Homological Aspects of Torsion Theories^{1}

Apostolos Beligiannis

Idun Reiten

Author address:

Department of Mathematics, University of the Aegean, 83200 Kar– lovassi, Samos, Greece

E-mail address: abeligia@mail.otenet.gr, abeligia@aegean.gr

Department of Mathematical Sciences, Norwegian University of Science and Technology, 7490 Trondheim, Norway

E-mail address: idunr@math.ntnu.no

115 May, 2001

Contents

Introduction 1

Chapter I. Torsion Theories in Abelian and Triangulated Categories 6 1. Torsion Theories in Abelian Categories 6 2. Torsion Theories in Triangulated Categories 8 3. Tilting Torsion Theories 15

Chapter II. Torsion Theories in Pretriangulated Categories 18 1. Pretriangulated Categories 18 2. Adjoints and Orthogonal Subcategories 23 3. Torsion Theories 27 4. Torsion Theories and Localization Sequences 30 5. Lifting Torsion Theories 32

Chapter III. Compactly Generated Torsion Theories in Triangulated

Categories 37

1. Torsion Theories of Finite Type 37 2. Compactly Generated Torsion Theories 38 3. The Heart of a Compactly Generated Torsion Theory 43 4. Torsion Theories Induced by Tilting Objects 48

Chapter IV. Hereditary Torsion Theories in Triangulated Categories 52 1. Hereditary Torsion Theories 52 2. Hereditary Torsion Theories and Tilting 57 3. Connections with the Homological Conjectures 61 4. Concluding Remarks and Comments 67

Chapter V. Torsion Theories in Stable Categories 69 1. Comparison of Subcategories 69 2. A Description of Torsion Theories 73 3. Good Pairs and Torsion Theories 76 4. Torsion Classes and Cohen-Macaulay Objects 80 5. Tilting Modules 84

Chapter VI. Triangulated Torsion(-Free) Classes in Stable Categories 88 1. Triangulated Subcategories 88 2. Triangulated Torsion(-Free) Classes 90 3. Good Triples and TTF-Theories 93

Chapter VII. Torsion Theories and Closed Model Structures 100 1. Preliminaries on Closed Model Categories 100 2. Approximation Sequences and Good Pairs 102 3. Good Pairs and Closed Model Structures 105 4. Closed Model Structures and (Pre)Torsion Theories 108 5. A Classification of (Pre)Torsion Theories 116

Chapter VIII. Generalized Tate-Vogel Cohomology 121 1. Hereditary Torsion Theories and Homological Functors 121 2. Torsion Theories and Generalized Tate-Vogel (Co-)Homology 124 3. Relative Homology and Generalized Tate-Vogel (Co)Homology 129 4. Good Triples 133 5. Big Cohen-Macaulay Modules 137

Bibliography 145

Abstract

In this paper we study torsion theories in the general setting of pretriangulated categories, an omnipresent class of additive categories which includes abelian, triangulated, stable, and more generally (homotopy categories of) closed model categories in the sense of Quillen, as special cases.

We explore the formal analogies of the concept of a torsion theory in the above settings, concentrating our study to the relationship between orthogonal subcategories and the existence of left or right adjoints in connection with their interaction with the structure of left/right triangles and exact sequences.

The main focus of our study lies to the investigation of the strong connections and the interplay between torsion theories and tilting theory in abelian, triangulated and stable categories. The proper setting for the formulation of these connections is via closed model structures. We also study (co)homological functors induced by torsion theories, thus generalizing the Tate-Vogel (co)homological functors, and we give applications to the structure of the representations of Artin algebras and Gorenstein rings.

2000 Mathematics Subject Classification. Primary 18E40, 18E30, 18E35, 18G55, 18G60; Secondary 16G10, 18E30, 18E10, 20C05, 20J05, 55U35 .

Key words and phrases. Torsion theories, Abelian categories, t-structures, Triangulated Categories, Compact objects, Tilting theory, Derived Categories, Contravariantly finite subcategories, Approximations, Stable categories, Reflective subcategories, Resolving subcategories, Closed Model Structures, Cohen-Macaulay Modules, Tate-Vogel Cohomology, Gorenstein Rings.

The main part of this work was done during summer 1998 at the University of Bielefeld where both authors were visitors and during winter 1999 and spring 2001 at the Norwegian University of Science and Technology, where the first named author was visiting the second. The first named author thanks his coauthor for the warm hospitality, and gratefully acknowledges support and hospitality from the Norwegian University of Science and Technology and the TMR-network “Algebraic Lie Representations”.

Introduction

The proper framework for the study of homological algebra and for large parts of representation theory is the context of abelian categories. In recent years triangulated categories, and in particular derived and stable categories, entered into the picture of homological representation theory in a very essential way, offering new invariants and classification limits, through the work of Happel [38], Rickard [69], Keller [52], Neeman [64], Happel-Reiten-Smalø [41], Krause [49] and others.

There is a formal analogy between abelian categories and triangulated categories. In the first case we have exact sequences and in the second case we have triangles, which can be regarded as a reasonable substitute for the exact sequences. In this paper we study pretriangulated categories, which can be regarded as a common generalization of abelian and triangulated categories. This generalization gains its importance from the fact that it incorporates at the same time also the stable categories, which are very useful for the study of the behaviour of several stable phenomena occurring in homological representation theory.

Recall that a pretriangulated category is an additive category C equipped with a pair (Σ, Ω) of adjoint endofunctors, where Ω is the loop functor and Σ is the suspension functor, and in addition with a class of left triangles Δ and a class ∇ of right triangles which are compatible with each other and with Σ and Ω. Important examples are abelian categories (in which case Σ = 0 = Ω) and triangulated categories (in which case Σ or Ω is an equivalence). An important source of examples of pretriangulated categories is an abelian category C with a functorially finite subcategory ω. Then the stable category C/ω becomes in a natural way a pretriangulated category; the functors Ω, Σ and the class of left and right triangles are defined via left and right ω-approximations of objects of C. As a special case we can choose C = mod(Λ), the category of finitely presented modules over an Artin algebra, and let ω be the additive subcategory generated by a (tilting or cotilting) module.

Torsion theories play an important role in the investigation of an abelian category. There is a natural analogous definition for triangulated categories which is closely related to the notion of a t-structure. Both concepts of torsion are connected with tilting theory, and there is an interesting interplay between them in the work of Happel, Reiten and Smalø [41], which led these authors to the investigation of quasitilted algebras. These concepts of torsion generalize naturally to the setting of pretriangulated categories. Our main interest lies in the investigation of these generalized torsion theories on pretriangulated, and especially on triangulated or stable categories. In this way we are provided with a convenient conceptual umbrella for the study of various aspects of torsion and their interplay. Our results indicate that torsion theories in this general setting can be regarded as generalized tilting theory.

On the other hand there are the notions of contravariantly and covariantly finite subcategories of an additive category as introduced in [10]. Special cases of these are subcategories for which there exist a right or left adjoint of the inclusion. Contravariantly or covariantly finite subcategories play a fundamental role in the representation theory of Artin algebras. They provide a convenient setting for the study of several important finiteness conditions in various settings and there is again a strong connection to tilting theory. For Artin algebras these categories occur in pairs (X , Y), under some natural additional assumptions, namely X is contravariantly finite, closed under extensions and contains the projectives and Y is covariantly finite, closed under extensions and contains the injectives. Then X and Y are in one-one correspondence, via the vanishing of Ext^{1} rather than of Hom. An important example of this correspondence is induced by a tilting (or cotilting) module T. It is well–known that if T is a classical tilting module (that is, the projective dimension of T is less than or equal to one), then T induces in a natural way a torsion theory in the module category, which is very useful for the study of representation-theoretic questions. However this is not the case if T is non-classical, but we show that T induces in a natural way an interesting torsion theory in the pretriangulated stable category obtained by factoring out the additive subcategory generated by T.

For abelian and triangulated categories (in the latter case first formulated in the language of t-structures), there is a close connection between X being a torsion class of a torsion theory and X admitting a right adjoint (and between Y being a torsion-free class of a torsion theory and Y admitting a left adjoint). There is a similar connection in the setting of pretriangulated categories, although the extension of the results does not work so smoothly in general. The main reason is that the compatibility of the left and right triangles is not so well behaved as in the abelian or triangulated case. However we prove the corresponding result under some additional assumptions. Of particular interest for us is the case of a stable category C/ω. Here we obtain at the same time the correspondence between contravariantly finite and covariantly finite subcategories via Ext^{1} discussed above and the torsion theory correspondence between the associated subcategories of C/ω, where ω = X ∩ Y. By a result of [19] such a stable category C/ω is the homotopy category of a closed model structure in C in the sense of Quillen, which is defined via left/right approximations of C by objects from ω. In this paper we show that there is a strong connection between the pairs of subcategories realizing the correspondence described above, and closed model structures in the sense of Quillen in C. Recently a similar connection from a different viewpoint was observed independently by Hovey [46].

In general it is of central importance for the structure of the pretriangulated category in question, that a reasonable subcategory is a torsion or torsion-free class. For instance when we have a torsion theory in a stable pretriangulated category of an abelian category, we define new universal (co)homology theories on the abelian category which are complete with respect to the torsion or torsion free class in an appropriate sense. These new (co)homology theories are generalizations of the Tate-Vogel (co)homology theories to the non-Gorenstein case. This procedure provides a useful interplay between torsion-theoretic properties of the stable category and homological properties of the abelian category. In this setting there is a strong connection between the relative homological algebra, in the formulation proposed by Butler-Horrocks and more recently by Auslander-Solberg, induced by the torsion theory and the behaviour of the (co)homology theories. This connection has important consequences for the homological structure of the abelian category.

The article is organized as follows.

In Chapter I, which is of preliminary nature, we recall briefly some concepts and results, most of them well-known, concerning (hereditary/cohereditary) torsion theories in an abelian or triangulated category, which serve as a motivating source for the results of the rest of the paper. Here we observe that in the triangulated case the concept of a torsion theory essentially coincides with the concept of a tstructure in the sense of Beilinson-Bernstein-Deligne [15]. We discuss briefly the connection between torsion theories in abelian and derived categories investigated in [41] and indicate the relationship with tilting theory.

The first section of Chapter II is devoted to reminding the reader of the basic results on left/right triangulated and pretriangulated categories. The constructions of this section provide a rather large source of examples for our later applications, since pretriangulated categories include abelian, triangulated and stable categories. In Section 2 we study the relationship between reflective/coreflective and covariantly/contravariantly finite subcategories in a pretriangulated category C, and we investigate their connection with orthogonal subcategories with respect to the vanishing of Hom. In this chapter we introduce the fairly general concept of torsion theory (X , Y) in a pretriangulated category, which is based on glueing left and right triangles with end terms in a pair of orthogonal subcategories. We show that if (X , Y) is a such a torsion theory, then X is a right triangulated subcategory of C and the inclusion of X in C admits a right adjoint and Y is a right triangulated subcategory of C and the inclusion of Y in C admits a left adjoint. Moreover the adjunction morphisms form a left and right triangle, and the pair (X , Y) is complete with respect to the vanishing of Hom. We show that this concept incorporates the classical concepts of torsion in abelian or triangulated categories discussed in Chapter I. Using the fact that to any left or right triangulated category there is associated in a universal way a triangulated category, called its stabilization and which reflects the left or right triangulated structure, we lift a hereditary or cohereditary torsion theory in a pretriangulated category to a hereditary or cohereditary torsion theory in its stabilization. This construction will be used in the final chapter as a tool for the definition of universal (co)homology theories induced by a torsion theory.

Chapters III to VIII, which are devoted to the investigation of torsion theories in triangulated and stable categories and their connection with closed model categories and universal (co)homology functors, form the main body of the article.

Chapters III and IV are devoted to constructions of torsion theories in a triangulated category large enough to contain all small coproducts. We show that any given set of compact objects in such a category generates in a natural way a torsion theory. We investigate its heart, which by the general theory of [15] is an abelian category, and we give conditions for this category to be a module category. We give applications to tilting and Morita theory in module and derived categories. Also any given set of compact objects generates a hereditary torsion theory and in many cases a TTF-theory. Following the fundamental work of Keller [52] on derived categories of DG-algebras, we describe the torsion class of the hereditary torsion theory generated by a set of compact objects in terms of derived categories of appropriate DG-algebras. These results provide simple torsion theoretic proofs of basic results of Happel [38], Rickard [69], and Keller [52]. We also construct interesting TTF-theories in this setting, which reduce to recollement situations in the sense of [15]. Finally we show that there is close relationship between certain homological conjectures in the representation theory of Artin algebras on one hand, and (hereditary) torsion theories in the unbounded derived category of the algebra on the other hand.

In Chapters V and VI we investigate torsion theories in a pretriangulated category which is obtained by factoring out an abelian category C by a functorially finite subcategory ω. Under rather mild assumptions we give necessary and/or sufficient conditions for the existence of a torsion theory in this setting, and we point out the close relationship with the correspondence of subcategories via Ext^{1} mentioned above [3], [7]. We also give applications to tilting theory in the setting of Artin algebras, characterizing a tilting or cotilting module in terms of the existence of a torsion theory in a naturally induced stable category. In Chapter VI we investigate the natural question of when a torsion or torsion-free class in a stable category is triangulated. We show that this happens iff the functorially finite subcategory ω is the category of projective or injective modules. In this case the torsion subcategory is related to the subcategory of Cohen-Macaulay modules and the torsion-free subcategory is related to the objects of finite projective or injective dimension. This leads us naturally to the class of Gorenstein algebras which is characterized by a host of equivalent conditions expressed in torsion theoretic terms. We characterize also TTF-theories in this setting and we show that they are induced from functorially finite resolving and coresolving subcategories of the module category.

In Chapter VII we investigate homotopy theoretic properties of torsion theories by studying the connections between torsion theories and closed model category structures in the sense of Quillen [66]. We characterize the closed model structures in an abelian category where natural full subcategories of fibrant and/or cofibrant objects induce a torsion theory in an appropriate stable category. When this happens the torsion class and the torsion-free class are Quillen homotopy categories. Actually we prove that in an abelian category there is a bijective correspondence between closed model structures in C which are compatible in an appropriate sense, and certain pairs of subcategories (X , Y) in C inducing a torsion theory in the stable category C/X ∩ Y; we call such pairs pretorsion theories in C. Note that pretorsion theories are omnipresent in various contexts and occur frequently in practice. The above results give a homotopy theoretic interpretation of the results of Chapters V and VI.

The last chapter is devoted to the construction and the investigation of universal (co)homology extension functors defined on an abelian category C, if the stable category C/ω of C by a functorially finite subcategory ω is endowed with a (co)hereditary torsion theory. Their definition involves the lifting of the torsion theory to a torsion theory in the stabilization of C/ω. We show that these universal (co)homology extension functors are natural generalizations of the Tate-Vogel (co)homology functors studied in homological group theory, commutative algebra and representation theory. They are characterized as the universal (co)homological extensions of the functor Ext^{∗} with the additional property that they are complete with respect to the torsion or torsion-free class in an appropriate sense. Moreover these functors are represented by an appropriate (co)homotopy functor in the sense of Eckmann-Hilton [42]. In case we work in the category of (not necessarily finitely generated) modules over an Artin algebra we show that in this way we can define a new cohomology theory which we call generalized Tate-Vogel or Cohen-Macaulay (co)homology. The rest of this chapter is devoted to the study of relative homological invariants emerging from the Cohen-Macaulay (co)homology theory. This leads us to the investigation of big Cohen-Macaulay modules and modules of virtually finite projective or injective dimension, which are defined in a torsion theoretic way.

Convention. Throughout the paper we compose morphisms in the diagrammatic order, i.e. the composition of f : A ⟶ B and g : B ⟶ C in a given category is denoted by f ∘ g. Our additive categories admit finite direct sums.

CHAPTER I Torsion Theories in Abelian and Triangulated Categories

In this chapter we recall some well-known concepts and results concerning torsion theories in an abelian or triangulated category, which serve as a motivating source for our later results. We also discuss an interesting interplay of the two concepts of torsion in abelian and triangulated categories.

1. Torsion Theories in Abelian Categories

In this section we recall some elementary results concerning (hereditary) torsion theories in an abelian category and we fix some notation.

If Z is a class of objects in an additive category C, we denote by

⊥Z := {X ∈ C | C(X, Z) = 0}

the left orthogonal subcategory of Z, and by

Z^{⊥} := {Y ∈ C | C(Z, Y) = 0}

the right orthogonal subcategory of Z.

Assume now that C is an abelian category. If X , Y are subcategories of C, we define their Gabriel product X ⋆Y to be the full subcategory of C consisting of all objects C which can be included in an exact sequence 0 ⟶ X ⟶ C ⟶ Y ⟶ 0 in C such that X ∈ X and Y ∈ Y. Then Z ⊆ C is closed under extensions iff Z = Z ⋆ Z.

Definition 1.1. A torsion theory in C is a pair (X , Y) of strict (i.e. closed under isomorphisms) full subcategories of C satisfying the following conditions:

(i) C(X, Y) = 0, ∀X ∈ X, ∀Y ∈ Y.

(ii) C = X ⋆ Y, i.e. for any object C ∈ C there exists a short exact sequence:

0 ⟶ X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶ 0  (1)

in C such that X_{C} ∈ X and Y^{C} ∈ Y. If (X , Y) is a torsion theory, then X is called a torsion class and Y is called a torsion-free class.

It is well–known and easy to see that for a torsion theory (X , Y) in C we have that X is closed under factors, extensions and coproducts and Y is closed under extensions, subobjects and products, and moreover: X^{⊥} = Y and ^{⊥}Y = X.

For a torsion theory (X , Y) in C the assignment R(C) = X_{C} extends to an additive functor R : C ⟶ X which is a right adjoint of the inclusion i : X ↪ C, that is X is a coreflective subcategory of C. Moreover the morphism f_{C} serves as the counit of the adjoint pair (i, R) evaluated at C. Dually the assignment L(C) = Y^{C} extends to an additive functor L : C ⟶ Y which is a left adjoint of the inclusion j : Y ↪ C, that is Y is a reflective subcategory of C. Moreover the morphism g^{C} serves as the unit of the adjoint pair (L, j) evaluated at C.

For comparison with our later results we include the following characterization of torsion or torsion-free classes. First we say that a full subcategory Z of C is closed under extensions of left exact sequences if for any left exact sequence 0 ⟶ Z_{1} ⟶ C ⟶ Z_{2} with Z_{1}, Z_{2} ∈ Z, the object C is in Z. Dually we say that Z is closed under extensions of right exact sequences if for any right exact sequence Z_{1} ⟶ C ⟶ Z_{2} ⟶ 0 with Z_{1}, Z_{2} ∈ Z, the object C is in Z.

Proposition 1.2. Let C be an abelian category and let X and Y be full subcategories of C closed under isomorphisms. Then we have the following.

(1) Y is a torsion-free class in C iff Y is closed under extensions of left exact sequences and the inclusion j : Y ↪ C admits a left adjoint L : C ⟶ Y.

(2) X is a torsion class in C iff X is closed under extensions of right exact sequences and the inclusion i : X ↪ C admits a right adjoint R : C ⟶ X.

