aa r X i v : . [ m a t h . R T ] M a y ON THE NON-EXISTENCE OF RIGHT ALMOST SPLIT MAPS
JAN ˇSAROCHAbstract.
We show that, over any ring, a module C is a codomain of a rightalmost split map if and only if C is a finitely presented module with localendomorphism ring; thus we give an answer to a 40 years old question byM. Auslander. Using the tools developed, we also provide a useful sufficientcondition for a class of modules to be non-precovering. Finally, we show a non-trivial application in the general context of morphisms determined by object. Introduction
Almost split sequences (also called Auslander–Reiten sequences) represent thecentral tool of Auslander–Reiten theory. They serve as stepping stones in the hardtask to understand the possible extensions in the category of finitely generatedmodules over an Artin algebra. Their utilization, however, is not restricted tothis very context. The theory of almost split sequences is developed in variousother categories, for instance, [10], in the category of complexes of modules (andcorrelatively, in its triangle version, in the homotopy category of modules), or forthe general case of exact categories, [12].The important question, common in various contexts, is whether, for a givenobject C , there exists an almost split sequence beginning or ending in C (cf. [3,questions (1) , (2) on pg. 4]). Since the almost split sequence comprises of two parts,the left almost split map and the right almost split map, we can ask even for themere existence of these maps having C as domain, codomain resp.In the category Mod- R , where R is any ring, it is easy to show that a necessarycondition on C is that End R ( C ) is local. In fact, by a result of Auslander, a finitelypresented module C is the codomain of a right almost split map in Mod- R if andonly if End R ( C ) is local. There are examples, however, where the domain of theright almost split map is necessarily non-finitely presented. Although it may happenthat there still is a right almost split map in mod- R having C as its codomain.These examples illustrate that the relation between the existence of right al-most split maps (and correlatively almost split sequences) in categories mod- R andMod- R is rather intricate. Suppose, on the other hand, that we want to use themachinery of almost split maps and the properties they provide us with to studymodules which are not finitely presented. Is the situation more clear in this case?Could we perhaps obtain some interesting new information on the possible exten-sions involving a particular countably, or even non-countably presented module?The answer to these questions, as it turns out, is ‘yes and no’. To be more precise:the main result of this paper, Theorem 4.4, states that every right almost split maphas to have a finitely presented codomain. This has been recently conjectured in a Date : September 5, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Right almost split map, tree module, non-existence of precovers,morphism determined by object.This research has been supported by grant GAˇCR 14-15479S. JAN ˇSAROCH slightly weaker form in [11], however, already in [3], Auslander asked for the precisedescription of modules appearing as the right-hand terms in almost split sequences.The proof of our result uses basically two main tools. The first one is a con-struction of a so-called tree module, i.e., a particular combinatorial object whichserves as a test module for splitting of a given epimorphism, in our case of a rightalmost split map f : B → C with a θ -presented module C where θ is an infinitecardinal. The tree module appears as the middle term in a short exact sequencewhich is subsequently used to show that C has to be, in fact, < θ -presented.This is done by incorporating the second main tool—a modified version ofHunter’s cardinal counting argument (Lemma 4.1), recently used in [2] as the keypart of the proof that the class of all flat Mittag–Leffler modules is not precoveringunless the underlying ring is right perfect.The structure of this paper is pretty straightforward. After a short sectionwhere we fix our notation and recall some of the known results, we describe theconstruction of tree modules in detail in §
3. Apart from this rather technicalconstruction, as a sort of byproduct, we answer a question by G. Bergman from [6]in Theorem 3.2.In §
4, we present the main theorem of our paper. We also discuss an applica-tion of the machinery at our disposal to the theory of approximations of modules.Finally, in § Preliminaries
Throughout the paper, R denotes an (associative) ring with enough idempotents.By a module, we mean a unitary right R -module (i.e. M such that M = M R ).The category of all modules is denoted by Mod- R , its full subcategory consistingof all finitely presented modules by mod- R and the category of all flat modules byFlat- R . In this case, Mod- R is a finitely accessible Grothendieck category. All ourresults are theorems in ZFC; bar Lemma 3.3, about elements of Mod- R .Let B, C ∈ Mod- R . A homomorphism f : B → C is called a right almost splitmap if, given any M ∈ Mod- R and k ∈ Hom R ( M, C ), the map k factorizes through f if and only if k is not a split epimorphism. A left almost split map is defineddually. We say that a short exact sequence0 −→ A m −→ B f −→ C −→ almost split sequence if m is left almost split and f is right almost split.It is not hard to show that a right almost split map appears as the epimorphismin an almost split sequence if and only if it is surjective and its kernel has got localendomorphism ring. Moreover, we have the following properties. Proposition 2.1.
