{'fraction_non_alphanumeric': 0.056538813276669, 'fraction_numerical': 0.013179837538329119, 'mean_word_length': 3.7604353393085788, 'pattern_counts': {'":': 0, '<': 20, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Baer and Mittag-Leffler modules over tame hereditary algebras

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones.

Since the fundamental work of Ringel [29], the study of infinite dimensional modules has become one of the challenging tasks of the representation theory of finite dimensional hereditary algebras. In the present paper, we consider in detail two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones.
Besides proving structure results, we investigate further the surprising analogy with modules over Dedekind domains discovered in [29]. Indeed, the current progress relies heavily on applications of the recent powerful set-theoretic and homological methods developed originally for modules over domains, cf. [3] and [15].
Baer modules can be defined in a rather general setting [26]: Let R be a ring and T be a torsion class in Mod−R. A module M ∈ Mod−R is a Baer module for T provided that Ext 1 R (M, T ) = 0 for all T ∈ T . In the particular case when R = Z, and more generally when R is a integral domain and T is the class of all torsion R-modules, we obtain thus the classical notions of a Baer group [8] and a Baer module [17]. It took quite a long time to prove that these notions actually coincide with the well-known notions of a free group and a projective module, respectively. Countable Baer groups were shown to be free already in 1936 by Baer [8], but the arbitrary ones only in 1969 by Griffith [16]. The projectivity of the classical Baer modules was also shown in two steps spreading over decades, however, in a different order. First, a reduction to countably presented modules was proved by set-theoretic methods by Eklof and Shelah [12] in 1990. Finally, the countably presented case has recently been settled in [3].
In the present paper, we consider Baer modules over tame hereditary algebras, so our T is the class of all torsion modules in the sense of Ringel [29] (see below for unexplained terminology). Baer modules in this sense have been studied since 1980's, notably by Okoh and Lukas in [24], [26], and [22]. Their focus was on countably presented modules, but their results already indicated the complexity of the general case.
Here we take a different approach. First, in Section 1, we apply set-theoretic methods to prove a reduction to finitely presented modules. We infer that the Baer modules are precisely the modules filtered by the finitely generated preprojective modules (Theorem 2). This reveals the main difference from the integral domain case: the role of the finitely generated projectives is taken by the much more complex class of the finitely generated preprojectives.
However, as shown in Section 2, the reduction is still sufficient to imply that Baer modules are locally projective, in the sense that their universal localizations, as introduced by Schofield in [31], are projective in case we localize with respect to a set of simple regular modules containing a complete clique (Theorem 5).
In Section 3 we employ infinite dimensional tilting theory in order to investigate the structure of the infinite dimensional torsion-free, and Baer modules. The key role is played here by an infinite dimensional tilting module, the so called Lukas tilting module L. It is a module which has no finite dimensional direct summands, not even pure-injective direct summands, but it is noetherian when viewed as a module over its endomorphism ring (Corollary 9).
Baer modules are characterized as the kernels of homomorphisms between modules in AddL, and torsion-free modules as the pure-epimorphic images of Baer modules with kernels in AddL (Corollaries 10 and 11).
We recover Lukas' theorem on the structure of countably presented Baer modules [22], and extend it to arbitrary Baer modules (Corollary 15); then we establish a bijective correspondence between equivalence classes of Baer modules and isomorphism classes of (infinite dimensional) preprojective modules (Corollary 18).
Also the Mittag-Leffler modules can be defined in a very general setting [30], [5]: Given a class of left R-modules Q, we call a module M Q-Mittag-Leffler if the canonical map M ⊗ R i∈I Q i → i∈I (M ⊗ R Q i ) is injective for any family {Q i } i∈I of left R-modules in Q. For Q = R-Mod, this is just the well-known notion of a Mittag-Leffler module from [27].
If R is a tame hereditary algebra then the connection with Baer modules is that every Baer module M is C-Mittag-Leffler where C denotes the class of all left R-modules without non-zero preinjective direct summands.
For artin algebras, it is known that Mittag-Leffler modules coincide with the separable modules, that is, the modules M whose finite subsets are always contained in finitely presented direct summands [35]; we observe that they also coincide with the strict Mittag-Leffler modules, that is, with locally pure-projective modules in the sense of [7]. We prove that in the setting of tame hereditary algebras, the torsion-free separable modules coincide with the pure submodules of products of indecomposable preprojective modules (Proposition 21), while the torsion reduced separable modules coincide with the locally split epimorphic images of direct sums of indecomposable finitely generated regular modules (Proposition 23).
