Dualities of locally compact modules over the rationals

Journal of Algebra 256 (2002) 433–466

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Dualities of locally compact modulesover the rationals✩

Dikran Dikranjana,∗ and Chiara Milanb

a Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze 206,33100 Udine, Italy

b Dipartimento di Matematica “U. Dini”, Università di Firenze, Viale Morgagni 67/A,50134 Firenze, Italy

Received 20 May 2001

Communicated by Kent R. Fuller

Dedicated to Adalberto Orsatti on his 65th birthday

Abstract

The concept ofcontinuity of a duality (i.e., involutive contravariant endofunctor) ofthe categoryLR of locally compact modules over a discrete commutative ringR, wasintroduced by Prodanov. Orsatti and the first-named author proved that the categoryLR

admits discontinuous dualities whenR is a large field of characteristic zero. We prove thatall dualities ofLR are continuous whenR = Q is the discrete field of rationals numbers,while this fails to be true for the discrete fieldsR andC of the real and of the complexnumbers, respectively. More generally, we describe the finitely closed subcategoriesL ofLQ such that all dualities ofL are continuous. All dualities of such a categoryL turn outto be naturally equivalent to the Pontryagin duality. This property extends toR andC. Thecontinuity of all dualities ofLQ is related to the fact that the adele ringAQ of the rationalshas no ring automorphisms beyond the identity. 2002 Elsevier Science (USA). All rights reserved.

Keywords:Topological module; Locally compact group; Pontryagin duality; Continuous duality;Discontinuous duality

✩ Work partially supported by Research Grant of the Italian MURST in the framework of theproject “Nuove prospettive nella teoria degli anelli, dei moduli e dei gruppi abeliani” 2000.

* Corresponding author.E-mail addresses:[email protected] (D. Dikranjan), [email protected] (C. Milan).

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0021-8693(02)00115-1

434 D. Dikranjan, C. Milan / Journal of Algebra 256 (2002) 433–466

1. Introduction

Let R be a locally compact commutative ring with unit and letLR be thecategory of locally compact topological unitaryR-modules. Following [15] wecall dualityof LR a contravariant functor# :LR → LR such that# ◦ # is naturallyequivalent to the identity ofLR and(rf )# = rf # for every morphismf :X → Y

in LR and r ∈ R (where, as usual,rf is the morphismX → Y defined by(rf )(x) = rf (x)). The classical example is the Pontryagin duality defined byX �→ X∗ := ChomZ(X,T), whereX ∈ LR,T = R/Z is the unit circle groupin additive notation andX∗ carries the natural structure ofR-module and thecompact-open topology.

Roeder [17] proved that the Pontryagin duality is the unique duality of thecategoryLZ of locally compact abelian groups. Prodanov, unaware of this fact,obtained much more general results and raised the questionhow manydualitiescan carryLR , and in particular, when the Pontryagin duality is theuniquedualityof LR [15,16]. Uniqueness of the Pontryagin duality in the case of acompactcommutative ringR was established by Stoyanov [19]. Gregorio and Orsatti[6,8] extended this result to the general case of a compact (not necessarilycommutative) ringR.

From now onR will be always equipped with thediscretetopology. Nowuniqueness may fail. LetCR (respectivelyDR) denote the full subcategory ofLR with objects all compact (respectively discrete) modules. Then every dual-ity # :LR → LR preserves exact sequences and sendsCR to DR . Moreover, thecompact moduleT = R# (the torus of the duality#) is an injective cogeneratorof CR with ChomR(T ,T )∼=R canonically. Vice versa, for every compact cogen-eratorT of CR with ChomR(T ,T ) ∼= R, the functorX �→ ∆T (X) is a dualitywhere, for everyX ∈ LR, ∆T (X) is the module ChomR(X,T ) of continuousR-module homomorphisms equipped with the compact-open topology. The nat-ural equivalenceω : 1LR

→ ∆T∆T is defined by the evaluation homomorphismωX :X → ∆T (∆T (X)) (i.e., for x ∈ X andχ ∈ ∆T (X),ωX(x)(χ) = χ(x)). Inother words, these two properties characterize the tori of dualities, so that in thesequel we calltorus any compact cogeneratorT of CR with ChomR(T ,T ) ∼= R

[2,16].In general, for every duality# :LR → LR the moduleX# is algebraically

isomorphic to∆T (X) for everyX ∈ LR (such a natural isomorphism can beobtained adapting the original construction of Morita [11,13,16]). The duality# is called continuousif for each X this isomorphism is also topological(i.e., the dualitiesX �→ ∆T (X) are continuous), otherwise# is discontinuous(an equivalent definition will be given below). Thus, continuous dualitiesare classified by their tori, hence by means of their discrete duals. Namely,the projective finitely generatedR-modulesV with EndR(V ) ∼= R, i.e., thecontinuous dualities can be classified through the Picard group Pic(R) of R.In particular, the unique continuous duality onLR is the Pontryagin duality if

D. Dikranjan, C. Milan / Journal of Algebra 256 (2002) 433–466 435

and only if Pic(R) = 0 [2, Theorem 5.17]. Prodanov proved that every dualityon LZ is continuous which, in view of Pic(Z) = 0, gives immediately Roeder’sresult [15], [3, Section 3.4]. This was extended to algebraic number ringsR in[1] as follows: all dualities onLR are continuous whenR is a totally real ringof algebraic numbers (i.e., all conjugates of the elements ofR are real), whilediscontinuous dualities exist for the ringR = Z[i] of Gauss integers.

It is more convenient to introduce an equivalent definition of continuitythat involves only the torus of a duality. Indeed, for a duality# :LR → LR

the isomorphismX# ∼= ∆T (X) allows to consider the elements ofX# ascontinuous homomorphismsX → T (characters), but the topology ofX#

need not be the compact-open one [2, Proposition 4.2]. Then there exists a(not necessarily continuous) automorphismκ :T → T , such that the naturalequivalenceE : 1LR

→ # ◦ # satisfiesEX(x) = κ ◦ ωX(x) for everyX ∈ LR andx ∈ X [2, Theorem 4.4]. Moreover,κ2 :T → T is a topological isomorphism, soa multiplication by an invertible elementr ∈ R. For this reasonκ is calledtheinvolution of #. The duality# is (dis)continuous whenκ is (dis)continuous. ThetorusT and the involutionκ determine the duality# up to natural equivalence;the question which involutions of a torusT are involutions of a duality with torusT is highly non-trivial [2]. Here we give a complete answer forR = Q,R, andC

making use of an appropriate simultaneous “parameterization” of involutions ofthe torus and subcategories ofLR (Theorems 1.11, 5.2).

Let L0R be the full subcategory ofLR with objects all modules having

a compact open submodule. This subcategory contains the subcategoriesCR andDR and every duality# :LR → LR sendsL0

R to L0R . This permits us to consider

the restriction of# to L0R as a duality ofL0

R . On the other hand, since bothLR and DR are contained inL0

R , we can associate to each duality# of L0R

a torus and an involution as we did forLR and speak of (dis)continuity of#.Then a compact moduleT is a torus inL0

R if and only if it is a torus inLR .Prodanov proved thatL0

R has always discontinuous dualities. SinceLZ does notadmit discontinuous dualities, we see in this way that dualities ofL0

Zneed not be

extendible to dualities of the wholeLZ. Orsatti and the first-named author noticedthatLR = L0

R for a fieldR of cardinality> c and char(R) = 0, hence one getsexamples of discontinuous dualities [2, Theorem 10.2].

Motivated by this result we study in this paper the dualities of the categoryLQ

whenR = Q is thesmallestfield with char(R)= 0 and consequently the categoryLQ is thelargestpossible (containing all other categoriesLR with R a field withchar(R) = 0). We also choose to consider the casesR = R andR = C to tastethe limits of the cardinality restraint|R| > c. ForR = Q the situation changessubstantially: here all dualities are continuous (cf. Theorem 1.5), whereas forR = R,C discontinuous dualities are available even ifLR �= L0

R (Theorem 1.13).More details about the main results of the paper are described in Sections 1.1–1.2,the proofs are given in Sections 2–5.


Continuous dualities in the non-commutative context were studied by Meniniand Orsatti [11] and Gregorio [7]. For a categorical treatment of the question see[5]. In both casesnon-involutivedualities are considered.

1.1. Notation and terminology

We denote byN andP the sets of positive naturals and primes, respectively;by Z the integers, byQ the rationals, byR the reals, byC the complex numbers,by Zp the ring ofp-adic integers and byQp the field ofp-adic numbers(p ∈ P).The (compact) Pontryagin dualQ∗ of the discrete groupQ is denoted byK.

If R is a commutative ring,X anR-module andr ∈ R, then we denote bymXr :X → X the multiplication byr. When no confusion is possible we writesimplymr .

For a setπ of prime numbers we denote bySπ the divisible hull of∏

p∈π Zp

equipped with the group topology that makes∏

p∈π Zp an open subgroup ofSπ(actually,Sπ becomes a locally compactQ-algebra). In particular,SP will beabbreviated simply toS.

We denote byL the arc component of zero inK (it will be proved in Section 2.1thatL coincides with the trace ofR in K, so thatL carries a natural structure ofan abstractR-module).

Let G be an abelian topological group. We denote byc(G) the connectedcomponent ofG. For p ∈ P an elementx ∈ G is quasi-p-torsion if either thecyclic subgroup〈x〉 of G generated byx is a finitep-group or〈x〉, when endowedwith the induced topology, is isomorphic toZ equipped with thep-adic topology([18], see also [3, Chapter 4]). The subset of quasi-p-torsion elements ofG isa subgroup and is denoted by tdp(G). We denote by td(G) the subgroup ofGconsisting of all elementsx such that eitherx is torsion or the induced topologyof 〈x〉 is non-discrete and linear (i.e., has a local base at 0 consisting of opensubgroups of〈x〉). ForX ∈ LQ, the subgroups td(X) and tdp(X) are functorialQ-submodules [3, Proposition 4.1.2].

1.2. The structure of the modules inLQ and finitely closed subcategories ofLQ

The structure of a moduleX ∈LQ is described as follows:

Theorem 1.1. For X ∈ LQ there exist integersn,np ∈ N ( for p ∈ P) and cardinalnumbersα,β such that

X ∼= Rn × Kα × Q(β) ×X0, (1)

whereX0 is a closed submodule ofX having a compact open essential subgroupK isomorphic to

∏p∈P Z

npp . Under this identification,

(a) Kα is the biggest compactQ-submodule ofX,


(b) c(X)= Rn × Kα ,(c) the closure oftd(X) coincides withKα ×X0.