Proof. (1) (⇐) If Y is closed under extensions of left exact sequences, then trivially Y is closed under extensions and subobjects. Let L : C ⟶ Y be the left adjoint of the inclusion j : Y ↪ C and let g^{C} : C ⟶ L(C) be the unit evaluated at C. Since Y is closed under subobjects, it is trivial to see that the inclusion Im(g^{C}) ↪ L(C) is invertible, hence g^{C} is epic. Let f_{C} : X_{C} ⟶ C be the kernel of g^{C}. Then X_{C} ∈ ^{⊥}Y. Indeed if α : X_{C} ⟶ Y is a morphism with Y ∈ Y, then in the push-out 0 ⟶ Y ⟶ Y′ ⟶^{h} L(C) ⟶ 0 of 0 ⟶ X_{C} ⟶ C ⟶ L(C) ⟶ 0 along α, the object Y′ is in Y since Y is closed under extensions. Since g^{C} is the reflection morphism of C in Y it follows directly that h splits, hence α factors through f_{C}, say via a morphism β : C ⟶ Y. Since Y ∈ Y, then β factors through g^{C}, and this implies trivially that β = 0, so α = 0. Hence we have X_{C} ∈ ^{⊥}Y. Then by definition we have that Y is the torsion-free class of the torsion theory (^{⊥}Y, Y) in C.

(⇒) Let Y be a torsion-free class and let 0 ⟶ Y_{1} ⟶^{α} C ⟶^{β} Y_{2} be a left exact sequence with Y_{1}, Y_{2} ∈ Y. Since Y is closed under subobjects, Im(β) ∈ Y, and since Y is closed under extensions, we have C ∈ Y. Hence Y is closed under extensions of left exact sequences and as noted before the inclusion Y ↪ C admits a left adjoint.

(2) The proof is dual.

Let (X , Y) be a torsion theory in C, let R be the right adjoint of the inclusion i : X ↪ C and let L be the left adjoint of the inclusion j : Y ↪ C. We recall that (X , Y) is said to be hereditary, resp. cohereditary, if the torsion class X is closed under subobjects, resp. the torsion–free class Y is closed under factors. It is well–known that (X , Y) is hereditary iff X is a Serre subcategory iff the (idempotent radical) functor iR : C ⟶ C is left exact [77]. Dually (X , Y) is cohereditary iff Y is a Serre subcategory iff the (idempotent coradical) functor jL : C ⟶ C is right exact. If V is a Serre subcategory of C, then we denote by C/V the induced Gabriel quotient, see [34] for details and more information.

We recall also that a torsion, torsion-free theory, TTF-theory for short, in the abelian category C is a triple (X , Y, Z) of strict full subcategories of C such that (X , Y) and (Y, Z) are torsion theories. If (X , Y, Z) is a TTF-theory in C, then obviously (X , Y) is cohereditary and (Y, Z) is hereditary. It follows that Y is a reflective and coreflective Serre subcategory of C. The following describes the Gabriel quotient C/Y and presents an interesting connection between the involved subcategories in a TTF-theory.

Proposition 1.3. [36] If (X , Y, Z) is a TTF-theory in C then there exists an equivalence: X ∩ Z ⟶^{≈} C/Y.

Proof. Let π : C ⟶ C/Y be the exact quotient functor. We show that the composition F : X ∩Z ↪ C ⟶^{π} C/Y is an equivalence. If f : U_{1} ⟶ U_{2} is a morphism in X ∩ Z such that π(f) = 0, then by the construction of C/Y we have that Im(f) lies in Y. Then Im(f) = 0 since U_{1} is in X. Hence f = 0 and consequently F is faithful. If α : π(U_{1}) ⟶ π(U_{2}) is a morphism in C/Y, where the U_{i} are in X ∩ Z, then α is represented by a diagram U_{1} ⟵^{s} U_{3} ⟶^{f} U_{2} where s is a morphism with kernel and cokernel in Y. Since (X , Y) = 0 = (Y, Z) we infer that s is invertible. Then it is easy to see that α = π(s^{−1} ∘ f). Hence F is full. Finally let π(C) be an object in C/Y. Consider the canonical exact sequences 0 ⟶ X_{C} ⟶ C ⟶ Y^{C} ⟶ 0 and 0 ⟶ Y_{C} ⟶ C ⟶ Z^{C} ⟶ 0. Applying the exact functor π to the above sequences, we have isomorphisms π(X_{C}) ⟶^{≅} π(C) ⟶^{≅} π(Z^{C}). Let U be the image of the composition X_{C} ⟶ C ⟶ Z^{C}. Then U lies in X ∩ Z since X is closed under factors and Z is closed under submodules. Since the composition π(X_{C}) ⟶ π(Z^{C}) is invertible, we have π(C) ≅ π(U). This shows that F is surjective on objects.

2. Torsion Theories in Triangulated Categories

Motivated by the concept of torsion in the abelian case we study in this section the analogous concept of a (hereditary) torsion theory in triangulated categories.

Let C be a triangulated category with suspension functor Σ and let Z, W be full subcategories of C. Then the Verdier product Z ⋆ W of Z, W is the full subcategory of C consisting of all objects C which can be included in a triangle Z ⟶ C ⟶ W ⟶ Σ(Z) with Z ∈ Z and W ∈ W. Then Z is closed under extensions iff Z = Z ⋆ Z.

A direct analogue of the definition of a torsion theory in an abelian category in the triangulated case is the following.

Definition 2.1. A torsion theory in C is a pair of strict full subcategories (X , Y) of C satisfying the following conditions:

(i) C(X , Y) = 0.

(ii) Σ(X) ⊆ X or Σ^{−1}(Y) ⊆ Y.

(iii) C = X ⋆ Y, i.e. for any C ∈ C there exists a triangle

X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{h^{C}} Σ(X_{C})  (2)

in C such that X_{C} ∈ X and Y^{C} ∈ Y. Then X is called a torsion class and Y is called a torsion-free class.

The following observation shows that condition (ii) above is symmetric.

Remark 2.2. Suppose that the pair (X , Y) satisfies (i), (iii) above.

(1) If Σ(X) ⊆ X, then: X^{⊥} = Y and Σ^{−1}(Y) ⊆ Y. Indeed by (i), Y ⊆ X^{⊥}. If C ∈ X^{⊥}, then by (iii) we have f_{C} = 0. This implies that Y^{C} ≅ C ⊕ Σ(X_{C}). Since Σ(X_{C}) ∈ X, we have trivially that C ≅ Y^{C} ∈ Y. For any Y ∈ Y we now have C(X , Σ^{−1}(Y)) ≅ C(Σ(X), Y) = 0. Hence Σ^{−1}(Y) ∈ X^{⊥} = Y, i.e. Σ^{−1}(Y) ⊆ Y.

(2) Dually if Σ^{−1}(Y) ⊆ Y, then: ^{⊥}Y = X and Σ(X) ⊆ X.

(3) Hence if (X , Y) is a torsion theory in C, then: ^{⊥}Y = X and X^{⊥} = Y.

The following result, first observed in [57], is a triangulated analogue of Proposition 1.2.

Proposition 2.3. Let C be a triangulated category and let X and Y be full subcategories of C closed under isomorphisms. Then we have the following.

(1) Y is a torsion-free class iff Y is closed under extensions and Σ^{−1}, and the inclusion j : Y ↪ C admits a left adjoint L : C ⟶ Y.

(2) X is a torsion class iff X is closed under extensions and Σ, and the inclusion i : X ↪ C admits a right adjoint R : C ⟶ X.

Proof. (1) (⇐) Let L : C ⟶ Y be the left adjoint of the inclusion j : Y ↪

C. For C ∈ C let X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} L(C) ⟶^{h^{C}} Σ(X_{C}) be a triangle in C where g^{C} is the counit of (L, j) evaluated at C. Applying the cohomological functor

C(−, Y) to the above triangle we get the long exact sequence C(ΣL(C), Y) ⟶^{(Σ(g^{C}),Y)}

C(Σ(C), Y) ⟶ C(Σ(X_{C}), Y) ⟶ C(L(C), Y) ⟶^{C(g^{C},Y)} C(C, Y) in which (g^{C}, Y) is

invertible. Since Y is closed under Σ^{−1}, then (Σ(g^{C}), Y) which is isomorphic to (g^{C}, Σ^{−1}(Y)), is also invertible. It follows that C(Σ(X_{C}), Y) ≅ C(X_{C}, Σ^{−1}(Y)) = 0. We set X := ^{⊥}Y, and it suffices to show that X_{C} ∈ X. Now let α : X_{C} ⟶ Y be a morphism with Y ∈ Y. Consider the following morphism of triangles:

X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} L(C) ⟶^{h^{C}} Σ(X_{C})
α↓  ^{∃ β}↓  ^{∥}↓  ^{Σ(α)}↓
Y ⟶^{f′} Y′ ⟶^{g′} L(C) ⟶^{h′} Σ(Y)

Then Y′ ∈ Y since Y is closed under extensions. Since g^{C} is the coreflection of C in Y, there exists a unique morphism ρ : L(C) ⟶ Y′ such that g^{C} ∘ ρ = β. Then α∘f′ = f_{C} ∘β = f_{C} ∘g^{C} ∘ρ = 0. Hence there exists a morphism τ : X_{C} ⟶ Σ^{−1}L(C) such that τ ∘ Σ^{−1}(h′) = α. Since C(X_{C}, Σ^{−1}(Y)) = 0, we have τ = 0 and then α = 0. We infer that X_{C} ∈ ^{⊥}Y = X.

(⇒) If Y is a torsion-free class with corresponding torsion class X, then by Remark 2.2, we have X = ^{⊥}Y which is closed under Σ and X^{⊥} = Y which is closed under Σ^{−1}. This implies trivially that Y is closed under extensions. Consider the

triangle X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{h^{C}} Σ(X_{C}) of Definition 2.1. Applying the functor C(−, Y) to this triangle and using that Σ(X) ⊆ X we see directly that C(g^{C}, Y) : C(Y^{C}, Y) ⟶ C(C, Y) is invertible, i.e. g^{C} is the coreflection morphism of C in Y. Hence setting L(C) = Y^{C} we obtain a left adjoint L : C ⟶ Y of the inclusion j : Y ↪ C.

(2) The proof is dual.

Remark 2.4. If (X , Y) is a torsion theory in C, then so is (Σ^{n}(X), Σ^{n}(Y)), ∀n ∈ ℤ. Indeed obviously we have C(Σ^{n}(X), Σ^{n}(Y)) = 0, Σ(Σ^{n}(X)) ⊆ Σ^{n}(X) and for any C ∈ C we have the triangle Σ^{n}RΣ^{−n}(C) ⟶ C ⟶ Σ^{n}LΣ^{−n}(C) ⟶ Σ(Σ^{n}RΣ^{−n}(C)).

Inspired by the characterization of hereditary torsion theories in abelian categories, we make the following definition.

Definition 2.5. A torsion theory (X , Y) in C is called hereditary, resp. cohereditary, if the idempotent functor iR : C ⟶ C is exact, resp. the idempotent functor jL : C ⟶ C is exact.

We recall that a thick subcategory of C is a full triangulated subcategory of C closed under direct summands. A thick subcategory Z ⊆ C is called localizing, resp. colocalizing, if the inclusion Z ↪ C admits a right, resp. left, adjoint. If Z is a thick subcategory of C then we denote by C/Z the induced Verdier quotient [79]. Thick subcategories can be regarded as triangulated analogues of Serre subcategories. The next result shows that hereditary torsion theories coincide with cohereditary torsion theories and includes a result of Bondal and Kapranov [25].

Proposition 2.6. If X and Y are full subcategories of C, then the following conditions are equivalent.

(i) (X , Y) is a hereditary torsion theory in C.

(ii) (X , Y) is a cohereditary torsion theory in C.

(iii) (X , Y) is a torsion theory in C and Σ^{−1}(X) ⊆ X.

(iv) (X , Y) is a torsion theory in C and Σ(Y) ⊆ Y.

(v) C(X , Y) = 0, X and Y are thick subcategories of C, and C is generated as a triangulated category by X and Y.

(vi) X is a localizing subcategory of C and there exists a short exact sequence:

0 ⟶ X ⟶^{i} C ⟶^{L} Y ⟶ 0

(vii) Y is a colocalizing subcategory of C and there exists a short exact sequence:

0 ⟶ Y ⟶^{j} C ⟶^{R} X ⟶ 0

Proof. (i) ⇔ (iii) If Σ^{−1}(X) ⊆ X, then X is a full triangulated subcategory of C. Hence i : X ↪ C is exact. It is well–known that left or right adjoints of exact functors between triangulated categories are exact, see [79]. Hence R is exact and then so is iR : C ⟶ C, i.e. (X , Y) is hereditary. If (X , Y) is hereditary, then iR is exact, in particular iR commutes functorially with Σ^{−1}. This implies trivially that Σ^{−1}(X) ⊆ X.

The proof of (ii) ⇔ (iv) is dual and the proof of (iii) ⇔ (iv) follows from the fact that if X^{⊥} = Y and ^{⊥}Y = X, then Σ^{−1}(X) ⊆ X iff Σ(Y) ⊆ Y.

(i) ⇔ (v) Obviously (i) implies (v). If (v) holds, then let D = X ⋆ Y and it suffices to show that D is a triangulated subcategory of C. Obviously D is closed under Σ, Σ^{−1}. Let α : D_{1} ⟶ D_{2} be a morphism in D, so that there are triangles

X_{i} ⟶^{f_{i}} D_{i} ⟶^{g_{i}} Y_{i} ⟶^{h_{i}} Σ(X_{i}) with X_{i} ∈ X and Y_{i} ∈ Y, i = 1, 2. Since C(X , Y) = 0

we have f_{1} ∘ α ∘ g_{2} = 0. Hence there exists a diagram

X_{1} ⟶^{f_{1}} D_{1} ⟶^{g_{1}} Y_{1} ⟶^{h_{1}} Σ(X_{1})
β↓  ^{α}↓  ^{∃ γ}↓  ^{Σ(β)}↓
X_{2} ⟶^{f_{2}} D_{2} ⟶^{g_{2}} Y_{2} ⟶^{h_{2}} Σ(X_{2})

By a result of Verdier [81], this can be completed to a 3 × 3−diagram of triangles which implies in particular that a cone D_{3} of α is included in a triangle X_{3} ⟶ D_{3} ⟶ Y_{3} ⟶ Σ(X_{3}) where X_{3} is a cone of β and Y_{3} is a cone of a morphism γ making the above diagram commutative. Since X , Y are triangulated, it follows that D_{3} ∈ D. Hence D is triangulated category.

(i) ⇔ (vi) Assume that (i) holds. Since X = ^{⊥}Y, it follows trivially that X is thick and by Proposition 2.3 we know that X is coreflective and Y is reflective with coreflection functor L. Consider the Verdier quotient C/X which is defined as the category of fractions C[𝔍^{−1}], where 𝔍 is the class of morphisms which admit a cone in X. This implies that L : C ⟶ Y sends 𝔍 to isomorphisms in Y. If F : C ⟶ E is an exact functor to a triangulated category with this property, then F factorizes uniquely through L by defining F^{∗} : Y ⟶ E, F^{∗} = F j. Indeed F^{∗}L = F jL is isomorphic to F, since the unit Id_{C} ⟶ jL lies in 𝔍 and F inverts 𝔍. Hence F^{∗}L = F. If G : C ⟶ E is another exact functor such that GL = F, then G = GLj = F j = F^{∗}. It follows that L : C ⟶ Y represents the quotient functor C ⟶ C[𝔍^{−1}] = C/X, i.e. it induces a triangle equivalence C/X ⟶ Y. Conversely since X is coreflective, by Proposition 2.3 we have a hereditary torsion theory (X , X^{⊥}) in C. Then by a well-known result of Verdier, see [79], we have triangle equivalences C/X ≈ Y = X^{⊥}.

(i) ⇔ (vii) The proof is dual and is left to the reader.

We recall that a Grothendieck category is an abelian category with a generator and exact filtered colimits. It is well–known that if G is a Grothendieck category, then a strict full subcategory X ⊆ G is a torsion class, resp. torsion-free class, iff X is closed under factors, resp. subobjects, extensions and coproducts, resp. products. The triangulated analogue of a Grothendieck category is a compactly generated triangulated category. We recall that a set U of objects of C is called a generating set, if U^{⊥} = 0 and U is closed under Σ and Σ^{−1}. An object T ∈ C is called compact, if the functor C(T, −) : C ⟶ Ab preserves coproducts. Then C is compactly generated if C has coproducts and admits a set of compact generators. This is equivalent to saying that C has coproducts and coincides with the smallest full triangulated subcategory which is closed under coproducts and contains the compact objects. The corresponding characterization of torsion or torsion-free classes in a compactly generated triangulated category is analogous, modulo some set-theoretic restrictions.

Corollary 2.7. Let C be a compactly generated triangulated category.

(1) For a strict full subcategory X of C, the following are equivalent.

(i) X is the torsion class of a (hereditary) torsion theory in C.

(ii) X is closed under Σ (and Σ^{−1}), extensions, coproducts and the right triangulated Verdier quotient C/X has small Hom sets.

(2) For a strict full subcategory Y of C, the following are equivalent.

(i) Y is the torsion-free class of a (cohereditary) torsion theory in C.

(ii) Y is closed under Σ^{−1} (and Σ), extensions, products and the left triangulated Verdier quotient C/Y has small Hom sets.

Proof. (1) (ii) ⇒ (i) By hypothesis X is a full right triangulated subcategory of C closed under coproducts. Then by Proposition 2.3 it suffices to show that the inclusion X ↪ C admits a right adjoint, and this follows from [19]. The converse follows trivially from Proposition 2.6.

(2) The proof is similar to (1) using Neeman’s Theorem [64] that in a compactly generated triangulated category C, a product preserving homological functor C ⟶ Ab is representable.

Definition 2.8. A torsion, torsion-free theory, TTF-theory for short, in C is a triple (X , Y, Z) of full subcategories of C such that the pairs (X , Y) and (Y, Z) are torsion theories. In this case Y is called a TTF-class.

If (X , Y, Z) is a TTF-theory in C, then we have that (X , Y) is cohereditary and (Y, Z) is hereditary. Hence by Proposition 2.6, X is a thick localizing subcategory of C, Z is a thick colocalizing subcategory of C, and Y is a thick localizing and colocalizing subcategory of C. As usual we denote by R : C ⟶ X the right adjoint of the inclusion i : X ↪ C, by T : C ⟶ Z the left adjoint of the inclusion k : Z ↪ C, and by L : C ⟶ Y, resp. S : C ⟶ Y the left, resp. right, adjoint of the inclusion j : Y ↪ C. Then we have adjoint pairs:

(i, R) : X ⟶^{i} ; ⟵^{R} C ⟶^{L} ; ⟵^{j} Y : (L, j) and (j, S) : Y ⟶^{j} ; ⟵^{S} C ⟶^{T} ; ⟵^{k} Z : (T, k)

and ∀C ∈ C we have the following functorial exact commutative diagram of triangles, where the involved morphisms are the adjunctions:

iR(C) ⟶^{=} iR(C)
↓  ↓
jS(C) ⟶ C ⟶ kT(C) ⟶ ΣjS(C)
∥↓  ↓  ↓  ^{∥}↓
jS(C) ⟶ jL(C) ⟶ jLkT(C) ⟶ ΣjS(C)
↓  ↓
ΣiR(C) ⟶^{=} ΣiR(C)

The second vertical triangle in the above diagram shows that the above reflection/coreflection functors are connected with a natural isomorphism:

jS iR ⟶^{≅} Σ^{−1}jL kT : C ⟶ C

In other words for any object C in C we have: Y_{X_{C}} ⟶^{≅} Σ^{−1}(Y^{Z^{C}}). Observe that for any two objects A, B in C we have natural isomorphisms:

C(L(A), B) ⟵^{≅} Y(L(A), S(B)) ⟶^{≅} C(A, S(B)).

