On an unpublished manuscript of ivan prodanov concerning locally compact modules and their...

This article was downloaded by: [University of Illinois Chicago]On: 16 April 2013, At: 09:20Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office:Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscriptioninformation:http://www.tandfonline.com/loi/lagb20

On an unpublished manuscript of ivan prodanovconcerning locally compact modules and theirdualitiesDikran Dikranjan a & Adalberto Orsatti ba Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, 1090,Bulgariab Dipartimento di Matematica pura ed applicata, via Belzoni 7, Padova, 35131,ItalyVersion of record first published: 29 Jul 2009.

To cite this article: Dikran Dikranjan & Adalberto Orsatti (1989): On an unpublished manuscript of ivan prodanovconcerning locally compact modules and their dualities , Communications in Algebra, 17:11, 2739-2771

To link to this article: http://dx.doi.org/10.1080/00927878908823873

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantialor systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that thecontents will be complete or accurate or up to date. The accuracy of any instructions, formulae, anddrug doses should be independently verified with primary sources. The publisher shall not be liablefor any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoevercaused arising directly or indirectly in connection with or arising out of the use of this material.

http://www.tandfonline.com/loi/lagb20

http://dx.doi.org/10.1080/00927878908823873

http://www.tandfonline.com/page/terms-and-conditions

COMMUNICATIONS IN ALGEBRA, 1 7 ( 1 1 ) , 2739-2771 ( 1 9 8 9 )

ON AN UNPUBLISHED MANUSCRIPT OF IVAN PRODANOV CONCERNING LOCALLY COMPACT MODULES

AND THEIR DUALITIES (*)

Dikran Dikranjan Institute of Mathematics

Bulgarian Academy of Sciences 1090 SOFIA (Bulgaria)

Adalberto Orsatti Dipartimento di Matematica pura ed applicata

via Belzoni 7 35 131 PADOVA (Italy)

Dedicated to the memory of Ivan Prodanov

Introduction.

This paper is based on an unpublished manuscript of Ivan Prodanov [PI, which was kindly offered to D. Dikranjan by Prodanov's wife, after the untimely death of its author in April 1985.

The main ideas and techniques developed here are essentially due to I. Prodanov. These ideas concern functorial dualities for locally compact modules over a discrete commutative ring from an original and interesting point of view.

Dualities of this kind were discussed in a seminar organized by Prodanov at the University of Sofia (1979-1982), where related topics were also discussed (e.g. Stone-type dualities [J]), inspiring a number of younger authors (Stojanov [S], Di- mov [Dl, Gregorio [G2], Dimov and Tholen [DT]).

In October 1984 a meeting between I. Prodanov and A. Orsatti occurred in Sofia leading to a fruitful exchange of ideas about Prodanov's results and those obtained by Menini and Orsatti [Moll , [MOz] on strictly related topics. It was decided to begin a scientific collaboration, which unfortunately failed.

Nevertheless, in the opinion of the present authors, Prodanov's ideas seem too important to be lost. For this reason we decided to give complete proofs of Prodanov's results and to answer to some of his conjectures.

(*) This work was partially supported by Grant n. 44 of Bulgarian Science Committee and by Italian Minister0 della Pubblica Istruzione.

Copyright O 1989 by Marcel Dekker, Inc.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13

2740 DIKRANJAN AND ORSATTI

Before illustrating the content of the paper, we need some definitions and notations.

Let R be a commutative discrete ring with identity; denote by LR the category of all unital locally compact Hausdorff R-modules, by DR the subcategory of all discrete modules and by CR the subcategory of all compact modules. For every X, Y E LR, denote by Chom R(X, Y) the R-module of all continuous R-morphisms of X into Y.

Assume that afunctorial duality (FD) * on LR is given, with the identity lr, naturally equivalent to * o * (for unexplained t e n s see Section 1). Then * maps DR onto CR and vice versa. The compact module T = R* is called the torus of the FD * and, for every X E LR, the elements of Chom X , T) are called the T-characters of X.

One of the main results of Prodanov is that * is naturally equivalent to an FD ', with canonical isomorphisms Ex : X -r X" ( X E LR), such that, for every X E LR, x E X and x E ChomR(X, T) , the following identity holds

(1) EX(X)(X) = Mx(x)

where M is an automorphism of T such that M~ is continuous (hence M~ is the multiplication by a unit of R). M is called the involution of *. In general M is not continuous, so that we subdivide the FD's on LR in two classes: the continuous FD's and the non continuous ones. Prodanov conjectured the existence of non continuous FD's and in this paper the conjecture is proved.

The present work is subdivided in 13 sections. The first two sections are intro- ductory. In the third we give a detailed description of all T E CR which can occur as tori of functorial dualities. Moreover it is proved that, for every X E LR, the weak topology of ChomR(X, T) does not depend on the choice of the torus T, because it coincides with the weak topology of Chomz (X , T ) , where T = R /Z.

Section 4 is concerned with the representation of * as an FD ' having property (1).

Section 5 is devoted to continuous dualities. The starting point is the study of the duality AT, where T is a torus and, for every X E LR, AT(X) is the module ChomR(X, T) endowed with the compact-open topology. Here it is established that an FD with torus T is continuous if and only if it is naturally equivalent to AT. Let r be the Pontryagin duality on R. A compact module T E CR is a torus if and only if T(T) is a (discrete) finitely generated projective R-module of rank 1. This fact leads us to prove the existence of a bijection between the set of all equivalence classes of continuous dualities on LR and the Picard group Pic ( R) of R.

In Section 6 we study an equivalence p : LR --t LR associated to a given FD * on LR with torus T and involution M: a well known result of Glicksberg [GI] guarantees that any locally compact R-module is uniquely determined by its T - characters. Then to every X E LR we associate the module X, E LR having as characters the maps of the form M o X, where x is a character of X. The assignment X cr X, defines an involutive equivalence p : LR 4 LR which is permutable with AT. Moreover p subordinates the identity on DR, leaves unchanged the continuous morphisms in LR and, for every X E CR, X is canonically isomorphic to X,.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13

LOCALLY COMPACT MODULES 2741

In Section 7 p is used to give another proof of a result of Roeder [R]: every FD on 4 is naturally equivalent to Pontryagin duality.

In Section 8 and 9, using again p, it is proved that any FD on LR is uniquely determined by its torus and its involution.

In Section 10 we study the FD's on the full subcategory L: of LR consisting of al l locally compact R-modules having an open compact submodule. The main result is that, for every torus T and for every involution M of T, there exists a (unique) FD on L: having torus T and involution M.

In Section 11, using the results of Section 10, we are able to prove the main conjecture of Prodanov, showing that there exist non continuous dualities on LR, for a suitable R. Namely, let R be a field of characteristic zero and cardinality strictly greater than 2 Then T = r( R) has discontinuous involutions and LR = L: . Fix any discontinuous involution M on T : then there exists a non continuous FD on LR = L: with torus T and involution M.

Section 12 is of a rather technical nature and, finally, Section 13 contains open questions and conjectures (some of them are partially solved) raised by Prodanov.

We conclude this introduction fixing some notations. We denote by N the positive integers and by Z, Q , R and Jp the rings and the underlying additive groups respectively of integers, rationals, reals and p-adic integers. T denotes the compact group R /Z with the usual topology.

By writing (X , r ) E LR we mean that the module X , with the topology r, is an object of LR.

1. Preliminaries.

Throughout this paper R denotes a commutative ring, with 1 + 0 , endowed with the discrete topology and LR denotes the category of all unital locally compact Hausdorff R-modules, with continuous R-morphisms. Unless otherwise expressly stated, all morphisms between modules in LR are continuous.