Since|Rn×X0| � c, Theorem 1.1 immediately yieldsLR = L0R for every field

R with char(R)= 0 and|R|> c [2, Theorem 11.1(a)].Obviously, the cardinalsα,β and the integersn,np determine the module

X up to isomorphism. The topological properties (a)–(c) guarantee uniquenessof the factorsKα , Rn × Kα , and Kα × X0 in the decomposition (1). As aconsequence, also the factorRn × Kα × X0 is uniquely determined as a sumof c(X) and the closure of td(X) (note that this is also the largest submoduleof X with no proper open submodules). None of the remaining factors has thisuniqueness property. One of the factors destroying uniqueness isKα . Indeed,consider a moduleX of the formX = X1 × Kα with α �= 0, 0 �= X1 ∈ LR , andα(X1) = 0. For every continuous homomorphismf :X1 → Kα,X decomposesalso asX =X2 × Kα , whereX2 ∼=X1 is a closed submodule ofX andX2 �=X1

wheneverf �= 0. Moreover, the product topologies onX1 × Kα andX2 × Kα

coincide (cf. Lemma 2.5). Since such anf always exists whenα �= 0 andX1 �= 0,the decompositionX = X1 × Kα is never uniquely determined. In particular, tosee that the first factor, the third one and the last one in the decomposition (1)are not uniquely determined, it suffices to take, e.g.,X1 = R,Q,D(Zp). Arguingas above, one can show that also the factorQ(β) destroys uniqueness, namely,a decomposition of the formX = X1 × Q(β), with β �= 0, 0 �= X1 ∈ LR , andβ(X1) = 0 is not uniquely determined (in such a case, it suffices to exploit non-zero continuous homomorphismsf :Q(β) →X1).

Finally, the factorRn ×X0 is a maximal submodule ofX that has no properopen submodules and no compact submodules beyond{0}, but it is not determinedby this property as a submodule ofX when α(X) �= 0. Furthermore,X0 hasthe additional property to be totally disconnected and non-discrete. Again, theseproperties do not uniquely determineX0 as a submodule ofX whenα(X) �= 0.

The proof of Theorem 1.1 will be given in Section 2.2 after a detaileddescription of the topological group structure ofK. In order to do that, werecall that there exists a topological group embeddingι :

∏p∈P Zp → K such that

K/H ∼= T, whereH = Im(ι) (cf. also [3, Proposition 3.6.2]). The embeddingιwill be fixed throughout the paper. Therefore, we fix also the closed subgroupH

of K, along with the subgroupsHπ = ι(∏

p∈π Zp) for everyπ ⊆ P. In particular,we setHp = ι(Zp) for everyp ∈ P. We give more details about the embeddingι

and its relation with the arc componentL of K in Section 2.1.

Definition 1.2. A full subcategoryL of LQ containingL0Q

is finitely closedif itis closed with respect to isomorphisms, taking quotients (with respect to closedsubmodules), finite products and closed sub-modules.


Example 1.3. (i) The subcategoryLconQ

of LQ consisting of all modulesX ∈ LQ

with np(X) = 0 for everyp ∈ P, is finitely closed. Equivalently, these are themodules of the formX ∼= Rn × Kα × Q(β), i.e., the connected component ofX isopen.

(ii) For every subsetπ ⊆ P the subcategoryL(π)Q

of LQ consisting of allmodulesX ∈ LQ, with n(X) = 0 andnp(X) = 0 for everyp ∈ P\π , is a finitely

closed subcategory ofLQ such thatR /∈ L(π)Q

. The assignmentπ �→ L(π)Q

is

monotone andQp ∈ L(π)Q

if and only if p ∈ π . In particular,L(∅)Q

= L0Q

and for

p ∈ P, the categoryL({p})Q

is minimal among the finitely closed subcategories

of LQ containing properlyL0Q

but not containing theQ-moduleR. If π ⊆ P is

finite,L(π)Q

coincides with the finitely closed subcategory ofL0Q

generated by allQp with p ∈ π . If π is infinite, this is not true.

A complete description of the finitely closed subcategories ofLQ is given inSection 4.1.

1.3. Dualities of finitely closed subcategories ofLQ and their extensions

Since Pic(Q) = 0,K is the only possible torus of a duality ofLQ, hence thePontryagin duality is the unique continuous duality ofLQ (cf. the introduction).For everyX ∈ LQ, we identify the Pontryagin dualX∗ with ∆K(X) via thecanonical isomorphismX∗ ∼= ChomQ(X,K). In particular, from now on, we omitthe subscript and write simply∆(X).

Let L be a finitely closed subcategory ofLQ and let # be a contravariantinvolutive functor# :L → L. Arguing as in [3, Proposition 3.4.7] one can easilyprove additivity of#. This automatically yields the axiom(rf )# = rf # for r ∈ Q

and every homomorphismf in L. This allows for a representation of the functor# as in [2, Sections 4.1–4.3], i.e., for everyX ∈ L, the dualX# coincides as anabstract module with the module∆(X), but the topology onX# need not be thecompact-open one. From now on we call such a functor# :L→ L a dualityof L.

It will be shown in Section 4.2 that the Pontryagin duality∆ sendsL to itself,so it defines a duality ofL which for brevity will be still denoted by∆.

Since every finitely closed subcategoryL of LQ containsL0Q

, one can prove

(arguing as in the case ofLQ) that if # :L → L is a duality, then there exists anautomorphismκ :K → K such that for the natural equivalenceE : 1L → # ◦ #,for every X ∈ L, x ∈ X, and χ ∈ X#, one hasEX(x)(χ) = κ(χ(x)), i.e.,EX(x)= κ ◦ωX(x) (so thatκ ◦χ ∈∆(X#) for everyχ ∈∆(X)). Moreover,κ2 is atopological automorphism ofK (i.e., the multiplication by some non zero rationalnumber). We say that# is (dis)continuousif and only if κ is (dis)continuous.

This motivates the following definition.


Definition 1.4. We call involution of K any (not necessarily continuous)automorphismκ :K → K such thatκ2 is continuous, i.e.,κ2 coincides with themultiplicationmK

s by some non-zero rationals ∈ Q.

The next theorem classifies the finitely closed subcategories ofLQ that admitdiscontinuous dualities.

Theorem 1.5. A finitely closed subcategoryL ofLQ does not admit discontinuousdualities if and only ifR × S ∈ L.

According to Theorem 1.5, the smallest finitely closed subcategoryL(∞) of LQ

having only continuous dualities is the one generated byR × S (clearly, a moduleX as in (1) belongs toL(∞) if and only if the sequence(np)p∈P is bounded). Inparticular, we have the following corollary.

Corollary 1.6. There exist discontinuous dualities onLconQ

and onL(π)Q

for everyπ ⊆ P.

Now we see that the discontinuous dualities (e.g., as in Corollary 1.6) do notadd essentially new dualities beyond the Pontryagin duality:

Theorem 1.7. Every finitely closed subcategoryL of LQ admits a unique, up tonatural equivalence, duality(namely the Pontryagin duality).

The proof of this theorem (given in Section 5.1) is based on specific propertiesof LQ. We do not know whether this remains true in general.

Question 1.8. Let R be a commutative ring. Is every duality ofLR naturallyequivalent to a continuous one? Is this true whenR is a PID?

The discontinuous duality ofLZ[i] produced in [1] is easily seen to be naturallyequivalent to a continuous one (namely, the Pontryagin duality as Pic(Z[i])= 0).

Theorem 1.5 will be proved in Section 5.1. Its proof is based on a techniqueof building discontinuous dualities developed in Theorem 1.11 (see belowinfra).More precisely, after introducing appropriate invariants of the involutionsκ of K

and of the finitely closed subcategoriesL of LQ (cf. Definition 1.9), it is possibleto prove that the above theorems are particular cases of a more general resultcharacterizing the involutionsκ :K → K that correspond to some duality# ofa finitely closed subcategoryL of LQ.

Definition 1.9. For a moduleX ∈ LQ the supportof X is the set supp(X) :={p ∈ P: np(X) > 0}.


(a) For a finitely closed subcategoryL of LQ setI(L) := {supp(X): X ∈ L}.(b) For an involutionκ of K setI(κ) := {π ⊆ P: κ |Hπ is continuous}.

Then bothI(L) andI(κ) are ideals of the Boolean algebra 2P = P(P) of allsubsets ofP (see Corollary 3.2 for the proof of the fact thatI(κ) is an ideal).

Conversely, to an idealI of 2P assign the finitely closed subcategoryX(I)

consisting of all modulesX ∈ LQ such that supp(X) ∈ I andn(X) = 0. It easilyfollows thatX(

⋂λ∈Λ Iλ)= ⋂

λ∈ΛX(Iλ) for every family of ideals{Iλ}λ∈Λ of 2P.Moreover, the inclusionL ⊆ X(I(L)) holds precisely for those finitely closedsubcategoriesL not-containing the moduleR. Otherwise,L is contained in thefinitely closed subcategory ofLQ generated byR andX(I(L)). To unify thesetwo cases, we give the following definition.

Definition 1.10. For every finitely closed subcategoryL of LQ, the finitely closedsubcategory sat(L) generated byL and X(I(L)) will be called thesaturationof L. We callL saturatedif L = sat(L).

It easily follows thatL(∞) is non-saturated (in fact, sat(L(∞)) = LQ), whileL(π) is saturated with idealI(L(π))= (π), i.e., the principal ideal of 2P generatedby π .

The next theorem will be used in the proof of Theorem 1.5.

Theorem 1.11. LetL be a finitely closed subcategory ofLQ and letκ :K → K bean involution. Then there exists a duality ofL with involutionκ if and only if thefollowing conditions are fulfilled:

I(κ)⊇ I(L) (2)

and

there existsρ ∈ R such that κ |L = mρ in caseR ∈ L. (3)

The above theorem allows us to answer all questions related to extensions ofa duality and to characterize, in particular, the non-extendible one’s.

Corollary 1.12. Let L be a finitely closed subcategory ofLQ. A duality # of Lwith involutionκ can be extended to a duality of a finitely closed subcategoryL′ ⊇ L if and only if I(L′) ⊆ I(κ) and κ |L = mρ , ρ ∈ R, in caseR ∈ L′. Inparticular, every duality ofL can be extended to a duality ofL′ ⊇ L if and only ifsat(L)= sat(L′).

It immediately follows from the above theorem that a finitely closed subcat-egoryL admits a duality that cannot be extended to any larger finitely closedsubcategory ofLQ if and only if L is saturated.


The proofs of Theorem 1.11 and Corollary 1.12 are given in Section 5.1. Thenecessity of (2) and (3) in Theorem 1.11 follows from properties of the modulesR andSπ established in Lemma 4.6. To prove them we use the fact that the adelering AQ = R × S of the rationals has no ring automorphisms beyond the identity.We feel that this fact must be probably known, but since we found no referencewe give a proof in Theorem 2.8.

Moreover, Theorem 1.11 gives rise to a Galois connection between the class offinitely closed subcategoriesL of LQ and all involutionsκ of K that may appearas an involution of some duality ofL. More details about this Galois connectionare given in Section 6.

In the sequelR and C are considered asdiscretefields. The next theoremanswers a question of Prodanov [16] and positively answers Question 1.8 forR = R,C.

Theorem 1.13. Both categoriesLR and LC admit discontinuous dualities.Moreover, for both categories the Pontryagin duality is the unique, up to naturalequivalence, duality.

The proof of Theorem 1.13, based on a counterpart of Theorem 1.11 forR, isgiven in Section 5.2.

2. The structure of the modules in LQ

2.1. Structure of the groupK

The structure of the compact groupK is described in terms of the embeddingι mentioned in Section 1.1 (or equivalently, in terms of its closed subgroupH ∼= ∏

p∈P Zp). The existence of embeddings with these properties is easy toobtain applying Pontryagin duality to the quotient mapQ → Q/Z (more detailsabout the relation between such embeddings will be given in Remark 2.10). Inorder to better explain how this embedding is related to the arc componentL

of K, we recall its construction from [3, Exercise 3.8.19]. Take the cartesianproductB = R × ∏

p∈P Zp and consider the element(1,1) ∈ B, where 1∈ R,1 = (. . . ,1p, . . .) and 1p is the identity ofZp for everyp ∈ P. Then the quotientB/〈(1,1)〉 is a compact, connected, torsion-free abelian group isomorphic toK

since the Pontryagin dual ofB/〈(1,1)〉 is topologically isomorphic toQ (being adivisible, torsion-free, discrete abelian group of rank 1). In the sequel we identifyK with this quotient and consider the canonical homomorphismψ :B → K.

