introduction - profs.scienze.univr.itprofs.sci.univr.it/~angeleri/tiltml-final.pdf · 2014. 10....

APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS

LIDIA ANGELERI HUGEL, JAN SAROCH, AND JAN TRLIFAJ

Abstract. Mittag-Leffler modules occur naturally in algebra, algebraic ge-

ometry, and model theory, [27], [20], [26]. If R is a non-right perfect ring, thenit 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 ≤ ℵ0, [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 cotorsionpair (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 withtilting 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 byGrothendieck as a sufficient condition for the exactness of the inverse limit func-tor, [20]. Flat Mittag Leffler modules then played an important role in the proofof the locality of the notion of a vector bundle in [27, Seconde partie]. The re-newed interest in Mittag-Leffler conditions stems from their role in the study ofroots of the Ext functor, see [3], [4], [12], and [29], as well as their connection togeneralized vector bundles [15], [18]. In [22], it was shown that flat-Mittag Lefflermodules coincide with the ℵ1-projective modules, studied already by Shelah et al.by means of set-theoretic homological algebra, [17]. A connection to model theorywas 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 globalstructure in the case when R is not right perfect. Similarly to the classes P0 and F0

of all projective and flat modules, the class FM of all flat Mittag-Leffler modules isclosed under transfinite extensions. However, unlike P0 and F0, it is not possible toobtain FM by transfinite extensions from any set of its elements. In other words,FM is not deconstructible, [22]. Moreover, unlike P0 and F0, the class FM doesnot provide for approximations when R is countable, [13].

Date: May 17, 2014.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 Hugel has been supported by DGI MICIIN MTM2011-28992-C02-

01, by Generalitat de Catalunya through Project 2009 SGR 1389, and by Fondazione Cariparo,

Progetto di Eccellenza ASATA. The research of Saroch and Trlifaj has been supported by grantGACR 14-15479S.

1

2 LIDIA ANGELERI HUGEL, JAN SAROCH, 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 wherethe tilting module T equals R). In the general case, the role of FM is played bythe class of locally T -free modules (Definition 2.1), that is, modules that can bewritten as a directed union of certain countably presented submodules from ⊥(T⊥).Unfortunately, most of the results obtained so far have the drawback of assumingthat either the ring, or the tilting module, satisfy restrictive conditions on their sizeor structure.

Our goal here is to remove these restrictions. We succeed in doing so in the caseof Mittag-Leffler modules by proving that FM is not precovering for an arbitrarynon-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 (Corollary8.2). In several particular cases, we show that this is further equivalent to theexistence of locally T -free precovers (Theorem 8.3), but the corresponding generalresult on precovering is still missing.

Some of the tools developed here appear to be of independent interest. We provethat every cotorsion pair (A,B) where B is closed under direct limits is a completecotorsion pair of countable type with B actually being definable (Theorem 6.1).This is a non-hereditary version of the Countable Telescope Conjecture for modulecategories 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 ofa 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 betested using certain small, countably presented modules, called Bass modules. Theterminology comes from the prototype example of a Bass module, namely the directlimit of the countable system

Rf0→ R

f1→ . . .fn−1→ R

fn→ Rfn+1→ . . .

where fn is the left multiplication by an in R, and Ra0 ⊇ Ra1a0 ⊇ . . . is adecreasing chain of principal left ideals in R. By a classic result of Bass, the directlimit is projective, if and only if the chain stabilizes. So non-right perfect rings arecharacterized by the existence of such Bass modules that are not projective. Weare going to see that this is a special instance of a more general phenomenon. Infact, some of our results can be viewed as a generalization of the famous TheoremP by Bass (cf. Corollary 9.4).

The paper is organized as follows. After some preliminaries in Section 2, we startout in Section 3 by discussing the case of flat Mittag-Leffler modules. Sections 4and 5 are devoted to relative Mittag-Leffler conditions and their role in connectionwith 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 countabletype. This is used in Sections 7 and 9 to characterize cotorsion pairs with bothclasses being closed under direct limits. Section 8 deals with the class of locallyT -free modules, in particular with deconstructibility and existence of precovers.Sections 10 and 11 discuss applications to pure-semisimple rings and hereditaryartin algebras.

2. Preliminaries

Let R be an (associative unital) ring. We denote by Mod-R the category of all(right R-) modules.

APPROXIMATIONS AND MITTAG-LEFFLER CONDITIONS 3

2.1. Bass modules. Given a class C of countably presented modules, we call amodule M a Bass module over C, provided that M is the direct limit of a countabledirect system

C0f0→ C1

f1→ . . .fn−1→ Cn

fn→ Cn+1fn+1→ . . .

where Cn ∈ C for each n < ω. All Bass modules over C are countably presented; infact, they are just the countable direct limits of the modules from C, cf. [31, §5].

2.2. Purity. For a (left, right) module M , we denote by M c = HomZ(M,Q/Z) thecharacter (right, left) module of M . A module M is pure-injective provided thatthe 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 propertyis well known to be inherited by pure submodules.

The pure-injective hull of a module M will be denoted by PE(M). A pure-injective module is discrete if it is isomorphic to the pure-injective hull of a directsum of indecomposable pure-injective modules.

Moreover, a module M is called Σ-pure-split if for all N ∈ Add(M), any pureembedding into N splits; here Add(N) denotes the class of all direct summands ofdirect sums of copies of N . Each

∑-pure-injective module is Σ-pure-split, but the

converse fails in general (see Section 9).

2.3. Roots of Ext. For a class of modules C ⊆ Mod-R, we define its left andright Ext-orthogonal classes by ⊥C = Ker Ext1R(−, C) and C⊥ = Ker Ext1R(C,−).For C = Mod-R, we have ⊥C = P0 and C⊥ = I0, the classes of all projective andinjective modules, respectively.

2.4. Tilting modules. A module T is tilting, provided T has finite projectivedimension, ExtiR(T, T (I)) = 0 for each i ≥ 1 and each set I, and there exista k < ω and an exact sequence 0 → R → T0 → · · · → Tk → 0 such thatTi ∈ Add(T ) for each i ≤ k. Each tilting module induces the tilting class T =T⊥∞ =

⋂1≤i Ker ExtiR(T,−). Moreover, T is called n-tilting in case T has projec-

tive dimension ≤ n.

2.5. Deconstructible classes. Given a module M and an ordinal number σ, wecall an ascending chainM = (Mα | α ≤ σ) of submodules of M a filtration of M , ifM0 = 0, Mσ = M , and M is continuous — that is,

⋃α<βMα = Mβ for each limit

ordinal β ≤ σ. Moreover, given a class of modules C, we call M a C-filtration ofM , provided that each of the consecutive factors Mα+1/Mα (α < σ) is isomorphicto 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 ofC consisting of all ≤ κ-presented and < κ-presented modules, respectively.

Let κ be an infinite cardinal. A class of modules C is κ-deconstructible pro-vided that each module M ∈ C is C<κ-filtered. For example, the class P0 is ℵ1-deconstructible by a classic theorem of Kaplansky. A class C is deconstructible incase it is κ-deconstructible for some infinite cardinal κ.

2.6. Cotorsion pairs. A pair of classes of modules C = (A,B) is a cotorsion pairprovided that A = ⊥B and B = A⊥. The class KerC = A ∩ B is the kernel of C.Notice that the class A is always closed under arbitrary direct sums and containsP0. Dually, the class B is closed under direct products and it contains I0. Ifmoreover Ext2R(A,B) = 0 for all A ∈ A and B ∈ B, then C is a hereditary cotorsionpair. 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(Pn,P⊥n ) and (⊥In, In), where Pn and In denote the classes of all modules ofprojective and injective dimension ≤ n, respectively.


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 consistingof countably (finitely) generated modules with countably (finitely) presented firstsyzygies, then C is said to be of countable (finite) type1. If R is right coherent, thenit 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 Cprovided that A = ⊥C.

For example, if T is a tilting module with the induced tilting class T , then thereis a hereditary cotorsion pair C = (⊥T , T ), called the tilting cotorsion pair inducedby T . Its kernel equals KerC = Add(T ), and C is of finite type. In particular, theclass ⊥T is ℵ1-deconstructible.

2.7. Approximations. A class C of modules is precovering, if for each module Mthere exists a morphism f ∈ HomR(C,M) with C ∈ C, such that each morphismf ′ ∈ HomR(C ′,M) with C ′ ∈ C factors through f . Such f is called a C-precover ofthe module M .

A precovering class of modules C is called special precovering provided thateach module M has a C-precover f : C → M which is surjective and satisfiesKer(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 automorphismof C, whenever g : C → C is an endomorphism of C with fg = f . Such f iscalled a C-cover of M . Dually, we define preenveloping, special preenveloping, andenveloping classes of modules.

We will also deal with preenveloping and precovering classes in the category ofall finitely presented modules – these classes are usually called covariantly finiteand contravariantly finite, respectively.

