem 2.5 was obtained by Motto Ros in [MR13, Corollary9.5] provided that κ is weakly compact.3. Existence of algebras with prescribed endomorphism ring
Some crucial results that will be used in the next sections are about the existenceof algebras with prescribed endomorphism ring.3.1. S -completions. We recall the basic definitions and some simple facts on S -completions following the treatise of [GT12, Chapter 1]. Notation 3.1.
In the remainder of this section R is a commutative ring with unit and S is a subset of R \ { } containing and closed under multiplication. Definition 3.2.
We say that R is S -reduced if T s ∈ S sR = 0 , and R is S -torsion-free if for all s ∈ S and r ∈ R , sr = 0 implies r = 0 . Further, we say that R is an S -ring provided that R is both S -reduced and S -torsion-free.In most of the applications S is assumed to be countable. In such case, we denote S by S as in [GT12, Chapter 1]. Examples of S -rings include every noetheriandomain R with S = { a n | n ∈ ω } , for any a ∈ R such that aR is a proper principalideal of R (see [GT12, Corollary 1.3]). Definition 3.3.
Let R be an S -ring and M be an R -module. We say that M is S -reduced if T s ∈ S sM = 0 , and M is S -torsion-free if for every s ∈ S and m ∈ M , sm = 0 implies m = 0 . F. CALDERONI
We denote by c M the S -completion of M , which is defined as follows. Given s, q ∈ S , we write q (cid:22) s if there is t ∈ S such that s = qt . Then we set c M := lim ←− s ∈ S M/sM, the inverse limit of inverse system of R -modules ( { M/sM } s ∈ S , { π sq } q (cid:22) s ) , where π sq : M/sM → M/qM, m + sM m + qM. Any R -module can be given the natural linear S -topology , i.e., the one generatedby { sM | s ∈ S } as a basis of neighborhoods of . A Cauchy net in M is a sequence ( m s | s ∈ S ) taking values in M , and such that m q − m qs ∈ qM , for all q, s ∈ S . Wesay that the Cauchy net ( m s | s ∈ S ) has limit m ∈ M if and only if m − m s ∈ sM ,for every s ∈ S . Finally, we say that M is S -complete if it is complete with respectto the S -topology, that is, every Cauchy net in M has a unique limit. If M is S -reduced and S -torsion-free, then c M is S -reduced, S -torsion-free and S complete(see [GT12, Lemma 1.6]).The canonical map η M : M → c M , m ( m + sM | s ∈ S ) , which is always a homomorphism of R -modules, is injective if and only if M is S -reduced; and it is a ring homomorphism, whenever R = M . Moreover, if M is S -complete, then η M is an isomorphism.For any R -module M , its S -completion c M carries a natural a b R -module struc-ture. I.e., given ¯ r = ( r s + sR | s ∈ S ) ∈ b R and ¯ m = ( m s + sM | s ∈ S ) ∈ c M , wedefine the scalar multiplication by ¯ r ¯ m := ( r s m s + sM | s ∈ S ) . Existence of abelian groups with prescribed endomorphism ring. If G is a Z -module (i.e., an abelian group) and S = N \ { } , then the S -topologyon G is usually called Z -adic topology and the S -completion of G is called the Z -adic completion . We refer to [Fuc70, Theorem 39.5] and [Fuc15, Section 2.7] ascomprehensive sources on Z -adic completions.The next theorem was pointed out by Przeździecki in [Prz14]. It states a slightlydifferent result of a classical theorem by Corner [Cor63, Theorem A], whose proofcan be adapted to show the following. Theorem 3.4 (Przeździecki [Prz14, Theorem 2.3]) . Let A be a ring of cardinalityat most ℵ such that its additive group is free. Then, there is a torsion-free abeliangroup M ⊆ b A such that (i) A ⊆ M as (left) A -modules, (ii) End M ∼ = A , (iii) | A | = | M | . A few comments on Theorem 3.4 may be of some help. We stress the fact that M is torsion-free. By construction M inherits the natural (left) A -module structurefrom b A , which is the one defined by setting(3.1) a ∗ m = η A ( a ) m, for every a ∈ A and m ∈ M . Moreover, condition (ii) of Theorem 3.4 is proved byshowing that for every h ∈ End M , there exists a ∈ A such that h ( m ) = a ∗ m forall m ∈ M . MBEDDABILITY ON TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE 7
