aa r X i v : . [ m a t h . L O ] S e p A DICHOTOMY FOR POLISH MODULES
JOSHUA FRISCH AND FORTE SHINKO
Abstract.
Let R be a ring equipped with a proper norm. We show that undersuitable conditions on R , there is a natural basis under continuous linear injectionfor the set of Polish R -modules which are not countably generated. When R is adivision ring, this basis can be taken to be a singleton. Introduction
The axiom of choice allows us to construct many abstract algebraic homomor-phisms between topological algebraic systems which are incredibly non-constructive.A longstanding theme in descriptive set theory is to study to what extent we can,and to what extent we provably cannot, construct such homomorphisms in a “defin-able” way. Here the notion of definability is context-dependent but often includescontinuous, Borel, or projective maps.A classical example of such an abstract construction, which provably cannot beconstructed with “nice” sets is the existence of a Hamel basis for R over Q . It iswell-known that such a basis cannot be Borel, or more generally, analytic. Similarphenomena show up when constructing Hamel bases for topological vector spaces, orconstructing an isomorphism of the additive groups of R and C .A more recent theme in descriptive set theory is that such undefinability criteriacan often be leveraged in order to gain, and hopefully utilize, additional structure.For example, Silver’s theorem [Sil80] and the Glimm-Effros dichotomy [HKL90] inter-pret the non-reducibility of Borel equivalence relations not as a pathology but ratheras the first step in the burgeoning theory of invariant descriptive set theory (see[Gao09] for background). Similarly, work starting with [KST99] studies and exploitsthe difference between abstract chromatic numbers and more reasonably definable(for example, continuous or Borel) chromatic numbers. A key feature in many ofthese theories (and all of the above examples) is the existence of dichotomy theo-rems, which state that either an object is simple, or there is a canonical obstructioncontained inside of it. This is usually stated in terms of preorders, saying that thereis a natural basis for the preorder of objects which are not simple (recall that a basis Date : September 15, 2020.The authors were partially supported by NSF Grant DMS-1950475. for a preorder P is a subset B ⊆ P such that for every p ∈ P , there is some b ∈ B with b ≤ p ).In this paper, we apply a descriptive set-theoretic approach to vector spaces andmore generally, modules, over a locally compact Polish ring . For a Polish ring R , a Polish R -module is a topological left R -module whose underlying topology is Polish.Given Polish R -modules M and N , we say that M embeds into N , denoted M ⊑ R N ,if there is a continuous linear injection from M into N . One particularly nice aspectof Polish modules is that the notion of “definable” reduction is much simpler than inthe general case. By Pettis’s lemma, any Baire-measurable homomorphism betweenPolish modules is in fact automatically continuous (see [Kec95, 9.10]). Thus there isno loss of generality in considering continuous homomorphisms rather than a priorimore general Borel homomorphisms.Our main results give a dichotomy for Polish modules being countably generated.More precisely, we give a countable basis under ⊑ R for Polish modules which are notcountably generated. While these results are stated in a substantial level of generality(they are true for all left-Noetherian countable rings and many Polish division rings),we feel that the most interesting cases are over some of the most concrete rings. Forexample, over Q , we show the existence of a unique (up to bi-embeddability) minimaluncountable Polish vector space ℓ ( Q ). We further show that nothing bi-embeddablewith ℓ ( Q ) is locally compact, and thus that every uncountable-dimensional locallycompact Polish vector space (for example, R ) is strictly more complicated than ℓ ( Q ).Another case of particular interest is the case of Z -modules, that is, abelian groups.We show that there is a countable basis of minimal uncountable abelian Polish groups(one for each prime number and one for characteristic 0). Furthermore, there exists amaximal abelian Polish group by [Shk99], as well as many natural but incomparableelements (for example, Q p and R are incomparable under ⊑ Q as are Q p and Q r for p = r ).Our dichotomy theorems will hold for rings equipped with a proper norm. A (complete, proper) norm on an abelian group A is a function k·k : A → [0 , ∞ )such that the map ( a, b )
7→ k a − b k is a (complete, proper) metric on A (recall thata metric is proper if every closed ball is compact). A norm on a ring R is a norm | · | on ( R, +) such that | rs | ≤ | r || s | for every r, s ∈ R . A proper normed ring is aring equipped with a proper norm. Every countable ring admits a proper norm (seeSection 3). Given a proper normed ring R , the R -module ℓ ( R ) is defined as follows: ℓ ( R ) = ( ( r k ) k ∈ R N : X k | r k | k ! < ∞ ) All rings will be assumed to be unital.