A cotorsion pair C = (A,B) is called complete, provided that A is a specialprecovering class (or, equivalently, B a special preenveloping class). For example,each cotorsion pair generated by a set of modules is complete; in particular, everytilting cotorsion pair is complete, and every deconstructible class is precoveringprovided it is closed under transfinite extensions. Further, C is closed, providedthat A = lim−→A. If C is closed and complete, then A is a covering class in Mod-R,

cf. [19, Part II].

2.8. 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 allmodules possessing a countable F-filtration. A module M is locally F-free provided

1In 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 thatfinite type of C implies definability of B.


that M contains an ℵ1-dense system of submodules from C. Notice that if M iscountably 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 pairC = (A,B). Then C = A≤ω, and a module is locally A≤ω-free if and only ifit admits an ℵ1-dense system of countably presented submodules from A. For adifferent 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), thenthe 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 flatMittag-Leffler modules [22]. If R is a hereditary artin algebra of infinite represen-tation type and T is the Lukas tilting module, then the locally T -free modules arecalled the locally Baer modules, [31].

2.9. Filter-closed classes. For every directed set (I,≤), there is an associatedfilter FI 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

FI = 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 letMx | x ∈ X be a set of modules. Set M =

∏x∈XMx. Then the F-product ΣFM

is the submodule of M such that

ΣFM = m ∈M | z(m) ∈ F

where for an element m = (mx | x ∈ X) ∈ M , we denote by z(m) its zero setx ∈ X | mx = 0.

For example, if F = P(X) then ΣFM = M is just the direct product, whileif F is the Frechet filter on X then ΣFM =

⊕x∈XMx is the direct sum. The

module M/ΣFM is called an F-reduced product. Note that for a, b ∈ M , we havethe equality a = b in the F-reduced product if and only if a and b agree on a set ofindices from the filter F.

In the case that Mx = My for every pair of elements x, y ∈ X, we speak of anF-power and an F-reduced power (of the module Mx) instead of an F-product andan F-reduced product, respectively. If moreover F is an ultrafilter on X, then theF-reduced power is called an ultrapower of Mx.

Finally, a nonempty class of modules G is called filter-closed, if it is closed underarbitrary F-products (for any set X and any arbitrary filter F on P(X)). Noticethat a class of modules is filter-closed in case it is closed under direct productsand direct limits (of direct systems consisting of monomorphisms), see [26, Lemma3.3.1].

3. Flat Mittag-Leffler modules and approximations

We start with a prototype of the notion of a locally T -free module studied sincethe classic works of Grothendieck [20] and Raynaud-Gruson [27], namely the notionof a flat Mittag-Leffler module. From its many facets, we choose the one involvingtensor products for a definition:

Definition 3.1. Let R be a ring. A module M is flat Mittag-Leffler providedthat the functor M ⊗R − : R-Mod → Mod-Z is exact, and the canonical grouphomomorphism ϕ : M ⊗R

∏i∈I Qi →

∏i∈IM ⊗R Qi defined by ϕ(m⊗R (qi)i∈I) =

(m⊗R qi)i∈I is monic for each family of left R-modules (Qi | i ∈ I).The class of all flat Mittag-Leffler modules will be denoted by FM.


As mentioned above, flat Mittag-Leffler modules coincide with the locally T -freemodules for T = R. If R is a right perfect ring, then they are just the projectivemodules, 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 rightartinian), 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 thatFM is not even precovering when R has cardinality ≤ ℵ0. The assumption of SCHhas recently been removed in [13].

Our goal in this section is to remove the cardinality restriction on R as well, andthus to obtain a basic example of a non-precovering class of locally T -free modulesfor an arbitrary non-right perfect ring R.

The following lemma is formulated for the more general setting of locally F-freemodules (see Definition 2.1):

Lemma 3.2. Let F be a class of countably presented modules, and L the class ofall locally F-free modules. Let N be a countable direct limit of modules from F∩P1.Assume that N ∈ P1, but N is not a direct summand in a module from L. ThenN 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 | ≤ 2κ = κω. 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 ∩P1. Clearly, L ∈ P1.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

0 0y y0 −−−−→ M

⊆−−−−→ Af−−−−→ N −−−−→ 0

η

y ε

y ∥∥∥0 −−−−→ E

⊆−−−−→ Pg−−−−→ N −−−−→ 0y y

Coker(η)∼=−−−−→ Coker(ε)y y

0 0.

By [31, Theorem 4.5], the class L is closed under transfinite extensions, whenceP ∈ 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 . ThenIm(h′) = M ∩ Im(h), and we have the exact sequence

0 −−−−→ Im(h′)⊆−−−−→ Im(h)

fIm(h)−−−−−→ N −−−−→ 0.

As E ∈ L⊥ and L ∈ P1, also Im(h′) ∈ L⊥. Thus, applying HomR(−, Im(h′)) to(1), we obtain the exact sequence

HomR(D, Im(h′))→ Ext1R(N, Im(h′))2κ

→ 0

where the first term has cardinality ≤ |M |κ ≤ 2κ, so the second term must be zero.That is, Im(h′) ∈ N⊥. Then f Im(h) splits, and so does f , a contradiction.


Now, we can easily prove

Theorem 3.3. Let R be an arbitrary ring. Then FM is precovering, if and onlyif R is right perfect.

Proof. The if part is well known: FM = P0 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<ω0 ) N such that N /∈ P0. By[22, Corollary 2.10(i)], the class FM coincides with the class of all locally F-free

modules for F = P≤ω0 . In particular, N /∈ FM, but N ∈ P1. So Lemma 3.2applies, and shows that N has no FM-precover.

We will present further applications of Lemma 3.2 in Section 8. First, we developseveral 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 willbe 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 vanishingof Ext. In Section 5 we will see that the classes B as above admit a pure-injectivecogenerator 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 = (Hi, hij | i ≤ j ∈ I) be an inverse systemof modules, with the inverse limit (H,hi | i ∈ I).

Then H is a Mittag-Leffler inverse system, provided that for each k ∈ I thereexists k ≤ j ∈ I such that Im(hkj) = Im(hki) for each j ≤ i ∈ I, that is, the termsof the decreasing chain (Im(hki) | k ≤ i ∈ I) of submodules of Hk stabilize. Ifmoreover the stabilized term Im(hkj) equals Im(hk), then H is called strict Mittag-Leffler. Notice that the two notions coincide when I is countable.

Let B be a module and M = (Mi, fji | i ≤ j ∈ I) be a direct system of finitelypresented modules, with the direct limit (M,fi | i ∈ I). Applying the contravariantfunctor HomR(−, B), we obtain the induced inverse system H = (Hi, hij | i ≤ j ∈I), where Hi = HomR(Mi, B) and hij = HomR(fji, 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 offinitely presented modules so that for each B ∈ B, the induced inverse system H isMittag-Leffler (strict Mittag-Leffler).

If B = B for a module B, we will use the notation B-stationary instead ofB-stationary, and similarly for the strict stationarity.

Remark 1. Let B be a module. Then the strict B-stationarity of M can equiv-alently be expressed as follows: for each l ∈ I, there exists l ≤ i ∈ I such thatIm(HomR(fil, B)) ⊆ Im(HomR(fl, B)), see [4, §8]. Moreover, the notions of a B-stationary and strict B-stationary module coincide in the case when M is countablypresented. 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 inversesystem H is Mittag-Leffler (strict Mittag-Leffler) for each presentation of M as thedirect 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].


First, we will deal with the strict B-stationarity of the modules in ⊥B. The fol-lowing 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 ∈ ⊥Bis strict B-stationary.

Proof. We will prove the following: if (M,fi | i ∈ I) is the direct limit of a directsystem M = (Mi, fji | i ≤ j ∈ I) consisting of finitely generated modules, then foreach l ∈ I there exists l ≤ i ∈ I such that for each B ∈ B and g ∈ HomR(Mi, B),we have gfil ∈ Im(HomR(fl, B)).

Suppose that the claim is not true, so there exists l ∈ I such that for eachl ≤ i ∈ I there exist Bi ∈ B and gi : Mi → Bi, such that gifil does not factorthrough fl. For i ∈ I such that l i, we let Bi = 0 and gi = 0. Put B =

∏i∈I Bi.

We define a homomorphism hji : Mi → Bj for each pair i, j ∈ I in the followingway: hji = gjfji if i ≤ j and hji = 0 otherwise. This family of maps givesrise to the canonical homomorphism h :

⊕k∈IMk → B. More precisely, if we

denote by πj : B → Bj the canonical projection and by νi : Mi →⊕

k∈IMk thecanonical inclusion, h is the (unique) map such that πjhνi = hji. Note that forevery i, j ∈ I such that i ≤ j, the set k ∈ I | hki = hkjfji is in the associatedfilter FI since it contains each k ≥ j. Hence, if we denote by ϕ the canonical pureepimorphism

⊕i∈IMi → M = lim−→i∈IMi (such that ϕνi = fi for all i ∈ I), then

h(Ker(ϕ)) ⊆ ΣFIB. So there is a well-defined homomorphism u from M to the FI -reduced product B/ΣFIB making the following diagram commutative (ρ denotesthe canonical projection):

Bρ−−−−→ B/ΣFIB −−−−→ 0

h

x u

x⊕i∈IMi

ϕ−−−−→ M −−−−→ 0.