Remark . The reader familiar with the bibliography may find our notation non-standard. In module theory people usually consider endomorphisms of R -modulesas acting on the opposite side from the scalars (e.g., see [GT12, Chapter 1]). Nev-ertheless, we prefer to follow the notation of [Prz14] and to have endomorphisms of M acting on the left.3.3. Existence of R -modules with prescribed endomorphism ring. An R -module M is S -cotorsion-free if it is S -reduced and Hom( b R, M ) = 0 . This definitionextends naturally to any R -algebra A by saying that A is S -cotorsion-free if A hasthis property as an R -module. It is shown in [GT12, Corollary 1.26] that whenever M is an R -module of size < ℵ , then M is S -cotorsion-free if and only if it is S -torsion-free and S -reduced. We recall one more theorem of existence of R -algebraswith prescribed endomorphism ring, that is a result of the same kind of Theorem 3.4. Theorem 3.6 (Göbel-Przeździecki [GP14, Corollary 4.5]) . Let R be an S -cotorsion-free ring such that | R | < ℵ . If A is an R -algebra of cardinality at most thecontinuum with a free additive structure over R , then there exists an R -module M such that: (i) A ⊆ M ⊆ b A as (left) A -modules, (ii) End R M ∼ = A , (iii) | A | = | M | . The embeddability relation between torsion-free abelian groups
In this section we focus on the embeddability relation between κ -sized torsion-free abelian groups and we prove Theorem 1.2. To this purpose we adapt theembedding from the category of graphs into the category of abelian groups definedin [Prz14]. For the sake of exposition we avoid the notion of colimit commonlyused in category theory, instead we use the classical notion of direct limit for directsystems of abelian groups which gives more insights on the possibility to define thereduction in a κ + -Borel way.Let Γ be a skeleton of the category of countable graphs; i.e., a full subcategoryof the category of countable graphs with exactly one object for every isomorphismclass. Without loss of generality, assume that every object in Γ is a graph over asubset of ω . Since we work under the assumption (2.1), there is a κ -sized universalgraph , i.e., a graph of cardinality κ , which contains all graphs of cardinality κ asinduced subgraphs. We denote by W κ the κ -sized universal graph on κ and by [ W κ ] κ the subspace of induced subgraphs of W κ of cardinality κ . We identify [ W κ ] κ withthe κ + -Borel subset of subsets of W κ of cardinality κ . Therefore we can consider [ W κ ] κ as the standard Borel κ -space of graphs of cardinality κ .For every graph T and every infinite cardinal λ , we denote by [ T ] <λ the set ofinduced subgraphs of T of cardinality < λ . Next, for every S ∈ [ W κ ] <ω , we fixan isomorphism θ S : S → σ ( S ) , where σ ( S ) denotes the unique graph in Γ which isisomorphic to S .Now define(4.1) A := Z (cid:2) Arw(Γ) ∪ { } ∪ P fin ( ω ) (cid:3) . That is, the free abelian group generated by the arrows in Γ , a distinguished element , and the finite subsets of ω . We endow A with a ring structure by multiplying F. CALDERONI the elements of the basis as follows, and then extending the multiplication to thewhole A by linearity. For every a, b ∈ Arw(Γ) ∪ P fin ( ω ) let ab = a ◦ b if ( a, b ∈ Arw(Γ) a and b are composable a ′′ b if ( b ⊆ dom aa ↾ b is an isomorphism otherwise(4.2) a a = a. (4.3) Remark . The definition of A in (4.1) differs from the one of [Prz14, Section 3]for including P fin ( ω ) in the generating set. These elements will play the crucialrole of embeddability detectors in Lemma 4.8.Now observe that the ring A has cardinality ℵ and its additive group is free.So let M be a group having endomorphism ring isomorphic to A as in Theorem 3.4.Notice that the elements of A act on M on the left as in (3.1). Definition 4.2.
