aa r X i v : . [ m a t h . C T ] F e b AUTOCOMPACT OBJECTS OF AB5 CATEGORIES
JOSEF DVOˇR ´AK AND JAN ˇZEMLIˇCKA
Abstract.
The aim of the paper is to describe autocompact objects in Ab5-categories, i.e. objects in cocomplete abelian categories with exactness pre-serving filtered colimits of exact sequences, whose covariant Hom-functor com-mutes with copowers of the object itself. A characterization of non-autocompactobject is given, a general criterion of autocompactness of an object via thestructure of its endomorphism ring is presented and a criterion of autocom-pactness of products is proven.
Introduction
An object C of an abelian category A closed under coproducts is said to be autocompact , if the corresponding covariant hom-functor A( C, −) with target cat-egory being the category of abelian groups commutes with coproducts C ( κ ) forall cardinals κ , i.e. there is a canonical abelian group isomorphism between ob-jects A( C, C ( κ ) ) and A( C, C ) ( κ ) . Note that it generalizes the profoundly treatednotion of compact objects whose covariant hom-functors commute with arbitrarycoproducts.A systematic study of compact objects in categories of modules began in late60’s with Hyman Bass remarking in [4, p.54] that the class of compact modulesextends the class of finitely generated ones. This observation was elaborated inthe work of Rudolf Rentschler [20], where he presented basic constructions andconditions of existence of infinitely generated compact modules. The attention toautocompact objects within the category of abelian groups was then attracted bythe work [3]. The later research was motivated mainly by progress in the structuraltheory of abelian groups [2, 5, 6] and modules [1, 7, 18]. Although the notions ofcompactness and autocompactness were in fact studied in various algebraic contextsand with heterogeneous motivation (structure of modules [15, 12], graded rings[14], representable equivalences of module categories [8], the structure of almostfree modules [21]), their overall categorial nature was omitted for a long time.Nevertheless, there have been several recent papers dedicated to the description ofcompactness in both non-abelian [17, 10] and abelian [16] categories published.The present paper follows the undertaking begun with [16] and its main goal isnot only to survey results concerning self-small abelian groups and modules fromthe standpoint of abelian categories, but it tries to deepen and extend some ofthem in a way that they could be applied back in the algebraic context. We initiatewith an investigation of the more general concept of relative compactness. Thesecond section summarizes some basic tools developed in [16], which allows for the Date : February 10, 2021.2000
Mathematics Subject Classification.
Key words and phrases. additive category, Ab5 category, autocompact object.This work is part of the project SVV-2020-260589. description of structure and closure properties of relative compactness, in particular,Proposition 3.12 shows that ⊕ M is ⊕ N -compact for finite families of objects M and N of an Ab5-category if and only if M is N -compact for all M ∈ M and N ∈ N .The third section presents a general criterion of an object to be autocompact viathe structure of its endomorphism ring (Theorem 4.4) and, as a consequence, adescription of autocompact coproducts (Proposition 4.5). The main result of thepaper presented in Theorem 5.4 which proves that ∏ M is an autocompact objectif and only if it is ⊕ M -compact.1. Preliminaries
A category with a zero object is called abelian if the following four conditionsare satisfied:(1) for each finite discrete diagram the product and coproduct exist and theyare canonically isomorphic,(2) each Hom-set has a structure of an abelian group such that the compositionof morphisms is bilinear,(3) with each morphism it contains its kernel and cokernel,(4) monomorphisms are kernels of suitable morphisms, while epimorphisms arecokernels of suitable morphisms.A category is said to be complete ( cocomplete ) if it contains limits (colimits) of allsmall diagrams; a cocomplete abelian category where all filtered colimits of exactsequences preserve exactness is then called an Ab5 category.Any small discrete diagram is said to be a family . Let M be a family of objectsfrom A ; then the corresponding coproduct (product) is denoted (⊕ M , ( ν M ∣ M ∈ M)) ( (∏ M , ( π M ∣ M ∈ M)) ) and ν M ( π M ) are called structural morphisms of thecoproduct (of the product). In case M = { M i ∣ i ∈ K } with M i = M for all i ∈ K ,where M is an object of A , we shall write M ( K ) ( M K ) instead of ⊕ M ( ∏ M ) andthe corresponding structural morphisms shall be denoted by ν i ∶ = ν M i ( π i ∶ = π M i resp.) for each i ∈ K .Let N be a subfamily of M . Following the terminology set in [16] the coprod-uct (⊕ N , ( ν N ∣ N ∈ N )) in A is called a subcoproduct and dually the product (∏ N , ( π N ∣ N ∈ N )) is said to be a subproduct . Recall there exists a unique canon-ical morphism ν N ∈ A (⊕ N , ⊕ M) ( π N ∈ A (∏ M , ∏ N ) ) given by the universalproperty of ⊕ N ( ∏ N ) satisfying ν N = ν N ○ ν N ( π N = π N ○ π N ) for each N ∈ N ,to which we shall refer as to the structural morphism of the subcoproduct (thesubproduct) over a subfamily N of M . If M = { M i ∣ i ∈ K } and N = { M i ∣ i ∈ L } where M i = M for an object M and for i from index sets L ⊆ K , the correspondingstructural morphisms are denoted by ν L and π L respectively. The symbol 1 M de-notes the identity morphism of an object M and the phrase the universal propertyof a limit (colimit) refers to the existence of unique morphism into the limit (fromthe colimit).For basic properties of introduced notions and unspecified terminology we referto [19, 13].Throughout the whole paper we assume that A is an Ab5 category.2. C -compact objects In order to capture in detail the idea of relative compactness, which is the centralnotion of this paper, let us suppose that M is an object of the category A and N is UTOCOMPACT OBJECTS OF AB5 CATEGORIES 3 a family of objects of A . Note that the functor A( M, −) on any additive categorymaps into Hom-sets with a structure of abelian groups, which allows for a definitionof the mapping Ψ N ∶ ⊕ (A( M, N ) ∣ N ∈ N ) → A( M, ⊕ N ) by the rule Ψ N ( ϕ ) = ν F ○ ν − ○ π F ○ τ where for the element ϕ = ( ϕ N ∣ N ∈ N ) of the abelian group ⊕ (A(
M, N ) ∣ N ∈ N ) the symbol F denotes the finite family { N ∈ N ∣ ϕ N ≠ } , the morphism ν ∈ A(⊕ F , ∏ F ) is the canonical isomorphism and τ ∈ A( M, ∏ N ) is the unique mor-phism given by the universal property of the product (∏ N , ( π N ∣ N ∈ N )) appliedon the cone ( M, ( ϕ N ∣ N ∈ N )) , i.e. π N ○ τ = ϕ N for each N ∈ N : M τ / / /o/o/o ϕ N " " ❉❉❉❉❉❉❉❉❉ ∏ N π F / / π N (cid:15) (cid:15) ∏ F ν − / / ⊕ F ν F / / ⊕ N N Recall a key observation regarding the algebraic concept of compactness:
Lemma 2.1. [16, Lemma 1.3]
For each family of objects N ⊆ A , the mapping Ψ N is a monomorphism in the category of abelian groups. Let M be an object and C a class of objects of the category A . In accordancewith [16], M is called C -compact if Ψ N is an isomorphism for each family N ⊆ C .For objects M, N ∈ Ac we say that M is N -compact (or relatively compact over N ) if it is an { N } -compact object and M is said to be autocompact whenever it is M -compact. Example . (1) If M and N are objects such that A( N, M ) =
0, then N is M -compact object, in particular Q is a Z -compact abelian group.(2) Self-small right modules over a unital associative ring, in particular finitelygenerated ones, are autocompact objects in the category of all right modules.Let us formulate an elementary but useful observation: Lemma 2.3.
