Journal of Mathematical Sciences, Vol. 120, No. 4, 2004

CONTINUOUS FUNCTORS AND DUALITY

M. B. Zvyagina UDC 512.58

Let Λ be an associative ring with identity, and let ΛM be the category of left unitary Λ-modules. A completecharacterization of continuous additive co- and contravariant functors ΛM →Z M is given. Such functors areeither representable, or equivalent to a tensor product, or trivial ones. The class of categories that are dual to ΛMand, therefore, are equivalent to the category of compact right Λ-modules is constructed by purely algebraic means. Acanonical category is singled out in this class. A purely algebraic structure that is equivalent to the topology-algebraicstructure of compact right Λ-modules is constructed. Algebraic analogs of connection and complete disconnectionare given. Bibliography: 6 titles.

Let Λ be an associative ring with identity, and let ΛM be the category of left unitary Λ-modules. In [1, 2], wediscussed a duality between some specific small subcategories of ΛM for different Λ. A natural question arises:How can one describe categories dual to ΛM in the whole? In general, such a category is well known owing tothe Pontryagin duality, namely, this is the category of compact right Λ-modules. However, compactness is atopological notion, and our aim is to construct a category dual to ΛM by purely algebraic means. Thus, we showhow the topological structure of a compact group on a set can be replaced by a purely algebraic structure.

The techniques used for the construction of a category dual to ΛM are taken from the original proofs ofthe theorems on continuous functors. For the most part, those theorems are well known but are not presentedsystematically. Thus, here we cite them not only with the aim of developing the techniques mentioned abovebut also for their systematization.

For notation, we refer the reader to [1, 2].

1. Classification of continuous functors of one argument

Proposition 1.1. Let F :ΛM →Z M be a covariant additive functor. The following statements are equivalent:

(a) F commutes with inductive limits;

(b) F is right faithful and commutes with direct sums.

Covariant functors satisfying the above conditions are said to be right continuous.

Proposition 1.2. Let F :Λ M →Z M be a contravariant additive functor. The following statements areequivalent:

(a) F takes inductive limits to projective limits;

(b) F is left faithful and takes direct sums to direct products.

Contravariant functors satisfying these equivalent conditions are said to be right continuous.

Proposition 1.3. Let F :ΛM →Z M be a covariant additive functor. The following statements are equivalent:

(a) F commutes with projective limits;

(b) F is left faithful and commutes with direct products.

Covariant functors satisfying these equivalent conditions are said to be left continuous.

Proposition 1.4. Let F :Λ M →Z M be a contravariant additive functor. The following statements areequivalent:

(a) F takes projective limits to inductive limits;

(b) F is right faithful and takes direct products to direct sums.

Contravariant functors satisfying these equivalent conditions are said to be left continuous.

Since Propositions 1.1–1.4 are given, for example, in [6], here we outline only the proof of Proposition 1.1.

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 281, 2001, pp. 186–209. Original article submitted June 21,2001.

1072-3374/04/1204-1591 c©2004 Plenum Publishing Corporation 1591

Proof. (a)⇒(b) Let F preserve inductive limits. Since F is an additive functor, it preserves finite direct sums;since an infinite direct sum is an inductive limit of finite sums, F commutes with infinite direct sums as well.

Further, if 0 → X1 → X2 → X3 → 0 is an exact sequence of left Λ-modules, then X3 is the cokernel of themorphism X1 → X2, whence F (X3) is the cokernel of the morphism F (X1) → F (X2), i.e., F (X1) → F (X2) →F (X3) → 0 is an exact sequence.

(b)⇒(a) Let F be a right faithful functor, and let it commute with direct sums. Let {Xi, fij}i,j∈I be aninductive system of left Λ-modules. Then we have the exact sequence

⊕i∈I

X(I)i → ⊕

i∈IXi → lim−→Xi → 0,

where the mapping ⊕i∈I

X(I)i → ⊕

i∈IXi is given as follows:

x ∈ X(I)i →

{fij(x) − x if i < j,

0 otherwise .

Therefore, ⊕i∈I

F (Xi)(I) → ⊕i∈I

F (Xi) → F (lim−→Xi) → 0 is also an exact sequence. Thus, we have the canonical

isomorphism lim−→F (Xi) = F (lim−→Xi).

2. Right continuous functors

Theorem 2.1. Let F :Λ M →Z M be an additive covariant functor. Then there exists a functorial morphismF (Λ)⊗Λ→ F , which is an isomorphism if and only if F is right continuous.

Remark. If F :ΛM →Z M is a right faithful, covariant functor, then F (Λ)⊗ΛX → F (X) is an isomorphism forfinitely representable ΛX. Hence, for any covariant functor F we have the canonical isomorphism F (Λ)⊗ΛX →L0F (X) if ΛX is finitely representable. On the other hand, if F :Λ M →Z M is a covariant functor commutingwith direct sums, then L0F is right continuous, whence F (Λ)⊗Λ → L0F is an isomorphism.

Corollary to Theorem 2.1 (functors with several arguments). Let Λ1, . . . , Λn be associative rings with iden-tities, and let F :Λ1 M× . . .×Λn M →Z M be an additive functor covariant in each argument. Then there existsa functorial morphism F (Λ1, . . . , Λn)⊗Λ1X1 ⊗ . . .⊗ΛnXn → F (X1, . . . , Xn), which is an isomorphism if andonly if F is right continuous in each argument.

Theorem 2.2. Let F : ΛM→ZM be an additive contravariant functor. Then there exists a functorial morphismF → HomΛ( , F (Λ)), which is an isomorphism if and only if F is right continuous.

Remark. Similarly to the case of a covariant functor, the morphism indicated above is an isomorphism forany finitely representable module ΛX only if F is right faithful. Hence, for any contravariant functor F wehave the isomorphism R0F (X) = HomΛ(X, F (Λ)) for finitely representable modules ΛX. On the other hand, ifa contravariant functor F takes direct sums to direct products, then R0F is right continuous, whence R0F =HomΛ( , F (Λ)).

