Mittag-Leffler conditions on modules

We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.


Introduction
In the last few years, Mittag-Leffler conditions on modules were successfully employed in a number of different problems ranging from tilting theory to commutative algebra, and to a conjecture originating in algebraic topology. Indeed, the translation of certain homological properties of modules into Mittag-Leffler conditions was a key step in solving the Baer splitting problem raised by Kaplansky in 1962 [3], as well as in proving that every tilting class is determined by a class of finitely presented modules [12] [14], and it is part of the strategy for tackling the telescope conjecture for module categories [6].
Motivated by these results, in this paper we undertake a systematic study of such conditions, and we give further applications of these tools. In fact, we give a new proof for the Date: February 1, 2008. The research of this paper was initiated while the second author was spending a sabbatical year at the Università di Padova (Italy). She thanks her host for its kind hospitality. She also thanks the Università dell'Insubria, Varese (Italy) for its hospitality in several visits while the paper was being written. The  [3]. Moreover, using the theory of matrix subgroups, we provide a new interpretation of certain finiteness conditions of a module over its endomorphism ring, in particular of endofiniteness. Furthermore, we show that Mittag-Leffler conditions appear naturally in the theory of cotorsion pairs, that is, pairs of classes of modules that are orthogonal to each other with respect to the Ext functor. As a consequence, we discover a new finiteness condition satisfied by tilting modules. Finally, we employ our investigations to discuss the pure-semisimplicity conjecture, developing an idea from [2].
Further applications of our work to finite-dimensional hereditary algebras, and to cotorsion pairs given by modules of bounded projective dimension will appear in [5] and [13], respectively.
We now give some details on the conditions we are going to investigate. Raynaud and Gruson studied in [32] the right modules M over a ring R having the property that the canonical map is injective for any family of left R-modules {Q i } i∈I . They showed that this is the case if and only if M is the direct limit of a direct system (F α , f β α ) β α∈Λ of finitely presented modules such that the inverse system (Hom R (F α , B), Hom R (f β α , B)) β α∈Λ satisfies the Mittag-Leffler condition for any right R-module B. Therefore such modules M are said to be Mittag-Leffler modules.
In this paper, we study relative versions of these properties by restricting the choice of the family {Q i } i∈I and of B. We thus consider the notions of a Q-Mittag-Leffler module and of a B-stationary module. Part of our work consists in developing these notions following closely [32].
While the definition of a Q-Mittag-Leffler module relies on the injectivity of the natural transformation ρ, the B-stationary modules are not "canonically" defined. We introduce the stronger notion of strict B-stationary modules. Again, this notion is inspired by [32]. Indeed, if B is the class of all right modules, then the strict B-stationary modules are precisely the strict Mittag-Leffler modules introduced by Raynaud and Gruson, and later studied by Azumaya [10] and other authors under the name of locally projective modules. We characterize strict B-stationary modules in terms of the injectivity of the natural transformation This relates our investigations to results on matrix subgroups obtained by Zimmermann in [36].
As mentioned above, our original motivation are the results in [12], where it was made apparent that for a countably presented module M the vanishing of Ext 1 R (M, B) for all modules B belonging to a class B closed under direct sums, can be characterized in terms of B-stationarity, see Theorem 3.11 for a precise statement. Furthermore, also the vanishing of lim ← − 1 , the first derived functor of the inverse limit lim ← − , can be interpreted in this way, see [18] and [3]. We believe that a thorough understanding of B-stationary modules and of their relationship with strict B-stationary and Q-Mittag-Leffler modules will provide a new insight in problems related to the vanishing of certain homological functors. The applications we present in this paper are oriented towards such developments.
Let us illustrate such applications by focussing on cotorsion pairs. Let S be a set of finitely presented modules, and let (M, L) be the cotorsion pair generated by S. In other words, L is the class of modules defined by the vanishing of Ext 1 R (S, −), while M is defined by the vanishing of Ext 1 R (−, L), see Definition 9.1. Denote further by C the class defined by the vanishing of Tor R 1 (S, −). We prove in Theorem 9.5 that a module is L-stationary if and only if it is C-Mittag-Leffler. Moreover, it turns out that every module in M is strict L-stationary.
In particular, this applies to cotorsion pairs arising in tilting theory (Corollary 9.8), yielding that every tilting module T is strict T -stationary. The latter property can be interpreted in terms of matrix subgroups and allows us to show that certain tilting modules are noetherian over their endomorphism ring, see Proposition 10.1 and [5].
The paper is organized as follows. In Section 1 we introduce Q-Mittag-Leffler modules, and we study the closure properties of the class Q and of the class of Q-Mittag-Leffler modules. For our applications to cotorsion pairs, it is relevant to note the good behavior of Mittag-Leffler modules with respect to filtrations established in Proposition 1.9. We revisit the topic of Q-Mittag-Leffler modules in Section 5, where we characterize them in the spirit of [32]: since the map ρ is bijective when M is finitely presented, and since every module is a direct limit of finitely presented modules, one has to determine the "gluing" conditions on the canonical maps u α , u β α in the direct limit presentation (M, (u α ) α∈I ) = lim − → (F α , u β α ) β α∈I that imply the injectivity of ρ. These conditions are called dominating with respect to Q. We introduce them in Section 4 where we also study their basic properties.
B-stationary modules are introduced in Section 3. Hereby we adopt the language of Hsubgroups from [39], which is the topic of Section 2. Our first aim is to give an intrinsic characterization of B-stationarity. This characterization, obtained in Theorem 4.8, is also given in terms of dominating maps. It will allow us to study the interplay between the conditions B-stationary and Q-Mittag-Leffler in Section 6.
The interrelationship between the different conditions is further pursued in Section 9, after having introduced and characterized the strict B-stationary modules in Section 8. Note that the condition strict B-stationary has again a good behavior under filtrations, cf. Proposition 8.12. This intertwines our investigations with the theory of cotorsion pairs. Our main results in this context are Theorem 9.5 and its application to tilting cotorsion pairs in Corollary 9.8, which we have already described above.
A further important application is given in Section 7 which is devoted to Baer modules over domains. A module M over a commutative domain R is said to be a Baer module if Ext 1 R (M, T ) = 0 for any torsion module T . Kaplansky in [30] raised the question whether Baer modules are projective. The last step in the positive solution of Kaplansky's problem was made in [3]. In the present work, we prove that a countably generated Baer module over an arbitrary commutative domain is always a Mittag-Leffler module. This yields another proof of the fact that Baer modules over commutative domains are projective.
Let us mention that the techniques introduced by Raynaud and Gruson have also been used by Drinfeld in [15]. We give in Corollary 5.5 a detailed proof of [15,Theorem 2.2].
Finally, as a last application, we consider left pure-semisimple hereditary rings in Section 10. In particular, we use Corollary 9.8 to study the tilting cotorsion pair generated by the preprojective right modules, following an idea from [2].
Acknowledgements. We would like to thank Javier Sánchez Serdà for valuable discussions on the paper [32], and Silvana Bazzoni for many comments on preliminary versions of the paper.
Notation. Let R be a ring. Denote by Mod-R the category of all right R-modules, and by mod-R the subcategory of all modules possessing a projective resolution consisting of finitely generated modules. R-Mod and R-mod are defined correspondingly.
For a right R-module M , we denote by M * = Hom Z (M, Q/Z) its character module. Instead of the character module we can also consider another dual module, for example, for modules over an artin algebra Λ we can take M * = D(M ) where D denotes the usual duality. If S is a class of modules, we denote by S * the corresponding class of all duals B * of modules B ∈ S.
For a class M ⊂ Mod-R we set We will say that a module M R with endomorphism ring S is endonoetherian if M is noetherian when viewed as a left S-module. If S M has finite length, then we say that M is endofinite.

Q-Mittag-Leffler modules
Definition 1.1. [34] Let M be a right module over a ring R, and let Q be 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. If Q just consists of a single module Q, then we say that M is Q-Mittag-Leffler.