Let M be an object and let B ⊆ C be families of objects of the category A . If M is C -compact, then it is B -compact. We shall need two basic structural observations concerning the category A for-mulated in [16], which express relationship between coproducts and products usingtheir structural morphisms. For the convenience of the reader we quote both of theresults, the first one is formulated for the special case of products coproducts ofcopies of M , while the second one is kept in the original form. Lemma 2.4. [16, Lemma 1.1]
Let M be an object of A and L ⊆ K be sets. If A contains products ( M L , ( π i ∣ i ∈ L )) and ( M K , ( π i ∣ i ∈ K )) , then (1) There exist unique morphisms ρ L ∈ A( M ( K ) , M ( L ) ) and µ L ∈ A( M L , M K ) such that ρ L ○ ν i = ν i , π i ○ µ L = π i for i ∈ L , and ρ L ○ ν i = , π i ○ µ L = for i ∉ L . (2) For each i ∈ K there exist unique morphisms ρ i ∈ A( M ( K ) , M ) and µ i ∈ A( M, M K ) such that ρ i ○ ν i = M , π i ○ µ i = M and ρ i ○ ν j = , π j ○ µ i = whenever i ≠ j . Denoting by ρ i and µ i the corresponding morphisms for i ∈ L , we have µ L ○ µ j = µ j and ρ L ○ ρ j = ρ j for all j ∈ L . JOSEF DVOˇR´AK AND JAN ˇZEMLIˇCKA (3)
There exists a unique morphism t ∈ A( M ( K ) , M K ) such that π i ○ t = ρ i and t ○ ν i = µ i for each i ∈ K . Lemma 2.5. [16, Lemma 1.1(i) and 1.2]
Let N ⊆ M be families of objects of A and let there exist products (∏ N , ( π N ∣ N ∈ N )) and (∏ M , ( π N ∣ N ∈ M)) in A . (1) There exist unique morphisms ρ N ∈ A(⊕ M , ⊕ N ) and µ N ∈ A(∏ N , ∏ M) such that ρ N ○ ν N = ⊕ N , π N ○ µ N = ∏ N and ρ N ○ ν M = , π M ○ µ N = for each M ∉ N . (2) There exist unique morphisms t ∈ A (⊕ N , ∏ N ) and t ∈ A (⊕ M , ∏ M) such that π N ○ t = ρ N and t ○ ν N = µ N for each N ∈ M , π N ○ t = ρ N and t ○ ν N = µ N for each N ∈ N . Furthermore, the diagram ⊕ N ν N / / t (cid:15) (cid:15) ⊕ M ρ N / / t (cid:15) (cid:15) ⊕ N t (cid:15) (cid:15) ∏ N µ N / / ∏ M π N / / ∏ N commutes. (3) Let κ be an ordinal and let (N α ∣ α < κ ) be a disjoint partition of M . For α < κ set S α ∶ = ⊕ N α , P α ∶ = ∏ N α and denote families of the correspondinglimits and colimits as S ∶ = ( S α ∣ α < κ ) , P ∶ = ( P α ∣ α < κ ) . Then ⊕ M ≃ ⊕ S and ∏ M ≃ ∏ P where both isomorphisms are canonical, i.e. for each object M ∈ M the following diagrams commute: M ν ( α ) M / / ν M (cid:15) (cid:15) S αν Sα (cid:15) (cid:15) ⊕ M ≃ / / /o/o/o ⊕ S ∏ P ≃ / / /o/o/o π Pα (cid:15) (cid:15) ∏ M π M (cid:15) (cid:15) P α π ( α ) M / / M Morphisms ρ L , ρ N , ( µ L , µ N ) from Lemma 2.4(1) and Lemma 2.5(1) are calledthe associated morphisms to the structural morphisms ν L , ν N ( π L , π N ) over thesubcoproduct (the subproduct) of M . The unique morphism t from Lemma 2.5(2)is said to be the compatible coproduct-to-product morphism. Note that in an Ab5-category t is a monomorphism by [19, Chapter 2, Corollary 8.10] and if K is finite,it is by definition an isomorphism.We translate now a general criteria [16, Lemma 1.4, Theorem 1.5] of categorial C -compactness to the description of N -compactness for an arbitrary object N : Theorem 2.6.