Corollary to Theorem 2.2 (functors with several arguments). Let Λ1, . . . , Λn be associative rings with iden-tities, and let F :Λ1 M × . . . ×Λn M →Z M be a functor additive and contravariant in each argument. Thenthere exists a functorial morphism

F (X1, . . . , Xn) → HomΛ1,... ,Λn

(X1⊗Z . . .⊗ZXn, F (Λ1, . . . , Λn)

),

which is an isomorphism if and only if F is right continuous in each argument (here HomΛ1,... ,Λn denotes the setof morphisms in the category of left polymodules Λ1,... ,ΛnY ).

We do not give here the proofs of the above theorems, because they are rather simple and can be found, forexample, in [4]. The theorems on left continuous functors are of primary interest, and we proceed to their proof.

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3. Covariant, left continuous functors. The statement and

beginning of the proof of the representability theorem

Theorem 3.1. Let F :ΛM→ZM be an additive covariant functor. Then there exists a functorial morphismF →HomΛ (HomFunct(F, Id), ), which is an isomorphism if and only if F is left continuous. (HomFunct(F, Id)denotes the Abelian group of functorial morphisms F → Id, where Id :ΛM→ZM is the forgetful functor.)

Similarly to the theorems in the previous section, this theorem is well known and its proof can be read in [4].As we have said above, we present its original proof with the aim of developing the techniques that will be usedin the sequel for the construction of a category dual to ΛM.

Proof. First we endow the Abelian group HomFunct(F, Id) with the structure of a left Λ-module in the followingway: for a functorial morphism ϕ : F → Id and for any left Λ-module X we set λϕ : F (X) → X, (λϕ)(y) = λϕ(y)(λ ∈ Λ). An easy verification shows that λϕ is a functorial morphism, λϕ : F → Id; moreover, (λµ)ϕ = λ(µϕ),whence Λ acts on HomFunct(F, Id) as a ring of left operators.

Next, we define a homomorphism F (X)→HomΛ(HomFunct(F, Id), X) as follows:

y ∈ F (X) → fy : HomFunct(F, Id) → X, fy(ϕ) =(ϕ(X)

)(y)

(where ϕ(X) : F (X) → X is the value of the functorial morphism ϕ on the object X). The fact that the abovemorphism is functorial is checked in the usual way.

If the morphism of functors F → HomΛ(HomFunct(F, Id), ) is an isomorphism, then, obviously, F is leftcontinuous (because HomΛ has the same property in the second argument). To prove the converse statement,we need some special lemmas.

4. Covariant, left continuous functors. Key lemmas and the

completion of the proof of the representability theorem

Lemma 4.1. Let F : ΛM →ZM be a covariant additive functor commuting with direct products. Then for any

ΛV and for a nonempty set I, F (V F(V )×I) is a monogenic left Γ(ΛV F(V )×I)-module. In particular, F (V F(V )) ismonogenic over F (ΛV F(V )). We recall that Γ(ΛX) denotes the endomorphism ring of the module ΛX.

Proof. We fix i0 ∈ I and define an element x0I ∈ F (V )F(V )×I (= F (V F(V )×I) as follows:

x0I = {x0

y,i}y∈F(V ),i∈I , x0y,i = yδi,i0 (δ is the Kronecker symbol).

We claim that the element x0I generates F (V )F(V )×I over Γ(ΛV F(V )×I). Indeed, let x = {xy,i} be an arbitrary

element of F (V )F(V )×I . Set γx : V F(V )×I → V F(V )×I , γx{vy,i} ={vxy,i,i0

}. Obviously, γx ∈ Γ(ΛV F(V )×I);

moreover, F (γx){zy,i} = {zxy,i,i0}. In other words, γx{zy,i} = {zxy,i,i0} (by the definition of the action of thering Γ(ΛV F(V )×I) on F (V F(V )×I)). We have γxx0

I ={x0

xy,i,i0

}= {xy,i} = x. Thus, an arbitrary element

x ∈ F (V )F(V )×I is represented in the form γxx0I, and thus x0

I generates F (V )F(V )×I over Γ(ΛV F(V )×I). Thelemma is proved.

Lemma 4.2. As above, let F :ΛM→ZM be a covariant functor commuting with direct products. Next, let ΛV

be an arbitrary module, and let I0 be the kernel of the epimorphism of left Γ(ΛV F(V ))-modules Γ(ΛV F(V )) ·x0

→F (V F(V )) that is constructed in Lemma 4.1 (for a one-element set I, x0 = {y}y∈F(V )). For any nonempty

set J we denote by IJ the kernel of the epimorphism of left Γ(ΛV F(V )×I0×J)-modules Γ(ΛV F(V )×I0×J)·x0

I0×J→F (V F(V )×I0×J) with x0

I0×J = {x0δi,0δj,j0} (j0 is a fixed element of the set J). Here, as above, we denote

{y}y∈F(V ) ∈ F (V )F(V ) by x0. Then the left ideal IJ of the ring Γ(ΛV F(V )×I0×J) is finitely generated (moreprecisely, it can be generated by two elements).

Proof. For brevity, we set Y = V F(V ) and Γ = Γ(ΛY I0×J). A morphism γ ∈ Γ lies in the ideal IJ if and only ifγ{x0δi,0δj,j0} = 0. We show that IJ is generated over Γ by two morphisms γ1 and γ2, where γ1{yi,j} = {i(y0,j0)}and γ2{yi,j} = {yi,j(1 − δi,0δj,j0)} (yi,j ∈ Y ). Obviously, γ1, γ2 ∈ IJ . For an arbitrary γ ∈ IJ we consider themorphism γ′ = γ − γγ2 = γ(id− γ2). We have γ′{yi,j} = γ{yi,jδi,0δj,j0}. We construct a morphism δγ such thatthe relation γ′ = δγγ1 holds. To this end, we define ρ0 : Y → Y I0×J , ρ0(y) = {yδi,oδj,j0}, and πi,j : Y I0×J → Y ,πi,j{yk,l} = yi,j (i.e., πi,j is the canonical projection of Y I0×J to the component with number (i, j)). Consider the

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composition πi,jγρ0 : Y → Y . We have πi,jγρ0(x0) = πi,jγ{x0δi,0δj,j0} = 0, because γ ∈ IJ . Hence, πi,jγρ0 ∈ I0

for any pair (i, j) ∈ I0 × J . We put δγ{yi,j} = {yπi,j γρ0, j0} and verify that γ′ = δγγ1:

δγγ2{yi,j} = δγ{i(yo,j0)} = {πi,jγρ0(yo,j0)} = γρ0(y0,j0) = γ{yi,jδi,0δj,j0} = γ′{yi,j}.