DICHOTOMY FOR POLISH MODULES 3 (here, k ! can be replaced with any summable sequence). Then k ( r k ) k k := P k | r k | k ! isa complete separable norm on ( ℓ ( R ) , +), turning ℓ ( R ) into a Polish R -module.The following theorems will be obtained as special cases of results in Section 5.A division ring is a ring R such that every nonzero r ∈ R has a two-sided inverse. Theorem 1.1.
Let R be a proper normed division ring and let M be a Polish R -vector space. Then exactly one of the following holds:(1) dim R ( M ) is countable.(2) ℓ ( R ) ⊑ R M . This seems to be new, even when R is a finite field, in which case ℓ ( R ) = R N .This also implies a special case of [Mil12, Theorem 24], which says that if dim R ( M )is uncountable, then there is a linearly independent perfect set (see Corollary 5.2).An analogous statement holds for a large class of discrete rings. A ring is left-Noetherian if every increasing sequence of left ideals stabilizes. Theorem 1.2.
Let R be a left-Noetherian discrete proper normed ring and let M bea Polish R -module. Then exactly one of the following holds:(1) M is countable.(2) ℓ ( S ) ⊑ R M for some nonzero quotient S of R . Note that this basis is countable since a countable left-Noetherian ring only hascountably many left ideals.For abelian Polish groups, we obtain an irreducible basis (see Theorem 4.3):
Theorem 1.3.
Let A be an uncountable abelian Polish group. Then one of thefollowing holds:(1) ℓ ( Z ) ⊑ Z A .(2) ( Z /p Z ) N ⊑ Z A for some prime p . Related statements have been shown by Solecki, see [Sol99, Proposition 1.3, The-orem 1.7].The theorems in Section 5 will be shown for a substantially broader class of mod-ules. In order to contextualize this, we remark that considering even very basicmodule homomorphisms (for example, the inclusion of Q into R as Q -vector spaces)naturally leads us to consider the broader class of quotients of Polish modules bysufficiently definable submodules. Such quotient modules are in general not Polish(they are not necessarily even standard Borel) but are still important objects ofdescriptive set-theoretic interest. They play a crucial role in [BLP20] in the formof “groups with a Polish cover”, and they also form some of the most classical ex-amples of countable Borel equivalence relations (for example, the commensurability DICHOTOMY FOR POLISH MODULES 4 relation on the positive reals naturally comes equipped with an abelian group struc-ture). The embedding order on quotient modules will be defined analogously to thehomomorphism reductions for Polish groups studied in [Ber14, Ber18].
Acknowledgments.
We would like to thank Alexander Kechris, S lawomir Solecki,and Todor Tsankov for several helpful comments and remarks.2.
Polish modules
Most Polish modules which cannot be written as direct sums, even over a field:
Proposition 2.1.