The following conditions are equivalent for objects M and N of thecategory A : (1) M is N -compact, (2) for each cardinal κ and f ∈ A ( M, N ( κ ) ) there exists a finite set F ⊂ κ anda morphism f ′ ∈ A ( M, N ( F ) ) such that f = ν F ○ f ′ . (3) for each cardinal κ and f ∈ A ( M, N ( κ ) ) there exists a finite set F ⊂ κ suchthat f = ∑ α ∈ F ν α ○ ρ α ○ f , (4) for each morphism ϕ ∈ A ( M, N ( ω ) ) there exists α < ω such that ρ α ○ ϕ = . (5) there exists a family of N -compact objects G and an epimorphism e ∈A (⊕ G , M ) such that for each countable family G ω ⊆ G there exists a non- N -compact object F and morphism f ∈ A ( F, M ) such that f c ○ e ○ ν G ω = for the cokernel f c of f . UTOCOMPACT OBJECTS OF AB5 CATEGORIES 5
Proof.
Equivalences ( ) ⇔ ( ) ⇔ ( ) follow immediately from [16, Lemma 1.4],while ( ) ⇔ ( ) ⇔ ( ) are consequences of [16, Theorem 1.5]3. Correspondences of compact objects
As the base step of our research we describe C -compact objects for a singleobject C of an Ab5 category A . Let us begin with the observation that we canstudy C -compactness of a suitable object instead of the compactness over a set ofobjects.Let us denote the classAdd A ( C ) = { A ∣ ∃ B, ∃ κ, ∀ α < κ, ∃ C α ∈ C ∶ A ⊕ B ≅ ⊕ α < κ C α } for every family C of objects of A and put Add A ( C ) ∶ = Add A ({ C }) . Lemma 3.1.
The following conditions are equivalent for an object M and a set ofobjects C of the category A : (1) M is ⊕ C -compact, (2) M is C -compact, (3) M is Add A (⊕ C ) -compact, (4) M is Add A ( C ) -compact.Proof. Since Add A (⊕ C ) = Add A ( C ) , the equivalence (3) ⇔ (4) is obvious. Implica-tions (3) ⇒ (1) and (4) ⇒ (2) are clear from Lemma 2.3.(2) ⇒ (4) Let ϕ ∈ A ( M, ⊕ D ) for a family D of objects of Add A (⊕ C ) . For each D ∈ D there exists a family C D of objects of C and a monomorphism ν D ∶ D → ⊕ C D ,hence there exists a monomorphism ν ∶ ⊕ D → ⊕ D ∈ D ⊕ C D . Since M is C -compact,the morphism νϕ factorizes through a finite subcoproduct by [16, Lemma 1.4],hence ϕ factorizes through a finite subcoproduct, so M is Add A ( C ) -compact by[16, Lemma 1.4] again.(1) ⇒ (3) Follows from the implication (2) ⇒ (4) where we take {⊕ C } instead of C . Corollary 3.2. If N ⊆ M are families of objects such that N contains infinitelymany nonzero objects, then ⊕ M is not N -compact, so it is not ⊕ N -compact. Since Add A ( M ) = Add A ( M ( n ) ) for any integer n , we have the following conse-quence: Corollary 3.3.
Let κ be a cardinal and M an autocompact object. Then M ( κ ) isautocompact if and only if κ is finite. The next result shows the correspondence between classes of compact objectsover different pairs of objects.
Lemma 3.4.
Let
A, B, M be objects of A and let there exist a cardinal λ and amonomorphism µ ∈ A ( A, B λ ) . If M is B -compact, then M is A -compact.Proof. Denote by ν α and ˜ ν α the corresponding structural morphisms of coproducts A ( ω ) and B ( ω ) , and by ρ α and ˜ ρ α their associated morphisms, respectively.Suppose that M is not A -compact. Then there exists ϕ ∈ A ( M, A ( ω ) ) such that ρ α ϕ ≠ α < ω by Theorem 2.6. Since µ is a monomorphism by assumption, weget that µρ α ϕ ≠
0, which implies that there exists β α < λ such that π β α µρ α ϕ ≠ JOSEF DVOˇR´AK AND JAN ˇZEMLIˇCKA each α < ω by the universal property of the product B λ . Put µ α = π β α µ ∈ A ( A, B ) and note we have proved that µ α ρ α ϕ is a nonzero morphism M → B for each α < ω .The universal property of the coproduct A ( ω ) implies that there exists a uniquelydetermined morphism ψ ∈ A ( A ( ω ) , B ( ω ) ) for which the diagram A µ α / / ν α (cid:15) (cid:15) B ˜ ν α (cid:15) (cid:15) A ( ω ) ψ / / /o/o/o B ( ω ) ˜ ρ γ / / B commutes, i.e. we have equalities ψν α = ˜ ν α µ α and ˜ ρ γ ψν α = ˜ ρ γ ˜ ν α µ α for each α, γ < ω . Hence for every α < ω we get ˜ ρ α ψν α = µ α and ˜ ρ γ ψν α = γ ≠ α byLemma 2.4(2). Note that it means that ˜ ρ γ ψν α = ˜ ρ γ ψν γ ρ γ ν α for all α, γ < ω By applying Theorem 2.6 again we need to show that ˜ ρ γ ψϕ ≠ γ < ω .The universal property of the coproduct A ( ω ) implies that for every γ < ω thereexists a unique morphism τ γ ∈ A ( A ( ω ) , B ) such that the diagram A ν α / / ν α (cid:15) (cid:15) A ( ω ) ˜ ρ γ ψ (cid:15) (cid:15) A ( ω ) τ γ / / /o/o/o/o B commutes for each α < ω . Since ˜ ρ γ ψν α = ˜ ρ γ ψν γ ρ γ ν α for all α, γ < ω , we get theequality ˜ ρ γ ψ = τ γ = ˜ ρ γ ψν γ ρ γ by the universal property of the coproduct A ( ω ) . Now,it remains to compute for every γ < ω ˜ ρ γ ψϕ = τ γ ϕ = ˜ ρ γ ψν γ ρ γ ϕ = µ γ ρ γ ϕ ≠ , so M is not B -compact by Theorem 2.6. Corollary 3.5.
Let M and N be objects such that there exists a cardinal λ and amonomorphism µ ∈ A ( M, N λ ) and M is N -compact, then M is autocompact. As another consequence of Lemma 3.4 we can observe that general compactnesscan be tested by a single object. Recall that the object E the category A is called cogenerator if the functor A (− , E ) is an embedding [13, Section 3.3]. It is wellknown that there is a monomorphism of A → E A( A,E ) for any object A and acogenerator E . Proposition 3.6.
