APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS

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1 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ Abstract. Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [27], [20], [26]. If R is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [22], and it does not provide for approximations when R has cardinality ℵ 0, [13]. We remove the cardinality restriction on R in the latter result, and we prove that these results are particular instances (for T = R) of much more general facts concerning tilting modules. For example, for every tilting module T, the class of all locally T -free modules is deconstructible, if and only if every pure submodule of a direct sum of copies of T is a direct summand. We also prove an extension of the Countable Telescope Conjecture [29]: a cotorsion pair (A, B) is of countable type whenever the class B is closed under direct limits. Further, we provide several conditions characterizing closure under direct limits of the class A. The proofs combine Mittag-Leffler conditions with tilting theory and set-theoretic homological algebra, and trace the facts to their ultimate, countable, origins in the properties of Bass modules. 1. Introduction The Mittag-Leffler condition for countable inverse systems was introduced by Grothendieck as a sufficient condition for the exactness of the inverse limit functor, [20]. Flat Mittag Leffler modules then played an important role in the proof of the locality of the notion of a vector bundle in [27, Seconde partie]. The renewed interest in Mittag-Leffler conditions stems from their role in the study of roots of the Ext functor, see [3], [4], [12], and [29], as well as their connection to generalized vector bundles [15], [18]. In [22], it was shown that flat-mittag Leffler modules coincide with the ℵ 1 -projective modules, studied already by Shelah et al. by means of set-theoretic homological algebra, [17]. A connection to model theory was discovered in [28]. The many facets of the notion of a Mittag-Leffler module, as well as its many applications, make it a key notion of contemporary algebra. Though locally projective, flat Mittag-Leffler modules have a very complex global structure in the case when R is not right perfect. Similarly to the classes P 0 and F 0 of all projective and flat modules, the class FM of all flat Mittag-Leffler modules is closed under transfinite extensions. However, unlike P 0 and F 0, it is not possible to obtain FM by transfinite extensions from any set of its elements. In other words, FM is not deconstructible, [22]. Moreover, unlike P 0 and F 0, the class FM does not provide for approximations when R is countable, [13]. Date: May 17, Key words and phrases. Mittag-Leffler conditions, approximations of modules, tilting module, cotorsion pair, pure-injective module, deconstructible class, locally T -free modules, Bass module, mono-orbit. The research of Angeleri Hügel has been supported by DGI MICIIN MTM C02-01, by Generalitat de Catalunya through Project 2009 SGR 1389, and by Fondazione Cariparo, Progetto di Eccellenza ASATA. The research of Šaroch and Trlifaj has been supported by grant GAČR S. 1

2 2 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ These results have recently been put in the framework of tilting theory in [31]. There, the Mittag-Leffler case is just the zero dimensional one (i.e., the case where the tilting module T equals R). In the general case, the role of FM is played by the class of locally T -free modules (Definition 2.1), that is, modules that can be written as a directed union of certain countably presented submodules from (T ). Unfortunately, most of the results obtained so far have the drawback of assuming that either the ring, or the tilting module, satisfy restrictive conditions on their size or structure. Our goal here is to remove these restrictions. We succeed in doing so in the case of Mittag-Leffler modules by proving that FM is not precovering for an arbitrary non-right perfect ring (Theorem 3.3). In the general case of locally T -free modules, we characterize deconstructibility by the property of T being Σ-pure split, that is, every pure submodule of a direct sum of copies of T is a direct summand (Corollary 8.2). In several particular cases, we show that this is further equivalent to the existence of locally T -free precovers (Theorem 8.3), but the corresponding general result on precovering is still missing. Some of the tools developed here appear to be of independent interest. We prove that every cotorsion pair (A, B) where B is closed under direct limits is a complete cotorsion pair of countable type with B actually being definable (Theorem 6.1). This is a non-hereditary version of the Countable Telescope Conjecture for module categories from [29, Theorem 3.5]. Furthermore, we characterize the locally A ω - free modules in terms of Mittag-Leffler conditions (Theorem 8.4). In the setting of a tilting cotorsion pair (A, B), we give a positive answer to a question by Enochs: the class A is covering if and only if it is closed under direct limits (Theorem 9.2). Surprisingly, both the non-deconstructibility and the non-precovering can be tested using certain small, countably presented modules, called Bass modules. The terminology comes from the prototype example of a Bass module, namely the direct limit of the countable system R f0 R f1... fn 1 R fn R fn+1... where f n is the left multiplication by a n in R, and Ra 0 Ra 1 a 0... is a decreasing chain of principal left ideals in R. By a classic result of Bass, the direct limit is projective, if and only if the chain stabilizes. So non-right perfect rings are characterized by the existence of such Bass modules that are not projective. We are going to see that this is a special instance of a more general phenomenon. In fact, some of our results can be viewed as a generalization of the famous Theorem P by Bass (cf. Corollary 9.4). The paper is organized as follows. After some preliminaries in Section 2, we start out in Section 3 by discussing the case of flat Mittag-Leffler modules. Sections 4 and 5 are devoted to relative Mittag-Leffler conditions and their role in connection with vanishing of Ext and pure-injectivity. In Section 6 we consider cotorsion pairs (A, B) where B is closed under direct limits and prove that they are of countable type. This is used in Sections 7 and 9 to characterize cotorsion pairs with both classes being closed under direct limits. Section 8 deals with the class of locally T -free modules, in particular with deconstructibility and existence of precovers. Sections 10 and 11 discuss applications to pure-semisimple rings and hereditary artin algebras. 2. Preliminaries Let R be an (associative unital) ring. We denote by Mod-R the category of all (right R-) modules.