Since B is filter-closed, ΣFIB ∈ B, whence Ext1R(M,ΣFIB) = 0. It follows thatthere exists g ∈ HomR(M,B) such that u = ρg.

For each i ∈ I, we have ρgfi = ρgϕνi = ρhνi; so gfi − hνi maps Mi into ΣFIB.Since Mi is finitely generated, there exists i ≤ j ∈ I such that πkgfi = πkhνi forall j ≤ k ∈ I. However, πkhνi = hki = gkfki. In particular, for i = l and k = j, weinfer that πjgfl = gjfjl. Thus, gjfjl factors through fl, 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 modulesuch that M ∈ ⊥G. Then the following is equivalent:

(1) M ∈ ⊥D for each module D isomorphic to a pure submodule of a productof modules from G;

(2) M ∈ ⊥D for each module D isomorphic to a countable direct sum of modulesfrom G;

(3) M is G-stationary.

Proof. Since direct sums are pure in the corresponding direct products, the impli-cation (1)⇒ (2) is trivial.

The implication (2) ⇒ (1) is exactly [29, Proposition 2.7] (for G replaced byits closure under countable direct sums). Its proof in [29] proceeds by showingthat (2) ⇒ (4) ⇒ (1), where (4) says that M is D-stationary for any module Disomorphic to a pure submodule of a product of modules from G. However, (4) isequivalent to (3) by [4, Corollary 3.9], whence (2)⇒ (3)⇒ (1).


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 sequenceof modules such that M ∈ ⊥(B(I))cc for each set I. Then N is strict B-stationaryif so is A.

Proof. According to [4, Theorem 8.11] (see also [35]), a necessary and sufficientcondition for the strict B-stationarity of N is that for each set I, the canonicalmap

ν : N ⊗R HomZ(B, (Q/Z)I)→ HomZ(HomR(N,B), (Q/Z)I)

defined by ν(n⊗ f)(g) = f(g(n)) is injective.

Since HomZ(B, (Q/Z)I) ∼= (B(I))c, we have TorR1 (M,HomZ(B, (Q/Z)I)) = 0 byour assumption on M . From the commutative diagram

0 −−−−→ N ⊗R HomZ(B, (Q/Z)I) −−−−→ A⊗R HomZ(B, (Q/Z)I)

ν

y α

yHomZ(HomR(N,B), (Q/Z)I) −−−−→ HomZ(HomR(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 in-jectives, and the elementary cogenerators. We will not deal with the model theoreticbackground of these notions; we just note that, by a classic theorem of Frayne, iftwo modules M and N are elementarily equivalent, then M is a pure submodule ofan ultrapower of N . The pure-injective hull PE(M) of a module M is elementarilyequivalent 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 byZimmermann [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, Proposition3.10]. It remains to prove that (3) implies (1).

Applying [4, Corollary 3.9], we get that all modules are B-stationary, where Bis any pure submodule of a pure-epimorphic image of a direct product of copies ofC. In particular, by [26, Lemma 3.3.1], this holds for any pure-injective moduleB which is elementarily equivalent to C (e.g., for B = PE(C)). Since B is pure-injective, 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 apure submodule of B.

We now turn to elementary cogenerators.

Definition 5.2. A pure-injective module E is called an elementary cogenerator, ifevery pure-injective direct summand of a module elementarily equivalent to Eℵ0 isa direct summand of a direct product of copies of E.


Notice that by [25, Corollary 9.36], for each module M there exists an elemen-tary cogenerator which is elementarily equivalent to M . This allows to prove thefollowing 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 directlimits. Then B contains a pure-injective module C, such that each module B ∈ Bis a pure submodule of a direct product of copies of C. Moreover, if B contains acogenerator, 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 closedunder direct summands. Now the closure properties of B imply that if B ∈ B iselementarily equivalent to a pure-injective module A, then alsoA ∈ 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 elemen-tary 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 iso-morphic to a pure submodule in a direct product of copies of C. Indeed, we haveBcc ∈ B; moreover Bcc is a pure-injective direct summand of (Bcc)ℵ0 which is amodule elementarily equivalent to Eℵ0 for some E ∈ S. Hence Bcc is a directsummand in a direct product, D, of copies of C (by Definition 5.2), and B is purein Bcc, and hence in D.

Moreover, if B ∈ B is a cogenerator for Mod-R, then C cogenerates Mod-R aswell.

Given a pure-injective cogenerator C as above, one can use the following lemmato find in any (strict) C-stationary module a rich supply of countably presentedC-stationary submodules.

Lemma 5.4. Let C be a pure-injective module which cogenerates Mod-R, and Mbe a strict C-stationary module. Then there exists an ℵ1-dense system L of strictC-stationary submodules of M such that

HomR(M,C)→ HomR(N,C) is surjective (†)

for every directed union N of modules from L.

Proof. Let Q = Cc. By [4, Proposition 8.14(2)], a module A is strict C-stationaryif 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 submodulesof M satisfying (2) from Definition 2.1 for λ = ℵ1: observe that the map v in thestatement of [4, Theorem 5.1(4)] is monic since the injectivity of v⊗RQ implies (†)(we use that C is a direct summand in Qc) and C is a cogenerator.

We extend F gradually by adding countable directed unions, noticing along theway that each newly added countably presented module N has the property thatN ⊆ M stays monic after applying − ⊗R Q (as TorR1 (−, Q) commutes with directlimits), and it is Q-Mittag-Leffler by [4, Corollary 5.2]. In this way, we eventuallyarrive at the desired ℵ1-dense system L of strict C-stationary submodules of M .Note that any directed union of modules from L satisfies (†) since (†) is impliedby the injectivity of the corresponding tensor map, which, in turn, is a propertypreserved by taking directed unions.

Remark 2. Lemma 5.4 holds with ℵ1 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 deconstructibleis provided by


Lemma 5.5. [29, Proposition 2.4] Let B be a filter-closed class of modules suchthat B cogenerates Mod-R. Then for each uncountable regular cardinal λ and eachmodule M ∈ ⊥B, there is a λ-dense system Cλ of submodules of M such thatM/N ∈ ⊥B for all N ∈ Cλ.

6. Countable type

Let C = (A,B) be a hereditary cotorsion pair in Mod-R. The Telescope Con-jecture for Module Categories asserts that C is of finite type whenever A and Bare closed under direct limits. This statement is known to be true in some specialcases [10]. On the other hand, the Countable Telescope Conjecture was proved infull generality in [29, Theorem 3.5]. It states that C is of countable type wheneverB is closed under unions of well-ordered chains.

As a first application of the tools developed above, we prove a version of theCountable 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 ofcountable type, and B is definable.

Proof. Let C be the pure-injective module constructed for B in Lemma 5.3.Let A0 = A≤ω. By induction on κ, we are going to prove that each κ-presented

module M ∈ A is A0-filtered. There is nothing to prove for κ ≤ ℵ0, so let κ beuncountable.

In the κ-presented module M ∈ A, we will construct by induction, for eachuncountable regular cardinal λ ≤ κ, a λ-dense system Cλ of submodules of M suchthat M/N ∈ A for each N ∈ Cλ, and

(2) Cλ ⊆ A.Our strategy is to start with a Cλ 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 everyN ∈ Cλ there exists L ∈ Cλ, such that N ⊆ L ∈ A. Then the family Cλ ∩ A is thedesired one, since each ascending chain in Cλ ∩A is actually an A-filtration and Ais closed under filtrations. Indeed, for each B ∈ B, if N ∈ Cλ and L ∈ A are suchthat 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 = ℵ1: Fix a free presentation of M

0 −−−−→ K −−−−→ R(κ) f−−−−→ M −−−−→ 0.

Let Cℵ1 be an ℵ1-dense system in M provided by Lemma 5.5. After taking theintersection with the set of all images f(R(X)) where X is a countable subset of κ,we can assume that Cℵ1 is compatible with this presentation, that is, each N ∈ Cℵ1has the form f(R(XN )) for a countable subset XN of κ.

Let K = Ker(f R(XN )) | N ∈ Cℵ1. Since B is closed under coproducts anddouble duals, it follows from Lemma 4.4 that K is strict C-stationary. Thus wecan use Lemma 5.4 to obtain another ℵ1-dense system, L, this time consisting ofsubmodules of K. Clearly, the system K ∩ L is ℵ1-dense as well. Our new Cℵ1 isdefined as N ∈ Cℵ1 | Ker(f R(XN )) ∈ L.

Notice that Cℵ1 ⊆ ⊥C. Indeed, given a module N ∈ Cℵ1 , each h : Ker(f R(XN )) → C can be extended to some h′ : R(XN ) → C by the property (†) fromLemma 5.4 and by the fact that M ∈ ⊥C.

Let N ∈ Cℵ1 . Since M/N ∈ A, N is the kernel of the epimorphism M → M/Nbetween two modules from A. By Lemma 4.2, M is strict C-stationary, so using


Lemma 4.4 again, we see that each N ∈ Cℵ1 is a countably presented C-stationarymodule from ⊥C. Using Proposition 4.3 (for G = C) together with the propertiesof C guaranteed by Lemma 5.3, we conclude that Cℵ1 ⊆ A.