Let M be an object and E be a cogenerator of A such that A contains the product E A( M,E ) . Then M is E -compact if and only if it is compact.Proof. Clearly, it is enough to prove the direct implication. Let M be E -compactand M be a family of objects. Since E is a cogenerator , there exists a cardinal λ and a monomorphism µ ∈ A (⊕ M , E λ ) . Then M is ⊕ M -compact by Lemma 3.4and so M -compact by Lemma 3.1.The rest of this section is dedicated to description of relative compactness overfinite coproducts of finite coproducts of objects. Lemma 3.7.
Let A be an object and M a finite family of objects. (1) If N is A -compact for each N ∈ M , then ⊕ M is A -compact. (2) If A is N -compact for each N ∈ M , then A is ⊕ M -compact. UTOCOMPACT OBJECTS OF AB5 CATEGORIES 7
Proof. (1) Assume that ⊕ M is not A -compact. Then by Theorem 2.6 there existsa morphism ϕ ∈ A (⊕ M , A ( ω ) ) with ρ n ϕ ≠ ρ n of A ( ω ) .Note that for each n < ω there exists some N ∈ M such that ρ n ϕν N ≠ ⊕ M , where ν N are the correspondingstructural morphisms of ⊕ M . Therefore there exists N ∈ N for which the set I = { n < ω ∣ ρ n ϕν N ≠ } is infinite and the morphism ˜ ϕ = ρ I ϕν N ensured by Lemma 2.4 satisfies ρ n ˜ ϕ = ρ n ρ I ϕν N ≠ n ∈ I . Now, Theorem 2.6 implies that N is not A -compact.(2) Put M = ⊕ M and denote by ρ i , ˜ ρ i and ρ N the corresponding associatemorphisms of coproducts M ( ω ) and N ( ω ) for each N ∈ M and i < ω . Denotefurthermore by ρ N ( ω ) ∈ A ( M ( ω ) , N ( ω ) ) the morphism given by Lemma 2.5 whichsatisfies ˜ ρ i ρ N ( ω ) = ρ N ρ i for each N ∈ M and i < ω . Assume that A is not ⊕ M -compact: there exists a morphism ϕ ∈ A ( A, M ( ω ) ) such that ρ n ϕ ≠ n by Theorem 2.6 and now using the same argument as in the proof of (1)we can find N ∈ M such that the set J = { i < ω ∣ ρ N ρ i ϕ ≠ } is infinite. Since ˜ ρ i ρ N ( ω ) ϕ = ρ N ρ i ϕ ≠ i ∈ J , the object A is not N -compact. Lemma 3.8.
Let M , N and A be objects of A and n be a natural number. If thereexists an epimorphism M ( n ) → N and M is A -compact, then N is A -compact.Proof. Assume that N is not A -compact. Then there exists a morphism ϕ ∈A ( N, A ( ω ) ) such that ρ α ϕ ≠ ρ α of A ( ω ) by The-orem 2.6. If µ ∈ A ( M ( n ) , N ) is an epimorphism, ρ i ϕµ ≠ i < ω , hence M ( n ) is not A -compact. Then M is not A -compact by Lemma 3.7(1).We can summarize the obtained necessary condition of autocompactness. Proposition 3.9.
Let M and N be objects of A such that there exists an epimor-phism M ( n ) → N for an integer n and a monomorphism N → M λ for a cardinal λ .If M is is autocompact, then N is autocompact as well.Proof. N is M -compact, as follows from Lemma 3.8. Hence it is N -compact, soautocompact by Corollary 3.5.The next consequence presents a categorial version of the classical fact that anendomorphic image of a self-small module is self-small. Corollary 3.10. If M is an autocompact object, such that there exist an epimor-phism ǫ ∈ A ( M, N ) and a monomorphism µ ∈ A ( N, M ) , then N is autocompact.Example . Let A be a self-small right modules over a ring, i.e. autocompactobject in the category of right modules. Denote K = { ker f ∣ f ∈ End ( A )} and let L ⊂ K . Then A / ⋂ L is a self-small module by Proposition 3.9 since there existmonomorphisms A / ⋂ L ↪ ∏ L ∈L A / L ↪ ∏ L ∈L A We conclude the section mentioning closure properties of relatively compactobjects.
JOSEF DVOˇR´AK AND JAN ˇZEMLIˇCKA
Proposition 3.12.
Let M and N be finite families of objects. Then ⊕ M is ⊕ N -compact if and only if M is N -compact for all M ∈ M and N ∈ N .Proof. (⇒) Since the associate morphism ρ N ∈ A (⊕ M , M ) is an epimorphismfor each M ∈ M and ⊕ M is ⊕ N -compact, each object M is ⊕ N -compact byLemma 3.8. As ν N ∈ A ( N, ⊕ N ) is a monomorphism for each N ∈ N , any object M ∈ M is N -compact by Lemma 3.4. (⇐) Lemma 3.7(1) implies that ⊕ M is N -compact for each N ∈ N and thenLemma 3.7(2) implies that ⊕ M is ⊕ N -compact.4. Description of autocompact objects
This section is dedicated mainly to the generalization of a classical autocom-pactness criteria [3] to an Ab5 category A .Assume M is an object such that the category A is closed under products M λ for all λ ≤ ∣ End A ( M )∣ and take I ⊆ End A ( M ) = A ( M, M ) . Then there exists aunique morphism τ I ∈ A ( M, M I ) satisfying π ι τ I = ι for each ι ∈ I by the universalproperty of the product M I . Let us denote by K ( I ) = ( K I , ν I ) the kernel of themorphism τ I and note that K ( I ) is defined uniquely up to isomorphism.For an object K of A consider a morphism ν ∈ A ( K, M ) . We will then set I ( K, ν ) = { ι ∈ End A ( M ) ∣ ιν = } . It is easy to see that I ( K, ν ) forms a left ideal of the endomorphism ring End A ( M ) .We say that a left ideal I of End A ( M ) is an annihilator ideal if I ( K ( I )) = I . Lemma 4.1.