3 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS Bass modules. Given a class C of countably presented modules, we call a module M a Bass module over C, provided that M is the direct limit of a countable direct system f 0 f 1 f n 1 f n f n+1 C 0 C1... Cn Cn+1... where C n C for each n < ω. All Bass modules over C are countably presented; in fact, they are just the countable direct limits of the modules from C, cf. [31, 5] Purity. For a (left, right) module M, we denote by M c = Hom Z (M, Q/Z) the character (right, left) module of M. A module M is pure-injective provided that the canonical embedding of M into M cc splits. Moreover M is -pure-injective, if every (countable) direct sum of copies of M is pure-injective. The latter property is well known to be inherited by pure submodules. The pure-injective hull of a module M will be denoted by P E(M). A pureinjective module is discrete if it is isomorphic to the pure-injective hull of a direct sum of indecomposable pure-injective modules. Moreover, a module M is called Σ-pure-split if for all N Add(M), any pure embedding into N splits; here Add(N) denotes the class of all direct summands of direct sums of copies of N. Each -pure-injective module is Σ-pure-split, but the converse fails in general (see Section 9) Roots of Ext. For a class of modules C Mod-R, we define its left and right Ext-orthogonal classes by C = Ker Ext 1 R(, C) and C = Ker Ext 1 R(C, ). For C = Mod-R, we have C = P 0 and C = I 0, the classes of all projective and injective modules, respectively Tilting modules. A module T is tilting, provided T has finite projective dimension, Ext i R(T, T (I) ) = 0 for each i 1 and each set I, and there exist a k < ω and an exact sequence 0 R T 0 T k 0 such that T i Add(T ) for each i k. Each tilting module induces the tilting class T = T = 1 i Ker Exti R(T, ). Moreover, T is called n-tilting in case T has projective dimension n Deconstructible classes. Given a module M and an ordinal number σ, we call an ascending chain M = (M α α σ) of submodules of M a filtration of M, if M 0 = 0, M σ = M, and M is continuous that is, α<β M α = M β for each limit ordinal β σ. Moreover, given a class of modules C, we call M a C-filtration of M, provided that each of the consecutive factors M α+1 /M α (α < σ) is isomorphic to a module from C. A module M is C-filtered, if it admits a C-filtration. Given a class C and a cardinal κ, we use C κ and C <κ to denote the subclass of C consisting of all κ-presented and < κ-presented modules, respectively. Let κ be an infinite cardinal. A class of modules C is κ-deconstructible provided that each module M C is C <κ -filtered. For example, the class P 0 is ℵ 1 - deconstructible by a classic theorem of Kaplansky. A class C is deconstructible in case it is κ-deconstructible for some infinite cardinal κ Cotorsion pairs. A pair of classes of modules C = (A, B) is a cotorsion pair provided that A = B and B = A. The class Ker C = A B is the kernel of C. Notice that the class A is always closed under arbitrary direct sums and contains P 0. Dually, the class B is closed under direct products and it contains I 0. If moreover Ext 2 R(A, B) = 0 for all A A and B B, then C is a hereditary cotorsion pair. The latter is equivalent to A being closed under kernels of epimorphisms. For example, for each n < ω, there exist hereditary cotorsion pairs of the form (P n, P n ) and ( I n, I n ), where P n and I n denote the classes of all modules of projective and injective dimension n, respectively.

4 4 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ The cotorsion pair C is said to be generated by a class S provided that B = S. In this case, if κ is an infinite regular cardinal such that each module in S is < κ- presented, then A is κ-deconstructible. If C is generated by a class S consisting of countably (finitely) generated modules with countably (finitely) presented first syzygies, then C is said to be of countable (finite) type 1. If R is right coherent, then it is equivalent to the statement C is generated by a class consisting of countably (finitely) presented modules. Dually, C is said to be cogenerated by a class C provided that A = C. For example, if T is a tilting module with the induced tilting class T, then there is a hereditary cotorsion pair C = ( T, T ), called the tilting cotorsion pair induced by T. Its kernel equals Ker C = Add(T ), and C is of finite type. In particular, the class T is ℵ 1 -deconstructible Approximations. A class C of modules is precovering, if for each module M there exists a morphism f Hom R (C, M) with C C, such that each morphism f Hom R (C, M) with C C factors through f. Such f is called a C-precover of the module M. A precovering class of modules C is called special precovering provided that each module M has a C-precover f : C M which is surjective and satisfies Ker(f) C. Moreover, C is called covering provided that each module M has a C- precover f : C M with the following minimality property: g is an automorphism of C, whenever g : C C is an endomorphism of C with fg = f. Such f is called a C-cover of M. Dually, we define preenveloping, special preenveloping, and enveloping classes of modules. We will also deal with preenveloping and precovering classes in the category of all finitely presented modules these classes are usually called covariantly finite and contravariantly finite, respectively. A cotorsion pair C = (A, B) is called complete, provided that A is a special precovering class (or, equivalently, B a special preenveloping class). For example, each cotorsion pair generated by a set of modules is complete; in particular, every tilting cotorsion pair is complete, and every deconstructible class is precovering provided it is closed under transfinite extensions. Further, C is closed, provided that A = lim A. If C is closed and complete, then A is a covering class in Mod-R, cf. [19, Part II] Locally free modules. The following less well known notation from [22] and [31] will be convenient: Definition 2.1. Let R be a ring and λ an infinite regular cardinal. A system S consisting of <λ-presented submodules of a module M satisfying the conditions (1) S is closed under unions of well-ordered ascending chains of length < λ, and (2) each subset X M such that X < λ is contained in some N S, is called a λ-dense system of submodules of M. Of course, M is then the directed union of these submodules. Let F be a set of countably presented modules. Denote by C the class of all modules possessing a countable F-filtration. A module M is locally F-free provided 1 In literature, the requirement is usually extended from the first one to all syzygies. However, this definition does not seem reasonable for non-hereditary cotorsion pairs, i.e. the ones we want to investigate. Our definition fits nicely into the crucial Theorem 6.1, while it still ensures that finite type of C implies definability of B.