1.1 Definition. A Functorial Duality (an FD, for short) on LR is a contravariant functor * : LR --t LR with the following properties:

(1) For every X , Y E LR, for every f , g E ChomR( X, Y) and for every r E R, we have

6) (f + g ) * = f + + g i (5) ( r f )* = rf* (2) There exists a natural equivalence e between the identity functor l ~ , on LR

and the functor * o *. This means that for every X E LR there exists a topological isomorphism ex : X --t X** such that, for every X , Y E LR and every f E ChomR(X, Y) the diagram D

ownl

oade

d by

[U

nive

rsity

of

Illin

ois

Chi

cago

] at

09:

20 1

6 A

pril

2013

f X - Y

DIKRANJAN AND O R S A T T I

is commutative.

1.2 It is easy to show that, for every X I Y E LR, the application

which sends f E ChOmR(X, Y) to the morphism ft is an isomorphism of abstract R-modules.

1.3 The definitions of monomorphism and epimorphism in LR are the usual cate- gorical ones. If f : X 4 Y is a morphism in La, then f is a monornorphism (epimorphism) if and only i f f is injective ( f (X) is dense in Y).

A morphism in LR which is monic and epic will be called a bimorphism. A morphism f : X + Y is an isomorphism if there exists g E ChomR(Y, X ) such that g o f = lx and f o g = ly.

Observe that a bimorphism f is an isomorphism if and only if f is open (or, equivalently, i f f is surjective).

Every FD sends monomorphisms to epimorphisms and vice versa.

1.4 Let f : X + Y be a morphism in LR. The kernel of f is a morphism h : N + X in LR such that f o h = 0 and, for every hl : Z X such that f o hl = 0, there exists a unique morphism g : Z -t N such that the diagram

is commutative. Clearly the kernel o f f is determined up to topological isomorphisms. It is easy

to see that setting N = ker ( f ) and taking as h the natural topological inclusion of ker ( f ) in X yields the kernel of f .

Dually, the cokernel of f is a morphism p : Y + C in LR such that p o f = 0 and, for every morphism pl : Y + Z such that pi o f = 0, there exists a unique morphism q : C 4 Z such that the following diagram

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13

LOCALLY COMPACT MODULES

- is commutative. Let f ( X ) be the closure off ( X ) in Y: setting C = Y/ f ( X ) with the quotient topology and taking as p the canonical projection, we get the cokernel o f f .

Clearly every FD on LR sends kernels to cokernels and vice versa. A sequence in LR

f B O 4 X - 4 Y 4 Z t O

is called exact i f f is the kernel of g and g is the cokernel o f f . Then f is a topological embedding so that f ( X ) is closed in Y; g is open and surjective and Y/ f ( X ) with the quotient topology is naturally isomorphic to Z in LR.

The observations above give the following important facts.

1.5 Proposition. Let * be an ED on LR. Then a sequence

in LR is exact if and only if the sequence

is exact.

1.6 Proposition. Let X E LR. men: (a) X is discrete if and only if every bimorphism Y -. X in LR is an isomor-

phism; (b) X is compact if and only if every bimorphism X + Y in LR is an isomor-

phism.

Proof (a) Clearly the condition is necessary. To show the converse, denote by Xd the module X endowed with the discrete topology and consider the bimorphism lX :xd'x.

(b) The condition is necessary. Conversely, let i : X -. Y be the Bohr compact- ification of X . It is well known that i is a bimorphism. Thus i is an isomorphism in LR.

Denote by DR (resp. CR) the full subcategory of LR consisting of all discrete (resp. compact) R-modules. Let L; be the full subcategory of LR consisting of all X E LR having an open compact submodule. Clearly X E L; if and only if there exists an exact sequence

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


1.7 Proposition. Let * be an FD on LR. Then

(a) X E DR (resp. X E CR) if and only if X * E CR (X * E DR); (b) X E L: ifandonlyifx* E L:.

2. The Pontryagin duality on LR.

2.1 Let R be the additive group of real numbers endowed with the usual topology and Z the additive group of integers. Set T = R /Z with the quotient topology.

For every X E LR denote by r ( X ) the group Chomz(X,T) endowed with the compact-open topology. This topology is defined by assuming as a basis of neighbourhoods of zero the sets of the form W( K; U) , where K is a compact subset of X, U is a neighbourhood of zero in T and

It is well known that T(X) is a locally compact Hausdorff group which assumes a natural structure of R-module, so that r ( X) E LR. The continuous Z -morphisms of X in T are called the characters of X and T( X) is the module of characters of X.

I f f : X +Y isamorphisminLRandifx E r ( X ) , s e t r ( f ) = x o f . Then T( f) : r ( Y ) + r ( X ) is a morphism in LR. The functor r : LR --t LR defined in this way is a FD over LR which is called the Pontryagin duality over LR. Let X E LR, x E X and define the Z-morphism Sby setting, for x E r ( X ) ,

Clearly 5 is a character of r ( X ) . Define the canonical morphism EX : X -+

TI'( X) by setting EX (x) = C. By classical Pontryagin duality, EX is a topological isomorphism in LR.

2.2 Definition. Let X E LR. For every x E X consider the morphism u, : R -+

X given by uz( r) = rx (r E R). Then u, E ChomR(R, X) . The assignment x w u, defines an R-isomorphism jx : X + ChomR( R, X) . It is clear thatjx is topological whenever ChomR( R, X) is endowed with the compact-open topology.

2.3 Theorem. Forevery X E LR the R-module Chom R( X, F(R)) , endowed with the compact-open topology, is topologically isomo~hic to T( X) . Proof. Let X, Y E LR and consider the natural isomorphism

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


First of all we show that p (X , Y) is topological, whenever both modules are endowed with the compact-open topology. Let V be an open neighbourhood of 0 in T , C be a compact subset of X and K a compact subset of r ( Y ) . Then W = W ( K; W(C; V) ) is a typical neighbourhood of zero in ChomR( r ( Y ) , r ( X ) ) . By the continuity of EY, there exists an open neighbourhood of zero in Y such that E~ ( U) C W ( K; V) . In other words, for every character x of Y, X( U) C V whenever x E K. This means that p (X,Y) (w(C; u)) C_ W. The continuity of ( p( X , Y) ) -' follows from duality. Now the conclusion is reached by the following canonical topological isomorphisms:

3. Tori.

3.1 Definition. A compact R-module T (T E CR) is called a torus if the following conditions hold:

(a) T is a cogenerator of CR; (b) ChomR(T, T) r R canonically.

3.2 Proposition. Let T E CR, T + 0 and let * be a FD on LR. Then T is a torus i f and only if T* is a (discrete) finitely generated projective R-module and End R( T*) r R canonically If T is a torus, then T has no small submodules.

Proof. Assume T is a torus: then T* is a generator of Mod-R = fDR and hence, by a standard argument, T* is a finitely generated projective module over its endomorphism ring which, in our case, is isomorphic to R. Conversely, assume that T* is a finitely generated projective module with End R(T*) S R. Since R is commutative and T* is faithful over R, then T* is a generator of Mod-R. Hence T = T** is a cogenerator of CR. Since EndR(T*) r R, it follows that ChomR(T, T ) % R. Finally, if T is a torus, then T(T) is finitely generated. Therefore T has no small submodules by [B].

3.3 Corollary. If * is an FD on LR, then R* is a t o m

From these results we get the following

3.4 Theorem. Let T E CR. Then T is a torus i f and only i f T fulfills the following three conditions:

(1) T is an injective cogenerator of CR; (2) T has no small submodules: (3) ChomR(T,T) R.