Let A be closed under products M λ for all λ ≤ ∣ End A ( M )∣ . Then I ( K, ν ) is an annihilator ideal of End ( M ) for all ν ∈ A ( K, M ) .Proof. Put I = I ( K, ν ) , ( K I , ν I ) = K ( I ) and ˜ I = IK ( I ) = I ( K I , ν I ) . Furthermore,denote by τ I ∈ A ( K, M I ) the morphism satisfying π ι τ I = ι for each ι ∈ I , i.e. ( K I , ν I ) is the kernel of τ I . Since τ I ν I =
0, we can easily compute that ιν I = π ι τ I ν I = ι ∈ I , which implies I ⊆ ˜ I .To prove the reverse inclusion ˜ I ⊆ I , let us note that by the universal propertyof the kernel ν I there exists a unique morphism α ∈ A ( K, K I ) such that all squaresin the diagram K ν / / α (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O M τ I / / M Iπ ι (cid:15) (cid:15) K I ν I / / M ι / / M commute for each ι ∈ I . Consider a morphism γ ∈ End ( M ) such that γ ∉ I . Then γν I α = γν ≠ I . Hence γν I ≠ γ ∉ ˜ I .Recall the concepts of exactness and inverse limits in Ab5 categories.The diagram A α Ð→ A α Ð→ . . . α n − Ð→ A n − α n Ð→ A n is said to be an exact sequence provided for each i = , . . . , n − α i + α i = K i together with morphisms ξ i ∈ A ( A i , K i ) and θ i ∈ A ( K i , A i ) such that ( K i , θ i ) is a kernel of α i + , ( K i , ξ i ) is a cokernel of α i and ξ i θ i = K i . In particular,the diagram 0 → A α Ð→ B β Ð→ C → short exact sequence provided α is akernel of β and β is a cokernel of α , hence α is a monomorphism and β is an UTOCOMPACT OBJECTS OF AB5 CATEGORIES 9 epimorphism. Recall that any monomorphism (epimorphism) can be expressed asthe first (second) morphism of some short exact sequence in an Ab5-category.A diagram
D = ({ M i } i < ω , { ν i,j } i < j < ω ) is called an ω -spectrum of M , if ν i,j ∈A ( M i , M j ) , ν j,k ν i,j = ν i,k for each i < j < k < ω , and there exist morphisms ν i ∈A ( M i , M ) for all i < ω such that ( M, { ν i } i < ω ) is a colimit of the diagram D (i.e. itis a direct limit of the spectrum D ). Lemma 4.2.
Let M be an object and M ( ω ) be a coproduct with structural mor-phisms ν i and associated morphisms ρ i , i < ω . Put n = { , . . . , n − } , [ n, ω ) = ω ∖ n = { i < ω ∣ i ≥ n } , let M ( n ) and M ([ n,ω )) be subcoproducts of M ( ω ) . Denote by ν ( n,m ) ∈ A ( M ( n ) , M ( m ) ) , ν < n ∈ A ( M ( n ) , M ( ω ) ) the structural morphisms and by ρ ( n,m ) ∈ A ( M ([ n,ω )) , M ([ m,ω )) ) , ρ ≥ n ∈ A ( M ( ω ) , M ([ n,ω )) ) the associated morphisms given by Lemma 2.5 for all n < m < ω . Then (1) for each n < m < ω all squares in the diagram with exact rows M n ∶ / / M ( n ) ν < n / / ν ( n,m ) (cid:15) (cid:15) M ( ω ) ρ ≥ n / / M ([ n,ω )) ρ ( n,m ) (cid:15) (cid:15) / / M m ∶ / / M ( m ) ν < m / / M ( ω ) ρ ≥ m / / M ([ m,ω )) / / commute, (2) the short exact sequence / / M ( ω ) id / / M ( ω ) / / / / with morphisms ( ν < n , id, ) forms a colimit of the ω -spectrum ({ M n } n , {( ν ( n,m ) , id, ρ ( n,m ) )} n < m ) in the category of complexes, (3) ρ i ν < n = ρ i if i < n and ρ i ν < n = otherwise.Proof. An easy exercise of application of Lemma 2.5 in a Ab5-category.Before we formulate the categorial version of [3, Proposition 1.1] we prove a moregeneral result:
Lemma 4.3.
Let for M ∈ A the category contain the products M ω . The followingconditions are equivalent for an object N ∈ A : (1) N is not M -compact, (2) there exists an ω -spectrum ({ N i } i < ω , { µ i,j } i < j < ω ) of N with colimit ( N, { µ i } i < ω ) such that all µ i and µ i,j for all i < j < ω are monomorphisms and for each i < ω there exists a nonzero morphism ϕ i ∈ A ( N, M ) satisfying ϕ i µ i = , (3) there exists an ω -spectrum with colimit ( N, { µ i } i < ω ) such that for each i < ω there exists a nonzero morphism ϕ i ∈ A ( N, M ) satisfying ϕ i µ i = .Proof. We will use the notation of Lemma 4.2 throughout the whole proof.(1) ⇒ (2) Let ϕ ∈ A ( N, M ( ω ) ) satisfying ρ i ϕ ≠ i < ω , which is ensured by(1) and Theorem 2.6. Furthermore, let us denote ϕ ≥ n = ρ ≥ n ϕ . Then ρ i ϕ = ρ i ϕ ≥ n for all i ≥ n . Now, for each n < ω denote by ( N n , µ n ) the kernel of the morphism ϕ ≥ n and note that by the universal property of the kernel there exists a morphism µ n,n + ∈ A ( N n , N n + ) such that all squares in the diagram with exact rows0 / / N n µ n / / µ n,n + (cid:15) (cid:15) N ϕ ≥ n / / M ([ n,ω )) ρ ( n,n + ) (cid:15) (cid:15) / / N n + µ n + / / N ϕ ≥ n + / / M ([ n + ,ω )) commute. Now let us define inductively for each n < m < ω morphisms µ n,m ∶ = µ m − ,m µ m − ,m − . . . µ n + ,n + µ n,n + ∈ A ( N n , N m ) . If we denote by ( X, { ξ i } i < ω ) the colimit of the ω -spectrum N = ({ N i } i < ω , { µ i,j } i < j < ω ) then from Lemma 4.2 we obtain the following commutative diagram with exact rows0 / / N n µ n / / µ n,m (cid:15) (cid:15) N ϕ ≥ n / / M ([ n,ω )) ρ ( n,m ) (cid:15) (cid:15) / / N m µ m / / ξ n (cid:15) (cid:15) N ϕ ≥ m / / M ([ m,ω )) (cid:15) (cid:15) / / X ξ / / N / / A is an Ab5-category. Thus ξ is an isomorphism, which implies that ( N, { µ i } i < ω ) is a colimit of the ω -spectrum N . Since all µ n ’s are kernel morphisms,they are monomorphisms. Furthermore, µ n,m are monomorphisms, because µ n = µ m µ n,m for all n < m < ω . Finally, put ϕ i ∶ = ρ i ϕ , which is nonzero by the hypothesis,and compute ϕ i µ i = ρ i ϕµ i = ρ i ϕ ≥ i µ i = ⇒ (3) This is clear.(3) ⇒ (1) Let us denote by τ ∈ A ( N, M ω ) the morphism satisfying π i τ = ϕ i , whichis (uniquely) given by the universal property of the product M ω . Recall that foreach n < ω we have denoted by π n ∈ A ( M ω , M n ) the corresponding structuralmorphism and we may identify objects M n and M ( n ) so we shall consider π n as amorphism in A ( M ω , M ( n ) ) .Put τ n ∶ = π n τ µ n . Since ρ i τ n = ρ i π n τ µ n = π i π n τ µ n we obtain that ρ i τ n = ϕ i µ n ≠ i < n and ρ i τ n = i ≥ n . Then the diagram N n τ n / / µ n,m (cid:15) (cid:15) M ( n ) ν ( n,m ) (cid:15) (cid:15) N m τ m / / M ( m ) commutes for every n < m < ω . Hence there exists ϕ ∈ A ( N, M ( ω ) ) such that thediagram N n τ n / / µ n (cid:15) (cid:15) M ( n ) ν < n (cid:15) (cid:15) N ϕ / / M ( ω ) UTOCOMPACT OBJECTS OF AB5 CATEGORIES 11 commutes for each n < ω by Lemma 4.2, as ( N, { µ i } i < ω ) is the colimit of the ω -spectrum ({ N i } i < ω , { µ i,j } i < j < ω ) and ({ M ( ω ) } i < ω , { µ ( i ) } i < ω ) is the colimit of the ω -spectrum ({ M ( i ) } i < ω , { ν ( i,j ) } i < j < ω ) in the Ab5-category A .Applying Theorem 2.6, it is enough to prove that ρ i ϕ ≠ i < ω . Wehave shown that ρ i τ n ≠ i < n , hence ρ i ϕµ n = ρ i ν < n τ n = ρ i τ n ≠ i < n , which implies ρ i ϕ ≠ Theorem 4.4.
Let M be an object such that A is closed under products M λ forall λ ≤ max (∣ End A ( M )∣ , ω ) . Then the following conditions are equivalent: (1) M is not autocompact, (2) there exists an ω -spectrum with colimit ( M, { µ i } i < ω ) such that for each i < ω there exists a nonzero morphism ϕ i ∈ End A ( M ) satisfying ϕ i µ i = , (3) there exists an ω -spectrum ({ M i } i < ω , { µ i,j } i < j < ω ) with colimit ( M, { µ i } i < ω ) such that { I ( M i , µ i )} i < ω forms a strictly decreasing chain of nonzero idealsof the ring End A ( M ) with ⋂ i < ω I ( M i , µ i ) = .Proof. (1) ⇔ (2) follows form Lemma 4.3 for N = M .(2) ⇒ (3) If ( M, { µ i } i < ω ) is the colimit which exists by (2), then { I ( M i , µ i )} i < ω is a decreasing chain of nonzero ideals of End A ( M ) . Suppose that γµ i = i < ω . Then γ =
0, since there exists unique such morphism by the universalproperty of the colimit ( M, { ν i } i < ω ) . Thus ⋂ i < ω I ( M i , µ i ) = I ( M i , µ i ) ≠ i . If we put J = { j < ω ∣ I ( M j , µ j ) ≠ I ( M j + , µ j + )} , then it is easy to see that ( M, { µ j } j ∈ J ) is the colimit of the ω -spectrum ({ M j } j ∈ J , { µ i,j } i < j ∈ J ) with a strictlydecreasing chain of nonzero ideals { I ( M j , µ j )} j ∈ J .(3) ⇒ (2) It is enough to choose ϕ i ∈ I ( M i , µ i ) ∖ I ( M i + , µ i + ) .The following criterion of autocompactness of finite coproducts generalizes re-sults [9, Proposition 5, Corollary 6] formulated in categories of modules. Proposition 4.5.
The following conditions are equivalent for a finite family ofobjects M and M = ⊕ M : (1) M is autocompact, (2) N is M -compact for each N ∈ M , (3) M is N -compact for each N ∈ M , (4) N is N -compact for each N , N ∈ M , (5) for each N , N ∈ M and any ω -spectrum ({ K i } i < ω , { µ i,j } i < j < ω ) of N withcolimit ( N , { µ i } i < ω ) and for each i < ω and nonzero ϕ ∈ A ( N , N ) , themorphism ϕµ i is nonzero.Proof. (1) ⇔ (4) This is proved in Proposition 3.12(2) ⇔ (3) ⇔ (4) These equivalences follow from Proposition 3.12 again, when ap-plied on pairs of families { M } , M and M , { M } .(4) ⇔ (5) This is an immediate consequence of Lemma 4.3.As a consequence, we can formulate the assertion of Corollary 3.3 more precisely. Corollary 4.6.
Let M be a family of nonzero objects. Then ⊕ M is autocompactif and only if M is finite and N is N -compact for each N , N ∈ M . The last direct consequence of Proposition 4.5 presents a categorial variant of[9, Corollary 7].
Corollary 4.7.
