Direct limits of modules of finite projective dimension

Lidia Angeleri Hügel and Jan Trlifaj ^{∗}

Dedicated to Paul Eklof on his 60th birthday

Abstract We describe in homological terms the direct limit closure of a class C of modules over a ring R. We also determine the closure of the cotorsion pair ℭ = (A, B) cogenerated by C. As an application, we solve a problem of Fuchs and Salce on the structure of direct limits of modules of projective dimension at most one over commutative domains. Then we consider the case when R is a right coherent ring and C = P^{<∞}, the class of all finitely presented modules of finite projective dimension. If findim R < ∞ then ℭ is a tilting cotorsion pair induced by a tilting module T. We characterize closure properties of A in terms of properties of T. Finally, we discuss an example where A is not closed under direct limits.

Let R be a ring. Denote by P the class of all modules of finite projective dimension, and by P^{<∞} the class of all finitely presented modules in P. For n < ω let P_{n} be the class of all modules of projective dimension at most n, and let P_{n}^{<∞} be the corresponding subclass of P^{<∞}.

In this paper, we study the categories \underrightarrow{lim} P_{n} and \underrightarrow{lim} P_{n}^{<∞} of all direct limits of modules in P_{n} and P_{n}^{<∞}, respectively. To this end, we consider the complete cotorsion pair (A_{n}, B_{n}) cogenerated by P_{n}^{<∞} and investigate the class A_{n}.

Our main tool is a homological description of \underrightarrow{lim} A_{n}. We show that in many cases the limit closure \underrightarrow{lim} A of the first component in a cotorsion pair ℭ = (A, B) consists of all modules X satisfying Ext_{R}^{1} (X, B) = 0 for each pure-injective module B ∈ B; there is also a characterization of \underrightarrow{lim} A in terms of vanishing of Tor (see Section 2).

This result allows us to discuss a more general question: What is the smallest complete cotorsion pair \overline{ℭ} = (\overline{A}, \overline{B}) containing ℭ with \overline{A} being closed under direct limits? The question is of particular interest because of a classical result of Enochs saying that in this case \overline{A} is a covering class, and \overline{B} an enveloping class, in ModR.

∗Research of the first author supported by a HWP-grant of LMU Munich. Second author supported by grants GAČR 201-00-0766 and MSM 113200007.

In Corollary 2.4, we answer the question for cotorsion pairs cogenerated by classes of finitely presented modules over right coherent rings.

Section 3 deals with an application to the particular case when n = 1 and R is a commutative domain. In Theorem 3.5, we solve a problem of Fuchs and Salce ([15, Problem 22 on p.246]) by showing that a module belongs to \underrightarrow{lim} P_{1} if and only if it has flat dimension at most one. Furthermore, we investigate the divisible modules of projective dimension at most one and answer a question related to the Fuchs’ divisible module δ ([14, Problem 6 in Chapter VI]).

In Section 4, we consider right coherent rings and continue our investigation started in [4] of the complete cotorsion pair (A, B) cogenerated by P^{<∞}. If findimR < ∞, then we know from [4] that there is a tilting module T such that B = T^{⊥}. Furthermore, A = P if and only if the category AddT is closed under cokernels of monomorphisms, and in this case the little and the big finitistic dimensions of R coincide: findim R = Findim R.

Our focus here is on the category \underrightarrow{lim} P^{<∞}. Note that A is always contained in \underrightarrow{lim} P^{<∞}. Moreover, if P is closed under direct limits, e. g. if R is right perfect and FindimR < ∞, then A ⊆ \underrightarrow{lim} P^{<∞} ⊆ P. Using the tilting module T from [4], we investigate closure properties of A. We characterize the cases A = \underrightarrow{lim} P^{<∞} and \underrightarrow{lim} P^{<∞} = P in Theorems 4.2 and 4.7. In Theorem 4.3, we determine when A is a definable class.

Finally, in Section 5, we study an important example in detail, namely the artin algebra introduced by Igusa, Smalø, and Todorov [19]. In this case, we show that A = \underrightarrow{lim} P^{<∞} fails while \underrightarrow{lim} P^{<∞} = P holds true.

1 Preliminaries

First, we fix our terminology and notation.

Let R be an arbitrary ring, ModR be the category of all (right) R-modules, and modR the subcategory of all finitely presented modules. For a subcategory M of ModR, we denote by AddM (respectively addM) the subcategory of all modules isomorphic to a direct summand of a (finite) direct sum of modules of M.

Following [24, p.210], we will say that a module M is FP_{n} provided that M has a projective resolution

⋯ → P_{k+1} → P_{k} → ⋯ → P_{1} → P_{0} → M → 0

where all P_{i} with i ≤ n are finitely generated. So FP_{0} stands for finitely generated, FP_{1} for finitely presented, and a module M is FP_{2} if and only if there are n < ω, a finitely presented module K, and a short exact sequence 0 → K → R^{n} → M → 0. Notice that P_{1}^{<∞} coincides with the class of all FP_{2} modules of projective dimension ≤ 1. If R is right coherent and M is finitely presented then M is FP_{n} for all n ≥ 1.

\underline{A. Precovers and preenvelopes.} Let M be a subcategory of ModR, and let A be a right R-module. A morphism f ∈ Hom_{R}(A, X) with X ∈ M is an M-preenvelope (or a left M-approximation) of A provided that the abelian group homomorphism Hom_{R}(f, M): Hom_{R}(X, M) ⟶ Hom_{R}(A, M) is surjective for each M ∈ M. An Mpreenvelope f ∈ Hom_{R}(A, X) of A is said to be special if f is a monomorphism and Ext_{R}^{1}(Coker f, M) = 0 for all M ∈ M. An M-envelope of A is an M-preenvelope f ∈ Hom_{R}(A, X) which is left minimal, that is, h is an automorphism of X whenever h ∈ End_{R}(X) satisfies hf = f. If it exists, an M-envelope is unique up to isomorphism.

The notions of an M-cover and a (special) M-precover are defined dually.

Finally, a subcategory C of modR is said to be covariantly (respectively, contravariantly) finite in modR if every finitely presented module has a C-preenvelope (respectively, a C-precover).

\underline{B. Closure under direct limits.} Let C be a class of modules. Denote by \underrightarrow{lim} C the class of all modules D such that D = \underrightarrow{lim}_{i∈I} C_{i} where \{C_{i} | i ∈ I\} is a direct system of modules from C.

In general, the class \underrightarrow{lim} C is not closed under direct limits:

Example 1.1 Let R = ℤ and let C = {A} where A is an indecomposable torsionfree abelian group of rank r ≥ 2 such that End(A) ≅ ℤ. (There is a proper class of such groups: by [10, XII.3.5], there exist arbitrarily large indecomposable torsion-free abelian groups such that End(A) ≅ ℤ.)

Consider the direct system {C_{n} | n < ∞} where C_{n} = A for all n < ∞ and f_{n} : C_{n} → C_{n+1} is the multiplication by n. Then \underrightarrow{lim}_{n<∞} C_{n} coincides with the injective envelope E(A) ≅ ℚ^{(r)} of A.