Proof. We apply Proposition 3.2 observing that if r ( T ) is projective, finitely generated and faithful over R, then r ( T ) is a generator of Mod-R.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13

2746 DIKRANJAN AND O R S A T T I

3.5 Let T be a torus. Since T is an injective cogenerator of CR, T has the following property:

(c) For every compact submodule C $ T of T there exists r E R \ (0) such that r C = 0 .

If * is an FD on LR, then property (c) is equivalent to the following:

(c*) For every submodule D of T*, D $ 0 , there exists r E R \ (0) such that rT* C D.

Indeed, consider the exact sequence

where i is the canonical inclusion and Y j 0. Let r E R \ (0) be such that ri = 0 . Dualizing (1) we get the exact sequence in CR

with ri* = 0. Then rIm i* = 0, i.e. rC* = 0 . Hence r(T*/D) = 0 , so that rT* D.

3.6 Corollary. Let R be an integral domain and let T be a torus in CR. nen, for every RY * on LR, T* is isomorphic to a (finitely generated and projective) ideal of R.

Proof. Let a E T*, a + 0, D = Ra C_ T*. There exists r E R \ (0) such that rT* C D. Since projective modules over integral domains are torsion free, we get D R and T* S rT*. Then rT* C_ D 2 R, so that T* is isomorphic to an ideal of R.

3.7 Proposition. LetT be a compact R-module with Chom R(T, T ) 2 R andlet * be an FD on La. men the following conditions are equivalent:

(a) T is a torus; (b) there exist n E N and continuous morphisms pi : T + R*, qi : R * - + T

such that ~ ~ ! , qi o pi = lT.

Proof. (a) =. (b) Since T* is projective and finitely generated, there exists a diagram

where F is a finite set and k o h = idp. For every fx E F, let px : R~ --, R be the canonical projection and ix : R -, RF the canonical inclusion. Then we have the following diagram

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


where hx = ?rx o h and kx = k o i x . We have

For every X E F define the morphisms

hence EX qx o PA = lT. (b) +- (a) Assume that the continuous morphisms pi : T + R*, qi : R * + T

such that xi qi o pi = lT are given ( i = 1, . . . , n). Define the morphisms

Then z:=l ki o hi = IT., so that T* is a direct summand of Rn. Therefore T* is a finitely generated projective R-module with endomorphism ring isomorphic to R. By Proposition 3.2, T is a torus.

3.8 Theorem. Let T, TI be two tori. Then there exist n E N and continuous mor- phismspi: T -+Tl,qi : T I + T ( i = 1, ..., n) such that

Proof. T(Tl ) is a finitely generated projective R-module and End R (F( Ti )) is isomorphic to R. Then we can repeat the proof of (a) =+- (b) of Proposition 3.7 using T(Tl) instead of R.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


3.9 Theorem. Let X E LR, T and TI be two tori. Let w and wl be the weak topologies of Chom X, T) and Chom X I TI ) respectively. Then w = wl . Proof. By Theorem 3.8 TI is a direct summand of Tn. Therefore wl c w. By the same argument, w C_ wl .

3.10 Corollary. Let X E LR andT be a torus. Let w and wo be the weak topologies ofChomR(X,T) andChomz(X,T). Thenw = WO.

Proof. In view of the above theorem, we can take T = T( R), without loss of generality. To show that wo > w, consider a net {xx : X E A ) in X and assume that lirn xx + 0 with respect to w. Then there exists a T( R) -character 11, : X + T ( R) such that limx +(xx) + 0 in T( R). Since T( R) is a compact group, there exists acharacterx E Chomz(F(R),T) such thatlimxx(ll,(xx)) + 0 i n T . Hence limx xx + 0 with respect to wo.

Vice versa, suppose that lirn xx = 0 in w and take x E Chom z ( X, T ) . Define nowX1 E ChOmR(X,F(R)) bysettingxl(x)(r) = x(rz) , forx E X andr E R. Since lirn zx = 0 in w, it follows that lim xl (xx) ( 1) = lim ,y(xx) = 0 in T , i.e. lirn x(zx) = 0 in wo. Thus wo c w.

3.11 Corollary. Let T be a torus. men, for everyX E LR, ChomR(X, T) separates the points of X . Proof. It follows from the well known fact that Chomz ( X , T ) separates the points of X.

3.12 Definition. Let ( X, r ) E LR. The weak topology rw of ( X I r ) is the weak topology of Chomz ( (X, r ) , T ) . By virtue of the above results, rw coincides with the weak topology of ChomR( ( X, r) , T) , for every torus T .

4. Representation of FD.

In all this section * is a fixed, but arbitrary, FD on LR; e is the natural equivalence between lL, and * o *. 4.1 Let X E LR and denote by ix : X * -+ Chom R(X, R*) the canonical algebraic isomorphism resulting from the composition of

v*(RJ8) X* ii; ChomR(Rl X*) - ChomR(X**,R*) r ChomR(X, R*)

LetX E LR and set X' = ChomR(X, R*). For every morphism f : X -r Y in LR, we &fine in an obvious way the trans-

posed morphism f' : Y1 -, XI. Here f is a continuous morphism, while f l , for the moment, is an abstract R-morphism.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


The following proposition is obvious.

4.2 Proposition. Far every mophism f : X + Y in LR, the following diagram

is commutative.

4.3 Let X E LR and consider the canonical algebraic morphism ix : X* -+ X'. En- dow X' with the topology such that ix is a topological isomorphism. Then X' E La. From now on X' will be always equipped with this topology. For every morphism f : X + Y in LB, consider the commutative diagram in LR

For every X E LR define the topological isomorphism Ex : X -t XI' by setting

In this way we get a natural equivalence between the functors * and ', since in the commutative diagram ( 1 ) ix is now a topological isomorphism. Therefore ' is a concrete representation of *. 4.4 Theorem. Let * be an FD on LR, nere exists a (non necessarily continuous) automorphism M : R* + R* such that, for every X E LR, for every x E X and for every^ E X' = ChOmR(XtR*)

Any other natural equivalence between lLR andN can be obtainedfiom (2) bymeans of the multiplication by an invertible element of R.

j ~ * Ed11 Proof. Set M = ER( 1 ) o jR. : R* 4 R' 4 R*, where jR. is the topological isomorphism defined in 2.2. Let us show that M is an automorphism of R*. For

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


this purpose it is enough to verify that ER( 1 ) is bijective. Indeed, since ER : R -+ R" is an isomorphism and 1 generates R, we have that ER( 1 ) generates R" = Chom R', R*) . Hence, being iil an element of ChomR( R', R*) , there exists r E R such that ii' = PER( 1 ) . Thus ER( 1 ) is an isomorphism.

To establish (2), let x E X and consider the following commutative diagram

Set f = j x (x ) . Then f l 'oER= Exo f s0that(fUoER)(1) = E x ( f ( 1 ) ) . On the other hand ( f " o ER) ( 1 ) = ER( 1 ) o f'. Recalling that f ( 1 ) = x we obtain:

Let x E X'. Then

Observe now that x o f = j p ( x ( x ) ) . Indeed, those being morphisms of R into R*, it is enough to see that they coincide at 1 E R. On one hand x ( f ( 1 ) ) = ~ ( x ) . On

I \

the other hand, jR. ( ~ ( x ) ) ( 1 ) = X ( x) by the definition of jR. Hence

This proves (2). Finally, let F be any natural equivalence between lL, and ". Consider the iso-

morphisms ER, FR : R -+ R"; then FR( 1 ) and ER( 1 ) are generators of R". Thus FR( 1 ) = XER( 1 ) for some invertible element X of R.

