aa r X i v : . [ m a t h . C T ] J un COMPACTNESS IN ABELIAN CATEGORIES
PETER K ´ALNAI AND JAN ˇZEMLIˇCKA
Abstract.
We relativize the notion of a compact object in an abelian categorywith respect to a fixed subclass of objects. We show that the standard closureproperties persist to hold in this case. Furthermore, we describe categoricaland set-theoretical conditions under which all products of compact objectsremain compact.
An object C of an abelian category A closed under coproducts is said to be compact if the covariant functor A( C, −) commutes with all direct sums, i.e. thereis a canonical isomorphism between A( C, ⊕ D) and ⊕ A( C, D) in the category ofabelian groups for every system of objects D . The foundations for a systematicstudy of compact objects in the context of module categories were laid in 60’s byHyman Bass [1, p.54] in 60’s. The introductory work on theory of dually slendermodules goes back to Rudolf Rentschler [13] and further research of compact objectshas been motivated by progress in various branches of algebra such as theory ofrepresentable equivalences of module categories [2, 3], the structure theory of gradedrings [9], and almost free modules [14].From the categorically dual point of view discussed in [7], commutativity of thecontravariant functor on full module categories behaves a little bit differently. Theequivalent characterizations of compactness split in this dual case into a hierarchyof strict implications dependent on the cardinality of commuting families. Thestrongest hypothesis assumes arbitrary cardinalities and it leads to the class of socalled slim modules (also known as strongly slender ), which is a subclass of themost general class of ℵ -slim modules (also called as slender ), which involves onlycommutativity with countable families. The authors proved that the cardinality ofa non-zero slim module is greater than or equal to any measurable cardinal (andpresence of such cardinality is also sufficient condition for existence of a non-zeroslim module) and that the class of slim modules is closed under direct sums. Thus,absence of a measurable cardinal ensures that there is at least one non-zero slimmodule and in fact, abundance of them. On the other hand, if there is a properclass of measurable cardinals then there is no such object like a non-zero slimmodule. This motivated the question in the dual setting, namely if the class ofcompact objects in full module categories (termed also as dually slender modules)is closed under direct products. Offering no surprise, set-theoretical assumptionshave helped to establish the conclusion also in this case.The main objective of the present paper is to refine several results on compact-ness. The obtained improvement comes from transferring behavior of modules tothe context of general abelian categories. In particular we provide a generalized Date : October 8, 2018.2000
Mathematics Subject Classification.
Key words and phrases. additive categories, compact object, Ulam-measurable cardinal. description of classes of compact objects closed under product that was initially ex-posed for dually slender modules in [10]. Our main result Theorem 3.4 shows thatthe class of C -compact objects of a reasonably generated category is closed underproducts whenever there is no strongly inaccessible cardinal. Note that this out-come is essentially based on the characterization of non- C -compactness formulatedin Theorem 1.5. Dually slender and self-small modules (which may be identicallytranslated as self-dually slender) form naturally available instances of compact andself-compact objects (see e.g. [5] and [4]).From now on, we suppose that A is an abelian category closed under arbitrarycoproducts and products. We shall use the terms family or system for any discretediagram, which can be formally described as a mapping from a set of indices to aset of objects. For unexplained terminology we refer to [8, 12].1. Compact objects in abelian categories
Let us recall basic categorical notions. A category with a zero object is called additive if for every finite system of objects there exist the product and coproductwhich are canonically isomorphic, every Hom-set has a structure of abelian groupsand the composition of morphisms is bilinear. An additive category is abelian ifthere exists kernel and a cokernel for each morphism, monomorphisms are exactlykernels of some morphisms and epimorphisms cokernels. A category is said tobe complete (cocomplete) whenever it has all limits (colimits) of small diagrams.Finally, a cocomplete abelian category where all filtered colimits of exact sequencespreserve exactness is
Ab5 . For further details on abelian category see e.g. [12].Assume M is a family of objects in A . Throughout the paper, the correspondingcoproduct is designated (⊕ M , ( ν M ∣ M ∈ M)) and the product (∏ M , ( π M ∣ M ∈ M)) . We call ν M and π M as the structural morphisms of the coproduct and theproduct, respectively.Suppose that N is a subfamily of M . We call the coproduct (⊕ N , ( ν N ∣ N ∈ N )) in A as the subcoproduct and dually the product (∏ N , ( π N ∣ N ∈ N )) as the sub-product . Note that there exist the unique canonical morphisms ν N ∈ A (⊕ N , ⊕ M) and π N ∈ A (∏ M , ∏ N ) given by the universal property of the colimit ⊕ N and thelimit ∏ N satisfying ν N = ν N ○ ν N and π N = π N ○ π N for each N ∈ N , to which werefer as the structural morphisms of the subcoproduct and the subproduct over asubfamily N of M , respectively. The symbol 1 M is used for the identity morphismof an object M .We start with formulation of two introductory lemmas which collects severalbasic but important properties of the category A expressing relations between thecoproduct and product over a family using their structural morphisms. Lemma 1.1.