We will need the following Lemma. Lemma 1.2. Let M R and R Q be a right and a left R-module, respectively. Assume that Q = lim − → (K α , f β α ) β α∈I . For any α ∈ I, let f α : K α → Q be the induced map. Let q 1 , . . . , q n ∈ Q and x 1 , . . . , x n ∈ M be such that Then there exist α 0 ∈ I and k 1 , . . . , k n ∈ K α0 such that . . , n, satisfy the desired properties.
Here are some closure properties of the class Q. Related results can be found in work of Rothmaler [ Proof. (i) Let {Q i } i∈I be a family of modules in Q, and let {Q ′ i } i∈I be a family of left R-modules such that for any i ∈ I the module Q ′ i is a pure submodule of Q i . For any i ∈ I, denote by ǫ i : Q ′ i → Q i the inclusion. As every ǫ i is a pure monomorphism, so is i∈I ǫ i . Then we have the commutative diagram (ii) is proved in [34, p. 39].
Let {Q i } i∈I be a family of modules such that, for any i ∈ I, Q i = lim − → (K i α , f i β α ) β α∈Ii and K i α ∈ Q for any α ∈ I i . For any i ∈ I and α ∈ I i , let f i α : K i α → Q i denote the canonical morphism.
We want to show that ρ : x j ⊗ (q i j ) i∈I be an element in the kernel of ρ. This means that, for any i ∈ I, n j=1 x j ⊗ q i j is the zero element of M ⊗ R Q i . By Lemma 1.2, for each i ∈ I, there exists α i ∈ I i and k i 1 , . . . , k i n ∈ K i αi such that n j=1 x j ⊗ k i j is the zero element of M ⊗ R K i αi and f i αi (k i j ) = q i j for every j = 1, . . . , n. Consider the commutative diagram Note that ρ ′ is injective because K αi ∈ Q for any i ∈ I. This shows that y = 0.
Proposition 1.4. Let R be a ring. The following statements hold true for M ∈ Mod-R.
(ii) Let Q 1 and Q 2 be two classes in R-Mod, and let Q be the class consisting of all extensions of modules in Q 1 by modules in Q 2 . Suppose that M is Q i -Mittag-Leffler for i = 1, 2, and that the functor M ⊗ − is exact on any short exact sequence with first term in Q 1 and end-term in Q 2 . Then M is Q-Mittag-Leffler.
Proof. (i). Let {Q i } i∈I be a family of modules in Q. For any j = 1, . . . , n, set is injective for any j = 1, . . . n, it follows that ρ is also injective.
(ii). Let {Q i } i∈I be a family of left modules such that, for any i ∈ I, there is an exact Then we have the commutative diagram where the bottom row is exact by assumption on M ⊗ −. As ρ 1 and ρ 2 are injective, ρ is also injective.
Corollary 1.5. Let R be a ring, and M ∈ Mod-R. Let (T , F ) be a torsion pair in R-Mod such that M is T -Mittag-Leffler and F -Mittag-Leffler. Assume further that the functor M ⊗ − is exact on any short exact sequence with first term in T and end-term in F . Then M is a Mittag-Leffler module.
Examples 1.6. (1) Let R be a commutative domain and denote by T and F the classes of torsion and torsionfree modules, respectively. Any flat R-module M which is T -Mittag-Leffler and F -Mittag-Leffler is a Mittag-Leffler module.
(2) Let Λ be a tame hereditary finite dimensional algebra over an algebraically closed field k, and let t be the class of all finitely generated indecomposable regular modules. It was shown by Ringel in [33, 4.1] that the classes (F , Gent) with F = t • = ⊥ t form a torsion pair, and for every module X ∈ Mod Λ there is a pure-exact sequence where tX = f ∈Hom(Y,X),Y ∈t Imf ∈ Gen t is the trace of t in X, and X/tX ∈ F. Thus a module M ∈ Mod Λ is Mittag-Leffler provided it is T -Mittag-Leffler and F -Mittag-Leffler.
(3) [36, 2.5] If Q is a left R-module satisfying the maximum condition for finite matrix subgroups (see Definition 8.6), for example an endonoetherian module, then every right R-module is Q-Mittag-Leffler.   Proof. Let {Q i } i∈I be a family of left S-modules. Since R → S is a ring epimorphism As M S is finitely presented this is isomorphic to This yields that the canonical map ρ : M ⊗ R i∈I Q i → i∈I M ⊗ R Q i is in fact an isomorphism.
It follows from Example 1.6(4) that M/N is also a Mittag-Leffler module with respect to the class S-Mod. Definition 1.8. Let M be a right R-module, and let τ denote an ordinal. An increasing chain (M α | α ≤ τ ) of submodules of M is a filtration of M provided that M 0 = 0, M α = β<α M β for all limit ordinals α ≤ τ and M τ = M .
Given a class C, a filtration (M α | α ≤ τ ) is a C-filtration provided that M α+1 /M α ∈ C for any α < τ . We say also that M is a C-filtered module.
We have the following result about the behavior of the Mittag-Leffler property with respect to filtrations. Proposition 1.9. Let S be a class of right R-modules that are Mittag-Leffler with respect to a class Q ⊆ S ⊺ . Then any module isomorphic to direct summand of an S ∪ Add R-filtered module is Q-Mittag-Leffler.
Proof. As projective modules are Mittag-Leffler and (S ∪Add R) ⊺ = S ⊺ , we can assume that S contains Add R . Moreover, since the class of Q-Mittag-Leffler modules is closed by direct summands we only need to prove the statement for S-filtered modules.
Let M be an S-filtered right R-module. Let τ be an ordinal such that there exists an S-filtration (M α ) α≤τ of M . We shall show that M is Q-Mittag-Leffler proving by induction that M α is Q-Mittag-Leffler for any α ≤ τ . Observe that for any β ≤ α ≤ τ , M α and M α /M β are S-filtered modules, so they belong to ⊺ Q.
As M 0 = 0 the claim is true for α = 0. If α < τ then, as Q ⊆ S ⊺ , we can apply an argument similar to the one used in Proposition 1.4 to the exact sequence Let α ≤ τ be a limit ordinal, and assume that M β is Q-Mittag-Leffler for any β < α. We shall prove that M α = β<α M β is Q-Mittag-Leffler. Let {Q i } i∈I be a family of modules in Q, and let x ∈ Ker (M α ⊗ R i∈I Q i → i∈I M α ⊗ R Q i ). There exists β < α and y ∈ M β ⊗ R i∈I Q i such that x = (ǫ β ⊗ R i∈I Q i ) (y), where ǫ β : M β → M α denotes the canonical inclusion. Considering the commutative diagram As ρ ′ is injective because M β is Q-Mittag-Leffler and, for any i ∈ I, ǫ β ⊗ Q i is also injective because Tor R 1 (M α /M β , Q i ) = 0, we deduce that y = 0. Therefore x = 0, and ρ is injective.

H-subgroups
We recall a notion from [39] which will be very useful in the sequel. For the following discussion it is important to keep in mind the following easy observations.
Recall that a homomorphism π : B → B ′′ is a locally split epimorphism if for each finite subset X ⊆ B ′′ there is a map ϕ = ϕ X : B ′′ → B such that x = πϕ(x) for all x ∈ X. Observe that every split epimorphism is locally split, and every locally split epimorphism is a pure epimorphism. Locally split monomorphisms are defined dually. Moreover, a submodule B ′ of a module B is said to be a locally split (or strongly pure [38]) submodule if the embedding B ′ ⊂ B is locally split.
Lemma 2.4. Let M and M ′ be right R-modules, and let v ∈ Hom R (M, M ′ ). Assume that M is finitely generated. Let further ε : B ′ → B be a pure monomorphism. If M ′ is finitely presented or ε is a locally split monomorphism, then Proof. The first case is treated in [12,Lemma 4.1] or [3,Lemma 2.8]. For the second case, we assume that ε is a locally split monomorphism. We show the inclusion ⊇. Consider f ∈ Hom R (M, B ′ ) such that ε f = h v for some h ∈ Hom R (M ′ , B). Choose a generating set x 1 , . . . , x n of M together with a map ϕ : In statement (3), we have that π : B → B ′′ is a pure epimorphism and M ′ is finitely presented, so For statement (5), we consider f v ∈ H v (B ′′ ), and choose a generating set x 1 , . . . , x n of M together with a map ϕ :
Let us specify the Mittag-Leffler condition for the case I = N.