Identify the subgroups{0}×∏p∈P Zp andR×{0} of B with

∏p∈P Zp andR,

respectively. Observe that the restrictionι of ψ to the subgroup∏

p∈P Zp of B is


injective since∏

p∈P Zp trivially intersects kerψ . Moreover,

K/Im(ι)∼= B/(

Z ×∏p∈P

Zp

)∼= R/Z = T.

Henceι is the desired embedding∏

p∈P Zp → K, so we write

Im(ι)=ψ

( ∏p∈P

Zp

)= H and ψ(Zp)= Hp for everyp ∈ P.

In particular, we consider1 and 1p as elements ofH ⊆ K.The embeddingι :

∏p∈P Zp → K can be (uniquely) extended to a continuous

group monomorphismi :S → K that necessarily sendsS onto the divisible hullD(H) of H in K. This allows us to identifySπ with D(Hπ) for everyπ ⊆ P.Since in the sequelD(H) (and its submodulesD(Hπ )) will also be consideredwith the topology induced byK, in this way we shall emphasize better which ofboth topologies is taken onD(H) (we recall that otherwiseS carries the locallycompact topology that makesH an open compact subgroup).

Analogously, the restrictionψ|R :R → K is injective sinceR triviallyintersects kerψ . Let L = ψ(R). Note thatK = L + H but this sum is notdirect as 0= ψ(1,0) + ψ(0,1) (in fact kerψ = 〈(1,1)〉), consequently,ψ(1,0)andψ(0,1) generate the same cyclic subgroupL ∩ H = 〈ψ(1,0)〉 = 〈ψ(0,1)〉.Observe that the subgroupL ∩ H is dense inH since〈1〉 is dense inH.

The next proposition characterizes the dense subgroups ofK.

Proposition 2.1 [3, Lemma 3.6.6].A subgroupG of K is dense inK if and onlyif for everyn ∈ N there existsχn ∈ G such thatnχn /∈ H. In particular, everynon-trivial divisible subgroup ofK is dense.

The proof of the next proposition is an immediate application of Pontryaginduality.

Lemma 2.2. For every pair of continuous homomorphismsψ1 :R → K andψ2 :R → K, there exists a real numberρ ∈ R such thatψ2 = ψ1 ◦ mρ . Inparticular, Im(ψ1)= Im(ψ2).

Remark 2.3. (a) Lemma 2.2 implies that Im(χ) = Im(ψ|R) = L for everycontinuous homomorphismχ :R → K. More precisely, there exists a real numberρ ∈ R such thatψ|R ◦ mρ = χ , wheremρ is the multiplication byρ. HenceL isthe unique image ofR in K which coincides with the arc component of zero (cf.[10, Theorem 8.30]).

(b) Moreover, asL is dense inK, every continuous endomorphism ofL extendsto a continuous endomorphism ofK. Since ChomZ(K,K)∼= Q, it follows that the


only continuous group endomorphisms ofL are the multiplicationsmLr by non-

zero rationalsr ∈ Q.

2.2. Canonical factorization inLQ

Every moduleX ∈ LQ is a divisible, torsion-free locally compact abeliangroup. In particular, ifK is a compact module inLQ, then it is isomorphic toa power ofK, i.e.,K ∼= Kα for some cardinalα. Indeed, sinceK is a divisible,torsion-free compact abelian group, its Pontryagin dualK∗ is a divisible, torsion-free group. Therefore,K∗ ∼= Q(α), with α = dimQK

∗, so we can conclude thatK ∼=K∗∗ ∼= Kα .

For every compact torsion-free abelian groupK the divisible hullD(K) of Kwill be endowed with the (locally compact) group topology that makesK an opensubgroup, so thatD(K) ∈ LQ.

Proof of Theorem 1.1. Take a moduleX ∈ LQ and consider a decompositionX ∼= Rn×Y , whereY is a closed subgroup ofX having a compact-open subgroupK [9]. SinceY is a direct summand ofX, it is divisible too, so thatY ∈ LQ. Theintegern� 0 is uniquely determined byX.

The divisible hullD(K) of K in Y is open inY , hence it splits; consequently,Y ∼=D(K)×YD topologically, whereYD ∼= Y/D(K) is discrete, torsion-free anddivisible. Therefore,YD ∼= Q(β) for some appropriate cardinalβ .

SinceK is a compact torsion-free abelian group, its Pontryagin dualK∗ isdivisible, hence its torsion part splits. Therefore,c(K) is a topological directsummand ofK, i.e.,K ∼= c(K)×K0 as topological groups, whereK0 ∼=K/c(K)

is compact and totally disconnected. On the other hand, sincec(K) is divisible,we getc(K)=D(c(K)). Consequently,D(K)∼= c(K)×D(K0).

As K0 is a torsion-free totally disconnected compact abelian group, itsPontryagin dualK∗

0 is a torsion divisible group. HenceK∗0

∼= ⊕p∈P Z(p∞)(np)

for some cardinalsnp . Applying Pontryagin duality, we getK0 ∼= ∏p∈P Z

npp . Let

us prove now that the cardinalsnp are non-negative integers.Indeed, for every fixed primep ∈ P, tdp(K0) is open in tdp(D(K0))

since tdp(K0) = K0 ∩ tdp(D(K0)) and K0 is an open subgroup ofD(K0).The multiplication by 1

p∈ Q is a topological isomorphism ofD(K0) and

1p

tdp(D(K0))⊆ tdp(D(K0)). In particular, the multiplication by1p

is continuousin tdp(D(K0)). Since tdp(K0) is open in tdp(D(K0)), there exists an openneighborhoodU of zero in tdp(D(K0)) such that1

p·U ⊆ tdp(K0). Consequently,

U ⊂ p · tdp(K0). In particular, it follows thatp · tdp(K0) is open in tdp(K0).On the other hand, tdp(K0) ∼= Z

npp , so that this is possible only ifnp < ∞.

This proves thatD(K) ∼= c(K)×D(K0), whereK0 is a compact open essentialsubgroup ofD(K0) isomorphic to

∏p∈P Z

npp , np ∈ N. On the other hand,c(K)

is a compact module inLQ, hence it is isomorphic to a powerKα of K for


some appropriate cardinalα. Hence the decomposition (1) holds, whereX0 isthe divisible hullD(K0) of K0 andc(X)∼= Rn × Kα .

It easily follows from [3, Proposition 4.1.6] that the closure of td(X) coincideswith Kα ×X0.

We also observe that no factor in the product (1), beyondKα , contributes withcompactQ-submodules, henceKα is the biggest compactQ-submodule ofX.In fact, Q(β) × Rn does not contain even non-trivial compactsubgroups, sinceits cyclic submodules are discrete. Suppose now thatC is a non-zero compactsubmodule ofX0 and letp :X0 →X0/K0 be the quotient homomorphism. ThenC is divisible, hence its imagep(C) in the discrete quotient groupX0/K0 isdivisible and finite (being compact). Therefore,p(C)= {0} so thatC is containedin the kernelK0 of p—a contradiction sinceK0 is reduced. ✷

The next corollary follows directly from Theorem 1.1.

Corollary 2.4. Q,R,K, andQp ( for all p ∈ P) are the only topologically simplemodules ofLQ.

It is helpful to note the (distinct) topological properties (discrete, connectedand non-compact, compact, quasi-p-torsion, respectively) distinguishing eachcase.

The following lemma is crucial for the proof of Theorem 1.11 and for theuniqueness of the decomposition (1) of a moduleX ∈ LQ (see the commentafter Theorem 1.1). LetX = X1 × K be an abelian group. There is a bijectivecorrespondence between direct summandsX2 of X that are complements toKand homomorphismsf :X1 → K (so that the graphΓf = {(x, f (x)): x ∈X1}of f is the direct summand ofX corresponding tof ). We consider alsothe automorphismγf of X associated tof (andX2) defined byγf (x, y) =(x, f (x)+ y) for everyx ∈X1, y ∈K. Note thatΓf =X2 = γf (X1); moreover,X2 �=X1 if and only if f �= 0.

Lemma 2.5. Let (X1, σ1) be a locally compact abelian group, let(K,σ) bea non-zero compact abelian group and letX be the groupX1 × K endowedwith the product topologyτ . Let X2 � X be a τ -closed complement ofK andlet f :X1 →K be the abstract homomorphism associated toX2. Then:

(a) the topologiesγf (σ1) and τ |X2 of X2 coincide(whereγf is the automor-phism associated toX2);

(b) the following conditions are equivalent:(b1) f :X →K is continuous;(b2) γf is a topological automorphism of(X, τ);(b3) τ coincides with the product topologyσ2 × σ of the abstract group

X =X2 ×K, whereσ2 = γf (σ1)= τ |X2.


Proof. (a) Let us note that the restrictionγf |X1 : (X1, σ1) → (X2, τ |X2) has asinverse the restrictionp1|X2, wherep1 : (X1 ×K,τ)→ (X1, σ1) is the canonicalprojection. Hencep1|X2 is continuous. Moreover, asK is compact,p1 is aclosed map by the Kuratowski’s closed projection theorem. SinceX2 is τ -closed,the restrictionp1|X2 :X2 → X1 is closed too. Being injective, it is also open.Therefore,γf |X1 is a homeomorphism between(X1, σ1) and(X2, τ |X2) so thatγf (σ1)= τ |X2.

(b1) → (b2) To prove that the automorphismγf : (X1 ×K,τ)→ (X1 ×K,τ)

is continuous, note that the compositionsp1 ◦ γf = p1 andp2 ◦ γf = f ◦p1 +p2

are continuous. The continuity of−f implies thatγ−f = γ−1f is continuous too,

henceγf is a topological automorphism of(X, τ).(b2) → (b3) Sinceγf : (X, τ)→ (X, τ) is a homeomorphism,γf (τ ) = τ . On

the other hand,X = (X1 × K,σ1 × σ) and γf = 〈γf |X1 × idK〉 :X1 ×K →X2 ×K , where γf |X1 : (X1, σ1) → (X2, γf (σ1)) and γf |K = idK : (K,σ) →(K,σ). Consequently,γf (τ ) = γf (σ1) × σ , so thatγf (σ1) × σ coincides withτ = γf (τ ).

(b3) → (b1) Observe that the homomorphismf :X1 → K can be viewed asthe composition of

(X1, σ1)γf |X1−−−→ (X2, σ2)

p2|X2−−−→K,

whereγf |X1 is a homeomorphism by the choice of the topology onX2 and therestrictionp2|X2 is continuous sinceτ = σ1 × σ = σ2 × σ by hypothesis. ✷2.3. The modulesSπ

Since the locally compactQ-modules of the formSπ , π ⊆ P, play a centralrole in the study of dualities of the finitely closed subcategories ofLQ, we collectin this section some results concerning their topological and algebraic propertiesthat will be crucial in the sequel.

We have already mentioned (cf. Notation and terminology and Section 2.1)that, for everyπ ⊆ P, the moduleSπ is identified with the submoduleD(Hπ)

of K provided with the group topology that makesHπ an open subgroup.Equipped with this topology,Sπ becomes a locally compact topological ring(more precisely, it has a structure ofQ-algebra) with identity element1π definedby setting

1π(p)={

0p whenp /∈ π ,

1p whenp ∈ π .

Moreover, the invertible elements of the ringSπ are those of the formξ =(ξp)p∈π ∈ Sπ with ξp �= 0 for everyp ∈ π andξp ∈ Hp\pHp for all but finitelymanyp ∈ π .

Let us denote byCπ the cyclicQ-submodule ofSπ generated by1π . Observethat the cyclic subgroup〈1π 〉 of Cπ is non-discrete, henceCπ is a non-discrete


module isomorphic toQ as an abstract module. Therefore, the closure ofCπ in Sπcontains the subgroupHπ , so coincides withSπ being divisible. This is why in thesequel we can considerQ as a submodule ofSπ identifying it with Cπ . Moreover,the introduction of the moduleCπ allows us to prove the following proposition.

Proposition 2.6. The continuous group automorphisms ofSπ are the multiplica-tions by the invertible elements ofSπ . In particular, they are topological isomor-phisms.

Proof. SinceSπ is a topological ring, the multiplications by invertible elementsare obviously continuous automorphisms. On the other hand, letα :Sπ → Sπbe a continuous group automorphism. If we defineα(1π ) := ξ , then obviouslyα|Cπ = mξ |Cπ . By the density ofCπ in Sπ we can conclude thatα = mSπ

ξ .Note that every continuous automorphism ofSπ is necessarily a topologicalisomorphism sinceSπ is σ -compact so that the open mapping theorem can beapplied [9, Theorem 5.29]. Consequently, the inverse ofα is continuous too, henceα−1 = mSπ

η , whereη = α−1(1π). Thusηξ = 1π , i.e., ξ is an invertible elementof Sπ . ✷Remark 2.7. If we consider the submoduleD(Hπ ) of K provided with theinduced topology,then the situation changes substantially. Indeed, ifh :D(Hπ)→D(Hπ) is a continuous group homomorphism with respect to the inducedtopology, then it extends to a continuous endomorphism ofK (sinceD(Hπ )

is dense inK by virtue of Proposition 2.1), hence it is the multiplication by anon-zero rational numberr. Thus the only continuous group homomorphismsD(Hπ) → D(Hπ) are those of the formmD(Hπ )

r , with r ∈ Q. This fact showsthat D(Hπ ) cannot be a topologicalHπ -module with respect to the naturalmultiplication (otherwise, the multiplications by all elements ofHπ would becontinuous too). The above observation also implies that the topology ofD(Hπ )

induced byK is not a ring topology, while this is true for the smaller subringHπ .

We give now the counterpart of Proposition 2.6 concerning the automorphismsof theabstract ringsSπ , π ⊆ P.

Theorem 2.8. Let π and π ′ be sets of primes and letϕ :Sπ → Sπ ′ be a ringisomorphism. Thenπ = π ′ andϕ = id.

Proof. The case when the setsπ = {p} andπ ′ = {q} are singletons seems to bea widely known fact from number theory. This is why we are giving here only abrief sketch of its proof that splits into two steps (the reader can find a proof of thetheorem in full detail in [4] in the more general setting for commutative rings).First one shows thatp = q since otherwise there exists an integern ∈ Z that is asquare in one of the fieldsQp andQq , butn is not a square in the other. Then, an


application of Hensel’s lemma entailsϕ(Zp) ⊆ Zp , hence continuity ofϕ. Nowthe density ofQ in Qp andϕ|Q = idQ, imply ϕ = idQp .

Now assumeπ = π ′ = P, i.e., ϕ ∈ Aut(S). Note that1π , for π ⊆ P, areallidempotents ofS and the principal ideal ofS generated by1π is preciselySπ with

S = Sπ × SP\π . (4)

Clearly,ϕ sends idempotents to idempotents, preserves the order of idempotents(recall thate � e′ for two idempotentse, e′ ∈ S when e · e′ = e) and fixes theminimal idempotents1p (by the first part of the argument). Henceϕ fixes allidempotents1π of S. Moreover,ϕ|Qp = idQp for all p ∈ P. SinceAp := SP\{p}is the annihilator of1p in S, we haveϕ(Ap) = Ap and

⋂p∈PAp = 0. Hence, to

verify x = ϕ(x) for everyx ∈ S, it suffices to check thatx − ϕ(x) ∈Ap for everyp ∈ P. Indeed, forx = (xp)p∈P ∈ S, the decomposition (4) withπ = {p} givesx = u+ v with u ∈ Sp , v ∈Ap . Consequently,ϕ(x)= u+ ϕ(v), asϕ(u)= u andϕ(v) ∈Ap. This givesx − ϕ(x)= v − ϕ(v) ∈Ap for everyp ∈ P.

In the general case one has to get firstπ = π ′ from the fact that1π ′ =ϕ(1π) = 1π (see above). Finally, to proveϕ = idSπ it suffices to combine (4)and Aut(S)= {1}. ✷

The next lemma follows from Theorem 1.1 and from the choice of the topologyon Sπ .

Lemma 2.9. (a) LetX ∈ LQ. ThenSsupp(X) is isomorphic to a closed submoduleofX0.

(b) LetX = X0 admit a continuous monomorphismi :X ↪→ K. Theni(X) �Ssupp(X).

Proof. (a) Follows from the definition ofX0 in (1).(b) Let K be a compact open subgroup ofX, so thatK ∼= ∏

p∈P Znpp and

X = D(K). Sincei is a monomorphism,np(X) � 1 for everyp ∈ P. Note thatthe restriction ofi to the compact subgroupK of X is still a monomorphism andconsequently,K ∼= ∏

p∈π Zp , whereπ = supp(X). LetN = i(K). ThenN ∼= Hπ

sinceHπ∼= K by the definition ofHπ . Consider the canonical homomorphism

f :K → K/H ∼= T. Being a totally disconnected compact subgroup ofT, f (N) isfinite, then there existsm ∈ N such thatm ·N ⊆ H. Since tdp(N)= 0 forp ∈ P\π ,this givesm ·N ⊆ Hπ . Now we can claim thati(X)=D(N) �D(Hπ)= Sπ . ✷Remark 2.10. With a slight modification of the proof of item (b) of the abovelemma, one can show that for any pair of embeddingsι, ι′ :

∏p∈P Zp → K

satisfying the conditionK/Im(ι)∼= T ∼= K/Im(ι′), there exists a non-zero rationalr such thatι′ = rι (nowπ = P and after provingm ·N ⊆ H as above, one has toexchange the roles ofN andH to get alsom′ · H ⊆N for somem′ ∈ N).


3. The involutions κ : K → K

The continuity of an involutionκ :K → K is a rather stringent condition.This is why we consider in this section some weaker versions of continuity, e.g.,continuity of therestrictionsof κ (to subgroups ofH in Section 3.1 and toL inSection 3.2). SinceK = L+S, it is not surprising that combining all these weakerversions we obtain continuity ofκ (cf. Theorem 3.9).

3.1. The ideal of an involution

Lemma 3.1. For an involutionκ of K and a non-empty subsetπ ⊆ P the followingare equivalent:

(1) κ |Hπ is continuous, i.e.,π ∈ I(κ);(2) κ |Sπ is continuous;(3) there exists an invertible elementξ = (ξp)p∈π ∈ Sπ such thatκ |Sπ = mξ .

Proof. (3) → (2) is obvious. Moreover, conditions (1) and (2) are equivalentsinceHπ is open inSπ . To prove the implication (2)→ (3), assume thatκ |Sπis continuous. Letp ∈ π . SinceSp = tdp(K) is a functorial subgroup ofK, onehasκ(Hp)� Sp , hence there exists a non-zeroξp ∈ Sp such thatκ |Sp = mξp . Byκ2 = s · idK, with s non-zero rational, we conclude thatξ2

p = s ∈ Sp . This entailsthatξp ∈ Hp\pHp for all but finitely manyp ∈ π . Thusξ = (ξp)p∈π ∈ Sπ is aninvertible element ofSπ . To conclude thatκ |Sπ = mξ , it suffices to note that thesubmoduleM = ⊕

p∈π Sp of Sπ is dense andκ |M = mMξ . ✷The next corollary directly follows from item (3) of the above lemma.

Corollary 3.2. I(k) is an ideal of2P for every involutionκ of K.

Remark 3.3. (a) Note that theξp ’s of the above lemma are all±1p whenκ2 = idK.

(b) If π is infinite andκ is supposed only to be an automorphism ofK,then the continuity ofκ |Hπ always implies thatκ |Sπ is continuous, but now theelementξ ∈ Sπ such thatκ |Sπ = mξ , need not be invertible. Indeed, ifκ is notan involution, one may haveκ(Sπ) < Sπ . For example, ifπ = P andξp = p forevery primep, thenκ is a topological isomorphism betweenS and its image. Aninvolutionκ cannot have such a “degenerate” behaviour.

For an idealI of 2P define thesupportof I as the set supp(I) := {p ∈ P:{p} ∈ I }. Clearly, the principal ideal of 2P generated by supp(I) containsI . Forthe idealI consider now the abstract submoduleAI := ∑{Sπ : π ∈ I } of K. Weshall equip this submodule with the topology induced bySsupp(I ).


Corollary 3.4. For an involutionκ of K and an idealI of 2P with Π = supp(I)the following are equivalent:

(1) there exists an invertible elementξ = (ξp)p∈Π ∈ SΠ such thatκ |AI = mξ ;(2) κ |AI is continuous;(3) I(κ)⊇ I .

Proof. The implications 1→ 2 → 3 are obvious. The implication 3→ 1 followsfrom the above lemma.✷Remark 3.5. Note that continuity of the restrictionκ |SΠ always impliescontinuity of the restrictionκ |AI but, as we shall see later (cf. the proof ofLemma 3.6), the converse is not true.

Note that the ideal of an involutionκ of K depends on its restriction toS.Therefore, one can define the ideal of any involutionκ of S.

Lemma 3.6. Let I be an ideal of2P. Then there exists an involutionκ of S suchthatI(κ)= I andκ(1)= −1.

Proof. SetΠ := supp(I) and observe thatS splits asS = SΠ × SP\Π . We shalldefine an involutionκ of S such that:

(a) κ restricted toSp is discontinuous for everyp ∈ P,p /∈Π ;(b) κ restricted toAI is continuous (this will guarantee the inclusionI(κ) ⊇ I

according to Corollary 3.4);(c) κ restricted toSπ is continuous for no infiniteπ ⊆Π such thatπ /∈ I ;(d) κ(1)= −1.

Since (a) involves only the restriction ofκ to SP\Π , we construct first thisrestriction ofκ . In Step 2 we build the restriction ofκ to SΠ , involving (b) and (c).

Step 1.To guarantee (a) fix a splittingSp = Cp ⊕ Dp for everyp ∈ P\Π . Nowdefineκ(1p)= −1p and letκ be the identity on the complementDp . This definesκ onM = ⊕

p∈P\Π Sp . If |P\Π | < ∞, thenM = SP\Π . If P\Π is infinite, thenM ∩ CP\Π = 0. SinceM is a direct summand ofSP\Π , we can find a complementN of M in SP\Π containingCP\Π . Now we can extendκ to the wholeSP\Π bysettingκ |N = −idN . In particular,κ(1P\Π) = −1P\Π , where1P\Π is defined asin Section 2.3.