The proof of the following analogue of Proposition 1.3, is a direct consequence of Proposition 2.6 and the above diagram.

Corollary 2.9. If (X , Y, Z) is a TTF-theory in C, then there are triangle equivalences: X ⟵^{≈} C/Y ⟶^{≈} Z. Moreover we have: C = X ⋆ Z = Z ⋆ X.

Note that the equivalence X ⟶^{≈} Z above is given explicitly by the functor Ti : X ⟶ Z which is given by X ↦ Z^{X}, with quasi-inverse the functor Rk : Z ⟶ X which is given by Z ↦ X_{Z}.

Remark 2.10. If (X , Y, Z) is a TTF-theory in an abelian category C, then in general it is not true that X and Z are equivalent. However if C has enough projectives and enough injectives, then it is not difficult to see that the analogous functors as above induce an equivalence A ⟶^{≈} B, where B is the Giraud subcategory of C corresponding to the hereditary torsion theory (Y, Z), and A is the co-Giraud subcategory of C corresponding to the cohereditary torsion theory (X , Y), see [77] for the concept of a (co-)Giraud subcategory corresponding to a (co-)hereditary torsion theory. Since there are equivalences A ⟵^{≈} C/Y ⟶^{≈} B, by Proposition 1.3 we have equivalences A ⟵^{≈} X ∩ Y ⟶^{≈} B.

With a proper definition of TTF-theory (X , Y, Z) in a stable category of an abelian category, we shall see in Chapter VI that the subcategories X and Z are equivalent up to projective or injective summands.

TTF-theories occur frequently in practice in a compactly generated triangulated category. In this setting we have the following result which will be useful later.

Proposition 2.11. Let C and D be compactly generated triangulated categories, and let F : D ⟶ C be a fully faithful exact functor which preserves coproducts and compact objects. If X = Im(F) is the essential image of F, then there exists a TTF-theory (X , Y, Z) in C, where Y := Ker(F), and Z := Y^{⊥}.

Proof. By [63] the functor F admits a right adjoint G : C ⟶ D which preserves coproducts, since F preserves compact objects. Since G preserves coproducts, it follows that G admits a right adjoint H : D ⟶ C. Hence we have an adjoint triple (F, G, H). Now it is easy to see that in such an adjoint triple, F is fully faithful iff H is fully faithful. Hence H is fully faithful [26]. Let Z := Im(H) be the essential image of H, and let Y := Ker(G). Observe that C(X , Y) = 0 and C(Y, Z) = 0. Indeed if X ∈ X, Y ∈ Y, and Z ∈ Z, then X = F (A) for some object A ∈ D, G(Y) = 0, and Z = H(B) for some object B in D. Then C(X, Y) = C(F (A), Y) ⟶^{≅} D(A, G(Y)) = 0. Similarly C(Y, Z) = C(Y, H(B)) ⟶^{≅} D(G(Y), B) = 0. Now let ε : F G ⟶ Id_{C} be the counit of the adjoint pair (F, G), and let δ : Id_{C} ⟶ HG be the unit of the adjoint pair (G, H). For any object C in C consider the functorial

triangles F G(C) ⟶^{ε_{C}} C ⟶ Y^{C} ⟶ Σ(F G(C)) and Y_{C} ⟶ C ⟶^{δ_{C}} HG(C) ⟶ Σ(Y_{C})

in C. Using standard properties of adjoint functors we have G(Y_{C}) = 0 = G(Y^{C}). Hence Y_{C}, Y^{C} ∈ Y. Since by definition F G(C) ∈ X and HG(C) ∈ Z, we infer that (X , Y) and (Y, Z) are torsion theories. Hence (X , Y, Z) is a TTF-theory in C.

Remark 2.12. In the situation of the above Proposition it is easy to see that iR = F G, kT = HG, jL is the cofiber of the counit ε : F G ⟶ Id_{C}, and jS is the fiber of the unit δ : Id_{C} ⟶ HG.

Torsion theories in triangulated categories are used in the literature mainly in the form of t-structures. Recall from [15] that a t-structure in a triangulated category C is a pair (T^{≤0}, T^{≥0}) of full subcategories such that setting T^{≤n} = Σ^{−n}(T^{≤0}) and T^{≥n} = Σ^{−n}(T^{≥0}), ∀n ∈ ℤ, the following are satisfied. First C(T^{≤0}, T^{≥1}) = 0. Second T^{≤0} ⊆ T^{≤1} and T^{≥1} ⊆ T^{≥0}. Third any object C ∈ C is included in a triangle C^{≤0} ⟶ C ⟶ C^{≥1} ⟶ Σ(C^{≤0}), where C^{≤0} ∈ T^{≤0} and C^{≥1} ∈ T^{≥1}.

The following result shows that torsion theories and t-structures essentially coincide.

Lemma 2.13. The maps

Φ : (X , Y) ⟼ (X , ΣY) and Ψ : (T^{≤0}, T^{≥0}) ⟼ (T^{≤0}, T^{≥1})

are mutually inverse bijections between torsion theories and t-structures in C.

Proof. If (X , Y) is a torsion theory in C, then set T^{≤0} = X and T^{≥0} = Σ(Y) and further T^{≤n} = Σ^{−n}(X) and T^{≥n} = Σ^{−n+1}(Y). Then C(T^{≤0}, T^{≥1}) = C(X , Y) = 0. Since Σ(X) ⊆ X, we have T^{≤0} = X ⊆ Σ^{−1}(X) = T^{≤1}. Since Σ^{−1}(Y) ⊆ Y, we have T^{≥1} = Y ⊆ Σ(Y) = T^{≥0}. Finally since (X , Y) is a torsion theory, ∀C ∈ C, there exists a triangle X_{C} ⟶ C ⟶ Y^{C} ⟶ Σ(X_{C}) with X_{C} ∈ T^{≤0} and Y^{C} ∈ T^{≥1}. So (X , Σ(Y)) is a t-structure in C. Conversely if (T^{≤0}, T^{≥0}) is a tstructure, then set X = T^{≤0} and Y = T^{≥1}. By definition C(X , Y) = 0 and Σ(X) = T^{≤−1} ⊆ T^{≤0} = X. Since ∀C ∈ C there exists a triangle X_{C} ⟶ C ⟶ Y^{C} ⟶ Σ(X_{C}) with X_{C} ∈ T^{≤0} and Y^{C} ∈ T^{≥1}, we infer that (T^{≤0}, T^{≥1}) is a torsion theory.

Let (X , Y) be a torsion theory in C. As usual we denote by R the right adjoint of the inclusion i : X ↪ C and by L the left adjoint of the inclusion j : Y ↪ C. Let T^{≤n} = Σ^{−n}(X) and T^{≥n} = Σ^{−n+1}(Y), ∀n ∈ ℤ. Then we have the functor τ^{≤n} := Σ^{−n}RΣ^{n} : C ⟶ T^{≤n} which is a right adjoint of the inclusion T^{≤n} ↪ C and the functor τ^{≥n} := Σ^{−n}LΣ^{n} : C ⟶ T^{≤n} which is a left adjoint of the inclusion T^{≥n} ↪ C. In other words the functors τ^{≤n} and τ^{≥n} are the truncation functors in the sense of [15] associated to the t-structure (X , Σ(Y)).

We recall form [15] that the heart of the torsion theory is defined to be the full subcategory H = X ∩ Σ(Y) = T^{≤0} ∩ T^{≥0} and the homology functors are defined by H^{n} := ΣLΣ^{−1}RΣ^{n} = RΣLΣ^{n−1} : C ⟶ H, ∀n ∈ ℤ. By [15], H is an abelian category and the functors H^{n} are indeed homological, i.e. they send triangles in C to long exact sequences in H.

Remark 2.14. It is not difficult to see that TTF-theories are in bijective correspondence with recollement situations in the sense of [15].

We close this section with the following result (possibly known to the experts) which describes connections between torsion theories in C and torsion theories in a Verdier-quotient C/V of C by a thick subcategory V of C, and between the corresponding hearts. It also can be used as a tool for constructing new torsion theories. In what follows we let Q : C ⟶ C/V be the localization functor of the Verdier quotient C/V with respect to the thick subcategory V.

Proposition 2.15. If (X , Y) is a (hereditary) torsion theory in C, then the pair (Q(X), Q(Y)) is a (hereditary) torsion theory in C/V iff for any object A in C which is isomorphic in C/V to an object from X, the object L(A) is in V.

If this is the case and H = X ∩ Σ(Y) is the heart of (X , Y) in C and K = Q(X) ∩ Σ(Q(Y)) is the heart of (Q(X), Q(Y)) in C/V, then U := X ∩ Σ(Y) ∩ V is a Serre subcategory of H and there exists a short exact sequence of abelian categories:

0 ⟶ U ⟶ H ⟶ K ⟶ 0

Moreover if V is a localizing, resp. colocalizing, subcategory of C, then the functor H ⟶ K admits a right, resp. left, adjoint.

Proof. It is clear that (Q(X), Q(Y)) is a torsion theory in C/V iff (Q(X), Q(Y)) = 0. Assume that the imposed condition holds, and let α : Q(X) ⟶ Q(Y) be a morphism in C/V. Then α is represented by a fraction X ⟵^{s} A ⟶^{ρ} Y, where in the triangle V ⟶^{κ} A ⟶^{s} X ⟶ Σ(V) the object V ∈ V. Consider the triangle R(A) ⟶^{f_{A}}

A ⟶^{g^{A}} L(A) ⟶ ΣR(A). Applying the Octahedral Axiom to the composition f_{A} ∘ s we see easily that there is a triangle L(A) ⟶ X′ ⟶ V ⟶ Σ(L(A) with X′ ∈ X and V ∈ V. Then by hypothesis L(A) is in V, hence f_{A} : R(A) ⟶ A is an isomorphism in C/V. This implies that the fraction X ⟵^{s} A ⟶^{ρ} Y is equal to the fraction X ⟵^{f_{A}∘s} R(A) ⟶^{0} Y which represents the zero morphism in C/V. Hence α = 0 in C/V. Conversely let (Q(X), Q(Y)) be a torsion theory in C/V, and let A be an object in C such that Q(A) ⟶^{≅} Q(X) where X is in X. Consider the triangle R(A) ⟶ A ⟶ L(A) ⟶ ΣR(A) in C. Then we have the triangle QR(A) ⟶ Q(A) ⟶ QL(A) ⟶ QΣR(A) in C/V. Since (Q(X), Q(Y)) is a torsion theory we have QL(A) ∈ Q(X) ∩ Q(Y)) = 0. This implies that L(A) lies in V.

Let A_{1} ↣ A_{2} ↠ A_{3} be a short exact sequence in H. Then there exists a morphism C ⟶ Σ(A) such that A_{1} ⟶ A_{2} ⟶ A_{3} ⟶ Σ(A_{1}) is a triangle in C. Applying Q we have a triangle Q(A_{1}) ⟶ Q(A_{2}) ⟶ Q(A_{3}) ⟶ ΣQ(A_{1}) in C/V with the Q(A_{i}) in K. Then the sequence Q(A_{1}) ↣ Q(A_{2}) ↠ Q(A_{3}) is short exact in K and we infer that the induced functor Q : H ⟶ K is exact. By definition Q is surjective on objects and Ker(Q) := U = X ∩ Σ(Y) ∩ V. Using the universal property of Q : C ⟶ C/V we infer that Q : H ⟶ K is the localization functor H ⟶ H/U. We leave the easy proof of the last assertion to the reader.

Observe that if the condition of the above result is true, then V is stable under the coreflection functor R. Hence by [15] we get an induced torsion theory (X ∩ V, Y ∩V) in V with heart precisely the category U. In this case the inclusion functor V ↪ C and the quotient functor C ⟶ C/V are t-exact in the sense of [15].

The above result can be used to construct interesting torsion theories:

Example. Let C be an abelian category and let H(C) be the homotopy category of, say unbounded, complexes over C. If X, resp. Y, denotes the full subcategory of complexes which are acyclic in degrees > 0, resp. < 1, then it is easy to see that (X , Y) is a torsion theory in H(C) which satisfies the condition in Proposition 2.15 with respect to the thick subcategory V of acyclic complexes. Then the quotient H(C)/V is the derived category D(C) of C and the induced torsion theory (Q(X), Q(Y)) is the natural torsion theory (= t-structure) (D^{≤0}(C), D^{≥0}(C)) of D(C), see [15]. In our case the exact sequence of the hearts in Proposition 2.15 reduces to the following exact sequence of abelian categories

0 ⟶ mod_{e}(C) ⟶ mod(C) ⟶ C ⟶ 0

where mod(C) is the category of contravariant coherent functors over C and mod_{e}(C) is the full subcategory consisting of all coherent functors F which admit a presentation C(−, A) ⟶ C(−, B) ⟶ F ⟶ 0 where A ⟶ B is an epimorphism. Note that the above exact sequence of abelian categories, which is due to Auslander [2], plays an important role in representation theory.

Example. Let Λ be a self-injective Artin algebra and let \underline{mod}(Λ) be its stable module category. Let D^{b}(mod(Λ)) be the bounded derived category and let (X , Y) be a torsion theory in D^{b}(mod(Λ)). Setting V = H^{b}(P_{Λ}) to be the full subcategory of perfect complexes and assuming that the condition of Proposition 2.15 holds, we get a torsion theory in \underline{mod}(Λ), since it is well-known that the Verdier quotient D^{b}(mod(Λ))/H^{b}(P_{Λ}) is triangle equivalent to \underline{mod}(Λ), see [57], [28], [70], [17].

3. Tilting Torsion Theories

In this section we discuss an interesting interplay between torsion theories in abelian and triangulated categories.

In classical tilting theory, the torsion theory associated with a tilting module of projective dimension less than or equal to one plays an important role in connection with homological or representation theoretic questions. Actually the tilting process can be made starting directly with a tilting torsion theory (T , F) in an abelian category C, in the sense that any object of C embeds in an object from T . This tilting process, which was initiated by Happel, Reiten and Smalø in [41], takes place in the bounded derived category D^{b}(C) and presents an interesting interplay between torsion theories in an abelian category and torsion theories in the bounded derived category, thus generalizing the classical tilting situation. More precisely let T and F be full subcategories of C closed under isomorphisms, and define full subcategories in D^{b}(C) as follows:

X (T ) := {C^{•} ∈ D^{b}(C) | H^{n}(C^{•}) = 0, ∀n > 0, H^{0}(C^{•}) ∈ T }
Y(F) := {C^{•} ∈ D^{b}(C) | H^{n}(C^{•}) = 0, ∀n < 0, H^{0}(C^{•}) ∈ F}

Then we have the following relationship.

Theorem 3.1. [41] The following conditions are equivalent.

(i) (T , F) is a torsion theory in C.

(ii) (X (T ), Y(F)) is a torsion theory in D^{b}(C).

Proof. For a proof that (i) ⇒ (ii) we refer to [41]. Assume that (ii) holds. We view C as the full subcategory of D^{b}(C) consisting of complexes concentrated in degree zero. If T ∈ T and F ∈ F, then T ∈ X (T ) and F ∈ Y(F). Hence C(T, F) ⟶^{≅} D^{b}(C)(T, F) = 0. Now let C be in C, and let X_{C} ⟶ C ⟶ Y^{C} ⟶ Σ(X_{C}) be the standard triangle associated with the torsion theory (X (T ), Y(F)) in D^{b}(C). Applying the usual homological functor H^{0} : D^{b}(C) ⟶ C to this triangle, we infer an exact sequence 0 ⟶ H^{0}(X_{C}) ⟶ C ⟶ H^{0}(Y^{C}) ⟶ 0 in C with H^{0}(X_{C}) ∈ T and H^{0}(Y^{C}) ∈ F. Hence (T , F) is a torsion theory in C.

If (T , F) is a torsion theory in C and if H is the heart of the torsion theory (X (T ), Y(F)) in D^{b}(C), then (Σ(F), T ) is a torsion theory in H [41]. Moreover if (T , F) is a tilting torsion theory then another application of the construction to D^{b}(H) recovers the original torsion theory. In addition if C has enough projectives or H has enough injectives, then there exists a derived equivalence D^{b}(H) ⟶^{≈} D^{b}(C), which extends the inclusion H ↪ C. The case of a classical tilting module is a special case of this construction. The map (T , F) ↦(X (T ), Y(F)), which establishes an injective function between the poset of torsion theories in C and the poset of torsion theories in D^{b}(C), seems to be a very useful tool for interchanging information between module categories and derived categories. For instance this generalized tilting process was used in an essential way in the classification of hereditary Noetherian categories with Serre duality by Reiten-Van den Bergh in [67] and the classification of hereditary categories with tilting object by Happel in [40].

It would be interesting to know under what conditions a torsion theory in D^{b}(C) can be recovered in some way from a torsion theory in C.

Example. Let (T , F) be the torsion theory generated by the injective envelope E(Λ) of Λ, where Λ is any ring, see [77]. Then T is a tilting torsion class, since it contains all the injectives. It follows that there is a triangle equivalence D^{b}(H) ⟶^{≈} D^{b}(Mod(Λ)), where H is the heart of the induced torsion theory in D^{b}(Mod(Λ)).

Using the above construction we can obtain, in some cases, a torsion theory in the stable module category of a self-injective algebra starting from a torsion theory in the module category.

Example. Let (T , F) be a torsion theory in mod(Λ), where Λ is a self-injective Artin algebra. If the induced torsion theory (X (T ), Y(F)) in D^{b}(mod(Λ)) satisfies the condition of Proposition 2.15 with respect to the thick subcategory H^{b}(P_{Λ}), then by the previous examples we get two torsion theories: one in H^{b}(P_{Λ}) and one in D^{b}(mod(Λ))/H^{b}(P_{Λ}) = \underline{mod}(Λ).

As noted above an important source of examples illustrating the above considerations emerges from tilting modules.

Let Λ be an Artin algebra and T be a tilting module with pd_{Λ}T = 1. Then (T , F) is a torsion theory in the category mod(Λ) of finitely generated Λ-modules, where:

T = {C ∈ mod(Λ) | Ext_{Λ}^{1}(T, C) = 0} and F = {C ∈ mod(Λ) | Hom_{Λ}(T, C) = 0}

The heart of the corresponding torsion theory (X (T ), Y(F)) in D^{b}(mod(Λ)) is equivalent to mod(End_{Λ}(T)) and Λ and End_{Λ}(T) are derived equivalent. See [41] for a generalization of this situation, which leads to the concept of a quasi-tilted algebra.

In general a tilting module T with pd_{Λ}T > 1, does not define a reasonable torsion theory in mod(Λ). However T induces important torsion theories in the derived category and in an appropriate stable category, see Chapters III, V.

CHAPTER II Torsion Theories in Pretriangulated Categories

Motivated by the results of the first chapter, we introduce and investigate in this chapter the concept of a torsion theory in the fairly general setting of a pretriangulated category. Pretriangulated categories provide a common generalization of abelian, triangulated and stable categories, so the choice to develop our theory in this more general setting is reasonable.