Then we prove the main result of Section 4, a structure theorem for Mittag-Leffler modules (Theorem 27).
All our rings are unital, and by the unadorned term (R-)module we mean a right module over the ring R.
We will use the following notation. For a class of modules C, we define = 0 for all C ∈ C and all i > 0}. Similarly, the classes C o , C ⊥1 and C ⊥ are defined.
We will need some notions related to purity. A monomorphism ι : N → M is said to be s-pure (or strongly pure or locally split) if for any finite subset S ⊆ N there is an R-homomorphism π : M → N such that πι(x) = x for each x ∈ S. According to [35], a module M is said to be locally pure-injective if every pure-monomorphism M → X with X ∈ Mod-R is s-pure.
An s-pure submodule N of a module M is a submodule with the property that the embedding ι : N → M is s-pure. Replacing the term "finite subset" by "countable subset", we obtain the notion of a c-pure submodule of a module M . Clearly, we have the implications: We will also use the dual notions. A homomorphism π : X → M is a locally split epimorphism if for each finite subset F ⊆ M there is a map ϕ = ϕ F : M → X such that m = πϕ(m) for all m ∈ F . A module M is said to be strict Mittag-Leffler (or locally pure-projective) if every pure-epimorphism X → M with X ∈ Mod-R is locally split, see [27] and [7].
Finally, recall that a module T is said to be a (1-) tilting module if it satisfies (T1) proj.dim(T ) ≤ 1; (T2) Ext 1 R (T, T (I) ) = 0 for each set I; and (T3) there is an exact sequence 0 → R → T 0 → T 1 → 0 where T 0 and T 1 are direct summands of a (possibly infinite) direct sum of copies of T .

Baer modules and p-filtrations
We start with a general setting and show that set-theoretic methods allow to reduce the structure of Baer modules to the countably generated ones.
Let σ be an ordinal. An increasing chain of submodules, M = (M α | α ≤ σ), of a module M is called a filtration of M provided that M 0 = 0, M α = β<α M β for all limit ordinals α ≤ σ and M σ = M .
Given a class of modules C and a module M , a filtration M is a C-filtration of M provided that M α+1 /M α is isomorphic to some element of C for each α < σ. In this case we say that M is C-filtered.
Given a cardinal κ, a filtration M is a κ-filtration of M provided that M α is < κ-generated for each α < σ. (1) M is a Baer module for T .
Proof. 1. implies 2. The case when ⊥ T consists of modules of finite (and hence bounded) projective dimension is a particular instance of [15,Theorem 4.3.10] (for ν = ℵ 0 and n = the common bound of the projective dimensions of all modules in ⊥ T ). If T consists of modules of finite injective dimension, then (as T is closed under arbitrary direct sums) there is n < ω such that T ⊆ I n where I n denotes the class of all modules of injective dimension ≤ n. We will proceed by a modification of the proof of [15, Theorem 10.1.5] (for µ = ℵ 0 ). Since the latter proof has an extra set-theoretic assumption of the Weak Diamond, we will give more details below indicating how to avoid this assumption by employing the fact that T is closed under arbitrary direct sums. For Denote by Q i a representative set of all countably generated modules in A i . By downward induction on i, we will prove that each module in A i is Q i -filtered (our claim 2. is then the case of i = 0, since ⊥ I 0 = Mod−R.) For i = n we have A i = ⊥ I n . From Baer's criterion of injectivity and the fact that all syzygies of cyclic modules can be taken countably generated (since R is ℵ 0 -noetherian) we deduce that the cotorsion pair ( ⊥ I n , I n ) is generated by a class of < ℵ 1 -presented modules. So the claim follows from [15,Theorem 4.2.11].
The induction step from i + 1 to i for i < n is proved by transfinite induction on the minimal number λ of R-generators of M ∈ A i as in [15,Theorem 10.1.5]. The only place where the Weak Diamond is used there is when λ is regular and uncountable, for selecting a subfiltration with consecutive factors in ⊥ T from a λ-filtration of M consisting of modules in ⊥ T . However, this is possible by [ From now on, unless stated otherwise, we will assume that R is an indecomposable tame hereditary artin algebra with standard duality D : mod−R → R−mod between right and left finitely generated R-modules, respectively.
We recall the notation of [28] and [29]: t and p will denote representative sets of all indecomposable finitely generated regular, and preprojective, modules, respectively. Further, is the torsion-free class of all torsion-free modules. Note that Since add t is closed under extensions, the corresponding torsion class, called the class of all torsion modules, is Gent, see [28, 3.5].