Case 2. λ weakly inaccessible: As in the previous step, we start with a familyCλ provided by Lemma 5.5. Since each N0 ∈ Cλ is < µ-presented for some regularuncountable cardinal µ < λ, we simply choose N ′0 ∈ Cµ ⊆ A containing N0 as asubmodule. Then there is N1 ∈ Cλ containing N ′0, etc. The union, N , of the chainN0 ⊆ N ′0 ⊆ N1 ⊆ N ′1 ⊆ · · · satisfies N ∈ Cλ. Moreover, N ∈ A since the chainN ′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 N0 from Cλ (not necessarily satisfying condition (2) above), we easilybuild a continuous chain N0 = (N0

α | α < ν) of modules from Cν ⊆ A so thatN0 ⊆

⋃N0. Again, the union is in A. We continue by choosing N1 ∈ Cλ containing

this union, and a chain N1 = (N1α | α < ν) in Cν such that N0

α ⊆ N1α for all

α < ν, and N1 ⊆⋃N1. We proceed further by taking N2 ∈ Cλ, etc. Clearly,

N =⋃i<ω Ni ∈ Cλ. Furthermore, the ascending chain⋃

i<ω

N i0 ⊆

⋃i<ω

N i1 ⊆ · · · ⊆

⋃i<ω

N iα ⊆ · · ·

of submodules from Cν 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 (να | α < µ) of infinite cardinalswhich is cofinal in ν, such that ν0 > µ. Let N0 be arbitrary module from thefamily Cλ given by Lemma 5.5. We will produce a similar ascending chain as inCase 3., however this time, we have to pick the modules from different classes Cδ,hence we lose continuity. To overcome this problem, we use a well known singularcompactness argument:

We gradually build the chainsNi = (N iα | α < µ), for i < ω, and pick the modules

Ni ∈ Cλ in an alternating way, so that the following conditions are satisfied for alli < ω:

(a) N iα ∈ Cν+

α;

(b)⋃Ni ⊆ Ni+1;

(c) the generators niα,β | β < να of the modules N iα are fixed;

(d) the generators niγ | γ < ν of Ni are fixed;

(e) N iα ⊇ ni−1δ,β | δ < µ & β < min(νδ, να) ∪ niγ | γ < να.

Then⋃i<ω Ni is in Cλ, and it is equal to the union of the chain H = (

⋃i<ω N

iα |

α < µ). However, this chain is continuous (by condition (e)), and provides thus anA-filtration of

⋃H.

Having constructed the families Cλ (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 easyto see that Cκ already contains an A-filtration of M . For κ singular, we apply [29,Lemma 3.2]. By inductive hypothesis, we conclude that M is A0-filtered.

It remains to show that the modules in A0 have countably presented first syzy-gies. We know from Lemma 4.4 that their first syzygies are strict B-stationary. Inparticular, the first syzygies are RR

c-stationary. By [4, Proposition 8.14(1)], theyare then RR-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 conditionsof Theorem 6.1, but are not hereditary.


Recall that for n < ω, a module M is an FPn-module provided that M has aprojective resolution

· · · → Pk+1 → Pk → · · · → P1 → P0 →M → 0

such that Pk is finitely generated for each k ≤ n, [14, §VIII.4]. Let FPn denotethe class of all FPn-modules. (Notice that FP0 is the class of all finitely generatedmodules, and FP1 the class of all finitely presented ones.) The classes (FPn |n < ω) form a decreasing chain whose intersection is the class of all stronglyfinitely presented modules, that is, the modules possessing a projective resolutionconsisting of finitely generated modules, see [14, Proposition VII.4.5].

Example 6.2. Let n ≥ 2 and let Rn be a ring such that there exists a moduleM ∈ FPn\FPn+1. Such rings were constructed by Bieri and Stuhler using integralrepresentation theory, see [14, §VIII.5] and the references therein; in fact, Rn canbe taken as the integral group ring ZGn for a suitable finitely generated group Gn,and M = Z. (For the particular case of n = 2, a different and simpler example ofR2 is constructed in [23, Example 1.4].)

Let Cn = (An,Bn) be the cotorsion pair generated by FPn. Since n ≥ 2, Bn isclosed under direct limits by [19, Lemma 6.6], so Theorem 6.1 applies.

In order to see that Cn is not hereditary, we observe that An is not closed underkernels of epimorphisms: indeed, the syzygy module Ω(M) does not belong to FPn,and hence Ω(M) /∈ An. To see the latter fact, notice that An consists of directsummands of FPn-filtered modules by [19, 6.14]. Since Ω(M) is finitely generated,if Ω(M) ∈ An then it is a direct summand of a finitely FPn-filtered module, by [19,Theorem 7.10] used for ℵ0 (see also Lemma 7.1 below). However, FPn is closedunder extensions and direct summands, whence Ω(M) ∈ FPn, a contradiction.

7. Closed cotorsion pairs

In this section, we will characterize the tilting cotorsion pairs C = (A,B) suchthat C is closed, that is, lim−→A = A. In fact, in Theorem 7.6 we will go far beyondthe tilting setting: we will not require C to be hereditary or A to have boundedprojective dimension. Further characterizations for the closure of C will be givenlater in Theorem 9.2 and Corollary 9.4.

First, we recall the following important result going back to Hill (in the formpresented in [19, Theorem 7.10], for example):

Lemma 7.1. Let λ be a regular infinite cardinal. Let S be a class of < λ-presentedmodules and M a module possessing an S-filtration (Mα | α ≤ σ). Then there is afamily F of submodules of M such that:

(1) Mα ∈ 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 isclosed under direct limits of direct systems of that size:

Lemma 7.2. Let A be an ℵ1-deconstructible class. Assume that (lim−→A)≤µ ⊆ Afor an infinite cardinal µ. Then lim−→D ∈ A for each direct system D of cardinality≤ µ such that D consists of modules from A.


Proof. In view of (the proof of) [19, Lemma 2.14], it suffices to prove the claimunder the extra assumption of D being well-ordered. Let D = (Aγ , fγδ | δ ≤ γ < λ)be such a system with ℵ0 ≤ λ ≤ µ, and let M = lim−→γ<λ

Aγ .

Since A is ℵ1-deconstructible, it is also λ+-deconstructible. Hence for each Aγ ,

we have a system Fγ from Lemma 7.1 (for S = A<λ+

). Let ν be an infinitecardinal such that all the modules Aγ (γ < λ), are ≤ ν-generated. Possibly allowingrepetitions, we enumerate the set of generators of Aγ as aαγ | α < ν.

We shall inductively build compatible A≤λ-filtrations, (Aαγ | α ≤ ν), of themodules Aγ as follows:

First, we let A0γ = 0 for all γ < λ. Assume that Aαγ ∈ Fγ are defined for all

γ < λ and all α < β (where β < ν is fixed) so that

• for all γ < λ,⋃δ<γ fγδ(A

αδ ) ⊆ Aαγ ;

• aα−1γ ∈ Aαγ provided that α is non-limit.

If β is a limit ordinal, we put Aβγ =⋃α<β A

αγ . If β is non-limit, we choose for

each δ < γ a subset Gβδ of Aβδ of cardinality ≤ λ whose image under the canonical

projection Aβδ → Aβδ /Aβ−1δ generates the factor module. Then the subset

aβ−1γ

∪⋃δ<γ

fγδ(Gβδ)

has cardinality at most λ, and by condition (4) of Lemma 7.1 we can find Aβγ ∈ Fγcontaining it together with the submodule Aβ−1γ ∈ Fγ .

For each α < ν, let Lα = lim−→γ<λAαγ . We claim that the (not necessarily strictly)

ascending chain (Lα | α < ν) forms an A-filtration of M .For each α < ν, denoting fγ = fγ+1γ , we have the direct system of short exact

sequences

0 −−−−→ Aαγ⊆−−−−→ Aα+1

γ −−−−→ Aα+1γ /Aαγ −−−−→ 0yfγAαγ yfγAα+1

γ

y0 −−−−→ Aαγ+1

⊆−−−−→ Aα+1γ+1 −−−−→ Aα+1

γ+1/Aαγ+1 −−−−→ 0yfγ+1A

αγ+1

yfγ+1Aα+1γ+1

y...

......

of modules from A where the right-hand terms are ≤ λ-presented, whose directlimit is the sequence 0 → Lα → Lα+1 → Lα+1/Lα → 0. So Lα+1/Lα is ≤ λ-presented, and our assumption on A gives Lα+1/Lα ∈ A. The chain (Lα | α < ν) iscontinuous by the construction above (namely, the step when β was limit). Finally,lim−→α<ν

Lα ∼= M , and the claim is proved.

In the setting of the next lemma, all modules in the classA are strict B-stationaryby 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 pure-epimorphic image of a module from A is B-stationary.