2. An inverse system of the form satisfies the Mittag-Leffler condition if and only if for any n ∈ N the chain of subsets of H n h n (H n+1 ) ⊇ · · · ⊇ h n · · · h n+k (H n+k+1 ) ⊇ · · · is stationary.
According to Raynaud and Gruson [32,p. 74] the following characterization of Mittag-Leffler inverse systems is due to Grothendieck as it is implicit in [27, 13.2.2]. We give a proof for completeness' sake.
For any m > n ≥ 1 set h nm = h n · · · h m−1 , and, for any n ≥ 1 let g n : lim ← − H i → H n denote the canonical map.
The inverse system H satisfies the Mittag-Leffler condition if and only if for any n ≥ 1 there exists ℓ(n) > n such that Proof. Observe that since, for any m > n ≥ 1, g n = h nm g m always Assume now that H satisfies the Mittag-Leffler condition. We only need to show that for any n ≥ 1 there exists ℓ(n) > n such that h n ℓ(n) (H ℓ(n) ) ⊆ g n (lim ← − H i ). To this aim fix n ≥ 1. Applying repeatedly that H satisfies the Mittag-Leffler condition we find a sequence of elements in N . In this fashion, the properties of the sequence ( * ) allow us to find a sequence a 0 = a, a 1 , . . . , a i , . . . such that a i ∈ H ni and h ni ni+1 (a i+1 ) = a i for any i ≥ 0.
The converse implication is clear because of the remarks at the beginning of the proof.
The above characterization does not extend to uncountable inverse limits; an example where this fails is implicit in Example 9.11.
We will be interested in inverse systems arising by applying the functor Hom R (−, B) on a direct system.
is an inverse system of left modules over the endomorphism ring of B, and Applying Lemma 2.3(1) to the situation of Remark 3.4 we obtain the following.
Lemma 3.5. Let (F α , u β α ) β α∈I be a direct system of right R-modules with direct limit M , and denote by u α : F α → M the canonical map. Let B be a right R-module.

This allows to interpret the Mittag-Leffler condition on inverse systems as in Remark 3.4 in terms of H-subgroups.
Lemma 3.6. Let (F α , u β α ) β α∈I be a direct system of right R-modules. Let α, β ∈ I with β ≥ α, and let B be a right R-module. The following statements are equivalent.
We adopt the following definition inspired by the terminology in [28].
(1) A direct system (F α , u β α ) β α∈I of right R-modules is said to be B-stationary provided that the inverse system (Hom R (F α , B), Hom R (u β α , B)) β α∈I satisfies the Mittag-Leffler condition, in other words, provided for any α ∈ I there exists β ≥ α such that the equivalent conditions in Lemma 3.6 are satisfied.
(2) A right R-module M is said to be B-stationary if there exists a B-stationary direct system of finitely presented modules (F α , u β α ) β α∈I such that M = lim − → F α . (3) Let B be a class of right R-modules. We say that a direct system (F α , u β α ) β α∈I or a right R-module M are B-stationary if they are B-stationary for all B ∈ B.
Let us start by discussing some closure properties of the class B.
Proposition 3.8. Let {B j } j∈J be a family of right R-modules. Let (F α , u β α ) β α∈I be a direct system of right R-modules. Then the following statements are equivalent.
If F α is finitely generated for any α ∈ I, then the above statements are further equivalent to Proof. We use the same arguments as in [3, 2.6]. (1) ⇔ (2). Statement (1) holds if and only if, for any α ∈ I there exists β such that Equivalently, if and only if (2) holds.
The proof of (2) ⇔ (3) follows in a similar way by observing that provided all F α are finitely generated.
Proof. The statements in Lemma 2.5 (2) and (3) imply statement (i). Statement (ii) is a direct consequence of Proposition 3.8 combined with (i), and (iii) is a special case of (ii). Proposition 3.10. Let (F α , u β α ) β α∈I be a direct system of right R-modules, and let B be a right R-module. Consider the following statements.
Proof. The fact that (1) and (1 ′ ) are equivalent statements follows directly from the definitions taking into account Example 3.2 and Remark 3.4.
We prove now (1) ⇒ (2). Assume for a contradiction that there exists α such that for any β ≥ α condition (1) in Lemma 3.6 fails. Now we construct a countable chain in I such that condition (1) fails.
For later reference, we recall the following result.
be a countable direct system of finitely presented right R-modules, and consider the pure exact sequence where φε n = ε n − ε n+1 u n and ε n : F n → ⊕ n∈N F n denotes the canonical morphism for every n ∈ N. Then the following statements are equivalent. (1) If F n belongs to A for all n ∈ N, then the following statement is further equivalent. (1) Every countable direct system of finitely presented modules in A has limit in A.
(2) Every countable direct system of finitely presented modules in A is B-stationary.
(3) Every direct system of finitely presented modules in A is B-stationary.
Obviously (3) implies (2). To see that (2) implies (1), let A = lim − → (F α , u β α ) β α∈I be such that I is countable and F α are finitely presented modules in A. Taking a cofinal set of I if necessary we may assume that I = N. Our hypothesis allows us to use 3.11 to conclude that A ∈ A.
In fact, M can be written as direct limit of a countable direct system as in 3.11, and for all modules B ∈ B the map Hom (2) Let B and A be as in (1). Assume that R is a right noetherian ring and B consists of modules of injective dimension at most one. Then every M ∈ A is B-stationary.
In fact, the additional assumption on B means that A is closed by submodules: Let N ≤ M ∈ A. For any B ∈ B, if we apply Hom R (−, B) to the exact sequence we obtain the exact sequence Hence, Ext 1 R (N, B) = 0. As R R is noetherian, any finitely generated submodule of M is finitely presented. Let I denote the directed set of all finitely generated submodule of M , then M = F ∈I F . If (3) Let M be a module with a perfect decomposition in the sense of [7], for example M a Σ-pure-injective module, or M a finitely generated module with perfect endomorphism ring. Let M be a class of finitely presented modules in AddM . Then every N ∈ lim − → M is Mod-R-stationary.
In fact, we can write N = lim − → F α where (F α , u β α ) β α∈I is a direct system of finitely presented modules in M. If we take a chain α 1 ≤ α 2 ≤ . . . in I, then (F αn , u αn+1 αn ) n∈N is a direct system in AddM with a totally ordered index set, so it follows from [7, 1.4] that the pure exact sequence ( * ) considered in Theorem 3.11 is split exact. In particular, Hom R (φ, B) is surjective for all modules B, hence (F αn , u αn+1 αn ) n∈N is Mod-R-stationary by 3.11. Now the claim follows from Proposition 3.10.
(4) Let B be a Σ-pure-injective module. Then every right R-module M is AddBstationary.
To see this, write M = lim − → F α where (F α , u β α ) β α∈I is a direct system of finitely presented modules. If we take a chain α 1 ≤ α 2 ≤ . . . in I and consider the direct system (F αn , u αn+1 αn ) n∈N , then for any B ′ ∈ AddB we know that Hom R (−, B ′ ) is exact on the pure exact sequence ( * ) considered in Theorem 3.11. So Hom R (φ, B ′ ) is surjective for all modules B ′ ∈ AddB, and the claim follows again by combining Theorem 3.11 and Proposition 3.10.

Dominating maps
From the characterization of Mittag-Leffler modules in [32], we know that a right module is Q-Mittag-Leffler for any left module Q if and only if it is B-stationary for any right module B. We will now investigate the relationship between the properties Q-Mittag-Leffler and B-stationary when we restrict our choice of Q and B to subclasses of R-Mod and Mod-R, respectively.
As a first step, in Theorem 4.8 we provide a characterization of when a module M is B-stationary which is independent from the direct limit presentation of M . To this end, we need the following notion which is inspired by the corresponding notion from [32].
For classes of modules Q and B in R-Mod and Mod-R, respectively, we say that v Bdominates u with respect to Q if v B-dominates u with respect to Q for any Q ∈ Q and any B ∈ B.
If Q = R-Mod, we simply say that v B-dominates u. If B = Mod-R, we say that v dominates u with respect to Q, and of course, this means that ker Finally, if Q = R-Mod and B = Mod-R, then we are in the case treated in [32, 2.1.1], and we say that v dominates u.