A module M is a Baer module provided that M is a Baer module for Gent. The class of all Baer modules is denoted by B. So Remark. Baer modules over tame hereditary algebras are torsion-free, and the same holds for the classical Baer modules over commutative integral domains. However, this fails in general: Let R be a ring and T be a tilting module. Take T = T ⊥ . This is the torsion class consisting of all modules generated by T , and the corresponding torsion-free class is T o . Thus T is a Baer module for T which is not torsion-free.
The modules having no non-zero homomorphism to p are called P ∞ -torsion, and their class is denoted by L. By the infinitary version of the Auslander-Reiten formula [11] L = o p = p ⊥ so L is a tilting torsion class. There is a countably infinitely generated tilting module generating L, called the Lukas tilting module, and denoted by L, cf. [18]. For a module M , we will denote by l(M ) the P ∞ -torsion submodule of M , that is, l(M ) is the trace of L in M . The torsion-free class corresponding to L will be denoted by P. This is the class of all (possibly infinitely generated) preprojective modules (Warning: in [21] and [22], preprojective modules are called 'P ∞ -torsionfree'). We have Moreover, if M is a module such that M / ∈ L, then M has a non-zero finitely generated preprojective direct summand (see [29,Corollary 2.2]). In particular, the preprojective indecomposable modules are precisely the copies of modules from p. Similarly, the finitely p-filtered modules are exactly the modules in addp.
In our particular setting, thanks to a result from [26], we can improve the reduction Theorem 1 and obtain the following characterization of Baer modules:

Theorem 2. Let R be a tame hereditary artin algebra and M be a module. Then M is a Baer module if and only if M is p-filtered.
Proof. Assume M ∈ B. By Theorem 1, M has a filtration M = (M α | α ≤ κ) such that M α+1 /M α is a countably generated Baer module for each α < κ. So w.l.o.g., we can assume that M is countably generated. Then, by [26, Theorem 1.2], M = n<ω M n where M n is finitely generated, and M n+1 /M n ∈ F for each n < ω. Since B ∩ mod−R = F ∩ mod−R = addp, we infer that M has a filtration with all consecutive factors finitely generated and preprojective, and hence M is p-filtered.
For the if-part, notice that since p ⊆ B, also addp ⊆ B, that is, all finitely generated preprojective modules are Baer. So the claim follows Eklof's Lemma [15, Remark. Theorem 2 is analogous to the corresponding result for integral domains. In that case, however, one has to replace "preprojective" by "projective", so the existence of the filtration M already yields a direct sum decomposition, and hence projectivity, of M , cf. [3] and [12].
Being in a hereditary ring, a submodule of a Baer module is again Baer. However, the existence of a filtration of M as in Theorem 2 implies existence of a large family of Baer submodules of M such that also all consecutive factors in the family are Baer: Proof. This follows by an application of the Hill lemma [15, Theorem 4.2.6] to the filtration M (for κ = ℵ 0 ).

Universal localizations of Baer modules
Though Theorem 2 does not yield direct sum decompositions of Baer modules, it can be used to prove that universal localizations of Baer modules, in the sense of Schofield [31], always decompose in case we localize with respect to a set of isomorphism classes of simple regular modules U containing at least one complete clique (that is, in case the universal localization R U is a hereditary noetherian prime ring, see [10, §4]). In fact, in this case the localized Baer modules are projective.
Recall that a clique is an equivalence class with respect to the equivalence relation induced on the set of isomorphism classes of simple regular modules by the relation In other words, two simple regular modules S and S ′ belong to the same clique if and only if they are in the same tube. All cliques are finite, and all but finitely many consist of exactly one simple regular module.
Let U be a set of isomorphism classes of simple regular modules. For every S ∈ U fix a presentation consisting of finitely generated projective modules. The ring R U can be described as the universal localization of R at the set of maps between projective modules Σ = {f S |S ∈ U}.
In [10, Theorem 4.2], Crawley-Boevey proved that R U is a hereditary noetherian prime ring provided that U contains, at least, one complete clique. Otherwise U is finite, and R U is again a tame hereditary artin algebra with the same center as R.
For discussing the behaviour of Baer modules under localization, we will need the following preliminary observations. Lemma 4. Let R be a tame hereditary algebra. Let U be a set of isomorphism classes of simple regular modules. Then: To prove (ii) we note that in the short exact sequence of left R-modules the module N is a directed union of left R-modules that are finitely V-filtered where V = {TrU | U ∈ U} and TrU denotes the transpose of U (see [32,Theorem 3]).