Let R be a Polish ring, let M be a Polish R -module, and let ( N x ) x ∈ R be a family of submodules of M such that the set { ( m, x ) ∈ M × R : m ∈ N x } is analytic. Then there are only countably many x ∈ R with N x nontrivial, and onlyfinitely many x ∈ R with N x uncountable.Proof. Let A n be the set of m ∈ M which can be written in the form P i Let R be a Polish ring with no nontrivial compact subgroups, andlet M be a locally compact Polish R -module. If M ⊑ R R N , then M is countablygenerated.Proof. Fix a continuous linear injection f : M ֒ → R N . Since R has no nontrivialcompact subgroups, the same holds for R N , and thus for M . Fix a complete norm k · k compatible with ( M, +). Let π n : R N → R n denote the projection to the first n coordinates, and let M n = ker( π n ◦ f ), which is a closed submodule of M . Fix ε such that the closed ε -ball around 0 ∈ M is compact, and let C = { m ∈ M : ε ≤k m k ≤ ε } . Then C ∩ T n M n = ∅ , so since C is compact, there is some n such that C ∩ M n = ∅ . We claim that M n is discrete. To see this, suppose that the ε -ballaround 0 ∈ M contained some nonzero m ∈ M n . Then the subgroup generated by m is not compact, so there is a minimal k ∈ N with k km k ≥ ε , and hence km ∈ C ,which is not possible. Thus M n is countable, so if we pick preimages ( m i ) i Every proper normed ring is locally compact and Polish. There are many examplesof proper normed rings: • The usual norms on Z , R , C and H are proper. • The p -adic norm on Q p is proper. • Every countable ring R admits a proper norm as follows. Let w : R → N bea function such that w (0) = 0, w ( r ) ≥ r = 0, and w ( r ) = w ( − r ). Weextend w to every term t in the language (+ , · ) ∪ R by w ( r + s ) = w ( r ) + w ( s ) DICHOTOMY FOR POLISH MODULES 6 and w ( r · s ) = w ( r ) w ( s ). Then let | r | be the minimum of w ( t ) over all terms t representing r . • Let R be a proper normed ring. If S ≤ R is a closed subring, then thereis a proper norm on S obtained by restricting the norm on R . If I ⊳ R is a closed two-sided ideal, then there is a proper norm on R/I given by | r + I | = min s ∈ r + I | s | .In general, we do not know if every locally compact Polish ring admits a compatibleproper norm.Given a closed two-sided ideal I ⊳ R , there is a natural quotient map ℓ ( R ) ։ ℓ ( R/I ) with kernel ℓ ( I ) := ℓ ( R ) ∩ I N .If R is finite proper normed ring, then ℓ ( R ) = R N , which in particular is homeo-morphic to Cantor space. For infinite discrete rings, there is also a unique homeomor-phism type. Recall that complete Erd˝os space is the space of square-summablesequences of irrational numbers with the ℓ -norm topology. Proposition 3.1. Let R be an infinite discrete proper normed ring. Then ℓ ( R ) ishomeomorphic to complete Erd˝os space. To show this, we will use a characterization due to Dijkstra and van Mill [DvM09,Theorem 1.1]. A topological space is zero-dimensional if it is nonempty and it hasa basis of clopen sets. Theorem 3.2 (Dijkstra-van Mill) . Let X be a separable metrizable space. Then X is homeomorphic to complete Erd˝os space iff there is a zero-dimensional metrizabletopology τ on X coarser than the original topology such that every point in X hasa neighbourhood basis (for the original topology) consisting of closed nowhere densePolish subspaces of ( X, τ ) .Proof of Proposition 3.1. We check the condition from Theorem 3.2. Let τ be theproduct topology on R N , which is zero-dimensional and metrizable. It is enough toshow that every closed ball is a closed nowhere dense Polish subspace of ( ℓ ( R ) , τ ).By translation, it suffices to consider balls of the form B = { m ∈ ℓ ( R ) : k m k ≤ ε } .Note that B is closed in R N . Thus ( B, τ ) is Polish, and B is closed in ( ℓ ( R ) , τ ).It remains to show that the complement of B is dense in ( ℓ ( R ) , τ ). Let U be anonempty open subset of ( ℓ ( R ) , τ ). We can