Thus, we have γ = γ′ + γγ2 = δγγ1 + γγ2 for any γ ∈ IJ . Therefore the left ideal IJ of the ring Γ is generatedby two elements γ1 and γ2. The lemma is proved.

Corollary 1. Let F:ΛM→ZM be a covariant functor commuting with direct products, and let ΛV be an arbitrarymodule. Then there exists a set I(F, V ) such that F (V )I(F,V )×J is a finitely representable left Γ(ΛV I(F,V )×J)-module for any nonempty set J .

Proof. We set I(F, V ) = F (V ) × I0, where I0 is the left ideal of Γ(ΛV F(V )) that is described in Lemma 4.2. ByLemmas 4.1 and 4.2, for any nonempty set J we have the resolvent of the Γ(ΛV I(F,V )×J)-module F (V )I(F,V )×J :

Γ(

ΛV I(F,V )×J)2

→ Γ(ΛV I(F,V )×J) → F (V )I(F,V )×J → 0.

Corollary 2. As above, let F be a covariant functor commuting with direct products. For an injective module

ΛV , the left Γ(ΛV (F,V )×J)-module F (V )F(V )×J is a V I(F,V )×J -reflexive module for any nonempty set J . Inother words, there exists a direct power V0 = V I(F,V ) of the module ΛV such that F (V0)J is a V J

0 -reflexive leftΓ(ΛV J

0 )-module for any nonempty set J .

Proof. It is sufficient to refer to the consequence of Lemma 2.1 from [1].

Lemma 4.3. Let F be a covariant functor commuting with direct products. For an arbitrary module ΛV andfor a nonempty set I, any homomorphism of left Γ(ΛV I)-modules F (V )I → V I is of the form ϕI for someϕ∈HomΓ(ΛV )(F (V ), V ). In other words, a monomorphism HV I (F (V I))→HV (F (V )) with ϕI →ϕ exists.

Proof. Let ϕI ∈ HV I (F (V )I); then the morphism ϕI commutes with all morphisms over Λ, V I γ→ V I , whencethe following commutative diagrams hold:

F (V )I ϕI−−−−→ V I

F(γ)

γ

F (V )I ϕI−−−−→ V I .

First let γ = γi : V I → V I be the projection to the ith factor. Then γi{vj} = {viδi,j}. We have F (γi){yj} ={yiδi,j}({yi} ∈ F (V )I). The commutativity of the above diagram for γ = γi implies that if ϕI{yi} = {vj},then ϕI{δi,jyj} = {δi,jvj} and, in particular, the ith component of the element ϕI(y) depends only on the ithcomponent of y ∈ F (V )I , i.e., ϕI = Π

i∈Iϕi, where ϕi ∈ HV (F (V )). Next, applying the above commutative

diagram to the morphism γ = γi,j that interchanges the ith and jth components of V I (i �= j), we have ϕi = ϕj

for any i, j ∈ I, i.e., ϕI = ϕI for some ϕ ∈ HV (F (V )). The lemma is proved.

Lemma 4.4. Let F be a covariant functor commuting with direct products, let ΛV be an injective cogenerator,and let V0 be the V I(F,V )-module constructed in Corollary 2 to Lemmas 4.1 and 4.2. Then for any ϕ ∈ HV0(F (V0))we have ϕI ∈ HV I

0(F (V I

0 )) for any nonempty set I. In other words, the monomorphism HV I0(F (V I

0 )) →HV0(F (V0)) with ϕI → ϕ constructed in the previous lemma is an isomorphism for any nonempty set I.

Proof. By Corollary 2 to Lemmas 4.1 and 4.2, F (V I0 ) is a V I

0 -reflexive left Γ(ΛV I0 )-module, whence the following

left Γ(ΛV I0 )-modules are isomorphic:

F (V I0 )

f(I)→ HomΛ(HV I0F (V I

0 ), V I0 ).

On the other hand, raising to the power I the isomorphism of left Γ(ΛV0)-modules F (V0)g→ HomΛ(HV0F (V0),

V0), we obtain the isomorphism F (V0)I gI

→ HomΛ(HV0F (V0), V I0 ). We denote by νV0,I the monomorphism

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constructed in Lemma 4.3, namely, HV I0F (V I

0 )ν

V0,I→ HV0F (V0), where νV0,I

(ϕI) = ϕ, and then we verify thecommutativity of the diagram

F (V0)I f(I)−−−−→ HomΛ(HV I0F (V I

0 ), V I0 )

(gI )−1

GV I0(νV0,I )↗

HomΛ(HV0(F (V0), V I0 ).

Indeed,f(I)(gI )−1(ψ → {ψ(yi)}) = f(I){yi} = (ϕI → {ϕ(yi)});

GV I0(νV0,I )(ψ→{ψ(yi)})=(ϕI → (ψ → {ψ(yi)}) ◦ ϕI =(ϕI →{ϕ(yi)}).

Thus, the above diagram is commutative, whence GV I0(νV0,I ) is an isomorphism of left Γ(ΛV I

0 )-modules.Next, denoting by C the cokernel of the monomorphism νV0,I of left Λ-modules, we have the exact sequence

0 → GV I0(C) → GV I

0(HV0F (V0))

GV I0

νV0,I

−→ GV I0(HV I

0F (V I

0 )) → 0 (by the injectivity of V ). Since GV I0νV0,I is an

isomorphism, we have GV I0(C) = 0, whence C = 0 (because V I

0 is a cogenerator), i.e., νV0,I is an isomorphism,as required.

Lemma 4.5. As above, let F commute with direct products, let ΛV be an injective cogenerator, and letV0 = V I(F,V ) be the module constructed in Corollary 2 to Lemmas 4.1 and 4.2. For any nonempty set I, thehomomorphism

η : HomFunct(F, Id) → HomΓ(ΛV I0 )(F (V I

0 ), V I0 ), η(ϕ) = ϕ(V I

0 ),

is an isomorphism of left Λ-modules. Moreover, the diagram

F (V I0 ) −−−−→ HomΛ(HomΓ(ΛV I

0 )(F (V I0 ), V I

0 )V I0 ) = GV I

0HV I

0F (V I

0 )

↘ HomΛ(η,id)

HomΛ(HomFunct(F, Id), V I

0 )

is commutative, where the oblique arrow is the value of the functorial morphism constructed in Sec. 3 on theobject V I

0 .