Now, consider the direct system {D_{n} | n < ∞} where D_{n} = E(A) for all n < ∞ and g_{n} : D_{n} → D_{n+1} is the projection on a fixed copy of ℚ in E(A). Then \underrightarrow{lim}_{n<∞} D_{n} ≅ ℚ.

On the other hand, ℚ ∉ \underrightarrow{lim} C. Namely, let (C_{i} | i ∈ I\} be a direct system with C_{i} = A for all i ∈ I, and let D = \underrightarrow{lim}_{i∈I} C_{i}. Since End(A) ≅ ℤ and A is torsion-free, all maps in the direct system are either monomorphisms or zero. It follows that either D contains a copy of A, or else D = 0. Anyway, D has rank ≠ 1, so D ≅ ℚ̸. □

There is an important case when \underrightarrow{lim} C is always closed under direct limits. The characterization goes back to Lenzing:

Lemma 1.2 [23] Let R be a ring, and let C be a full additive subcategory of modR which is closed under isomorphisms and direct summands. The following statements are equivalent for a module A.

(1) A ∈ \underrightarrow{lim} C.

(2) There is a pure epimorphism ∐_{i∈I} X_{i} → A for a sequence (X_{i} | i ∈ I) of modules from C.

(3) Every homomorphism h : F → A where F is finitely presented factors through a module in C. In particular, \underrightarrow{lim} C is closed under direct limits, and \underrightarrow{lim} C ∩ modR = C.

Crawley-Boevey and Krause observed that Lenzing’s result implies a characterization of when \underrightarrow{lim} C is a definable class of modules. Recall that a subcategory M of ModR is definable provided it is closed under direct limits, direct products and pure submodules.

Theorem 1.3 [9, 4.2] [21, 3.11] Let R be a ring, and let C be a full additive subcategory of modR which is closed under isomorphisms and direct summands. The following statements are equivalent.

(1) C is covariantly finite in modR.

(2) \underrightarrow{lim} C is closed under products.

(3) Every right R-module has a \underrightarrow{lim} C-preenvelope.

(4) \underrightarrow{lim} C is definable.

For example, if R is a left coherent and right perfect ring, then \underrightarrow{lim} P_{1}^{<∞} = P_{1} is closed under products, so P_{1}^{<∞} is covariantly finite in modR, cf. [4], [18].

\underline{C. Cotorsion pairs.} Next, we recall the notion of a cotorsion pair. This is the analog of the classical (non-hereditary) torsion pair where Hom is replaced by Ext^{1}.

For a class of modules M ⊆ ModR, we set M^{⊥_{1}} = {X ∈ ModR | Ext_{R}^{1}(M, X) = 0 for all M ∈ M} and ^{⊥_{1}}M = {X ∈ ModR |Ext_{R}^{1}(X, M) = 0 for all M ∈ M}.

By the well-known properties of Ext collected below in Lemma 1.4, the class M^{⊥_{1}} is definable if M consists of finitely presented modules over a right coherent ring, and the class ^{⊥_{1}}M is closed under direct limits if M consists of pure-injective modules.

Lemma 1.4 [13, Lemma 10.2.4], [5, Chap 1, Proposition 10.1] Let R be a ring, M an R-module, and {(N_{α}, f_{αβ}) | α ≤ β ∈ I} an arbitrary direct system of modules. Then the following hold true for each n < ω.

(1) Ext_{R}^{n} (M, \underrightarrow{lim}_{α∈I} N_{α}) ≅ \underrightarrow{lim}_{α∈I} Ext_{R}^{n} (M, N_{α}) provided that M is FP_{n+1}.

(2) Ext_{R}^{n} (\underrightarrow{lim}_{α∈I} N_{α}, M) ≅ \underleftarrow{lim}_{α∈I} Ext_{R}^{n} (N_{α}, M) provided that M is pure-injective.

Let A, B ⊆ ModR be classes of modules. Then (A, B) is said to be a cotorsion pair if A = ^{⊥_{1}}B and B = A^{⊥_{1}}. The class A ∩ B is called the kernel of the cotorsion pair (A, B).

The basic relation between cotorsion pairs and approximations goes back to Salce [27]. It may be viewed as a substitute for the non-existence of a duality for arbitrary modules:

Lemma 1.5 [27, Corollary 2.4.] Let R be a ring and (A, B) be a cotorsion pair. The following are equivalent:

(1) Every module has a special A-precover.

(2) Every module has a special B-preenvelope.

In this case, the cotorsion pair (A, B) is called complete.

Moreover, we say that a cotorsion pair (A, B) is closed if A is closed under direct limits. The importance of this notion comes from the following result of Enochs: If (A, B) is a complete and closed cotorsion pair, then every module has an A-cover and a B-envelope [13, 7.2.6].

Complete and/or closed cotorsion pairs occur quite frequently. For a class of modules C, let (A, B) be the cotorsion pair cogenerated by C, that is, let B = C^{⊥_{1}} and A = ^{⊥_{1}}(C^{⊥_{1}}). Then we know from [11] that (A, B) is complete provided that the isomorphism classes of modules in C form a set. In this case, there is a useful description of the modules in A. Recall that for an ordinal σ, a chain of modules (M_{α} | α ≤ σ) is said to be continuous provided that M_{α} ⊆ M_{α+1} for all α < σ and M_{α} = ⋃_{β<α} M_{β} for all limit ordinals α ≤ σ. Moreover, if S is a class of modules, a module M is S-filtered provided that there is a continuous chain (M_{α} | α ≤ σ) consisting of submodules of M such that M = M_{σ}, and each of the modules M_{0}, M_{α+1}/M_{α} (α < σ), is isomorphic to an element of S. Now, if the isomorphism classes of C form a set S, then A = ^{⊥_{1}}(C^{⊥_{1}}) consists of all direct summands of S ∪ {R}-filtered modules, [28, Theorem 2.2].

Dually, let (A, B) be the cotorsion pair generated by C, that is, let A = ^{⊥_{1}}C and B = (^{⊥_{1}}C)^{⊥_{1}}. Then we know from [12] that (A, B) is complete and closed provided that C consists of pure injective modules.

\underline{D. Examples of complete cotorsion pairs.} Let n < ω. Denote by P_{n} (I_{n}) the class of all modules of projective (injective) dimension at most n, and by F_{n} the class of all modules of flat (= weak) dimension at most n. Then (P_{n}, (P_{n})^{⊥_{1}}) and (^{⊥_{1}}I_{n}, I_{n}) are complete cotorsion pairs, cf. [13, 7.4.6] and [28, 2.1]. Moreover, (F_{n}, (F_{n})^{⊥_{1}}) is complete and closed, [12].

For N ∈ Mod-R, let N^{c} = Hom_{ℤ}(N, ℚ/ℤ) be the dual module of N. Denote by H the class of all dual modules of all left R-modules. It is well-known that the class PI of all pure-injective modules consists of all direct summands of modules in H, and that the cotorsion pair (F_{0}, (F_{0})^{⊥_{1}}) is generated by H (and by PI), cf. [13].