assume that there is a finite sequence( r k ) k For a general Polish ring R , we do not know much about the preorder ⊑ R , includingthe following: DICHOTOMY FOR POLISH MODULES 7 Problem 4.1. Is there a maximum Polish R -module under ⊑ R ?This is known for some particular rings, which we mention below.4.1. Principal ideal domains. Recall that a principal ideal domain (PID) isan integral domain in which every ideal is generated by a single element. There isan irreducible basis for uncountable Polish modules over a PID: Theorem 4.2. Let R be a proper normed discrete PID and let M be a Polish R -module. Then exactly one of the following holds:(1) M is countable.(2) There a prime ideal p ⊳ R such that ℓ ( R/ p ) ⊑ R M .Moreover, the ℓ ( R/ p ) are ⊑ R -incomparable for different p .Proof. Suppose that M is not countable. By Theorem 1.2, there is some proper ideal I ⊳ R such that ℓ ( R/I ) ⊑ R M . Then since R is a PID, there is some prime ideal p ⊳ R and some nonzero s ∈ R such that I = p s . Then the linear injection R/ p ֒ → R/I defined by r rs induces a continuous linear injection ℓ ( R/ p ) ֒ → ℓ ( R/I ).It remains to show that if p and q are prime ideals with ℓ ( R/ p ) ⊑ R ℓ ( R/ q ),then p = q . Fix a continuous linear injection ℓ ( R/ p ) ֒ → ℓ ( R/ q ). Since R/ p is anintegral domain, the annihilator of any nonzero element of ℓ ( R/ p ) is p , and similarlyfor q . Then for any nonzero x ∈ ℓ ( R/ p ), its image in ℓ ( R/ q ) must have the sameannihilator since the map is injective, and thus p = q . (cid:3) Abelian groups. Applying Theorem 4.2 with R = Z gives an irreducible basisfor uncountable abelian groups: Theorem 4.3. Let A be an uncountable abelian Polish group. Then one of thefollowing holds:(1) ℓ ( Z ) ⊑ Z A .(2) ( Z /p Z ) N ⊑ Z A for some prime p . By [Shk99], there is a ⊑ Z -maximum abelian Polish group A max . So the preorder ⊑ Z on uncountable abelian Polish groups looks like the following: DICHOTOMY FOR POLISH MODULES 8 ℓ ( Z ) R Q p ( Z / Z ) N ( Z / Z ) N ( Z / Z ) N · · ·· · · · · · · · · · · · · · ·· · · · · · · · · A max Q -vector spaces. Fix a proper norm on Q . By Proposition 2.2, a ⊑ Q -minimumuncountable Polish Q -vector space cannot be locally compact. By Theorem 1.2, wehave ℓ ( Q ) ⊏ Q R , where the strictness is due to ℓ ( Q ) being totally disconnected.However, it is open as to whether there is an intermediate vector space: Problem 4.4. Is there a Polish Q -vector space V such that ℓ ( Q ) ⊏ Q V ⊏ Q R ?4.4. Real vector spaces. We consider the order ⊑ R on uncountable-dimensionalPolish R -vector spaces. By Theorem 1.1, there is a minimum element ℓ ( R ), whichis bi-embeddable with the usual space ℓ of absolutely summable sequences. ByProposition 2.2, any uncountable-dimensional locally compact Polish R -vector spacemust be strictly above ℓ . By [Kal77], there is a maximum Polish R -vector space V max . 5. Proof of the main theorems Every abelian Polish group A has a compatible complete norm defined by k a k = d ( a, d is an invariant metric on A (see [BK96, 1.1.1, 1.2.2]). If B ⊆ A is aBaire-measurable subgroup, then by Pettis’s lemma, B is either open or meager (see[Kec95, 9.11]).Setting N = 0 in the following theorem recovers Theorem 1.1. Theorem 5.1. Let R be a proper normed division ring, let M be a Polish R -vectorspace, and let N ⊆ M be an analytic vector subspace. Then exactly one of thefollowing holds:(1) dim R ( M/N ) is countable.