Proof. First we observe that η(ϕ) = ϕ(V I0 ) : F (V I

0 ) → V I0 is a homomorphism of left Γ(ΛV I

0 )-modules for anyfunctorial morphism F

ϕ→ Id. Moreover, η is a homomorphism of left Λ-modules. Show that it is an isomorphism.Let ψI ∈ HomΓ(ΛV I

0 )(F (V I0 ), V I

0 ), where ψ ∈ HomΓ(ΛV0)(F (V0), V0). The morphism ϕ(V J0 ) = ψJ : F (V J

0 ) →V J

0 is functorial for the restrictions of the functors F and Id to the complete subcategory of ΛM whose objectsare left Λ-modules of the form V J

0 . Indeed, the diagrams

F (V J0 )

F(γ)−−−−→ F (V J0 )

ψJ

ψJ

V J

0γ−−−−→ V J

0

are commutative for γ ∈ Γ(ΛV J0 ) because, by Lemma 4.4, ψJ ∈ HomΓ(ΛV J

0 )(F (V J0 ), V J

0 ), and this implies thecommutativity of the diagrams

F (V J0 )

F(γ)−−−−→ F (V0)

ψJ

ψ

V J

0γ−−−−→ V0

for γ ∈ HomΛ(V J0 , V0) and the diagrams

F (V J0 )

F(γ)−−−−→ F (V K0 )

ψJ

ψK

V J

0γ−−−−→ V K

0

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for γ ∈ HomΛ(V J0 , V K

0 ), which means that the morphism ϕ(V J0 ) = ψJ is functorial.

Thus, starting from the homomorphism ψI ∈ HomΓ(ΛV I0 )(F (V I

0 ), V I0 ), we constructed the morphism ϕ =

ν0(ψI ) of the restrictions of the functors F and Id to the complete subcategory of ΛM whose objects are modulesof the form V J

0 . Since any left Λ-module has a unique resolvent, up to homotopy, composed of modules ofthe form V J

0 and F is a left faithful functor, the morphism mentioned above can be extended to a functorialmorphism ν(ψ) : F → Id. Obviously, ν : HomΓ(ΛV I

0 )(F (V J0 ), V I

0 ) → HomFunct(F, Id) is the homomorphism of leftΛ-modules, inverse to η, and thus, η is an isomorphism.

The commutativity of the diagram for the lemma is verified directly, which proves the lemma.

Completion of the proof of Theorem 3.1. Let F be a covariant, left continuous functor. We need to prove thatthe functorial morphism F → HomΛ(HomFunct(F, Id), ) constructed in Sec. 3 is an isomorphism. By Lemma 4.5,this morphism on objects of the form V I

0 can be expressed as a composition of two isomorphisms and, thus, itis an isomorphism (as before, V0 denotes the injective cogenerating left Λ-module constructed in Corollary 2 toLemmas 4.1 and 4.2). Consider a resolvent of the form 0 → X → V I

0 → V J0 for any ΛX. Applying the 5-lemma

to the commutative diagram

0 −−−−→ F (X) −−−−→ F (V I0 ) −−−−→ F (V J

0 ) 0 −−−−→ HomΛ(HomFunct(F, Id), X) −−−−→ HomΛ(HomFunct(F, Id), V I

0 ) −−−−→ HomΛ(HomFunct(F, Id), V J0 )

(which can be applied, because the functors F and HomΛ(HomFunct(F, Id), ) are left faithful), we obtain an isomor-phism F (X)→HomΛ(HomFunct(F, Id), X) induced on X by the morphism of functors F→HomΛ(HomFunct(F, Id), ).Thus, this functorial morphism is in fact an isomorphism, and the theorem is proved.

Remark. If F :Λ M →Z M is a covariant functor commuting with direct products, then the right derivativefunctor R0F is left continuous and, thus, is representable, R0F = HomΛ(HomFunct(F, Id), ).

5. Left continuous, contravariant functors

Theorem 5.1. Let F :ΛM →ZM be a left continuous, contravariant additive functor. Then F = 0.

Remark 5.1. On a category dual to ΛM, nonzero contravariant, left continuous functors exist; namely, thefunctors dual to A⊗Λ for AΛ �= 0 are such functors.

Remark 5.2. To prove the above theorem, it is sufficient to show that F (V ) = 0, where ΛV is a certain (fixed)injective cogenerator.

Proof. Since F takes direct products to direct sums, from F (V ) = 0 it follows that F (V I) = 0 for any nonemptyset I. Any module ΛX is embedded in V I for some I, 0 → X → V I , and from the fact that F is right faithful,we have the exact sequence F (V I) → F (X) → 0, from which it follows that F (X) = 0.

Remark 5.3. To prove the above theorem for any Λ, it is sufficient to prove it for Λ = Z.

Proof. For an arbitrary Abelian group ZY , consider the covariant functor ΛM→ ZM with X→HomZ(F (X), Y ).This functor is left continuous and, thus, by the theorem in the previous section, it is represented by a leftΛ-module G(Y ), HomZ(F (X), Y ) = HomΛ(G(Y ), X). The usual verification shows that G(Y ) depends on Y ina cotravariant way; moreover, G :Z M →Λ M is a left continuous functor. Hence, if the above theorem is validfor Λ = Z, then G = 0, whence HomZ(F (X), ) = 0 for any ΛX and, therefore, F = 0. Thus, the theorem underconsideration is true for any Λ.