Let A be a complete abelian category, M a family of objects of A with all coproducts and N ⊆ M . Then (i) There exist unique morphisms ρ N ∈ A(⊕ M , ⊕ N ) and µ N ∈ A(∏ N , ∏ M) such that ρ N ○ ν M = ν M , π M ○ µ N = ν M if M ∈ N and ρ N ○ ν M = , π M ○ µ N = if M ∉ N . (ii) For each M ∈ M there exist unique morphisms ρ M ∈ A(⊕ M , M ) and µ M ∈ A( M, ∏ M) such that ρ M ○ ν M = M , π M ○ µ M = M and ρ M ○ ν N = , π N ○ µ M = whenever N ≠ M . If ρ M and µ M denote the correspondingmorphisms for M ∈ N , then µ N ○ µ N = µ N and ρ N ○ ρ N = ρ N for all N ∈ N . OMPACTNESS IN ABELIAN CATEGORIES 3 (iii)
There exists a unique morphism t ∈ A(⊕ M , ∏ M) such that π M ○ t = ρ M and t ○ ν M = µ M for each M ∈ M .Proof. (i) It suffices to prove the existence and the uniqueness of ρ N , the secondclaim has a dual proof.Consider the diagram ( M ∣ M ∈ M) with morphisms (̃ ν M ∣ M ∈ M) ∈ A( M, ⊕ N ) where ̃ ν M = ν M for M ∈ N and ̃ ν M = (⊕ M , ( ν M ∣ M ∈ M)) .(ii) Note that for the choice
N ∶ = ⊕( M ) ≃ M we have ν M = M and the claimfollows from (i).(iii) We obtain the requested morphism by the universal property of the product (∏ M , ( π M ∣ M ∈ M)) applying on the cone (⊕ M , ( ρ M ∣ M ∈ M)) that is providedby (ii). Dually, there exists a unique t ′ ∈ A(⊕ M , ∏ M) with t ′ ○ ν M = µ M . Then π M ○ ( t ○ ν M ) = ρ M ○ ν M = M = π M ○ µ M = π M ○ ( t ′ ○ ν M ) , hence t ○ ν M = µ M by the uniqueness of the associated morphism µ M and t = t ′ because t ′ is the only one satisfying the condition for all M ∈ M .We call the morphism ρ N ( µ N ) from (i) as the associated morphism to thestructural morphism ν M ( π M ) over the subcoproduct (the subproduct) over N .For the special case in (ii), the morphisms ρ M ( µ M ) from (ii) as the associatedmorphism to the structural morphism ν M ( π M ). Let the unique morphism t becalled as the compatible coproduct-to-product morphism over a family M . Notethat this morphism need not be a monomorphism, but it is so in case A being anAb5-category [12, Chapter 2, Corollary 8.10]. Moreover, t is an isomorphism if thefamily M is finite. Lemma 1.2.
Let us use the notation from the previous lemma. (i)
For the subcoproduct over N , the composition of the structural morphismof the subcoproduct and its associated morphism is the identity. Dually forthe subproduct over N , the composition of the associated morphism of thesubproduct and its structural morphism is the identity, i.e. ρ N ○ ν N = ⊕ N and π N ○ µ N = ∏ N , respectively. (ii) If t ∈ A (⊕ N , ∏ N ) and t ∈ A (⊕ M , ∏ M) denote the compatible coproduct-to-product morphisms over N and M respectively, then the following dia-gram commutes: ⊕ N ν N / / t (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ⊕ M ρ N / / t (cid:15) (cid:15) ⊕ N t (cid:15) (cid:15) (cid:15)O(cid:15)O(cid:15)O ∏ N µ N / / ∏ M π N / / ∏ N (iii) Let (N α ∣ α < κ ) be a disjoint partition of M and for α < κ let S α ∶ = ⊕ N α , P α ∶ = ∏ N α , and denote families of the limits and colimits like S ∶ = ( S α ∣ α < κ ) , P ∶ = ( P α ∣ α < κ ) . Then ⊕ M ≃ ⊕ S and ∏ M ≃ ∏ P where theboth isomorphisms are canonical, i.e. for every object M ∈ M the diagrams PETER K´ALNAI AND JAN ˇZEMLIˇCKA 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 Proof. (i) The equality ρ N ○ ν N = ⊕ N is implied the uniqueness of the universalmorphism and the equalities ( ρ N ○ ν N ) ○ ν N = ρ N ○ ν N = ν N and 1 ⊕ N ○ ν N = ν N forall N ∈ N . The equality π N ○ µ N = ∏ N is dual.