Next, we give a criterion for equality of two cotorsion pairs. Recall that a class M ⊆ ModR is resolving (coresolving) if it is closed under extensions, kernels of epimorphisms (cokernels of monomorphisms), and it contains all projective (injective) modules. For example, the class^{⊥}M = {X ∈ ModR |Ext_{R}^{i}(X, M) = 0 for all M ∈ M and all i > 0} is resolving, while M^{⊥} = {X ∈ ModR | Ext_{R}^{i}(M, X) = 0 for all M ∈ M and all i > 0} is coresolving.

Lemma 1.6 Let (E, D) be a complete cotorsion pair such that E is resolving. Let further (X , Y) be a cotorsion pair with E ⊆ X. Then the two cotorsion pairs coincide if and only if X ∩ D ⊆ E ∩ D.

Proof : It is enough to verify X ⊆ E in case X ∩ D ⊆ E ∩ D. Take X ∈ X and consider a special D-preenvelope 0 ⟶ X ⟶ D ⟶ E ⟶ 0. Since E ∈ E ⊆ X and X is closed under extensions, we have D ∈ X ∩ D ⊆ E ∩ D. Thus X ∈ E because E is resolving. □

Finally, we consider cotorsion pairs induced by a tilting module. Recall from [3] that a module T is a tilting module provided that (T1) pdT < ∞;

(T2) Ext_{R}^{i}(T, T^{(I)}) = 0 for each i > 0 and all sets I;

(T3) There is r ≥ 0 and a long exact sequence 0 ⟶ R_{R} ⟶ T_{0} ⟶ ⋯ ⟶ T_{r} ⟶ 0

with T_{i} ∈ AddT for each 0 ≤ i ≤ r. In this case (^{⊥}(T^{⊥}), T^{⊥}) is a complete cotorsion pair with the kernel AddT, and ⊥(T^{⊥}) ⊆ P_{n} where n =pdT, see [3, Section 2].

2 The closure of a cotorsion pair

We now consider the natural partial order ≤ on the class of all cotorsion pairs induced by inclusion of the first components. Observe that ≤ is a complete lattice order, the least element being 𝔖 = (P_{0}, Mod-R), the largest 𝔏 = (Mod-R, I_{0}), and the meet of the cotorsion pairs {(A_{α}, B_{α}) | α ∈ I} being (⋂_{α∈I} A_{α}, (⋂_{α∈I} A_{α})^{⊥_{1}}).

Since 𝔏 is closed, and meets of closed cotorsion pairs are likewise closed, each cotorsion pair ℭ is contained in the smallest closed one, the closure of ℭ.

The interesting case is when the closure is complete, hence provides for envelopes and covers of modules. We will show that this always occurs when ℭ is cogenerated by a class of finitely presented modules over a right coherent ring (see Corollary 2.4 below).

For a class of modules C we denote by C̃ the class of all pure epimorphic images of elements of C. Clearly, C ∩ modR = C̃∩ modR provided that C is closed under direct summands.

For example, if (A, B) is a complete cotorsion pair, then the class Ã is easily seen to coincide with the class of all modules M such that each (or some) special A-precover of M is a pure epimorphism.

Define C^{⊺} = Ker Tor_{1}^{R}(C, −) for a class C ⊆ModR, and ^{⊺}D = Ker Tor_{1}^{R}(−, D) for a class D ⊆ RMod. For a class C ⊆ModR, we define Ĉ = ^{⊺}(C^{⊺}).

Note that \underrightarrow{lim} C ⊆ Ĉ, and C̃ ⊆ Ĉ, since Ĉ is obviously closed under direct limits and pure epimorphic images. Moreover, we have

Lemma 2.1 Let R be a ring, C be a class of modules, and (A, B) be the cotorsion pair cogenerated by C.

1. Ĉ = ^{⊥_{1}}(B ∩ H) = Â.

2. Assume that C is closed under arbitrary direct sums. Then \underrightarrow{lim} C ⊆ C̃ ⊆ Ĉ.

3. Assume that C consists of FP_{2} modules. Then M ∈ B if and only if M^{cc} ∈ B for any module M. In particular, Ĉ = ^{⊥_{1}}(B ∩ PI).

Proof : 1. Let M ∈ ModR. The well-known Ext-Tor relations yield M ∈ Ĉ iff M ∈ ^{⊥_{1}}(N^{c}) for all N ∈ C^{⊺}. Moreover, N ∈ C^{⊺} iff N^{c} ∈ C^{⊥_{1}} ∩ H = B ∩ H. Taking C = A, we get in particular Â = ^{⊥_{1}}(B ∩ H).

2. This is clear since C̃ is closed under direct limits in this case.

3. Let M ∈ ModR. In this setting, the Ext-Tor relations yield M ∈ B iff M^{c} ∈ C^{⊺} iff M^{cc} ∈ B. Since each pure-injective module M is a direct summand in M^{cc}, ⊥_{1}(B ∩ H) = ^{⊥_{1}}(B ∩ PI), and the assertion follows by part 1. □

Lemma 2.2 Let R be a ring. Let ℭ = (A, B) be a complete cotorsion pair such that B is closed under taking double dual modules. Then Ã = Â.

Proof : Let M ∈ Â. By Lemma 2.1.1, M ∈ ^{⊥_{1}}(B ∩ H). Let 0 ⟶ B ⟶^{μ} A ⟶ M ⟶ 0 be an exact sequence with A ∈ A and B ∈ B. Consider the canonical pure embedding ν : B → B^{cc}, and take the push-out of μ and ν:

0 ⟶ B ⟶^{μ} A ⟶ M ⟶ 0
ν↓  ^{η}↓  ∥
0 ⟶ B^{cc} ⟶^{τ} N ⟶ M ⟶ 0

By assumption, B^{cc} ∈ B ∩ H, so the bottom row splits. It follows that ν factors through μ, hence μ is pure, and M ∈ Ã. □

By Lemma 2.1, each cotorsion pair ℭ = (A, B) is contained in the complete and closed cotorsion pair (Â, Â^{⊥_{1}}) generated by the class B ∩ H. We now investigate whether the latter is the closure of ℭ.

Theorem 2.3 Let R be a ring. Let C be a class consisting of FP_{2} modules such that C is closed under extensions and direct summands and R ∈ C. Then \underrightarrow{lim} C = Ĉ. If (A, B) denotes the cotorsion pair cogenerated by C then \underrightarrow{lim} C = \underrightarrow{lim} A = Ã = Â.

Proof : From Lemmas 2.1.1, 2.1.3 and 2.2 we get that Ã = Â = Ĉ.

Next, we show that A ⊆ \underrightarrow{lim} C. The proof is a generalization of a particular case considered in [4, 2.1]. First, the isomorphism classes of C form a set, so A consists of all direct summands of C-filtered modules. By Lemma 1.2, \underrightarrow{lim} C is closed under direct limits, hence under direct summands. So it suffices to prove that \underrightarrow{lim} C contains all C-filtered modules.