Step 2.WhenΠ ∈ I , to guarantee (b), letκ |AI coincide with−idAI . Now (c) isvacuously satisfied andκ(1Π)= −1Π .


We assume from now on thatΠ /∈ I . To guarantee (b) in this case, letκ |AI bethe identity ofAI .

Our leading idea to guarantee (c) will be the following. Consider an infinitesubsetπ ⊆ Π such thatπ /∈ I . Thenκ |Sπ is continuous if and only if it is acontinuous extension of its restriction on the direct sum

⊕p∈π Sp where it is

already determined as the identity. Hence, to ensure discontinuity ofκ |Sπ for someinfinite π ⊆ Π , it suffices to haveκ(1π) �= 1π . We consider now familiesF ofinfinite subsetsπ ⊆ Π such thatπ /∈ I . For such a familyF setEF = ∑{Cπ :π ∈ F}. We shall be interested in familiesF that satisfy

EF ∩AI = (0). (5)

It is easy to see that the singleton-familyF = {Π} satisfies (5). Indeed, assumethat k · 1Π ∈ AI for some non-zero integerk. Then there existsπ ∈ I such thatk · 1Π ∈ Sπ and consequentlyΠ ⊆ π ∈ I . This contradicts the choice ofΠ .

Since (5) is checked on finite subsets ofF , one can easily show that thefamiliesF satisfying (5) form an inductively ordered (with respect to inclusion)set. Hence we can apply Zorn’s lemma to claim that there exists amaximalfamilyF containing{Π} and satisfying (5). Then, by definition, the sumB =EF +AI isdirect. In order to extendκ toB, defineκ(x)= −x for everyx ∈EF . In particular,κ(1Π)= −1Π . Moreover, forx ∈ B

κ(x)= x iff x ∈AI . (6)

Indeed, if x = y + z, with y ∈ EF and z ∈ AI , then κ(x) = −y + z, so thatκ(x)= x yieldsy = 0.

Let us prove next thatB = (∑

π⊆Π Cπ) + AI . Indeed, the inclusion “⊆”follows fromEF ⊆ ∑

π⊆Π Cπ , so it remains to prove the other inclusion. SinceAI contains1π for all π ∈ I , it suffices to show that1π ∈B for all infiniteπ ⊆Π

such thatπ /∈ I . If π ∈ F then clearly1π ∈ B. Assume thatπ /∈ F . Then, bythe maximality ofF , there must exist a finite subsetF0 = {π1, . . . , πn} ⊆F suchthat{π}∪F0 does not satisfy (5). SinceF0 satisfies (5), this yields that there existnon-zero integersk, ki such thatk ·1π −∑n

i=1 ki ·1πi ∈ Sπ ′ for someπ ′ ∈ I . Thisyields1π ∈B.

To see thatκ defined in this way onB satisfies (c), take infiniteπ ⊆ Π

such thatπ /∈ I . Then1π ∈ B and1π /∈ AI . Hence (6) applied tox = 1π givesκ(1π) �= 1π and consequentlyκ |Sπ is discontinuous. This ensures (c) as waspointed out above.

Our next goal is to extendκ to the whole moduleSΠ . To this end, observethat the restriction ofκ to the direct summandB of SΠ has already been definedabove. Now we can extendκ to the wholeSΠ by settingκ |Z = idZ whereZ isa complement ofB in SΠ . This completes Step 2.

Now it remains to extendκ to the whole moduleS so that (d) holds. This ispossible sinceS splits asS = SΠ × SP\Π , so that the involutionκ is completelydetermined by Steps 1 and 2. Moreover, condition (d) is satisfied asκ(1Π)= −1Πandκ(1P\Π)= −1P\Π .


Items (a)–(c) entail thatκ is an involution ofS with idealI(κ)= I . ✷Corollary 3.7. LetI be an ideal of2P. Then there exists an involutionκ of K suchthatI(κ)= I .

3.2. L-continuous involutions

Definition 3.8. An involutionκ :K → K is calledL-continuousif the compositionκ ◦ χ is continuous for every continuous homomorphismχ :R → K.

It follows immediately that if the restriction ofκ to L is continuous, thenκ isL-continuous, so this latter property is weaker than continuity of the restrictionκ |L. According to Lemma 2.2,κ is L-continuous if and only if for everycontinuous homomorphismχ :R → K there existsρχ ∈ R such thatκ ◦ χ =ψ ◦ mR

ρχ.

The following theorem shows the importance of this new notion.

Theorem 3.9. Letκ be an involution ofK. Then the following are equivalent:

(a) κ is continuous;(b) κ is L-continuous andI(κ)= 2P.

Proof. Obviously, (a) implies (b). Assume thatκ is L-continuous andI(κ)= 2P.One can introduce onL the structure of anR-module by definingρ · x = ψ(ρz)

for ρ ∈ R andx = ψ(z), z ∈ R. In such a case, theL-continuity of κ impliesthatκ |L coincides with the multiplicationmL

ρ for someρ ∈ R. On the other hand,by Lemma 3.1, there exists an invertible element(ξp)p∈P ∈ S such that fory =(yp)p∈P ∈ S the involutionκ :S → κ(S) ⊆ K is defined byy �→ κ(y) = (ξpyp).By L ∩ S = CP, κ(L)= L (sinceκ is L-continuous) andκ(S)= S, we concludethatκ(〈1〉)⊆ CP. In particular,κ(1)= ρ ·1 yieldsρ ∈ Q since theR-moduleL istorsion-free. On the other hand,κ(1)= (ξp) = ρ · 1. This means thatξp = ρ foreveryp ∈ P. Henceκ = mK

ρ sinceK = L + S. To finish the proof it is enough toobserve that this obviously entails continuity ofκ . ✷Corollary 3.10. Assume thatI(κ)= 2P. Thenκ is continuous if and only if it isL-continuous.

Remark 3.11. (a) Let π ⊆ P. By Proposition 2.6, a continuous involutionαof Sπ (i.e., the continuous automorphismsα :Sπ → Sπ such thatα2 = idSπ ) is the

multiplicationmSπε by someε = (εp)p∈π ∈ Sπ such thatε2 = 1π , i.e.,εp = ±1p

for everyp ∈ π . We callmSπε thestandard involutionof Sπ associated toε.

In the same way, for everyπ �= P and every involutionmSπε of Sπ , we callstan-

dard involutionof L + Sπ (associated toε) the involutionκε :L + Sπ → L + Sπ


that coincides onSπ with the standard involution ofSπ associated toε and withidL on L.

(b) Let us see that if the restriction ofκ to the subgroupL of K is continuouswith I(κ) �= P, then this substantially improves the global properties ofκ , whichis due to the fact that the real numberρ in (3) is actually rational. Indeed, beinga continuous homomorphism,κ |L sendsL to L so it extends to a continuoushomomorphism̃κ :K → K. Thenκ(L) = κ̃(L) = L and κ̃ = mK

r for some non-zero rationalr. On the other hand, by the definition of involution ofK, there existsa non-zero rationals ∈ Q such thatκ2 = s · idK and consequently,r2 = s. Now,

for everyp ∈ π ∈ I(κ), the restrictionκ |Sp coincides withmSpξp

for some non-

zero ξp ∈ Sp . Moreover,ξ2p = s = r2. Therefore, for everyp ∈ π , there exists

εp = ±1p such thatξp = εpr. Thusκ |L+Sπ = r · κε, whereκε is the standardinvolution ofL + Sπ associated toε = (εp)p∈π .

(c) If κ is an involution ofK in the usual sense, i.e.,κ2 = idK (see Remark 5.1),then we have the equalityξ2

p = s = 1p , wherep ∈ π ∈ I(κ), for free without anycontinuity assumption on the restrictionκ |L. So that in this case we can claim thatκ itself is a standard involution ofK.

Theorem 3.12. Let I be an ideal of2P. Then:

(a) there exists an involutionκ of K such that I(κ) = I and κ is notL-continuous;

(b) there exists anL-continuous involutionκ of K such thatI(κ)= I ;(c) if I �= 2P, there exists anL-continuous involutionκ of K such thatI(κ)⊇ I

andκ is discontinuous.

Proof. By Lemma 3.6 there exists an involutionκ of S such thatI(κ) = I andκ(1)= −1. Therefore, in order to define a nonL-continuous involution ofK withidealI , it suffices to extendκ onL so thatκ |L �= −idL. To define anL-continuousinvolution ofK, it is enough to extendκ onL in such a way thatκ |L = −idL. Thissettles (a) and (b).

(c) Note thatP /∈ I since by assumptionI �= 2P. This yields L ∩ AI ⊆L ∩ S ∩ AI ⊆ 〈1〉 ∩ AI = (0). Therefore, the direct summandL of K admitsa complementN containingAI . Let us defineκ |L = −idL andκ |N = idN , sothatI(κ) ⊇ I andκ is L-continuous. Note finally thatκ is not continuous sincethe only continuous automorphisms ofK are the multiplicationsmK

r by non-zerorationalsr ∈ Q while κ |L = −idL andκ |N = idN . ✷


4. Finitely closed subcategories of LQ and their dualities

4.1. The ideal of a finitely closed subcategory ofLQ

In the sequel we consider dualities of finitely closed subcategoriesL of LQ,i.e., contravariant involutive functorsL→ L. In order to see that any such dualitypreserves the smaller finitely closed subcategories ofLQ, we define now a secondinvariant of such subcategories ofLQ. The first one was the idealI(L) of theBoolean algebra 2P defined in Section 1.2 (having as elements the sets of theform supp(X) whenX varies inL). We observe thatP ∈ I(L) if and only if Lcontains theQ-moduleS.

Remark 4.1. To an ideal I of 2P we may assign also the finitely closedsubcategoryY(I) generated byDQ,CQ and allSπ with π ∈ I . ThenY(2P) isthe finitely closed subcategory ofLQ generated byS and

Y(I(L)

) ⊆ L⊆ sat(L)

holds for every finitely closed subcategoryL. This shows that the assignmentL �→ I(L) is not one-to-one as all three categories have the same ideal. Inparticular, one cannot describe all finitely closed subcategories ofLQ by meansof the idealsI(L).

In order to describe all finitely closed subcategories ofLQ, one has to considerideals of the following latticeL instead of ideals of the Boolean algebra 2P.Let P0 = {0} ∪ P and L = NP0 = N × NP equipped with the lattice structuregiven by the natural order ofN. For X = Rn × Kα × Q(β) × D(

∏p∈P Z

npp ),

let φ(X) = (n, (np)) ∈ L. WhenL is a finitely closed subcategory, then the set{φ(X): X ∈L} is an ideal ofL that we denote byJ (L). The idealJ (L) uniquelydetermines the subcategoryL. Indeed, for every idealI of L one can definethe finitely closed subcategoryLI in the following way. Forf ∈ I, let Xf be

the moduleRf (0) × D(∏

p∈P Zf (p)p ) and now letLI := {Xf × D × C: f ∈ I,

D ∈ DQ, C ∈ CQ}. Varying the idealI of L, we get throughLI all finitely closedsubcategories ofLQ, i.e., one hasL = LJ (L). This proves the next lemma.

Lemma 4.2. The correspondenceL �→ J (L), I �→ LI defines a bijectionbetween finitely closed subcategories ofLQ and ideals of the latticeL.