Since, by Lemma 4, Tor R 1 (P α , R U ) = 0 for each α < κ, the R U -module M U = M ⊗ R R U is a direct limit of the continuous direct system of modules it remains to investigate the universal localization of the indecomposable preprojective R-modules.
We start with the case where U does not contain a complete clique. Then R U is again a tame hereditary algebra, but with a smaller rank of the Grothendieck group, see [10,Theorem 4.2]. Proceeding by induction, we can assume that U = {S} for some simple regular module S in a tube of rank at least two. Then, as shown in [14, 10.1], [33, 6.9], the preprojective component So we conclude that M U is filtered by finitely generated preprojective R U -modules, hence it is a Baer R U -module. Now assume that U contains at least one complete clique. We prove that in this case the universal localization of any indecomposable preprojective R-module P is a nonzero projective R U -module. This will show that M U is a projective Let P be an indecomposable preprojective module. By assumption, there is a tube Θ whose mouth is a subset of U. Then the embedding of P into its injective envelope E(P ) factorizes through a finite direct sum U = i≤m U i of elements of Θ. Making a pullback of a resolution of U consisting of finitely generated projective modules with the inclusion ε : P → U we obtain a commutative diagram where Q is projective because ε ′ is injective and the ring is hereditary. As, for each i ≤ m, all simple regular composition factors of U i are in U, an inductive argument on the regular composition length of U shows that U ⊗ Let us now investigate the universal localization of the Lukas tilting R-module L. If U contains at least one complete clique then the localization is a projective R U -module by Theorem 5. The interesting case is the one when U does not contain a complete clique, so R U is again a tame hereditary algebra: Theorem 6. Let R be a tame hereditary algebra with the Lukas tilting R-module L. Let U be a set of isomorphism classes of simple regular modules which does not contain a complete clique. Denote by R U the universal localization of R with respect to U. Then the universal localization L U = L ⊗ R R U of L is equivalent to the Lukas tilting R U -module L ′ .
Proof. As in the proof of Theorem 5 we can assume that U = {S} for a simple regular module S in a tube of rank at least two. Let p U denote a representative set of all indecomposable preprojective R U -modules.
First, we claim that Ext 1 RU (P ′ , L U ) = 0 for each P ′ ∈ p U . By the infinitary version of the Auslander-Reiten formula, it suffices to prove that Hom RU (L U , P ′ ) = 0 for each P ′ ∈ p U . Since the map R → R U is a ring epimorphism [31, p. 56], this is equivalent to Hom R (L U , P ′ ) = 0 for each P ′ ∈ p U . By Lemma 4, the sequence of right R-modules 0 → L → L U → L ⊗ R N → 0 is exact.
We know from [14, 10.1] that p U = p ∩ Mod−R U . Thus P ′ is also preprojective as right R-module, so Hom R (L, P ′ ) = 0. Similarly, Hom R (L ⊗ R N, P ′ ) ∼ = Hom R (L, Hom R (N, P ′ )) = 0 since N ∼ = R U /R (as a right R-module) is a directed union of finitely U-filtered modules (that have no non-zero homomorphisms to P ′ ) by [32,Theorem 3]. Thus Hom R (L U , P ′ ) = 0, and our claim is proved. Now, we prove that L U is a tilting module.
U ) = 0 for each finitely generated preprojective R U -module P ′ . Since P ′ is a finitely presented R U -module, the latter is equivalent to Ext 1 RU (P ′ , L U ) = 0, and (T2) follows from our claim above. For condition (T3), consider an exact sequence 0 → R → L 0 → L 1 → 0 where L 0 and L 1 are direct summands in some L (J) . By Lemma 4, the sequence 0 U . This shows that L U is a tilting module; it is also a Baer R U -module by Theorem 5, thus L U ∈ ⊥ (L ′ ⊥ ). Moreover, by our claim above, Ext 1 RU (L ′ , L U ) = 0 because L ′ is p U -filtered. So L U ∈ Add(L ′ ), hence L U is equivalent to L ′ . Theorem 6 will be used in a forthcoming paper for describing all infinite dimensional tilting modules over a tame hereditary artin algebra.

The structure of Baer modules
Throughout this section, let R be a tame hereditary artin algebra. We will need further properties of the (infinitely generated) R-modules.