Let f : B → C be a right almost split map. Then: (1) The endomorphism ring of C is local. (2) The map f is surjective if and only if C is not a projective module. The main theorem on the existence of right almost split maps is due to Auslander.It is a partial converse of (1) above.
Theorem 2.2. ([5, Theorem 4])
Let C be a finitely presented module. There existsa right almost split map with codomain C if and only if the module C has got localendomorphism ring. Moreover, if C is non-projective, then there is even an almostsplit sequence ending in C . In what follows, given a set X , we denote by | X | the cardinality of X . Formally, | X | denotes the set of all ordinal numbers smaller than the cardinal number | X | N THE NON-EXISTENCE OF RIGHT ALMOST SPLIT MAPS 3 (which is an ordinal as well). For example, if X is a finite set of n elements,then | X | = { , , . . . , n − } = n . As usual, | X | + | Y | denotes the cardinality ofthe disjoint union of X and Y . If α, β are ordinals, we use interchangeably thenotations α < β, α ∈ β in the obvious meaning ‘ α is less than β ’. Finally, for aninfinite cardinal θ , cf( θ ) denotes the cofinality of θ ,i.e., the least cardinality of aset of smaller ordinals which converge to θ . We always have θ ≥ cf( θ ). An infinitecardinal θ is called regular if θ = cf( θ ), otherwise it is called singular . Note thatcf( θ ) is always a regular cardinal. By ω or ℵ , we denote the countable cardinal.For any two sets X, Y , we denote by X Y the set of all functions from X to Y .Moreover, if λ, µ are cardinals, then λ <µ denotes the cardinality of the set <µ λ = { η : α → λ | α < µ, η a function } . Similarly, λ µ denotes the cardinality of theset µ λ . It comes in handy to view the elements of µ λ as subsets of µ × λ .For a module M = Q i ∈ I M i and an infinite cardinal µ , we denote by Q <µi ∈ I M i the submodule of M consisting of all elements with support of cardinality < µ . Wecall this submodule a µ -bounded product of the modules M i .We say that a module M is finitely presented if the functor Hom( M, − ) commuteswith direct limits. For an infinite cardinal θ , we call a module M θ -presented ( < θ -presented , resp.) provided that M is the direct limit of a direct system of cardinality ≤ θ ( < θ , resp.) consisting of finitely presented modules.We finish by recalling a useful classic result. We say that a well-ordered directsystem of modules ( M α , m βα | α ≤ β < µ ), where µ is a regular infinite cardinal, is continuous provided that, for each δ < µ limit, M δ = lim −→ ( M α , m βα | α ≤ β < δ ). Lemma 2.3.
Let θ be an infinite cardinal and M a θ -presented module. Then M is the direct limit of a continuous well-ordered direct system of cardinality cf( θ ) consisting of < θ -presented modules.Proof. For a suitable directed poset ( I, ≤ ), we express the module M as the directlimit of a system F = ( F i , f ji : F i → F j | i ≤ j ∈ I ) consisting of finitely presentedmodules. Moreover, we can w.l.o.g. assume that | I | = θ . If θ = ℵ , there is a cofinalcountable well-ordered subsystem of F , so we can assume that θ is uncountable.Let ( β γ | γ < cf( θ )) be a continuous increasing sequence of infinite ordinalssmaller than θ converging to θ . From this system, we easily build an ⊆ -increasingsequence ( I γ | γ < cf( θ ) , | I γ | = | β γ | ) of directed subposets of ( I, ≤ ) such that I δ = S γ<δ I γ for δ limit, and I = S γ< cf( θ ) I γ (cf. [1, Lemma 1.6]).It remains to define M γ = lim −→ ( F i , f ji : F i → F j | i ≤ j ∈ I γ ) and, for all γ ≤ δ < cf( θ ), let m δγ be the canonical colimit factoring map. Then M γ is < θ -presented, for each γ < cf( θ ), and M = lim −→ ( M γ , m δγ | γ ≤ δ < cf( θ )). (cid:3) Construction of tree modules
Let θ be an infinite cardinal. We call a module M finitely θ -separable if it is thedirected union of a system consisting of < θ -presented direct summands of M . Wedenote by S θ the class of all finitely θ -separable modules.In what follows, we construct a particular type of these modules, and we showthat S θ forms a test class for splitting of epimorphisms with θ -presented codomain.By this, we mean that, given any epimorphism f : B → C with θ -presentedcodomain, there is a module L ∈ S θ such that f is a split epimorphism if andonly if Hom R ( L, f ) is surjective.The construction presented in this section is not entirely new. It is an uncount-able variant of a well-known technique, cf. [14, § pure-projective if it is a direct summand in a direct sumof finitely presented modules. If a module M has the property that each of its finite JAN ˇSAROCH subsets is contained in a pure-projective submodule of M which is pure in M , wecall M a Mittag–Leffler module . We denote the class of all Mittag–Leffler modulesby ML . The modules in ML have been studied a lot in the last 40 years, andmany equivalent characterizations are known of this class (cf. [8, Theorem 3.14]).We have chosen this one since it immediately yields S ℵ ⊆ ML .As the first step towards the announced construction, we recall an instance ofthe classic result on embedding of direct limits into reduced products, cf. the proofof [13, Theorem 3.3.2]. Proposition 3.1.