For a torsion theory (X , Y) in an abelian or triangulated category C we have seen that the pair (X , Y) is complete with respect to the vanishing of Hom and that the inclusion X ↪ C, resp. Y ↪ C, admits a right, resp. left, adjoint, and hence X is contravariantly finite, resp. Y is covariantly finite. We investigate more closely these conditions in a pretriangulated category, we provide localization sequences connecting the torsion and the torsion free class of a torsion theory, and finally we lift torsion theories from pretriangulated categories to triangulated categories.

1. Pretriangulated Categories

In this section we discuss briefly the concept of a pretriangulated category which is the working setting for the rest of the paper. We also give many examples, illustrating the wide variety of contexts that are covered by the notion of a pretriangulated category.

Left and Right Triangulated Categories. First we need to recall basic definitions and results about left, right and pretriangulated categories. For details and more information on one-sided triangulated categories we refer to [16], [19], [57]. Let C be an additive category equipped with an additive endofunctor Ω : C ⟶ C. Consider the category LT (C, Ω) whose objects are diagrams of the form Ω(C) ⟶^{h} A ⟶^{g} B ⟶^{f} C and where the morphisms are indicated by the following diagram:

Ω(C) ⟶^{h} A ⟶^{g} B ⟶^{f} C
Ω(γ)↓  ^{α}↓  ^{β}↓  ^{γ}↓
Ω(C′) ⟶^{h′} A′ ⟶^{g′} B′ ⟶^{f′} C′

A left triangulation of the pair (C, Ω) is a full subcategory Δ of LT (C, Ω) which satisfies all the axioms of a triangulated category, except that Ω is not necessarily an equivalence. Then the triple (C, Ω, Δ) is called a left triangulated category, the functor Ω is the loop functor and the diagrams in Δ are the (left) triangles.

Dually if C is an additive category equipped with an additive endofunctor Σ : C ⟶ C, consider in C the category RT (C, Σ) with objects diagrams of the form A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) and morphisms indicated by the following diagram:

A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A)
α↓  ^{β}↓  ^{γ}↓  ^{Σ(α)}↓
A′ ⟶^{f′} B′ ⟶^{g′} C′ ⟶^{h′} Σ(C′)

A right triangulation of the pair (C, Σ) is a full subcategory ∇ of RT (C, Σ) which satisfies all the axioms of a triangulated category, except that Σ is not necessarily an equivalence. Then the triple (C, Σ, ∇) is called a right triangulated category, Σ is the suspension functor and the diagrams in ∇ are the (right) triangles.

Exact and Homological Functors. The most important functors defined on a left or right triangulated category are the homological functors and the (left or right exact) functors. We recall here the basic definitions.

If (C, Ω, Δ) is a left triangulated category, then an additive functor F : C ⟶ Ab is called homological, if for any triangle Ω(C) ⟶ A ⟶ B ⟶ C in Δ, the sequence ⋯ ⟶ F Ω(C) ⟶ F (A) ⟶ F (B) ⟶ F (C) is exact. It is easy to see that ∀E ∈ C, the functor C(E, −) : C ⟶ Ab is homological. We recall that a morphism g : A ⟶ B is a weak kernel of f : B ⟶ C, if the sequence of functors

C(−, A) ⟶^{(−,g)} C(−, B) ⟶^{(−,f)} C(−, C) is exact. It follows that any morphism in a left triangle is a weak kernel of the next one. Similarly an additive functor F : C^{op} ⟶ Ab is called cohomological, if for any triangle Ω(C) ⟶ A ⟶ B ⟶ C in Δ, the sequence F (C) ⟶ F (B) ⟶ F (A) ⟶ F Ω(C) ⟶ ⋯ is exact.

Dually if (C, Σ, ∇) is a right triangulated category, then an additive functor F : C^{op} ⟶ Ab is called cohomological, if for any triangle A ⟶ B ⟶ C ⟶ Σ(A) in ∇, the sequence ⋯ ⟶ F Σ(A) ⟶ F (C) ⟶ F (B) ⟶ F (A) is exact. It is easy to see that ∀E ∈ C, the functor C(−, E) : C^{op} ⟶ Ab is cohomological. We recall that a morphism g : B ⟶ C is a weak cokernel of f : A ⟶ B if the sequence of functors

C(C, −) ⟶^{(g,−)} C(B, −) ⟶^{(f,−)} C(A, −) is exact. It follows that any morphism in a right triangle is a weak cokernel of the previous one. Similarly an additive functor F : C ⟶ Ab is called homological, if for any triangle A ⟶ B ⟶ C ⟶ Σ(A) in ∇, the sequence F (A) ⟶ F (B) ⟶ F (C) ⟶ F Σ(C) ⟶ ⋯ is exact.

If (C_{1}, Σ_{1}, ∇_{1}), (C_{2}, Σ_{2}, ∇_{2}) are right triangulated categories, then a functor F : C_{1} ⟶ C_{2} is called right exact if there exists a natural isomorphism ξ : F Σ_{1} ⟶^{≅} Σ_{2}F

such that for any triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ_{1}(A) in C_{1}, the diagram F (A) ⟶^{F (f)}

F (B) ⟶^{F (g)} F (C) ⟶^{F (h)∘ξ_{C}} Σ_{2}(F (A)) is a triangle in C_{2}. Similarly one defines left exact functors between left triangulated categories.

Pretriangulated Categories. If an additive category is left and right triangulated, then usually the left and right structures are compatible in a nice way. Following [19] we formalize this situation in the following definition.

Definition 1.1. Let C be an additive category. A pre-triangulation of C consists of the following data:

(i) An adjoint pair (Σ, Ω) of additive endofunctors Σ, Ω : C ⟶ C. Let ε : ΣΩ ⟶ Id_{C} be the counit and let δ : Id_{C} ⟶ ΩΣ be the unit of the adjoint pair.

(ii) A collection of diagrams Δ in C of the form Ω(C) ⟶ A ⟶ B ⟶ C, such that the triple (C, Ω, Δ) is a left triangulated category.

(iii) A collection of diagrams ∇ in C of the form A ⟶ B ⟶ C ⟶ Σ(A), such

that the triple (C, Σ, ∇) is a right triangulated category. (iv) For any diagram in C with commutative left square:

A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A)
α↓  ^{β}↓  ^{∃γ}↓ ^{Σ(α)∘ε_{C′}} ↓
Ω(C′) ⟶^{f′} A′ ⟶^{g′} B′ ⟶^{h′} C′

where the upper row is in ∇ and the lower row is in Δ, there exists a morphism γ : C ⟶ B′ making the diagram commutative.

(v) For any diagram in C with commutative right square:

A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A)
δ_{A}∘Ω(α)↓  ^{∃γ}↓  ^{β}↓  ^{α}↓
Ω(C′) ⟶^{f′} A′ ⟶^{g′} B′ ⟶^{h′} C′

where the upper row is in ∇ and the lower row is in Δ, there exists a morphism γ : B ⟶ A′ making the diagram commutative.

A pretriangulated category is an additive category together with a pre-triangu-lation, and is denoted by (C, Σ, Ω, ∇, Δ, ϵ, δ).

If C is a pretriangulated category and if F : C ⟶ D is a functor to a right, resp. left, triangulated category D, then F is called right, resp. left, exact, if F is a right, resp. left, exact functor of right, resp. left, triangulated categories. Let H : C_{1} ⟶ C_{2} be a functor between pretriangulated categories. Then H is called exact if H is left and right exact. More generally we define a morphism of pretriangulated categories to be an adjoint pair (F, G) of additive functors F : C_{1} ⟶ C_{2}, G : C_{2} ⟶ C_{1}, such that F is right exact and G is left exact.

Example. (1) Triangulated categories are pretriangulated. Here ∇ = Δ and Ω = Σ^{−1} and the notions of right or left exactness coincide (with the usual notions).

(2) Any additive category with kernels and cokernels (in particular any abelian category) is pretriangulated, with Ω = Σ = 0 and Δ the class of left exact sequences and ∇ the class of right exact sequences. The notions of right or left exactness have their usual meaning. However there are abelian pretriangulated categories for which Ω and Σ are not zero. An example is the category of ℤ−graded vector spaces.

(3) The homotopy category of an additive closed model category in the sense of Quillen [66] is pretriangulated.

Homologically Finite Subcategories and Triangulations. One important source of examples of pretriangulated categories arises from stable categories, as we now explain.

Let C be an additive category with split idempotents and let X be a full additive subcategory of C closed under direct summands and isomorphisms.

The induced stable category is denoted by C/X. We recall that the objects of C/X are the objects of C. If A, B ∈ C, then C/X (A, B) is defined to be the quotient C(A, B)/I_{X} (A, B), where I_{X} (A, B) is the subgroup of C(A, B) consisting of all morphisms factorizing through an object of X. We denote by π : C ⟶ C/X the natural projection functor and we set π(A) = \underline{A} and π(f) = \underline{f}.

We recall from [10] that a morphism f_{A} : X_{A} ⟶ A in C is called a right X - approximation of A if X_{A} is in X and any morphism X ⟶ A with X ∈ X factors through f_{A}. The subcategory X is called contravariantly finite [10] if any object of C admits a right X -approximation. The dual notions are left X -approximation and covariantly finite. The subcategory X is called functorially finite if it is both contravariantly finite and covariantly finite.

Assume now that C is abelian and X is contravariantly finite in C. By [16] the subcategory X induces on the stable category C/X a left triangulated structure (Ω_{X} , Δ_{X}), where Ω_{X} : C/X ⟶ C/X is the loop functor and Δ_{X} is the triangulation. The loop functor Ω_{X} : C/X ⟶ C/X is defined as follows. Let A ∈ C and let 0 ⟶ K_{A} ⟶^{κ_{A}} X_{A} ⟶^{χ_{A}} A be a sequence in C, where χ_{A} is a right X -approximation of A and κ_{A} is the kernel of χ_{A}. Then in C/X : Ω_{X} (\underline{A}) = \underline{K}_{A}. If f : A ⟶ B is a morphism in C, then we have the following commutative diagram:

K_{A} ⟶^{κ_{A}} X_{A} ⟶^{χ_{A}} A
κ_{f} ↓  ^{χ_{f}} ↓  ^{f}↓
K_{B} ⟶^{κ_{B}} X_{B} ⟶^{χ_{B}} B

It is easy to see that setting Ω(\underline{f}) = \underline{κ}_{f}, we obtain an additive functor Ω_{X}. The

triangulation Δ_{X} is defined as follows. Let 0 ⟶ C ⟶^{g} B ⟶^{f} A be a sequence in C, where f has the property that C(X , f) : C(X , A) ⟶ C(X , B) ⟶ 0 is surjective and g = ker(f). Then we have a commutative diagram:

K_{A} ⟶^{κ_{A}} X_{A} ⟶^{A} A
κ_{f} ↓  ^{χ_{f}} ↓  ^{||}↓
C ⟶^{g} B ⟶^{f} A

Hence in C/X we have a diagram Ω_{X} (\underline{A}) ⟶^{\underline{h}} \underline{C} ⟶^{\underline{g}} \underline{B} ⟶^{\underline{f}} \underline{A}. The triangulation Δ_{X} consists of all diagrams Ω_{X} (\underline{H}) ⟶ \underline{F} ⟶ \underline{G} ⟶ \underline{H} which are isomorphic in C/X to diagrams of the above form.

Remark 1.2. A morphism f : A ⟶ B is said to be an X -epic if the morphism C(X , f) : C(X , A) ⟶ C(X , B) is surjective. Then the above construction works in the more general situation in which any X -epic in C has a kernel [16]. For instance let A be a full subcategory of an abelian category C with enough projectives, and assume that X contains the projectives and is closed under extensions and kernels of epimorphisms. If P is the full subcategory of projectives of C, then the stable category X /P is a left triangulated category, in fact a left triangulated subcategory of C/P.

Dually a morphism f : A ⟶ B is said to be an X -monic if the morphism C(f, X) : C(B, X) ⟶ C(A, X) is surjective. If Y is covariantly finite in C and C is abelian, or more generally if any Y-monic has a cokernel in C, then the stable category C/Y admits a right triangulated structure (C/Y, Σ_{Y}, ∇_{Y}), where Σ_{Y} is the suspension functor and ∇_{Y} is the right triangulation. The construction is completely dual using left Y-approximations. For instance if C has enough injectives and Y contains the injectives and is closed under extensions and cokernels of monomorphisms, then the stable category Y/I is a right triangulated category, in fact a right triangulated subcategory of C/I, where I is the full subcategory of injective objects of C.

Our basic source of examples of pretriangulated categories is the following.

Example. Let X be a functorially finite subcategory of an additive category C and assume that C is abelian, or more generally that any X -epic has a kernel and any X -monic has a cokernel in C. Then C/X admits a right triangulated structure (Σ_{X} , ∇_{X}) and a left triangulated structure (Ω_{X} , Δ_{X}). Moreover by [17] we have an adjoint pair (Σ_{X} , Ω_{X}) and the compatibility conditions (iv), (v) in the above definition hold. Hence the stable category C/X is a pretriangulated category. Note that by [19], C/X is the homotopy category of a naturally defined closed model structure in C.

Actually the above example also covers abelian categories (just take X = 0) and unbounded derived categories of module categories (take C to be the category of homotopically projective complexes [52] and X to be the full subcategory of contractible complexes).

An Example from Artin Algebras. There is a class of examples of pretriangulated stable categories for which the loop and the suspension functor can be computed rather explicitly.

Let Λ be an Artin algebra and let mod(Λ) be the category of finitely generated right Λ-modules. Let P_{Λ} be the full subcategory of finitely generated projective right Λ-modules and I_{Λ} the full subcategory of finitely generated injective right Λmodules. Since P_{Λ} is functorially finite, the stable category mod(Λ)/P_{Λ} := \underline{mod}(Λ) is pretriangulated with loop functor the usual first syzygy functor Ω and suspension functor Σ_{P} = TrΩTr, where Tr : \underline{mod}(Λ) ⟶ \underline{mod}(Λ^{op}) is the transpose duality functor [4]. The left P_{Λ}-approximation p^{A} : A ⟶ P^{A} of A is computed as follows. Let d = Hom_{Λ}(−, Λ) denote both Λ-dual functors. If Q ⟶ d(A) is a projective cover, then P^{A} = d(Q) and p^{A} is composition A ⟶ d^{2}(A) ⟶ d(Q) where the first morphism is the natural one, and there is an exact sequence

0 ⟶ Ext_{Λ}^{1}(Tr(A), Λ) ⟶ A ⟶^{p^{A}} P^{A} ⟶ Σ_{P}(A) ⟶ 0

Similarly, since I_{Λ} is functorially finite, the stable category mod(Λ)/I_{Λ} = \overline{mod}(Λ) is pretriangulated with suspension functor the usual first cosyzygy functor Σ, and loop functor Ω_{I} = DTrΩTrD, where D is the usual duality for Artin algebras. The right I_{Λ}-approximation i_{A} of A is included in an exact sequence

0 ⟶ Ω_{I}(A) ⟶ N^{+}(P) ⟶^{i_{A}} A ⟶ DExt_{Λ}^{1}(TrD(A), Λ) ⟶ 0

where N^{+} = Dd is the Nakayama functor, N^{−} = dD its right adjoint, P ⟶ N^{−}(A) is a projective cover and i_{A} is the composition N^{−}(P) ⟶ N^{+}N^{−}(A) ⟶ A where the last morphism is the counit of the adjoint pair (N^{+}, N^{−}).

It is easy to see that the adjoint pair (d, d) of Λ-dual functors induces a morphism of pretriangulated categories d : \underline{mod}(Λ) ⇆ \underline{mod}(Λ^{op}) : d. It follows that the adjoint pair (N^{+}, N^{−}) of Nakayama functors induces a morphism of pretriangulated categories N^{+} : \underline{mod}(Λ) ⇆ \overline{mod}(Λ) : N^{−}.

More generally let T ∈ mod(Λ) and let ω = add(T) be the full subcategory consisting of all direct summands of finite direct sums of copies of T. It is wellknown that ω is functorially finite, so the stable category \underline{mod}_{T} (Λ) := mod(Λ)/ω is pretriangulated.

Other Examples from Ring Theory. The previous example can be generalized to cover the stable module category (of not necessarily finitely generated modules) modulo projectives or injectives, for certain classes of rings that we discuss below.

Let Λ be a ring and let Mod(Λ) be the category of right Λ-modules. Let P_{Λ} be the category of projective right Λ-modules and I_{Λ} the category of injective right Λ-modules. By [19] we have the following.

• If Λ is left coherent and right perfect, then P_{Λ} is functorially finite, hence the stable category \underline{Mod}(Λ) modulo projectives is pretriangulated.

• If Λ is right Noetherian, then I_{Λ} is functorially finite, hence the stable cat-

egory \overline{Mod}(Λ) modulo injectives is pretriangulated. More generally let T ∈ Mod(Λ) and let ω = Add(T) be the full subcategory consisting of all direct summands of arbitrary direct sums of copies of T. We denote the stable category Mod(Λ)/ω by \underline{Mod}_{T} (Λ). In a given category with coproducts, we denote by C^{(I)} the coproduct of copies of the object C indexed by the set I. Assume that T has the property that the canonical map (T, T)^{(I)} ⟶ (T, T^{(I)}) is invertible for any index set I, and that the endomorphism ring End(T) is left coherent and right perfect. It is not difficult to see that ω is functorially finite, so the stable category \underline{Mod}_{T} (Λ) is pretriangulated. Also the category \underline{Mod}_{T} (Λ) is pretriangulated if the module T is endofinite in the sense that T has finite length as a module over its endomorphism ring. More generally \underline{Mod}_{T} (Λ) is pretriangulated if T is product-complete in the sense that any product of copies of T is a direct summand of a coproduct of copies of T, see [50] for more information on endofinite or product-complete modules.

For more examples we refer to Chapter V.

2. Adjoints and Orthogonal Subcategories

In this section we examine more closely the connections between orthogonal, with respect to the vanishing of Hom, subcategories of a pretriangulated category, and the existence of left or right adjoints for the corresponding inclusion functors. This connection will be useful when discussing torsion theories.

Throughout this section we fix a pretriangulated category C = (C, Σ, Ω, Δ, ∇). Let X , Y be full additive subcategories of C. We assume tacitly that X , Y are

closed under direct summands and isomorphisms. We consider the left orthogonal subcategory ^{⊥}Y = {A ∈ C | C(A, Y) = 0} of Y and the right orthogonal subcategory X^{⊥} = {B ∈ C | C(X , B) = 0} of X.

We begin with some elementary necessary conditions for a subcategory to be a left or right orthogonal subcategory.

Lemma 2.1. (1) Assume that ^{⊥}Y = X. If Y is closed under the loop functor Ω (for instance if Y is a left triangulated subcategory of C), then X is a right triangulated subcategory of C, closed under extensions of right triangles.

(2) Assume that X^{⊥} = Y. If X is closed under the suspension functor Σ (for instance if X is a right triangulated subcategory of C), then Y is a left triangulated subcategory of C, closed under extensions of left triangles.

Proof. (1) If Ω(Y) ⊆ Y, then ∀X ∈ C, C(Σ(X), Y) ≅ C(X, Ω(Y)) = 0. Hence Σ(X) ∈ X and X is closed under suspension. Let X_{1} ⟶ C ⟶ X_{2} ⟶ Σ(X_{1}) be a right triangle with X_{1}, X_{2} ∈ X. Since the functor C(−, Y) is cohomological with respect to right triangles, the sequence ⋯ ⟶ C(Σ(X_{1}), Y) ⟶ C(X_{2}, Y) ⟶ C(C, Y) ⟶ C(X_{1}, Y) is exact. It follows that C ∈ ^{⊥}Y = X, so X is closed under extensions of right triangles. This implies that X is a right triangulated subcategory of C. The proof of (2) is dual.