We note some properties of dominating maps. (1) "B-dominating with respect to Q" is translation invariant on the right. That is, if v B-dominates u with respect to Q and t : X → M is a homomorphism, then vt B-dominates ut with respect to Q. Proof.
Hence, if v B-dominates u with respect to Q we deduce that also mv B-dominates u with respect to Q.
We recall the following property of direct limits.
Let F be a finitely presented module and u ∈ Hom R (F, M ). Since Hom R (F, M ) is canonically isomorphic to lim − → Hom R (F, S γ ), there exist γ ∈ I and v : F → S γ such that u = u γ v where u γ : S γ → M denotes the canonical morphism.
To prove the converse, write M = lim − → F α where (F α , u β α ) β α∈I is a direct system of finitely presented right R-modules. By hypothesis, for each α ∈ I there exists S α ∈ S, v α : F α → S α and t α : S α → M such that the canonical map u α : , there exists β ≥ α and a commutative diagram: The next result will provide us with a tool for comparing the relative Mittag-Leffler conditions. In fact, we will see in  (1). There is a direct system of finitely presented right R-modules (F α , u β α ) β α∈I with M = lim − → (F α , u βα ) β,α∈I having the property that for any α ∈ I there exists β ≥ α such that u βα B-dominates the canonical map u α : F α → M with respect to Q.
(2). Every direct system of finitely presented right R-modules (F α , u β α ) β α∈I with M = lim − → (F α , u βα ) β,α∈I has the property that for any α ∈ I there exists β ≥ α such that u βα B-dominates the canonical map u α : F α → M with respect to Q.
(3). For any finitely presented module F (belonging to S) and any homomorphism u : F → M there exist a module S ∈ S and a homomorphism v : F → S such that u factors through v, and v B-dominates u with respect to Q.
Then u factors through v and, by Lemma 4.2 (2), v B-dominates u with respect to Q.
Similarly, to see that condition (3) restricted to modules F belonging to S implies (1), we proceed as in (3) ⇒ (2) but considering a direct system of finitely presented right R-modules Observe that the condition M ∈ lim − → S in the hypothesis of Proposition 4.4 is also necessary. This can be deduced from condition (3) by employing Lemma 4.3.
We will need the following result.
If coker(u) is finitely presented, the following statement is further equivalent.
(iii) h factors through u.
We can now interpret the property "B-dominates" in terms of H-subgroups.
This shows the claim.
For the converse implication, assume that v B-dominates u and coker(u) is finitely presented. Let h ∈ H v (B). Then ker(u ⊗ R Q) ⊆ ker(h ⊗ R Q) for any left R-module Q. By 4.5 this means that h ∈ H u (B).
Lemma 4.7. Let B be a right R-module and let Q be a left R-module. Let further (F α , u β α ) β α∈I be a direct system of finitely presented right R-modules with M = lim − → (F α , u βα ) β,α∈I . For α, β ∈ I with β ≥ α, the following statements hold true. Proof. (1) To show the only-if-part, fix γ ≥ α. As u α = u γ u γα , for any left R-module Q Therefore u βα B-dominates u γα with respect to Q. The converse implication is clear from the properties of direct limits.
(2) By (1), u βα B-dominates u α if and only if u βα B-dominates u γα for any γ ≥ α. As coker u γα is finitely presented, we know from Proposition 4.6 that the latter is equivalent to H u βα (B) ⊆ H uγα (B) for any γ ≥ α. But this means H u βα (B) = γ≥α H uγα (B) by Lemma 3.6.
From Lemma 4.7 and Definition 3.7, we immediately obtain the announced characterization of B-stationary modules. (2). There is a direct system of finitely presented right R-modules (F α , u β α ) β α∈I with M = lim − → (F α , u βα ) β,α∈I having the property that for any α ∈ I there exists β ≥ α such that u βα B-dominates the canonical map u α : F α → M .
(3). For any finitely presented module F (belonging to S) and any homomorphism u : F → M there exist a module S ∈ S and a homomorphism v : F → S such that u factors through v, and v B-dominates u.
We close this section with some closure properties of the class B in the definition of "B-dominating". Let us first prove the following preliminary result. (1) v B-dominates u.
(2) For any finitely presented left R-module Q (3) For any (finitely presented) left R-module Q Proof. We follow the idea in the proof of [32, Proposition 2.
Recall that it will stay a push-out diagram when we apply the functor − ⊗ R Q for any left module Q. Hence we have the exact sequence This shows that, for any left module Q, ker(u ⊗ R Q) ⊆ ker(h ⊗ R Q) if and only if ker(u ′ ⊗ R Q) = 0, that is, if and only if u ′ is a pure monomorphism. Since a morphism is a pure monomorphism if and only if it is a monomorphism when tensoring by finitely presented modules, we deduce that (1) and (2) are equivalent statements.
To prove that (2) and (3) are equivalent, note that h∈Hv (B) ker(h ⊗ R Q) is the kernel of the product map induced by all homomorphisms h ⊗ R Q with h ∈ H v (B). When Q is finitely presented, the natural morphism ρ : and the statement is verified. To obtain the statement for arbitrary Q, proceed as in the proof of (1) ⇔ (2).  Proof. (i) Let B ∈ B, and assume that the inclusion ǫ : C → B is a pure monomorphism. If h ∈ H v (C), then ǫh ∈ H v (B), and ker(h ⊗ Q) = ker(ǫh ⊗ Q) contains ker(u ⊗ Q).
(ii) By (i) it is enough to consider modules of the form i∈I B i where {B i } i∈I is a family of modules in B. Let Q be a finitely presented module. As the canonical morphism ρ : Then the claim follows from 4.9.

Q-Mittag-Leffler modules revisited
As a next step towards establishing a relationship between Q-Mittag-Leffler and Bstationary modules, we provide a characterization of Q-Mittag-Leffler modules in terms of dominating maps. It is inspired to work of Azumaya and Facchini [11,Theorem 6]. (1) M is Q-Mittag-Leffler.
(2) Every direct system of finitely presented right R-modules (F α , u β α ) β α∈I with M = lim − → (F α , u βα ) β,α∈I has the property that for any α ∈ I there exists β ≥ α such that u βα dominates the canonical map u α : F α → M with respect to Q. Proof.
(2) ⇒ (1). Fix a direct system of finitely presented right R-modules (F α , u β α ) β α∈I with M = lim − → (F α , u βα ) β,α∈I . Let {Q k } k∈K be a family of modules of Q, and let x ∈ kerρ where ρ : . Let β ≥ α be such that u βα dominates the canonical map u α with respect to Q. The commutativity of the diagram By Proposition 4.4, we already know that (2) and (3) are equivalent statements.
Set S 0 = 0 and let v 0 be the zero map. Let n ≥ 0 and assume as inductive hypothesis that S m and v m have been constructed for any m ≤ n. Let u : S n ⊕ R → M be defined as u(g, r) = v n (g) + x n+1 r for any (g, r) ∈ S n ⊕ R. By (3), there exist S n+1 ∈ S, v : S n ⊕ R → S n+1 , and v n+1 : S n+1 → M such that u = v n+1 v and ker(v ⊗ Q) = ker(u ⊗ Q) for all Q ∈ Q. Let ε : S n → S n ⊕ R denote the canonical inclusion and set f n = v • ε. Then v n = uε = v n+1 (v ε) = v n+1 f n . This completes the induction step. Note moreover that also ker(v n ⊗ Q) = ker(f n ⊗ Q) for all Q ∈ Q.
Set N = lim − → S n and v = lim − → v n . Then N is countably presented. As for any Q ∈ Q, ker(v ⊗ Q) = lim − → ker(v n ⊗ Q) and ker(v n ⊗ Q) = ker(f n ⊗ Q), we deduce that v ⊗ Q is injective.
To show that N is Q-Mittag-Leffler we verify that N satisfies (2), as we already know that (1) and (2) are equivalent. By Proposition 4.4 it is enough to check the condition for the direct system (S n , f n ) n≥0 and the canonical maps u n : S n → M .