Proof. First, the cotorsion pair C is complete and A is ℵ1-deconstructible by The-orem 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 → Bu →


Auπu→ Mu → 0 indexed by an inverse tree Iκ such that πu is a special A-precover

of Mu, and π = lim−→uπu : A → M , where A does not necessarily belong to A.

However, by our assumption on B, we have Ker(π) ∈ B. Since M is a pure-epimorphic image of a module from A, the epimorphism π is pure, too. By theproof of [29, Theorem 5.11], in this setting, the following holds true for each directsystem (Ki, kji | i, j ∈ J & i ≤ j) consisting of finitely presented modules withlim−→i∈J Ki = M : for each i ∈ J there exists s(i) > i such that ks(i)i factors through

some Au (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 (Lp, lqp | p, q ∈ I & p ⊆ q), derived from (Ki, kji | i, j ∈ J & i ≤ j),such that lim−→p∈I Lp

∼= 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 everynonempty 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 ∈ Iwith |p| < n (1 < n < ω). For q ∈ I with |q| = n, we let

δ(q) = s(cδ(p) | ∅ 6= p ( q).

Next, we let Lp = Kδ(p) and lqp = kδ(q)δ(p). It follows from our constructionthat lim−→p∈I Lp

∼= M and that, for every q ) p ∈ I, the homomorphism lqp factors

through a module Au from A.By our assumption on A, the direct limit of each countable subsystem of (Lp, lqp)

belongs to A. Since all modules in A are B-stationary by Lemma 4.2, we deducethat any countable subsystem of (Lp, lqp) is B-stationary, and so M is B-stationaryby [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 undercountable direct limits. Assume that M is a countably presented pure-epimorphicimage of a module from A.

Then there exists a direct system (Fn, fnm | m ≤ n < ω) of finitely presentedmodules such that M = lim−→n<ω

Fn, and fn+1,n factors through a module from Afor 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 overA≤ω (A<ω).

Proof. Let D = (Fn, fnm | m < n < ω) be a direct system of finitely presentedmodules with the direct limit (M,fn | n < ω). We can expand D to a direct systemof special A-precovers πn : An → Fn of the modules Fn (n < ω) so that the diagram

. . . −−−−→ Angn−−−−→ An+1 −−−−→ . . .

πn

y πn+1

y. . .

fn,n−1−−−−→ Fnfn+1,n−−−−→ Fn+1

fn+2,n+1−−−−−−→ . . .

is commutative. Then π = lim−→n<ωπn is a pure epimorphism: indeed, as Ker(π) ∈ B

by our assumption on B, the presentation of M as a pure-epimorphic image of amodule from A factors through π. Since the modules Fn are finitely presented,by possibly dropping some of them, we can assume that fn+1,n = πn+1νn whereνn ∈ HomR(Fn, An+1) for each n < ω. Then (M,fnπn | n < ω) is the direct limitof the direct system (An, gnm | m ≤ n < ω) where gnm = νn−1πn−1 . . . νmπm forall m < n < ω.


If C is of countable type, then each An is A≤ω-filtered. We use Lemma 7.1,for λ = ℵ1, to build inductively, for each n, a submodule A′n ∈ A≤ω of An whichcontains (at most countable) generating sets of Im(νn−1), gn−1(A′n−1) as well as ofa finitely generated module G such that G + Ker(πn) = An. We replace each Anby A′n, and πn and gn by their restrictions π′n and g′n to A′n, respectively. So thediagram

. . .g′n−1−−−−→ A′n

g′n−−−−→ A′n+1

g′n+1−−−−→ . . .

π′n

y π′n+1

y. . .

fn,n−1−−−−→ Fnfn+1,n−−−−→ Fn+1

fn+2,n+1−−−−−−→ . . .

is commutative. As above, (M,fnπ′n | n < ω) is the direct limit of the direct system

(A′n, g′nn | m ≤ n < ω) where g′nm = νn−1π

′n−1 . . . νmπ

′m for all m < n < ω. In

particular, M is a Bass module over A≤ω.If C is of finite type, then each An 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 canw.l.o.g. assume that in the special A-precover πn : An → Fn, the module An isA<ω-filtered. Using Lemma 7.1 for λ = ℵ0, we replace each An by its submoduleA′n ∈ A<ω, and πn and gn by their restrictions π′n and g′n to A′n, respectively,so that Im(νn) ⊆ A′n+1, and the diagram above is commutative. As above, weconclude that M is a Bass module over A<ω.

Finally, we recall a useful description of the class of pure-epimorphic images ofmodules 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 thatB is closed under direct limits, and let B′ be the class of all pure-injective modulesfrom B. Then ⊥B′ coincides with the class A of all pure-epimorphic images ofmodules from A.

We can now characterize the cotorsion pairs (A,B) with both classes closed underdirect 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 suchthat 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 pure-epimorphic images. In order to verify closure under pure submodules, we take apure submodule X of a module A from A and show that HomR(X,−) is exact onthe short exact sequence 0→ C → E(C)→ Z → 0 given by the injective envelopeof C. Since the first cosyzygy Z of the pure-injective module C is pure-injective (seee.g. [19, Lemma 6.20]), every f ∈ HomR(X,Z) can be extended to a homomorphismf ′ ∈ HomR(A,Z). Now f ′ factors through E(C) as A ∈ A = ⊥C. Restricting toX, 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 A = ⊥(∏S), where A denotes


the class of all pure-epimorphic images of modules from A. The direct product canbe replaced by the pure-injective hull, C, of

⊕S, which is a discrete pure-injective

direct summand in∏S. Further, we can assume w.l.o.g. that C cogenerates Mod-R

(possibly replacing it by C ⊕Q where Q is an injective cogenerator).

Let M ∈ A = ⊥C be ≤ κ-presented. By induction on κ, we will prove thatM ∈ A (then A = ⊥C, and (1) will hold). The case of κ = ℵ0 follows from (4) byLemma 7.4.

Assume that κ is uncountable and all <κ-presented modules from A are in A.Consider a free presentation of M ,

0 −−−−→ K −−−−→ R(κ) f−−−−→ M −−−−→ 0.

Since R(κ) is (strict) C-stationary, so is K by Lemma 4.4 and Lemma 7.5. Considerthe family L for K provided by Lemma 5.4. By Lemma 7.3, M is (strict) C-stationary as well. Using Lemma 5.4 again, we can build in M an ℵ1-dense systemof countably presented submodules, w.l.o.g. of the form f(R(I)) for a countablesubset I of κ. We denote this system by H, and for every H ∈ H, take a countablesubset IH ⊆ κ such that f(R(IH)) = H.

Let M = H ∈ H | Ker(f R(IH)) ∈ L. Then M is ℵ1-dense and consists of

modules from A, by the property of the modules in L and by the assumption ofM ∈ ⊥C = A. So M⊆ A by the inductive premise for κ = ℵ0.

Next, we fix a continuous strictly ascending chain (δγ | γ < cf(κ)) of infiniteordinals < κ which is cofinal in κ. Then we can easily build a continuous ascendingchain (Mγ | γ < cf(κ)) consisting of ⊆-directed subsets ofM such that |Mγ | = |δγ |and

⋃γ<cf(κ)Mγ = M. We claim that the ascending chain F = (

⋃Mγ | γ <

cf(κ)) is actually an A-filtration of M (proving that M ∈ A).

Indeed, all modules in F are elements of A. Moreover, the property (†) fromLemma 5.4 is preserved when taking directed unions. It follows that all consecutivefactors in F belong to A, and hence to A by the inductive premise.

If C = (A,B) is a cotorsion pair of finite type, then the class B is definable (cf.[19, Example 6.10]), so Theorem 7.6 applies. In fact, the Bass modules over A<ωare sufficient in this case (see Lemma 7.4).

Corollary 7.7. Let R be a ring, and C = (A,B) be a cotorsion pair of finite type.Then C is closed if and only if A contains all Bass modules over A<ω.

8. Locally free modules and approximations

Now we can present several consequences for the structure of locally free modules.We start with the deconstructibility.

Let (A,B) be a cotorsion pair of countable type and let L denote the class of alllocally A≤ω-free modules. Assume there exists a Bass module N over A≤ω suchthat N /∈ A. Then, by [31, Theorem 6.2], the class L is not deconstructible. Thus,Theorem 7.6 gives

Theorem 8.1. Let R be a ring and C = (A,B) a cotorsion pair with lim−→B = B.

Let L be the class of all locally A≤ω-free modules. Then L is deconstructible, if andonly if A contains all Bass modules over A≤ω.

Corollary 8.2. If T is a tilting module, then the class all locally T -free modules isdeconstructible, if and only if T is

∑-pure-split, if and only if A contains all Bass

modules over A<ω.

Proof. Every tilting cotorsion pair C is of finite type, thus Corollary 7.7 applies.Further, by [19, Proposition 13.55], C is closed if and only if T is Σ-pure-split.


Remark 3. Corollary 8.2 provides for a common generalization of several resultsfrom [31, §6], where the classes of locally T -free modules were shown not to bedeconstructible for various instances of non-

∑-pure split tilting modules T . In

particular, the assumption in [31, §6] of T being a direct sum of countably presentedmodules, turns out to be redundant.