For every C ∈ Γ , let G C := id C ∗ M. Notice that G C is a subgroup of M , for all C ∈ Γ . Moreover, if C, D ∈ Γ and γ : C → D , then γ ∈ A and thus induces a group homomorphism Gγ from G C to G D by left-multiplication(4.4) Gγ : G C → G D id C ∗ m γ ∗ ( id C ∗ m ) . We make sure that such map is well defined as γ ∗ ( id C ∗ m ) = id D ∗ ( γ ∗ ( id C ∗ m )) ,which is clearly an element of G D .Now fix any T ∈ [ W κ ] κ . For every S, S ′ ∈ [ T ] <ω such that S ⊆ S ′ , the inclusionmap i SS ′ : S → S ′ induces a map γ SS ′ from σ ( S ) to σ ( S ′ ) , the one that makes thediagram below commute. S S ′ σ ( S ) σ ( S ′ ) i SS ′ θ S θ S ′ γ SS ′ The map γ SS ′ is in Γ and thus it induces functorially a group homomorphism Gγ SS ′ as described in (4.4). For all S, S ′ ∈ [ T ] <ω such that S ⊆ S ′ , let τ SS ′ = Gγ SS ′ . Weclaim that ( { G σ ( S ) } , { τ SS ′ } S ⊆ S ′ (cid:1) S,S ′ ∈ [ T ] <ω is a direct system of torsion-free abeliangroups indices by the poset [ T ] <ω , which is ordered by inclusion. Definition 4.3.
For every T ∈ [ W κ ] κ , let (4.5) GT := lim −→ S ∈ [ T ] <ω G σ ( S ) . As the referee kindly pointed out the notation GT instead of G ( T ) or G T is nonstandard indescriptive set theory. Nevertheless we prefer to stick to the common practice in category theoryto denote functors by juxtaposition. MBEDDABILITY ON TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE 9
For the sake of definiteness, every element of the direct limit in (4.5) is regardedas the equivalence class [( m, S )] of an element of the disjoint union F S ∈ [ T ] <ω G σ ( S ) factored out by the equivalence relation ∼ T , which is defined by setting ( m, S ) ∼ T ( m ′ , S ′ ) provided that there is S ′′ ⊇ S, S ′ such that τ SS ′′ ( m ) = τ S ′ S ′′ ( m ′ ) . Suchcharacterization for (4.5) holds because the poset of indexes is directed (see [Rot09,Corollary 5.31]).Notice that for every T ∈ [ W κ ] κ , the group GT is abelian by definition, andit is torsion-free as torsion-freeness is preserved by taking subgroups and colimits.Moreover, we claim that GT has cardinality κ . It is clear that | GT | is bounded by | F κ M | = κ . Further, we observe that each GT has at least κ distinct elements.To see this, consider id C , where C stands for the unique graph with one vertexand no edges in Γ . For sake of definiteness, suppose that C is the graph with noedges whose unique vertex is . For every α < κ , let { α } denote the subgraph of T with the only vertex α . It is clear that id C ∈ G σ ( { α } ) . Moreover, for any distinct α, β ∈ κ , we have that ( id C , { α } ) and ( id C , { β } ) represent two distinct elementsof G T . For, if S ⊇ { α } , { β } , then one has τ { α } S ( id C ) = γ { α } S id C = (0 θ S ( α )) τ { β } A ( id C ) = γ { β } S id C = (0 θ S ( β )) , which are not equal as θ S is bijective.The next lemma basically states that G can be defined in a κ + -Borel way. Lemma 4.4.