4.5 Theorem. Let * be an FD on LR. Then there exists a unique natural equivalence F between lLR and" such that (2) holds for M = i i l 0 jR*.

Proof. As observed in 4.4 there exists a uniquely determined invertible element r E R such that i i l = rER(l) . Therefore ER(l ) o jR* = r-I '-I o jp. Setting F = rE, we obtain the required equivalence.

From now on we shall use the equivalence E between lr , and " for which M = i i l o j;.

4.6 Remark. Observe that in general M is discontinuous. In fact iil is continuous, so that M is continuous if and only if j81 is continuous. This happens precisely when the topology of R' coincides with the compact-open topology.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


4.7 Definitions. We say that the FD * is continuous if M is continuous. In this case we have the identity

E x ( x ) ( x ) = x(x) up to a multiplication by an invertible element of R. If M is not continuous we say that * is discontinuous. In both cases, R* will be called the torus of *. 4.8 Theorem. Let * be an FD on LR, T = R* its torus and M the automorphism associated to * according to Theorem 4.4. l3en M 2 is a continuous automorphism of T, i. e. M is the multiplication by a unit of R.

Proof. Since ChomR(T, T) % R and T is compact, it is enough to show that M 2 is continuous. Let {xx : X E A ) be a net in T with limx xx = 0. Then limx ET(xx) = 0 in TI'. Consider the element IT of T' = ChomR(T,T); then ET!( IT) is a character of TI', thus limx ET!( IT) (&-(xx)) = 0. Applying (2) we get

for each X E A. Thus limx M 2 ( xA) = 0 and so M2 is continuous.

Definition. M will be called the involution of the FD *. 5. Continuous dualities.

5.1 Let T be a fixed, but arbitrary, torus in LR. For every X E LR a T-character (or simply a character if no confusion can arise) of X is a continuous morphism of X into T.

For every X E LR, denote by AT(X) the module ChomR(X,T) endowed with the compact-open topology. We define the canonical morphism wx : X --t

A T A ~ ( X ) by setting, for every x E X , w( x) = j.", where Z ( x ) = X( x) , for every x E AT(X) . In this way we obtain a functorial morphism w : 1~~ -, AT o AT. If no confusion can arise we write A instead of AT. Clearly A( R) T , the isomorphism being canonical and topological.

5.2 Theorem. The assignmentx H AT(X) defines a continuousFD on LR, with torus T, natural equivalence w and involution IT.

Proof. The proof consists of the following steps. 1) A( X ) is locally compact. By Theorem 3.8 there are continuous morphisms pi : T + r ( R ) , qi : r ( X ) -+

T ( i = 1 , . . . , n), such that I:=, qipi = IT. For every X E LR define the morphisms:

by setting

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Then the morphisms Pi, Qi are continuous and satisfy

Consider the following commutative diagram

where P = ( P I , . . . , P,) and Q = Cy=l Q;. Then P is a topological embedding, since the inverse of the continuous injection P : A( X ) + P( A( X ) ) is given by Q ~ P ( A ( X ) ) . Moreover P ( A ( X ) ) is closed in r ( X ) , , being the kernel of Q. Hence A( X ) is locally compact.

2) Let f : X -+ Y be a morphism in LR. Then the transposed morphism

is continuous The proof is standard. 3) Let X E LR, x E X . men Z : A( X ) -+ T is continuous Indeed, consider A ( X ) as a submodule of the compact module T~ endowed with

the topology of pointwise convergence w. Then Fis continuous, since it coincides with the resmction to A( X ) of the canonical projection T ~ . But the compact-open topology is finer than w , so that Zis continuous on A ( X ) .

4) Let X E LR. Then wx : AA( X ) is bijective. Injectivity follows from Corollary 3.11. We show now that wx is surjective. Let

II, E AA( X ) and consider the commutative diagram

There exists xij E X such that II,ij(x) = x ( x i j ) , for every x E r ( X ) , since +ij is a character of the Pontryagin dual of X .

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Let cp E A ( X ) and set X i = Pi(p) . Then:

On the other hand, +ij( P i ( p ) ) = P i ( v ) ( x i j ) = p i ( p ( x i j ) ) and SO

Summing with respect to i gives

Since ChomR( T , T ) = R we have:

for suitable rij E R. This proves that

Hence wx is surjective. 5) wx is a topological isomorphism For every X E LR, let E = E X : X --t r r ( X ) the canonical isomorphism. Let

us prove that the following diagram commutes:

where the horizontal row Pi is relative to A ( X ) . In fact, for every x E X and for every p E A ( X ) , we have:

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


On the other hand:

Hence: P i o w = T ( P i ) o ~

Multiplying on the left by Qi and summing with respect to i we get

so that w is continuous. Let us prove that w i ' is continuous. For this purpose, let pi : T ( R ) -+ T =

A ( R ) , qi : T 4 T ( R) ( i = I , . . . , n) be continuous morphisms with Ci qipi = lr(R) (see Theorem 3.8). Define the morphisms

Pi : r ( X ) 4 A ( X ) and Q , : A ( X ) -+ r ( x ) by setting:

Fix x E X and consider the morphism

suggested by the following diagram

We show that

from which it will follow that w-' is continuous. We have

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


C ~ ( P , ) 0 Qi O W ( % ) = Cr(~i ) ( ~ i 0 w ( x ) ) = C Q ~ O W ( % ) 0 Pi.

Applying this morphism to ( E r ( X ) we get:

Thus (1) is proved.

A straightforward calculation gives the following

5.3 Proposition. For everymorphism f : X -+ Y in LR, we have the commufative diagram

f X - Y

5.4 Proposition. Every torus T E LR is an injective object in LR.

Proof. Clearly the duality A is an FD. Therefore, for every exact sequence

in LR, the sequence

is exact in LR.

Our next objective is to show that every continuous duality on LR is a duality of type AT. The main tool is the following

5.5 Theorem. (Glicksberg [ G I ] ) Let X E & , r the topology of X , rw the weak topology of T -characters of ( X , r) . Then ( X , r) and ( X , 7,) have the same compact subsets,

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


This result has a number of important corollaries.

5.6 Corollary. Let X E LR and denote by X, the module X endowed with its weak topology. Let T be a torus in LR . Then AT ( X ) is topologically isomorphic to the module Chom ( X, , T) endo wed with the compact-open topology.

5.7 Corollary. Let X be an R-module, T and a two topologies on X such that both ( X, r) and ( X , a) belong to LR. Let r, and a, be the respective weak topologies. Then

r = a if and only if rw = a,

Proof. Suppose 7, = a,. By Corollary 5.6, r ( X, r) is topologically isomorphic to (XI a ) . Therefore ( X I r ) is topologically isomorphic to ( X, a ) .

It follows that a = r since both ( X , r ) and ( X , a ) are canonically isomorphic to their second dual.

5.8 Corollary. Let ( X, r ) , (Y, a) E LR and f : X + Y be a morphisrn of abstract R-modules. Then f : (X , r ) -t ( Y, a) is continuous if and only i f f : ( X I 7,) + ( Y, a,) is continuous.

Proof. If f : (X , r,) -+ (Y, a,) is continuous, then the morphism T ( f ) : T(Y, a,) -, r ( X , 7,) is continuous too. By Corollary 5.6, T(Y, a,) = r ( Y , a) and r ( X , 7,) = r ( X , r ) , SO that r( f ) : r ( X , r) -+ T(Y, a ) is continuous. Du- alizing once more and making use of the natural isomorphisms EX and EY, we get the continuity of f : ( X , r) + ( Y, a). The verification of the other implication is mvial.