We denote by G the generic module. By [29, 5.5], for each M ∈ F there are a unique cardinal κ, called the (torsion-free) rank of M , and an exact sequence Following [23], we call E = G ⊥ the class of all cotorsion modules. In analogy with the Dedekind domain case, Okoh proved that all torsion-free cotorsion modules are pure-injective, cf. [23, p.265]. The first part of the following Proposition will give this as a consequence of (infinite dimensional) cotilting theory.
Let We will denote by (C, D) the cotorsion pair in R-Mod studied in [28]. It is generated by a representative set t ′ of all indecomposable regular left R-modules, that is, D = (t ′ ) ⊥ is the class of all divisible left R-modules. Moreover, (C, D) is cogenerated by a representative set q ′ of all indecomposable preinjective left R-modules, that is, C = ⊥ (q ′ ) is the class of all left modules without non-zero preinjective summands. Finally, (C, D) is a tilting and cotilting cotorsion pair (co)generated by the (co)tilting module W which we call the Ringel tilting left Rmodule. Note that W is the direct sum of the generic left R-module G ′ with a representative set of the Prüfer left R-modules. Thus D(W ) is the direct product of G with a representative set of the adic right R-modules.

Proposition 7. Let R be a tame hereditary algebra.
(1) (F , E) is the cotorsion pair generated by G. This is also the cotilting cotorsion pair cogenerated by D(W ). In particular, F ∩ E = ProdD(W ), so all torsion-free cotorsion modules are pure-injective. (2) (B, L) is the tilting cotorsion pair generated by the tilting module L. It is also generated by the set p. In particular, B ∩ L = AddL, and L has no non-zero finitely generated direct summands.
. So, we deduce from [6, 2.3] that F is the cotilting torsion-free class cogenerated by the cotilting module D(W ). This proves the first two claims. That F ∩ E = ProdD(W ) follows from well-known properties of cotilting cotorsion pairs (see e.g. [15, §8.1]). As dual modules (and also cotilting modules) are pure injective, we deduce that torsion-free cotorsion modules are pure-injective. . Finally, if F ∈ AddL is finitely generated, then F ∈ B ∩ mod−R = add(p) ⊆ P, so F ∈ P ∩ L = P ∩ o p = 0.
Example 8. We observed on page 4 that B ⊆ lim − → addp = F , and the finitely generated modules in both classes coincide (with the finitely generated preprojective modules). However, the two classes are different -that is, G ∈ F \ B -because the class of all cotorsion modules is not closed under arbitrary direct sums by [23] (or because there exist preprojective modules which are not Baer, see below). In particular, B is not closed under direct limits.
Note furthermore that also the adic modules belong to F \ B. In fact, let S λ , λ ∈ P, be a representative set of simple regular modules, and let S λ [−∞], λ ∈ P, be the corresponding adic modules. If we assume that S λ [−∞] belongs to ⊥ (p ⊥ ), then we obtain from [22 Remark. Notice the analogies of Proposition 7 with the case when R is an integral domain: 1. In general, there are three different cotorsion pairs of R-modules: (T F, RE) where T F is the cotilting class of all torsion-free modules, (F 0 , EE) where F 0 is the class of all flat modules, and (SF, ME), the cotorsion pair generated by the quotient field Q of R. Always SF ⊆ F 0 (the equality holds iff all proper factors of R are perfect rings, [9]) and F 0 ⊆ T F (the equality holds iff R is a Prüfer domain, [13]). The cotilting module C generating T F can be taken of the form δ * where δ is the Fuchs divisible (tilting) module (see e.g. [15, Example 5.1.2]). The three cotorsion pairs coincide iff R is a Dedekind domain; in this case, C can be taken of the form T * = Hom R (T, Q/R) where T = Q ⊕ Q/R is the divisible tilting module analogous to the Ringel tilting module. (In the tame hereditary case, the analogs of the first and third cotorsion pairs coincide by Proposition 7.1, but the second is the trivial cotorsion pair (P 0 , Mod−R)).
2. All Baer R-modules are projective by [3], so the analogue of the tilting cotorsion pair (B, L) from Lemma 7.(2) is the trivial tilting cotorsion pair (P 0 , Mod−R), and the analogue of the Lukas tilting module is R. This is reflected in the behaviour of (B, L) under universal localization at tubes, see Theorem 5.
We now employ Proposition 7 to investigate the structure of the torsion-free R-modules. Let us start with the Lukas tilting module.