Let ( C α , f βα : C α → C β | α ≤ β < µ ) be a well-ordered directsystem of modules indexed by an infinite regular cardinal µ . Then there is anembedding of pure short exact sequences −−−−→ Q <µα<µ C α ⊆ ∗ −−−−→ Q α<µ C α −−−−→ Q α<µ C α / Q <µα<µ C α −−−−→ x ρ x σ x −−−−→ F ⊆ ∗ −−−−→ L α<µ C α −−−−→ lim −→ α<µ C α −−−−→ where the second row is the canonical presentation of the direct limit, σ is a puremonomorphism, and for all α < µ and x ∈ C α , we have ρ ( x )( β ) = f βα ( x ) if α ≤ β < µ , and ρ ( x )( β ) = 0 otherwise. Applications of the embedding above are not very frequent in the literature. Asa starter, we show the following interesting test for whether a module is cotorsion,answering [6, Question 33]. Recall that a module M is cotorsion provided thatExt R ( F, M ) = 0 whenever F is a flat module. Theorem 3.2.
Let R be a countable unital ring with R ω flat and Mittag–Leffler,and let us denote by ι : R ( ω ) → R ω the pure inclusion. Then a module M iscotorsion if and only if Hom R ( ι, M ) is surjective.Proof. First, if M is cotorsion, R ω /R ( ω ) flat yields Ext R ( R ω /R ( ω ) , M ) = 0. ThusHom R ( ι, M ) is onto.Now assume that M is not a cotorsion module. Since R is countable, thereexists a countably presented flat module F such that Ext R ( F, M ) = 0. By Lazard’stheorem, F is the direct limit of a countable well-ordered system of free modulesof finite rank. Hence, it is a pure submodule in R ω /R ( ω ) by Proposition 3.1. Let σ denote this pure embedding, and let π : R ω → R ω /R ( ω ) be the canonical (pure)epimorphism.Forming the pullback of σ and π , 0 0 x x Coker( ε ) Coker( σ ) x x −−−−→ R ( ω ) ι −−−−→ R ω π −−−−→ R ω /R ( ω ) −−−−→ (cid:13)(cid:13)(cid:13) ε x σ x −−−−→ R ( ω ) ⊆ −−−−→ N −−−−→ F −−−−→ x x , N THE NON-EXISTENCE OF RIGHT ALMOST SPLIT MAPS 5 we see that N is isomorphic to a pure submodule of a flat Mittag–Leffler module,hence it is flat and Mittag–Leffler. Moreover, N is countably presented, and soit is a projective module; in particular, Ext R ( N, M ) = 0. Since Ext R ( F, M ) = 0,there is a homomorphism h : R ( ω ) → M which cannot be extended to an elementof Hom R ( N, M ), from which it readily follows that there is no extension R ω → M of h either. (cid:3) Remark . The countable unital rings R for which R ω is flat and Mittag–Lefflerare precisely the left coherent ones satisfying the additional condition that inter-sections of finitely generated left submodules of R R ( ω ) are finitely generated (cf. [9,Theorem 4.7]). In particular, all countable left noetherian unital rings satisfy thehypothesis of Theorem 3.2.The hypothesis on the cardinality of the ring is necessary. As a counterexample,take R = End K ( K ( ω ) ) where K is a field. This is a self-injective von Neumannregular ring which is not right hereditary. At the same time R ∼ = R ω , and sothe projective dimension of R ω /R ( ω ) is at most one. It follows that there existsa non-cotorsion module M such that Ext R ( R ω /R ( ω ) , M ) = 0. The construction.