(ii) We need to show that t ○ ν N = µ N ○ t . For all N ∈ N , ( π N ○ t )○ ν N = ρ N ○ ν N = N by (iii) and by (ii). But π N ○ ν N = N , hence µ N = t ○ ν N by the uniqueness of µ N .If µ N ∈ A ( N, ∏ N ) denotes the unique homomorphism ensured by (ii), then thelast argument proves that µ N = t ○ ν N . Thus ( t ○ ν N ) ○ ν N = t ○ ( ν N ○ ν N ) = t ○ ν N = µ N = µ N ○ µ N = µ N ○ ( t ○ ν N ) == ( µ N ○ t ) ○ ν N and the claim follows from the universal property of the coproduct (⊕ N , ( ν N ∣ N ∈N )) . The dual argument proves that π N ○ t = t ○ ρ N .(iii) A straightforward consequence of the universal properties of the coproductsand products.Applying the categorical tools we have introduced we are ready to present thecentral notion of the paper. Let us suppose that C is a subclass of objects of A , M is an object in A and N is a system of objects of C . As the functor A ( M, −) onany additive category maps into Hom-sets with a structure of abelian groups wecan define a mappingΨ N ∶ ⊕ ( A ( M, N ) ∣ N ∈ N ) → A ( M, ⊕ N ) in the following way:For a family of mappings ϕ = ( ϕ N ∣ N ∈ N ) in ⊕ ( A ( M, N ) ∣ N ∈ N ) let us denoteby F a finite subfamily such that ϕ N = N ∉ F and let τ ∈ A ( M, ∏ N ) be the unique morphism given by the universal property of the product (∏ N , ( π N ∣ N ∈ F )) applied on the cone ( M, ( ϕ N ∣ N ∈ N )) , i.e. π N ○ τ = ϕ N for every N ∈ N .Then Ψ N ( ϕ ) = ν F ○ ν − ○ π F ○ τ where ν ∈ A (⊕ F , ∏ F ) denotes the isomorphism provided by Lemma 1.1(iii).Note that the definition Ψ N ( ϕ ) does not depend on choice of F and recall anelementary observation which plays a key role in the definition of a compact object. Lemma 1.3.
The mapping Ψ N is a monomorphism in the category of abeliangroups for every family of objects N .Proof. If Ψ N ( σ ) =
0, then σ = ( ρ N ○ σ ) N = ( ) N , hence Ker Ψ N = M is said to be C -compact if Ψ N is an isomorphism for every family N ⊆ C , M is compact in the category A if it is A o -compact for the class of allobject, and M is self-compact assuming { M } -compact. Note that every object is { } -compact.First we formulate an elementary criterion of identifying C -compact object. OMPACTNESS IN ABELIAN CATEGORIES 5
Lemma 1.4. If M is an object and a class of objects C , then it is equivalent: (1) M is C -compact, (2) for every N ⊆ C and f ∈ A ( M, ⊕ N ) there exists a finite subsystem F ⊆ N and a morphism f ′ ∈ A ( M, ⊕ F ) such that f = ν F ○ f ′ . (3) for every N ⊆ C and every f ∈ A ( M, ⊕ N ) there exists a finite subsystem F contained in N such that f = ∑ F ∈ F ν F ○ ρ F ○ f .Proof. ( ) → ( ) : Let N ⊆ C and f ∈ A ( M, ⊕ N ) . Then there exists a Ψ N -preimage ϕ of f , hence there can be chosen a finite subsystem F ⊆ N such that f = Ψ N ( ϕ ) = ν F ○ ν − ○ π F ○ τ, where we use the notation from the definition of the mapping Ψ N . Now it remainsto put f ′ = ρ F ○ f and utilize Lemma 1.1(ii) to verify that ν F ○ f ′ = ν F ○ ρ F ○ f = ν F ○ ρ F ○ ν F ○ ⊕F ○ ν − ○ π F ○ τ = f. ( ) → ( ) : Since ρ F ○ ν F = ⊕ F by Lemma 1.1(ii) we obtain that ν F ○ ρ F ○ f = ν F ○ ρ F ○ ν F ○ f ′ = ν F ○ f ′ = f. Moreover, ν F ○ ρ F = ∑ F ∈ F ν F ○ ρ F , hence f = ν F ○ ρ F ○ f = ∑ F ∈ F ν F ○ ρ F ○ f. ( ) → ( ) : If we put ϕ F ∶ = ρ F ○ f for F ∈ F and ϕ N ∶ = N ∉ F and take ϕ ∶ = ( ϕ N ∣ N ∈ N ) , then it is easy to see that f = Ψ N ( ϕ ) hence Ψ N is surjective.Now, we can prove a characterization, which generalizes equivalent conditionswell-known for the categories of modules. Note that it will play similarly impor-tant role for categorical approach to compactness as in the special case of modulecategories. Theorem 1.5.