5 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 5 that M contains an ℵ 1 -dense system of submodules from C. Notice that if M is countably presented, then M is locally F-free, if and only if M C. We will mainly be interested in the case when F = A ω for a cotorsion pair C = (A, B). Then C = A ω, and a module is locally A ω -free if and only if it admits an ℵ 1 -dense system of countably presented submodules from A. For a different description of locally A ω -free modules, see Theorem 8.4. If T is a tilting module with the induced tilting cotorsion pair C = (A, B), then the locally A ω -free modules are simply called the locally T -free modules. For example, if T = R, then the locally T -free modules coincide with the flat Mittag-Leffler modules [22]. If R is a hereditary artin algebra of infinite representation type and T is the Lukas tilting module, then the locally T -free modules are called the locally Baer modules, [31] Filter-closed classes. For every directed set (I, ), there is an associated filter F I on (P(I), ), namely the one with a basis consisting of the upper sets i = {j I j i} for all i I, that is F I = {X I ( i I)( i X)}. Definition 2.2. Let F be a filter on the power set P(X) for some set X, and let {M x x X} be a set of modules. Set M = x X M x. Then the F-product Σ F M is the submodule of M such that Σ F M = {m M z(m) F} where for an element m = (m x x X) M, we denote by z(m) its zero set {x X m x = 0}. For example, if F = P(X) then Σ F M = M is just the direct product, while if F is the Fréchet filter on X then Σ F M = x X M x is the direct sum. The module M/Σ F M is called an F-reduced product. Note that for a, b M, we have the equality ā = b in the F-reduced product if and only if a and b agree on a set of indices from the filter F. In the case that M x = M y for every pair of elements x, y X, we speak of an F-power and an F-reduced power (of the module M x ) instead of an F-product and an F-reduced product, respectively. If moreover F is an ultrafilter on X, then the F-reduced power is called an ultrapower of M x. Finally, a nonempty class of modules G is called filter-closed, if it is closed under arbitrary F-products (for any set X and any arbitrary filter F on P(X)). Notice that a class of modules is filter-closed in case it is closed under direct products and direct limits (of direct systems consisting of monomorphisms), see [26, Lemma 3.3.1]. 3. Flat Mittag-Leffler modules and approximations We start with a prototype of the notion of a locally T -free module studied since the classic works of Grothendieck [20] and Raynaud-Gruson [27], namely the notion of a flat Mittag-Leffler module. From its many facets, we choose the one involving tensor products for a definition: Definition 3.1. Let R be a ring. A module M is flat Mittag-Leffler provided that the functor M R : R-Mod Mod-Z is exact, and the canonical group homomorphism ϕ : M R i I Q i i I M R Q i defined by ϕ(m R (q i ) i I ) = (m R q i ) i I is monic for each family of left R-modules (Q i i I). The class of all flat Mittag-Leffler modules will be denoted by FM.

6 6 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ As mentioned above, flat Mittag-Leffler modules coincide with the locally T -free modules for T = R. If R is a right perfect ring, then they are just the projective modules, and form a covering class of modules by a classic theorem by Bass. However, if R is not right perfect (e.g., if R is right noetherian, but not right artinian), then the class FM is not deconstructible [22, Corollary 7.3]. In [30, Theorem 1.2(iv)], the Singular Cardinal Hypothesis (SCH) was used to prove that FM is not even precovering when R has cardinality ℵ 0. The assumption of SCH has recently been removed in [13]. Our goal in this section is to remove the cardinality restriction on R as well, and thus to obtain a basic example of a non-precovering class of locally T -free modules for an arbitrary non-right perfect ring R. The following lemma is formulated for the more general setting of locally F-free modules (see Definition 2.1): Lemma 3.2. Let F be a class of countably presented modules, and L the class of all locally F-free modules. Let N be a countable direct limit of modules from F P 1. Assume that N P 1, but N is not a direct summand in a module from L. Then N has no surjective L-precover. Proof. Assume that f : A N is a surjective L-precover of N. Let M = Ker(f), and let κ be an infinite cardinal such that R κ and M 2 κ = κ ω. By [31, Lemma 5.6], there exists a short exact sequence (1) 0 D L N (2κ) 0 such that L L and D is a direct sum of κ modules from F P 1. Clearly, L P 1. Let η : M E be a special {L} -preenvelope of M with an {L}-filtered cokernel (such η exists, see e.g. [19, Theorem 6.11(a)]). Consider the pushout M η 0 E = A ε P Coker(η) Coker(ε) f N 0 g N By [31, Theorem 4.5], the class L is closed under transfinite extensions, whence P L. Since f is an L-precover, there exists h : P A such that fh = g. Then f = gε = fhε, whence M + Im(h) = A. Let h = h E : E M. Then Im(h ) = M Im(h), and we have the exact sequence 0 Im(h ) Im(h) f Im(h) N 0. As E L and L P 1, also Im(h ) L. Thus, applying Hom R (, Im(h )) to (1), we obtain the exact sequence Hom R (D, Im(h )) Ext 1 R(N, Im(h )) 2κ 0 where the first term has cardinality M κ 2 κ, so the second term must be zero. That is, Im(h ) N. Then f Im(h) splits, and so does f, a contradiction.