This lemma shows that the class of all finitely closed subcategories ofLQ is infact a set which splits in two disjoint subsets. Namely, the setF of all finitelyclosed subcategoriesL of LQ not-containing theQ-moduleR (i.e., such thatn(X) = 0 for everyX ∈ L), and the setF′ of all finitely closed subcategoriesL of LQ containing theQ-moduleR (i.e., such thatn(X) �= 0 for someX ∈L).


If we order the finitely closed subcategoriesL ∈ F by inclusion, then thecategoryL(P) = {X ∈ LQ: n(X) = 0} turns out to be the top element ofFwhile the bottom element is obviously the categoryL0

Q. Moreover, the setF is

a complete lattice where the join∨

λ∈ΛLλ of a family(Lλ)λ∈Λ in F is the finitelyclosed subcategoryL = 〈Lλ: λ ∈Λ〉 generated by allLλ, λ ∈Λ. It is easy to seethatL coincides with the subcategory of all modulesZ ∈ LQ for which there existλ1, . . . , λn ∈ Λ, Xi ∈ Lλi , closed submodulesY � X1 × · · · × Xn, andY1 � Y

such thatZ ∼= Y/Y1. In particular, it follows thatI(∨

λ∈ΛLλ) coincides with theideal of 2P generated by allI(Lλ), λ ∈Λ.

As we have mentioned above, one cannot describe all finitely closed sub-categories ofF by means of the idealsI(L). Nevertheless, the correspondenceL �→ I(L), I �→ X(I) restricted to the subclassS of F of saturated finitely closedsubcategories, defines a bijection between the classS and the complete lattice ofall ideals of the Boolean algebra 2P. In particular, the finitely closed subcategoryL(P) is the top element ofS with idealI(L(P))= (P)= 2P.

Analogously, one establishes that also the setF′ is a complete lattice withtop element the whole categoryLQ and bottom element the finitely closedsubcategoryLcon. According to Theorem 1.1, every moduleX ∈ L ∈ F′ admitsa canonical factorization of the formX ∼= Rn × Kα × Q(β) × X0. Since all theconsidered subcategories are finitely closed, this implies that the assignments

Fτ

F′σ

defined by

L �→ τ (L) := 〈L,R〉, L′ �→ σ(L′) := {X/Rn(X): X ∈L′}

determine an isomorphism between the latticesF andF′ = τ (F). In particular,τ (L(P))= LQ andτ (L0

Q)= Lcon.

The next diagram represents the finitely closed subcategories ofLQ as thedisjoint unionF∪F′, where the saturated ones are placed on the vertical segmentsS and τ (S) denoted by double parallel lines, whileL (respectivelyτ (L))represents a generic member ofF (respectivelyF′):

LQ

F′τ (sat(L)) L(P)

Fτ (S) τ (L) sat(L)

Lcon S L

L0Q


4.2. Dualities of a finitely closed subcategory ofLQ

Let L be a finitely closed subcategory ofLQ. Thenφ(X∗) = φ(X) for X =Rn×Kα×Q(β)×X0 ∈ LQ, sinceX∗ ∼= Rn×Kβ ×Q(α)×X0. By Lemma 4.2 thesubcategoryL is determined by its idealJ (L), so this yields that the Pontryaginduality ∗ preservesL. This can be obtained also from the following proposition.

Proposition 4.3. Let L be a finitely closed subcategory ofLQ and let# :L → Lbe a duality. Then:

(a) # sendsDQ to CQ and vice versa; in particular, K# ∼= Q andQ# ∼= K;(b) if R ∈ L, thenR# ∼= R;(c) if Qp ∈L for somep ∈ P, thenQ#

p∼= Qp;

(d) n(X#) = n(X), np(X#) = np(X), α(X#) = β(X), and β(X#) = α(X) foreveryX ∈ L and everyp ∈ P;

(e) X# ∼=X wheneverX =X0 ∈L.

Proof. We note first that due to the closure properties of finitely closedsubcategories ofLQ, a homomorphismf :X → Y in L is an epimorphism if andonly if f (X) is dense inY andf is a monomorphism if and only iff is injective.Therefore,# sends kernels to cokernels and vice versa (see [2, Proposition 1.5]). Inparticular,# sends embeddings to quotient maps and vice versa, preserves shortexact sequences and finite direct products. Hence# sends topologically simplemodules to topologically simple modules.

(a) The first part follows from [2, Proposition 1.6]; the isomorphismsK# ∼= Q

andQ# ∼= K follow from Corollary 2.4.(b) SinceR is topologically simple, non-compact and non-discrete, alsoR#

has the same properties. IfR# were isomorphic toQp for some primep, then thering R = ChomQ(R,R) would be isomorphic to the ringQp = ChomQ(Qp,Qp)

—a contradiction. By Corollary 2.4,R# ∼= R. This proves (b).(c) SinceQp is topologically simple as aQ-module,Q#

p is topologically simpletoo. Q#

p cannot be discrete or compact, hence we getQ#p

∼= R or Q#p

∼= Qq forsome primeq ∈ P. By item (b)R# Qp, henceQ#

p R## ∼= R. Therefore, weconclude thatQ#

p∼= Qq for some primeq ∈ P. This gives rise to the following

ring isomorphism:

ChomQ

(Q#p,Q

#p

) ∼= ChomQ(Qq,Qq)∼= Qq .

On the other hand, by the properties of a duality

ChomQ

(Q#p,Q

#p

) ∼= ChomQ(Qp,Qp)∼= Qp.

By Theorem 2.8 the isomorphismQp∼= Qq implies p = q . This gives an

isomorphismQ#p

∼= Qp of topologicalQ-modules.


To prove (d) note that if there exists an embeddingRm ↪→ X, then we geta quotient mapX# → Rm in view of (b). This proves thatn(X#) � n(X).SinceX ∼=X##, we getn(X#)= n(X). Analogously, every embeddingQm

p ↪→X

produces a quotient mapX# → Qmp by (c). Hencenp(X#) � np(X) and we

conclude as before thatnp(X#)= np(X) for everyp ∈ P. In the same way, everyembeddingKα ↪→ X (Q(β) ↪→X), gives a quotient mapX# → Q(α) (respectivelyX# → Kβ ). Henceβ(X#) � α(X) (respectivelyα(X#) � β(X)). Arguing asabove, we getβ(X#)= α(X) andα(X#)= β(X).

To prove (e) note that ifX = X0, then n(X) = 0 and X is completelydetermined by its invariantsnp(X). Now (d) applies to giveX# ∼=X. ✷

The next corollary directly follows from Lemma 4.2 and the above proposition.It shows that smaller subcategories may easier have discontinuous dualities.

Corollary 4.4. For finitely closed subcategoriesL ⊇ L′ of LQ, every duality# :L → L sendsL′ to itself. In particular, ifL admits a discontinuous duality,then so doesL′.

Proof. To see that# sendsL′ to itself, it suffices to note thatφ(X) determines amoduleX ∈L′ up to isomorphism, hence every module of the formX = Rn×X0

is autodual, i.e.,X# ∼= X ∈ L′. To conclude, observe that (dis)continuity ischecked by the restriction of# to L0

Q⊆ L′. ✷

4.3. The equivalenceµ of a duality of a finitely closed subcategory ofLQ

Now we prove that ifL is a finitely closed subcategory ofLQ and# :L → Lis a duality with involutionκ :K → K, then there exists a concrete equivalenceµ :L → L (i.e.,µ(X) andX have always the same underlying module) havingthe same properties as the natural equivalence associated to a duality of the wholecategoryLQ introduced by Prodanov (cf. also [2, Section 6]):

Lemma 4.5. Let L be a finitely closed subcategory ofLQ and let # :L → Lbe a duality. Then there exists an involutive covariant concrete equivalenceµ :L→ L associated to#, such that:

(a) µ is exact and preserves finite products;(b) µ is identical onDQ;(c) for everyX ∈ L a homomorphismχ :X → K is continuous if and only if

κ ◦ χ :µ(X)→ K is continuous;(d) a homomorphismf :X → Y is continuous if and only if the homomorphism

f :µ(X)→ µ(Y ) is continuous.


Proof. We argue as in [2, Theorem 6.2] to defineµ(X) via property (c) andVaropoulos’ Theorem [20]. According to this theorem, one can recover thetopology of a locally compact moduleX knowing its charactersχ :X → K, i.e.,if X,Y ∈ LQ have the same underlying abstract module and the same characters,then they coincidetopologically. More precisely, the topological moduleµ(X)will be the moduleX equipped with a locally compactQ-module topology suchthat theK-characters ofµ(X) are precisely the module homomorphisms of theform κ ◦ χ whereχ is anyK-character ofX. To see that there exists a locallycompactQ-module topology onX with these continuous characters, just chooseY such thatX =∆(Y ) (up to isomorphism,Y is the Pontryagin dual ofX). Thenthe abstract moduleX is the carrier of both∆(Y ) andY #. Now it suffices to notethat everyχ ∈∆(X)=∆(∆(Y )) has the formχ = ωY (y) for somey ∈ Y , so thatκ ◦χ = κ ◦ωY (y)=EY (y) holds by the definition ofκ . This proves the existenceof µ(X) and gives a concrete covariant functorµ :L → LQ. Finally, note thatthe modulesX andµ2(X) have the same underlying abstract module and thesameK-characters (sinceκ2 is continuous), therefore they coincide topologically.Henceµ is an involution withµ(CQ)= CQ, µ(DQ)=DQ. Observe now as in [2,Theorem 6.2] that, up to composition with the natural equivalenceω,µ can beviewed as a composition of two dualities, namely# and the Pontryagin duality∆.Indeed, if Y = ∆(Z), then the topology induced onZ through the algebraicisomorphismωZ :Z → ∆(Z)# is precisely that ofµ(Z). This yields thatµsatisfies the properties (a)–(d) andL is invariant underµ by Proposition 4.3. ✷

In the sequel, we denote the moduleµ(X) also byXµ.The next lemma gives some necessary conditions that the equivalenceµ

associated to a duality must satisfy.

Lemma 4.6. For a finitely closed subcategoryL of LQ, a duality# :L → L withinvolutionκ and associated equivalenceµ, the following hold:

(a) if Sπ ∈Lπ for someπ ⊆ P, thenµ(Sπ)= Sπ andπ ∈ I(κ);(b) if Qp ∈L for somep ∈ P, thenµ(Qp)= Qp;(c) if R ∈ L, thenµ(R)= R;(d) n(µ(X))= n(X) andnp(µ(X))= np(X) for everyX ∈L and everyp ∈ P;(e) more generally, ifX =X0 ∈L, thenµ(X)=X.