The following conditions are equivalent for an object M and a classof objects C : (1) M is not C -compact, (2) there exists a countably infinite system N ω of objects from C and ϕ ∈A ( M, ⊕ N ω ) such that ρ N ○ ϕ ≠ for every N ∈ N ω , (3) for every system G of C -compact objects and every epimorphism e ∈ A (⊕ G , M ) there exists a countable subsystem G ω ⊆ G such that f c ○ e ○ ν G ω ≠ for thecokernel f c of every morphism f ∈ A ( F, M ) where F is a C -compact object.Proof. ( ) → ( ) : Let N be a system of objects from C for which there exists amorphism ϕ ∈ A ( M, ⊕ N ) ∖ ImΨ N . Then it is enough to take N ω as any countablesubsystem of the infinite system ( N ∈ N ∣ ρ N ○ ϕ ≠ ) . ( ) → ( ) Let G be a family of C -compact objects and e ∈ A (⊕ G , M ) an epi-morphism. If N ∈ N ω , then ( ρ N ○ ϕ ) ○ e ≠
0, hence by the universal property ofthe coproduct ⊕ G applied on the cone ( N, ( ρ N ○ ϕ ○ e ○ ν G ∣ G ∈ G )) there exists G N ∈ G such that A ( G N , N ) ∋ ρ N ○ ϕ ○ e ○ ν G N ≠
0. Put G ω = ( G N ∣ N ∈ N ω ) , whereevery object from the system G is taken at most once, i.e. we have a canonicalmonomorphism ν G ω ∈ A (⊕ G ω , ⊕ G ) .Assume to the contrary that there exist a C -compact object F and a morphism f ∈ A ( F, M ) such that f c ○ e ○ ν G ω = f c ∈ A ( M, cok ( f )) is the cokernelof the morphism f . Let N ∈ N ω and, furthermore, assume that ρ N ○ ϕ ○ f = PETER K´ALNAI AND JAN ˇZEMLIˇCKA
Then the universal property of the cokernel ensures the existence of a morphism α ∈ A ( cok ( f ) , N ) such that α ○ f c = ρ N ○ ϕ , i.e. that commutes the diagram: F f (cid:15) (cid:15) ⊕ G ω ν G ω / / ⊕ G e / / M ϕ / / f c (cid:15) (cid:15) ⊕ N ωρ N (cid:15) (cid:15) cok ( f ) α / / /o/o/o/o N Thus ( ρ N ○ ϕ )○ e ○ ν G ω = ( α ○ f c )○ e ○ ν G ω =
0, which contradicts to the constructionof G ω . We have proved that ρ N ○ ( ϕ ○ f ) ≠ N ∈ N ω , hence ϕ ○ f ∈A ( F, ⊕ N ) ∖ ImΨ N ω . We get the contradiction with the assumption that F is C -compact, thus f c ○ e ○ ν G ω =≠ ( ) → ( ) : If M is C -compact itself, then the system G = ( M ) and the identitymap e on M are counterexamples for the condition ( ) . Corollary 1.6. If A contains injective envelopes E ( U ) for all objects U ∈ C , thenan object M is not compact if and only if there exists a (countable) system ofinjective envelopes E in A of objects of C for which Ψ N is not surjective for somesubsystem N of C .Proof. By the previous proposition, it suffices to consider the composition of ϕ ∈A ( M, ⊕ N ω )∖ ImΨ N ω where N ω witnesses that M is not C -compact and the canon-ical morphism ι ∈ A (⊕ N ω , ⊕ E ) , where we put E ∶ = ( E ( N ) ∣ N ∈ N ω ) .2. Classes of compact objects
Let us denote by A a complete abelian category and C a class of some objects of A . First, notice that closure properties of the class of C -compact objects are similarto closure properties of classes of dually slender modules since their follows by thefact that the contravariant functor A (− , ⊕ N ) commutes with finite coproducts andit is left exact. We present a detailed proof of the fact that the class of all C -compactobjects is closed under finite coproducts and cokernels using Theorem 1.5. Lemma 2.1.
The class of all C -compact objects is closed under finite direct sumsand all cokernels of morphisms α ∈ A ( M, C ) where C is C -compact and M isarbitrary.Proof. Suppose that ⊕ ni = M i is not C -compact. Then by Theorem 1.5 there exista sequence ( N i ∣ i < ω ) of objects and a morphism ϕ ∈ A (⊕ ni = M i , ⊕ j < ω N j ) suchthat ρ j ○ ϕ ≠ j < ω . Since ω = ⋃ ni = { j < ω ∣ ρ j ○ ϕ ○ ν i ≠ } there exists i for which the set { j < ω ∣ ρ j ○ ϕ ○ ν i ≠ } is infinite, hence M i is not C -compact byapplying Theorem 1.5.Similarly, suppose that α c is the cokernel of α ∈ A ( M, C ) , where cok ( α ) is not C -compact, and ϕ ∈ A ( cok ( α ) , ⊕ j < ω N j ) for ( N i ∣ i < ω ) satisfies ρ j ○ ϕ ≠ j < ω . Then, obviously, ρ j ○ ϕ ○ π ≠ j < ω and so C is not C -compact againby Theorem 1.5. Lemma 2.2. If M is an infinite system of objects in A satisfying that for each M ∈ M there exists C ∈ C such that A ( M, C ) ≠ , then ⊕ M is not C -compact. OMPACTNESS IN ABELIAN CATEGORIES 7
Proof.
It is enough to take
N = ( C M ∣ M ∈ M ) where A ( M, C M ) ≠ Corollary 2.3.