Let M be a finite family of autocompact objects satisfying thecondition A ( N , N ) = whenever N ≠ N . Then ⊕ M is autocompact. If M is a finite family of objects, then ⊕ M and ∏ M are canonically isomorphic(cf. Lemma 2.5), so the Proposition 4.5 holds true in case we replace any ⊕ by ∏ there. Although there is no autocompact coproduct of infinitely many nonzeroobjects by Corollary 3.2, the natural question that arises is, under which conditionsthe products of infinite families of objects are autocompact. The following exampleshows that the straightforward generalization of the claim does not hold true ingeneral. Example . Denote by P the set of all prime numbers and consider the full sub-category T of the category of abelian groups Ab consisting of all torsion abeliangroups. If A is a torsion abelian group and A p denotes its p -component for each p ∈ P , then the decomposition ⊕ p ∈ P A p forms both the coproduct and product of thefamily A = { A p ∣ p ∈ P } . Indeed, if B is a torsion abelian group and τ p ∈ Ab ( B, A p ) for p ∈ P , then for every b ∈ B there exist only finitely many p ∈ P for which τ p ( b ) ≠
0, hence the image of the homomorphism f ∈ Ab ( B, ∏ p A p ) given by theuniversal property of the product ∏ p A p is contained in ⊕ p ∈ P A p , hence ⊕ p ∈ P A p isthe product of A in the category T .Thus, e.g. ⊕ p ∈ P Z p is the product of the family { Z p ∣ p ∈ P } in T , which is notautocompact in T by Corollary 4.6, however Z p is Z q -compact for every p, q ∈ P .5. Which products are autocompact?
Although the final section tries to answer the question formulated in its title, westart with one more closure property.
Lemma 5.1. If → A → B → C → is a short exact sequence such that an object M is A -compact and C -compact, then it is B -compact.Proof. Proving indirectly, assume that M is not B -compact. Then by Lemma 4.3there exists a colimit ( M, { µ i } i < ω ) of some ω -spectrum ({ M i } i < ω , { µ i,j } i < j < ω ) andnonzero morphisms ϕ i ∈ A ( M, B ) such that ϕ i µ i = i < ω . If we suppose that M is C -compact and consider the short exact sequence0 / / A α / / B β / / C / / , then βϕ i µ i = i ∈ ω , hence there exists n such that βϕ i = i ≥ n by Lemma 4.3. By the universal property of the kernel α of (the cokernel) β thereexist ψ i satisfying αψ i = ϕ i ≠ i ≥ n . As α is a monomorphism, ψ i ≠ i ≥ n , hence M is not A -compact by Lemma 4.3 again, a contradiction. Corollary 5.2. If → A → B → C → is a short exact sequence such that theobject B is A -compact and C -compact, then B is autocompact. The previous corollary is a partial answer to the concluding question raised in[9]. As the next example shows, its assertion cannot be reversed.
Example . If we consider the short exact sequence 0 → Z → Q → Q / Z → Q is self-small, i.e. autocompact abelian groupand Z -compact, but it is not Q / Z -compact. UTOCOMPACT OBJECTS OF AB5 CATEGORIES 13
Now, we can formulate a criterion for autocompact objects which generalizes [11,Theorem 3.1].
Theorem 5.4.
Let M be a family of objects such that the product M = ∏ M existsin A and put S = ⊕ M . Then the following conditions are equivalent: (1) M is autocompact, (2) M is S -compact, (3) M is ⊕ C -compact for each countable family C ⊆ M .Proof. (1) ⇒ (2) Since M is Add A ( M ) -compact by Lemma 3.1 and S = ⊕ M ∈
Add A ( M ) , it is S -compact by Lemma 2.3.(2) ⇒ (3) This is an easy consequence of Proposition 3.12.(3) ⇒ (1) Assume on contrary that M is not autocompact. Then by Lemma 4.3there exist an ω -spectrum ({ M i } i < ω , { µ i,j } i < j < ω ) of M with the colimit ( M, { µ i } i < ω ) such that µ i is a monomorphism for all i < ω and for each i < ω there existsa nonzero morphism ϕ i ∈ A ( M, M ) with ϕ i µ i =
0. Then for each i < ω thereexists N i ∈ M such that π N i ϕ i ≠
0. Put
C = { N i ∣ i < ω } and denote by ˜ ν N i the structural morphisms of the coproduct ⊕ C . Since ˜ ν N i π N i ϕ i ∈ A ( M, ⊕ C ) suchthat ˜ ν N i π N i ϕ i µ i =
0, there exists n for which ˜ ν N n π N n ϕ n = π N i ϕ i ≠ i < ω . Corollary 5.5.
Let M be an object and I be a set. Then M I is autocompact ifand only if M I is M -compact. Let us make a categorial observation about transfer of ω -spectra via morphisms. Lemma 5.6.
Let G and M be objects of A and α ∈ A ( G, M ) . If ({ M i } i < ω , { µ i,j } i < j < ω ) is an ω -spectrum of M with the colimit ( M, { µ i } i < ω ) such that all µ i ’s are monomor-phisms, (1) then there exists an ω -spectrum ({ G i } i < ω , { γ i,j } i < j < ω ) of G with the colimit ( G, { γ i } i < ω ) where γ i are monomorphisms for all i and there exist mor-phisms α i ∈ A ( G i , M i ) such that the diagram G i γ i / / α i (cid:15) (cid:15) G α (cid:15) (cid:15) M i µ i / / M commutes for each i < ω . (2) If G is A -compact for an object A and t i ∈ A ( M, A ) are morphisms satis-fying t i µ i = for each i < ω , then there exists n such that t i α = for each i ≥ n .Proof. (1) If we denote by c i ∈ A ( M, T i ) the cokernel of µ i and γ i ∈ A ( G i , G ) thekernel of c i α for every i < j < ω , then µ i is the kernel of c i and by the univer-sal property of the kernel, there exists a morphism α i ∈ A ( G i , M i ) such that thediagram with exact rows0 / / G i γ i / / α i (cid:15) (cid:15) G c i α / / α (cid:15) (cid:15) T i / / M i µ i / / M c i / / T i / / commutes for each i < ω . Furthermore, if we construct morphisms γ i,j , i < j < ω using the universal property of the kernels as in the proof of Lemma 4.3 (1) ⇒ (2),then we get the following commutative diagram G i γ i / / γ i,j (cid:15) (cid:15) GG j γ j / / G and checking that ( G, { γ i } i < ω ) is a colimit of the ω -spectrum ({ G i } i < ω , { γ i,j } i < j < ω ) of G is easy.(2) From (1) we have an ω -spectrum ({ G i } i < ω , { γ i,j } i < j < ω ) with the colimit ( G, { γ i } i < ω ) and morphisms α i ∈ A ( G i , M i ) such that the diagram0 / / G i γ i / / α i (cid:15) (cid:15) G α (cid:15) (cid:15) / / M i µ i / / M t i / / A commutes for every i < ω . Since t i αγ i = t i µ i α i = A -compact object G and morphisms t i α , i < ω , give us n such that t i α = i ≥ n . Lemma 5.7.