Notice that vu n = v n . Therefore, for any Q ∈ Q from which we conclude that f n dominates u n with respect to Q.
we then have that ρ ′ is injective because N is Q-Mittag-Leffler, and k∈K (v⊗Q k ) is injective by assumption on v. This shows that x = 0. Assume now that R ∈ Q. To show (3) ⇒ (5), we proceed as in the proof of (2) ⇒ (3) in [11,Theorem 6]. We take an epimorphism p : F → M 0 from a finitely generated free module F , set u = ǫ p, and construct v as in condition (3). Note that keru = kerv since Q contains R. We thus obtain w : M 0 → S and t : S → M such that v = w p and ǫ = t w. To show ker(ǫ ⊗ R Q) = ker(w ⊗ R Q) for all Q ∈ Q it is enough to verify the inclusion ⊆. So, take a left R-module Q ∈ Q and y ∈ ker(ǫ ⊗ Q). Note that y = (p ⊗ Q)(x) for some x ∈ F ⊗ Q.
Condition (4) in Theorem 5.1 gives the following characterization of Q-Mittag-Leffler modules.
Corollary 5.2. Let Q be a class of left R-modules, and let S be a class of finitely presented right R-modules. For a fixed right R-module M ∈ lim − → S denote by C the class of its countably generated submodules N such that N is Q-Mittag-Leffler and the inclusion N ⊆ M remains injective when tensoring with any module Q ∈ Q.
Then M is Q-Mittag-Leffler if and only if M is a directed union of modules in C.
Moreover, if R ∈ Q, the modules in C can be taken countably presented and in lim − → S.
Proof. For the only-if implication, we follow the notation of Theorem 5.1 (4). We only have to prove that v(N ) is a Q-Mittag-Leffler module and that the inclusion ε : v(N ) → M remains injective when tensoring with any module Q ∈ Q. Let {Q k } k∈K be a family of modules in Q. Consider the commutative diagram, and (εv ⊗ k∈K Q k )(y) = 0. Since ρ (εv ⊗ k∈K Q k ) = ( k∈K (v⊗Q k )) ρ 1 is an injective map, we infer y = 0, so x = 0. This shows that ε⊗ k∈K Q k is injective. Then also k∈K (ε ⊗ Q k ) ρ 2 = ρ (ε ⊗ k∈K Q k ) is injective, and so is ρ 2 . To prove the converse implication proceed as in the proof of (4) ⇒ (1) of Theorem 5.1. The statement for the case when R ∈ Q is clear because then the map v in Theorem 5.1 is injective, so N is isomorphic to v(N ).  Proof. Fix α ∈ I, and denote by u α : F α → M the canonical map. As M is Bstationary, we infer from Proposition 4.8 that there exists β ≥ α such that u βα B-dominates the canonical map u α , that is for all left R-modules Q. Our assumption implies that ker(u α ⊗ R Q) ⊆ ker(u βα ⊗ R Q) for all Q ∈ Q, so Theorem 5.1 gives the desired conclusion.
Before we continue our discussion of the general case, let us notice the following projectivity criteria for countably generated flat modules that improves Proof. To see that a countably generated projective module is R-stationary use for example Theorem 3.11.
Assume that M is countably generated, flat and R-stationary. Then M is also R (N)stationary by 3.9. Let (F α , u βα ) α β∈I be a direct system of finitely generated free modules such that M = lim − → F α . Notice that for each β ∈ I we have a split monomorphism t β : F β → R (N) , hence t β ⊗ Q is a split monomorphism for any left R-module Q. This implies that the criterion of Lemma 5.4 is fulfilled for any left R-module, hence M is a Mittag-Leffler module. Now we can conclude either by using [32, 2.2.2] or arguing that then M is R-Mittag-Leffler, hence countably presented by Corollary 5.5, and then use [3, Propostion 2.5]. We thus recover a characterization due to Goodearl of the modules that are Mittag-Leffler with respect to the class of flat modules [26,Theorem 1]. In particular, if R is right noetherian, then (5 ′ ) is trivially satisfied, and so any right R-module is Mittag-Leffler with respect to the class of flat modules (cf. [26]).

Relating B-stationary and Q-Mittag-Leffler modules
Throughout this section, we fix a right R-module M together with a direct system of finitely presented modules (F α , u β α ) β α∈I such that M = lim − → F α .
Lemma 6.1. Let B be a class of right R-modules closed under direct sums, and let Q be a class of left R-modules. Assume that M is B-stationary. If for any pair α, β ∈ I with β ≥ α and for any Q ∈ Q there exists B = B β α (Q) ∈ B such that then M is a Q-Mittag-Leffler module.
Proof. Let Q ′ = {Q k } k∈K be any family of modules in Q. To prove the statement, we verify that Q ′ satisfies the assumption of Lemma 5.4 for By hypothesis and by the construction of B, if we fix a pair α, β ∈ I with β ≥ α ∈ I, then for all Q ∈ Q ′ As M is B-stationary, we conclude from Lemma 5.4 that M is Q ′ -Mittag-Leffler. Proposition 6.2. Let B be a class of right R-modules closed under direct sums, and let Q be a class of left R-modules. Assume that M is B-stationary. 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.
Proof. We verify the condition in Lemma 6.1. Let β ≥ α in I and Q ∈ Q. By hypothesis, there is f β : and the reverse inclusion is always true.
We have seen several conditions implying that a B-stationary module is Q-Mittag-Leffler. Let us now discuss the reverse implication. We will need the following notion. (1) v B-dominates u.
Statements (1) and (2) are equivalent, and statement (3) implies (1) and (2). Moreover, if there is a class of left R-modules Q such that the character module B * ∈ Q and f ⊗ R Q is a monomorphism for all Q ∈ Q, then all three statements are equivalent to (4) v dominates u with respect to Q.
Proposition 6.5. Let B be a class of right R-modules. Assume that for every α ∈ I there exists a B-preenvelope f α : Proof. For any α ∈ I, denote by u α : F α → M the canonical map. By Theorem 4.8 we must show that there exists β ≥ α such that u βα B-dominates u α , which means ker(u α ⊗ R B * β ) ⊆ ker(u βα ⊗ R B * β ) by Lemma 6.4. So, it is enough to find β ≥ α such that To this end, we take a generating set (x k ) k∈K of ker(u α ⊗ R Q) and consider the diagram for all k ∈ K, and we conclude ker(u α ⊗ R Q) ⊆ ker(u βα ⊗ R Q).
The previous observations are subsumed in the following result. Theorem 6.6. Let B be a class of right R-modules closed under direct sums, and let Q be a class of left R-modules. Assume that for every finitely presented module F there exists a B-preenvelope f : F → B such that the character module B * ∈ Q and f ⊗ R Q is a monomorphism for all Q ∈ Q. Then the following statements are equivalent for a right R-module M .
Proof.   Hom R (f, B) is an epimorphism, thus applying Hom Z (−, Q/Z) and using Hom-⊗-adjointness, we see that f ⊗ R Q is a monomorphism for all Q ∈ Q. So the claim follows immediately from 6.6.
Further applications of Theorem 6.6 are given in Section 9.

Baer modules
Throughout this section, let R be a commutative domain. A module M is said to be a Baer module if Ext 1 R (M, T ) = 0 for any torsion R-module T . Kaplansky in 1962 proposed the question whether the only Baer modules are the projective modules. He was inspired by the analogous question raised by Baer for the case of abelian groups, which was solved by Griffith in 1968.
In the general case of domains, a positive answer to Kaplansky's question was recently given by the authors in joint work with S. Bazzoni [3]. The proof uses an important result of Eklof, Fuchs and Shelah from 1990 which reduces the problem to showing that countably presented Baer modules are projective (cf. [24,Theorem 8.22]).