7 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 7 Now, we can easily prove Theorem 3.3. Let R be an arbitrary ring. Then FM is precovering, if and only if R is right perfect. Proof. The if part is well known: FM = P 0 whenever R is a right perfect ring. Assume R is not right perfect. By a classic result of Bass, there exists a countably presented flat module (= a Bass module over P 0 <ω ) N such that N / P 0. By [22, Corollary 2.10(i)], the class FM coincides with the class of all locally F-free modules for F = P ω 0. In particular, N / FM, but N P 1. So Lemma 3.2 applies, and shows that N has no FM-precover. We will present further applications of Lemma 3.2 in Section 8. First, we develop several technical tools involving Mittag-Leffler conditions and pure injectivity. 4. Stationarity and the vanishing of Ext In order to deal with the general case, we will have to work with relative Mittag- Leffler conditions. More precisely, we will use B-stationary modules, where B will be a class closed under direct products and direct limits. In the present Section, we review this concept with a special emphasis on its relationship with vanishing of Ext. In Section 5 we will see that the classes B as above admit a pure-injective cogenerator C, and we will thus focus on C-stationarity. Let us first recall the classic notions related to inverse systems of modules (see [4], [27]): Definition 4.1. Let R be a ring and H = (H i, h ij i j I) be an inverse system of modules, with the inverse limit (H, h i i I). Then H is a Mittag-Leffler inverse system, provided that for each k I there exists k j I such that Im(h kj ) = Im(h ki ) for each j i I, that is, the terms of the decreasing chain (Im(h ki ) k i I) of submodules of H k stabilize. If moreover the stabilized term Im(h kj ) equals Im(h k ), then H is called strict Mittag- Leffler. Notice that the two notions coincide when I is countable. Let B be a module and M = (M i, f ji i j I) be a direct system of finitely presented modules, with the direct limit (M, f i i I). Applying the contravariant functor Hom R (, B), we obtain the induced inverse system H = (H i, h ij i j I), where H i = Hom R (M i, B) and h ij = Hom R (f ji, B) for all i j I. Let B be a class of modules. A module M is B-stationary (strict B-stationary), provided that M can be expressed as the direct limit of a direct system M of finitely presented modules so that for each B B, the induced inverse system H is Mittag-Leffler (strict Mittag-Leffler). If B = {B} for a module B, we will use the notation B-stationary instead of {B}-stationary, and similarly for the strict stationarity. Remark 1. Let B be a module. Then the strict B-stationarity of M can equivalently be expressed as follows: for each l I, there exists l i I such that Im(Hom R (f il, B)) Im(Hom R (f l, B)), see [4, 8]. Moreover, the notions of a B- stationary and strict B-stationary module coincide in the case when M is countably presented. They also coincide for B (locally) pure-injective by [21, Proposition 1.7]. In fact, if M is B-stationary (strict B-stationary), then the induced inverse system H is Mittag-Leffler (strict Mittag-Leffler) for each presentation of M as the direct limit of a direct system M of finitely presented modules, see [4]. Also, FM coincides with the class of all flat R-stationary modules, cf. [27].

8 8 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ First, we will deal with the strict B-stationarity of the modules in B. The following result is a mix of Lemmas 2.3 and 2.5 from [29]; for the reader s convenience, we provide a detailed proof here: Lemma 4.2. Let B be a filter-closed class of modules. Then every module M B is strict B-stationary. Proof. We will prove the following: if (M, f i i I) is the direct limit of a direct system M = (M i, f ji i j I) consisting of finitely generated modules, then for each l I there exists l i I such that for each B B and g Hom R (M i, B), we have gf il Im(Hom R (f l, B)). Suppose that the claim is not true, so there exists l I such that for each l i I there exist B i B and g i : M i B i, such that g i f il does not factor through f l. For i I such that l i, we let B i = 0 and g i = 0. Put B = i I B i. We define a homomorphism h ji : M i B j for each pair i, j I in the following way: h ji = g j f ji if i j and h ji = 0 otherwise. This family of maps gives rise to the canonical homomorphism h : k I M k B. More precisely, if we denote by π j : B B j the canonical projection and by ν i : M i k I M k the canonical inclusion, h is the (unique) map such that π j hν i = h ji. Note that for every i, j I such that i j, the set {k I h ki = h kj f ji } is in the associated filter F I since it contains each k j. Hence, if we denote by ϕ the canonical pure epimorphism i I M i M = lim M i (such that ϕν i = f i for all i I), then i I h(ker(ϕ)) Σ FI B. So there is a well-defined homomorphism u from M to the F I - reduced product B/Σ FI B making the following diagram commutative (ρ denotes the canonical projection): B h i I M i ρ B/Σ FI B 0 u ϕ M 0. Since B is filter-closed, Σ FI B B, whence Ext 1 R(M, Σ FI B) = 0. It follows that there exists g Hom R (M, B) such that u = ρg. For each i I, we have ρgf i = ρgϕν i = ρhν i ; so gf i hν i maps M i into Σ FI B. Since M i is finitely generated, there exists i j I such that π k gf i = π k hν i for all j k I. However, π k hν i = h ki = g k f ki. In particular, for i = l and k = j, we infer that π j gf l = g j f jl. Thus, g j f jl factors through f l, a contradiction. We will also need the following variant of [29, Proposition 2.7]. Proposition 4.3. Let G be a class of modules and M a countably presented module such that M G. Then the following is equivalent: (1) M D for each module D isomorphic to a pure submodule of a product of modules from G; (2) M D for each module D isomorphic to a countable direct sum of modules from G; (3) M is G-stationary. Proof. Since direct sums are pure in the corresponding direct products, the implication (1) (2) is trivial. The implication (2) (1) is exactly [29, Proposition 2.7] (for G replaced by its closure under countable direct sums). Its proof in [29] proceeds by showing that (2) (4) (1), where (4) says that M is D-stationary for any module D isomorphic to a pure submodule of a product of modules from G. However, (4) is equivalent to (3) by [4, Corollary 3.9], whence (2) (3) (1).