From Example 8 and the classification of the indecomposable pure-injective Rmodules we conclude that L has no pure-injective direct summand. In particular, L is not locally pure-injective, see [35, 2.1(5)]. On the other hand, L satisfies the following finiteness condition.
Corollary 9. L is noetherian over its endomorphism ring.
Proof. Since L is torsion-free without preprojective summands, D(L) is divisible without preinjective summands, thus D(L) ∈ C ∩ D = AddW , cf. [5, 9.4]. This shows that D(L) is Σ-pure-injective since so is W by [28, 10.1]. Now we apply [5, 9.9] which states that a tilting module is noetherian over its endomorphism ring if and only if its dual module is Σ-pure-injective.

Corollary 10. Let R be a tame hereditary algebra. A module M is a Baer module if and only if there is an exact sequence
Proof. This follows from Proposition 7.(2) and from the well-known characterization of modules forming tilting cotorsion pairs (see e.g. [15,Proposition 5.1.9]). Next, we look at the relation between B and P. Since P ∩ mod−R = addp, we have B ∩ mod−R = P ∩ mod−R. Moreover, all preprojective modules of countable rank are Baer [22, 4.3(4)]:

Lemma 12. Let M be a preprojective module of countable rank. Then M is a countably generated Baer module.
Proof. First, let F be a submodule of M of finite rank. We prove that F is finitely generated. Indeed, if F = 0 then F / ∈ L, so F = P 1 ⊕ F 1 where 0 = P 1 ∈ addp. Similarly, if F 1 = 0 then F 1 = P 2 ⊕ F 2 for 0 = P 2 ∈ addp etc. Since F has finite rank, there is n < ω such that F n = 0, so F = i≤n P i ∈ addp is finitely generated.
So w.l.o.g., we can assume that M has countably infinite rank. Consider an exact sequence 0 → M → G (ω) . Then M n = M ∩ G (n) (n < ω) are submodules of M of finite rank, hence finitely generated, with the property that M = n<ω M n , and the factors M n+1 /M n ⊆ G belong to F ∩ mod−R = addp. So M is countably generated, and M ∈ B by Theorem 2.
On the other hand, clearly, the countably generated tilting module L is Baer, but not preprojective. In fact, L is the only obstacle for the classes of countable rank Baer and preprojective modules to coincide. This comes from the following A weaker statement holds for arbitrary Baer modules. In general, the P ∞ -torsion submodule l(M ) of a Baer module M is not a direct summand, but we are going to see that is always a c-pure submodule. Actually, we prove a slightly stronger result: Proof. By Proposition 13, we can w.l.o.g. assume that M is not countably generated. Let M be an addp-filtration of M coming from Theorem 2. Let Y be a family of Baer submodules of M corresponding to M by Corollary 3. Let Y c denote the set of all countably generated modules from Y. By Proposition 13, for each Y ∈ Y c , the exact sequence . By conditions (H1) and (H3), the module M/Y is Baer, so we have the exact sequence Now, if C is a countable subset of l(M ) then there exist countably many maps f i ∈ Hom R (L, M ) (i < ω) such that C ⊆ S = i<ω Im(f i ). Since L is countably generated, by conditions (H2) and (H4), there exists Y ∈ Y c such that S ⊆ Y , and hence C ⊆ S ⊆ l(Y ). By the first part of the proof, D = l(Y ) is a direct summand in M .
Remark. Notice the following peculiar property of Baer modules: each Baer module M possess a p-filtration M = (M α | α ≤ κ) as in Theorem 2, and in fact M has many other p-filtrations by Corollary 3. In particular, M is the well-ordered directed union of the M α 's (α < κ). The exact sequences 0 → l(M α ) − → M α − → M α /l(M α ) → 0 (α ≤ κ) also form a well-ordered direct system. But unlike its middle term, M, this system is not continuous in general, that is, a sequence indexed by a limit ordinal α need not be the direct limit of the sequences indexed by all β < α. This is the price payed for M/l(M ) being preprojective, since direct limits of preprojective modules need not be preprojective.
This phenomenon is best seen for the Lukas tilting module L: it is a directed union of a countable chain (P n | n < ω) where all P n 's are finitely generated preprojective, so l(P n ) = 0 for each n < ω, but l(L) = L. Corollary 11 shows that any preprojective module occurs as the P ∞ -torsion-free part of some Baer module. Such presentation is non-trivial in the uncountable rank case, since there are preprojective modules which are not Baer: Example 16. Let Q be the direct product of a set of representatives of the isoclasses from p. Then Q is preprojective (since P is closed under arbitrary direct products), but not Baer (see [26,Proposition 3.1]). So, B is not closed under direct products.