Let M be a system of objects of A . Then ⊕ M is C -compact if andonly if all M ∈ M are C -compact and there exists C M ∈ C such that A ( M, C M ) ≠ for only finitely many M ∈ M . Let us confirm that relativized compactness behaves well under taking finiteunions of classes and verify with an example that this closure property can not beextended to an infinite case.
Lemma 2.4.
Let C , . . . , C n be a finite number of classes of objects and let C ∈ A .Then C is ⋃ ni = C i -compact if and only it C is C i -compact for every i ≤ n .Proof. The direct implication is trivial. If C is not ⋃ ni = C i -compact, there existsa sequence ( C i ∣ i < ω ) of objects of ⋃ ni = C i with a morphism ϕ ∈ A ( C, ⊕ j < ω B j ) such that ρ j ○ ϕ ≠ j < ω by Theorem 1.5. Since there exists k ≤ n forwhich infinitely many C i ’s belong to C j we can see that C is not C j -compact byTheorem 1.5. Example . Let R be a ring over which there is an infinite set of non-isomorphicsimple right modules. Any non-artinian Von Neumann regular ring serves as anexample where the property holds. Suppose that A is the full subcategory ofcategory consisting of all semisimple right modules, which is generated by all simplemodules. Fix a countable sequence S i , i < ω , of pair-wisely non-isomorphic simplemodules. Then the module ⊕ i < ω S i is { S i } -compact for each i but it is not ⋃ i < ω { S i } -compact.Relative compactness of an object is preserved if we close the class under allcogenerated objects. Lemma 2.6.
Let
Cog ( C ) be the class of all objects cogenerated by C . Then every C -compact object is Cog ( C ) -compact.Proof. Let us suppose that an object C is not Cog ( C ) -compact and fix a sequence B ∶ = ( B i ∣ i < ω ) of objects of Cog (C) and a morphism ϕ ∈ A( C, ⊕ B) such that ρ j ○ ϕ ≠ j < ω which exists by Theorem 1.5. Since Cog (C) is closedunder subobjects we may suppose that ρ j ○ ϕ are epimorphisms. Furthermore, forevery j < ω there exists a non-zero morphism τ j ∈ A( B j , T j ) with T j ∈ C . Formthe sequence T ∶ = ( T i ∣ i < ω ) . Let τ be the uniquely defined morphism from A(⊕ B , ⊕ T ) satisfying τ ○ ν j = ν j ○ τ j . Then ρ j ○ τ ○ ν i = ρ j ○ ν i ○ τ i which is equalto τ i whenever i = j and it is zero otherwise, hence ρ i ○ τ ○ ν i ○ ρ i = ρ i ○ τ by theuniversal property of ⊕ B . Finally, since ρ i ○ ϕ is an epimorphism and τ i is non-zero τ i ○ ρ i ○ ϕ ≠ ρ j ○ τ ○ ϕ = ρ i ○ τ ○ ν i ○ ρ i ○ ϕ = ρ i ○ ν i ○ τ i ○ ρ i ○ ϕ = τ i ○ ρ i ○ ϕ ≠ i < ω . Thus the composition τ ○ ϕ witnesses that C is not C -compact againby Theorem 1.5.A complete abelian category A is called C -steady , if there exists an A -projective C -compact object G which finitely generates the class of all C -compact objects,i.e. for every C -compact object F there exists n ∈ N and a homomorphism h ∈ PETER K´ALNAI AND JAN ˇZEMLIˇCKA A( G ( n ) , F ) such that cokernel of h is isomorphic to F . A is said to be steady whenever it is an A o -steady category for the class A o of all objects of A . Example . If R is a right steady ring and A = Mod - R is the category of all right R -modules, then C -compact objects are precisely dually slender modules and A isa steady category.Furthermore, in [9, Theorem 1.7] it was proved that a locally noetherian Grothendieckcategory is steady.Recall that an object A is simple if for every B and any non-zero morphismfrom A( A, B ) is a monomorphism and an object is semisimple if it is isomorphicdo direct sum of simple objects. We characterize steadiness of categories such thatall their object are semisimple. Lemma 2.8.
Let A be generated by a class of simple objects S . Then A is C -steadyif and only if there are only finitely many non-isomorphic simple objects S such thatHom ( S, C ) ≠ for some object C ∈ C .Proof. If there exist infinitely many non-isomorphic simple objects S such that A ( S, T S ) ≠ T S ∈ C , it is enough to take countably infinite family ofsuch simple objects S ∶ = ( S i ∣ i < ω ) and non-zero morphisms τ i ∈ A ( S i , T S i ) where T S i ∈ C are collected into T ∶ = ( T S i ∣ i < ω ) . Then the morphism τ ∈ A (⊕ S , ⊕ T ) defined by the universal property of a direct sum witnesses that ⊕ S is not C -compact. Since it is easy to see that for every epimorphism from A ( A, ⊕ S ) , theobject A is not C -compact as well, yielding that the category A is not steady. Example . Let A be a category semisimple right modules over ring with infiniteset of non-isomorphic simple right modules as in Example 2.5. Then A is notsteady, and if the ring R is right steady, which is true for example for each countablecommutative regular ring, the category of all right R -modules Mod - R steady.We say that a complete abelian category A is ∏ C -compactly generated if thereis a set G of objects of A that generates A and the product of any system of objectsin G is C -compact. Note that G consists only of C -compact objects. Lemma 2.10. If E is a C -compact injective generator of A such that there existsa monomorphism m ∈ A ( E ( ω ) , E ) , then A is ∏ C -compactly generated.Proof. It follows immediately from Theorem 1.5(iii).