Corollary 2.2. Assume that X^{⊥} = Y and ^{⊥}Y = X. Then X is a right triangulated subcategory of C iff Y is a left triangulated subcategory of C.

It is trivial to see that in an arbitrary category, any coreflective, resp. reflective, subcategory is contravariantly, resp. covariantly, finite. In the setting of pretriangulated categories we have the following connection between coreflective/reflective, contravariantly/covariantly finite and right/left orthogonal subcategories. This connection can be regarded as a stable analogue of Wakamatsu’s Lemma [7], which we state below for later use. First we recall that a morphism α : X ⟶ C is called right minimal, if any endomorphism ρ of X such that ρ ∘ α = α, is invertible. A minimal right X -approximation is a right minimal, right X -approximation. Left minimal morphisms and minimal left approximations are defined dually. Examples of minimal right approximations are the coreflections and examples of minimal left approximations are the reflections.

Wakamatsu’s Lemma. Let X be a full extension closed subcategory of an abelian category C.

(i) If X_{C} ⟶^{f_{C}} C is a minimal right X -approximation, then

Ext^{1}(X , Ker(f_{C}))= 0.

(ii) If C ⟶^{g^{C}} X^{C} is a minimal left X -approximation, then

Ext^{1}(Coker(g^{C}), X )= 0.

We have the following result.

Lemma 2.3. (1) If Σ(X) ⊆ X, then the following are equivalent.

(i) The inclusion i : X ↪ C admits a right adjoint R.

(ii) X is contravariantly finite and ∀C ∈ C, there exists a left triangle Ω(C) ⟶

Y_{C} ⟶ X_{C} ⟶^{f_{C}} C in Δ, where f_{C} is a right X -approximation and Y_{C} ∈ X^{⊥}.

(2) If Ω(Y) ⊆ Y, then the following are equivalent.

(i) The inclusion j : Y ↪ C admits a left adjoint L.

(ii) Y is covariantly finite and ∀C ∈ C, there exists a right triangle C ⟶^{g^{C}} Y^{C} ⟶ X^{C} ⟶ Σ(C) in ∇, where g^{C} is a left Y-approximation and X^{C} ∈ ^{⊥}Y.

Proof. We prove only (1), since the proof of (2) is dual.

(i) ⇒ (ii) Let C ∈ C and let f_{C} be the coreflection of C. Then trivially f_{C} is a right X -approximation, hence X is contravariantly finite. Consider a left triangle Ω(C) ⟶ Y_{C} ⟶ R(C) ⟶^{f_{C}} C in C. Then applying the homological functor C(X , −), we have the long exact sequence

⋯ ⟶ C(X , ΩR(C)) ⟶ C(X , Ω(C)) ⟶ C(X , Y_{C}) ⟶ C(X , R(C)) ⟶ C(X , C).

Since the morphism C(X , f_{C}) : C(X , R(C)) ⟶ C(X , C) is invertible, the sequence C(X , ΩR(C)) ⟶ C(X , Ω(C)) ⟶ C(X , Y_{C}) ⟶ 0 is exact. Using the adjoint pair (Σ, Ω) it follows that the morphism C(X , Ω(f_{C})) : C(X , ΩR(C)) ⟶ C(X , Ω(C)) is isomorphic to C(Σ(X), f_{C}) : C(Σ(X), R(C)) ⟶ C(Σ(X), C). Since Σ(X) ⊆ X, the above isomorphism shows that the morphism C(X , Ω(f_{C})) is also invertible. From the long exact sequence above we infer that C(X , Y_{C}) = 0. Hence Y_{C} ∈ X^{⊥}.

(ii) ⇒ (i) It suffices to show that f_{C} is the coreflection of C in X. If α : X ⟶ C is a morphism with X ∈ X which admits two liftings β, γ : X ⟶ X_{C} through f_{C}, then β − γ factors through the weak kernel Y_{C} ⟶ X_{C} of f_{C}. Since Y_{C} ∈ X^{⊥}, it follows that β = γ and this implies that X_{C} ⟶ C is the coreflection of C in X.

The following result shows that in some cases contravariant/covariant finiteness implies coreflectivity/reflectivity. First we recall that an additive category is called a Krull-Schmidt category if any of its objects is a finite coproduct of objects with local endomorphism ring. If X is a skeletally small additive category, then Mod(X) denotes the category of additive functors X^{op} ⟶ Ab.

Proposition 2.4. Assume that X is a skeletally small Krull-Schmidt right, resp. left, triangulated subcategory of C. Then the following are equivalent.

(i) The inclusion functor X ↪ C has a right, resp. left, adjoint.

(ii) X is contravariantly, resp. covariantly, finite in C.

Proof. We prove only that (ii) ⇒ (i) for the contravariantly finite case. Consider the functor S : C ⟶ Mod(X) defined by S(C) = C(−, C)|_{X}. Then S is a homological functor with respect to the left triangulation Δ of C. Let f_{C} : X_{C} ⟶ C be a right X -approximation of C ∈ C and let Ω(C) ⟶ Y_{C} ⟶ X_{C} ⟶^{f_{C}} C be a left triangle in C. Then applying S to this triangle and using that X is closed under Σ, we have a short exact sequence 0 ⟶ S(Y_{C}) ⟶ S(X_{C}) ⟶ S(C) ⟶ 0 in Mod(X), which shows that S(C) is finitely generated functor, since S(X_{C}) = X (−, X_{C}) is a finitely generated projective functor. Since X is a Krull-Schmidt category, it is well-known that any finitely generated functor over X has a projective cover. Let r_{C} : X (−, R(C)) ⟶ S(C) be the projective cover. Now we claim that S(C) is a flat functor over X. Indeed since the functor Tor_{∗}^{X} (S(C), −) commutes with filtered colimits, it suffices to show that Tor_{1}^{X} (S(C), F) = 0, for any finitely presented functor F : X ⟶ Ab. Let X (X_{1}, −) ⟶ X (X_{0}, −) ⟶ F ⟶ 0 be a finite presentation of F and let (T): X_{0} ⟶ X_{1} ⟶ X_{2} ⟶ Σ(X_{0}) be a right triangle in X. Since X is a right triangulated subcategory of C, (T) is also a right triangle in C. Then the sequence ⋯ ⟶ X (Σ(X_{0}), −) ⟶ X (X_{2}, −) ⟶ X (X_{1}, −) ⟶ X (X_{0}, −) ⟶ F ⟶ 0 is a projective resolution of F, hence Tor_{1}^{X} (S(C), F) is the homology of the complex S(C) ⊗_{X} X (X_{2}, −) ⟶ S(C) ⊗_{X} X (X_{1}, −) ⟶ S(C) ⊗_{X} X (X_{0}, −) which is isomorphic to C(X_{2}, C) ⟶ C(X_{1}, C) ⟶ C(X_{0}, C). However the last complex is exact since it is the result of the application of the cohomological functor C(−, C) to the right triangle (T) ∈ ∇. We infer that S(C) is flat. By a well-known result of Bass, the only flat functors admitting a projective cover are the projective ones. Hence S(C) is projective, i.e. r_{C} : X (−, R(C)) ⟶ S(C) = C(−, C)|_{X} is invertible. We infer that the morphism r_{C}(1_{R(C)}) : R(C) ⟶ C is the coreflection of C in X.

Example. Let D^{b}(mod(Λ)) be the bounded derived category of finitely generated modules over an Artin algebra Λ. Then gl.dimΛ < ∞ iff H^{b}(P_{Λ}) is contravariantly finite in D^{b}(mod(Λ)) iff H^{b}(I_{Λ}) is covariantly finite in D^{b}(mod(Λ)). We prove only the assertion for H^{b}(I_{Λ}). Indeed if gl.dimΛ < ∞, then H^{b}(I_{Λ}) = D^{b}(mod(Λ), hence the assertion is trivial. If H^{b}(I_{Λ}) is covariantly finite, then by Proposition 2.4, H^{b}(I_{Λ}) is reflective, so by Proposition I.2.3 it is a torsion-free class in D^{b}(mod(Λ)). Since trivially ^{⊥}H^{b}(I_{Λ}) = 0, we infer that H^{b}(I_{Λ}) = D^{b}(mod(Λ)), and this obviously implies that gl.dimΛ < ∞.

Viewing an abelian category as a pretriangulated category (with Ω = 0 = Σ), we have the following well-known observation.

Example. Let C be abelian and X a skeletally small full additive Krull-Schmidt subcategory of C, closed under quotient objects, resp. subobjects. Then X is a right, resp. left, triangulated subcategory of C. It follows that X is coreflective, resp. reflective, iff X is contravariantly, resp. covariantly, finite.

Now we search for conditions such that a full coreflective subcategory X of C is pretriangulated. We also study when the right adjoint of the inclusion is left exact.

Lemma 2.5. (1) Let X be a right triangulated subcategory of C, closed under the loop functor Ω and assume that the inclusion X ↪ C admits a right adjoint R. Then X is a pretriangulated subcategory of C.

(2) Let Y be a left triangulated subcategory of C, closed under the suspension functor Σ and assume that the inclusion Y ↪ C admits a left adjoint L. Then Y is a pretriangulated subcategory of C.

Proof. We prove only the assertion for X. First by definition X is closed under Ω. Let X_{2} ⟶ X_{1} be a morphism in X and let Ω(X_{1}) ⟶ A ⟶ X_{2} ⟶ X_{1} be a left triangle in C. It is easy to see that if (F, G) is an adjoint pair of functors between pretriangulated categories, then F preserves weak cokernels and G preserves weak kernels. It follows that R preserves weak kernels, hence we have a complex ⋯ ⟶ RΩ(X_{2}) ⟶ RΩ(X_{1}) ⟶ R(A) ⟶ R(X_{2}) ⟶ R(X_{1}) in C in which any morphism is a weak kernel of the next one, and a commutative diagram

RΩ(X_{2}) ⟶ RΩ(X_{1}) ⟶ R(A) ⟶ R(X_{2}) ⟶ R(X_{1})
f_{Ω(X_{2})}↓  ^{f_{Ω(X_{1})}}↓  ^{f_{A}}↓  ^{f_{X_{2}}} ↓  ^{f_{X_{1}}} ↓
Ω(X_{2}) ⟶ Ω(X_{1}) ⟶ A ⟶ X_{2} ⟶ X_{1}

in which the coreflection morphisms f_{Ω(X_{2})}, f_{Ω(X_{1})}, f_{X_{2}}, f_{X_{1}} are invertible. Let mod(C) be the category of finitely presented contravariant additive functors over C and let 𝕐 : C ↪ mod(C) be the Yoneda embedding. Since C has weak kernels, mod(C) is abelian and the image under 𝕐 of the above diagram in mod(C) is exact. By the 5−Lemma, it follows that the map 𝕐(f_{A}) : 𝕐(R(A)) ⟶ 𝕐(A) is invertible. Since 𝕐 is fully faithful, f_{A} is invertible, so A ∈ X. It follows that X is a left triangulated subcategory of C. It is easy to see that the compatibility conditions of definition 1.1 hold for X, hence X is a pretriangulated subcategory of C.

Note that if R exists in the above Lemma, then the suspension functor Σ : X ⟶ X admits the functor RΩ : X ⟶ X as a right adjoint, even if X is not closed under Ω. This follows from the natural isomorphisms X (Σ(X), X′) ≅ C(X, Ω(X′)) ≅ X (X, RΩ(X′)) for all X, X′ ∈ X. This suggests that X is closed under Ω iff the functors R, Ω commute. The following result which will be useful later in connection with hereditary torsion theories, shows that this is the case.

Proposition 2.6. Assume that the inclusion functor i : X ↪ C has a right adjoint R and X is closed under the suspension functor Σ. Then there exists a natural morphism α : RΩ ⟶ ΩR and the following are equivalent:

(i) α : RΩ ⟶ ΩR is an isomorphism.

(ii) Ω(X) ⊆ X.

If (ii) holds and in addition X is closed under extensions of left triangles, then R : C ⟶ X and the idempotent functor iR : C ⟶ C are left exact.

Proof. Set X^{⊥} := Y and consider the counit f_{C} : R(C) ⟶ C of the adjoint pair (i, R). Let Ω(C) ⟶^{h_{C}} Y_{C} ⟶^{g_{C}} R(C) ⟶^{f_{C}} C be a triangle in Δ. By Lemma 2.3,

we have that Y_{C} ∈ Y. Consider the rotation ΩR(C) ⟶^{Ω(f_{C})} Ω(C) ⟶^{h_{C}} Y_{C} ⟶^{g_{C}} R(C) of the above triangle, and let f_{Ω(C)} : RΩ(C) ⟶ Ω(C) be the counit evaluated at Ω(C). Since Y_{C} ∈ Y, the composition f_{Ω(C)} ∘ h_{C} = 0. Hence there exists a morphism α_{C} : RΩ(C) ⟶ ΩR(C) such that α_{C} ∘ Ω(f_{C}) = f_{Ω(C)}. The morphism α_{C} is unique with this property, since if α′ : RΩ(C) ⟶ ΩR(C) is another morphism with α′ ∘ Ω(f_{C}) = f_{Ω(C)}, then the morphism α_{C} − α′ factors through Ω(Y_{C}). By Lemma 2.1, Ω(Y_{C}) ∈ Y and then obviously α_{C} = α′, since (X , Y) = 0. It is not difficult to see that the morphism α_{C} is the component of a natural morphism α : RΩ ⟶ ΩR. (i) ⇒ (ii) If the natural morphism α : RΩ ⟶ ΩR is invertible, then ∀C ∈ C we have that ΩR(C) ∈ X. This implies trivially that Ω(X) ⊆ X.

(ii) ⇒ (i) Assume that Ω(X) ⊆ X. Then ΩR(C) ∈ X. Hence there exists a unique morphism β : ΩR(C) ⟶ RΩ(C) such that β∘f_{Ω(C)} = Ω(f_{C}). Then we have: α_{C} ∘β ∘f_{Ω(C)} = f_{Ω(C)} and β ∘α_{C} ∘Ω(f_{C}) = Ω(f_{C}). By the uniqueness properties of the involved morphisms, we conclude that α_{C} ∘ β = 1_{RΩ(C)} and β ∘ α_{C} = 1_{ΩR(C)}. Hence the natural morphism α : RΩ ⟶ ΩR is invertible.

The easy proof of the last assertion is left to the reader.

3. Torsion Theories

In this section we introduce the fairly general concept of torsion theory in a pretriangulated category and we derive the main properties.

As in the previous section, throughout C denotes a pretriangulated category. Let X , Y be two full additive subcategories of C, closed under direct summands and isomorphisms. Inspired by the definition of a torsion theory in an abelian or triangulated category, we introduce the following concept of torsion theory in a pretriangulated category.

Definition 3.1. The pair (X , Y) is called a torsion theory in C, if:

(i) C(X , Y) = 0.

(ii) Σ(X) ⊆ X or Ω(Y) ⊆ Y.

(†) [The glueing condition]: ∀C ∈ C, there are triangles

\{ \begin{matrix}Δ(C) : Ω(Y^{C}) ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} & ∈ Δ \\ ∇(C) : X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{f^{C}} Σ(X_{C}) & ∈ ∇ \end{matrix}  (Glueing Triangles)

with X_{C} ∈ X , Y^{C} ∈ Y.

If (X , Y) is a torsion theory in C, then X, resp. Y, is called a torsion class, resp. torsion-free class. A TTF-theory is an ordered triple (X , Y, Z), where (X , Y) and (Y, Z) are torsion theories in C. In this case Y is called a TTF-class.

The following remark shows that condition (ii) in the definition is symmetric.

Remark 3.2. If the pair (X , Y) satisfies (i) and the glueing condition, then Σ(X) ⊆ X iff Ω(Y) ⊆ Y. Indeed if Σ(X) ⊆ X and C ∈ X^{⊥}, then f_{C} = 0. Then

from the right triangle X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{f^{C}} Σ(X_{C}), we have that g^{C} is split monic, so C ∈ Y. It follows that X^{⊥} = Y. Then by Lemma 2.1, we have that Ω(Y) ⊆ Y. Conversely Ω(Y) ⊆ Y implies that Σ(X) ⊆ X.

We record the following immediate consequence of the above definition.

Proposition 3.3. Assume that the pair (X , Y) is a torsion theory in C.

(1) X = ^{⊥}Y is a right triangulated subcategory of C closed under extensions of right triangles and the inclusion i : X ↪ C admits a right adjoint R : C ⟶ X.

(2) Y = X^{⊥} is a left triangulated subcategory of C closed under extensions of left triangles, and the inclusion j : Y ↪ C admits a left adjoint L : C ⟶ Y.

(3) C is generated as a left or right triangulated category by X , Y.

(4) We have a “direct sum decomposition” C = X ⊕ Y of C, in the sense that KerR := {C ∈ C | R(C) = 0} = Y and KerL := {C ∈ C | L(C) = 0} = X and there exist sequences

0 ⟶ Y ⟶^{j} C ⟶^{R} X ⟶ 0 and 0 ⟶ X ⟶^{i} C ⟶^{L} Y ⟶ 0

such that Rj = 0, Li = 0, Ri = Id_{X} , Lj = Id_{Y}.

(5) If (X , Y, Z) is a TTF-theory in C, then Y is a pretriangulated subcategory of C, the inclusion Y ↪ C admits a left and a right adjoint and finally the following relations are true: X = ^{⊥}Y = ^{⊥⊥}Z, Z = Y^{⊥} = X^{⊥⊥}, Y = X^{⊥} = ^{⊥}Z.

Proof. Obviously X ⊆ ^{⊥}Y. If C ∈ ^{⊥}Y, then from the triangle Δ(C) it follows that C ≅ X_{C}, so C ∈ X. Hence X = ^{⊥}Y. Now the remaining assertions of (1) follow from Lemmas 2.1, 2.3. The proof of (2) is dual and the proof of the remaining parts follow directly from (1), (2) and the definitions.

In analogy with the definitions of Chapter I, we define a torsion theory (X , Y) in C to be hereditary, resp. cohereditary, if the idempotent functor iR : C ⟶ C is left exact, resp. the idempotent functor jL : C ⟶ C is right exact. If (X , Y) is hereditary then by Lemma 2.5 and Proposition 2.6, X is a pretriangulated subcategory of C and dually if (X , Y) is cohereditary, then Y is a pretriangulated subcategory of C.

The class 𝕋(C) of (hereditary or cohereditary) torsion theories on C is partially ordered if we define: (X , Y) ≺ (X′, Y′) iff X ⊆ X′ (or equivalently Y′ ⊆ Y). The torsion theory (0, C) is the minimum element and the torsion theory (C, 0) is the maximum element of 𝕋(C).

The following examples show that a torsion theory in a pretriangulated category is a common generalization of a torsion theory in an abelian category and of a tstructure in a triangulated category.

Example. (1) Let C be abelian considered as a pretriangulated category. Then the concept of a torsion theory in C in the sense of the above definition, coincides with the concept of a torsion theory in the (usual) sense of Chapter I. Also the concept of a TTF-theory reduces to the well–known concept studied in ring theory.