We proceed by induction on the length, δ, of the filtration. The cases when δ = 0 and δ is a limit ordinal are clear (the latter by Lemma 1.2). Let δ be non-limit, so we have an exact sequence 0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0 with A ∈ \underrightarrow{lim} C and C ∈ C. We will apply Lemma 1.2 to prove that B ∈ \underrightarrow{lim} C.

Let h : F → B be a homomorphism with F finitely presented. Since C is FP_{2}, there is a presentation 0 ⟶ G ⟶ P ⟶^{p} C ⟶ 0 with P finitely generated projective and G finitely presented. There is also q : P → B such that p = gq. We have the commutative diagram

0 ⟶ F′ ⟶^{f′} F ⊕ P ⟶^{(gh)⊕p} C ⟶ 0
h′↓  ^{h⊕q}↓  ∥
0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0

Considering the pull-back of p and (gh) ⊕ p, we see that the pull-back module U is an extension of G by F ⊕ P, and F′ is isomorphic to a direct summand in U. So U, and F′, are finitely presented. Since A ∈ \underrightarrow{lim} C, Lemma 1.2 provides for a module C′ ∈ C and maps σ′ : F′ → C′, τ′ : C′ → A such that h′ = τ′σ′. Consider the push-out of f′ and σ′:

0 ⟶ F′ ⟶^{f′} F ⊕ P ⟶^{gh⊕p} C ⟶ 0
σ′↓  ^{σ}↓  ∥
0 ⟶ C′ ⟶^{ρ} D ⟶ C ⟶ 0

By assumption, D ∈ C. By the push-out property, there is τ : D → B such that τσ = h ⊕ q, hence τσ ↾ F = h. So h factors through D, and B ∈ \underrightarrow{lim} C.

Now, since \underrightarrow{lim} C is closed under pure epimorphic images by Lemma 1.2, we infer that Ã ⊆ \underrightarrow{lim} C. So \underrightarrow{lim} C = Ã = \underrightarrow{lim} A. □

Corollary 2.4 Let R be a ring and ℭ = (A, B) be a cotorsion pair cogenerated by a class of FP_{2} modules. (For example, let R be right coherent and ℭ be cogenerated by a subclass of modR.) Then the closure \overline{ℭ} = (\overline{A}, \overline{B}) of ℭ is generated by the class B ∩ PI. In particular, \overline{ℭ} is complete, and \overline{A} = \underrightarrow{lim} A = Â.

Proof : If ℭ is cogenerated by a class of FP_{2} modules D, we let C be the smallest class of modules closed under extensions and direct summands which contains D ∪ {R}. Then C also consists of FP_{2} modules, and it cogenerates ℭ, so Lemma 2.1.3 and Theorem 2.3 apply. □

Corollary 2.5 Let R be a ring and C be a class consisting of FP_{2} modules. Assume R ∈ C. Then the class ^{⊥_{1}}(C^{⊥_{1}}) consists of all direct summands of C-filtered modules while ^{⊺}(C^{⊺}) consists of all pure-epimorphic images of C-filtered modules.

One of the ingredients in the proof of Theorem 2.3 was Lemma 1.4.1. Part 2 of that Lemma yields another case of coincidence of the classes \underrightarrow{lim} A and Â:

Proposition 2.6 Let R be a ring and ℭ = (A, B) be a cotorsion pair generated by a class of dual modules. Then A = \underrightarrow{lim} A = Â.

Proof : By Lemma 1.4.2, A = \underrightarrow{lim} A. By Lemma 2.1.2, it suffices to prove that Â ⊆ A. But Â = ^{⊥_{1}}(B ∩ H) by Lemma 2.1.1. Since C ⊆ B ∩ H, we have Â ⊆ ^{⊥_{1}}C = A. □

As an application, we consider the classes of modules of bounded flat dimension:

Corollary 2.7 Let R be a ring and n < ∞. Let H_{n} = {Ω^{−n}(M) | M ∈ H} where Ω^{−n}(M) denotes an n-th cosyzygy of M. Then F_{n} = ^{⊥_{1}} H_{n} = F̂_{n}.

Proof : First, F_{0} = ^{⊺}(RMod) = ^{⊥_{1}}H_{0}. Let n > 0. For a module N, denote by Ω_{n}(N) the n-th syzygy (in a projective resolution) of N. We have N ∈ F_{n} iff Ω_{n}(N) ∈ F_{0} iff Ext_{R}^{1} (Ω_{n}(N), H_{0}) = 0 iff Ext_{R}^{n+1}(N, H_{0}) = 0 iff Ext_{R}^{1} (N, H_{n}) = 0. This proves that F_{n} = ^{⊥_{1}}H_{n}. Finally, all cosyzygies of a dual module can be taken to be dual modules as well, so Proposition 2.6 applies. □

3 Modules of projective dimension at most one over commutative domains

In this section, R will denote a commutative domain and Q its quotient field.

We start by reviewing some properties of the class DI of all divisible modules. Recall that DI = CP^{⊥_{1}} where CP = {R/rR | r ∈ R} denotes a set of representatives of all cyclically presented modules. It is well-known that the complete cotorsion pair (^{⊥_{1}}DI, DI) is cogenerated by a tilting module of projective dimension one, namely the Fuchs’ divisible module δ, cf. [15, §VII.1], [28].

Denote by HD the class of all h-divisible modules, that is, of all modules that are homomorphic images of direct sums of copies of Q. Clearly, HD ⊆ DI, and the equality holds true if and only if R is a Matlis domain, that is, pdQ = 1, cf. [15, §VII.2].

For any domain R, we have P_{1} = ^{⊥_{1}}HD by [15, VII.2.5], so the complete cotorsion pair (P_{1}, (P_{1})^{⊥_{1}}) is generated by the class of all h-divisible modules.

Let us look more closely at the case when R is a Prüfer domain.

Example 3.1 Assume R is Prüfer. Then (^{⊥_{1}}DI, DI) coincides with (P_{1}, (P_{1})^{⊥_{1}}) and also with the cotorsion pair (A, B) cogenerated by P^{<∞}, so the Fuchs’ divisible module δ is just the tilting module considered in [4].

Indeed, in this case we have P^{<∞} = P_{1}^{<∞} = modR, hence A ⊆ P_{1}, cf. [4, Section 2]. So, we have a chain of complete cotorsion pairs (^{⊥_{1}}DI, DI) ≤ (A, B) ≤ (P_{1}, (P_{1})^{⊥_{1}}). On the other hand, P_{1} ⊆ ^{⊥_{1}}DI by [14, VI.3.9], and the claim follows.

Note that, in contrast to the artin algebra case [4, 4.2], the fact that P^{<∞} is contravariantly finite in modR does not force A to coincide with \underrightarrow{lim} P^{<∞}. Indeed, A is not closed under direct limits unless R is a Dedekind domain. □

In particular, the above observations show that for all Matlis and all Prüfer domains, all divisible modules of projective dimension at most one belong to Addδ, cf. [14, VI.3.10]. Problem 6 of [14, Chapter VI] asks whether this is true for any domain. The following result provides for an answer.