Let us now consider the existence of locally free precovers. The prototype caseof flat Mittag-Leffler modules has already been treated in Section 3; we now givean analogous answer in several other cases. Note that we have no bound on thecardinality of R, but in each of the cases (i)-(iii) below, the tilting modules T haveprojective dimension ≤ 1:

Theorem 8.3. Let R be a ring, T a tilting module, and C = (A,B) the tiltingcotorsion pair induced by T . Moreover, assume that either (i) R is right hereditary,or (ii) R is 1-Iwanaga-Gorenstein, or (iii) T is 1-tilting and R is right perfect.

Then the class of all locally T -free modules is (pre)covering if and only if T isΣ-pure-split.

Proof. By Corollary 8.2, if T is not Σ-pure-split, then there exists a Bass moduleN over A<ω such that N /∈ A. Notice that each of our assumptions on R impliesthat lim−→P

<ω1 ⊆ P1, so N ∈ P1. Since A≤ω coincides with the class of all countably

presented locally T -free modules, N is not (a direct summand of) a locally T -freemodule. So N satisfies the assumptions of Lemma 3.2, whence N has no precoverin the class of all locally T -free modules.

Conversely, if T is Σ-pure split, then the class L of all locally T -free modulescoincides with A. Indeed, we have A ⊆ L by [31, Theorem 4.5] and [19, Theorem7.13], and the other inclusion follows by Corollaries 7.7 and 8.2, in particular thefact that A = lim−→A. Thus L is covering by e.g. [32, Theorem 2.2.8].

We can also express the local freeness in terms of B-stationarity:

Theorem 8.4. Let (A,B) be a cotorsion pair with B = lim−→B. Then a module M

is locally A≤ω-free, if and only if M is a B-stationary pure-epimorphic image of amodule from A.

In particular, the class of all locally A≤ω-free modules is closed under pure sub-modules.

Proof. The local A≤ω-freeness of M just says that M possesses an ℵ1-dense systemof submodules from A≤ω. Since M is the directed union of these submodules, it is apure-epimorphic image of their direct sum. Moreover, all modules in A are (strict)B-stationary by Lemma 4.2, whence M is B-stationary by [21, Corollary 2.6(5)].

For the if-part, we use Lemma 5.3 to obtain a pure-injective module C ∈ B suchthat C is a cogenerator for Mod-R, and each module B ∈ B is a pure submoduleof a direct product of copies of C.

Next, we consider a free presentation of M ,

0 −−−−→ K −−−−→ R(X) f−−−−→ M −−−−→ 0.

Since M is a pure-epimorphic image of a module in A, it follows from Lemma 7.5and Lemma 4.4 that M ∈ ⊥C and K is strict B-stationary. Applying Lemma 5.4,we obtain an ℵ1-dense system L consisting of strict C-stationary submodules of K.Since M is B-stationary, it is (strict) C-stationary by Remark 1. Using Lemma 5.4again, this time for M and C, we obtain an ℵ1-dense system H of submodules inM , where each H ∈ H is w.l.o.g. of the form f(R(XH)) for a countable subset XH

of X.Now, the set M = N ∈ H | Ker(f R(XH)) ∈ L is an ℵ1-dense system

of submodules in M , and it consists of strict C-stationary countably presented


modules from ⊥C (use (†) for L and M ∈ ⊥C). By Proposition 4.3 and thedefinition of C, we infer M⊆ A≤ω, whence M is locally A≤ω-free.

The final claim follows from the fact that a module M is B-stationary, if and onlyif it is (strict) C-stationary, see [4, Corollary 3.9] and Remark 1. The latter propertyis inherited by pure submodules by [4, Corollary 8.12(1)]. Moreover, the class of allpure-epimorphic images of modules from A equals ⊥C by Lemma 7.5, and as such,it is always closed under pure submodules (cf. the proof of Theorem 7.6).

9. Covers and pure-injectivity

In the late 1990s, Enochs asked whether each covering class of modules A isclosed under direct limits (see e.g. [19, Open Problems 5.4]). This problem is stillopen in general. We will give a positive answer for the case when A fits in acotorsion pair C = (A,B) such that B is closed under direct limits. In particular,we have a positive answer when C is any tilting cotorsion pair.

We will approach the problem via an extension of Theorem 7.6 by further equiv-alent conditions. First, we need a proposition of independent interest:

Proposition 9.1. Let (A,B) be a cotorsion pair such that B(ω) ∈ B for everyB ∈ B. Assume that h : A→M is an A-cover. Then h is an isomorphism, if andonly if the embedding Ker(h) ⊆ A is locally split, and M (ω) has an A-cover.

Proof. The only-if part is trivial. For the if-part, consider x ∈ Ker(h). By thehypothesis, there is a homomorphism g : A → Ker(h) such that g(x) = x. Notethat h is a special A-precover by the Wakamatsu lemma [19, Lemma 5.13]. By ourassumption on B, the coproduct map h(ω) : A(ω) → M (ω) is a special A-precover.We can now use [32, Theorem 1.4.7] (since the assumption of M (ω) having an A-cover, which is missing from its statement, is satisfied here) and obtain that h(ω) isan A-cover. By [32, Theorem 1.4.1], we conclude that there exists m < ω such that0 = gm(x) = x. This proves that Ker(h) = 0, whence h is an isomorphism.

Theorem 9.2. Let R be a ring and C = (A,B) be a cotorsion pair in Mod-R suchthat B is closed under direct limits. Then the following conditions are equivalent:

(1) C is closed;(2) every module (in B) has an A-cover;(3) Ker(C) is closed under countable direct limits;(4) (lim−→A)≤ω consists of (strict) B-stationary modules;

(5) Ker(C) consists of Σ-pure-split modules;

(6) The class L of all locally A≤ω-free modules coincides with the class A ofall pure-epimorphic images of modules from A.

Proof. First, we recall that by Theorem 6.1, the cotorsion pair C is of countabletype, hence it is complete, and B is a definable class. Also, by Lemmas 5.3 and 7.5,there is a pure-injective module C with A = ⊥C and such that all modules from Bare isomorphic to pure submodules in a product of copies of C.

(1)⇒ (2). This is well known: each precovering class closed under direct limitsis covering, cf. [32, Theorem 2.2.8].

(2)⇒ (3). Let M be a countable direct limit of modules from Ker(C) = A ∩ B.Then there exist modules Mi ∈ Ker(C), morphisms gi : Mi → Mi+1, and a pureexact sequence

0→⊕i<ω

Mig→⊕i<ω

Mih→M → 0

such that g Mi = idMi− gi for each i < ω, e.g. by [19, Lemma 2.12]. It is easy to

see that g is locally split.


By our assumption on B, h is a special A-precover of M , and M ∈ B. By (2)and [32, Theorem 1.2.7], there is a direct summand A of

⊕i<ωMi such that h A

is A-cover of M , and A ∩ Ker(h) is a direct summand in Ker(h). Note that theinclusion A ∩ Ker(h) ⊆ A inherits the property of being locally split from g. ByProposition 9.1, we conclude that M ∼= A ∈ A ∩ B.

(3) ⇒ (4). By Lemma 7.4, each M ∈ (lim−→A)≤ω is a direct limit of a countable

direct system, (An | n < ω), of modules from A. We expand this direct system,to a direct system of short exact sequences induced by special B-preenvelopes. Itsdirect limit is a short exact sequence 0 → M → B → A → 0 where B ∈ A ∩ B by(3), and A ∈ lim−→A. Then B is strict B-stationary by Lemma 4.2, and A ∈ ⊥D foreach pure-injective module D ∈ B by Lemma 7.5. So M is strict B-stationary byLemma 4.4.

(4) ⇒ (1). Since (lim−→A)≤ω ⊆ ⊥C, it follows from (4) and Proposition 4.3 that

(lim−→A)≤ω ⊆ A. So C is closed by Theorem 7.6.

(1) ⇒ (5). This is clear, since B is definable, and A is closed under pure-epimorphic images by Theorem 7.6.

(5)⇒ (6). First, notice that B ∩ A ⊆ A: Indeed, if M ∈ B ∩ A, and f : A→Mis a special A-precover of M , then f is a pure epimorphism and A ∈ A ∩ B, hencef splits by (5).

Next, for a module N ∈ A, we form a special B-preenvelope g : N → B.Then B ∈ A by the previous argument. In particular, B is strict B-stationary byLemma 4.2, and N is strict B-stationary by Lemma 4.4. So N ∈ L by Theorem8.4. Conversely, L ⊆ lim−→A

≤ω ⊆ A, whence (6) holds.

(6) ⇒ (4). Let M ∈ (lim−→A)≤ω. By (6), M ∈ L, so M is B-stationary byTheorem 8.4.

Remark 4. Notice that by the proof above, the condition in (4) can be relaxedfurther to assuming the stationarity with respect to a single (pure-injective) moduleC.

In what follows, we will see that statements contained in the preceding theoremactually generalize (part of) the famous Theorem P by Hyman Bass. The nextlemma sheds more light on the structure of Ker(C), showing that it is quite similarto the tilting setting:

Lemma 9.3. Let C = (A,B) be a cotorsion pair with B closed under direct limits.Then Ker(C) = Add(K) for a module K.