There is a κ + -Borel map [ W κ ] κ → X κ TFA T
7→ G T such that, for every T ∈ [ W κ ] κ , the group G T is isomorphic to GT .Proof. Let (cid:22) be a well-ordering of B = F S ∈ [ W κ ] <ω G σ ( S ) . First consider the map f : [ W κ ] κ → B , T G S ∈ [ T ] <ω G σ ( S ) . To see that f is κ + -Borel consider the subbasis of B given by the sets { x : B → | x (( m, S )) = 1 } and { x : B → | x (( m, S )) = 0 } , for every ( m, S ) ∈ B . For anyfixed ( m , S ) ∈ B , one has f − ( { x : B → | x (( m , S )) = 1 } ) = { T ∈ [ W κ ] κ | S ⊆ T } which is κ + -Borel.Then let g : Im f → B be the map defined by mapping f ( T ) to the subset of f ( T ) which is obtained by deleting all of the ( m, S ) that are ∼ T -equivalent (i.e.,equivalent in the relation used to define the direct limit indexed by [ T ] <ω ) to somepoint appearing before in the well-ordering (cid:22) . One has g ( f ( T ))(( m, S )) = 1 ⇐⇒ S ⊆ T ∧ ∀ ( m ′ , S ′ ) ≺ ( m, S )(( m ′ , S ′ ) ≁ T ( m, S )) where ( m ′ , S ′ ) ≁ T ( m, S ) is a shorthand for ∄ S ′′ ⊇ S, S ′′ ( τ SS ′′ ( m ) = τ S ′ S ′′ ( m ′ )) . Then, for every T , we define a group G T with underlying set κ and operation ⋆ T by setting α ⋆ T β = γ if and only if the product of the α -th element and the β -thelement in g ( f ( T )) according to (cid:22) is ∼ T -equivalent to the γ -th element in g ( f ( T )) . Notice that there is a unique element in g ( f ( T )) which is ∼ T -equivalent to suchproduct, thus the map T
7→ G T is well defined and is κ + -Borel. (cid:3) Next lemma is derived essentially as in [Prz14, Lemma 3.6].
Lemma 4.5. If T, V ∈ X κ GRAPHS and T ⊑ κ GRAPHS V , then GT ⊑ κ TFA GV .Proof. We first claim that if
C, D ∈ Γ and γ : C → D is an embedding then Gγ : G C → G D is one-to-one. Notice that by (i) of Theorem 3.4 and the definitionof G C one obtains Z [Γ C ∪ P fin ( C )] ⊆ G C ⊆ V Z [Γ C ∪ P fin ( C )] . Acting by left-multiplication, γ induces the injective map h γ i : Z [Γ C ∪ P fin ( C )] → Z [Γ D ∪ P fin ( D )] , a γa, which in turn induces the injective map on the Z -adic completions(4.6) c h γ i : V Z [Γ C ∪ P fin ( C )] → V Z [Γ D ∪ P fin ( D )] , ¯ a γ ∗ ¯ a. Comparing (4.6) with (4.4) it follows that Gγ is indeed the restriction of c h γ i on G C , which implies that Gγ is injective because so is c h γ i .Now let ϕ : T → V be a graph embedding. Then there exists a group homomor-phism Gϕ : GT → GV, [( g, S )] [( Gγ Sϕ ′′ S ( g ) , ϕ ′′ S )] , where ϕ ′′ S is the point-wise image of S through ϕ and γ Sϕ ′′ S : σ ( S ) → σ ( ϕ ′′ S ) is themap induced by ϕ ↾ S , which is clearly a graph embedding. We are left to prove that Gϕ is one-to-one. So fix any [( g, S )] , [( g ′ , S ′ )] ∈ GT such that [( g, S )] = [( g ′ , S ′ )] .By directedness of [ T ] <ω we can assume that S = S ′ without any loss of generality.One has Gϕ ([( g, S )]) = [( Gγ Sϕ ′′ S ( g ) , ϕ ′′ S )] ,Gϕ ([( g ′ , S )]) = [( Gγ Sϕ ′′ S ( g ′ ) , ϕ ′′ S )] , which are different elements of GV because Gγ Sϕ ′′ S is injective. (cid:3) Now we are left to prove that GT ⊑ κ TFA GV implies that T ⊑ κ GRAPHS V . Givenany linear combination P k i ϕ i , k i ∈ Z and ϕ i ∈ Hom(