5.9 Remark. Denote by TB the category of precompact R-modules, i.e. all Haus- dorff topological R-modules whose topology coincides with the weak topology of their T-characters. By the above rcsuits, the assignment ( X , r ) H ( X , r,), for every (XI r) E LR is a full and faithful functor cp : LR -t !PR. Moreover, if X E cp(LR), there is one and only one Y E LR such that cp(Y) = X. It would be interesting to give an explicit description of p(LR).

5.10 Proposition. Let * be an FD on LR, wjth torus T and involution M, and let A = AT, men, for every X E LR, a morphism f : X' + T is conthuous if and only if M-' o f : A( X) + T is continuous.

Proof. Define X', X", Ex as in 4.3 and recall that A(X) = ChomR(X, T) and X' coincide as abstract R-modules. Suppose that f : X' + T is continuous. Then f E ChomR(X, T) = X u (as abstract modules). There exists x E X such that f = Ex(x). ByTheorem4.4,Ex(x) = Mowx(x),andthus M-' of = wx(x) : A(X) + T is continuous.

Conversely, assume that M-' o f : A(X) -+ T is continuous. Then M-' o f E A( A( X ) ) , so that M-' of = wx ( x ) for some x f X . Then, by the above remark, f = M o wx ( x) = Ex (x) : X' -t T is continuous.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


5.11 Remark. The above proposition says that the T-characters of X' are precisely the morphisms M o X, where x is any T-character of A(X).

5.12 Theorem. Let * be a continuous m) on LR with torus T. Set A = AT. Then, for every X E LR, the canonical isomoqhism ix is a topological isomorphsm of X* onto A(X) . Proof. By Proposition 5.10, an R-morphism f : X' + T is continuous if and only of f : A(X) -r T is continuous. This means that the locally compact modules X' and A(X) , which coincide as abstract modules, have the same T-characters. Hence, by Theorem 3.9 and Corollary 3.10, they have the same weak topology. By Corollary 5.7, X' = A( X) topologically.

5.13 Theorem. Let * be an FD on LR with torus T and involution M. I f* is naturally equivalent to A = AT, then * is continuous.

Proof. By means of the natural isomorphism jT, we identify the torus A( R) of A with T. Recall that M : T + T is continuous if and only if the topology of R' coincides with the compact-open topology (see Remark 4.6). Since this topology is defined by the algebraic isomorphism iR : T = R* + R', it is enough to show that i~ : T + Chom R( R, T) is continuous whenever Chom R( R, T) is endowed with the compact-open topology. By the definition of iR, we have only to show that cp,( R, T) : ChOmR(R, T) 4 Chom R(T*, T) is continuous (see 4.1) with respect to the compact-open topologies.

Let 7 : * + A be a natural equivalence. For ( E ChomR( R, T) we have the following commutative diagram

Hence (* = o A(€) o TT, and therefore

v*(RlT) =7i1 OPA(R,T) O ~ T .

The continuity of pA( R, T) can be established in the same way as in Section 2.

5.14 Corollary. An FD * on LR with torus T is continuous if and only if * is naturally equivalent to the duality AT.

Now we show that the continuous dualities on LR can be described by means of the Picard group of the ring R.

5.15 Definition A finitely generated projective R-module V E Mod-R has rank 1 if, for every prime ideal 9' of R, the localization Vp of V at T is a free Rp -module of rank 1.

We shall denote as usual by Spec (R) the set of prime ideals of R.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13

2758 DIKRANJAN AND ORSATTZ

5.16 Lemma. Let V be a finitely generated projective R-module. Then V has rank 1 if and only ifEnd V ) r R canonically.

Proof. Assume that V has rank 1 . Observe that V is faithful over R. In fact, assume that rV = 0 for some r E R \ ( 0 ) and let P E Spec( R ) be such that AnnR( r ) C T . Then rRp + 0, while Vp g Rz yields r Rp = 0 -a contradiction.

Set R' = End R( V ) : then R L, R1 canonically and, for every T E Spec( R ) , Rp g Rh , so that ( R1/R)p = 0. Then R'/R = 0 and R' = R.

Assume that End R( V ) g R. For every T E Spec ( R) , Vp is flat as an R- module. Since V is finitely generated, we have that Vp % End R, ( Vp ) . Since Vp is a free Rp -module, this implies Vp E? Rp . Therefore V has rank 1.

Denote by Pic( R) the set of the isomorphism classes of finitely generated projective modules having rank 1. For every such V, denote by [ V ] the isomorphism class of V . Pic ( R ) becomes an abelian group by setting

The zero of Pic ( R) is [ R] and the opposite of [ V ] is [ 91, where we have set P = Hom R( V , R) . In fact, if v is finitely generated of rank 1, it is v 8~ GI g R.

The group Pic ( R) is called the Picard group of R.

5.17 Theorem. The equivalence classes of continuous FD on LR are in bijective correspondence with the elements ofPic ( R) . Consequently, the unique continuous FD on LR is the Ponhyagin duality if and only if Pic ( R) = 0.

Proof. Follows from Proposition 3.2 and the above lemmas.

6. The functorial topology p.

6.1 Lemma. Let X, Y be two precompact modules in TR and let T be any torus in LR. men an R-morphism f : X -t Y is continuous if and only if; for every T -character ( of Y, the rnorphism E o f is continuous.

6.2 Theorem. Let c be an FD on LR with torus T and involution M . For every ( X, r) E LR, there exists a unique topology I-, on X such that ( X, 7,) E LR and

(a) p is a T-character of ( X , 7,) if and only if M-' o p is a T-character of ( X , T ) .

Moreover:

(b) for every ( X, T ) , (Y, a) E LR, any R-morphism f : X --+ Y is continuous with respect to T and a if and only i f f is continuous with respect to I-, and UP.

Proof. Consider the continuous duality A = AT. Then every X E LR has the form X = A(Y) , with Y E LR unique up to topological isomorphisms (Y g A ( X ) ) . Consider the topological module Y1 defined in 4.3. We have the identities of abs trac t

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


R-modules X = Y' = A(Y). We know by Proposition 5.10 that the T-characters of Y' are exactly the maps of the form M o X, where x is a character of A(Y). Let r be the topology of X. We endow X with the topology of Y' and denote by 7; this topology. Thus ( X I 7,) E LR and T, is uniquely determined by property (a), in view of Corollary 5.7.

The weak topology of ( X I 7,) coincides with the weak topology of the morphisms of the form M o X, where x is a character of A(Y) = ( X , 7). Therefore property (b) follows from Corollary 5.8.

6.3 Definitions. Because of property (b), the topology r, will be called thefuncto- rial topology.

Let ( X I r ) E LR. We shall often write X, instead of ( X , 7;) ; X = X, will mean that r = 7,. Moreover, we express property (b) of Theorem 6.2 by saying that f is continuous if and only if it is p-continuous.

We denote by p : LR 4 LR the covariant functor which associates to every X E LR the module Xu and leaves unchanged the morphisms.

6.4 Remark. Let * be an FD on LR with torus T and involution M and let X E LR. From the considerations above, it follows that rp is the unique topology on X for which the isomorphism wx : X 4 A(X) ' is topological. Looking at the isomorphism ~ A ( x ) : A(X)* -t A(X)', we have the diagram

so that r, is the unique topology on X for which the isomorphism w j l o is topological. Therefore, up to the natural equivalence w , p can be thought as the composition of the dualities AT and ' and thus p is an equivalence LR -+ LR. This remark makes obvious the following

6.5 Theorem. The functorp : LR + LR associated to an FZ) * on LR as in 6.2 has the following properties:

(a) p is an involution; (b) p sends CR onto CR; (c) p coincides with the identityfunctor on DR; (d) p preserves tinite products and exact sequences.