Proof. Put µ = |R|+ ℵ0, and let K be the direct sum of a representative set of all≤µ-presented modules in Ker(C). Let us denote this representative set by K. Wehave to show that each M ∈ Ker(C) is (isomorphic to) a direct sum of ≤µ-presentedmodules, which is equivalent to M being K-filtered.

We will prove this by induction on λ where λ is the minimal cardinal such thatM is λ-presented. It holds trivially for λ ≤ µ. Assume that λ > µ is regular.Since M ∈ A, M is A<λ-filtered by Theorem 6.1 and [19, Theorem 7.13]. Fix onesuch filtration of M and use Lemma 7.1 with S = A<λ to obtain a family F ofsubmodules of M . Note that F ⊆ A. We build a continuous chain (Mα | α ≤ λ)of submodules of M such that M0 = 0,Mλ = M and Mα are <λ-presented, for allα < λ, as follows:

For α limit, we put Mα =⋃β<αMβ . Let α = β + 1. If α is odd, we define Mα

as a member from F given by Lemma 7.1(4) for N = 0 and X = Mβ .Now, let α be even. We choose Mα as a pure submodule of M containing Mβ

with |Mα| ≤ |Mβ | + µ < λ (it is possible by [19, Lemma 2.25(a)]). Then Mα ∈ B


since M ∈ B and B is definable. Note, however, that Mα need not be a member ofF .

Consider the subchain C = (Mα | α ≤ λ, α is limit). Using the properties ofB and F , we see that each member of C as well as each of its consecutive factorsbelongs to Ker(C). The consecutive factors are <λ-presented, and we can use theinductive assumption to deduce that M is K-filtered.

If λ is singular, we use [19, Theorem 7.29]. As the sets Sκ, µ < κ < λ regular,witnessing κ-K-freeness of M , we choose F ∩ B<κ, where F is the family given byLemma 7.1 used for the regular cardinal κ and S = A<κ. It is straightforward toverify that Sκ has the desired properties stated in [19, Definition 7.27].

Recall that a module M has a perfect decomposition if it has a decompositionin modules with local endomorphism ring, and every module N ∈ Add(M) has asemiregular endomorphism ring SN = EndR(N), i.e., idempotents lift modulo theJacobson radical J(SN ), and SN/J(SN ) is a von Neumann regular ring.

The notion of perfect decomposition has many equivalent definitions, cf. [9]. Inparticular, M has perfect decomposition if and only if every pure monomorphismfrom any direct sum

⊕α∈I Nα into a module from Add(M) splits whenever all

of its finite subsums split. For example, every Σ-pure-split module has a perfectdecomposition.

Our next corollary extends (part of) Bass’ Theorem P (which is the case ofC = (P0,Mod-R) and K = R), and more in general, it also covers the tiltingsetting (for K = T a tilting module).

Corollary 9.4. Let R be a ring, and C = (A,B) be a cotorsion pair with B =lim−→B. Let K be a module such that Ker(C) = Add(K) (see Lemma 9.3). Then thefollowing conditions are equivalent:

(1) A = lim−→A;

(2) every module (in B) has an A-cover;(3) every module in Ker(C) has a semiregular endomorphism ring;(4) K has perfect decomposition;(5) K is Σ-pure-split;(6) every module (in B) has a Ker(C)-cover.

Proof. The first two conditions are the same as in Theorem 9.2. Condition (5) isequivalent to the same condition there. (4)⇒ (3) is trivial.

(5)⇒ (4) follows by the equivalent definition mentioned above.(3)⇒ (6) follows from [1, Proposition 4.1] and the fact that Ker(C) = Add(K).(6) ⇒ (2). Observe that any Ker(C)-cover of a module B ∈ B is, in fact, an

A-cover: this follows, for instance, from [32, Theorem 1.2.7] applied to a specialA-precover (hence Ker(C)-precover) of B.

The condition (5) above cannot be replaced by ‘K is Σ-pure-injective.’ Whileeach Σ-pure-injective module is Σ-pure-split, the converse is not true in general,even for tilting modules: If R is right perfect, then K = R is certainly Σ-pure-split,but it need not be Σ-pure-injective. In fact, [34, §2] contains an example of a rightartinian ring which is not right pure-injective.

There is, however, a case where Σ-pure splitting and Σ-pure-injectivity coincide,namely when the cotorsion pair (A,B) has the property that A<ω is covariantlyfinite. This is always true when R is left noetherian and A<ω consists of modulesof projective dimension ≤ 1 (see [6, Proposition 2.5]). A further case is providedby the following observation.


Lemma 9.5. Let R be a left hereditary ring, and let C be a class consisting of FP2-modules, such that C is closed under extensions, direct summands, and contains R.Then C is covariantly finite.

Proof. By [19, Theorem 8.40], we can compute lim−→C as Ker TorR1 (−,A) where A =

Ker TorR1 (C,−). Now A is closed under direct limits, and by the Ext-Tor-relations,it is the left-hand class of a cotorsion pair of left R-modules. By the assumption onR and [19, Lemma 9.7], A = lim−→A

<ω, whence lim−→C = Ker TorR1 (−,A<ω). It yieldsthat lim−→C is closed under direct products. By the classic result of Crawley-Boevey,this amounts to C being covariantly finite in the class of all finitely presented rightmodules.

We state the next auxiliary result in a more general setting:

Lemma 9.6. Let C be a covariantly finite subcategory of the category of all finitelypresented modules, and T ∈ lim−→C be a module. Assume that every countable directsystem in C is T -stationary. Then T is Σ-pure-injective.

Proof. We will verify condition (4) of Lemma 5.1. Let M be an arbitrary countablypresented module. We express M as the direct limit of a countable direct system(Fn, fn | n < ω) of finitely presented modules. Since C is covariantly finite, we canexpand this direct system into a direct system of C-preenvelopes pn : Fn → Cn.

Now, apply the contravariant Hom-functor HomR(−, T ). From T ∈ lim−→C, it

follows (using Lenzing’s result characterizing modules in lim−→C) that HomR(pn, T )

are surjective maps. Since the inverse system (HomR(Cn, T ))n<ω is Mittag-Lefflerby our assumption, the inverse system (HomR(Fn, T ))n<ω is the epimorphic imageof a Mittag-Leffler inverse system. As such, it must be Mittag-Leffler as well by [20,Proposition 13.2.1]. In other words, M is T -stationary, and Lemma 5.1 applies.

Corollary 9.7. Let C = (A,B) be a cotorsion pair of finite type. Assume that A<ωis covariantly finite in the category of all finitely presented modules.

Then the equivalent conditions of Theorems 7.6 and 9.2 are further equivalent tothe condition that Ker(C) consists of Σ-pure-injective modules, which amounts toKer(C) = Add(K) for a product-complete module K.

Proof. If Ker(C) consists of Σ-pure-injective modules, then (5) from Theorem 9.2trivially follows. For the opposite direction, assume condition (4) of Theorem 9.2and use Lemma 9.6 for C = A<ω. (Notice that Ker(C) ⊆ A ⊆ lim−→C because C is of

finite type.)Finally, assume that C is closed. Since C is of finite type, we have A = lim−→A

<ω,and we can use Crawley-Boevey’s result to deduce that A is closed under directproducts. Thus Ker(C) is closed under direct products as well. Taking K as inLemma 9.3, we see that Ker(C) = Add(K) for a product-complete module K.

10. An application: pure-semisimple hereditary rings

Our previous results allow us to give various characterizations of hereditary ringswhich are related to pure-semisimplicity.

Theorem 10.1. Let R be right hereditary. The following statements are equivalent.

(1) Every cotorsion pair in Mod-R is tilting.(2) Every cotorsion pair in Mod-R is of finite type.(3) All cotorsion pairs in Mod-R form a set.(4) Every class of right R-modules closed under transfinite extensions and direct

summands and containing R is deconstructible.


(5) Every class of right R-modules closed under transfinite extensions and directsummands and containing R is (special) precovering.

(6) Every 1-tilting right R-module is Σ-pure-split.

Moreover, under these conditions, R is right artinian.

Proof. (1)⇒ (2) holds by [19, Theorem 13.46]. (2)⇒ (3) is trivial.(3)⇒ (4) follows by contradiction: if such class C is not deconstructible, then(

⊥(C≤λ⊥), C≤λ⊥) | λ infinite cardinal

is not a set. (Note that the closure properties of C imply that ⊥(C≤λ⊥

)⊆ C for all

λ, by [19, Corollary 6.13].)(4)⇒ (5). The condition (4) yields that any such class is the left-hand class of a

cotorsion pair generated by a set (cf. [19, Corollary 6.13]), hence the class is specialprecovering by [19, Theorem 6.11(b)].

(5) ⇒ (6). By Theorem 3.3, it follows from (5) that R is right perfect. UsingTheorem 8.3 (iii) and [31, Theorem 4.5], we conclude that all 1-tilting modules haveto be Σ-pure-split.