Proposition 3.2 The following are equivalent for a commutative domain R.

(1) All divisible modules of projective dimension at most one belong to Addδ.

(2) The cotorsion pairs (^{⊥_{1}}DI, DI) and (P_{1}, (P_{1})^{⊥_{1}}) coincide.

(3) DI = (^{⊥_{1}}HD)^{⊥_{1}}.

(4) A module has projective dimension at most one if and only if it is a direct summand of a CP-filtered module.

Proof : We always have 𝔇 = (^{⊥_{1}}DI, DI) ≤ (P_{1}, (P_{1})^{⊥_{1}}) = 𝔓. Condition (1) states that P_{1} ∩ DI =Addδ = ^{⊥_{1}}DI ∩ DI. This is equivalent to condition (2) by Lemma 1.6. Further, (2) is equivalent to (3), since the cotorsion pair 𝔓 is generated by HD. On the other hand, the cotorsion pair 𝔇 is cogenerated by the set CP, hence ⊥_{1}DI consists of all direct summands of CP-filtered modules, cf. section 1.C. So (2) is equivalent to (4). □

Next, we consider the class T F of all torsion-free modules, that is, T F = (CP)^{⊺}. There is a duality between torsion-free and divisible modules: a module N is torsionfree iff N^{c} is divisible, [28, §4]. By a result of Warfield, (T F, Q^{⊥_{1}} ∩ I_{1}) is a closed cotorsion pair, cf. [28, §2].

First, we show that (T F, Q^{⊥_{1}} ∩ I_{1}) is generated by the class of all pure-injective modules of injective dimension at most one:

Lemma 3.3 Let R be a commutative domain. Then T F = ^{⊥_{1}}(I_{1} ∩ PI) = (F_{1})^{⊺}.

Proof : Since Q is a flat module, Q^{⊥_{1}} contains all pure-injective modules, so Q^{⊥_{1}} ∩ I_{1} ∩ PI = I_{1} ∩ PI and T F ⊆ ^{⊥_{1}}(I_{1} ∩ PI). Consider M ∈ ^{⊥_{1}}(I_{1} ∩ PI). Let N ∈ F_{1}. Then N^{c} ∈ I_{1}, so 0 = Ext_{R}^{1}(M, N^{c}) ≅ (Tor_{1}^{R}(M, N))^{c}. It follows that ⊥_{1}(I_{1} ∩ PI) ⊆ (F_{1})^{⊺}. Finally, since R/rR ∈ F_{1}, we get (F_{1})^{⊺} ⊆ T F. □

Similarly, the cotorsion pair (F_{1}, (F_{1})^{⊥_{1}}) is generated by the class of all divisible pure-injective modules. In fact, there is a more general result (where, for a class of modules C, ^{⊥_{n}}C denotes the class {M ∈ ModR | Ext_{R}^{n}(M, C) = 0 for all C ∈ C}):

Lemma 3.4 Let R be a commutative domain and n > 0. Then F_{n} = ^{⊥_{n}}(DI ∩ PI). In particular, F_{1} = ĈP = ^{⊥_{1}}(DI ∩ PI).

Proof : First, F_{1} = ĈP by Corollary 2.7 and Lemma 3.3. Let n > 0. By Corollary 2.7, F_{n} = ^{⊥_{1}}H_{n} = ^{⊥_{n}}H_{1}. Since H_{1} ⊆ DI, we have F_{n} ⊇ ^{⊥_{n}}(DI ∩ PI).

Conversely, let M ∈ F_{n}, and let N be a module such that N^{c} ∈ DI. Then N ∈ T F = (F_{1})^{⊺} by Lemma 3.3. So 0 = Tor_{1}^{R}(Ω^{n−1}M, N) ≅ Tor_{n}^{R}(M, N). This shows that M ∈ ^{⊥_{n}}(DI∩H). Finally, by Lemma 2.1.3,^{⊥_{n}}(DI∩H) = ^{⊥_{n}}(DI∩PI). □

Altogether, we have the following chain of complete cotorsion pairs

(^{⊥_{1}}DI, DI) ≤ (P_{1}, (P_{1})^{⊥_{1}}) ≤ (F_{1}, (F_{1})^{⊥_{1}}) = (ĈP, (ĈP)^{⊥_{1}}).

By Corollary 2.4 and Lemma 3.4, (F_{1}, (F_{1})^{⊥_{1}}) is the closure of (^{⊥_{1}}DI, DI) and hence of (P_{1}, (P_{1})^{⊥_{1}}).

Problem 22 in [15, p.246] asks for the structure of the modules which are direct limits of modules in P_{1}. The answer has already been known for Prüfer domains: \underrightarrow{lim} P_{1} = ModR, cf. Example 3.1. Since Prüfer domains are characterized as the domains of weak global dimension at most one, \underrightarrow{lim} P_{1} then coincides with the class of all modules of flat dimension at most one. The latter description extends to any commutative domain:

Theorem 3.5 Let R be a commutative domain. Then

\underrightarrow{lim} P_{1}^{<∞} = \underrightarrow{lim} P_{1} = F_{1}.

Proof : Theorem 2.3 applied to the cotorsion pair (A_{1}, B_{1}) cogenerated by P_{1}^{<∞} and combined with Lemma 2.1.3 yields \underrightarrow{lim} P_{1}^{<∞} = { \atop P_{1}^{<∞}} = ^{⊥_{1}}(B_{1} ∩ PI). We now claim that ^{⊥_{1}}(B_{1} ∩ PI) = F_{1}. By Lemma 3.4, it suffices to show that B_{1} ∩ H = DI ∩ H. Now, for a module N ∈ R-Mod, we have N^{c} ∈ B_{1} = (P_{1}^{<∞})^{⊥_{1}} iff N ∈ (P_{1}^{<∞})^{⊺}. By Lemma 3.3, the latter is equivalent to N ∈ T F, and hence to N^{c} ∈ DI.

So, the claim is proven, and we obtain F_{1} = \underrightarrow{lim} P_{1}^{<∞} ⊆ \underrightarrow{lim} P_{1}. On the other hand, P_{1} ⊆ F_{1}, and \underrightarrow{lim} P_{1}^{<∞} is closed under direct limits by Lemma 1.2. Thus \underrightarrow{lim} P_{1} = F_{1}. □

We don’t know whether this result can be extended to higher dimensions, that is, whether the limit closure of P_{n} always coincides with F_{n}.

4 Direct limits of finitely presented modules of finite projective dimension

Throughout this section, R will denote a right coherent ring, and (A, B) will be the (complete) cotorsion pair cogenerated by P^{<∞}. From Section 2 we obtain

(A, B) ≤ (\underrightarrow{lim} P^{<∞}, (\underrightarrow{lim} P^{<∞})^{⊥})

where the right-hand term is the closure. In general, (A, B) is not closed.