Assume we are given a well-ordered direct system C = ( C α , f βα : C α → C β | α ≤ β < µ ) indexed by an infinite regular cardinal µ (as in the statementof Proposition 3.1), and a cardinal λ such that λ <µ = λ . Then we use a simple Lemma 3.3.
There is a T ⊂ µ λ such that | T | = λ µ and each two distinct elements η, ζ ∈ T coincide on an initial segment of µ , i.e., Dom( η ∩ ζ ) ∈ µ .Proof. We embed µ λ into µ ( <µ λ ) via the assignment ν : η ( α η ↾ α ). Usingthe assumption on λ , we can fix a bijection ι : <µ λ → λ . We define T as the set { ι ◦ ( ν ( η )) | η ∈ µ λ } . Now, for two distinct η, ζ ∈ µ λ , we consider the least α suchthat η ( α ) = ζ ( α ). From the definition of ν , it follows that Dom( ν ( η ) ∩ ν ( ζ )) = α ,whence also Dom( ι ◦ ( ν ( η )) ∩ ι ◦ ( ν ( ζ ))) = α . (cid:3) Elements of the set T we have obtained in this way form branches of length µ of the forest S T . We are going to decorate these branches uniformly using ourwell-ordered direct system C . This is done via an enhancement of Proposition 3.1.Recall that we view the elements of T as subsets of µ × λ .Set C = lim −→ C . For all ( α, β ) ∈ S T , put C α,β = C α . We have the followingcommutative diagram with exact rows (where π denotes the canonical projection)0 −−−−→ Q <µ ( α,β ) ∈ S T C α,β ⊆ ∗ −−−−→ Q ( α,β ) ∈ S T C α,β π −−−−→ Im( π ) −−−−→ x ρ x σ x −−−−→ F ( T ) ⊆ ∗ −−−−→ ( L α<µ C α ) ( T ) τ −−−−→ C ( T ) −−−−→ ρ isdefined as follows: for each η ∈ T , let ν η : L α<µ C α → ( L α<µ C α ) ( T ) be thecanonical embedding onto the η th coordinate. For α < µ and x ∈ C α , we set ρν η ( x ) = k where k ( β, η ( β )) = f βα ( x ) if α ≤ β < µ , and k ( β, γ ) = 0 otherwise.Notice that, for each η ∈ T , the diagram from Proposition 3.1 embeds into thediagram above. In the second row, this is just the coproduct embedding corre-sponding to η . The first row embeds as follows (where Q = Im( π ↾ Q α<µ C α,η ( α ) )):0 −−−−→ Q <µα<µ C α,η ( α ) ⊆ ∗ −−−−→ Q α<µ C α,η ( α ) π ↾ Q α<µ C α,η ( α ) −−−−−−−−−−−→ Q −−−−→ . The partial order is defined by ( α, β ) < ( γ, δ ) ⇔ α < γ & ( ∃ η ∈ T )( η ( α ) = β & η ( γ ) = δ ). JAN ˇSAROCH
Using this observation, from Proposition 3.1, we know that ρν η is a monomor-phism for each η ∈ T (since ρ is), and also that ρν η ↾ F maps into the µ -boundedproduct. Thus ρ ↾ F ( T ) maps into the µ -bounded product as well, whence we getthe unique σ completing the diagram above.In what follows, for a class X ⊆
Mod- R , Sum( X ) denotes the class of all modulesisomorphic to a direct sum of modules from X . Lemma 3.4.