Example . Let R be a right self-injective, purely infinite ring. Then E ∶ = R isan injective generator and there is an embedding 0 → R ( ω ) → R . By the previouslemma, the category Mod - R is ∏ C -generated.3. Products of compact objects
We start the section by an observation that the cokernel of the compatiblecoproduct-to-product morphism over a countable family is C -compact where C is aclass of objects in an abelian category A . This initial step will be later extendedto families regardless of their cardinality. Lemma 3.1.
Let A be ∏ C -compactly generated and let M be a countable fam-ily of objects in A . If t ∈ A (⊕ M , ∏ M ) is the compatible coproduct-to-productmorphism, then cok ( ν ) is C -compact. OMPACTNESS IN ABELIAN CATEGORIES 9
Proof.
As for a finite M there is nothing to prove, suppose that M = ( M n ∣ n < ω ) .Let G be a family of objects of A such that every product of a system of objectsin G is C -compact and let e ∈ A (⊕ G , ∏ M ) be an epimorphism, which exists bythe hypothesis. Let t c be the cokernel of t . Then both t c and e ′ ∶ = t c ○ e areepimorphisms and t c ○ t =
0. We will show that for every countable subsystem G ω of G there exists a C -compact object F and a morphism f ∈ A ( F, cok ( t )) such that A (⊕ G ω , cok ( f )) ∋ f c ○ e ′ ○ ν G ω = f c ∈ A ( cok ( t ) , cok ( f )) . ByTheorem 1.5 this yields that cok ( t ) is C -compact.Since for any finite G ω ⊆ G it is enough to take F ∶ = ⊕ G ω and f ∶ = e ′ ○ ν G ω , wemay fix a countably infinite family G ω = ( G n ∣ n < ω ) ⊆ G . For each n < ω put G n = ( G i ∣ i ≤ n ) and let π G n ∈ A (∏ G ω , ∏ G n ) and π M n ∈ A (∏ M , M n ) denote thestructural morphisms, and let u − ∈ A (∏ G n , ⊕ G n ) be the inverse of the compatiblecoproduct-to-product morphism u ∈ A (⊕ G n , ∏ G n ) that exists for finite families.First, let us fix n ∈ ω and we prove that ν G k = ν G n ○ u − ○ π G n ○ µ G k . Let ν G k ∈ A ( G k , ⊕ G n ) be the structural morphism of the coproduct ⊕ G n and let µ G k ∈A ( G k , ∏ G n ) be the associated morphism to the product ∏ G n . Since ν G n ○ ν G k = ν G k and µ G k = u ○ ν G k (by Lemma 1.1(iii)), then we immediately infer the followingequalities from Lemma 1.2(ii) : ν G k = ν G n ○ ν G k = ν G n ○ ( u − ○ u ○ ν G k ) = ( ν G n ○ u − ) ○ u ○ ν G k == ( ν G n ○ u − ) ○ ( π G n ○ u ○ ν G n ) ○ ν G k = ( ν G n ○ u − ) ○ π G n ○ u ○ ν G k == ( ν G n ○ u − ) ○ π G n ○ µ G k Now, if we employ the universal property of the product (∏ M , ( π M n ∣ n < ω )) with respect to the cone (∏ G ω , ( π M n ○ e ○ ν G n ○ u − ○ π G n ∣ n < ω )) , then there existsa unique morphism α ∈ A (∏ G ω , ∏ M ) such that the middle non-convex pentagonin the following diagram commutes : G k µ Gk " " ν Gk / / Gk ⊕ G ω u / / ∏ G ωπ G n (cid:15) (cid:15) π Mn ○ α " " ❋❋❋❋❋❋❋❋ α ! ! .n -m -m ,l +k *j )i (h 'g &f $d ⊕ M t (cid:15) (cid:15) G k ν Gk / / Gk ⊕ G nν G n O O u ≃ / / ν G ω ○ ν G n (cid:15) (cid:15) ∏ G n M n ∏ M t c (cid:15) (cid:15) π Mn o o G k ̃ ν Gk / / ⊕ G π Mn ○ e ❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ e cok ( t ) Then for each k ≤ n we deduce that π M n ○ ( α ○ µ G k − e ○ ̃ ν G k ) = π M n ○ ( α ○ µ G k − e ○ ν G ω ○ ν G n ○ u − ○ π G n ○ µ G k ) == ( π M n ○ α − π M n ○ e ○ ν G ω ○ ν G n ○ u − ○ π G n ) ○ µ G k = α ○ µ G n = e ○ ̃ ν G n for every n < ω is yielded as the number n was fixed. Notethat ∏ G ω is C -compact by the hypothesis. Now, consider f c the cokernel of themorphism f = t c ○ α ∈ A (∏ G ω , cok ( t )) . Then0 = f c ○ t c ○ ( e ○ ̃ ν G k − α ○ µ G n ) == f c ○ t c ○ e ○ ̃ ν G n − f c ○ t c ○ α ○ µ G n = f c ○ e ′ ○ ̃ ν G n hence 0 = f c ○ e ′ ○ ̃ ν G n = f c ○ e ′ ○ ν G ω ○ ν G k for every n < ω , which finishes the proof.Let I ≠ ∅ be a system of subsets of a set X . We recall that I is said to