(6) ⇒ (1). Let (A,B) be a cotorsion pair. Since R is right hereditary, thecotorsion pair C generated by A<ω is 1-tilting. By [19, Lemma 9.7], A ⊆ lim−→A

<ω.

The latter class, however, is the left-hand class of C by (6) and Corollary 9.4, whenceC = (A,B).

Finally, we have already seen that R must be right perfect when (1)–(6) holdtrue. Further, the cotorsion pair generated by the class of all finitely presentedmodules is 1-tilting, and so, by (6), its left-hand class is closed under direct limits.It follows that the tilting class coincides with the class of all injective modules whichis therefore closed under direct limits. This proves that R is right noetherian.

Remark 5. The implications (1)⇒ (2)⇒ (3)⇒ (4)⇒ (5)⇒ (6) hold true over anyring R. If R is left hereditary, then (6) implies that every 1-tilting right R-moduleis Σ-pure-injective and even product-complete, see Corollary 9.7.

Recall that a ring R is left pure-semisimple if all left R-modules are (Σ-)pure-injective. This is equivalent to the fact that every left R-module is a direct sum offinitely generated modules. Further, a module M is a splitter if Ext1R(M,M) = 0.

Proposition 10.2. [2, Proposition 4.2], [16, Remark 4.1(2)] Let R be a left pure-semisimple hereditary ring. Then every cotorsion pair in R-Mod is generated by afinitely generated tilting module, and every indecomposable (finitely generated) leftR-module is a splitter.

Observe that the second property above then also holds for right modules.

Remark 6. Let R be a left pure-semisimple hereditary ring. Then every indecom-posable finitely generated right R-module is a splitter. Indeed, every such moduleA is endofinite, and by [33, Lemma 5.1(2)] it follows that A ∼= A++, where +denotes the local dual. Now we know that A+, being an indecomposable finitelygenerated left module, is a direct summand of a tilting module T with cotorsionpair C = (A,B), so it is in the kernel of C. By [4, Lemma 9.4 (3) and (5)], it followsthat A ∼= A++ is in the kernel of the dual cotilting cotorsion pair (C,D) in Mod-R,so it is isomorphic to a direct summmand in a direct product of copies of a cotiltingmodule, and thus it is a splitter.

As for the first property, we don’t know whether it is left-right-symmetric. Infact, this would imply the long-standing Pure-Semisimplicity Conjecture.


Proposition 10.3. Let R be a left pure-semisimple hereditary ring. The followingstatements are equivalent.

(1) R has finite representation type.(2) R satisfies (one of) the conditions in Theorem 10.1.(3) There are only finitely many tilting left modules up to equivalence.(4) There are only finitely many tilting right modules up to equivalence.

Proof. By [5, Theorem 2.10] and [2, Theorem 5.6], the ring R has finite repre-sentation type if and only if a certain tilting left R-module W is endofinite, orequivalently, a certain tilting right R-module T satisfies the descending chain con-dition on cyclic EndR(T )-submodules. But the latter follows from condition (6) inTheorem 10.1, since T is then even Σ-pure-injective, cf. Remark 5.

(3)⇒(1). Every tilting left module is equivalent to a finitely generated one,so there are only finitely many indecomposable modules up to isomorphism thatoccur as direct summands in a tilting module. Since every (finitely generated)indecomposable left module is a splitter, thus a direct summand in some tiltingmodule, the claim follows.

(4)⇒(3). Since any tilting left module is product-complete, it is also cotilting,and thus equivalent to T c for some tilting right module T by [19, Theorems 15.31and 15.18(a)]. The rest follows by [19, Theorem 15.18(b)].

11. An example: Bass modules for Lukas’ tilting over hereditaryartin algebras

We have seen that non-∑

-pure-split tilting modules T give rise to non-deconstruc-tible, and even non-precovering, classes of all locally T -free modules. The point isthe existence of a Bass module N over A<ω such that N /∈ A (see Corollary 8.2and Theorem 8.3).

We will now present a concrete construction of such a Bass module for theparticular case when R is a hereditary artin algebra of infinite representation typeand T is the Lukas tilting module (see [19, Example 13.7(b)] and [31, §7]).

In this case, it is well known that the representative set of all indecomposablefinitely generated modules can be divided in three parts: p, q, and t formed by thethe indecomposable preprojective, preinjective, and regular modules, respectively.The modules in the class add(p) (add(q), add(t)) are called the finitely generatedpreprojective (preinjective, regular) modules.

In our setting, A<ω = add(p), while A is the class of all Baer (= p-filtered)modules, L the class of all locally Baer modules, and lim−→A = lim−→ p coincides with

the torsion-free class F in the torsion pair (T ,F) generated by t, cf. [7, 8, 31].In this section, a strictly increasing chain of finitely generated modules

P : 0 = P0 ( P1 ( · · · ( Pn ( Pn+1 ( . . .

is called special provided that P consists of preprojective modules, but Pn+1/Pn ∈is regular for each 0 < n < ω.

Lemma 11.1. Let P be a special chain. Then the module N =⋃n<ω Pn is a Bass

module over A<ω such that N /∈ A.

Proof. Clearly, N ∈ lim−→ωA<ω.

Assume N is a Baer module. As shown in [7], N has a p-filtration (Qn | n < ω)consisting of finitely generated preprojective modules. Let i < ω be such thatP1 ⊆ Qi. Then there is an epimorphism N/P1 → N/Qi. However, this contradictsthe fact that in the torsion pair (T ,F), the torsion-free class F contains all Baer


modules, so N/Qi ∈ F , while T contains all finitely generated regular modules, soN/P1 ∈ T . This proves that N /∈ A.

So it suffices to construct the special chains. We will distinguish the tame casefrom the wild one. For the latter, we will employ the notion of a mono orbit dueto Dagmar Baer (see [11] and [24]):

Definition 11.2. Let R be a wild hereditary artin algebra, and P ∈ p. The τ−1-orbit of P (that is, the set O(P ) := τ−n(P ) | n < ω where τ−1 = TrD is theAuslander-Reiten translation) is called a mono orbit provided that

(a) for each X ∈ O(P ) and each Y ∈ p, all non-zero homomorphisms f : X →Y are injective; and

(b) every non-zero homomorphism between elements of O(P ) has a regularcokernel.

Baer proved that for each hereditary artin algebra of wild representation type,there exists an indecomposable projective module P whose τ−1-orbit is a monoorbit (see [11, Proposition 2.2]). Of course, O(Q) is then a mono orbit for eachQ ∈ O(P ).

Lemma 11.3. (1) Assume R is tame. Let P be a non-zero finitely generatedpreprojective module, and Sn | 0 < n < ω be any sequence of simpleregular modules from (not necessarily distinct) homogenous tubes. Thenthere exists a special chain P such that P1 = P , and Pn+1/Pn ∼= Sn foreach 0 < n < ω.

(2) Assume R is wild. Let P ∈ p be such that O(P ) is a mono orbit. ThenO(P ) contains a special chain such that P1 = P .

Proof. (1) Let P1 = P , and assume that Pn is defined for some 0 < n < ω. Wecan factorize the embedding of Pn into its injective envelope through the homoge-nous tube containing Sn. Since the elements of the tube are Sn-filtered, nec-essarily HomR(Pn, Sn) 6= 0. As τ(Sn) = Sn, the Auslander-Reiten formula gives

Ext1R(Sn, Pn) 6= 0. So there is a non-split extension 0→ Pn → Pn+1gn→ Sn → 0.

It remains to prove that Pn+1 is preprojective. Clearly, Pn+1 cannot have anynon-zero preinjective direct summands. If Tn is a non-zero regular direct summandof Pn+1, then gn(Tn) = Sn, because Sn is simple regular. Since R is tame, thecategory of all finitely generated regular modules is abelian, so the kernel of gn Tnis the zero submodule of Pn. Thus gn splits, a contradiction.

(2) (Kerner) Consider 0 < k < ω such that HomR(P, τ−kP ) 6= 0. Since O(P )is a mono orbit, there is an exact sequence 0 → P → τ−kP → X → 0 where Xis regular. Iterated application of the Auslander-Reiten translation τ−1 yields theexact sequences 0 → τ−nkP → τ−(n+1)kP → τ−nX → 0 for all 0 < n < ω. Sincethe modules τ−nX (n < ω) are regular, it suffices to put P1 = P and Pn+1 = τ−nkPfor each 0 < n < ω.

Acknowledgement : We thank Otto Kerner for suggesting the use of mono orbitsfor constructing special chains in the wild case.

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Dipartimento di Informatica - Settore di Matematica, Universita degli Studi di

Verona, Strada le Grazie 15 - Ca’ Vignal, 37134 Verona, Italy

E-mail address: [email protected]

Charles University, Faculty of Mathematics and Physics, Department of Algebra,

Sokolovska 83, 186 75 Praha 8, Czech RepublicE-mail address: [email protected]

Charles University, Faculty of Mathematics and Physics, Department of Algebra,Sokolovska 83, 186 75 Praha 8, Czech Republic

E-mail address: [email protected]

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