With the notation as above, we have: (1) σ is a monomorphism. (2) There is an exact sequence −→ D ⊆ −→ L g −→ C ( T ) −→ where L = Im( ρ ) and D = Ker( π ↾ L ) . Moreover, if λ ≥ | R | , and the system C consists of λ -presented modules, then | D | ≤ λ . (3) For each finite subset S of T , the module L S = P η ∈ S Im( ρν η ) is a directsummand in L , and L S ∈ Sum( { C α | α < µ } ) . Furthermore, we have L = S { L S | S ⊂ T finite } . (4) For each S ⊆ T with | S | ≤ µ , the module L S = P η ∈ S Im( ρν η ) decomposesas L ′ S ⊕ K ′ S where g ↾ L ′ S = 0 and K ′ S ∈ Sum( { C α | α < µ } ) .Proof. (1). Pick an arbitrary element x = x + · · · + x n ∈ ( L α<µ C α ) ( T ) such that0 = x i = ν η i ( y i ), for all i = 1 , . . . , n , where η , . . . , η n ∈ T are pairwise distinct.Assume that ρ ( x ) belongs to the µ -bounded product. Then there is a γ < µ such that η ( γ ) , . . . , η n ( γ ) are pairwise distinct and ρ ( x )( β, η i ( β )) = 0 for each i = 1 , . . . , n and β ≥ γ . The former condition and the way ρ is defined imply thatthe latter condition can be rephrased as ρν η i ( y i )( β, η i ( β )) = 0 for each i = 1 , . . . , n and β ≥ γ .Since σ from Proposition 3.1 is a monomorphism, we infer (using the observationwith the embedding of diagrams) that y i ∈ F for each i = 1 , . . . , n . It follows that x ∈ F ( T ) . Since x was arbitrary, we conclude that σ is a monomorphism.The first part of (2) follows from (1) by putting g = σ − ( π ↾ L ) where σ − is theinverse of the isomorphism σ : C ( T ) → Im( σ ). Let us prove the moreover clause.Since µ ≤ λ and S T ⊆ µ × λ , we have | S T | ≤ λ . By the assumption, thecardinality of modules in the system C is at most λ . The assumption λ = λ <µ then implies that the cardinality of the µ -bounded product is at most λ , too. Thus | D | ≤ λ , since D is a submodule in the µ -bounded product.(3). Let S = { η i | i < | S |} be a finite subset in T . Put n = | S | . For each ζ ∈ T \ S , let ψ ( ζ ) denote the least ordinal β < µ such that ( β, ζ ( β )) S S . For i < n , we put ψ ( η i ) = min { β < µ | ( β, η i ( β )) S j ψ ( γ )) k ( δ, η γ ( δ )) = 0 } . Then L ′ S = E ∩ L S is contained in the µ -bounded product, hence g ↾ L ′ S = 0.On the other hand, the module K ′ S = M γ<κ ρν η γ ( M ψ ( γ ) <α<µ C α ) ∼ = M γ<κ M ψ ( γ ) <α<µ C α ! is a complement of L ′ S in L S . Indeed, we readily check that L ′ S ∩ K ′ S = 0, and foreach γ < κ , α ≤ ψ ( γ ) and y ∈ C α , we can write ρν η γ ( y ) = ρν η γ ( y − f ( ψ ( γ )+1) α ( y )) + ρν η γ ( f ( ψ ( γ )+1) α ( y )) where the first term is in L ′ S and the second in K ′ S . (cid:3) Remark . By the construction, τ = gρ . It follows that g is a pure epimorphism.In fact, it is even µ -pure since τ is, i.e., any homomorphism from a < µ -presentedmodule into C ( T ) factorizes through τ (and hence through g ).The module L from the short exact sequence in Lemma 3.4(2) is the tree moduleconstructed from the data C , λ, T . Choosing this data a little bit more carefully, wecan impose further properties on L and the short exact sequence it fits in. Proposition 3.5.
With the notation as above. Let θ be an infinite cardinal with cf( θ ) = µ . Assume that C consists of < θ -presented modules. Then the followinghold. (1) The module L is finitely θ -separable. (2) If e ∈ End R ( L ) is an idempotent with e ↾ D = 0 , and End R (Im( e )) isa local ring, then Im( e ) is < θ -presented.Proof. (1). By Lemma 3.4(3), the module L is the directed union of the system( L S | S ⊂ T finite). We replace this system by ( L S,β | γ S ≤ β < µ, S ⊂ T finite)defined as follows:Let a finite S ⊂ T be fixed, and let ψ : T → µ be defined as in the proof ofLemma 3.4(3). Put γ S = max { ψ ( η ) | η ∈ S } and L S,β = M η ∈ S ρν η ( M ψ ( η ) ≤ α ≤ β C α ) ∼ = M η ∈ S M ψ ( η ) ≤ α ≤ β C α ! . This is a direct sum of < cf( θ ) of < θ -presented modules, hence it is < θ -presented.Moreover, L S,β is also a direct summand in L S with the complement K S,β where K S,β = M η ∈ S ρν η ( M β<α<µ C α ) ∼ = ( M β<α<µ C α ) S . JAN ˇSAROCH