Example 4.1 Let R be a von-Neumann-regular ring. Then P^{<∞} consists of the finitely generated projective modules, so A = P_{0}, while \underrightarrow{lim} P^{<∞} = ModR. □

In order to investigate when (A, B) is a closed cotorsion pair, we will assume findimR < ∞ and use the tilting module T from [4] satisfying B = T^{⊥}. Moreover, we will deal with the property that all pure submodules of a given module M are direct summands. Modules M with such property are called pure-split in [7]. We will say that M is ∑-pure-split if all modules in AddM are pure-split. For example, every ∑-pure-injective module is ∑-pure-split.

Theorem 4.2 Let R be a right coherent ring with findimR < ∞, and let T be a tilting module with B = T^{⊥}. Then the following statements are equivalent.

(1) A is closed under direct limits (that is, A = \underrightarrow{lim} P^{<∞}).

(2) T is ∑-pure-split.

Proof : (1)⇒(2): If A is closed under direct limits then it is closed under pureepimorphic images by Theorem 2.3. Recall further that B is always closed under pure submodules as it is definable. So, if 0 ⟶ X ⟶ T′ ⟶ Y ⟶ 0 is a pure-exact sequence with T′ ∈ Add T, we have Y ∈ A and X ∈ B, which shows that the sequence splits. For the implication (2)⇒(1), we first observe that (\underrightarrow{lim} P^{<∞}, (\underrightarrow{lim} P^{<∞})^{⊥_{1}}) is a cotorsion pair such that \underrightarrow{lim} P^{<∞} = Ã contains the resolving class A, cf. Theorem 2.3 and Corollary 2.4. So, by Lemma 1.6, it suffices to prove that \underrightarrow{lim} P^{<∞} ∩ B ⊆ A ∩ B = Add T. For this purpose, consider M ∈ \underrightarrow{lim} P^{<∞} ∩ B and let 0 → B → A → M → 0 be an exact sequence with A ∈ A and B ∈ B. Then A ∈ Add T. Further, since M ∈ Ã, the exact sequence above is pure-exact. Hence it splits by assumption, and M ∈ Add T. □

Closure under direct limits of A can thus be interpreted as a pure-injectivity property of the module T. We now exhibit a further closure property of A that can be tested on T. Recall that a module M with AddM being closed under products is said to be product-complete, see [22].

Theorem 4.3 Let R be a right coherent ring with findimR < ∞, and let T be a tilting module with B = T^{⊥}. The following statements are equivalent.

(1) A is definable.

(2) A is closed under direct products.

(3) T is product-complete.

(4) A is closed under direct limits, and P^{<∞} is covariantly finite in modR.

Proof : (1)⇒(2) is clear. Further, (2) implies (3) because AddT = A∩B, see Section 1.D. The implication (4)⇒(1) holds by Theorems 1.3 and 2.3. (3)⇒(4): Observe that a module A belongs to A if and only if there is a long exact sequence 0 → A → T_{0} → T_{1} → … → T_{n} → 0 with n = findim R and T_{i} ∈ Add T for all 0 ≤ i ≤ n. In fact, the if-part holds since A is resolving, and the only-if-part is shown as for the special case A = R in the proof of [3, 4.1]. This shows that A is closed under direct products if so is AddT. Moreover, T is Σ-pure-injective by [22] and thus Σ-pure-split, hence A = \underrightarrow{lim} P^{<∞} by Theorem 4.2. From Theorem 1.3 it now follows that the category P^{<∞} is covariantly finite in modR. □

Note that with the results above we can refine [4, 3.4 and 3.7]:

Corollary 4.4 Let R be a right coherent ring with findim R < ∞, and let T be a tilting module with B = T^{⊥}. Assume that P^{<∞} is covariantly finite (this happens when R is a two-sided coherent and right perfect ring with FindimR = 1, or more generally, with P = \underrightarrow{lim} P^{<∞}, see [4]). Then A is definable if and only if it is closed under direct limits. In other words, T is product-complete if and only if T is ∑-pure-split.

Recall that a family (M_{i})_{i∈I} is called locally (right) t-nilpotent if for each sequence of non-isomorphisms M_{i_{1}} ⟶^{f_{1}} M_{i_{2}} ⟶^{f_{2}} M_{i_{3}} … with indices (i_{n})_{n∈IN} from I, and each element x ∈ M_{i_{1}}, there exists m = m_{x} ∈ IN such that f_{m} f_{m−1} … f_{1}(x) = 0.

Corollary 4.5 Let R be a right coherent ring with findim R < ∞, and assume that A is closed under direct limits. If T is a tilting module with B = T^{⊥}, then there is a locally t-nilpotent family (T_{i})_{i∈I} of modules with local endomorphism ring such that T = ⨁_{i∈I} T_{i}. In particular, the direct sum M of a representative set of all indecomposable modules from A ∩ B is a tilting module with B = M^{⊥}.

Proof : Recall that a submodule X of a module Y is called a local direct summand of Y if there is a decomposition X = ∐_{k∈K} X_{k} with the property that ∐_{k∈K_{0}} X_{k} is a direct summand of Y for every finite subset K_{0} ⊂ K. Of course, X then is a pure submodule of Y. So, we can now conclude from Theorem 4.2 that all local direct summands of modules in AddT are direct summands. By [16, 2.3] this implies that T has a decomposition T = ⨁_{i∈I} T_{i} in modules with local endomorphism ring. By [17, 7.3.5] and [1, 2.7] we further know that the family (T_{i})_{i∈I} is locally t-nilpotent. The statement on M then follows from Azumaya’s Decomposition Theorem. □

The next result shows that we can test contravariant finiteness of P^{<∞} on the indecomposable modules in P^{<∞} ∩ B. Note however that in general P^{<∞} ∩ B can be zero, see Remark 5.6.

Corollary 4.6 The following statements are equivalent for an artin algebra R with findim R < ∞.

(a) P^{<∞} is contravariantly finite in modR.

(b) P^{<∞} is covariantly finite in modR, and the direct sum U of a representative set of all indecomposable modules from P^{<∞} ∩ B is a ∑-pure-split tilting module with U^{⊥} = B.

Proof : Of course, if P^{<∞} ∩ B ≠ 0, then U ∈ A ∩ B = Add T. Now, we know from [4, 4.2 and 4.3] that P^{<∞} is contravariantly finite if and only if T is finitely presented, and that in this case A = P is closed under direct limits. This proves that (a) implies the second statement in (b), while the first is shown in [18]. Conversely, if (b) is satisfied, then we know from Theorems 4.2 and 4.3 that U is product-complete, which implies by [2, 5.2] that U is finitely presented, hence (a) holds true. □

If Findim R < ∞ and P is closed under direct limits (e. g. if R is right perfect), we also have

(\underrightarrow{lim} P^{<∞}, (\underrightarrow{lim} P^{<∞})^{⊥}) ≤ (P, P^{⊥}).

Our next aim is a criterion for P = \underrightarrow{lim} P^{<∞}. More generally, for an integer n ≥ 0, we will consider the cotorsion pair (A_{n}, B_{n}) cogenerated by P_{n}^{<∞} and describe when P_{n} = \underrightarrow{lim} P_{n}^{<∞}. Our criterion is in terms of properties of pure-injective modules:

Theorem 4.7 Let R be a right perfect and right coherent ring.