(2) Let C be a triangulated category considered as a pretriangulated category. Then the concept of a torsion theory in C in the sense of the above definition, coincides with the concept of a torsion theory (= t-structure) in the sense of Chapter I and the TTF-theories reduce to recollement situations [15].

Motivated by the results of Chapter I we introduce the concept of the heart of a torsion theory in C. There are actually two, not necessarily equivalent, hearts, reflecting the fact that the suspension and the loop functor of C are not necessarily quasi-inverse equivalences.

Definition 3.4. Let (X , Y) be a torsion theory in a pretriangulated category C. Then its left heart is H_{l} := X ∩ ΣY and its right heart is H_{r} := ΩX ∩ Y.

Remark 3.5. (1) Let (X , Y) be a torsion theory in an abelian category C viewed as a pretriangulated category with Ω = 0 = Σ. Then the hearts of a torsion theory (X , Y) in C are trivial: H_{l} = 0 = H_{r}.

(2) If C is triangulated, then by [15] the hearts of a torsion theory (X , Y) (= t-structure), are abelian and H_{r} = Σ^{−1}H_{l}. Observe that if (X , Y) is hereditary, then: H_{r} = 0 = H_{l}.

We refer to Chapter III for a study of the heart in the triangulated case and to Chapter V for an example of a torsion theory in a pretriangulated category which is not abelian or triangulated such that the left heart is non-zero abelian and the right heart is zero.

We close this section by a characterization of torsion theories. First we need a simple lemma. We recall that in a pretriangulated category any morphism has a weak kernel and a weak cokernel.

Lemma 3.6. Let f be a morphism in C. If f is a weak kernel (weak cokernel), then f is a weak kernel (weak cokernel) of its weak cokernel (weak kernel).

Proof. Let A ⟶^{f} B ⟶^{g} C ⟶ Σ(A) be a triangle in C. Then g is a weak cokernel of f. Let α : B ⟶ D be a morphism such that f is a weak kernel of α. Then f ∘α = 0, hence there exists a morphism β : C ⟶ D such that g ∘β = α. Now let κ : M ⟶ B be a morphism such that κ ∘ g = 0. Then κ ∘ g ∘ β = κ ∘ β = 0. Since f is a weak kernel of α, there exists a morphism λ : M ⟶ A such that λ ∘ f = κ. Hence f is a weak kernel of g. The parenthetical case is dual.

The following characterizes when a pair of subcategories form a torsion theory.

Proposition 3.7. If C(X , Y) = 0, then the following are equivalent:

(i) (α) The inclusion i : X ↪ C has a right adjoint R and the counit f_{C} : R(C) ⟶ C is a weak kernel, ∀C ∈ C.

(β) If R(C) ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{h^{C}} ΣR(C) is a triangle in ∇, then Y^{C} ∈ Y.

(γ) Ω(Y) ⊆ Y and if Ω(Y^{C}) ⟶ A ⟶ C ⟶^{g^{C}} Y^{C} is a triangle in Δ, then

C(Σ(A), Y^{C}) = 0.

(ii) (α) The inclusion j : Y ↪ C has a left adjoint L and the unit g^{C} : C ⟶ L(C) is a weak cokernel, ∀C ∈ C.

(β) If ΩL(C) ⟶^{h_{C}} X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} L(C) is a triangle in Δ, then X_{C} ∈ X. (γ) Σ(X) ⊆ X and if X_{C} ⟶^{f_{C}} C ⟶ B ⟶ Σ(X_{C}) is a triangle in ∇, then

C(X_{C}, Ω(B)) = 0.

(iii) The pair (X , Y) is a torsion theory in C.

Proof. We prove that (i) ⇔ (iii). The proof of (ii) ⇔ (iii) is similar. Since (iii) ⇒ (i) follows directly from Proposition 3.3, it remains to show that (i) ⇒ (iii). For any C ∈ C, let f_{C} : R(C) ⟶ C be the counit of the adjoint pair (i, R), and let (∗): R(C) ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{h^{C}} ΣR(C) be a triangle in ∇. If Ω(Y^{C}) ⟶^{β} A ⟶^{α}

C ⟶^{g^{C}} Y^{C} is a triangle in Δ, then since f_{C} ∘ g^{C} = 0, there exists a morphism κ : R(C) ⟶ A such that: κ ∘ α = f_{C}. By Lemma 3.6, f_{C} is a weak kernel of g^{C}, hence since α∘g^{C} = 0, there exists a morphism λ : A ⟶ R(C) such that λ∘f_{C} = α.

Consider now the left triangle Ω(C) ⟶^{h_{C}} Y_{C} ⟶^{g_{C}} R(C) ⟶^{f_{C}} C in Δ. By the axioms of a left triangulated category, there are morphisms ϕ : Y_{C} ⟶ Ω(Y^{C}) and ψ : Ω(Y^{C}) ⟶ Y_{C} such that the following diagram commutes:

Ω(C) ⟶^{h_{C}} Y_{C}  ⟶^{g_{C}} R(C) ⟶^{f_{C}} C
||↓  ^{∃ ϕ}↓  ^{κ}↓  ^{||}↓
Ω(C) ⟶^{Ω(g^{C})} Ω(Y^{C}) ⟶^{β} A ⟶^{α} C
||↓  ^{∃ ψ}↓  ^{λ}↓  ^{||}↓
Ω(C) ⟶^{h_{C}} Y_{C}  ⟶^{g_{C}} R(C) ⟶^{f_{C}} C
||↓  ^{∃ ϕ}↓  ^{κ}↓  ^{||}↓
Ω(C) ⟶^{Ω(g^{C})} Ω(Y^{C}) ⟶^{β} A ⟶^{α} C

Since κ∘λ∘f_{C} = f_{C}, we have that κ∘λ = 1_{R(C)}. By the axioms of a left triangulated category, this implies that also ϕ ∘ ψ = 1_{Y_{C}}. Further we have that λ ∘ κ ∘ α = α, so (1_{A} − λ ∘ κ) ∘ α = 0. Hence there exists a morphism σ : A ⟶ Ω(Y^{C}) such that σ ∘ β = 1_{A} − λ ∘ κ. By condition (γ), C(Σ(A), Y^{C}) ≅ C(A, Ω(Y^{C})) = 0. Hence σ = 0 and then 1_{A} = λ ∘ κ. It follows that A ≅ R(C). But then also Y_{C} ≅ Ω(Y^{C}). Since by hypothesis Y^{C} ∈ Y and Ω(Y) ⊆ Y, we have that Ω(Y^{C}) ∈ Y. Now setting X_{C} := R(C) we have that in the following diagram

Ω(Y^{C}) ⟶^{h_{C}} X_{C} ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{h^{C}} Σ(X_{C})

the three morphisms on the right constitute a right triangle and the three morphisms on the left constitute a left triangle. Hence the pair (X , Y) is a torsion theory.

4. Torsion Theories and Localization Sequences

Associated to a (co)hereditary torsion theory in an abelian or triangulated category there are localization sequences connecting the ambient category with the torsion and the torsion-free subcategory. In this section we discuss an analogous situation in the pretriangulated case, which will be useful in the investigation of torsion theories in a stable category in Chapter VI.

Let (X , Y) be a torsion theory in the pretriangulated category C. Our aim here is to construct localization sequences relating the categories X , Y and C in analogy with the localization sequences in the triangulated case, cf. Proposition I.2.6.

Since X is a right triangulated subcategory of C, closed under direct summands and extensions of right triangles, we can construct the localization category C/X as in the triangulated case, see [35], [79] for details. Recall that the category C/X is defined as the category of fractions C[R_{X}^{−1}], where the class R_{X} consists of all

morphisms f : A ⟶ B such that in any right triangle A ⟶^{f} B ⟶ X ⟶ Σ(A), the object X lies in X. It is not difficult to see that the class of morphisms R_{X} admits a calculus of left fractions in the sense of [35]. Then the right triangulated structures of X and C induce a right triangulated structure in the localization category C/X in such a way that the canonical functor C ⟶ C/X is right exact and we have an exact sequence of right triangulated categories 0 ⟶ X ⟶ C ⟶ C/X ⟶ 0.

Dually since Y is a left triangulated subcategory of C, closed under direct summands and extensions of left triangles, we can construct the localization category C/Y which is the category of fractions C[L_{Y}^{−1}], where the class L_{Y} consists of all

morphisms f : A ⟶ B such that in any left triangle Ω(B) ⟶ Y ⟶ A ⟶^{f} B, the object Y lies in Y. Then the class of morphisms L_{Y} admits a calculus of right fractions and the left triangulated structures of Y and C induce a left triangulated structure in the localization category C/X in such a way that the canonical functor P : C ⟶ C/Y is left exact. In this case we have an exact sequence of left triangulated categories 0 ⟶ Y ⟶ C ⟶ C/Y ⟶ 0.

We have the following analogue of Proposition I.2.6.

Proposition 4.1. (i) If Ω(Y) = Y, then Y is right triangulated and the functor L : C ⟶ Y induces an equivalence C/X ⟶^{≈} Y. In other words we have an exact sequence of right triangulated categories: 0 ⟶ X ⟶^{i} C ⟶^{L} Y ⟶ 0.

(ii) If Σ(X) = X, then X is left triangulated and the functor R : C ⟶ Y induces an equivalence C/Y ⟶^{≈} X. In other words we have an exact sequence of left triangulated categories: 0 ⟶ Y ⟶^{j} C ⟶^{R} X ⟶ 0.

Proof. We prove only part (i). Consider the class of morphisms R_{X} defined above. We call an object M ∈ C left R_{X} −closed if for any A ⟶^{f} B ∈ R_{X}, the morphism C(f, M) : C(B, M) ⟶ C(A, M) is invertible. Let A ⟶^{f} B be in R_{X} and let A ⟶^{f} B ⟶ X ⟶ Σ(A) be a right triangle in C with X ∈ X. Then for any Y ∈ Y, we have the long exact sequence ⋯ ⟶ C(Σ(X), Y) ⟶ C(Σ(B), Y) ⟶ C(Σ(A), Y) ⟶ C(X, Y) ⟶ C(B, Y) ⟶ C(A, Y), which shows that C(Σ(f), Y) : C(Σ(B), Y) ⟶ C(Σ(A), Y) or equivalently C(f, Ω(Y)) : C(B, Ω(Y)) ⟶ C(A, Ω(Y)), is invertible. Hence for any Y ∈ Y, the object Ω(Y) is left R_{X} −closed. Since Ω(Y) = Y, it follows that any object of Y is left R_{X} −closed. Conversely if M is left R_{X} −closed, then M lies in Y. Indeed let C ∈ C and consider the right

triangle R(C) ⟶^{f_{C}} C ⟶^{g^{C}} Y^{C} ⟶^{h^{C}} ΣR(C), where f_{C} is the coreflection of C in X. Since ΣR(C) is in X, it follows that the morphism g^{C} is in R_{X}. Hence for any left R_{X} −closed object M we have C(g^{C}, M) : C(Y^{C}, M) ⟶^{≅} C(C, M). Now if C ∈ X, the morphism f_{C} is invertible, hence g^{C} = 0. It follows that C(X, M) = 0, for any X ∈ X, i.e. M ∈ X^{⊥} = Y. We infer that Y coincides with the class of left R_{X} −closed objects. Now let C ∈ C and consider the right triangle

R(C) ⟶^{f_{C}} C ⟶^{g^{C}} L(C) ⟶^{h^{C}} ΣR(C), in which the morphism g^{C} : C ⟶ L(C) is in R_{X} and the object L(C) is left R_{X} −closed. By [75], the localization functor C ⟶ C/X admits a fully faithful right adjoint C/X ↪ C and the composite functor Y ⟶^{j} C ⟶ C/X is an equivalence with inverse the composite C/X ⟶ C ⟶^{L} Y. Since C/X is right triangulated, so is Y.

The above result can be applied when the torsion class X is triangulated or the torsion-free class Y is triangulated. For concrete examples of this situation we refer to Chapter VI.

5. Lifting Torsion Theories

To any left or right triangulated category there is associated in a universal way a triangulated category called the stabilization and which keeps track of the complexity of its left or right triangulation. Our aim in this section is to study the relationship between the stabilizations of the torsion or torsion-free class of a torsion theory defined in a pretriangulated category. Indeed we shall show that if the torsion theory is (co)hereditary, then it can be lifted to a torsion theory in the stabilization. This lifting construction will be used in the investigation of universal cohomology theories with respect to (co)hereditary torsion theories in Chapter VIII.

Throughout this section we fix a pretriangulated category C and a torsion theory (X , Y) in C. As usual we denote by R : C ⟶ X the right adjoint of the inclusion i : X ↪ C and by L : C ⟶ Y the left adjoint of the inclusion j : Y ↪ C.

Stabilizations. Before we study the lifting of torsion theories in the left and right stabilization of C, we need to recall some facts about the stabilization process. We investigate also when the left or right stabilization of C can be realized by the torsion or torsion-free subcategory, via the (co)reflection functor.

Recall from [17], [57] that with a given left triangulated category C, we can associate in a universal way a triangulated category T (C) which reflects many important homological properties of C. More precisely there exists a left exact functor P : C ⟶ T (C) such that for any left exact functor F : C ⟶ T to a triangulated category T , there exists a unique up to isomorphism exact functor F^{∗} : T (C) ⟶ T such that F^{∗}P ≅ F. The category T (C) is called the stabilization of C and the functor P is called the stabilization functor. We refer to [17], [57], for details. Here we need only the following facts.

The objects of T (C) are pairs (C, n), where C is an object in C and n ∈ ℤ. The space of morphisms T (C)[(A, n), (B, m)] is identified with the direct limit lim_{⟶}_{k≥n,k≥m}C[Ω^{k−n}(A), Ω^{k−m}(B)]. The loop functor Ω : T (C) ⟶ T (C) is defined

by Ω(C, n) = (C, n − 1) and the stabilization functor P is defined by P(C) = (C, 0). Finally the extension of the left exact functor F above is defined by F^{∗}(C, n) = Ω^{−n}F (C). Note that T (C) = 0 iff Ω is locally nilpotent, that is for any C ∈ C, Ω^{n}(C) = 0 for some n ≥ 0. Dually any right triangulated category admits its stabilization which has a dual description and dual properties.

The stabilizations are very natural objects of study:

Example. Let (X , Y) be the natural torsion theory (= t-structure) in the the unbounded derived category D(C) of an abelian category. Then T_{r}(X) = D^{−}(C) is the derived category of right bounded complexes and T_{l}(Y) = D^{+}(C) is the derived category of left bounded complexes.

The pretriangulated category C as a left and right triangulated category admits its stabilization P_{l} : C ⟶ T_{l}(C) when considered as a left triangulated category and its stabilization P_{r} : C ⟶ T_{r}(C) when considered as a right triangulated category. In general the stabilizations T_{l}(C) and T_{r}(C) are not connected in a nice way, in particular they are rarely equivalent. It is important to have computable descriptions of the left and right stabilizations of C. We now study this problem in case C admits a torsion theory.

Consider the torsion theory (X , Y) in C. Since X is right triangulated and Y is left triangulated, it is natural to ask what is the relationship between the stabilizations of X , Y and C. In this respect the following subcategories are useful:

X̂ = {C ∈ C | ∃n ≥ 0 : Ω^{n}(C) ∈ X },
Ỹ = {C ∈ C | ∃n ≥ 0 : Σ^{n}(C) ∈ Y}

Similarly we introduce the following full subcategories of C:

P^{<∞}(C) := {C ∈ C | ∃n ≥ 0 : Ω^{n}(C) = 0},
I^{<∞}(C) := {C ∈ C | ∃n ≥ 0 : Σ^{n}(C) = 0}

It follows from [17] that KerP = P^{<∞}(C). The category P^{<∞}(C) is the prototype of the stable category of modules of finite projective dimension modulo projectives, see Chapter V. It is clear that P^{<∞}(C) is a left triangulated subcategory of C and its stabilization is trivial: T_{l}(P^{<∞}(C)) = 0. Moreover any left triangulated subcategory of C with trivial stabilization is contained in P^{<∞}(C). Dual remarks are applied for the category I^{<∞}(C).

The following gives a useful connection between the above subcategories.

Lemma 5.1. (i) If X is closed under Ω, then: Y ⊆ P^{<∞}(C) iff X̂ = C. (ii) If Y is closed under Σ, then: X ⊆ I^{<∞}(C) iff Ỹ = C.

Proof. (i) Assume that Y ⊆ P^{<∞}(C). If C ∈ C, then consider the standard triangle Δ(C) : Ω(C) ⟶^{h_{C}} Y_{C} ⟶^{g_{C}} R(C) ⟶^{f_{C}} C. Since Y_{C} ∈ Y, there exists n ≥ 0 such that Ω^{n}(Y_{C}) = 0. Consider the n−th rotation Ω^{n+1}(C) ⟶ Ω^{n}Y_{C} ⟶ Ω^{n}R(C) ⟶ Ω^{n}(C) of the above triangle. Then Ω^{n+1}(f_{C}) : Ω^{n+1}R(C) ⟶^{≅} Ω^{n+1}(C) is an isomorphism, since Ω^{n}(Y_{C}) = 0. Since R(C) ∈ X and X is closed under Ω, it follows that Ω^{n+1}(C) ∈ X. Hence C ∈ X̂ and then C = X̂. Conversely if this equality holds, then fix an object C ∈ C and let n ≥ 0 be such that Ω^{n}(C) ∈ X. Since by Proposition 2.6, R commutes with Ω, it follows easily that Ω^{n}R(C) ≅ RΩ^{n}(C). Then the morphism Ω^{n}(f_{C}) : Ω^{n}R(C) ⟶ Ω^{n}(C) is invertible. From the triangle Δ(C) above we infer that Ω^{n+1}(Y_{C}) = 0. Hence for any object C in C there exists n ≥ 0, such that Ω^{n}(Y_{C}) = 0. Let now Y ∈ Y be an arbitrary object. Then in the standard triangle Ω(Y) ⟶^{h_{Y}} Y_{Y} ⟶^{g_{Y}} R(Y) ⟶^{f_{Y}} Y we have that f_{Y} = 0. This implies trivially that Y_{Y} = R(Y) ⊕ Ω(Y). Since R(Y) ∈ X and Y_{Y} ∈ Y we have R(Y) = 0. Hence Y_{Y} = Ω(Y). By the above argument there exists n ≥ 0, such that Ω^{n}(Y_{Y}) = 0. Hence Ω^{n+1}(Y) = 0, i.e. Y ⊆ P^{<∞}(C).

The following result characterizes when the coreflection functor R is the stabilization of C with respect to its left triangulation, and when the reflection functor L is the stabilization of C with respect to its right triangulation, in terms of properties of the subcategories introduced above.

Proposition 5.2. Using the above notation, we have the following.

(i) The following are equivalent:

(a) R : C ⟶ X is the stabilization functor with respect to the left triangulation of C.

(b) X is triangulated and X̂ = C (or equivalently Y = P^{<∞}(C)). (ii) The following are equivalent:

(a) L : C ⟶ Y is the stabilization functor with respect to the right triangulation of C.

(b) Y is triangulated and Ỹ = C (or equivalently X = I^{<∞}(C)).