We have proved (1), since L S splits in L by Lemma 3.4(3).For (2), put H = Im( e ), and let h : C ( T ) → H be the epimorphism such that hg = e . We can assume that H = 0; otherwise the conclusion trivially holds.Pick any non-zero u ∈ H . Then there is a finite subset S of T such that e ( u ) = hε S π S g ( u ) = u where π S and ε S are the canonical projection, embedding resp.,between C ( T ) and C S . Using that End R ( H ) is local, we see that hε S π S g ↾ H is anautomorphism of H . Thus we can w.l.o.g. assume that e factorizes through π S g .From Lemma 3.4(3), we know that L = L S ⊕ K S where K S ⊆ P ζ ∈ T \ S Im( ρν ζ ).We readily check that K S ⊆ Ker( π S g ) ⊆ Ker( e ). It follows that id H = e ↾ H factorizes through L/K S ∼ = L S which is a direct sum of < θ -presented modules.So H is a direct summand in a direct sum of < θ -presented modules. However,End R ( H ) is local whence H must be < θ -presented itself. (cid:3) Remark . Fix a class F of finitely presented modules such that F is closed un-der finite direct sums. We can relativize the construction in this section to thesubcategory lim −→ F as follows.Assume that C ∈ lim −→ F . By [8, Lemma 2.13], lim −→ F is closed under taking directlimits and pure submodules. It follows that, once we choose the modules C α , α < µ ,from the class lim −→ F (which can be easily done, cf. the proof of Lemma 2.3), themodules D and L belong to lim −→ F as well—use Remark 2 and Lemma 3.4(3).4. The main theorem
We start with a general observation.
Lemma 4.1.
Let λ be an infinite cardinal, → D → L g → C (2 λ ) → a short exactsequence, and f : B → Y be an epimorphism. Assume that | Ker( f ) | ≤ λ and | D | ≤ λ . Then the following holds. Hom R ( C, f ) is onto if and only if Im(Hom R ( g, Y )) ⊆ Im(Hom R ( L, f )) . Moreover, the group
Im(Hom R ( g, Y )) / Im(Hom R ( L, f )) ∩ Im(Hom R ( g, Y )) iseither trivial or of cardinality ≥ λ .Proof. Put A = Ker( f ). We have the following commutative diagram with exactrows and columns:0 −−−−→ Hom R ( C (2 λ ) , B ) Hom R ( g,B ) −−−−−−−→ Hom R ( L, B ) Hom R ( C (2 λ ) ,f ) y Hom R ( L,f ) y −−−−→ Hom R ( C (2 λ ) , Y ) Hom R ( g,Y ) −−−−−−−→ Hom R ( L, Y ) ε y ξ y Hom R ( D, A ) δ −−−−→ Ext R ( C (2 λ ) , A ) Ext R ( g,A ) −−−−−−−→ Ext R ( L, A ) . Suppose that Hom R ( C, f ) is surjective. Then Hom R ( C (2 λ ) , f ) is surjective aswell, and we get Im(Hom R ( g, Y )) ⊆ Im(Hom R ( L, f )) using the commutativity ofthe upper rectangle.If Hom R ( C, f ) is not surjective, then Im( ε ) ∼ = (Hom R ( C, Y ) / Im(Hom R ( C, f ))) λ implies | Im( ε ) | ≥ λ . On the other hand, 2 λ = (2 λ ) λ ≥ | A | | D | ≥ | Hom R ( D, A ) | ≥| Im( δ ) | . Using this cardinality discrepancy, the exactness of the third row and thecommutativity of the lower rectangle, we obtain a set W ⊆ Hom R ( C (2 λ ) , Y ) of N THE NON-EXISTENCE OF RIGHT ALMOST SPLIT MAPS 9 cardinality 2 λ such that the restriction of the map Ext R ( g, A ) ε = ξ Hom R ( g, Y )to W is one-one. It follows thatIm( ξ ↾ Im(Hom R ( g, Y ))) ∼ = Im(Hom R ( g, Y )) / Im(Hom R ( L, f )) ∩ Im(Hom R ( g, Y ))is a group of cardinality at least 2 λ . (cid:3) The part (1) in the following theorem says that the class, S θ , of all finitely θ -separable modules is a test class for splitting of epimorphisms with θ -presentedcodomain. The second part constitutes the core of the proof of our main result. Theorem 4.2.