1. Let n < ∞. Then P_{n} = ^{⊥_{1}}(P_{n}^{⊥} ∩ H).

Moreover, P_{n} = \underrightarrow{lim} P_{n}^{<∞} iff B_{n} ∩ PI = P_{n}^{⊥} ∩ PI iff P_{n} ^{⊺} = (P_{n}^{<∞})^{⊺}.

2. Assume Findim R < ∞. Then P = ^{⊥_{1}}(P^{⊥} ∩ H). Moreover, P = \underrightarrow{lim} P^{<∞} iff B ∩ PI = P^{⊥} ∩ PI iff P^{⊺} = (P^{<∞})^{⊺}.

Proof : 1. By assumption, P_{n} = F_{n}, so Corollary 2.7 and Lemma 2.1.1 give the first assertion.

Assume P_{n} = \underrightarrow{lim} P_{n}^{<∞}. Then B_{n} ∩PI = P_{n}^{⊥} ∩PI by Lemma 1.4.2. Furthermore, if B_{n} ∩ PI = P_{n}^{⊥} ∩ PI, then B_{n} ∩ H = P_{n}^{⊥} ∩ H, so P_{n} ^{⊺} = (P_{n}^{<∞})^{⊺}. The latter implies P_{n} = ^{⊺}(P_{n} ^{⊺}) = ^{⊺}(P_{n}^{<∞})^{⊺} = \underrightarrow{lim} P_{n}^{<∞} by Corollary 2.7 and Theorem 2.3.

2. By assumption, P = P_{n} and P^{<∞} = P_{n}^{<∞} for some n < ∞, and part 1. applies. □

5 On an example of Igusa, Smalø, and Todorov

Examples 3.1 and 4.1 already yield cases where P = \underrightarrow{lim} P^{<∞} and A = ^{⊥_{1}}(P^{<∞} ^{⊥_{1}}) is not closed under direct limits. We now show that the same can happen over an artin algebra. To this end, we study the following example from [19].

Let k be an algebraically closed field and R the finite-dimensional algebra given by the quiver

[figure: 1; 2; γ; α; β]

with the relations αγ = βγ = γα = 0.

It was shown in [19] that FindimR = findimR = 1, but P^{<∞} is not contravariantly finite in modR. We then know from [4] that there is a P^{<∞}-filtered tilting module T of projective dimension 1 which is not finitely generated and satisfies T^{⊥} = B.

Moreover, each module of finite projective dimension is a direct limit of elements of P^{<∞}.

We are now going to show that A is not closed under direct limits. Let us start by collecting further information on the algebra R.

Conventions: We write R-modules by specifying the vector space V_{i} corresponding to the vertex , i = 1, 2, together with three maps f_{α}, f_{β}, f_{γ} corresponding to the three arrows.

For each of the two vertices i = 1, 2, we denote by P_{i}, I_{i}, and S_{i} the corresponding projective, injective or simple module respectively, and set I = I_{1} ⊕ I_{2}.

R has a factor algebra isomorphic to the Kronecker algebra which we will denote by Λ. The R-modules where the map f_{γ} corresponding to the arrow γ is zero are also Λ-modules and will be called Kronecker modules.

Note that every R-module X admits a canonical short exact sequence 0 ⟶ P_{1}^{(I)} ⟶ X ⟶ X̄ ⟶ 0 where I is a set and X̄ is a Kronecker module.

Recall the classification of the indecomposable finite length Kronecker modules:

i) The preprojectives D_{n}, n ∈ IN_{0}: Set V_{1} = k^{n+1}, V_{2} = k^{n}, f_{β} = (E, 0) and f_{α} = (0, E) where E denotes the unit in K^{n×n}; ii) The preinjectives M_{n}, n ∈ IN_{0}: Set V_{1} = k^{n}, V_{2} = k^{n+1}, f_{β} = (E, 0)^{t} and f_{α} = (0, E)^{t}; iii) The simple regulars: A family R_{λ} indexed by λ ∈ k defined as V_{1} = V_{2} = k, f_{β} the multiplication with λ ∈ k, f_{α} the identity, and further a module R_{∞} defined as V_{1} = V_{2} = k, f_{β} the identity, and f_{α} = 0; iv) Every simple regular module R_{λ} with λ ∈ k ∪ {∞} moreover defines a tube T_{λ}, that is a chain of indecomposable modules X_{i} starting in the simple regular X_{0} linked by non-split exact sequences 0 ⟶ X_{i} ⟶ X_{i−1} ⊕ X_{i+1} ⟶ X_{i} ⟶ 0 which are almost split in modΛ. Any finite length indecomposable regular module occurs in this way.

The following is implicitly proven in [19].

Proposition 5.1 An indecomposable finite length Kronecker module X has finite projective dimension if and only if it lies in ⋃_{λ∈k} T_{λ}. In particular, P^{<∞} consists of the finitely {P_{1}} ∪ ⋃_{λ∈k} T_{λ} -filtered modules.

Proof : Since preprojectives and preinjectives have odd dimension, they are not in P^{<∞}, see [19]. So X ∈ P^{<∞} must be regular, and taking into account the shape of the regular components (see iv), we see that all other modules in the tube X belongs to will be in P^{<∞} too. But since the simple regular R_{∞} is isomorphic to P_{2}/Soc P_{2}, where Soc P_{2} ∉ P^{<∞}, we have R_{∞} ∉ P^{<∞}, hence X belongs to one of the other tubes. For the if-part, note that R_{λ} ≅ P_{2}/P_{1} for each λ ∈ k. Again, we have that all modules in the correponding tube are then in P^{<∞}. □

Lemma 5.2 [6] The finitely generated modules in B are precisely the finitely genIfiltered modules. In particular, S_{2}, and all the finite length indecomposable regular modules in T_{∞} are in B.

Proof : S_{2} ≅ I_{2}/Rad I_{2} and R_{∞} ≅ I_{2}/S_{2} belong to genI, hence to B. Since B is extension-closed, all the tube T_{∞} lies in B. □

Lemma 5.3 Let X, Y be Kronecker modules. Assume that either Y ∈ A, or Hom_{R}(S_{2}, X) = 0. Then Ext_{Λ}^{1} (Y, X) = 0 if and only if Ext_{R}^{1} (Y, X) = 0.