Aim of this section is to give a new proof for the fact that every countably generated Baer module is projective, which uses our previous results. In fact, we are going to see that a countably generated Baer module is Mittag-Leffler. Then the result follows because countably generated flat Mittag-Leffler modules are projective (cf. [ So, let us consider a countably presented Baer module M . By Kaplansky's work, we know that M is flat of projective dimension at most one. So M can be written as a direct system of the form where, for each n ≥ 1, F n is a finitely generated free R-module. As the class of torsion modules is closed under direct sums, it follows from [12], see 3.11, that M is a Baer module if and only if for any torsion module T the inverse system (Hom R (F n , T ), Hom R (f n , T )) satisfies the Mittag-Leffler condition, in other words, iff M is T -stationary. (i) Let Q be a finitely generated torsion module. For any n ≥ 0 there exist a torsion module T and a homomorphism h : R n → T such that h ⊗ Q is injective. (ii) Let Q be a finitely generated torsion-free R module. For any n ≥ 0 there exists a torsion module T such that Proof. (i). If n = 0, the claim is trivial (we assume that 0 is torsion). Fix n ≥ 1. As Q is finitely generated and torsion I = ann R (Q) is a nonzero ideal of R. So that, T = (R/I) n is a torsion module. Note that R/I ⊗ Q ∼ = Q/IQ = Q. Hence the canonical projection R n → T satisfies the desired properties.
(ii). First we show that for any 0 = x ∈ R n ⊗ Q there exist a torsion module T x and h : R n → T x such that (h ⊗ Q)(x) = 0. Let K denote the field of quotients of R. As Q is torsion-free and finitely generated, it can be identified with a finitely generated submodule of K m for some m. Moreover, as Q is finitely generated, multiplying by a suitable nonzero element of R, we can assume that Q ≤ R m .
The claim is trivial for n = 0. Fix n ≥ 1. As 0 = x ∈ R n ⊗ Q = Q n there is i ∈ {1, . . . , n} such that the i-th component of x is nonzero. Let π i : R n → R denote the projection on the i-th component. As (π i ⊗ Q)(x) = 0, we only need to prove the statement for n = 1.
. . , m} be such that x j = 0. As R is a domain, there exists 0 = t ∈ R such that tR x j R. Hence x ∈ tQ ⊆ tR m . That is, if Theorem 7.2. If R is a commutative domain then any countably presented Baer module over R is Mittag-Leffler. Therefore any Baer module is projective.
Proof. Let M R be a countably presented Baer module. Then M R is flat, hence a direct limit of finitely generated free modules.
Denote by T and F the classes of torsion and torsion-free modules, respectively. By Theorem 3.11, M R is T -stationary. Since T is closed under direct sums, the previous Lemma 7.1(i) together with 6.2 implies that M is Q-Mittag-Leffler where Q is the class of all finitely generated modules from T . By Theorem 1.3, it follows that M is T -Mittag-Leffler.
Next, we show that M is also F -Mittag-Leffler. Again by Theorem 1.3, it is enough to show that M is Mittag-Leffler with respect to the class of finitely generated torsion-free modules. So, let Q be a finitely generated torsion-free module, and let u ∈ Hom R (F, F ′ ) be a morphism between finitely generated free modules F, F ′ . By Lemma 7.1(ii) there exists a torsion module T such that h∈Hom(F ′ ,T ) ker(h ⊗ Q) = 0. So, if x ∈ F ⊗ R Q and y = (u ⊗ R Q)(x) = 0, then there must exist h ′ ∈ Hom R (F ′ , T ) such that (h ′ ⊗ R Q)(y) = 0, which means (h ′ u ⊗ R Q)(x) = 0 and shows that x ∈ h∈Hu(T ) ker(h ⊗ R Q). Thus we deduce that ker(u ⊗ R Q) = h∈Hu(T ) ker(h ⊗ R Q). Our claim then follows from Lemma 6.1.
Since M is flat, we now conclude from Corollary 1.5 that M is Mittag-Leffler and thus projective.

Matrix subgroups
In [32] Raynaud and Gruson also studied modules satisfying a stronger condition, which they called strict Mittag-Leffler modules. In this section, we investigate the relative version of this condition and interpret it in terms of matrix subgroups. Hereby we establish a relationship with work of W. Zimmermann [36].
We start out with a stronger version of Proposition 4.4.
Proposition 8.1. Let B be a right R-module, and let S be a class of finitely presented right R-modules. For a right R-module M ∈ lim − → S, the following statements are equivalent.
(2). Every direct system of finitely presented right R-modules (F α , u β α ) β α∈I with M = lim − → (F α , u βα ) β,α∈I has the property that for any α ∈ I there exists β ≥ α such that the canonical map u α : F α → M satisfies H uα (B) = H u βα (B). In view of the characterization of B-stationary modules given in Theorem 4.8, we introduce the following terminology. If B is a class of right R-modules, then we say that M is strict B-stationary if it is strict B-stationary for every B ∈ B. Proposition 8.4. Let {B j } j∈J be a family of right R-modules. Let (F α , u β α ) β α∈I be a direct system of finitely presented right R-modules and M = lim − → F α . Then the following statements are equivalent.
(1) M is strict j∈J B j -stationary.
We now recall some notions from [36].
Definition 8.6. Given two right R-modules A, B, an integer n ∈ N, and an element a = (a 1 , . . . , a n ) ∈ A n , we consider the EndB-linear map  N, and a = (a 1 , . . . , a n ) ∈ A n . (1) If a 1 , . . . , a n is a generating set of A, then ε a is a monomorphism. Proof. Is left to the reader.
We will need some further terminology from [36]. pair (A, a) consisting of a right R-module A and an element a = (a 1 , . . . , a n ) ∈ A n will be called an n-pointed module. A morphism of n-pointed modules Consider now a direct system of right R-modules (F α , u β α ) β α∈I with direct limit M = lim − → (F α , u βα ) β,α∈I and canonical maps u α : F α → M . If for every α ∈ I the elements x α ∈ F α n are chosen in such a way that the u βα : (F α , x α ) → (F β , x β ) are morphisms of n-pointed modules, then ((F α , x α ), u βα ) β,α∈I is called a direct system of n-pointed modules. Setting m = u α (x α ) for some α ∈ I, we have that also the u α : (F α , x α ) → (M, m) are morphisms of n-pointed modules. We then write (M, m) = lim − → (F α , x α ).
We now show that the strict B-stationary modules are precisely the modules studied by Zimmermann in [36, 3.2]. Like the Mittag-Leffler modules, they can be characterized in terms of the injectivity of a natural transformation. Let S B R be and S-R-bimodule, and let S V be a left S-module. For any right R-module M R there is a natural transformation Notice that when B = V and S B R is faithfully balanced, then ν is induced by the evaluation map of M inside its bidual. If M R is finitely presented and S V is injective then ν is an isomorphism (cf. [19, Theorem 3.2.11]). The case that ν is a monomorphism for all injective modules S V was studied by Zimmermann in [36, 3.2]. We are going to see below that this happens precisely when M is strict B-stationary. So, like Q-Mittag-Leffler modules, strict B-stationary modules can be characterized in terms of the injectivity of a natural transformation, in this case the injectivity of ν. Let us first discuss how the injectivity of ν behaves under direct sums.
If M = ⊕ i∈I M i then, for each i ∈ I, the canonical inclusion M i → M induces an inclusion

This family of inclusions induces an injective map
given by the rule This allows us to deduce that (1) M is strict B-stationary.
Here are some consequences of the previous theorem. Proposition 8.12. Let S be a class of right R modules that is strict B-stationary with respect to a class B ⊆ S ⊥ . Then any module isomorphic to direct summand of an S ∪Add Rfiltered module is strict B-stationary.
Proof. As projective modules are strict ModR-stationary and S ⊥ = (S ∪ Add R) ⊥ , we can assume that S contains Add R . By Corollary 8.11, the class of strict B-stationary modules is closed by direct summands. So, we only need to prove the statement for Sfiltered modules. Also by Corollary 8.11, we know that arbitrary direct sums of modules in S are strict B-stationary.
Let M be an S-filtered right R-module. Let τ be an ordinal such that there exists an S-filtration (M α ) α≤τ of M . Observe that for any β ≤ α ≤ τ , M α and M α /M β are S-filtered modules, so they belong to ⊥ B by [25, 3.1.2]. For the rest of the proof we fix B ∈ B, a ring S such that S B R is a bimodule, and an injective left S-module V . We shall show that M is strict B-stationary proving by induction that for any α ≤ τ the canonical map is injective.