Proof. We prove only part (i). Assume that X is triangulated and X̂ = C. Then the coreflection functor R and the inclusion functor i are left exact. If F : C ⟶ D is a left exact functor to a triangulated category D, then F (Y) = 0. Indeed, since X̂ = C, by Lemma 5.1 we have Y ⊆ P^{<∞}(C), hence for any Y ∈ Y, there exists n ≥ 0 such that Ω^{n}(Y) = 0. Since F commutes with Ω, we have Ω^{n}F (Y) = 0 where Ω is the loop functor in D. Then F (Y) = 0, since Ω is invertible in D.

Now consider the functorial left triangle ΩL(C) ⟶^{h_{C}} R(C) ⟶^{f_{C}} C ⟶^{g^{C}} L(C) in C, where f_{C} : iR(C) ⟶ C is the coreflection of C in X. Then we have a triangle

ΩF L(C) ⟶ F R(C) ⟶^{F (f_{C})} F (C) ⟶ F L(C) in D. Since F (Y) = 0, it follows that F (f_{C}) : F R(C) ⟶ F (C) is invertible. Define a functor F^{∗} : X ⟶ D by F^{∗} = F i. Then F^{∗} is obviously left exact and the above isomorphism shows that F (f) : F^{∗}R ⟶^{≅} F. If G : X ⟶ D is another exact functor endowed with a natural isomorphism ξ : GR ⟶^{≅} F, then ξi : G ≅ GRi ⟶^{≅} F i = F^{∗}. Hence F^{∗} is the unique up to isomorphism left exact functor which extends F. Then R : C ⟶ X represents X as the stabilization of C. Conversely assume that R represents X as the stabilization of C with respect to its left triangulation. By [17], we know that the kernel of the stabilization functor is P^{<∞}(C). Since KerR = Y, we infer that P^{<∞}(C) = Y or equivalently by Lemma 5.1, X̂ = C.

Lifting Torsion Theories. Our aim now is to show that in some cases the exact localization sequences of left or right triangulated categories in Proposition 4.1 can be lifted to exact sequences in the stabilizations, thus providing a lifting of the torsion theory from the pretriangulated category to the triangulated stabilization. Assume throughout that the torsion theory (X , Y) is hereditary, the cohereditary case being treated in a dual way.

Since (X , Y) is hereditary we have that X is pretriangulated, and the inclusion functor i : X ↪ C and the coreflection functor R : C ⟶ X are left exact. Consider the stabilization functors P : C ⟶ T_{l}(C) and P : X ⟶ T_{l}(X) of C and X when these are considered as left triangulated categories. Since i and R are left exact, there exist unique exact functors i^{∗} : T_{l}(X) ⟶ T_{l}(C) and R^{∗} : T_{l}(C) ⟶ T_{l}(X), such that the following diagrams commute:

\begin{matrix}X & ⟶^{i} & C \\ ^{P}↓ & & ^{P}↓ \\ T_{l}(X) & ⟶^{i^{∗}} & T_{l}(C) \end{matrix}  \begin{matrix}C & ⟶^{R} & X \\ ^{P}↓ & & ^{P}↓ \\ T_{l}(C) & ⟶^{R^{∗}} & T_{l}(X) \end{matrix}

Proposition 5.3. There exists a hereditary torsion theory (T_{l}(X), T_{l}(Y)) in T_{l}(C), that is, there exists a short exact sequence of triangulated categories

0 ⟶ T_{l}(Y) ⟶^{j^{∗}} T_{l}(C) ⟶^{R^{∗}} T_{l}(X) ⟶ 0

The functor i^{∗} is a left adjoint of R^{∗} and the functor j^{∗} admits a left adjoint.

Proof. It suffices to show that the functor R^{∗} admits a fully faithful left adjoint and KerR^{∗} = T_{l}(Y). By Proposition 3.4 of [17] we have that the left exact functor j : Y ↪ C induces a fully faithful exact functor j^{∗} : T_{l}(Y) ↪ T_{l}(C). Let (C, n) be in T_{l}(C) such that R^{∗}(C, n) = 0. Then Ω^{−n}P(R(C)) = (R(C), n) = 0 and this means that there exists k with k + n ≥ 0 such that Ω^{k+n}R(C) = 0. Since R commutes with the loop functor Ω we have RΩ^{k+n}(C) = 0. Hence Ω^{k+n}(C) lies in Y and then P(Ω^{k+n}(C)) = Ω^{n+k}P(C) lies in T_{l}(Y). Since the latter is a triangulated subcategory of T_{l}(C) we have that Ω^{−2n−k}Ω^{n+k}P(C) = Ω^{−n}P(C) = (C, n) lies in Y. On the other hand if (Y, n) is in T_{l}(Y) then R^{∗}(Y, n) = (R(Y), n) = 0 since R(Y) = 0. We infer that KerR^{∗} = T_{l}(Y). Since Ri = Id_{X} it follows that R^{∗}i^{∗} = Id_{T_{l}(X)}. Consider now the coreflection morphism f : iR ⟶ Id_{X}. It is not difficult to see that the lifting i^{∗}R^{∗} ⟶ Id_{T_{l}(X)} of f in T_{l}(C) serves as the coreflection morphism for the adjoint pair (i^{∗}, R^{∗}) and of course the functor i^{∗} is fully faithful, since i is so. Now standard arguments [79] show that j^{∗} admits a left adjoint.

Dually when the torsion theory is cohereditary, then Y is pretriangulated, and the inclusion functor j : Y ↪ C and the reflection functor L : C ⟶ Y are right exact. Consider the stabilizations Q : C ⟶ T_{r}(C) and Q : Y ⟶ T_{r}(Y) of C and Y when these are considered as right triangulated categories. Since j and L are right exact, there exist unique exact functors j^{∗} : T_{r}(X) ⟶ T_{r}(C) and L^{∗} : T_{r}(C) ⟶ T_{r}(Y), such that the following diagrams commute:

\begin{matrix}Y & ⟶^{j} & C \\ ^{Q}↓ & & ^{Q}↓ \\ T_{r}(Y) & ⟶^{j^{∗}} & T_{r}(C) \end{matrix}  \begin{matrix}C & ⟶^{L} & Y \\ ^{Q}↓ & & ^{Q}↓ \\ T_{r}(C) & ⟶^{L^{∗}} & T_{r}(Y) \end{matrix}

The following is a dual version of Proposition 5.3.

Proposition 5.4. If (X , Y) is cohereditary, then there exists a hereditary torsion theory (T_{r}(X), T_{r}(Y)) in T_{r}(C), that is, there exists a short exact sequence of triangulated categories

0 ⟶ T_{r}(X) ⟶^{i^{∗}} T_{r}(C) ⟶^{L^{∗}} T_{r}(Y) ⟶ 0

The functor j^{∗} is a right adjoint of L^{∗} and the functor i^{∗} admits a right adjoint.

We have the following direct consequence.

Corollary 5.5. (1) If (X , Y) is hereditary, then the functor R^{∗} : T_{l}(C) ⟶ T_{l}(X) is a triangle equivalence iff Y ⊆ P^{<∞}(C). If this is the case and X is triangulated, then R^{∗} : T_{l}(C) ⟶ X is a triangle equivalence and R : C ⟶ X is the stabilization functor with respect to the left triangulation of C.

(2) If (X , Y) is cohereditary, then the functor L^{∗} : T_{r}(C) ⟶ T_{r}(Y) is a triangle equivalence iff X ⊆ I^{<∞}(C). If this is the case and Y is triangulated, then L^{∗} : T_{r}(C) ⟶ Y is a triangle equivalence and L : C ⟶ Y is the stabilization functor with respect to the right triangulation of C.

Grothendieck Groups. We close this chapter with an application of the lifting of torsion theories to K-theory.

If C is a left triangulated category with loop functor Ω and (left) triangulation Δ, then its Grothendieck group K_{0}(C) is defined as the quotient of the free abelian group on the isoclasses of objects of C, modulo the relations (A) − (B) + (C), for each triangle Ω(C) ⟶ A ⟶ B ⟶ C in Δ. The definition of the Grothendieck group of a right triangulated category is dual. If C is pretriangulated, then we denote by K_{0}^{Δ}(C) the Grothendieck group when C is considered as a left triangulated category, and by K_{0}^{∇}(C) the Grothendieck group when C is considered as a right triangulated category. Of course if C is triangulated, then K_{0}^{Δ}(C) = K_{0}^{∇}(C) := K_{0}(C) is the usual Grothendieck group of C.

Corollary 5.6. (1) Let (X , Y) be a hereditary torsion theory in C. Then the coreflection functor R : C ⟶ X induces an isomorphism

K_{0}^{Δ}(C) ⟶^{≅} K_{0}^{Δ}(X) ⊕ K_{0}^{Δ}(Y)

In particular K_{0}^{Δ}(C) ≅ K_{0}^{Δ}(X) if Y ⊆ P^{<∞}(C).

(2) Let (X , Y) be a cohereditary torsion theory in C. Then the reflection functor L : C ⟶ Y induces an isomorphism

K_{0}^{∇}(C) ⟶^{≅} K_{0}^{∇}(X) ⊕ K_{0}^{∇}(Y)

In particular K_{0}^{∇}(C) ≅ K_{0}^{∇}(Y) if X ⊆ I^{<∞}(C).

Proof. We prove (1) since (2) is dual. By a well-known result of Grothendieck the exact sequence of triangulated categories in Proposition 5.3 induces an exact sequence K_{0}(T_{l}(Y)) ⟶ K_{0}(T_{l}(C)) ⟶ K_{0}(T_{l}(X)) ⟶ 0 which is a split short exact sequence, since the functor j^{∗} admits a left adjoint. Now the assertion follows from a result of [17] which says that the Grothendieck group is invariant under stabilization.

The following consequence was first observed in [67].

Corollary 5.7. Let (X , Y) be a hereditary torsion theory in the triangulated category C. Then the assignment [C] ⟼([R(C)], [L(C)]) induces an isomorphism

K_{0}(C) ⟶^{≅} K_{0}(X) ⊕ K_{0}(Y).

CHAPTER III Compactly Generated Torsion Theories in Triangulated Categories

Throughout this chapter we fix a triangulated category C. Our purpose here is to develop a technique for constructing torsion theories in C induced by a given set of objects, and to give some applications to tilting theory by investigating the heart. In particular we are interested in finding conditions such that the heart is a module category. This will be important in connection with classical tilting theory and the construction of derived equivalences. As it happens in the case of abelian categories, for the construction of torsion theories generated by a set of objects, we have to assume that the category C is big enough to allow certain infinite constructions.

1. Torsion Theories of Finite Type

It is well-known that an important and well-behaved class of torsion theories in an abelian category G with exact filtered colimits, is the class of torsion theories of finite type. We recall from [50] that a torsion theory (T , F) in G is said to be of finite type if the torsion free class F is closed under filtered colimits (in case G is locally coherent this is equivalent to the condition that the torsion subfunctor commutes with filtered colimits). These torsion theories are important in connection with the Ziegler spectrum and the localization theory of abelian categories, see [50] for a detailed analysis from a representation theoretic point of view.

Filtered colimits rarely exist in a triangulated category. However all small coproducts, which are special filtered colimits, exist in many interesting triangulated categories, for instance the unbounded derived category of an abelian category with exact coproducts, the stable homotopy category of spectra, and the stable module category of a group algebra. So we are led to introduce and investigate the analogous concept of torsion theories of finite type in triangulated categories.

Throughout this section C denotes a triangulated category containing all small coproducts. In analogy with the abelian case we make the following definition.

Definition 1.1. A torsion theory (X , Y) in C is called of finite type if the torsion free class Y is closed under all small coproducts.

Let (X , Y) be a torsion theory in C, let R : C ⟶ X be the right adjoint of the inclusion i : X ↪ C, and let L : C ⟶ Y be the left adjoint of the inclusion j : Y ↪ C. The following criterion will be useful.

Lemma 1.2. (1) The torsion theory (X , Y) is of finite type iff R preserves coproducts. In this case the functors i : X ↪ C and L : C ⟶ Y preserve compact objects.

(2) L preserves products iff X is closed under products.

Proof. (1) If R preserves coproducts and {Y_{i}; i ∈ I} is a set of objects of Y, then: C(X , ⊕_{i∈I}Y_{i}) ≅ C(X , R(⊕_{i∈I}Y_{i})) = C(X , ⊕_{i∈I}R(Y_{i})) = 0. Hence ⊕_{i∈I}Y_{i} ∈ X^{⊥} = Y and Y is closed under coproducts. Conversely if Y is closed under coproducts and {C_{i}; i ∈ I} is a set of objects of C, then consider the triangle

⊕_{i∈I}R(C_{i}) ⟶^{⊕f_{i}} ⊕_{i∈I}C_{i} ⟶^{⊕g_{i}} ⊕_{i∈I}L(C_{i}) ⟶^{⊕h_{i}} Σ(⊕_{i∈I}R(C_{i})) which is a coprod-

uct of the triangles R(C_{i}) ⟶^{f_{i}} C_{i} ⟶^{g_{i}} L(C_{i}) ⟶^{h_{i}} ΣR(C_{i}). Since Y is closed under coproducts, we have ⊕_{i∈I}L(C_{i}) ∈ Y. Then any morphism α : X ⟶ ⊕_{i∈I}C_{i} with X ∈ X factors through ⊕f_{i} since α ∘ ⊕g_{i} = 0. Moreover the factorization is unique since for any two of them, their difference factorizes through Σ^{−1}(⊕_{i∈I}L(C_{i})) ∈ Y, so it is zero. We infer that ⊕_{i∈I}R(C_{i}) is the coreflection of ⊕_{i∈I}C_{i} in X. This implies that the canonical morphism ⊕_{i∈I}R(C_{i}) ⟶ R(⊕_{i∈I}C_{i}) is invertible.

If X is compact in X, and {C_{i} | i ∈ I} is a family of objects in C, then the following isomorphisms show that X is compact in C:

⊕_{i∈I}C(X, C_{i}) ⟶^{≅} ⊕_{i∈I}X (X, R(C_{i})) ⟶^{≅} X (X, ⊕_{i∈I}R(C_{i}))

⟶^{≅} X (X, R(⊕_{i∈I}C_{i})) ⟶^{≅} C(X, ⊕_{i∈I}C_{i}).

Now since Y is closed under coproducts, the inclusion j : Y ↪ C preserves coproducts. Then working as above we have that the reflection functor L preserves compact objects.

The proof of (2) is dual and is left to the reader.

Note. It is easy to see that the above proof works in any pretriangulated category with coproducts, resp. products. This more general result will be useful in the study of the torsion theory induced by Cohen-Macaulay modules in the stable module category of an Artin algebra, see the last chapter.

Example. Let (T , F) be a torsion theory in a Grothendieck category G and assume that (T , F) is of finite type, or more generally that F is closed under coproducts. The tilting construction of Section I.3 (which as easily seen works for unbounded complexes) produces a torsion theory of finite type (X (T ), Y(F)) in the unbounded derived category D(G). For instance let G be a compact object in G and let (T , F) be the torsion theory in G generated by G, i.e. F = G^{⊥} and T = ^{⊥}(G^{⊥}). Then the torsion theory (X (T ), Y(F)) in D(G) is of finite type.

It would be very interesting to have a classification of all torsion theories of finite type in C. This seems to be very difficult; see the remarks in the last section of the next chapter. Thus we concentrate on a special class of torsion theories of finite type, namely the torsion theories generated by compact objects. Already this class has important applications.

2. Compactly Generated Torsion Theories

Throughout this section we fix a triangulated category C. Our purpose here is to construct a torsion theory in C generated by a set of objects, and to give, if possible, a convenient description of the torsion or the torsion-free class. This will be important in connection with having a nice description of the heart in the next section.

We fix a set of objects P ⊆ C. To get a torsion theory (X_{P}, Y_{P}) in C generated by P it is necessary by Proposition II.3.3 that X_{P} is right triangulated and Y_{P} is left triangulated. As in the case of torsion theories in abelian categories a reasonable candidate for the pair (X_{P}, Y_{P}) is obtained by defining:

Y_{P} := {C ∈ C | C(Σ^{n}(P), C) = 0, ∀P ∈ P, ∀n ≥ 0} and X_{P} := ^{⊥}Y_{P}.

The above definition is also similar to the definition of the natural torsion theory (= t-structure) in the derived category of an abelian category. We are interested in finding sufficient conditions for the pair (X_{P}, Y_{P}) to be a torsion theory in C. First we record the following immediate consequence of the above definition.

Lemma 2.1. X_{P} is a right triangulated subcategory of C closed under coproducts and extensions, and Y_{P} is a left triangulated subcategory of C closed under products and extensions.

From now on we assume that: C contains all small coproducts and the set P consists of compact objects of C. Throughout we use the following notation:

Q := {Σ^{n}(P) | P ∈ P, n ≥ 0}

so Q remains a set of compact objects and by construction Q is closed under Σ. We denote by Add(Q) the full subcategory of C consisting of direct summands of arbitrary coproducts of objects of Q. Observe that Add(Q)^{⊥} = Q^{⊥}.

The above setup and notation will be fixed throughout this chapter.

We have the following useful result.

Lemma 2.2. Add(Q) is contravariantly finite in C and for any C ∈ C, there exists a triangle Q_{0} ⟶^{f_{0}} C ⟶^{g_{0}} Y_{0} ⟶^{h_{0}} Σ(Q_{0}) in C with the following properties:

(i) f_{0} is a right Add(Q)-approximation of C.

(ii) 0 ⟶ C(Q, Σ^{−n−1}(Y_{0})) ⟶ C(Q, Σ^{−n}(Q_{0})) ⟶ C(Q, Σ^{−n}(C)) ⟶ 0 is a short

exact sequence, ∀n ≥ 0. (iii) The morphism C(Σ^{n}(g_{0}), Y_{P}) : C(Σ^{n}(Y_{0}), Y_{P}) ⟶ C(Σ^{n}(C), Y_{P}) is invertible,

∀n ≥ 0.

Proof. For any C ∈ C, consider the set of morphisms I_{C} := {P ⟶ C | P ∈ Q} and let Q_{0} = ⊕_{λ∈I_{C}} P_{λ}. The set I_{C} induces a morphism f_{0} : Q_{0} ⟶ C which by construction has the property that (Q, f_{0}) is epic. This implies that (A, f_{0}) is epic, ∀A ∈ Add(Q), i.e. f_{0} is a right Add(Q)-approximation of C. If (T) : Q_{0} ⟶^{f_{0}} C ⟶ Y_{0} ⟶ Σ(Q_{0}) is a triangle in C, then the morphism C(Q, Σ^{−n}(f_{0})) : C(Q, Σ^{−n}(Q_{0})) ⟶ C(Q, Σ^{−n}(C)) is isomorphic to C(Σ^{n}(Q), f_{0}) : C(Σ^{n}(Q), Q_{0}) ⟶ C(Σ^{n}(Q), C), ∀n ≥ 0. Hence assertion (ii) follows if we apply C(Q, −) to (T), using that Σ(Q) ⊆ Q. Since Y_{P} = Q^{⊥}, applying C(−, Y_{P}) to the triangle (T), the last assertion follows.

In the following we use a version of Bousfield’s localization, called finite localization, popularized by Miller [60] in case C is the stable homotopy category of spectra. Our result is more general since in our case C is an arbitrary triangulated category with coproducts whereas everything in sight in [60] is closed under Σ, Σ^{−1}.