Proof : The if-part is always true. For the only-if-part, observe that S_{2} = M_{0} is a simple injective Λ-module, and write X = X′ ⊕ S_{2}^{(J)} with Hom_{R}(S_{2}, X′) = 0. If Y ∈ A, then Ext_{R}^{1} (Y, S_{2}^{(J)}) = 0 by 5.2, so we can assume that we are in the second case, that is X = X′. To verify Ext_{R}^{1} (Y, X) = 0, we now show that every extension E_{R} of X by Y is actually a Kronecker module. In fact, consider a short exact sequence 0 ⟶ X_{R} ⟶^{f} E_{R} ⟶^{g} Y_{R} ⟶ 0, and assume that E_{R} contains a submodule isomorphic to P_{1}. Since P_{1} is uniserial and its socle S_{2} is not contained in Imf, we deduce that g |_{P_{1}} is a monomorphism. But this is not possible because Y is a Kronecker module. □

For each tube T_{λ} with λ ∈ k ∪ {∞}, we denote by Y_{λ} the corresponding Prüfer module, that is, the direct limit of the chain of inclusions X_{0} ⊂ X_{1} … in the tube. It follows from 5.1 that the Y_{λ} with λ ∈ k are P^{<∞}-filtered and therefore belong to A.

Proposition 5.4 (1) Let Y = ⨁_{λ∈k} Y_{λ}. Then B = Y ^{⊥}.

(2) A Kronecker module X belongs to B if and only if Ext_{Λ}^{1} (R_{λ}, X) = 0 for all λ ∈ k.

Proof : (1) The inclusion ⊆ is clear. For the other inclusion, let X be a module in Y^{⊥}. For any λ ∈ k and any A ∈ T_{λ} we have an exact sequence 0 ⟶ A ⟶ Y ⟶ C ⟶ 0 where C must belong to P and thus must have projective dimension at most one. This shows that Ext_{R}^{1} (A, X) = 0. Then we infer from 5.1 that Ext_{R}^{1} ( , X) = 0 vanishes on all modules in P^{<∞}, that is X ∈ B.

(2) The only-if-part is clear since R_{λ} ∈ P^{<∞} ⊆ A for all λ ∈ k. For the if-part, we deduce from 5.3 that Ext_{R}^{1} (R_{λ}, X) = 0. Then Ext_{R}^{1} (−, X) = 0 vanishes also on all modules in T_{λ} and on the T_{λ}-filtered module Y_{λ}, see [13, 7.3.4]. Hence Ext_{R}^{1} (Y, X) = 0, and even X ∈ Y^{⊥} since Y ∈ P = P_{1}. The claim now follows from (1). □

We will need some further notions from [25]. For a Λ-module X, we denote by τ X the sum of its finitely generated submodules without nonzero preprojective direct summand. If X = τ X, then X is said to be torsion, and if τ X = 0, or equivalently, if Hom_{Λ} (R_{λ}, X) = 0 for all λ ∈ k ∪ {∞}, then X is said to be torsionfree. Moreover, X is called divisible if Ext_{Λ}^{1} (R_{λ}, X) = 0 for all λ ∈ k ∪ {∞}. It is shown in [25] that the divisible modules are precisely the direct sums of preinjectives, Prüfer modules and copies of a module Q which is the unique indecomposable torsionfree divisible module. Finally, X is called regular if it does not have any preprojective, nor any preinjective summand.

We have the following immediate consequence of 5.4.

Corollary 5.5 All Kronecker modules X which are divisible over Λ belong to B. In particular, this shows that Y belongs to A ∩ B = Add T.

Remark 5.6 (1) The finite length indecomposable Kronecker modules in B are precisely the preinjectives and the regular modules in T_{∞}. Indeed, if X is a finite length indecomposable Kronecker modules in B, then we know from 5.2 that there is a nonzero map f : I_{i} → X with i = 1 or i = 2. In case i = 1, the module X is preinjective since I_{1} = M_{1}. In case i = 2, the map f cannot be a monomorphism and therefore factors through I_{2}/Soc I_{2} ≃ R_{∞}, which shows that X is preinjective or regular in T_{∞}. In particular, this shows that there are no non-zero modules in P^{<∞} ∩ B. For Kronecker modules, this follows immediately from 5.1. But then no other module X can belong to P^{<∞} ∩ B, because otherwise we would obtain a contradiction from the canonical exact sequence 0 ⟶ P_{1} ^{n} ⟶ X ⟶ X̄ ⟶ 0 where n ≥ 0 and X̄ is a Kronecker module.

(2) It is well known that Q is a direct summand of a product of copies of Y. Hence Q belongs to P and is a direct limit of modules in P^{<∞}. Note however that Q is not P^{<∞}-filtered. In fact, it follows from 5.1 that a Kronecker module which is P^{<∞}-filtered is always filtered by finite length regular modules and is therefore torsion.

Let us draw a further consequence from Proposition 5.4. For each tube T_{λ} with λ ∈ k ∪ {∞}, we denote by Z_{λ} the corresponding adic module, that is, the inverse limit of the chain of epimorphisms … X_{1} → X_{0} in the tube.

Corollary 5.7 The adic module Z_{λ} belongs to B if and only if λ = ∞.

Proof : We know from [20, proof of 9.3] that for each λ there is a universal exact sequence 0 ⟶ Z_{λ} ⟶ Q_{λ} ⟶ Y_{λ} ⟶ 0 where Q_{λ} is a direct sum of copies of Q. This shows that the Z_{λ} with λ ∈ k do not belong to Y^{⊥} = B. On the other hand, applying Hom_{R}(Y, ) on the sequence for λ = ∞, we deduce from long exact sequence … 0 = Hom_{R}(Y, Y_{∞}) → Ext_{R}^{1} (Y, Z_{∞}) → Ext_{R}^{1} (Y, Q_{∞}) = 0 … that Z_{∞} ∈ B. □

We are now in a position to decide whether A is closed under direct limits.

Theorem 5.8 T is not product-complete, hence A ≠ P, and A is not closed under direct limits.

Proof : Assume that T is product-complete. Recall that Q is a direct summand of a product of copies of Y, and Y lies in AddT by Corollary 5.5. So, we infer that Q must belong to A. On the other hand, we have just seen that the adic module Z_{∞} is in B, and it is well-known that Z_{∞} is pure-injective but not Σ-pure-injective. Then there is a cardinal β such that Z_{∞} ^{(β)} is not pure-injective, but still is a torsionfree regular module in B. Now since Q ∈ A, we have Ext_{Λ}^{1} (Q, Z_{∞} ^{(β)}) = 0. But it was shown by Okoh in [26, Prop.1 and Remark on p.265] that a torsionfree regular module belongs to Ker Ext_{Λ}^{1} (Q, −) if and only if it is pure-injective. So, we obtain a contradiction. Thus T is not product-complete. From Corollary 4.4 we then conclude that A is not closed under direct limits, and in particular A ≠ P. □

Acknowledgment We would like to thank Birge Huisgen-Zimmermann, Robert El Bashir, Mike Prest and Sverre Smalø for interesting comments.

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Address of the authors:

Lidia Angeleri-Hügel Jan Trlifaj Mathematisches Institut der Universität, Katedra algebry MFF UK, Theresientrasse 39, Sokolovská 83, 186 75 Prague 8, D-80333 München, Germany Czech Republic e-mail: angeleri@rz.mathematik.uni-muenchen.de email: trlifaj@karlin.mff.cuni.cz Current address: Università degli Studi dell’Insubria Dipartimento di Informatica e Comunicazione Via Mazzini 5, I - 21100 Varese, Italy