Remark. Let * be an FD on LR with torus T and involution M and let p : LR 4 LR be the equivalence associated to *. Then * is continuous if and only if T = T,. Indeed M : T, -+ T is continuous by the definition of p. Therefore T, = T implies that M : T 4 T is continuous.

6.6 Let T be a torus in LR and let M be an involution in T. Let p : LR -t LR be an involutive equivalence such that, for every X E LR, a morphism cp : X, -, T is

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


continuous if and only if M-' o cp is a T-character of X. Furthermore we assume that ,u satisfies conditions (a), (b), (c) and (d) of Theorem 6.5. Set A = AT. For every X E LR define the R-morphism

by setting

6.7 Theorem. q = {qx : X E LR) is a natural equivalence between the functors p o A a n d A o p .

Proof. It is clear that qx is an isomorphism for all X E LR. Let us prove that qx is topological. By Corollary 5.8, qx is continuous if and only if qx is weakly continuous. Thus, to prove continuity, we can use Lemma 6.1. Let x be a character of A(X),: then x = M o C, where C is a character of A(X) , so that there exists a unique x E X such that < = wx ( x) . Therefore:

Let a E A(X,). Then a = M o <, where < is a character of X. Thus:

Hencexoqx = w x , ( x ) , s o t h a t ~ o q ~ E A(X,). Let us prove the continuity of q,' : A(X), -+ A(X,). Note that, for x E

A( X),, we have q;l(X) = M o X. Take a character < : A(X9) + T. Then E = wx, ( x) , for some x E X. We show now that

Since M o wx(x) is clearly a character of A(X),, (1) will prove that T&' is weakly continuous, and thus that qjjl is continuous.

Take x E A(X),: then x E A(X). We have

Finally, for every morphism h : X + Y in LR, the diagram

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


is commutative. Indeed, for every p E A(Y,) , we have

7. Dualities on b.

7.1 Lemma. Let * be an FD on LR and let p be the functorial topology associated to *. Let ( X , 7) E LR and suppose that there exists a topological isomorphism f : X, 4 X of R-modules. Denote by A theringChomR( X, X ) = ChomR(X,, X,) endowed with the topology of pointwise convergence. Suppose that:

(a) eveq automorphism of the abstract ring A is continuous; (b) there exists an element s o E X and a continuous R-module morphism a :

X + Asuchthat,foreachx E X , x = cu(x)(xo). Then T, = T.

Proof. For each 5 E A = ChomR(X,, X,) set p ( a ) = f o o o f-' E A = ChOmR(X, X). Hence cp is an automorphism of A and so it is continuous by (a). We have the commutative diagram

For each x E X , it is

Set now x = s o and yo = f (so). Then (1) yields

Then, for a = a (%) , (2) and (b) imply

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Let {xx : X E A ) be a net in X with limx xx = 0 . By the continuity of a , we have limx a ( x ~ ) = 0 in A. Now the continuity of p yields limx p (a (xx) ) (yo) = 0 in X . So (3 ) gives limx f (xx) = 0 in X . Therefore the morphism f : X -t X is continuous with respect to T. Since f : X, -, X is a topological isomorphism, this implies that T 2 T,. To show that r 2 T,, recall that (( X , 7,) , T,) = ( X , 7). Consequently, f : (X, T) -t ( X , 7,) is a topological isomorphism and, arguing as above, r E 7,.

7.2 From now on, * will be an FD on & , with torus T and involution M. Since T* is a projective ideal of Z , it must be T r T (see Corollary 3.6).

7.3 Proposition. Let T be the usual topology of R . Then r, = 7.

Proof. ( R , T,) is a locally compact group. By a structure theorem [HR, Theorem 24.30, p. 3891, ( R , T,) E R" x X, where nis a non negative integer and X E $, i.e. X has a compact open subgroup. By Theorem 6.5, it is clear that X, E $. Therefore R " ((R , T,) , 7,) 2 R * x X I where X1 E g. Since R contains no non trivial compact subgroup, XI is discrete. On the other hand, R is connected, so that X I = 0 . Thus R E R* implies n = 1. Therefore ( R , T,) is topologically isomorphic to R . By Lemma 7.1, T, = r.

7.4 Proposition. T, = T

Proof. Applying the functor p to the exact sequence

O - t Z - t R - t T 4 O

in & gives the exact sequence

O - t Z - t R +T,-+O

so that T is topologically isomorphic to T, by mean of the identity morphism, hence T, = T .

7.5 Theorem. The FD * on & is continuous. Therefore the unique FZ) on .& is, up to natural equivalence, the Ponpyagin duality.

Proof. Consider IT : T -, T . This is a character of T , so that M = M o IT : T, + T is a character of T,. By the previous Proposition, T, = T . Therefore M : T -t T is continuous.

7.6 Remark. The uniqueness of the Pontryagin duality on Lz was established by Roeder in [R].

8. Functorial homomorphisms.

8.1 Definition. Afunctorial homomorphism M : CR -+ CR is given by assigning to every C E CR a homomorphism (in general discontinuous) of R-modules Mc : C + C such that, for every continuous morphism f : C -t C1 (C1 E CR), the diagram

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


commutes.

8.2 Proposition. Let T E CR be a torus. For every functorial homomorphism M : CR + CR consider the homomorphism MT : T + T . Then the assignment M H

MT defines a bijection between functorial homomorphisms M : CR --+ CR and endomorphisms of T .

This comspondence preserves the operations between homomorphisms. In particular MT is an involution if and only if; for the corresponding M : CR -+ CR, every Mc is an involution.

Proof. Set * = AT and let C E CR. Then C* is discrete so that ChomR(C*, T) = Hom R(C*, T) . Fix an endomorphism p of T and define

by setting

(1) M c ( x ) = ( P O X ( X E C**) (we identify C** with C ) . Let us show that, for every continuous morphism f : C -, C1 (CI E CR), the diagram

Mc C** --L C**

commutes. In fact, let x E C**. Then

Mc1f**(x) = M c l ( x o f C ) = ( ( ~ o x ) ( f * )

= f ** (p 0 X ) = f * * M c ( x ) Hence

Mcl o ft'= f+ 'oMc. Therefore, by means of formula (I), we assigned to the endomorphism p of T the functorial homomorphism M : CR + CR. It is clear that MT = p.

The other statements follow from the following observations: (I) the correspondent of IT is lc, for every C E C'; (2) Let MT, NT be two endomorphisms of T: then MT o NT gives rise to M o N .

8.3 Proposition. Let * be an on LR with torus T and involution MT and let M : CR + CR be the functorial homomorphism associated to Mr. Then, for each C E CR the moqhism Mc : C, -t C is a topological isomorphism.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Proof. Set A = AT and X = A(C). Then X E DR so X = X,. Consider now the topological isomorphism qx : A(X) -+ A(X), defined in 6.6. Then qx is a topological isomorphism of C onto C, and Mc = T$ , by the definitions of the maps Mc and qx . 9. Uniqueness of FD with given torus and involution. 9.1 Let * and " be two FD on LR having the same torus T and the same involution M. For every X E LR we have the algebraic canonical isomorphisms

(1) X*=Ch0mR(X,T), Xw=Ch0mR(X,T) and there exist the natural isomorphisms

(2) e x : X + X * * , u x : X + X W given by

ex(x)(x) = Mx(x); ax (z ) (x ) = Mx(x) for every X E LR, x E X and x E Chom X, T) .