be– an ideal if it is closed under subsets (i.e. if A ∈ I and B ⊆ A , then B ∈ I )and under finite unions, (i.e. if A, B ∈ I , then A ∪ B ∈ I ),– a prime ideal if it is a proper ideal and for all subsets A , B of X , A ∩ B ∈ I implies that A ∈ I or B ∈ I ,– a principal ideal if there exists a set Y ⊆ X such that I = P ( Y ) .The set I ∣ Y = { Y ∩ A ∣ A ∈ I } is called a trace of on ideal I on Y .Note that the trace of an ideal is also an ideal and that I is a prime ideal if andonly if for every A ⊆ X , A ∈ I or X ∖ A ∈ I . Moreover, a principal prime ideal on X is of the form P ( X ∖ { x }) for some x ∈ X .Dually, a system F ≠ ∅ of non-empty subsets of X is said to be– a filter if it is closed under finite intersections and supersets,– an ultrafilter if it is a filter which is not properly contained in any otherfilter on X ,We say that a filter F is λ -complete , if ⋂ G ∈ F for every subsystem
G ⊆ F suchthat ∣ G ∣ < λ and F is countably complete , if it is ω -complete.Note that there is a one-to-one correspondence between ultrafilters and primeideals on X defined by I ↦ P ( X ) ∖ I for an ideal I .Now, we are able to generalize [10, Lemma 3.3] Proposition 3.2.
Let A be a ∏ C -compactly generated category, M a family of C -compact objects of A and N = ( N n ∣ n < ω ) a countable family of objects of C .Suppose that Ψ N is not surjective and fix ϕ ∈ A (∏ M , ⊕ N ) ∖ Im Ψ N . If we denote I n = { J ⊆ M ∣ ρ N k ○ ϕ ○ µ J = ∀ k ≥ n } and I = ⋃ n < ω I n ⊆ P ( M ) , then the followingholds: (i) I n is an ideal for each n , (ii) I is closed under countable unions of subfamilies, (iii) there exists n < ω for which I = I n , (iv) there exists a subfamily U ⊆ M such that the trace of I on U forms anon-principal prime ideal.Proof. Let G be a set of C -compact objects satisfying that every product of a systemof objects in G is C -compact, which is guaranteed by the hypothesis.(i) Obviously, ∅ ∈ I n and I n is closed under subsets. The closure of I n underfinite unions follows from Lemma 1.2(iii) applied on the disjoint decomposition J ∪ K = J ∪ ( K ∖ J ) , i.e. from the canonical isomorphism ∏ J ∪ K ≅ ∏ J × ∏ K ∖ J .(ii) First we show that I is closed under countable unions of pairwise disjointsets. Let K j , j < ω be pair-wisely disjoint subfamilies of I and put K = ⊍ j < ω K j .Let K i ∶ = ∏ K i . We show that there exists k < ω such that K j ∈ I k for each j < ω .Assume that for all n < ω there exist possibly distinct i ( n ) such that K i ( n ) ∉ I n .Hence ρ N l ( n ) ○ ϕ ○ µ K i ( n ) ≠ l ( n ) ≥ n and there is a C -compact generator G n ∈ G and a morphism f n ∈ A ( G n , K i ( n ) ) with ρ N l ( n ) ○ ϕ ○ µ K i ( n ) ○ f n ≠
0. Set K ′ ∶ = ( K i ( n ) ∣ n < ω ) , P ∶ = ( K n ∣ n < ω ) .Put G ω ∶ = ( G j ∣ j < ω ) and denote by (∏ G ω , ( π G j ∣ j < ω )) the product of G ω andby µ G j ∈ A ( G j , ∏ G ω ) , j < ω , the associated morphisms given by Lemma 1.1(i).Then the universal property of the product ∏ K ′ applied to the constructed cone OMPACTNESS IN ABELIAN CATEGORIES 11 gives us a morphism f ∈ A (∏ G ω , ∏ K ′ ) such that f n ○ π G n = π K i ( n ) ○ f , hence f n = f n ○ π G n ○ µ G n = π K i ( n ) ○ f ○ µ G n = π K i ( n ) ○ µ K i ( n ) ○ f n Since ∏ G ω is C -compact by the hypothesis there exists arbitrarily large m < ω suchthat ρ N l ( m ) ○ ϕ ○ µ K ′ ○ f = µ K ′ ∈ A (∏ K ′ , ∏ M ) is the associated morphismto π K ′ ∈ A (∏ M , ∏ K ′ ) over the subcoproduct of K ′ . Hence ρ N l ( m ) ○ ϕ ○ ( µ K i ( m ) ○ f m ) = ρ N l ( m ) ○ ϕ ○ µ K ′ ○ f ○ µ G m = , a contradiction.We have proved that there is some n < ω such that ρ N k ○ ϕ ○ µ K j = k ≥ n and j < ω , without loss of generality we may suppose that n =
0. Denote by t c the cokernel of the compatible coproduct-to-product morphism t ∈ A (⊕ K , ∏ K ) .As ϕ ○ µ K ○ t =