9 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 9 The next lemma on strict B-stationarity is inspired by [21, Lemma 3.15]. Lemma 4.4. Let B be a module and 0 N A M 0 a short exact sequence of modules such that M (B (I) ) cc for each set I. Then N is strict B-stationary if so is A. Proof. According to [4, Theorem 8.11] (see also [35]), a necessary and sufficient condition for the strict B-stationarity of N is that for each set I, the canonical map ν : N R Hom Z (B, (Q/Z) I ) Hom Z (Hom R (N, B), (Q/Z) I ) defined by ν(n f)(g) = f(g(n)) is injective. Since Hom Z (B, (Q/Z) I ) = (B (I) ) c, we have Tor R 1 (M, Hom Z (B, (Q/Z) I )) = 0 by our assumption on M. From the commutative diagram 0 N R Hom Z (B, (Q/Z) I ) A R Hom Z (B, (Q/Z) I ) ν α Hom Z (Hom R (N, B), (Q/Z) I ) Hom Z (Hom R (A, B), (Q/Z) I ), we infer that ν is injective, because α is injective by our assumption on A. 5. Stationarity and pure-injectivity The classes of pure-injective modules especially relevant here are the Σ-pure injectives, and the elementary cogenerators. We will not deal with the model theoretic background of these notions; we just note that, by a classic theorem of Frayne, if two modules M and N are elementarily equivalent, then M is a pure submodule of an ultrapower of N. The pure-injective hull P E(M) of a module M is elementarily equivalent to M by a theorem of Eklof and Sabbagh, and so is its double dual M cc, cf. [25]. The relation between stationarity and -pure-injectivity goes back to work by Zimmermann [35]. We will need a slight extension of his results: Lemma 5.1. For a module C, the following conditions are equivalent: (1) C is Σ-pure-injective; (2) all modules are strict C-stationary; (3) all modules are C-stationary. (4) all countably presented modules are C-stationary. Proof. The equivalence of the first two conditions comes from [35, Theorem 3.8], and (2) (3) (4) are trivial. For the implication (4) (3), see [4, Proposition 3.10]. It remains to prove that (3) implies (1). Applying [4, Corollary 3.9], we get that all modules are B-stationary, where B is any pure submodule of a pure-epimorphic image of a direct product of copies of C. In particular, by [26, Lemma 3.3.1], this holds for any pure-injective module B which is elementarily equivalent to C (e.g., for B = P E(C)). Since B is pureinjective, it follows from Remark 1 that all modules are even strict B-stationary, and B is Σ-pure-injective by the above. Thus C is Σ-pure-injective, because C is a pure submodule of B. We now turn to elementary cogenerators. Definition 5.2. A pure-injective module E is called an elementary cogenerator, if every pure-injective direct summand of a module elementarily equivalent to E ℵ0 is a direct summand of a direct product of copies of E.

10 10 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ Notice that by [25, Corollary 9.36], for each module M there exists an elementary cogenerator which is elementarily equivalent to M. This allows to prove the following result which will play an important role in the sequel. Lemma 5.3. Let B be a class of modules closed under direct products and direct limits. Then B contains a pure-injective module C, such that each module B B is a pure submodule of a direct product of copies of C. Moreover, if B contains a cogenerator, then C is a cogenerator for Mod-R. Proof. First of all, it is a well-known fact that B = lim B yields that B is closed under direct summands. Now the closure properties of B imply that if B B is elementarily equivalent to a pure-injective module A, then also A B. In particular, B is closed under taking pure-injective hulls, and double duals. Thus we can choose among the modules from B a representative set for elementary equivalence, S, consisting of elementary cogenerators. Let C = S. Then C B is pure-injective, and it has the property that every module B B is isomorphic to a pure submodule in a direct product of copies of C. Indeed, we have B cc B; moreover B cc is a pure-injective direct summand of (B cc ) ℵ0 which is a module elementarily equivalent to E ℵ0 for some E S. Hence B cc is a direct summand in a direct product, D, of copies of C (by Definition 5.2), and B is pure in B cc, and hence in D. Moreover, if B B is a cogenerator for Mod-R, then C cogenerates Mod-R as well. Given a pure-injective cogenerator C as above, one can use the following lemma to find in any (strict) C-stationary module a rich supply of countably presented C-stationary submodules. Lemma 5.4. Let C be a pure-injective module which cogenerates Mod-R, and M be a strict C-stationary module. Then there exists an ℵ 1 -dense system L of strict C-stationary submodules of M such that Hom R (M, C) Hom R (N, C) is surjective ( ) for every directed union N of modules from L. Proof. Let Q = C c. By [4, Proposition 8.14(2)], a module A is strict C-stationary if and only if it is Q-Mittag-Leffler. Repeatedly using [4, Theorem 5.1(4)] for M, we obtain a -directed set F of countably presented Q-Mittag-Leffler submodules of M satisfying (2) from Definition 2.1 for λ = ℵ 1 : observe that the map v in the statement of [4, Theorem 5.1(4)] is monic since the injectivity of v R Q implies ( ) (we use that C is a direct summand in Q c ) and C is a cogenerator. We extend F gradually by adding countable directed unions, noticing along the way that each newly added countably presented module N has the property that N M stays monic after applying R Q (as Tor R 1 (, Q) commutes with direct limits), and it is Q-Mittag-Leffler by [4, Corollary 5.2]. In this way, we eventually arrive at the desired ℵ 1 -dense system L of strict C-stationary submodules of M. Note that any directed union of modules from L satisfies ( ) since ( ) is implied by the injectivity of the corresponding tensor map, which, in turn, is a property preserved by taking directed unions. Remark 2. Lemma 5.4 holds with ℵ 1 replaced by any regular uncountable cardinal. However, we won t need it in this generality. A useful tool for proving that many classes of the form B are deconstructible is provided by