As M 0 = 0 the claim is true for α = 0. If α < τ then the exact sequence and the fact that B ⊆ S ⊥ yields a commutative diagram with exact rows The natural map ν is injective because M α+1 /M α is strict B-stationary. So, if ν α is injective, then ν α+1 is also injective.
Next, we investigate the relationship between relative Mittag-Leffler modules and strict stationary modules. It was shown by Azumaya [10,Proposition 8] that a module M is strict Mittag-Leffler if and only if every pure-epimorphism X → M , X ∈ Mod-R, is locally split. We will now see that also the dual property plays an important role in this context. According to [38], we will say that a right R-module B is locally pure-injective if every pure-monomorphism B → X, X ∈ Mod-R, is locally split.
Moreover, in the following, for a right R-module B, we will indicate by B • a left Rmodule which is obtained from B by some duality, that is, by taking a ring S such that S B R is a bimodule together with an injective cogenerator S V of S-Mod, and setting R B • = Hom S (B, V ). For example, B • can be the character module B * of B. But it can also be the local dual B + of B, which is obtained as above by choosing S = End R B. For a left R-module C, the notation C • is used correspondingly.  Proof.
(2) The first part of the statement is shown by Zimmermann [36, 3.3(2)(a)]. For the converse, assume that B is locally pure-injective. Let R B • = Hom S (B, V ) where S is a ring such that S B R is a bimodule and S V is an injective cogenerator of S-Mod. Consider a ring T such that S V T is a bimodule, let U T be an injective cogenerator of Mod-T , and assume w.l.o.g. that V T ⊆ U T . Then R B • T is also a bimodule, and we can consider B • • = Hom T (B • , U ). By (1), M is strict B • • -stationary. Furthermore, the evaluation map B → B • • is a pure monomorphism (see e.g. [39, 1.2(4)]), hence locally split. By Corollary 8.5(i) it follows that M is strict B-stationary.
Example 8.14. Let B be a locally pure-injective right R-module with the property that all finite matrix subgroups of B are finitely generated over the endomorphism ring of B. Then a right R-module M is strict B-stationary if and only if it is B-stationary. In particular, this applies to the case when B is a pure-projective right R-module over a left pure-semisimple ring R.
Proof. The statement follows by combining Example 6.7 with Proposition 8.13 (2). When R is a left pure-semisimple ring, all finitely presented right R-modules are endofinite [29]. Hence every pure-projective right R-module B is locally pure-injective by [38, 2.4], and endonoetherian by [40]. So, the assumptions are satisfied in this case.
Restricting to local duals, we can employ recent work of Dung and Garcia [17] to obtain a criterion for endofiniteness of finitely presented modules.
Proposition 8. 15. Let R C be a finitely presented left R-module. The following statements are equivalent.
Proof. (1)⇒ (2): C is endofinite if and only if it satisfies the descending and the ascending chain condition on finite matrix subgroups. Hence C is Σ-pure-injective, and every right R-module is C-Mittag-Leffler, see Example 1.6(3).
(2)⇒ (3): By Proposition 8.13 the module C + is strict C + -stationary. By condition (3) in Theorem 8.10 for n = 1, it follows that all matrix subgroups of C + of the form H C + , m (C + ) with m ∈ C + are finite matrix subgroups, and of course, the matrix subgroups of such form are precisely the cyclic EndC + -submodules of C + .
(3)⇒ (1): Since C is Σ-pure-injective, the module C + satisfies the ascending chain condition on finite matrix subgroups, see [40,Proposition 3]. Furthermore, every finitely generated EndC + -submodule of C + is a finite sum of cyclic submodules, hence a finite matrix subgroup, because the class of finite matrix subgroups is closed under finite sums, see e.g. [39, 2.5]. So, we conclude that C + satisfies the ascending chain condition on finitely generated EndC + -submodules, in other words, C + is endonoetherian, see also [37]. Now the claim follows from [17, 4.2], where it is shown that a finitely presented module is endofinite provided its local dual is endonoetherian.
Using [16, 4.1], we obtain the following observation. Before turning in more detail to pure-semisimple rings, let us apply our results to the following setting.

Cotorsion pairs
In this section, we shall see that the theory of relative Mittag-Leffler modules and (strict) stationary modules fits very well into the theory of cotorsion pairs. (3) We will say that a cotorsion pair (M, L) is of finite type provided it is generated by a set of modules S ⊆ mod-R. Note that we can always assume S = M ∩ mod-R.
(4) Dually, if S is a set of right R-modules, we obtain a cotorsion pair (M, L) by setting M = ⊥ S and L = ( ⊥ S) ⊥ . It is called the cotorsion pair cogenerated by S.
For more information on cotorsion pairs, we refer to [25].
Certain classes of complete cotorsion pairs provide a good setting for relative stationarity and Mittag-Leffler properties. As a first approach we give the following result.  Proof. The hypotheses in (1) imply that any right R-module X fits into an exact sequence 0 → X → L → M → 0 where L ∈ L and M ∈ M. Hence the statement follows from Proposition 6.2.
To prove (2) observe first that a countably presented module in M is strict L-stationary by Example 3.13(1) and Remark 8.3 (3). Now if M ∈ M then, by [25,Corollary 3.2.4], M is a direct summand of a module N filtered by countably presented modules in M. By the observation above, N is filtered by strict L-stationary modules. Then M is strict L-stationary by Proposition 8.12. As the cotorsion pair is complete by 9.1(2), we infer from (1) that M is also C-Mittag-Leffler.
(  (2). For the converse implication, let M be a countably generated module in lim − → S which is L-stationary. Statement (1) implies that M is C-Mittag-Leffler. As R ∈ C, we deduce from Corollary 5.3 that M is countably presented. Therefore, and because L is closed under direct sums, we can apply Theorem 3.11 to conclude that Ext 1 R (M, L) = 0 for any L ∈ L. Thus M ∈ M.
¿From [35, 1.9] we immediately obtain the following consequence. In the following results, we use again the notation B • to indicate a module obtained from B by some duality, like the character module, or the local dual of B. For a class of modules S, we write S • in order to indicate a class consisting of modules that are obtained by some duality from the modules of S. Note that we are not assuming a functorial relationship between S and S • .   (2), and also statements (3) -(5). For statement (6), we assume that the cotorsion pair (C, D) is of finite type. Then X ∈ D iff Ext(C, X) = 0 for all C ∈ C ∩ mod R, which is equivalent to Tor(X • , C) = 0 for all C ∈ C ∩ mod R. But since C ⊆ lim − → (C ∩ mod R) by [8, 2.3], and Tor commutes with direct limits, the latter means that X • ∈ ⊺ C = lim − → S. Statement (7) is [8, 2.4]. As an application of our previous results, we obtain Theorem 9.5. Let (M, L) be a cotorsion pair of finite type in Mod-R. Set S = M∩mod-R and C = M ⊺ , and denote by L ′ the class of all locally pure-injective modules from L. Then the following statements are equivalent for a right R-module M .
a finite matrix subgroup.
(2) We know from [40,Proposition 3] that T • is Σ-pure-injective if and only if T satisfies the ascending chain condition on finite matrix subgroups. By (1), the latter means that S T is noetherian.
If (M, L) is a cotorsion pair of finite type, then it follows from Theorem 9.5 that M is contained in the class of strict L-stationary modules. If S = M∩mod-R, then the countably generated modules in lim − → S that are strict L-stationary are precisely the countably generated modules in M, and they also coincide with the countably generated modules in lim − → S that are L-stationary. Raynaud and Gruson in [32, p. 76] provide examples showing that, in general, M is properly contained in the class of modules in lim − → S that are strict L-stationary, and the latter class is also properly contained in the class of L-stationary modules. We explain these examples for completeness' sake. First we prove the following Lemma 9.10. Let R be a ring, and let F 1 and F 2 be flat right R-modules such that there exists an exact sequence Proof. Statement (i) follows from Examples 1.6(4).
(ii) Assume that F 1 is strict R-stationary. Let S be the class of finitely generated free modules. By Proposition 8.1 and since F 1 ∈ lim − → S, there exist n > 0 and a homomorphism v : R → R n , such that u = tv for some t : R n → F 1 and H u (R) = H v (R). Since u is a pure monomorphism so is v, but being a pure monomorphism between finitely generated projective modules v splits. Therefore the identity map belongs to H v (R) = H u (R), thus u also splits.