Proof of the theorem. In accordance with the above remarks, it is sufficient to consider the case where Λ = Z

and to prove that F (V ) = 0 for an injective cogenerating Abelian group ZV . We fix an injective cogeneratorZV and an infinite set I. Consider the canonical embedding 0→ V (I) → V I . Since F is right faithful, we havean epimorphism F (V I)→F (V (I))→0. The canonical morphism F (V )(I)→F (V I) is an isomorphism (we meanprecisely this isomorphism when we say that F takes direct products to direct sums). Hence, the compositionF (V )(I)→F (V I)→F (V (I)) is an epimorphism. On the other hand, this composition is a monomorphism as theinductive limit of the monomorphisms F (V )n→F (V (I)). Hence, F (V I)→F (V (I)) is an isomorphism. Since V (I)

is an injective Abelian group, the exact sequence 0→V (I)→V I →V I/V (I)→ 0 splits, whence F (V I/V (I))=0,and by the fact that F is right faithful and the embedding V → V I/V (I), where v ∈ V →{v}i∈I mod V (I), is“diagonal,” we have F (V )=0. The theorem is proved.

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6. Characterization of continuous functors with several arguments

Theorem 6.1. Let Λ1, . . . , Λm, Γ1, . . . , Γn, ∆1, . . . , ∆k be associative rings with identities, and let F :Λ1

M×. . .×ΛmM×Γ1M×. . .×ΓnM×∆1M×. . .×∆kM →Z M be a nonzero additive functor with n + m + k argumentsthat is contravariant and right continuous for the first m arguments, covariant and right continuous for the nextn arguments, and, finally, covariant and left continuous for the last k arguments. Then the following three caseshold:

(1) m = k = 0 and F (Y1, . . . , Yn) = F (Γ1, . . . , Γn) ⊗Γ1 Y1 ⊗ . . .⊗Γn Yn;

(2) n=k=0 and F (X1, . . . , Xm)=HomΛ1,... ,Λm(X1 ⊗Z . . .⊗Z Xm, F (Λ1, . . . , Λm));(3) n = 0, k = 1 (∆1 = ∆), and

F (X1, . . . , Xm, Z) = HomΛ1,... ,Λm

(X1 ⊗Λ1 . . .⊗Λm Xm, Hom∆(HomFunct(F (Λ1, . . . , Λm), ), Id), Z

).

Proof. (1) Let F :∆1 M ×∆2 M →Z M be a covariant functor, left continuous for both arguments. Then F = 0.Indeed, such a functor is representable, for definiteness, in the second argument; namely,

F (Z1, Z2) = Hom∆2(HomFunct(F (Z1, ), Id), Z2).

Moreover, HomFunct(F (Z1, ), Id) must be a contravariant functor in the argument Z1 and left continuous, whenceit is equal to zero (by Theorem 5.1), so that F = 0.

(2) Let F :Γ M×∆ M →Z M be a covariant functor in two arguments, right continuous in the first argument,and left continuous in the second argument. Then F = 0.

Indeed, in this case F (Y, Z) = F (Γ, Z)⊗∆ Y , and if F (Γ, Z) �= 0, then the functor under consideration cannotbe left continuous in the argument Z, because ⊗∆Y does not commute with direct products if Y is not a finitelygenerated module. Therefore, F (Γ, Z) = 0, whence F = 0.

(3) Let F :Λ M⊗∆ M →Z M be a right continuous functor in two arguments that is contravariant in the firstargument and covariant in the second argument. Then F = 0.

Indeed, F (X, Y ) = HomΛ(X, F (Λ, Γ) ⊗Γ Y ) and if F (Λ, Γ) �= 0 then this functor cannot be right continuousin the argument Y (because HomΛ(X, ) does not commute with direct sums for infinitely generated modulesΛX). Hence, F (Λ, Γ) = 0 and F = 0.

Combining cases (1), (2), and (3), we prove the theorem.

7. Restatement of the results obtained above in terms of categories of functors

We denote by ΛRgCo, ΛRgCt, ΛLfCo, and ΛLfCt, respectively, the categories of right continuous covariant,right continuous contravariant, left continuous covariant, and left continuous contravariant functors ΛM →Z M

and functorial morphisms. The theorems proved in Secs. 1–5 can be restated as follows:

(1) the category ΛRgCo is equivalent to MΛ;(2) the category ΛRgCt is equivalent to ΛM;(3) the category ΛLfCo is dual to ΛM;(4) ΛLfCt is the trivial category.

The most intriguing (and least trivial) result is concerned with the duality between ΛM and ΛLfCo. Togetherwith the Pontryagin duality, this result yields an equivalence between ΛLfCo and the category CompΛ ofcompact right Λ-modules. In a highly nonstrict sense, a compact right Λ-module is a left continuous, covariantfunctor ΛM →Z M. Thus, we have already obtained the pure algebraic category ΛLfCo dual to ΛM. However,the objects of this category are described in a nonconstructive way. The category to which the remaining partof the present paper is devoted is built much more constructively than the category ΛLfCo equivalent to it,although in a noninvariant way, because it depends on the choice of an injective cogenerator ΛV . However,the presence in the category ΛM of the canonical injective cogenerator Cont(Λ) = HomZ(Λ, R/Z) ensures theconstruction of a category canonically dual to ΛM. As was promised, this construction shows how one can replacethe algebraic-topological structure of a compact Λ-module on a set by a purely algebraic structure.

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8. ΛV -multistructural Abelian groups

Let ΛV be an arbitrary module. An Abelian group Y is said to be ΛV -multistructural if on groups of the formY I , structures of left Γ(ΛV I)-modules consistent with each other are given. This means that to any nonemptyset I corresponds a ring homomorphism α(I) : Γ(ΛV I) → End Z(Y I); moreover, we have commutative diagramsof the form (

Γ(ΛV I)(J))J −−−−→ Γ(ΛV I×J)

(α(I)(J))J

α(I×J)

(End Z(Y I)(J)

)J −−−−→ End Z(Y I×J)

with natural horizontal arrows

{γk,j}k,j∈J →({yj}j∈J →

{ ∑k∈J

γk,j(yk)}

j∈J

),

where γk,j ∈ Γ(ΛV I) and yj ∈ V I for the upper row and γk,j ∈ End Z(Y I ) and yj ∈ Y I for the lower row.

By a ΛV -multimorphism we mean a homomorphism Y1ϕ→Y2 of ΛV -multistructural Abelian groups such that for

any nonempty set I, Y I1

ϕI

→ Y I2 is a homomorphism of left Γ(ΛV I)-modules. The category of ΛV -multistructural

Abelian groups and ΛV -multimorphisms is denoted by ΛV -Mult.