(2) ℓ ( R ) ⊑ R M/N . DICHOTOMY FOR POLISH MODULES 9 Proof. Suppose that the dimension of M/N is uncountable. Then N is not open, so N is meager, i.e., we have N ⊆ S k F k for some increasing sequence ( F k ) k of closednowhere dense sets. Fix a complete norm k·k compatible with ( M, +). For every k , we define ε k > m k ∈ M such that the image of ( m k ) k in M/N is linearlyindependent over R/I . We proceed by induction on k . Choose ε k > ε k < ε i for every i < k ,(ii) for every ( r i ) i Thus P k r k m k is well-defined with (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k r k m k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) < (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k Corollary 5.2 (Miller) . Let R be a proper normed division ring, and let M be aPolish R -module. If dim R ( M ) is uncountable, then there is a linearly independentperfect subset of M .Proof. By Theorem 1.1, we can assume that M = ℓ ( R ). Fix an enumeration ( q n ) n ∈ N of Q . For every x ∈ R , define χ x ∈ ℓ ( R ) by( χ z ) n = ( q n < x χ x ) x ∈ R is an uncountable linearly independent Borel subset of ℓ ( R ), so weare done by taking any perfect subset of this. (cid:3) There is an analogous generalization of Theorem 1.2. Theorem 5.3. Let R be a left-Noetherian discrete proper normed ring, let M be aPolish R -module, and let N ⊆ M be a Baire-measurable submodule. Then exactlyone of the following holds:(1) M/N is countable.(2) ℓ ( R ) /ℓ ( I ) ⊑ R M/N for some proper two-sided ideal I ⊳ R . In particular,there is a linear injection ℓ ( R/I ) ֒ → M/N .Proof. Suppose that M/N is not countable. Then N is not open, and thus meager.Let ( U k ) k be a descending neighborhood basis of 0 ∈ M , and let I k = { r ∈ R : rU k ⊆ N } . Then ( I k ) k is an increasing sequence of ideals, so since R is left-Noetherian, thissequence stabilizes at some I = I n . Note that I is a proper ideal, since otherwise U n ⊆ N , a contradiction to N being meager. Note also that I is a two-sided ideal,since if r ∈ R , then there is some k > n with rU k ⊆ U n , and thus IrU k ⊆ IU n ⊆ N ,and thus Ir ⊆ I . By replacing M with the submodule generated by U n (which By proper, we mean a proper subset (no relation to proper norms). DICHOTOMY FOR POLISH MODULES 11 is analytic non-meager, and therefore open), we can assume that for every open V ⊆ M , we have { r ∈ R : rV ⊆ N } = I . Then for every r / ∈ I , the subgroup { m ∈ M : rm ⊆ N } is not open, and therefore meager. Thus more generally, if m ′ ∈ M , then { m ∈ M : rm ∈ N + m ′ } is meager.Fix a complete norm k·k compatible with ( M, +). Let ( F k ) k be an increasingsequence of closed nowhere dense sets with N ⊆ S k F k . For every k , we define ε k > m k ∈ M such that the image of ( m k ) k in M/N is linearly independentover R/I . We proceed by induction on k . Choose ε k > ε k < ε i for every i < k ,(ii) for every ( r i ) i For every i , we have k r n + i m n + i k < ε n + i , and thus k r n + i m n + i k < i +1 ε n by induc-tively using ε k +1 < ε k . Thus k r n + i m n + i k < i +1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X k R/I , if n is sufficientlylarge, then P k J. Symb. Log. , 79(4):1148–1183, 2014.[Ber18] Konstantinos A. Beros. 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Studia Math. ,61(2):161–191, 1977.[Kec95] Alexander S. Kechris. Classical descriptive set theory . Springer, 1995.[KST99] A. S. Kechris, S. Solecki, and S. Todorcevic. Borel chromatic numbers. Adv. Math. ,141(1):1–44, 1999.[Mil12] Benjamin D. Miller. The graph-theoretic approach to descriptive set theory. Bull. SymbolicLogic , 18(4):554–575, 2012.[Shk99] S. A. Shkarin. On universal abelian topological groups. Mat. Sb. , 190(7):127–144, 1999.[Sil80] Jack H. Silver. Counting the number of equivalence classes of Borel and coanalytic equiv-alence relations. Ann. Math. Logic , 18(1):1–28, 1980. DICHOTOMY FOR POLISH MODULES 13 [Sol99] S lawomir Solecki. Polish group topologies. In Sets and proofs (Leeds, 1997) , volume 258 of London Math. Soc. Lecture Note Ser. , pages 339–364. Cambridge Univ. Press, Cambridge,1999. Department of Mathematics, California Institute of Technology, 1200 E. Cali-fornia Blvd., Pasadena, CA 91125 E-mail address : [email protected] Department of Mathematics, California Institute of Technology, 1200 E. Cali-fornia Blvd., Pasadena, CA 91125 E-mail address ::