By (1) X* and Xv coincide as abstract R-modules, but not necessarily as topological modules, so that the modules X** and X'' may be, a priori, distinct.

9.2 Lemma. In the above situation, X* = X 'andXt* = X "as topologicalmod- ules, up to natural isomorphisms in LR.

Proof. Let E E X**. By (2) there exists x E X such that E = ex (x). Then, for every x E X*, we have

Since E = ax (x), we have E E X"". In a similar way we can prove that X " C X**. Therefore the modules X* and Xv have the same characters, hence the same weak topology. Then, by Corollary 5.7, X* and Xv have the same topology. Thus X* = X" as topological modules. On the other hand X** = X"' as abstract modules. Having the same characters, they coincide as topological modules too.

It is clear that, for every X E LR, there exists a natural isomorphism vx : X* 4

X" in LR. Therefore we have the following

9.3 Theorem. Two ED on LR having the same torus and the same involution are naturally equivalent.

9.4 For the moment, we have shown that, if threre exists an FD on LR with torus T and involution M, this FD is unique. Let a and 8 be full subcategories of LR. The notion of functorial duality between A and 23 is clear.

In the following we shall show not only the uniqueness, but also the existence of FD between relevant subcategories of LR.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


10. Dualities on L: . 10.1 Recall that L: is the subcategory of LR consisting of all modules X E LR for which there exists an exact sequence of the form

with C E CR and D E DR.

In this section we shall prove the following

10.2 Theorem. Let T be a torus in LR and M any involution of T. Then there exists an FD on L: having torus T and involution M .

From this theorem it follows that, in general, there exist discontinuous dualities on Lo . For example this is true for g. In general, however, a discontinuous duality 3 on LR cannot be extended to an FD on LR: indeed every FD on & is continuous and equivalent to the Pontryagin duality.

If, for a suitable R, we have LR = L: and if there exists a torus T in LR having a non continuous involution, then there is a non continuous FD on LR.

10.3 We need some remarks before the proof of Theorem 10.2. Let T be a torus in LR and let MT be an involution of T. By Proposition 8.2 there exists a unique functorial homomorphism M : CR 4 CR associated to Mr. Let (C , r ) E CR and consider the isomorphism Mc : C 4 C. Endow C with the topology r, in such a way that Mc becomes a topological isomorphism of (C, I-,,) onto (C, r ) . Clearly the T-characters of the compact module (C, r,) are exactly the maps of the form M o X, where x is a T-character of (C, r). Therefore an R-morphism f : C -+ C' (C, Cf E CR) is continuous if and only i f f : C, -+ Ch is continuous (we are using the same notations as in Section 6).

If there exists on LR a (unique!) FD with torus T and involution M, then the topology r, is exactly the one defined by Theorem 6.2.

In this way we obtain a covariant additive equivalence p : CR + CR. We shall now extend p to an involutive equivalence on L: .

If ( X , r ) E L:, consider the exact sequence (1) and let d be the relative topology on C. Define (X, r,) as the module in L: having (C, r,) as a compact open submodule. This definition of (X , 7,) does not depend on the choice of C. In fact, let C' be another open compact submodule of (X, r ) . Since C + C' is an open compact submodule of X , we can assume without loss of generality that C C'. If i is the inclusion of C in C', then i : C, c, Ch is a topological embedding. Therefore Cf gives the same topology r, on X. The functor p : L: -+ L: has the following properties:

(1) p is a covariant involutive equivalence on L: ; (2) p is the identity on DR (in fact 0 is a compact open submodule of any

discrete module); (3) p is exact and preserves finite products.

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Set A = AT and for every X E LR define the isomorphism ijX : A(X,) --t

A( X), setting

as in 6.6. The same proof usedin Theorem 6.7 guarantees that ij' = { i jx : X E L: ) is a natural equivalence between p o A and A o p.

Observe that, if X E DR, then f x = ~ 2 : ~ ) .

Proof of Theorem 10.2 Define a duality * on L: by setting, for every X E L;,

Note that R* = A( R,) = A( R) = T. Define, for every X E L; , the canonical isomorphism ex : X + X** in the following way. Since i j x : A(X,) + A ( X ) , is a topological isomorphism, then i j x : A(X,), + (A(x ) , ) , is an isomorphism

too. Therefore A ( f x ) sends A((A(x) , ) , )= A A ( X ) onto A(A(x,) ,) = X**.

Thus ex = A( i jx ) o wx is a natural topological isomorphism X --t X**. This proves that * is an FD with torus T. Let us see that the involution of * is M-' . In fac t , i fx E Xr,andx E X,then

10.4 Let H1 : BR -+ CR, H2 : CR 4 DR be a pair of contravariant functors such that ( H i , Hz) is an FD. Set T = HI ( R ) : then T is a torus in CR. Indeed, since R is a finitely generated projective generator of DR = Mod-R, then T is a cogenerator of CR with ChomR(T, T ) g R canonically. Set A = AT and let X E CR. Then we have the canonical isomorphisms

Thus Hz is naturally equivalent to A 1 cR.

On the other hand, for Y E DR, there exists X E CR such that Y = A( X ) and we have the canonical isomorphisms

which define a natural equivalence between Hi and A I DR.

By Theorem 10.2, for every torus T E CR and for every involution M of T , there exists a unique FD * on L: with torus T and involution M. Then * subordinates an

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


FD ( HI , Hz ) between DR and CR as above, so that HI A1 c, and Hz g A 1 The involution M seems to be disappeared. Actually it appears in the following commutative diagrams of categories and functors

where ,u is the involutory equivalence associated to * (recall that X is topologically isomorphic to X,, for all X E CR).

This fact shows how Prodanov's results on dualities between discrete and compact modules agree with those of [MOs] in the case of commutative rings.

11. Examples of discontinuous dualities.

11.1 Theorem. Let R be a field of characteristic zero and cardinality strictly greater than 2 Then:

(a) L R = L OR; (b) L R admits discontinuous dualities.

Proof. Let a = IRI, the cardinality of R. If we consider R as a discrete abelian group, we have R % Q("), so that T (R) 2 T(Q)" . Therefore IT(R)I = 2" and T ( R) , as a vector space over R , has at least 2" distinct involutions. On the other hand the continuous automorphisms of r ( R) are the multiplications by non-zero elements of R and so they are precisely a. Hence the torus F ( R) has discontinuous involutions. Now we have only to prove (a) and apply Theorem 10.2.

First of all observe that, for every positive integer m, R m $ L R. In fact, assume R E L R. Then, for each r E R \ (01, the multiplication f ( r ) by r in R is a continuous automorphism of the group R m, and then f ( r ) is also an automorphism of the R -space R m. We get in this way a ring homomorphism f : R -+ EndR ( R m). Since R is a field, f is injective, so that I RI 2 2N0-a contradiction. Thus R $! L R.

Let X E L R, X + 0 and let us prove that X contains an open compact subgroup. Indeed X has an open subgroup of the form R x K , with m 2 0 and K a compact subgroup of X. Let KO be the connected component of 0 in K . and set A = R x K O . Then A is the connected component of 0 in R x K . The latter is open in X and so A is the connected component of 0 in X, whence A is an R- submodule of X. Observe now that KO is the greatest compact subgroup of A, since R " contains no non-zero compact subgroups. Therefore KO is invariant under continuous endomorphisms of A, so that KO is an R-submodule of A. It follows that R AIKo E L R and this implies m = 0 . Thus K is an open compact subgroup of X.