Moreover, considering the exact sequence 0 → N → B → Q → 0 corresponding to Q by Corollary 11, we see that B is an uncountably generated Baer module whose P ∞ -torsion submodule is pure in B, but does not split. So Proposition 13 cannot be extended to uncountably generated Baer modules.  Proof. Assume B 1 ∼ B 2 , so P = B 1 /l(B 1 ) ∼ = B 2 /l(B 2 ) ∈ P. Considering the pull-back of the epimorphisms B 1 → P and B 2 → P , we obtain the following commutative diagram 0 0 and L 1 = l(B 2 ) belong to AddL. By Proposition 7.(2), the middle row and the middle column split, so The reverse implication follows by factoring out the P ∞ -torsion submodule of

Corollary 18. There is a bijective correspondence between equivalence classes of Baer modules and isomorphism classes of preprojective modules.
Proof. The correspondence is defined on representatives as follows: B → B/l(B) for a Baer module B; given a preprojective module P , we first consider a special B-precover B → P , and assign P → B. If B ′ → P is another special B-precover of P then B ∼ B ′ (see Lemma 17), and the claim is clear.
Since general preprojective modules are not classified, Corollary 18 leaves little hope for a complete classification of all Baer modules.

The structure of Mittag-Leffler modules
Recall that a module M over an arbitrary ring is said to be separable if every finite subset of M is contained in a finitely presented direct summand of M .
The torsion-free separable modules over a tame hereditary algebra were studied by Lenzing and Okoh in [25] and [20]. We will see that they are precisely the torsion-free (strict) Mittag-Leffler R-modules. This will then enable us to give a complete classification of all Mittag-Leffler R-modules.
We start by recalling some notions. If M is a right R-module, and Q is a class of left R-modules, we say that M is a Q-Mittag-Leffler module if the canonical map is injective for any family {Q i } i∈I of modules in Q. For Q = R-Mod we obtain the notion of a Mittag-Leffler module from [27].
The notion of relative Mittag-Leffler modules appears naturally in connection with cotorsion pairs. For example, if R is a tame hereditary algebra and C denotes the class of all left R-modules without non-zero preinjective summands then, from a general result on cotorsion pairs [5, 9.5], we get the following result Proof. In order to show (i) ⇒ (ii) let π : X → M be a pure epimorphism. If F is a finite subset of M then, by hypothesis, F is contained is a finitely presented direct summand N of M . Denote by ε : N → M the inclusion and by f : M → N any map such that f ε = Id. As N is pure projective, there exists g : N → X such that πg = ε. Set ϕ = gf . Then πϕ(x) = π(gf )ε(x) = ε(x) = x for any x ∈ F . This show that π is a locally split epimorphism. Therefore, M is strict Mittag-Leffler.
The implication (ii) ⇒ (iii) is due to Raynaud and Gruson [27]. The remaining part of the statement is due to Zimmermann [35, 2.11].
We now aim at a classification of all Mittag-Leffler modules over a tame hereditary algebra R. This is obtained in several steps. We start with a classification of the torsion-free Mittag-Leffler modules. A partial result in this direction was already achieved in [25, 3.1]. Proof. Let P be the direct sum of the modules in p. It is well known that P is noetherian over its endomorphism ring. This follows from the fact that its dual module is Σ-pure-injective [19, 4.6] [25,Introduction]. A characterization of pure-projective preprojectives can be found in [25, 2.4].
(2) The module L is an example of a Baer module that is not Mittag-Leffler, because L is countably generated and not pure-projective, see [27, 2.2.2].
Recall that every right R-module is a direct sum X ⊕ Y of a reduced module X and a divisible module Y , see [29]. Next, we investigate the torsion reduced Mittag-Leffler modules. Let us give a further characterization of the torsion-free, and torsion reduced modules, respectively, that are Mittag-Leffler. To this aim, we need some preliminaries. Recall that an inverse system of sets (H α , h α γ ) α γ∈I is said to satisfy the Mittag-Leffler condition if for any α ∈ I there exists β ≥ α such that It is shown in [27] that a right R-module M is Mittag-Leffler if and only if there exists a direct system of finitely presented modules (F α , u β α ) β α∈I with M = lim − → F α such that the inverse system (Hom R (F α , B), Hom R (u β α , B)) β α∈I satisfies the Mittag-Leffler condition for any right R-module B.