T, V ) , one can define a grouphomomorphism Ψ( P k i ϕ i ) : GT → GV as follows. For any ϕ i and S ∈ [ T ] <ω , let δ Si be the function such that the diagram commutes S ϕ ′′ i Sσ ( S ) σ ( ϕ ′′ i S ) ϕ i ↾ Sθ S θ ϕ ′′ i S δ Si MBEDDABILITY ON TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE 11
Since δ Si is an arrow in Γ , it induces a group homomorphism Gδ Si : G σ ( S ) → G σ ( ϕ ′′ i S ) m δ Si ∗ m. as observed in (4.4). Then we define Ψ( X k i ϕ i ) : GT → GV [( m, S )] X k i [( Gδ Si ( m ) , ϕ ′′ i S )] . Theorem 4.6 (Przeździecki [Prz14, Theorem 3.14]) . There is a natural isomor-phism
Ψ : Z [Hom( T, V )] ∼ = −→ Hom(
GT, GV ) . Remark . Theorem 4.6 states that G is an almost-full embedding, according tothe terminology of [Prz14, GP14]. It can be proved arguing as in [Prz14, Section3]. Now we come to the point where our modification from (4.2) becomes crucial.Since A contains the finite subsets of ω , we use them and the property of almost-fullness of G to detect an embedding among ϕ , . . . , ϕ n when Ψ( P i ≤ n k i ϕ i ) isone-to-one. Lemma 4.8.
For every two graphs T and V in X κ GRAPHS , if GT ⊑ κ GROUPS GV holdsthen T ⊑ κ GRAPHS V .Proof. Let
T, V be as in the hypothesis and h : GT → GV a group embedding. ByTheorem 4.6 we have h = Ψ( X i ∈ I k i ϕ i ) , for some linear combination of graph homomorphisms ϕ i ∈ Hom(
T, V ) . We claimthat there must be some i ∈ I such that ϕ i is a graph embedding from T into V .Suppose that it is not true, aiming for a contradiction. Since [ T ] <ω is directed,there is some finite S ∈ [ T ] <ω such that, for every i ∈ I , the restriction map ϕ i ↾ S is not one-to-one or does not preserve non-edges. Call d the vertex set of σ ( S ) .Such d is a finite subset of ω and is an element of G σ ( S ) because d = id σ ( S ) ∗ d .Now consider [( d, S )] , the element of GT represented by d ∈ G σ ( S ) . Then [( d, S )] is a nontrivial element and h ([( d, S )]) = X k i [( Gδ Si ( d ) , ϕ ′′ i S )] == X k i [( δ Si ∗ d, ϕ ′′ i S )] = 0 because if ϕ i ↾ S is not an embedding then neither is the induced map δ Si . Thiscontradicts the fact that h is one-to-one. (cid:3) Summing up the results of this section we can prove the main theorem.
Proof of Theorem 1.2.
In view of the Lemma 4.4, we can assume that G is κ + -Borel. By Lemma 4.5, G is a homomorphism from ⊑ κ GRAPHS to ⊑ κ TFA , and Lemma4.8 yields that G is a reduction. (cid:3) Corollary 4.9.
For every uncountable κ such that κ <κ = κ , the embeddabilityrelation ⊑ κ TFA on the κ -space of κ -sized torsion-free abelian groups is a complete Σ quasi-order. Proof.
Combining Theorem 1.2 with Theorem 2.5, it follows that ⊑ κ TFA is a complete Σ quasi-order provided that κ <κ = κ holds. (cid:3) It is worth mentioning that the analogue of Corollary 4.9, where κ = ω , has beenproved recently in [CT].5. The embeddability relation between R -modules In this section we use the second Corner’s type theorem stated in section 3 toprove that the quasi-order of embeddability between R -modules, for any S -ring R of cardinality less than the continuum, is a complete Σ quasi-order. Proof of Theorem 1.3.