Now we show that K = KO, so that K is a submodule of X and X E L i . Set X I = X I KO and K1 = KI KO. Then X I is a totally disconnected locally compact

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


R-module, while K1 is an open totally disconnected compact subgroup of Xi. Since R has characteristic zero, K1 is torsion-free. By the Pontryagin duality for compact abelian groups,

where P is the set of prime numbers, Jp is the group of p-adic integers, op is a suitable cardinal number and the isomorphism is topological.

For every p E P set

tdp(X1) = {x E X1 : lim pnx = 0). nEN

Clearly tdp(Xl) is a closed R-submodule of X1 such that

We show now that each op is finite. Fix p E P . Then (2) implies that J? is an open subgroup of tdp(X1), since K1 is open in XI . Since 1 /p E R, the multiplication by 1 / p in tdp(X1) is a continuous automorphism. Now J? is open in tdp(Xl), SO

that there exists an open neighbourhood U of zero in td p( Xi ) such that U J?, whence U C - p ~ ? . Therefore p ~ ? is an open subgroup of J? and this is possible only if op is finite.

By the definition of tdp(X1) and since J? is open in tdp(X1), the factor group tdp(X1 ) / J? is a p-group. Indeed the condition lim pnx = 0 in the discrete group td ,(Xi ) / J,bp becomes pmx = 0 for some rn E N .

Suppose a, $ 0 . Since tdp( K1) = J? is torsion-free, J? has the same torsion- free rank as tdp(X1). Since up is finite, I tdp(X1) 1 = 2'0. On the other hand tdp(X1) is a non-zero torsion-free R-module so it contains a copy of R. Thus 1 tdp(X1 ) I > 2 No-a contradiction. This proves that up = 0 for all p E P. Conse- quently K1 = 0 by (1).

12. Discontinuous dualities.

12.1 Throughout this section any FD * on L R with torus T and involution MT takes its concrete form; namely we assume that, for every X E L R , the canonical isomorphism ex : X -t X** is given, for x E X and x E ChomR(X*, T) , by

12.2 Theorem. Let M : C R + C R be a functorial homomorphism. Assume that there exists a torus Tl such that the morphism MTl : TI -t TI given by M is the involution of an FD 'on L R with torus TI . Then, for every torus T E C R, the pair ( T, MT) is associated to an FD * on L R

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Proof. There exists continuous morphisms pi : T -t T I , qi : Ti 4 T , ( n E N, i = 1, . . . , n) such that Cy=l qi o pi = IT. For each X E L R consider the abstract module X* = Chom R ( X , T ) . Define, for a = 1, . . . , n, the continuous morphisms

by setting Pi(X) = pi 0 X and Q i ( p ) = qi o p, for X E X* and p E X'. Then Cy=l Qi 0 Pi = lx*.

Arguing as in Section 5, we show that X* is a retract of ( X v ) " . Endow X* with the relative topology so that X* E L R . We shall now see that the assignment X w X* defines the desired FD, provided we define the action of * on morphisms in the usual way. To show that * : L -+ L is a functor, consider a morphism h : X -t Y in L R. We have the following commutative diagram

For every x E Y * we have

hence h* = xi QioKoPi is continuous. Therefore * : L R -t L R is a contravariant functor.

Let us &fine now the morphism Ex : X -+ X** by setting, for each x E X and each x E X*

E x ( x ) ( x ) = M T ( x ( x ) ) .

We have to show that Ex is a topological isomorphism. For every X E L R &note by ex the canonical morphism X -+ Xv". Consider

now the commutative diagram

where $ E X** and $ij E X w . Hence there exist elements xij E X such that $ij = ex ( ~ i j ) and SO $ij(x) = M'(x( x i j ) ) for each x E X v , where M' = Mn .

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


Summing with respect to i we get

Thus

From the given functorial homomorphism M we get the commutative diagram

MT T - T

so that $( p) = MT xi qjpip( xij). Since Horn R ( T ~ , Ti ) = R, there exist elements rij E R such that

This proves that Ex is surjective. On the other hand ChomR(X, T ) separates the points of X, so that Ex is injective. Arguing as in 5.2 and writing * instead of A and ' instead of I', we can conclude.

A functorial homomorphism satisfying the condition of Theorem 12.2 will be called admissible.

13. Conjectures and open questions proposed by I. Prodanov.

13.1 Discontinuous dualities for R discrete and commutative. (a) Describe all the admissible functorial homomorphisms. (b) Characterize those rings R for which every FD on LR is continuous. Pro-

danov conjectured that all rings of algebraic numbers have this property. Recently D. Dikranjan proved this conjecture (to appear). He has proved also that, for R = R

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13


(the field of real numbers) and R = C (the field of complex numbers), there exist discontinuous dualities, thus answering to a question of Prodanov.

13.2 FD over a locally compact commutative ring R. (a) L. Stojanov [S] proved that if R is compact then the unique FD on LR is the

Pontryagin duality. This result was extended by E. Gregorio to the non commutative case.

(b) For R a locally compact field, Prodanov conjectured that every FD is continuous. The question is open.

13.3 FD's over a non commutative (discrete) ring R In this case a FD may be defined between the category RL of left locally com-

pact R-modules and the category LR of right ones (see [Gz]). Clearly, in this case, condition (ii) in 1.1 must be omitted.

Another way is to weaken the definition of a torus T, requiring only that T be a cogenerator of RC. Then, setting A = ChomR(T, T) , one can study FD between LA and RL . This type of dualities between Mod-A and have been extensively studied by Menini and Orsatti in [Moll.

Acknowledgement. The authors are deeply indebted to E. Gregorio for a number of useful suggestions, in particular for Section 5.

References

[Bl S. Bazzoni, Pontryagin type dualities over commutative rings, Ann. Mat. Pura Appl. (4) 121 (1979). 373-385; Widem 123 (1980), 403404.

[Dl G. Dimov, An axiomatic characterization of Stone duality, Serdica 10 (1984), 165-173. [DT] G. Dimov, W. Tholen, A characterization of representable dualities. (in preparation) [Gll I. Glicksberg, Unvorm boundedness for groups, Canad. J. Math. 17 (1962), 419435. [Gz I E. Gregorio, Dualities over compact rings, Rend. Sem. Mat. Univ. Padova (to appear). [HR] E. Hewitt, K. Ross, "Abstract harmonic analysis I," Springer, Berlin, Heidelberg, New York,

1963. [J] Johnstone, "Stone spaces," Cambridge University Press, Cambridge, 1982. [ M o l l C. Menini, A. Orsatti, Good dualities and strongly quasi-injective modules, Ann. Mat. Pura

Appl. (4) 127 (1981). 182-230. [MOzl C. Menini, A. Orsatti, Dualities between categories of topological modules, Comrn. Algebra

11 (1983), 21-66. [MO, I C. Menini, A. Orsatti, Representable equivalences between categories of modules and appli-

cations, Rend. Sem. Mat. Univ. Padova (to appear). [PI I. Prodanov, Pontryagin dualities. (unpublished manuscript) [Rl D.W. Roeder, Functorial characterization of Pontryagin duality, Trans. Amer. Math. Soc. 154

(1971). 151-175. [S] L. Stojanov, Dualities over compact commutative rings, Rend. Accad. Naz. Sci. XL, Mem. Mat.

vn (1983),155-176.

Received: June 1988

Dow

nloa

ded

by [

Uni

vers

ity o

f Il

linoi

s C

hica

go]

at 0

9:20

16

Apr

il 20

13

Date post:	08-Dec-2016
Category:	Documents
Upload:	adalberto
View:	212 times
Download:	0 times

On an unpublished manuscript of ivan prodanov concerning locally compact modules and their...

Documents