Example. If X is a left Λ-module, then the functor HomΛ(X, ) generates a ΛV -multistructure on the Abeliangroup HomΛ(X, V ) (induced by the system of natural isomorphisms HomΛ(X, V I) = HomΛ(X, V )I ). In partic-ular, we have the canonical ΛV -multistructure on V = HomΛ(Λ, V ).

In the next section, we show that for an injective cogenerator ΛV , the above example is the general one, i.e.,any ΛV -multistructural Abelian group is isomorphic to HomΛ(X, V ) for some ΛX.

9. A theorem on the representativity of a ΛV -multistructure

Theorem 9.1. Let Y be a ΛV -multistructural Abelian group. Then there exists a ΛV -morphism

Y → HomΛ(HomΛV -Mult(Y, V ), V ),

which is an isomorphism if ΛV is an injective cogenerator.

Proof. Since the proof of the theorem under consideration is an exact copy of the proof of Theorem 3.1 (to-gether with the techniques used), we restrict ourselves to the construction of the ΛV -multimorphism Y

θY→HomΛ(HomΛV -Mult(Y, V ), V ) and to the statement of some lemmas. The morphism θY is given as follows:(θY (y))(ψ) = ψ(y) for y ∈ Y and ψ ∈ HomΛV -Mult(Y, V ).

Lemma 9.1. Let ΛV be an arbitrary module, and let Y be a ΛV -multistructural Abelian group. Then for anynonempty set I, Y Y ×I is a monogenic left Λ-module generated by an element x0

I ={yδi,i0

}y∈Y,i∈I

for fixed

i0 ∈ I. In particular, Y Y is generated over Γ(ΛV Y ) by the element x0 = {y}y∈Y .

Lemma 9.2. As before, let ΛV be an arbitrary module and let Y be a ΛV -multistructural Abelian group. We

denote by I0 the kernel of the epimorphism of left Γ(ΛV Y )-modules Γ(ΛV Y ) ·x0

→ Y Y constructed in Lemma

9.1 and by IJ the kernel of the epimorphism of Γ(ΛV Y ×I0×J)-modules Γ(ΛV Y ×I0×J)·x0

I0×J→ Y Y ×I0×J withx0

I0×J = {x0δi,0δj,j0} and x0 = {y}y∈Y (for any nonempty set J and fixed j0 ∈ J). Then the left ideal IJ of the

ring Γ(ΛV Y ×I0×J ) is finitely generated (more precisely, is generated by two elements).

Corollary 1. Let Y be a ΛV -multistructural Abelian group. Then there exists a set I(Y ) such that Y I(Y )×J isa finitely representable left Γ(ΛV I(Y )×J)-module for any nonempty set J .

Corollary 2. Let ΛV be an injective module, Y be a ΛV -multistructural Abelian group, and I(Y ) be the setconstructed in Corollary 1. Next, put V0 = V I(Y ) and Y0 = Y I(Y ). Then the Γ(ΛV J

0 )-module Y J0 is a V J

0 -reflexivemodule for any nonempty set J .

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Lemma 9.3. Let ΛV be an arbitrary module, and let Y1 and Y2 be ΛV -multistructural Abelian groups. Forany nonempty set I, every homomorphism of left Γ(ΛV I)-modules Y I

1 → Y I2 is of the form ϕI for some ϕ ∈

HomΓ(ΛV )(Y1, Y2). In particular, setting Y2 = V we have a monomorphism HV I (Y I) → HV (Y ) with ϕI → ϕ.

Lemma 9.4. Let ΛV be an injective cogenerator, Y be a ΛV -multistructural Abelian group, V0 = V I(Y ) be themodule constructed in Corollary 2 to Lemmas 9.1 and 9.2, and Y0 = Y I(Y ). Then for any ϕ ∈ HV0(Y0) we haveϕI ∈ HV I

0(Y I

0 ) for any nonempty set I. In other words, the monomorphism HV I0(Y I) → HV0(Y0) constructed in

Lemma 9.3 is an isomorphism for any nonempty set I.

Lemma 9.5. As before, let ΛV be an injective cogenerator, Y be a ΛV -multistructural Abelian group, andY0 and V0 be the modules constructed in Corollary 2 to Lemmas 9.1 and 9.2. Then the homomorphism η :HomΛV -Mult(Y, V ) → HomΓ(ΛV I )(Y I

0 , V I0 ) with η(ϕ) = ϕI is an isomorphism of left Λ-modules; moreover, we

have the commutative diagram

Y I0 −−−−→ HomΛ(HomΓ(ΛV I

0 )(Y I0 , V I

0 ), V I0 )

θIY0↘ HomΛ(η,id)

HomΛ(HomΛV -Mult(Y, V ), V I

0 ).

We believe that the above five lemmas, which are proved in precisely the same way as the lemmas in Sec. 4(because in the proofs of those lemmas we used only the fact that a functor F that commutes with direct sumsinduces a ΛV -multistructure on an arbitrary Abelian group F (V )) prove Theorem 9.1.Remark 9.1. The theorem proved above provides an efficient way for constructing ΛV -multistructural groupsfor injective cogenerators ΛV . Indeed, if Y = HomΛ(X, V ), then applying the functor GV to the free resolventΛ(J) → Λ(I) → X → 0 of a left Λ-module X we have the exact sequence 0 → Y → V I θ→ V J , wherethe homomorphism θ is given by a matrix with finitary columns {λi,j} ∈ (Λ(I))J and θ{vi} =

{ ∑i∈I

λijvi

}.

Thus, an arbitrary ΛV -multistructural Abelian group is isomorphic to the kernel of a homomorphism θ of theform indicated above with the structures of Γ(ΛV K)-modules on Y K that are induced by the homomorphisms

(V K)I θK

→ (V K)J of left Γ(ΛV K)-modules.Henceforth we call homomorphisms V I → V J given by the matrices {λi,j} ∈ (Λ(I))J

ΛV -continuous homo-morphisms. Thus, an Abelian group Y admits a ΛV -multistructure if and only if it is isomorphic to the kernelof a ΛV -continuous homomorphism. Since the free resolvent ΛX : Λ(J) → Λ(I) → 0 is determined up to homo-topic equivalence, the ΛV -continuous resolvent 0 → V (I) → V (J) is determined up to ΛV -continuous homotopicequivalence (i.e., homotopic equivalence of complexes in the category whose objects are modules of the form V I

and the morphisms are ΛV -continuous). Thus, the specification of a ΛV -structure on Y is equivalent to theindication of a ΛV -continuous homotopic class of ΛV -continuous resolvents Y : 0 → V I → V J .