11 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 11 Lemma 5.5. [29, Proposition 2.4] Let B be a filter-closed class of modules such that B cogenerates Mod-R. Then for each uncountable regular cardinal λ and each module M B, there is a λ-dense system C λ of submodules of M such that M/N B for all N C λ. 6. Countable type Let C = (A, B) be a hereditary cotorsion pair in Mod-R. The Telescope Conjecture for Module Categories asserts that C is of finite type whenever A and B are closed under direct limits. This statement is known to be true in some special cases [10]. On the other hand, the Countable Telescope Conjecture was proved in full generality in [29, Theorem 3.5]. It states that C is of countable type whenever B is closed under unions of well-ordered chains. As a first application of the tools developed above, we prove a version of the Countable Telescope Conjecture for not necessarily hereditary cotorsion pairs. Theorem 6.1. Let C = (A, B) be a cotorsion pair with lim B = B. Then C is of countable type, and B is definable. Proof. Let C be the pure-injective module constructed for B in Lemma 5.3. Let A 0 = A ω. By induction on κ, we are going to prove that each κ-presented module M A is A 0 -filtered. There is nothing to prove for κ ℵ 0, so let κ be uncountable. In the κ-presented module M A, we will construct by induction, for each uncountable regular cardinal λ κ, a λ-dense system C λ of submodules of M such that M/N A for each N C λ, and (2) C λ A. Our strategy is to start with a C λ given by Lemma 5.5 for G = B, and then select a suitable subfamily in A. Note that it is enough to ensure that for every N C λ there exists L C λ, such that N L A. Then the family C λ A is the desired one, since each ascending chain in C λ A is actually an A-filtration and A is closed under filtrations. Indeed, for each B B, if N C λ and L A are such that N L, then all homomorphisms from N to B extend to M, and hence to L; thus L/N {B}. We will distinguish the following four cases: Case 1. λ = ℵ 1 : Fix a free presentation of M 0 K R (κ) f M 0. Let C ℵ1 be an ℵ 1 -dense system in M provided by Lemma 5.5. After taking the intersection with the set of all images f(r (X) ) where X is a countable subset of κ, we can assume that C ℵ1 is compatible with this presentation, that is, each N C ℵ1 has the form f(r (X N ) ) for a countable subset X N of κ. Let K = { Ker(f R (X N ) ) N C ℵ1 }. Since B is closed under coproducts and double duals, it follows from Lemma 4.4 that K is strict C-stationary. Thus we can use Lemma 5.4 to obtain another ℵ 1 -dense system, L, this time consisting of submodules of K. Clearly, the system K L is ℵ 1 -dense as well. Our new C ℵ1 is defined as {N C ℵ1 Ker(f R (X N ) ) L}. Notice that C ℵ1 C. Indeed, given a module N C ℵ1, each h : Ker(f R (X N ) ) C can be extended to some h : R (X N ) C by the property ( ) from Lemma 5.4 and by the fact that M C. Let N C ℵ1. Since M/N A, N is the kernel of the epimorphism M M/N between two modules from A. By Lemma 4.2, M is strict C-stationary, so using

12 12 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ Lemma 4.4 again, we see that each N C ℵ1 is a countably presented C-stationary module from C. Using Proposition 4.3 (for G = {C}) together with the properties of C guaranteed by Lemma 5.3, we conclude that C ℵ1 A. Case 2. λ weakly inaccessible: As in the previous step, we start with a family C λ provided by Lemma 5.5. Since each N 0 C λ is < µ-presented for some regular uncountable cardinal µ < λ, we simply choose N 0 C µ A containing N 0 as a submodule. Then there is N 1 C λ containing N 0, etc. The union, N, of the chain N 0 N 0 N 1 N 1 satisfies N C λ. Moreover, N A since the chain N 0 N 1 is an A-filtration of N. Case 3. λ a successor of a regular cardinal: λ = ν + for a regular cardinal ν. For each N 0 from C λ (not necessarily satisfying condition (2) above), we easily build a continuous chain N 0 = (Nα 0 α < ν) of modules from C ν A so that N 0 N 0. Again, the union is in A. We continue by choosing N 1 C λ containing this union, and a chain N 1 = (Nα 1 α < ν) in C ν such that Nα 0 Nα 1 for all α < ν, and N 1 N 1. We proceed further by taking N 2 C λ, etc. Clearly, N = i<ω N i C λ. Furthermore, the ascending chain N0 i N1 i Nα i i<ω i<ω of submodules from C ν forms an A-filtration of N, showing that N A. Case 4. λ a successor of a singular cardinal: λ = ν + where µ = cf(ν) < ν. We choose a strictly increasing continuous chain (ν α α < µ) of infinite cardinals which is cofinal in ν, such that ν 0 > µ. Let N 0 be arbitrary module from the family C λ given by Lemma 5.5. We will produce a similar ascending chain as in Case 3., however this time, we have to pick the modules from different classes C δ, hence we lose continuity. To overcome this problem, we use a well known singular compactness argument: We gradually build the chains N i = (N i α α < µ), for i < ω, and pick the modules N i C λ in an alternating way, so that the following conditions are satisfied for all i < ω: (a) N i α C ν + α ; (b) N i N i+1 ; (c) the generators {n i α,β β < ν α} of the modules N i α are fixed; (d) the generators {n i γ γ < ν} of N i are fixed; (e) N i α {n i 1 δ,β δ < µ & β < min(ν δ, ν α )} {n i γ γ < ν α }. Then i<ω N i is in C λ, and it is equal to the union of the chain H = ( i<ω N i α α < µ). However, this chain is continuous (by condition (e)), and provides thus an A-filtration of H. Having constructed the families C λ (λ κ), we can use them to build an A- filtration of M consisting of < κ-presented modules. If κ is regular, then it is easy to see that C κ already contains an A-filtration of M. For κ singular, we apply [29, Lemma 3.2]. By inductive hypothesis, we conclude that M is A 0 -filtered. It remains to show that the modules in A 0 have countably presented first syzygies. We know from Lemma 4.4 that their first syzygies are strict B-stationary. In particular, the first syzygies are R R c -stationary. By [4, Proposition 8.14(1)], they are then R R-Mittag-Leffler, and the claim follows from [4, Corollary 5.3]. Finally, B is definable by [19, Theorem 13.41]. We finish this section by examples of cotorsion pairs which satisfy the conditions of Theorem 6.1, but are not hereditary. i<ω