Let θ be an infinite cardinal and f : B → C be an epimorphismwhere C is a θ -presented module. Then the following hold. (1) There exists L ∈ S θ such that Hom R ( L, f ) is onto if and only if f splits. (2) If f is a right almost split map, then C is < θ -presented.Proof. Let us denote µ = cf( θ ). We have a short exact sequence0 −→ A m −→ B f −→ C −→ C is the direct limit of a well-ordered direct system C = ( C α , f βα : C α → C β | α ≤ β < µ ) consisting of < θ -presented modules (cf. Lemma 2.3).Let λ be an infinite cardinal such that | R | + θ ≤ λ = λ <µ < λ µ = 2 λ ≥ | A | .For the cardinals λ, θ and the system C , we use Lemma 3.4 and Proposition 3.5 toobtain a short exact sequence0 −→ D ⊆ −→ L g −→ C (2 λ ) −→ | D | ≤ λ and L ∈ S θ (recall that | T | = λ µ = 2 λ ).Proving (1), we have the trivial implications: f splits ⇒ Hom R ( L, f ) is onto ⇒ Im(Hom R ( g, C )) ⊆ Im(Hom R ( L, f )). Using Lemma 4.1 with C = Y , we seethat the last inclusion implies that f splits.Meanwhile, the assumption in (2) implies that f does not split. By the precedingparagraph, it follows that there is a d ∈ Hom R ( C (2 λ ) , C ) such that dg does notfactorize through f . Using that f is right almost split, we deduce that dg is a splitepimorphism. By Proposition 2.1(1), C has got local endomorphism ring. If wedenote by e ∈ End R ( L ) an idempotent which factorizes through dg , then C ∼ = Im( e )is a < θ -presented module by Proposition 3.5(2). (cid:3) Remark . As the cardinal λ in the proof, we can pick for instance i µ ( | R | + | A | + θ ).Here, i (Beth) is the function defined for all cardinals κ inductively by putting i ( κ ) = κ , i α +1 ( κ ) = 2 i α ( κ ) and i α ( κ ) = P β<α i β ( κ ) for α limit.The main scope of application of Theorem 4.2(1) is in proving that a particularclass of modules is not precovering. Recall, that a class B of modules is a precoveringclass if, for any M ∈ Mod- R , there exist B ∈ B and f ∈ Hom R ( B, M ) such that,for all B ′ ∈ B , the map Hom R ( B ′ , f ) is surjective. The homomorphism f is thencalled a B -precover of M . Corollary 4.3.
Let B be a precovering class of modules closed under direct sum-mands and F a class of finitely presented modules closed under finite direct sums.Assume that B contains a generator, and that there exists an infinite cardinal θ such that S θ ∩ lim −→ F ⊆ B . Then B contains all θ -presented modules from lim −→ F .Proof. Since B contains a generator, any B -precover must be an epimorphism. Itremains to use Theorem 4.2(1) together with Remark 3. (cid:3) JAN ˇSAROCH
If the ring R is not right perfect, then there exists a countably presented flatmodule which is not Mittag–Leffler (by the famous result of H. Bass). By Corol-lary 4.3 and the fact that S ℵ ⊆ ML , we readily see, taking for F the class ofall free modules of finite rank, that the class of all flat Mittag–Leffler modules isnot precovering in this case (cf. [2, § ML is not precovering unless R is right pure semisimple (where the equality ML = Mod- R holds). The point is that over a ring R which is not right pure-semisimple, there exists a countably presented module which is not Mittag–Leffler,cf. [2, Lemma 5.1].We can prove the main result of our paper. Theorem 4.4.
Let R be a ring and C be a module. Then C is a codomain ofa right almost split map if and only if C is a finitely presented module with localendomorphism ring.Proof. The if part follows from Theorem 2.2. Assume that f : B → C is a rightalmost split map. Then C has got local endomorphism ring by Proposition 2.1(1).By the part (2) of the same proposition, f non-surjective yields C projective. How-ever, C projective and End R ( C ) local immediately imply that C is finitely presented(it is even a direct summand in the regular module R R ).If f is an epimorphism and C is not finitely presented, then there is a leastinfinite cardinal θ such that C is θ -presented. We use Theorem 4.2(2) to get thecontradiction. (cid:3) Remark . By [11], the if part of Theorem 4.4 holds in the general setting of finitely accessible additive categories (also called locally finitely presented additivecategories in [7]). So does the only-if part: indeed, by [7, Theorem 1.1], any finitelyaccessible additive category is equivalent to the category of flat modules over a ringwith enough idempotents. By Remark 3, choosing F as the class of all finitelygenerated projective modules, we know that our construction relativizes to thecategory Flat- R .Theorem 4.4 has an immediate consequence also for the first term of any almostsplit sequence. Recall that a module is called pure-injective if it is injective relativeto pure embeddings. Corollary 4.5.