10. The main result on duality

Theorem 10.1. Let ΛV be an injective cogenerator. Then the category ΛV -Mult is dual to the category ΛM.

Proof. The desired duality is given by the functors

ΛM−→←Λ V -Mult, ΛX → HomΛ(X, V ), HomV -Mult(Y, V ) ← Y.

The results in Sec. 9 show that the above functors are inverse to each other.

Remark. For the canonical injective cogenerator V = Cont(Λ) = HomZ(Λ, R/Z), we obtain the canonicalduality between ΛM and ΛCont(Λ)-Mult.

At the same time, we present the functors realizing the Pontryagin duality between ΛM and ΛLfCo and theequivalence of the categories ΛLfCo and ΛV -Mult.

The restriction of the Pontryagin duality to the category ΛM:ΛM−→← CompΛ is ΛX →(the compact right

Λ-module HomZ(X, R/Z) = HomΛ(X, Cont(Λ))) and (a compact right module YΛ) → HomCompZ(Y, R/Z) =

HomCompΛ(Y, Cont(Λ)) (in the latter formula, Cont(Λ) = HomZ(Λ, R/Z) has structure of a compact right

Λ-module ensured by the classical Pontryagin duality).The duality between ΛM and ΛLfCo is given in the following way:

ΛY → HomΛ(Y, ) and (F :ΛM →Z M) → HomFunct(F, Id).

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The equivalence between ΛLfCo and ΛV -Mult is given as follows:(F :Λ M →Z M) → (the ΛV -multistructural Abelian group F (V ) with structures of left Γ(ΛV I)-mo-dules on F (V )I = F (V I) induced by the functor F ) and (a ΛV -multistructure Y ) → (the functorHomΛ(Hom

ΛV -Mult(Y, V ), ) : ΛM →Z M).

11. Replacement of a topological structure by an algebraic structure

From the results in Sec. 10, it follows that the specification of the structure of a compact right Λ-module onan Abelian group Y is equivalent, to the specification of a left continuous, covariant functor ΛM →Z M takingthe value Y on the module ΛCont(Λ), or what is equivalent to setting a ΛCont(Λ)-multistructure on Y .

Namely, if YΛ is a compact right Λ-module and γ ∈ HomΛ(Cont(Λ)I , Cont(Λ)I), then the morphism γ :

Y I → Y I must commute with all powers of continuous morphisms Y I ϕI

→ U I , where ϕ ∈ HomCompΛ(Y, U)), ofcompact right Λ-modules (because for equivalent topological and purely algebraic structures, the notions of acontinuous homomorphism of compact right Λ-modules and a ΛCont(Λ)-multimorphism must coincide). Hence,a continuous embedding Y

η→ Cont(Λ)Y ∗, where η(y) = {x(y)} (Y ∗ = HomCompΛ

(Y, Cont(Λ))), must ensure thecommutativity of the diagrams of the form

Y I ηI

−−−−→ Cont(Λ)Y ∗×I

γ

γ

Y I ηI

−−−−→ Cont(Λ)Y ∗×I

(naturally we put γ : Cont(Λ)Y ∗×I → Cont(Λ)Y ∗×I = γY ∗: (Cont(Λ)I )Y ∗ → (Cont(Λ)I)Y ∗

). Thus, we seethat γηI (Y I ) ⊂ ηI(Y I) and the formula γ{yi}i∈I = (ηI)−1γηI{yi}i∈I determines the desired Cont(Λ)-structureon Y .

Conversely, if YΛ is a Cont(Λ)-multistructural Abelian group, then, taking as base neighborhoods of zero inY the sets of the form f−1(U), where f ∈ Hom

ΛCont(Λ)-Mult(Y, Cont(Λ)) and U is a base neighborhood of zero inCont(Λ), we obtain the structure of a compact right Λ-module on Y .

Proposition 11.1. The structure of a compact right Λ-module is equivalent to a ΛCont(Λ)-multistructure.

In particular, the structure of a compact Abelian group is equivalent to a ZR/Z-multistructure.

Proposition 11.2. (a) The structure of a connected compact right Λ-module on Y is equivalent to a ΛCont(Λ)-multistructure such that Hom

ΛCont(Λ)-Mult(Y, Cont(Λ)) is a left Λ-module without elements of finite order.

(b) The structure of a totally disconnected compact right Λ-module on Y is equivalent to a ΛCont(Λ)-multistructure such that Hom

ΛCont(Λ)-Mult(Y, Cont(Λ)) consists of elements of finite order.

The above propositions and similar ones that describe topological properties of compact groups in purelyalgebraic terms are consequences of the Pontryagin duality theorem (see [5, Chap. 6, Sec. 38]).

12. The rational skeleton and completion

We may compactify an Abelian group Y in a more efficient way than this was done in the previous section.Namely, if an Abelian group Y can be compactified, then it admits the structure of a left Γ(ZR/Z)-module,because Y = HomZ(X, R/Z) with X = HomCompZ

(Y, R/Z). Moreover, the Abelian group HomZ(X, Q/Z) iscanonically isomorphic to IY , where I is a right ideal of Γ(ZR/Z) that consists of homomorphisms of Abeliangroups R/Z → R/Z with images in Q/Z ⊂ R/Z. Since both pairs of functors Z-Mult −→← Z R/Z-Mult, ZM

−→← Z

Q/Z-Mult and ZX → HomZ(X, R/Z), ZX → HomZ(X, Q/Z) realize the equivalence of the respective categories,it follows that the composite functors ZQ/Z-Mult −→←Z R/Z-Mult also yield the equivalence of the categories.Hence, the specification of the structure of a compact set on a left Γ(ZR/Z)-module Y is equivalent to thespecification of a Q/Z-multistructure on IY .

We call the Abelian group IY the rational skeleton of a left Γ(ZR/Z)-module Y . If Y is compactified, thenthe corresponding topological space is the completion of the rational skeleton of Y (in the induced topology).