The easiest instance of tilting cotorsion pair is the one generated by the class S of finitely generated free modules. Then S ⊥ = Mod-R, and ⊥ (S ⊥ ) = P is the class of all projective modules. The ring R is a tilting module that generates the tilting cotorsion pair (P, Mod-R). Note that lim − → S = F is the class of flat modules. The relative Mittag-Leffler, strict stationary and stationary modules associated to this cotorsion pair in Corollary 9.8 are the Mittag-Leffler, strict Mittag-Leffler and (Mod-R)-stationary modules, respectively. If F 1 is a flat Mittag-Leffler module and Ext 1 R (F 1 , R) = 0, then there is a non split exact sequence In view of Lemma 9.10, F 2 is a Mittag-Leffler module that is not strict Mittag-Leffler. An example for this situation is the following: Example 9.11. For any set I, the abelian group Z I is a flat strict Mittag-Leffler abelian group. If I is infinite, there exist nonsplit extensions of Z I by Z, hence there are flat Mittag-Leffler abelian groups that are not strict Mittag-Leffler.
Proof. Of course, Z I is flat. Since any finitely generated submodule of Z I is contained in a finitely generated direct summand of Z I [22, Proof of Theorem 19.2], we deduce from Proposition 8.1 that Z I is strict Mittag-Leffler.
When I is infinite, Z I is not a Whitehead group, that is, Ext 1 Z (Z I , Z) = 0 [23, Proposition 99.2]. Then the claim follows by the remarks above. 10. The pure-semisimplicity conjecture Throughout this section, we assume that R is a twosided artinian, hereditary, indecomposable, left pure semisimple ring. It is well known that then every indecomposable finitely generated non-projective left R-module is end-term of an almost split sequence in R-Mod consisting of finitely presented modules, and every indecomposable finitely generated noninjective right R-module is the first term of an almost split sequence in Mod-R consisting of finitely presented modules.
We adopt the notation A = τ C and C = τ − A if 0 −→ A −→ B −→ C −→ 0 is an almost split sequence, and define inductively τ n resp. τ −n . We know from [9] that there is a preprojective component p in Mod-R, that is, a class of finitely generated indecomposable right R-modules satisfying the following conditions.
(1) For any X ∈ p there are a left almost split morphism X → Z and a right almost split morphism Y → X in Mod-R with Z, Y being finitely generated. (2) If X → Y is an irreducible map with one of the modules lying in p, then both modules are in p. (3) The Auslander-Reiten-quiver of p is connected and has no oriented cycles. (4) For every Z ∈ p there is m ≥ 0 such that τ m Z is projective.
Similarly, there is a preinjective component in R-Mod, i. e. a class of finitely generated indecomposable left R-modules with the dual properties. Moreover, the two components are related by the local duality, that is, there is a bijection q → p, R A → A + R . The modules in addp are called preprojective, the modules in addq are called preinjective.
In [2], the cotorsion pair (M, L) in Mod-R generated by p and the cotorsion pair (C, D) in R-Mod cogenerated by q are investigated. In particular, it is shown that there is a finitely generated product-complete tilting and cotilting left R-module W such that C = CogenW = ⊥ W and D = GenW = W ⊥ . Note that C = M ⊺ = p ⊺ by Lemma 9.4(2). Moreover, (M, L) is a tilting cotorsion pair in Mod-R with corresponding cotilting cotorsion pair (C, D) in R-Mod. But (C, D) is also a tilting cotorsion pair in R-Mod, and the corresponding cotilting cotorsion pair in Mod-R is (lim − → addp, E), see [2, 5.2 and 5.4]. We now apply our previous results to this setup, specializing to the case where B • denotes the local dual of a module B. Let us fix a tilting module T such that T ⊥ = L.
Proposition 10.1. The following statements hold true.
(1) T is noetherian over its endomorphism ring.
(2) All right R-modules in M are strict T -stationary (and hence W -Mittag-Leffler).
Proof. (1) Any left R-module is pure-injective, thus Σ-pure-injective. In particular, T + is Σ-pure-injective. By Corollary 9.9 we conclude that T is noetherian over its endomorphism ring.
As shown in [2], the validity of the Pure-Semisimplicity Conjecture is related to the question whether W is endofinite. We obtain the following criteria for endofiniteness of W . (1) W is endofinite.
(3) Every (countable) direct limit of preprojective right R-modules is W -Mittag-Leffler. (4) Every (countable) direct system of preprojective right R-modules is T -stationary. (5) Every (countable) direct system of preprojective right R-modules has limit in M. (6) If A is direct limit of a (countable) direct system of preprojective right R-modules, and L is a locally pure-injective module from L, then Ext 1 R (A, L) = 0.
Proof. (1)⇒(5) follows from [2, 5.6], which asserts that W is endofinite if and only if the class M is closed under direct limits.  (6). For the converse implication, observe first that L is a definable class, hence it is closed under pure-injective envelopes. So, every module L ∈ L is isomorphic to a pure submodule of a (locally) pure-injective module in L. Note further that the class L ′ of all locally pure-injective modules from L is closed under direct sums, because this is true for the tilting class L and for the class of locally pure-injective modules, see [38, 2.4]. Consider now a module A which is direct limit of a countable direct system of preprojective right R-modules. Then A is countably presented, and it follows from (6) and [12, 2.5] that A ⊥ contains all pure submodules of modules in L ′ . We conclude that Ext 1 R (A, L) = 0 for all L ∈ L, which proves A ∈ M.
Remark 10.3. It is well known that every cotilting class is a torsion-free class. So, let us consider the torsion pair defined by the cotilting class lim − → addp, denoting by t the corresponding torsion radical. Assume the following condition holds true (cf. [31, 4.1]): If N is a finitely generated submodule of W + , then t(W + /N ) is finitely generated.
Then it follows that W is endofinite. In fact, since W + ∈ lim − → addp, all its finitely generated submodules are in M. Moreover, for every finitely generated submodule N of W + there is a finitely generated submodule N ′ ⊆ W + for which N ⊆ N ′ and W + /N ′ ∈ lim − → addp. To see this, choose N ′ such that N ′ /N = t(W + /N ) and use that N ′ /N is finitely generated. So, we conclude that W + is a directed union of finitely generated submodules N ′ that belong to M and satisfy W + /N ′ ∈ lim − → addp. But then W + is W -Mittag-Leffler by Corollary 9.6, which means that W is endofinite by Proposition 10.2.
We close with a criterion for R being of finite representation type.
Proposition 10.4. Let P be the direct sum of a set of representatives of the isomorphism classes of the modules in p. Then the following statements are equivalent.
Proof. It is clear that (1) implies (2). Indeed, R is of finite representation type if and only if all right (and left) R-modules are Mittag-Leffler, hence stationary with respect to any module [11].
Assume (2) holds. To prove that R has finite representation type, it suffices to show that p is finite, see [2, 3.5]. Assume on the contrary that p is infinite. By applying [40,Theorem 9] to p, we know that there are an infinite family (P i ) i∈N of pairwise non-isomorphic modules in p and a sequence of homomorphisms (f i : P i → P i+1 ) i∈N such that f i . . . f 0 = 0 for any i ∈ N.
is (strict) p-stationary, see Corollary 8.11 (3). But by our assumption it is even P -stationary, and since AddP = Addp, we infer from Corollary 3.9(3) that it is Addp-stationary. By Corollary 3.9(2) it follows that for any n ∈ N there exists m ≥ n such that H fm···fn (P i ) = l≥n H f l ···fn (P i ) for all i ∈ N. On the other hand, as our modules are preprojective, for any i ∈ N there exists l i such that Hom R (P l , P i ) = 0 for any l > l i , hence H f l ···fn (P i ) = 0 for any l ≥ l i . So, we deduce that H fm···fn (P i ) = 0 for all i ∈ N. In particular, we have f m · · · f n = Id Pm+1 f m · · · f n ∈ H fm···fn (P m+1 ) = 0, which contradicts the choice of the sequence (f i ) i∈N . Thus we conclude that p is finite.