Let A and B be objects of A and A ( A, B ) = . If α ∈ A ( A ∏ B, B ) then there exists τ ∈ A ( B, B ) for which α = τ π B .Proof. Since 0 → A ν A Ð→ A ⊕ B ρ B Ð→ B → αν A =
0, theclaim follows from the universal property of the cokernel ρ B and by applying thecanonical isomorphism A ⊕ B ≅ A ∏ B .Recall that G is a projective generator of A , if for any nonzero object B in A , A ( G, B ) ≠ A , B , any epimorphism π ∈ A ( A, B ) and any morphism ϕ ∈ A ( G, B ) there exists τ ∈ A ( G, A ) such that ϕ = πτ .The following assertion is a categorial version of [22, Proposition 1.6] (cf. also[3, Corollary 1.3]). Call an A -compact object briefly compact object. Theorem 5.8.
Let M be a family of objects, A contain a compact projective gener-ator and the product M = ∏ M . Denote M N = ∏ ( M ∖ { N }) and let A ( M N , N ) = for each N . Then M is autocompact if and only if N is autocompact for each N ∈ M .Proof. ( ⇒ ) Since M ≅ N ⊕ M N for every N ∈ M , the assertion follows from Propo-sition 4.5.( ⇐ ) First note that M N is a trivial example of an N -compact module (cf. Ex-ample 2.2), so M is N -compact for every N ∈ M by Proposition 4.5.Assume that M is not M -compact, hence by Lemma 4.3 there exists an ω -spectrum ({ M i } i < ω , { µ i,j } i < j < ω ) with the colimit ( M, { µ i } i < ω ) such that for each i < ω there exists a nonzero ˜ ϕ i ∈ A ( M, M ) and N i ∈ M for which ˜ ϕ i µ i = π N i ˜ ϕ i ≠
0. Put ϕ i = π N i ˜ ϕ i for each i < ω and C = { N i ∣ i < ω } and note that thereexist ψ i ∈ A ( N i , N i ) satisfying ψ i π N i = ϕ i by Lemma 5.7 applied on M N i ∏ N i foreach i < ω . UTOCOMPACT OBJECTS OF AB5 CATEGORIES 15 If C is finite, then M is ∏ C -compact by Proposition 3.12 applied on { M } and C , hence there exists n such that ϕ n = π N n π C ϕ = π N n ϕ =
0, which contradicts thefact that ϕ i ≠ i < ω .Thus C is infinite and we may assume w.l.o.g. that N i ≠ N j whenever i ≠ j .Denote the cokernel of the composition π N i µ i by σ i ∈ A ( N i , T i ) for i < ω . Then wehave a commutative diagram0 / / M i µ i / / M π Ni / / ϕ i (cid:15) (cid:15) N i σ i / / ψ i (cid:15) (cid:15) T i ˜ µ Ti (cid:15) (cid:15) T i / / N i N i ∏ j T j ˜ π Ti / / T i / / i < ω , where ˜ π T i and ˜ µ T i denote the structural and associated morphismsof the product ∏ j T j . Since ψ i π N i µ i = ϕ i µ i = ψ i ≠
0, the morphism π N i µ i is not an epimorphism and so T i ≠
0. As G is a projective generator, there exists ζ i ∈ A ( G, N i ) satisfying σ i ζ i ≠ i < ω . Then by the universal property ofthe product ∏ C , there is ζ ∈ A ( G, ∏ C ) such that ˆ π N i ζ = ζ i , hence σ i ˆ π N i ζ = σ i ζ i ≠ i < ω . If we define t i = ˜ µ T i σ i π N i and denote by µ C ∈ A ( ∏ C , M ) the associatedmorphism, we can easily compute t i µ i = ˜ µ T i σ i π N i µ i = t i µ C ζ = ˜ µ T i σ i π N i µ C ζ = ˜ µ T i σ i ζ i ≠ µ T i for i < ω is a monomorphism, which contradicts the hypothesis that G iscompact by Lemma 5.6(2).The following example shows that the existence of the compact projective gen-erator cannot be removed from the assumptions of the last assertion. Example . Consider the category of all torsion abelian groups T from Ex-ample 4.8. Then M = ⊕ q ∈ P Z q is the product of the family { Z q ∣ q ∈ P } and M p = ⊕ q ≠ p Z p is the product of the family { Z q ∣ q ∈ P ∖ { p }} for all p ∈ P in the cat-egory T . Although Hom T ( M p , Z p ) = Z p is autocompact in T for each p ∈ P , M is not autocompact. Let us remark that the category T contains no compactgenerator. [22, Corollary 1.8]).We conclude with a well-known example of an autocompact product. Example . Any finitely generated free abelian group is a compact projectivegenerator in the category of abelian groups and the family { Z q ∣ q ∈ P } satisfies thehypothesis of Proposition 5.8 by [22, Lemma 1.7], hence ∏ q ∈ P Z q is autocompact(cf.[22, Corollary 1.8]). References [1] Albrecht U., Breaz, S.: A note on self-small modules over RM-domains,
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Commun. Algebra (2008), No. 7, 2570–2576. CTU in Prague, FEE, Department of mathematics, Technick´a 2, 166 27 Prague 6 &MFF UK, Department of Algebra, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
Email address : [email protected] Department of Algebra, Charles University, Faculty of Mathematics and PhysicsSokolovsk´a 83, 186 75 Praha 8, Czech Republic
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