13 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 13 Recall that for n < ω, a module M is an FP n -module provided that M has a projective resolution P k+1 P k P 1 P 0 M 0 such that P k is finitely generated for each k n, [14, VIII.4]. Let FP n denote the class of all FP n -modules. (Notice that FP 0 is the class of all finitely generated modules, and FP 1 the class of all finitely presented ones.) The classes (FP n n < ω) form a decreasing chain whose intersection is the class of all strongly finitely presented modules, that is, the modules possessing a projective resolution consisting of finitely generated modules, see [14, Proposition VII.4.5]. Example 6.2. Let n 2 and let R n be a ring such that there exists a module M FP n \FP n+1. Such rings were constructed by Bieri and Stuhler using integral representation theory, see [14, VIII.5] and the references therein; in fact, R n can be taken as the integral group ring ZG n for a suitable finitely generated group G n, and M = Z. (For the particular case of n = 2, a different and simpler example of R 2 is constructed in [23, Example 1.4].) Let C n = (A n, B n ) be the cotorsion pair generated by FP n. Since n 2, B n is closed under direct limits by [19, Lemma 6.6], so Theorem 6.1 applies. In order to see that C n is not hereditary, we observe that A n is not closed under kernels of epimorphisms: indeed, the syzygy module Ω(M) does not belong to FP n, and hence Ω(M) / A n. To see the latter fact, notice that A n consists of direct summands of FP n -filtered modules by [19, 6.14]. Since Ω(M) is finitely generated, if Ω(M) A n then it is a direct summand of a finitely FP n -filtered module, by [19, Theorem 7.10] used for ℵ 0 (see also Lemma 7.1 below). However, FP n is closed under extensions and direct summands, whence Ω(M) FP n, a contradiction. 7. Closed cotorsion pairs In this section, we will characterize the tilting cotorsion pairs C = (A, B) such that C is closed, that is, lim A = A. In fact, in Theorem 7.6 we will go far beyond the tilting setting: we will not require C to be hereditary or A to have bounded projective dimension. Further characterizations for the closure of C will be given later in Theorem 9.2 and Corollary 9.4. First, we recall the following important result going back to Hill (in the form presented in [19, Theorem 7.10], for example): Lemma 7.1. Let λ be a regular infinite cardinal. Let S be a class of < λ-presented modules and M a module possessing an S-filtration (M α α σ). Then there is a family F of submodules of M such that: (1) M α F for all α σ. (2) F is closed under arbitrary sums and intersections. (3) For each N, P F such that N P, the module P/N is S-filtered. (4) For each N F and a subset X M of cardinality < λ, there is P F such that N X P and P/N is < λ-presented. Next, we show that if A contains modules from lim A of a certain size, then it is closed under direct limits of direct systems of that size: Lemma 7.2. Let A be an ℵ 1 -deconstructible class. Assume that (lim A) µ A for an infinite cardinal µ. Then lim D A for each direct system D of cardinality µ such that D consists of modules from A.

14 14 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ Proof. In view of (the proof of) [19, Lemma 2.14], it suffices to prove the claim under the extra assumption of D being well-ordered. Let D = (A γ, f γδ δ γ < λ) be such a system with ℵ 0 λ µ, and let M = lim A γ. γ<λ Since A is ℵ 1 -deconstructible, it is also λ + -deconstructible. Hence for each A γ, we have a system F γ from Lemma 7.1 (for S = A <λ+ ). Let ν be an infinite cardinal such that all the modules A γ (γ < λ), are ν-generated. Possibly allowing repetitions, we enumerate the set of generators of A γ as {a α γ α < ν}. We shall inductively build compatible A λ -filtrations, (A α γ α ν), of the modules A γ as follows: First, we let A 0 γ = 0 for all γ < λ. Assume that A α γ F γ are defined for all γ < λ and all α < β (where β < ν is fixed) so that for all γ < λ, δ<γ f γδ(a α δ ) Aα γ ; a α 1 γ A α γ provided that α is non-limit. If β is a limit ordinal, we put A β γ = α<β Aα γ. If β is non-limit, we choose for each δ < γ a subset G β δ of Aβ δ of cardinality λ whose image under the canonical projection A β δ Aβ δ /Aβ 1 δ generates the factor module. Then the subset {a β 1 } ( γ f γδ G β) δ δ<γ has cardinality at most λ, and by condition (4) of Lemma 7.1 we can find A β γ F γ containing it together with the submodule Aγ β 1 F γ. For each α < ν, let L α = lim A α γ<λ γ. We claim that the (not necessarily strictly) ascending chain (L α α < ν) forms an A-filtration of M. For each α < ν, denoting f γ = f γ+1γ, we have the direct system of short exact sequences 0 A α γ A α+1 γ A α+1 γ /A α γ 0 fγ Aα γ f γ A α+1 γ 0 A α γ+1 A α+1 γ+1 Aα+1 γ+1 /Aα γ+1 0 fγ+1 Aα γ+1 f γ+1 A α+1 γ+1.. of modules from A where the right-hand terms are λ-presented, whose direct limit is the sequence 0 L α L α+1 L α+1 /L α 0. So L α+1 /L α is λ- presented, and our assumption on A gives L α+1 /L α A. The chain (L α α < ν) is continuous by the construction above (namely, the step when β was limit). Finally, lim L α = M, and the claim is proved. α<ν In the setting of the next lemma, all modules in the class A are strict B-stationary by Lemma 4.2. We show that also all their small pure factors are B-stationary: Lemma 7.3. Let C = (A, B) be a cotorsion pair with B = lim B. Assume that (lim A) <κ A for an uncountable cardinal κ. Then any κ-presented pureepimorphic image of a module from A is B-stationary. Proof. First, the cotorsion pair C is complete and A is ℵ 1 -deconstructible by Theorem 6.1. Let M be a κ-presented pure-epimorphic image of a module from A. In view of Lemma 7.2, the assumption of (lim A) <κ A enables us to use [29, Lemma 5.10] to build a direct system of short exact sequences 0 B u.