0, the universal property of the cokernel ensures the existence of themorphism τ ∈ A ( cok ( t ) , ⊕ N ) such that ϕ ○ µ K = τ ○ t c . Hence there exists n < ω such that ρ N k ○ ϕ ○ µ K = k ≥ n since cok ( t ) is C -compact by Lemma 3.1,which proves that K ⊆ I n .To prove the claim for whatever system ( J j ∣ j < ω ) in I is chosen, it remains toput J = K and J i = K i ∖ ⋃ j < i K j for i > I ≠ I j for every j < ω . Then there exists a countable sequenceof families of objects ( J j ∈ I ∖ I j ∣ j ∈ ω ) . By (ii) we get J ∶ = ⋃ j < ω J j ∈ I and thereis some n < ω such that J ∈ I n . Having J n ⊆ J ∈ I n leads us to a contradiction.(iv) We will show that there exists a family U ⊆ M such that for every K ⊆ U , K ∈ I or U ∖ K ∈ I . Assume that such U does not exist. Then we may constructa countably infinite sequence of disjoint families ( K i ∣ i < ω ) where K i are non-empty for i > K = ∅ and J = M . There exist disjointsets J i + , K i + ⊂ J i such that J i = J i + ∪ K i + where J i + , K i + / ∈ I . Now, for each n ≥ G n ∈ G and a morphism f n ∈ A ( G n , ∏ K n ) such that ρ N k ○ ϕ ○ µ K n ○ f n ≠ k > n which contradicts to the fact that ∏ n < ω G n is C -compact (hence ρ N k ○ ϕ ○ µ K n ○ f n ○ π n = k < ω ).The trace of I on U is a prime ideal and assume that it is principal, i.e. itconsists of all subfamilies of U excluding one particular index U ∈ U , so I ∣ U = P ( U /{ U }) ∈ I . On the other hand, U is C -compact itself, which implies { U } ∈ I .This yields I ∣ U containing U , a contradiction.As a consequence of Proposition 3.2 we can formulate a generalization of [10,Theorem 3.4]: Corollary 3.3.
Let A be a ∏ C -compactly generated category. Then the followingholds: (i) A product of countably many C -compact objects is C -compact. (ii) If there exists a system M of cardinality κ of C -compact objects such thatthe product ∏ M is not C -compact, then there exists an uncountable cardinal λ < κ and a countable complete nonprincipal ultrafilter on λ .Proof. (i) An immediate consequence of Proposition 3.2(iii).(ii) Let M be a system of cardinality κ of C -compact objects and suppose that ∏ M is not a C -compact object. Then there exists a countable family N such thatΨ N is not surjective. By Lemma 3.2(iv) there exists a subfamily U ⊆ M suchthat the trace of I on U forms a non-principal prime ideal which is closed under countable unions of families by Lemma 3.2(ii). If we define V = P ( U ) ∖ ( I ∣ U ) then V forms a countable complete non-principal ultrafilter on U . It is uncountable byapplying (i).Before we formulate the main result of this section which answers the questionfrom [6] in Abelian categories, let us list several set-theoretical notions and theirproperties guaranteeing that the hypothesis of the theorem is consistent with ZFC.A cardinal number λ is said to be measurable if there exists a λ -complete non-principal ultrafilter on λ and it is Ulam-measurable if there exists a countablycomplete non-principal ultrafilter on λ . A regular cardinal κ is strongly inaccessible if 2 λ < κ for each λ < κ . Recall that ● [15, Theorem 2.43.] every Ulam-measurable cardinal is greater or equal tothe first measurable cardinal; ● [15, Theorem 2.44.] every measurable cardinal is strongly inaccessible; ● [11, Corollary IV.6.9] it is consistent with ZFC that there is no stronglyinaccessible cardinal. Theorem 3.4.
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OMPACTNESS IN ABELIAN CATEGORIES 13 [15] Zelenyuk, E.G., Ultrafilters and topologies on groups, de Gruyter Expositions in Mathematics50, de Gruyter, Berlin 2011.
Department of Algebra, Charles University in Prague, Faculty of Mathematics andPhysics Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
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