Proof. (a) From item (e) of Proposition 4.3 we getS#π

∼= Sπ . Therefore,µ(Sπ) ∼= Sπ topologically sinceµ is the composition of two dualities. Letj :µ(Sπ) → Sπ be a topological isomorphism. To finish the proof we argueas in [1, Lemma 2.4]. Indeed, for everyξ ∈ Sπ , let us consider the followingcommutative diagram:


µ(Sπ)j

mξ

Sπ

h

µ(Sπ)j

Sπ

(7)

whereh = j ◦ mξ ◦ j−1. Sincemξ :Sπ → Sπ is continuous and the underlyinggroup ofµ(Sπ) is Sπ , from item (d) of Lemma 4.5 it follows thatmξ :µ(Sπ)→µ(Sπ) is continuous too. Henceh :Sπ → Sπ is continuous as a compositionof continuous maps. Then there exists an elementη ∈ Sπ such thath is themultiplication byη, i.e.,h= mη. Therefore,

j (ξx)= j(mξ (x)

) = h(j (x)

) = ηj (x) (8)

for everyx ∈ Sπ . Clearly,η is uniquely determined byξ and the mapϕ :Sπ → Sπdefined byϕ(ξ) = η is a ring automorphism ofSπ . Therefore,ϕ = idSπ byTheorem 2.8. Now, settingx = 1π in (8), we getj (ξ)= ξj (1π) for everyξ ∈ Sπ .Hencej preserves the topology ofSπ being the multiplication byj (1π) in Sπ . Inother words,µ(Sπ)= Sπ . To proveπ ∈ I(κ), we have to see that the restriction ofκ to Sπ is continuous. This follows from (c) of Lemma 4.5, sinceκ |Sπ coincideswith the composition ofκ with the continuous inclusion characterSπ ↪→ K.

(b) It follows from (a) withπ = {p}.(c) By Proposition 4.3 we can conclude as above thatµ(R)∼= R topologically.

Now argue as in (a) to find a ring automorphism ofR. To finish the proof, itsuffices to observe that the only ring endomorphism ofR is the identity (sincethe positive reals are squares inR and consequently, every endomorphism ofR

is order-preserving, hence continuous with respect to the archimedean absolutevalue).

(d) Follows from (b) and (c).(e) Let X = D(

∏p Z

npp ). By (c) of Lemma 4.5 it suffices to see that for

an arbitrary continuous homomorphismχ :X → K the compositionκ ◦ χ iscontinuous too. Indeed, let us factorizeχ through the inclusioni :χ(X) ↪→ K,i.e., one hasχ = i ◦f wheref :X → χ(X). In the sequel we considerχ(X) withits quotient topology, so thatχ(X) ∈ L sinceX ∈ L. Let π = supp(χ(X)). By(b) of Lemma 2.9,χ(X) � Sπ . The restriction ofκ to Sπ is continuous by (a),hence the restriction ofκ to χ(X) is continuous too. Therefore,κ ◦ χ = κ ◦ i ◦ fis continuous too. This proves thatµ(X)=X. ✷

We see now thatµ does not alter the modulesX that have no compactsubmodules.

Corollary 4.7. If X ∈ L andα(X)= 0, thenµ(X)=X.


5. Proofs of the main theorems

5.1. The caseR = Q

Proof of Theorem 1.11. (a) Here we prove the necessity of (3) and (2).AssumeR ∈L. By Lemma 4.6µ(R)= R hence item (c) of Lemma 4.5 applied

to ψ :R → K, yields that the compositionφ = κ ◦ ψ :R → K is continuous.By Lemma 2.2 there exists a real numberρ ∈ R such thatφ = ψ ◦ mρ , i.e.,κ(x)= ρ · x for everyx ∈ L. This proves the necessity of (3).

The necessity of (2) was proved in item (a) of Lemma 4.6 (indeed, ifπ ∈ I(L),thenSπ ∈L so thatπ ∈ I(κ), loc. cit.).

(b) To prove the sufficiency, assume that the involutionκ :K → K satisfiesconditions (3) and (2).

We shall define the duality through its natural equivalenceµ, so that we startthe construction ofµ on CQ. Following [2, Section 10], we note that there existsa unique functorial homomorphism

κ̃ :CQ → CQ

of the categoryCQ associated toκ . Namely, since every compact module(C, τ ) ∈CQ has the formKα for some cardinalα, for such a module set

κ̃C = κα :Kα → Kα.

Now equip C with the topology τµ that makesκ̃C : (C, τµ) → (C, τ )

a topological isomorphism and denote(C, τµ) by µ(C) or simplyCµ. Clearly,theK-characters of the compact moduleµ(C) are exactly the homomorphisms ofthe formκ ◦ χ , whenχ is aK-character ofC.

Therefore, aQ-homomorphismf between compact modulesC,C′ ∈ CQ iscontinuous if and only iff :Cµ → C′

µ is continuous. In this way we get anadditive covariant equivalence

µ :CQ → CQ

that we will extend to an involutive equivalenceµ of L.To this end, it suffices to observe that every moduleX ∈ L decomposes as

X = X1 × Kα , whereX1 ∈ L has no compact submodules beyond 0 (i.e.,X1 =Rn × Q(β) ×X0 or, equivalently,α(X1)= 0). The construction of the restrictionof µ to CQ was described above so it remains to define the effect ofµ on modulesof the formX1. For these modules we setµ to be identical, i.e.,µ(X1) = X1.This means thatµ does not alter the topologyσ1 of X1. Consequently, for everymoduleX ∈L, we have

µ(X) = µ(Rn

) ×µ(Q(β)

) ×µ(X0)×µ(Kα

)= Rn × Q(β) ×X0 ×µ

(Kα

).


Therefore, we get a concrete functor

µ :L→L, X �→ µ(X)=X1 ×µ(Kα

),

whereµ(X) is equipped with the product topologyσ1 × τµ (andτ , as before, isthe topology ofKα).

Let us see that the definition ofµ(X) does not depend on the decomposition ofthe moduleX. Indeed, suppose the moduleX decomposes also asX =X2 × Kα ,whereα(X2)= 0. Letσ2 be the induced topology ofX2 and letσ1 × τ, σ2 × τ bethe product topologies onX1 × Kα andX2 × Kα , respectively. Sinceσ1 × τ =σ2 × τ , by the implication(b3) → (b1) of Lemma 2.5, the homomorphismf :X1 → Kα associated to the direct summandX2 is continuous. Let us showthat the topologiesσ1 × τµ andσ2 × τµ coincide. Indeed, since the composition(κα)−1 ◦f : (X1, σ1)→ (µ(Kα), τµ) is continuous, the graphΓ(κα)−1◦f =X2 is aclosed submodule ofµ(X). So we are in position to apply Lemma 2.5 and by theimplication(b1)→ (b3), we can conclude that the topologiesσ1 × τµ andσ2× τµcoincide. Therefore, the definition ofµ(X) does not depend on the decompositionof the moduleX.

The functorµ has the following properties:

(1) µ(R)= R if R ∈L, andµ(Qp)= Qp if Qp ∈L;(2) µ(X0)=X0 for everyX =X0 ∈L;(3) µ is an involutive covariant equivalence ofL;(4) µ is identical onDQ (actually,µ(X)=X for everyX ∈L with α(X) = 0);(5) µ is exact and preserves finite products.

To define onL a duality with involutionκ and equivalenceµ, we need to checkthe condition (c) from Lemma 4.5. To this end, let us consider the full subcategoryL′ of L consisting of all modulesX satisfying this condition (i.e., for everycontinuous characterχ :X → K alsoκ ◦ χ :µ(X) → K is continuous) and notethatL′ containsDQ andCQ. Moreover,X1,X2 ∈ L′ impliesX = X1 ×X2 ∈ L′since every characterχ :X1×X2 → K is the sum of the restrictionsχ1 :X1 → K,χ2 :X2 → K, andµ(X) = µ(X1) × µ(X2). Furthermore, ifR ∈ L, then alsoR ∈ L′ (henceRn ∈ L′ for everyn ∈ N). Indeed, by Lemma 2.2, there existsσ ∈ R

such thatχ = ψ ◦ mσ while, by condition (3),κ ◦ ψ = ψ ◦ mρ for someρ ∈ R.Therefore,κ ◦χ = κ ◦ψ ◦mσ = ψ ◦mρ ◦mσ =ψ ◦mρσ is continuous. Analogousargument shows thatL′ contains also every moduleX = X0 ∈ L. Indeed, ifχ :X → K is continuous, thenχ factorizes through the inclusionSπ ↪→ K (as inthe proof of Lemma 4.6(e)). Moreover, according to condition (2), the restrictionof κ to Sπ is continuous. This proves that the compositionκ ◦ χ is continuoustoo. Therefore,L′ = L.

Let us consider now the Pontryagin duality∆ and lets be the rational numbersuch thatκ2 = s · idK. We have seen that for everyX ∈ L one hasχ ∈ ∆(X) ifand only ifκ ◦ χ ∈∆(Xµ), so we get an isomorphism of abstract modules


ηX :∆(X)→∆(Xµ)

by settingηX(χ) = s−1κ ◦ χ for everyχ ∈ ∆(X). Since∆(X)µ has the sameunderlying abstract module as∆(X), one can see (as in [2, Theorem 6.7]) thatηX :∆(X)µ → ∆(Xµ) is a topological isomorphism (with inverseη−1

X given byξ �→ κ ◦ ξ for everyξ ∈ ∆(Xµ)) andη = {ηX: X ∈ L} is a natural equivalencebetweenµ ◦∆ and∆ ◦µ.

Define a duality# onL by settingX# :=∆(Xµ) for everyX ∈L.Note thatQ# = ∆(Qµ) = ∆(Q) = K. For everyX ∈ L define the canonical

isomorphismeX :X →X## as follows. SinceηX :∆(X)µ →∆(Xµ) is a topolog-ical isomorphism, thenηX : (∆(X)µ)µ = ∆(X) → ∆(Xµ)µ is an isomorphismtoo. Therefore,∆(ηX) sends∆(∆(Xµ)µ) = X## isomorphically onto∆∆(X).Consequently,eX = ∆(ηX)

−1 ◦ ωX :X → X## is a natural topological isomor-phism. This proves that# is an involutive duality onL with torusK. It remains tosee that the involution of# is κ . In fact, if χ ∈X# andx ∈X, then

eX(x)(χ) = (∆(ηX)

−1 ◦ ωX)(x)(χ)= (

∆(ηX)−1(ωX(x)))(χ)

= (ωX(x) ◦ η−1

X

)(χ)

and (ωX(x) ◦ η−1

X

)(χ)= ωX(x)

(η−1X (χ)

) = ωX(x)(κ ◦ χ)= κ(χ(x)

). ✷

Remark 5.1. It is proved in [12] that every duality# admits a representationsuch that the involutionκ :K → K satisfies the stronger propertyκ2 = idK.With such an involutionκ the numberρ ∈ R in (3) can be proved to be±1,i.e., κ |L = ±idL (actually, this is proved in [12] for dualities# :LR → LR witharbitrary commutative ringR).

Proof of Corollary 1.12. The first part of the assertion directly follows fromTheorem 1.11. Moreover, it implies that every duality ofL can be extended toa duality ofL′ if both finitely closed subcategories have the same saturation.Now assume that this fails to be true, i.e.,I = I(L) < I(L′). By Corollary 3.7we can build an involutionκ of K such thatI(κ) = I . Moreover, if R ∈ L,item (b) of Theorem 3.12 ensures thatκ can be chosen to be alsoL-continuous.By Theorem 1.11 there exists a duality# of L with involution κ , but there is noduality onL′ with involutionκ . Hence# cannot be extended toL′. ✷Proof of Theorem 1.5. Let L be a finitely closed subcategory ofLQ such that alldualities ofL are continuous. We will show thatL contains theQ-moduleR × S,i.e., the necessity of the conditionR×S ∈ L. SinceL is finitely closed, it sufficesto prove thatS ∈ L andR ∈ L. The proof of the necessity splits in two steps thatseparately dealS andR.