Conversely, if Y1 is a ZQ/Z-multistructural group, then we may set topology on it, taking as base neighborhoodsof zero the sets of the form f−1(U), where f ∈ Hom

Q/Z-Mult(Y1, Q/Z) and U is a base neighborhood of zero in

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Q/Z. The completion Y1 of the topological group Y1 is compact; moreover, Y1 is the rational skeleton of Y1. Thegroup HomZ

(Hom

Q/Z-Mult(Y1, Q/Z), R/Z)

is the ZR/Z-multistructural group corresponding to Y1.The rational skeleton of a compact Abelian group is a dense subgroup of it, which may coincide with the

entire compact set. The coincidence holds, for example, for compact groups Zp and, in general, for all totallydisconnected compact Abelian groups (and only for them). Indeed, Y is a totally disconnected group if and onlyif Y = HomZ(X, R/Z), where X is the torsion group of it. Accordingly, in the above and only in the above casewe have IY = HomZ(X, Q/Z) = HomZ(X, R/Z) = Y .

Combining all what we have said in this section, we obtain the following proposition.

Proposition 12.1. (a) The structure of a compact set on a left Γ(ZR/Z)-module is equivalent to a ZQ/Z-multistructure on its rational skeleton;

(b) any ZQ/Z-multistructural Abelian group Y1 is the rational skeleton of its completionHomZ

(Hom

Q/Z-Mult(Y1Q/Z), R/Z);

(c) the rational skeleton of a compact Abelian group Y is a dense subgroup of it, which coincides with Y ifand only if Y is a totally disconnected group.

13. An efficient way of constructing a set with an algebraic structure

that is equivalent to the structure of a compact Abelian group

Proposition 13.1. (a) An Abelian group can be compactified if and only if it is isomorphic to the kernel ofa ZR/Z-continuous homomorphism. In this case, the specification of compact topology on an Abelian group Yis equivalent to the indication of an R/Z-continuous homotopic class of resolvents of Y , i.e., of R/Z-continuoushomomorphisms (R/Z)I → (R/Z)J with kernel isomorphic to Y ;

(b) continuous homomorphisms of compact Abelian groups are induced precisely by ZR/Z-continuous mor-phisms of their ZR/Z-resolvents;

(c) the rational skeleton of a compact Abelian group Y is the preimage of (Q/Z)I under the embedding0 → Y → (R/Z)I that is the kernel of the resolvent of Y ;

(d) an Abelian group admits the structure of the rational skeleton of a compact set if and only if it is isomorphicto the kernel of a ZQ/Z-continuous homomorphism. The specification of the topology of the rational skeletonof a compact set on an Abelian group Y1 is equivalent to the indication of a Q/Z-continuous homotopic class of

ZQ/Z-resolvents of Y1, i.e., ZQ/Z-continuous morphisms (Q/Z)I → (Q/Z)J with kernel that is isomorphic to Y1.

The above statements follow from the results in Secs. 10–12 and from Remark 9.1, and thus they need noproof.

Proposition 13.2. (a) A compact Abelian group Y is connected if and only if for an arbitrary diagram of ZR/Z-

continuous morphisms

(R/Z)I −−−−−→ (R/Z)JR/Z

, where the horizontal arrow is an ZR/Z-continuous resolvent of Y ,

the fiber sum (R/Z)J⊔

(R/Z)I

R/Z is equal either to R/Z or to zero;

(b) a compact Abelian group Y is totally disconnected if and only if, under the above assumptions, the fibersum indicated above is always different from zero;

(c) the rational skeleton Y1 of a compact Abelian group has a connected completion if and only if for an

arbitrary diagram of ZQ/Z-continuous morphisms of the form

(Q/Z)I −−−−−→ (Q/Z)JQ/Z

the horizontal arrow of

which is the ZQ/Z-resolvent of Y1, the fiber sum (Q/Z)J⊔

(Q/Z)I

Q/Z is equal either to Q/Z or to zero;

(d) the rational skeleton Y1 of a compact Abelian group has a disconnected completion and, therefore, is itselfa totally disconnected Abelian group if and only if, under the same assumptions, the above fiber sum is alwaysdifferent from zero.

Proof. We prove only statement (a), because the other statements are proved similarly. By the Pontryaginduality theorem, Y is a connected compact Abelian group if and only if Y = HomZ(X, R/Z) for a torsion free

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Abelian group X (see [5]). Take a free resolvent 0 ← X ← Z(I) ← Z(J) of the Abelian group X; X is a torsionfree group if and only if any monogenic subgroup Z(I) of it either lies in the kernel of the epimorphism Z(I) → X(i.e., comes from Z(J)) or its intersections with the kernel is zero. In terms of diagrams, the above condition

means that for any diagramZ(I) ←−−−−− Z(J)Z

that consists of Abelian groups, the fiber product Z(J)∏

Z(I)

Z is

equal either to Z or to zero.Since GR/Z is a faithful functor and it carries finite direct products to finite direct sums, it carries finite pro-

jective limits to finite inductive limits (see the proof of propositions in Sec. 1). In particular, GR/Z

(Z(J)

∏Z(I)

Z)

=

(R/Z)J⊔

(R/Z)I

R/Z. Thus, applying the functor GR/Z to the above resolvent condition that X is a torsion free

group, we obtain the connection of Y that is stated in (a).

The author is grateful to A. V. Yakovlev and A. A. Suslin for their interest in the present work and for carefulreading of the manuscript, to A. F. Ivanov for constant attention to this work and useful discussions, and to B.B. Lur’e for friendly support.

Translated by V. V. Ishkhanov.

REFERENCES

1. M. B. Zvyagina, “A theorem on local duality,” Zap. Nauchn. Semin. POMI, 227, 66–73 (1995).1. M. B. Zvyagina, “Local duality for modules over Noetherian commutative rings,” Zap. Nauchn. Semin. POMI,

265, 169–188 (1999).3. S. MacLane, Homology, Springer-Verlag (1963).4. C. E. Watts, “Intrinsic characterizations of some additive functors,” Proc. Amer. Math. Soc., 11, 5–8 (1960).5. L. S. Pontryagin, Continuous Groups [in Russian], Nauka, Moscow (1973).6. K. Feis, Algebra: Rings, Modules, and Categories [Russian translation], Vol. 1, Mir, Moscow (1977).

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