Let Γ , W κ , and σ be as in section 4. Define(5.1) A := R (cid:2) Arw(Γ) ∪ { } ∪ P fin ( ω ) (cid:3) . That is, the free R -module generated by the arrows in Γ , a distinguished element and the finite subsets of ω . The R -module A can be endowed a ring structure bydefining a multiplication on the element of its basis as in (4.2). Such multiplicationis compatible with the R -module structure, therefore we can regard A as an R -algebra. Notice that A has cardinality the continuum so we apply Theorem 3.6which yields the existence of an R -module M ∼ = End R A . We continue defining G similarly to how we did in Section 4. That is, for every C ∈ Γ , G C := id C ∗ M, and for every T ∈ [ W κ ] κ , let GT := lim −→ S ∈ [ T ] <ω G σ ( S ) . Since | M | = | A | , for every T ∈ [ W κ ] κ , GT has size κ . Then one can argue asin Lemmas 4.4, 4.5, and 4.8 to prove that G is a Borel reduction from ⊑ κ GRAPHS to ⊑ κR - MOD . In this case the almost-fullness for G was proved essentially in [GP14,Theorem 3.16] with the same argument used in [Prz14]. (cid:3) Corollary 5.1.
For every uncountable cardinal κ such that κ <κ = κ and everyring R as in the statement of Theorem 1.3, the embeddability relation ⊑ κR - MOD onthe κ -space of κ -sized R -module is a complete Σ quasi order. The isomorphism problem
At the end of his paper Przeździecki posed the question if every two isomorphicgroups in the target of the functor have isomorphic inverse images (see [Prz14,Section 8]). We still do not know whether after our modification the answer ispositive, i.e., whether the map G defined in Definition 4.3 is a reduction for theisomorphisms. Then we ask a more general question in terms of Borel reducibility. Question 6.1.
Is there any Borel reduction from isomorphism ∼ = κ GRAPHS on κ -sizedgraphs to isomorphism ∼ = κ TFA on κ -sized torsion-free abelian groups? Question 6.1 is still open even in the case κ = ω , where a positive answer will yieldthat the relation ∼ = TFA of isomorphism on countable torsion-free abelian groups is amaximum up to Borel reducibility among all the equivalence relations induced by aBorel action of S ∞ on a standard Borel space. A remarkable result in such directionis one by Hjorth, who proved in [Hjo02] that ∼ = TFA is not Borel. In fact, this was
MBEDDABILITY ON TORSION-FREE ABELIAN GROUPS OF UNCOUNTABLE SIZE 13 extended by Downey-Montalbán [DM08], who showed that ∼ = TFA is complete Σ asa set of pairs. We ought to mention that there are several results in classical Borelreducibility concerning the isomorphism relation on torsion-free abelian groups withfinite rank. For every n < ω , denote by ∼ = TFA n the isomorphism on countabletorsion-free abelian group of rank n . An old result by Baer [Bae37] establishes that ∼ = TFA is essentially E (i.e., it is Borel bi-reducible with E ). Moreover Thomasproved in [Tho03] that for every n ≥ , ( ∼ = TFA n ) < B ( ∼ = TFA n +1 ) .Now we go back to Question 6.1. Some results of [FHK14] and [HM17] usecertain model theoretic properties of complete theories to obtain information aboutthe isomorphism relation between the models of those. First let us mention that in[HK15] the authors give the following example, among many others, of a complete Σ equivalence relation in the constructible universe. Definition 6.2.
Let E κω the equivalence relation defined on κ κ by x E κω y ⇐⇒ { α < κ | x ( α ) = y ( α ) } contains an ω -club . In [HK15, Theorem 7], under the assumption
V = L , the equivalence relation E κω is shown to be complete Σ for every inaccessible cardinal κ . Then, in [HM17,Definition 5.4] the authors defined the orthogonal chain property ( OCP ) for stabletheories and proved the following result.
Theorem 6.3 (Hyttinen-Moreno [HM17, Corollary 5.10]) . Assume that κ is inac-cessible. For every stable theory T with OCP , the equivalence relation E κω reducescontinuously to ∼ = κT . As it was kindly pointed out by Hyttinen to the author of this paper, the theoryof Z p (i.e., the group of p -adic integers) has OCP and is stable. Thus one obtainsTheorem 1.4 as a corollary of Theorem 6.3. This gives an affirmative partial answerto Question 6.1.
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