Step 1. Assume thatS /∈ L. Then I = I(L) �= 2P. According to (c) ofTheorem 3.12, there exists anL-continuous involutionκ of K such thatI(κ)⊇ I


andκ is discontinuous. Since conditions (3) and (2) of Theorem 1.11 are fulfilled,there exists a duality# of L with involutionκ . As κ is discontinuous, the duality# is discontinuous too—a contradiction. This proves thatS ∈ L.

Step 2.Assume now thatR /∈ L, i.e.,n(X) = 0 for everyX ∈ L. Hence we canassume without loss of generality thatL coincides with the biggest finitely closedsubcategoryL(P) of LQ defined with the conditionn(X) = 0 for everyX ∈ L.Then I = I(L) = 2P. By (a) of Theorem 3.12, there exists an involutionκ ofK such thatI(κ) = I andκ is not L-continuous. This ensures condition (2) ofTheorem 1.11 and the proof proceeds as above to produce a discontinuous dualityof L. This proves the necessity of the condition given in Theorem 1.5.

To prove the sufficiency, suppose thatL is a finitely closed subcategory ofLQ containing theQ-moduleR × S. Then, clearly,R ∈ L and S ∈ L. Let #

be a duality ofL with involution κ . By Theorem 1.11,κ is L-continuous andI(κ) ⊇ I(L) = 2P. Hence, by Theorem 3.9,κ is continuous. By definition, theduality # is continuous. ✷Proof of Theorem 1.7. Let # be a duality ofL with involution κ :K → K andnatural equivalenceµ. By Lemma 4.6, the action ofµ is essentially onCQ, i.e., forX ∈L with X = Rn ×Q(β) ×X0 ×Kα , one hasXµ = Rn ×Q(β)×X0 ×µ(Kα).Let us recall now that̃κ :µ(Kα)→ Kα , defined as in the proof of Theorem 1.11,is a topological isomorphism. Therefore, the isomorphismρX :Xµ → X definedby ρX = idRn × idQ(β) × idX0 × κ̃Kα is topological. It is easy to prove that{∆(ρX): X ∈ L} defines a natural equivalence between the Pontryagin duality∆ and the duality#. ✷5.2. The caseR = R,C

Let us consider now the caseR = R. To enhance the fact thatR carries thediscrete topology when considered as a ring, we write sometimesRd or simplyR.In this case, the torusT =R∗ is isomorphic toKc as a topological group. Arguingas in the case ofQ, we get a continuous monomorphismΨ :R → T with denseimageL such that every non-trivial continuous homomorphismR → T in LR hasimageL.

More precisely, the counterpart of Lemma 2.2 holds true. Finitely closedsubcategoriesL of LR can be introduced analogously. Furthermore, for everyduality # :L → L, one can introduce the involutionκ :T → T . The counterpartof Theorem 1.11, characterizing the involutionsκ :T → T that correspond todualities of finitely closed subcategories ofLR , has the following simple form:

Theorem 5.2. LetL be a finitely closed subcategory ofLR containingR and letκ :T → T be an involution. Then there exists a duality ofL with involutionκ ifand only ifκ |L = mLρ for someρ ∈ R.


Proof. The necessity of the conditionκ |L = mLρ , ρ ∈ R, is verified as in theproof of Theorem 1.11. To prove the sufficiency, one has to build a dualitycorresponding to a given involutionκ :T → T with this property.

Note that no power ofQp may belong toLR for any primep ∈ P. Indeed,if Qn

p ∈ LR , then there would exist a ring embeddingj :R ↪→ Mn(Qp) =ChomQ(Q

np,Q

np). Since every scalar multiplicationmρ , ρ ∈ R, commutes with

the continuous group endomorphisms,j (R) is contained in the center of the ringMn(Qp). This is impossible since the center is isomorphic toQp . Indeed, choosean integera > 0 that is not a quadratic residue modulop. Thena is not a square inQp , whilea = (

√a )2 with

√a ∈ R—a contradiction. Therefore, every module in

LR has the formRn × T α ×R(β). Thus the construction of Theorem 1.11 worksagain to produce a natural equivalenceµ defined by settingµ(Rn×T α ×R(β))=Rn ×µ(T α)×R(β), where, as before, the topology ofµ(T α) is defined as the onetransported fromT α by the functorial isomorphismκα . As in Theorem 1.11, thisgives the required duality ofLR . ✷

We observe that Remark 5.1 applies also toR = R, i.e., the numberρ ∈ R inTheorem 5.2 can be proved to be±1 so thatκ |L = ±idL.

The same argument and observation apply to the caseR = C. We are not givingthe formulations explicitly.

Proof of Theorem 1.13. We begin with the caseR = Rd . From the proofof Theorem 5.2, it follows that there are no proper finitely closed subcate-gories ofLR . Since the subcategoryL0

R always admits discontinuous dualities[2, Theorem 10.2], we can assume without loss of generality thatL = LR . As wehave mentioned above, the torusT = R∗ is topologically isomorphic toKc so ithas size 2c. Note that the submoduleL of T splits andT = L⊕M algebraically,with |M| = |T | = 2c. SinceM admits at least|M|> |R| involutions, there existsone, sayκ ′, that is not a multiplication by any real numberρ ∈R. Since all contin-uous endomorphisms ofT are multiplications by someρ ∈ R, we conclude thatκ ′ cannot be the restriction of a continuous endomorphism ofT . Furthermore, onecan choose the involutionκ :T → T such thatκ |M = κ ′ andκ |L = idL. Then, byTheorem 5.2, there exists a duality# :LR → LR with involutionκ :T → T . As κis discontinuous, the duality# is discontinuous too.

To finish the proof of the caseR = Rd , argue as in the proof of Theorem 1.7 toshow that every duality ofLR is naturally equivalent to Pontryagin’s duality.

The same argument works forR = C. ✷

6. The Galois correspondence related to dualities of finitely closedsubcategories of LQ

The above theorems suggest to define a Galois connection between the classof all finitely closed subcategoriesL of LQ and all involutionsκ of K that may


appear as an involution of some duality ofL. We will split the construction ofsuch a Galois correspondence in two steps.

Let us consider first the setF of all finitely closed subcategoriesL of LQ

non-containing theQ-moduleR introduced in Section 4.1. LetI be the set of allinvolutions ofK.

Definition 6.1. Given a subcategoryL ∈ F and an involutionκ ∈ I, we call thepair (L, κ) compatibleif κ appears as an involution of some duality ofL. In sucha case, we say thatL is compatible withκ andκ is compatible withL.

Roughly speaking, the Galois correspondence is obtained by assigning to eachL ∈ F the setϕ(L) of all involutionsκ compatible withL and by assigning toeach involutionκ ∈ I the set of all subcategoriesL ∈ F that are compatible withκ (note thatX(I(κ)) is the largest saturated finitely closed subcategory ofF whereκ may appear as an involution of some duality, cf. Corollary 1.12).

For every idealI of 2P define now the sets:

LI := {L ∈ F: I(L)⊆ I

}and TI := {

κ ∈ I: I(κ)⊇ I}.

Then, according to Theorem 1.11, for everyL ∈ LI andκ ∈ TI the pair(L, κ)is compatible. Conversely, for every compatible pair(L, κ), there exists an idealI of 2P such thatL ∈ LI andκ ∈ TI (it suffices to consider any idealI such thatI(L)⊆ I ⊆ I(κ)).

Now we discuss the Galois correspondence in these terms. IfL ∈ F, thenϕ(L) coincides withTI(L) ∈ P(I) and we note that the applicationϕ :F →P(I)is monotonically decreasing and “continuous,” i.e.,ϕ(

∨λ∈ΛLλ) = ⋂

λ∈Λ ϕ(Lλ)

when (Lλ)λ∈Λ is a family of elements ofF (indeed, an involutionκ ∈ϕ(

∨λ∈ΛLλ) if and only if I(κ) ⊇ I(Vλ∈ΛLλ) if and only if I(κ) ⊇ I(Lλ) for

everyλ ∈Λ if and only if κ ∈ ⋂λ∈Λ ϕ(Lλ)).

On the other hand, for everyκ ∈ I let ψ(κ) := X(I(κ)) and extendψ to anapplicationψ :P(I)→ F by setting

ψ(W) :=⋂κ∈W

ψ(κ)

for W ⊆ I. Obviously,ψ commutes with intersections.By Corollary 3.7, for everyL ∈ F there existsκ0 ∈ I such thatI(κ0) = I(L),

henceκ0 ∈ ϕ(L). Consequently,I(κ)⊇ I(κ0) for everyκ ∈ ϕ(L). Therefore,

ψ(ϕ(L)

) =⋂

κ∈ϕ(L)ψ(κ)=

⋂κ∈ϕ(L)

X(I(κ)

) = X(I(κ0)

) = X(I(L)

)= sat(L)

since the assignmentI → L(I) is monotone. This proves the following proposi-tion.


Proposition 6.2. The assignments

Fϕ

P(I)ψ

define a Galois correspondence between the complete latticesF andP(I) suchthat the Galois closure of anyL ∈ F is preciselysat(L). In particular, the abovecorrespondence induces a Galois equivalence betweenS and the image ofϕ.

Remark 6.3. At this point we note that the continuity of the applicationϕ permitsus to extend it toP(F) by setting

ϕ̂({Lλ}λ∈Λ

) :=⋂λ∈Λ

ϕ(Lλ)

and this intersection coincides withϕ(∨

λ∈ΛLλ).On the other hand, for every finitely closed subcategoryL ∈ F one can consider

the principal ideal

CL := {L′ ∈ F: L ⊇ L′}of the latticeF and observe that the assignmentL → CL defines an embeddingof F into P(F).

If we defineψ̂(κ) := Cψ(κ) for everyκ ∈ I, as before we can extend̂ψ to anapplicationψ̂ :P(I)→P(F) by setting

ψ̂(W) :=⋂κ∈W

ψ̂(κ)

for W ⊆ I. Obviously,ψ̂ commutes with intersections.This implies that the Galois correspondence defined in Proposition 6.2 can

be extended to a higher level (as far asF is concerned) by considering theassignments

P(F)ϕ̂

P(I).ψ̂

Let us consider now the setF′ of all finitely closed subcategoriesL of LQ

containing theQ-moduleR described in Section 4.1. Ifκ ∈ I andL ∈ F′, then, asin Definition 6.1, we can introduce compatibility of the pair(L, κ). According toTheorem 1.11, the pair(L, κ) is compatible if and only ifI(L) ⊆ I(κ) andκ isL-continuous.

Moreover, sinceI(L) = I(σ (L)) for everyL ∈ F′ (whereσ is the isomor-phism introduced in Section 4.1), the definition ofϕ can be extended toF′ bysettingϕ(L) := ϕ(σ(L)), for everyL ∈ F′.

In order to obtain a Galois correspondence related to dualities of arbitraryfinitely closed subcategories ofLQ, it remains to consider only those contained


in F′. In particular, for these finitely closed subcategories it suffices to restrain ourattention to the subsetC ⊆ I of all L-continuous involutions ofK and define

ϕ̃(L) := ϕ(L)∩ C for everyL ∈ F′ and

ψ̃(κ) := ⟨ψ(κ),R

⟩for every involutionκ ∈ C.

Acknowledgments

It is a pleasure to thank A. McIntyre and U. Zannier for helpful discussionsrelated to the adele ringAQ. Our thanks go also to the referee for her/his verycareful reading and substantially useful suggestions.

References

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Dualities of locally compact modules over the rationals

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