In the next Proposition we present a refinement of this result from [5]. It will allow us to establish a situation in which we can restrict to special choices of B for proving that a module is Mittag-Leffler. Proposition 24. [5, 6.2] Let R be a ring. Let B be a class of right R-modules closed under direct sums, and let Q be a class of left R-modules. Assume there exists a direct system of finitely presented modules (F α , u β α ) β α∈I with M = lim − → F α such that the inverse system (Hom R (F α , B), Hom R (u β α , B)) β α∈I satisfies the Mittag-Leffler condition for any right R-module B ∈ B.
If for every Q ∈ Q and every α ∈ I there exists a map f α : F α → B α such that B α ∈ B and f α ⊗ R Q is a monomorphism, then M is a Q-Mittag-Leffler module.
Proposition 25. Let R be a ring, and let S be a class of finitely presented modules. Let B be a module such that S ⊆ Add (B). If M is a module in lim − → S, then the following statements are equivalent: (1) M is Mittag-Leffler.
Corollary 26. Let R be a tame hereditary artin algebra. Let P be the direct sum of the modules in p, and let T be the direct sum of the modules in t.
(1) A torsion-free module M is Mittag-Leffler if and only if there exists a direct system of finitely presented modules (F α , u β α ) β α∈I with M = lim − → F α such that the inverse system (Hom R (F α , P ), Hom R (u β α , P )) β α∈I satisfies the Mittag-Leffler condition.
(2) A torsion reduced module M is Mittag-Leffler if and only if there exists a direct system of finitely presented modules (F α , u β α ) β α∈I with M = lim − → F α such that the inverse system (Hom R (F α , T ), Hom R (u β α , T )) β α∈I satisfies the Mittag-Leffler condition.
Proof. The statements are immediate consequences of Proposition 25, because the torsion-free modules are the modules in lim − → add p, and the torsion reduced modules belong to lim − → add t, see [28, 3.4].
We are now ready to give the announced classification.
Theorem 27. Let R be a tame hereditary artin algebra. The (strict) Mittag-Leffler modules over R are precisely the modules of the form X ⊕ Y where Y is a direct sum of indecomposable preinjective modules, and there exists a unique pure-exact sequence ε : 0 → X ′ ֒→ X → X ′′ → 0 such that X ′ is a locally split epimorphic image of a direct sum of indecomposable finitely generated regular modules, and X ′′ is an (s-)pure submodule of a product of indecomposable preprojective modules.
Proof. First of all, we check that every module of the stated form is (strict) Mittag-Leffler. The module Y is pure-projective, hence strict Mittag-Leffler. The modules X ′ and X ′′ are strict Mittag-Leffler by Propositions 21 and 23. Since the class of strict Mittag-Leffler modules is closed under pure extensions [5, 8.12], it follows that X and X ⊕ Y are strict Mittag-Leffler. Conversely, if M is a (strict) Mittag-Leffler module, we write M = X ⊕ Y as direct sum of a reduced module X and a divisible module Y . Recall that divisible modules are direct sums of indecomposable preinjective modules and of indecomposable infinite dimensional pure-injective modules, see [29]. But no indecomposable infinite dimensional module is separable. Hence Y is a direct sum of indecomposable preinjective modules. As for X, we know from [29, 4.1] that there is a unique pure-exact sequence 0 → tX → X ν − → X/tX → 0 where tX = f ∈HomR(Y,X),Y ∈t Im f ∈ Gen t is the trace of t in X, and X/tX ∈ F. Since the class of reduced modules is closed under submodules, and the class of Mittag-Leffler modules is closed under pure submodules, we deduce that X ′ = tX is torsion reduced Mittag-Leffler, hence it has the stated form by Proposition 23. It remains to prove that X ′′ = X/tX is Mittag-Leffler. Then it will have the stated form by Proposition 21. So, we check that X/tX is separable. Let 0 = z ∈ X/tX with z ∈ X. Then z is contained in a finitely generated direct summand of X, so we can write X = A ⊕ A ′ ⊕ B with A finitely generated torsion-free, A ′ finitely generated torsion, and z = a + a ′ ∈ A ⊕ A ′ . Since Hom R (tX, A) = 0, we have tX ⊆ A ′ ⊕ B, which shows that X/tX = ν(A) ⊕ ν(A ′ ⊕ B). Thus ν(A) is a finitely generated direct summand of X/tX which contains z = ν(z) = ν(a). This completes the proof.