15 APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 15 π A u u Mu 0 indexed by an inverse tree I κ such that π u is a special A-precover of M u, and π = lim π u : A M, where A does not necessarily belong to A. u However, by our assumption on B, we have Ker(π) B. Since M is a pureepimorphic image of a module from A, the epimorphism π is pure, too. By the proof of [29, Theorem 5.11], in this setting, the following holds true for each direct system (K i, k ji i, j J & i j) consisting of finitely presented modules with lim K i = M: for each i J there exists s(i) > i such that k i J s(i)i factors through some A u (which belongs to A by construction). This defines a function s : J J. Let I denote the set of all nonempty finite subsets of J. We are going to build a direct system (L p, l qp p, q I & p q), derived from (K i, k ji i, j J & i j), such that lim L p p I = M and every lqp, p q, factors through a module from A. To this purpose, we first recursively define a monotone function δ : (I, ) (J, ). For technical reasons, we also consider a function c : I J which assigns to every nonempty finite set of elements from J their common upper bound in (J, ). We start by putting δ({i}) = i for all i J. Assume δ(p) is defined for all p I with p < n (1 < n < ω). For q I with q = n, we let δ(q) = s(c{δ(p) p q}). Next, we let L p = K δ(p) and l qp = k δ(q)δ(p). It follows from our construction that lim L p p I = M and that, for every q p I, the homomorphism lqp factors through a module A u from A. By our assumption on A, the direct limit of each countable subsystem of (L p, l qp ) belongs to A. Since all modules in A are B-stationary by Lemma 4.2, we deduce that any countable subsystem of (L p, l qp ) is B-stationary, and so M is B-stationary by [4, Proposition 3.10]. In the countable case, we will use Lemma 7.4. Let C = (A, B) be a complete cotorsion pair with B closed under countable direct limits. Assume that M is a countably presented pure-epimorphic image of a module from A. Then there exists a direct system (F n, f nm m n < ω) of finitely presented modules such that M = lim F n, and f n+1,n factors through a module from A n<ω for each n < ω. In particular, M is a countable direct limit of modules from A. If moreover C is of countable type (finite type), then M is a Bass module over A ω (A <ω ). Proof. Let D = (F n, f nm m < n < ω) be a direct system of finitely presented modules with the direct limit (M, f n n < ω). We can expand D to a direct system of special A-precovers π n : A n F n of the modules F n (n < ω) so that the diagram g n... A n An+1... π n π n+1... f n,n 1 Fn f n+1,n Fn+1 f n+2,n+1... is commutative. Then π = lim n<ω π n is a pure epimorphism: indeed, as Ker(π) B by our assumption on B, the presentation of M as a pure-epimorphic image of a module from A factors through π. Since the modules F n are finitely presented, by possibly dropping some of them, we can assume that f n+1,n = π n+1 ν n where ν n Hom R (F n, A n+1 ) for each n < ω. Then (M, f n π n n < ω) is the direct limit of the direct system (A n, g nm m n < ω) where g nm = ν n 1 π n 1... ν m π m for all m < n < ω.

16 16 LIDIA ANGELERI HÜGEL, JAN ŠAROCH, AND JAN TRLIFAJ If C is of countable type, then each A n is A ω -filtered. We use Lemma 7.1, for λ = ℵ 1, to build inductively, for each n, a submodule A n A ω of A n which contains (at most countable) generating sets of Im(ν n 1 ), g n 1 (A n 1) as well as of a finitely generated module G such that G + Ker(π n ) = A n. We replace each A n by A n, and π n and g n by their restrictions π n and g n to A n, respectively. So the diagram g n 1... A g n n A g n+1 n+1... π n π n+1 f n,n 1... Fn f n+1,n Fn+1 f n+2,n+1... is commutative. As above, (M, f n π n n < ω) is the direct limit of the direct system (A n, g nn m n < ω) where g nm = ν n 1 π n 1... ν m π m for all m < n < ω. In particular, M is a Bass module over A ω. If C is of finite type, then each A n is a direct summand in a A <ω -filtered module with a complement in Ker(C) = A B (see [19, Corollary 6.13(b)]). Thus, we can w.l.o.g. assume that in the special A-precover π n : A n F n, the module A n is A <ω -filtered. Using Lemma 7.1 for λ = ℵ 0, we replace each A n by its submodule A n A <ω, and π n and g n by their restrictions π n and g n to A n, respectively, so that Im(ν n ) A n+1, and the diagram above is commutative. As above, we conclude that M is a Bass module over A <ω. Finally, we recall a useful description of the class of pure-epimorphic images of modules from A, cf. [19, Lemmas 8.38 and 8.39] and [29, Proposition 5.12]. Lemma 7.5. Let R be a ring and C = (A, B) a cotorsion pair in Mod-R such that B is closed under direct limits, and let B be the class of all pure-injective modules from B. Then B coincides with the class à of all pure-epimorphic images of modules from A. We can now characterize the cotorsion pairs (A, B) with both classes closed under direct limits among those which satisfy the condition only on the right-hand side. The characterization shows that when testing for lim A = A, it suffices to check only the Bass modules over A ω : Theorem 7.6. Let R be a ring and C = (A, B) a cotorsion pair in Mod-R such that B is closed under direct limits. Then the following conditions are equivalent: (1) C is cogenerated by a (discrete) pure-injective module; (2) A is closed under pure-epimorphic images (and pure submodules); (3) C is closed (i.e., lim A = A); (4) A contains all Bass modules over A ω. Proof. (1) (2). Since A = C and C is pure-injective, A is closed under pureepimorphic images. In order to verify closure under pure submodules, we take a pure submodule X of a module A from A and show that Hom R (X, ) is exact on the short exact sequence 0 C E(C) Z 0 given by the injective envelope of C. Since the first cosyzygy Z of the pure-injective module C is pure-injective (see e.g. [19, Lemma 6.20]), every f Hom R (X, Z) can be extended to a homomorphism f Hom R (A, Z). Now f factors through E(C) as A A = C. Restricting to X, we obtain the desired factorization. The implications (2) (3) and (3) (4) are trivial. (4) (1). By Theorem 6.1, C is of countable type (whence C is complete), and B is definable. We can apply [29, Proposition 5.12] and obtain a set S of indecomposable pure-injective modules such that à = ( S), where à denotes