Let → A m → B → C → be an almost split sequence. Then C isfinitely presented and A is pure-injective.Proof. The first part trivially follows from Theorem 4.4. Assume that A is notpure-injective. Then the pure embedding of A into its pure-injective envelope doesnot split, and hence factorizes through m . It follows that the almost split sequenceis pure. However, it would split in such a case since C is finitely presented. (cid:3) Morphisms determined by objects
In his famous paper [4], M. Auslander studied closely the general notion ofa morphism determined by object.
Definition 5.1.
Let C be a module and f : B → Y a homomorphism. We saythat f is right C -determined if the following holds:For any B ′ ∈ Mod- R and h ∈ Hom R ( B ′ , Y ), the map h factorizes through f ifand only if Im(Hom R ( C, h )) ⊆ Im(Hom R ( C, f )).Of course, the direct implication in Definition 5.1 always holds. The non-trivialand highly restrictive part is the converse. It turns out that the notion of a rightalmost split map is just a special case of this concept where C = Y , End R ( C ) islocal and Im(Hom R ( C, f )) is the Jacobson radical of End R ( C ), cf. [4, § II.2].
N THE NON-EXISTENCE OF RIGHT ALMOST SPLIT MAPS 11
Using the machinery developed, we can prove the following theorem. Recall that,given an infinite cardinal θ , an epimorphism f : B → Y is called θ -pure providedthat any homomorphism from a < θ -presented module into Y factorizes through f .Thus the notion of ℵ -pure epimorphism coincides with the usual notion of pureepimorphism. Theorem 5.2.
Let θ be an infinite regular cardinal and C a θ -presented module.Assume that f ′ : B → Y ′ is a right C -determined map. Let f : B → Im( f ′ ) denotethe epimorphism which coincides with f ′ . Then f is not a θ -pure epimorphismunless f splits.Proof. Assume that f is a non-split epimorphism. We aim to prove that it is not θ -pure. For this, set µ = cf( θ ) = θ, A = Ker( f ) and Y = Im( f ). Fix a well-ordereddirect system C = ( C α , f βα : C α → C β | α ≤ β < µ ) consisting of < θ -presentedmodules such that C = lim −→ C . As in Theorem 4.2, let λ be an infinite cardinal suchthat | R | + θ ≤ λ = λ <µ < λ µ = 2 λ ≥ | A | , and0 −→ D ⊆ −→ L g −→ C (2 λ ) −→ | D | ≤ λ and L ∈ S θ , obtained for the data λ, θ, C by Lemma 3.4 and Proposition 3.5.Observe that the epimorphism f is right C -determined, too. Using this andour assumption that f does not split, we get that Hom R ( C, f ) is not onto. ByLemma 4.1, we obtain an element d ∈ Hom R ( C (2 λ ) , Y ) such that the map dg doesnot factorize through f . Again, since f is right C -determined, it follows that thereis an h ∈ Hom R ( C, L ) such that dgh does not factorize through f .Since C is θ -generated, there is a set S ⊆ T of cardinality at most θ (= µ ) suchthat Im( h ) ⊆ L S = P η ∈ S Im( ρν η ). Let L S = L ′ S ⊕ K ′ S be the decomposition fromLemma 3.4(4). We can write g = g g , where g : L → L/L ′ S is the canonicalprojection. Then Im( g h ) ⊆ L S /L ′ S ∼ = K ′ S .Since dg g h does not factorize through f , neither does dg ↾ L S /L ′ S . However, L S /L ′ S is isomorphic to a direct sum of < θ -presented modules, and so f cannotbe θ -pure. (cid:3) Let us record an immediate corollary of the above result.
Corollary 5.3.
Let C be a countably presented module and f : B → Y be a right C -determined homomorphism. Then f : B → Im( f ) either splits or is not a pureepimorphism. In particular, f : B → Im( f ) splits whenever R has got weak globaldimension ≤ and Y is a flat module. Acknowledgements.
I would like to express my gratitude to Dolors Herbera forinviting me to participate on the project MTM2011-28992-C02-01 of DGI MINECO(Spain). Significant part of the material in this paper was written during my stayat Universitat Aut`onoma de Barcelona from 20 March to 1 April 2015.Many thanks also to Jan Trlifaj and Ivo Herzog for reading and discussing vastmajority of the text.
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