Theorem 2.3. Let C be a triangulated category with coproducts and let P ⊆ C be a set of compact objects. Then (X_{P}, Y_{P}) is a torsion theory of finite type in C, where as before Y_{P} := {Σ^{n}(P) | n ≥ 0}^{⊥} and X_{P} := ^{⊥}Y_{P}.

Proof. By Lemma 2.1, X_{P} is a full subcategory of C, closed under Σ and extensions, and Y_{P} is a full subcategory of C, closed under Σ^{−1} and extensions. Let Q_{0} ⟶^{f_{0}} C ⟶^{g_{0}} Y_{0} ⟶^{h_{0}} Σ(Q_{0}) be the triangle constructed in Lemma 2.2. Inductively ∀n ≥ 0, we can construct triangles Q_{n} ⟶^{f_{n}} Y_{n−1} ⟶^{g_{n}} Y_{n} ⟶^{h_{n}} Σ(Q_{n}) in C, where Y_{−1} = C and f_{n} is a right Add(Q)−approximation of Y_{n−1} satisfying the properties of Lemma 2.2. Consider the tower of objects

C ⟶^{g_{0}} Y_{0} ⟶^{g_{1}} Y_{1} ⟶^{g_{2}} Y_{2} ⟶ ⋯ ⟶ Y_{n} ⟶^{g_{n+1}} Y_{n+1} ⟶ ⋯  (Π)

and let holim_{⟶}Y_{n} be its homotopy colimit which is defined by the triangle

⨁_{n≥0} Y_{n} ⟶^{1−g} ⨁_{n≥0} Y_{n} ⟶^{α} holim_{⟶}Y_{n} ⟶^{β} Σ ⨁_{n≥0} Y_{n}

where the morphism 1 − g is induced by Y_{n−1} ⟶^{(1_{Y_{n−1}} ,−g_{n})} Y_{n−1} ⊕ Y_{n} ↪ ⊕_{n≥0}Y_{n}.

Since Q consists of compact objects, it is well-known that the canonical morphism lim_{⟶}C(Q, Y_{n}) ⟶ C(Q, holim_{⟶}Y_{n}) is invertible [18]. Since by construction C(Q, g_{n}) = 0, ∀n ≥ 0, the short exact sequence

0 ⟶ ⨁_{n≥0} C(Q, Y_{n}) ⟶^{1−(Q,g)} ⨁_{n≥0} C(Q, Y_{n}) ⟶ lim_{⟶}C(Q, Y_{n}) ⟶ 0

shows that 0 = lim_{⟶}C(Q, Y_{n}) ≅ C(Q, holim_{⟶}Y_{n}). Hence holim_{⟶}Y_{n} ∈ Y_{P}, and since Y_{P}

is closed under Σ^{−1}, we have that Σ^{−1}holim_{⟶}Y_{n} ∈ Y_{P}. The tower of objects (Π) induces a morphism g_{C} : C ⟶ holim_{⟶}Y_{n}. Let

Σ^{−1}holim_{⟶}Y_{n} ⟶^{h_{C}} X ⟶^{f_{C}} C ⟶^{g_{C}} holim_{⟶}Y_{n}  (∗)

be a triangle in C. We claim that g_{C} is the reflection of C in Y_{P}. By construction if we apply C(−, Y_{P}) to the tower of objects (T), we get an inverse tower

⋯ ⟶ C(Y_{2}, Y) ⟶^{(g_{2},Y_{P})} C(Y_{1}, Y_{P}) ⟶^{(g_{1},Y_{P})} C(Y_{0}, Y_{P}) ⟶^{(g_{0},Y_{P})} C(C, Y_{P}) where the

involved morphisms are invertible. Hence lim_{⟵}C(Y_{n}, Y_{P}) ≅ C(C, Y_{P}). Now by [18], we have a short exact sequence

0 ⟶ lim_{⟵}^{(1)}C(Σ(Y_{n}), Y_{P}) ⟶ C(holim_{⟶}Y_{n}, Y_{P}) ⟶ lim_{⟵}C(Y_{n}, Y_{P}) ⟶ 0

where lim_{⟵}^{(1)} denotes the first right derived functor of the inverse limit functor [81].

By part (iii) of Lemma 2.2, in the inverse system ⋯ ⟶ C(Σ(Y_{2}), Y_{P}) ⟶^{(Σ(g_{2}),Y_{P})}

C(Σ(Y_{1}), Y_{P}) ⟶^{(Σ(g_{1}),Y_{P})} C(Σ(Y_{0}), Y_{P}) the involved morphisms are invertible. Then by the Mittag-Leffler condition [81], we have lim_{⟵}^{(1)}C(Σ(Y_{n}), Y_{P}) = 0. Hence C(holim_{⟶}Y_{n}, Y_{P}) ≅ lim_{⟵}C(Y_{n}, Y_{P}) ≅ C(C, Y_{P}). We infer that g_{C} : C ⟶ holim_{⟶}Y_{n}

induces an isomorphism C(g_{C}, Y_{P}) : C(holim_{⟶}Y_{n}, Y_{P}) ⟶^{≅} C(C, Y_{P}), i.e. g_{C} is the reflection of C ∈ Y_{P}. Hence by Proposition I.2.3, f_{C} : X ⟶ C is the coreflection of C in X_{P} and then (X_{P}, Y_{P}) is a torsion theory in C. Since Y_{P} = Q^{⊥} and Q consists of compact objects, it follows trivially that Y_{P} is closed under coproducts, so (X_{P}, Y_{P}) is of finite type.

It is now natural to make the following definition.

Definition 2.4. A torsion theory (X , Y) in C is generated by a set of compact objects or it is compactly generated if there exists a set of compact objects P in C such that Y = {Σ^{n}(P) | n ≥ 0}^{⊥} and X = ^{⊥}Y. Such a torsion theory is always denoted by (X_{P}, Y_{P}).

By the above theorem it follows in particular that any compact object T ∈ C generates a torsion theory (X_{T} , Y_{T}) of finite type in C.

Having a torsion theory (X_{P}, Y_{P}) in C generated by a set compact objects P, we are also interested in having a convenient description of the torsion class X_{P}.

We denote by Loc^{+}(P) the smallest full subcategory of C which is closed under extensions, coproducts, Σ, and contains P. Then automatically Loc^{+}(P) is a right triangulated subcategory of C. Clearly the torsion class X_{P} contains Loc^{+}(P). It is rather useful to know under what conditions we have Loc^{+}(P) = X_{P}. In this generality we can prove that the equality holds under an additional assumption. If U ⊆ C, we denote by U^{⋆n} = U ⋆ U ⋆ ⋯ ⋆ U (n−factors) the iterated Verdier product of U.

We need the following easy observation whose direct proof is left to the reader.

Lemma 2.5. If C(P, Σ^{t}(P)) = 0, ∀t ≥ 1, then C(P, Σ^{t}(C)) = 0, ∀t ≥ 1, for any C ∈ Add(Q)^{⋆n} and n ≥ 1.

Using this lemma we get a description of the torsion class X_{P} under a mild assumption.

Proposition 2.6. If C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, then X_{P} = Loc^{+}(P).

Proof. Since X_{P} is closed under coproducts, extensions, Σ and contains P, it follows by construction that Loc^{+}(P) ⊆ X_{P}. Let C ∈ C and consider the triangles Q_{n} ⟶^{f_{n}} Y_{n−1} ⟶^{g_{n}} Y_{n} ⟶ Σ(Q_{n}), n ≥ 0 constructed in Theorem 2.3, where C = Y_{−1}. Setting χ_{0} = f_{0} and Q_{0} = T_{0}, we can construct inductively a tower of objects T_{0} ⟶^{τ_{0}} T_{1} ⟶^{τ_{1}} T_{2} ⟶ ⋯ and the following tower of triangles:

T_{0} ⟶^{χ_{0}} C ⟶^{g_{0}} Y_{0} ⟶^{h^{C}} Σ(T_{0})
τ_{0}↓  ^{∥}↓  ^{g_{1}}↓  ^{Σ(τ_{0})}↓
T_{1} ⟶^{χ_{1}} C ⟶^{g_{0}∘g_{1}} Y_{1} ⟶ Σ(T_{1})
τ_{1}↓  ^{∥}↓  ^{g_{2}}↓  ^{Σ(τ_{1})}↓
T_{1} ⟶^{χ_{2}} C ⟶^{g_{0}∘g_{1}∘g_{2}} Y_{2} ⟶ Σ(T_{2})
↓  ↓  ↓  ↓
⋮  ⋮  ⋮  ⋮

Using the Octahedral Axiom in each step it is not difficult to see that each object T_{n} lies in Add(Q)^{⋆n} [18]. Consider the homotopy colimit holim_{⟶}T_{n} of the left vertical tower, which is equipped with a morphism χ_{C} : holim_{⟶}T_{n} ⟶ C induced by the above diagram, and let

holim_{⟶}T_{n} ⟶^{χ_{C}} C ⟶^{g} Ĉ ⟶^{h} Σ(holim_{⟶}T_{n})  (T)

be a triangle in C. By construction holim_{⟶}T_{n} lies in Loc^{+}(P). Using Lemma 2.2, we

have short exact sequences 0 ⟶ C(Q, Σ^{−1}(Y_{n})) ⟶ C(Q, T_{n}) ⟶ C(Q, C) ⟶ 0, ∀n ≥ 0. Taking direct limits we deduce a short exact sequence 0 ⟶ lim_{⟶}C(Q, Σ^{−1}(Y_{n})) ⟶ lim_{⟶}C(Q, T_{n}) ⟶ C(Q, C) ⟶ 0. Since Q consists of compact objects, we have

lim_{⟶}C(Q, Σ^{−1}(Y_{n})) ≅ C(Q, holim_{⟶}Σ^{−1}(Y_{n})) = 0 and lim_{⟶}C(Q, T_{n}) ≅ C(Q, holim_{⟶}T_{n}). It follows that the morphism χ_{C} induces an isomorphism C(Q, χ_{C}) : C(Q, holim_{⟶}T_{n}) ⟶

C(Q, C). Since Σ(Q) ⊆ Q, it follows that C(Q, Σ^{−1}(χ_{C})) : C(Q, Σ^{−1}(holim_{⟶}T_{n}))

⟶ C(Q, Σ^{−1}(C)) is invertible. This implies that C(Q, Σ^{−1}(Ĉ)) = 0, i.e. Ĉ ∈ Σ(Y_{P}). Assume now that C ∈ X_{P}. Then Ĉ ∈ X_{P} since holim_{⟶}T_{n} lies in X_{P} and

the latter is right triangulated. It follows that Ĉ ∈ X_{P} ∩ Σ(Y_{P}). Applying the functor C(P, −) to the triangle (T), we have an exact sequence C(P, holim_{⟶}T_{n}) ⟶

C(P, C) ⟶ C(P, Ĉ) ⟶ C(P, Σ(holim_{⟶}T_{n})). Since P consists of compact objects we have lim_{⟶}C(P, Σ(T_{n})) ≅ C(P, Σ(holim_{⟶}T_{n})), and by Lemma 2.5, C(P, Σ(holim_{⟶}T_{n})) =

0. Since C(P, χ_{C}) is invertible we infer that C(P, Ĉ) = 0, hence Ĉ ∈ Y_{P}. Then Ĉ = 0, since Ĉ lies also in X_{P}. Hence C = holim_{⟶}T_{n} ∈ Loc^{+}(P), i.e. Loc^{+}(P) = X_{P}.

Remark 2.7. The assumption C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, in Proposition 2.6 can be removed, in case C admits a model, for example if C is the stable module category of a quasi-Frobenius ring or the stable homotopy category of spectra or the unbounded derived category of an AB4 category, that is an abelian category with exact coproducts. Indeed in any of the above cases, the tower of triangles constructed above induces a triangle holim_{⟶}T_{n} ⟶ C ⟶ holim_{⟶}Y_{n} ⟶ Σ(holim_{⟶}T_{n}) in C. Then as above we infer that if C ∈ X_{P}, then C = holim_{⟶}T_{n}, since holim_{⟶}Y_{n} ∈ Y_{P}. The problem is that in a general triangulated category it is not known if the homotopy colimit of a tower of triangles is a triangle, see [18].

Trivially the assumption can also be removed if P is closed under Σ^{−1}. In this case the torsion class X_{P} coincides with the smallest triangulated subcategory of C which is closed under coproducts and contains P.

Using the above result we get a description of the torsion class X_{P}, not only of the torsion-free class Y_{P}. This will be useful for a nice description of the heart of the torsion theory in the next section. First we recall from Chapter I that a set of objects U generates a triangulated category C, or that is a generating set, if {Σ^{n}(U) | n ∈ ℤ}^{⊥} = 0.

Proposition 2.8. If C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, then
X_{P} ⊆ {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≥ 1}.

Moreover the following statements are equivalent:

(i) P generates C and C(P, Σ^{n}(P)) = 0, ∀n ≥ 1.

(ii) X_{P} = {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≥ 1}.

Proof. By Proposition 2.6, the hypothesis implies that X_{P} = Loc^{+}(P) is the smallest full right triangulated subcategory of C which is closed under coproducts and contains P. Since obviously {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n > 0} is a right triangulated subcategory of C, closed under coproducts and contains P, it follows that X_{P} ⊆ {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≥ 1}.

(i) ⇒ (ii) It suffices to show that {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≥ 1} ⊆ X_{P}. Let C be in C such that C(P, Σ^{t}(C)) = 0, ∀t > 0, and consider the glueing triangle R(C) ⟶ C ⟶ L(C) ⟶ ΣR(C)) in C. By Proposition 2.6 we can write R(C) = holim_{⟶}T_{n}, where T_{n} ∈ Add(Q)^{⋆n}. Applying C(Σ^{−t}(P), −) to the above

triangle and using Lemma 2.5, we see directly that C(Σ^{−t}(P), L(C)) = 0, ∀t > 0. Since L(C) ∈ Y_{P}, we infer that C(P, Σ^{t}L(C)) = 0, ∀t ∈ ℤ. Since P generates C, we have L(C) = 0. This implies that C ≅ R(C) ∈ X_{P}.

(ii) ⇒ (i) Since P ⊆ X_{P}, it suffices to show that P is a generating set. So let C be an object in C such that C(P, Σ^{t}(C)) = 0, ∀t ∈ ℤ. Then obviously C ∈ Y_{P}. Moreover the description of X_{P} implies that C ∈ X_{P}. Hence C ∈ X_{P} ∩ Y_{P} = 0. We infer that P is a generating set.

The following examples illustrate the above constructions.

Example. (1) Let C = D(Mod(Λ)) be the unbounded derived category of right Λ-modules over a ring Λ. Let P = {Λ} consist only of the regular module Λ concentrated in degree zero. Then P satisfies the conditions of Proposition 2.8, and the torsion theory generated by Λ is the natural torsion theory (= t-structure) in D(Mod(Λ)). The torsion class coincides with the smallest right triangulated subcategory of D(Mod(Λ)) which is closed under coproducts and contains Λ.

(2) Let Ho(Sp) be the stable homotopy category of spectra and let P = {S^{0}} consist only of the sphere spectrum. It is well-known that S^{0} is compact and {S^{0}} satisfies the conditions of Proposition 2.8 [59]. Then the torsion theory generated by S^{0} is the natural torsion theory (= t-structure) in Ho(Sp). The torsion class X_{P} is known in algebraic topology as the category of connective spectra.

3. The Heart of a Compactly Generated Torsion Theory

We continue to assume that C is a triangulated category with small coproducts. Having a torsion theory (X_{P}, Y_{P}) in C compactly generated by P, it is essential to know the structure of its heart in view of the important role which it plays in tilting theory, see for instance [41]. In particular we are interested in finding sufficient conditions such that the heart is a module category. This will be applied to tilting theory in the next section. A similar approach was considered independently also in [44].

For the rest of this section we denote by H(P) = X_{P} ∩ Σ(Y_{P}) the heart of the torsion theory (X_{P}, Y_{P}). As in Chapter I, H(P) is an abelian category and the functor H^{0} = ΣLΣ^{−1}R = RΣLΣ^{−1} : C ⟶ H(P) is homological. The higher homological functors are denoted by H^{n} : C ⟶ H(P) and are defined by H^{n} := H^{0}Σ^{n} = ΣLΣ^{−1}RΣ^{n} = RΣLΣ^{n−1}, ∀n ∈ ℤ.

If A is an abelian category then we call a set of objects T ⊆ A a set of weak generators if A(T, A) = 0, ∀T ∈ T implies that A = 0. We denote by H^{0}(P) the full subcategory {H^{0}(P) | P ∈ P} of C.

We have the following result which will be useful later.

Lemma 3.1. (i) (a) C(P, H) ⟶^{≅} H(P)(H^{0}(P), H), ∀H ∈ H(P).

(b) H^{n}(P) = 0, ∀n ≥ 1. (ii) H(P) has exact coproducts, i.e. it is AB4. (iii) H^{0}(P) is a set of compact weak generators in H(P).

Proof. (i) Since P ⊆ X_{P}, we have: H^{0}(P) = ΣLΣ^{−1}R(P) = ΣLΣ^{−1}(P). Then for any object H in H(P) we have: H(P)(H^{0}(P), H) = H(P)(ΣLΣ^{−1}(P), H). Since H lies in H(P), we have H = Σ(Y) for some Y ∈ Y_{P}. Then H(P)(H^{0}(P), H)

= Y_{P}(L(Σ^{−1}(P), Y) ⟶^{≅} C(Σ^{−1}(P), Y) = C(P, Σ(Y)) = C(P, H) and this proves (a). Since Σ^{n}(P) ∈ X_{P} for n ≥ 0, we have H^{n}(P) = ΣLΣ^{n−1}(P). If n ≥ 1, then LΣ^{n−1}(P) = 0, since Σ^{t}(P) lies in X_{P} for t > 0. This proves (b).

(ii) Since the torsion theory (X_{P}, Y_{P}) is of finite type, the heart H(P) has all small coproducts, so it is cocomplete. If 0 ⟶ A_{i} ⟶ B_{i} ⟶ C_{i} ⟶ 0 is a set of short exact sequences in H(P) indexed by I, then there are triangles A_{i} ⟶ B_{i} ⟶ C_{i} ⟶ Σ(A_{i}) in C. Since the coproduct of triangles is a triangle, we have a triangle ⊕_{i∈I}A_{i} ⟶ ⊕_{i∈I}B_{i} ⟶ ⊕_{i∈I}C_{i} ⟶ Σ(⊕_{i∈I}A_{i}) in C. Then the sequence 0 ⟶ ⊕_{i∈I}A_{i} ⟶ ⊕_{i∈I}B_{i} ⟶ ⊕_{i∈I}C_{i} ⟶ 0 is short exact in H(P). Hence coproducts are exact and H(P) is an AB4 category.

(iii) Let H be an object in the heart such that H(P)(H^{0}(P), H) = 0. Then by (i) we have C(P, H) = 0. This implies that C(Σ^{n}(P), H) = 0, ∀n ≥ 0 since H lies also in Σ(Y_{P}). Hence H ∈ X_{P} ∩ Y_{P} = 0. It follows that H^{0}(P) is a set of weak generators of the heart. It remains to prove that the set H^{0}(P) consists of compact objects in H(P). However this follows directly from the isomorphism in (i)(a) and the fact that P consists of compact objects in C.

We would like to know when the set H^{0}(P) consists of compact projective generators in H(P). The following result gives a sufficient condition.

Lemma 3.2. H^{0}(P) is a set of projectives in H(P) iff LΣ^{−1}(P) ⊆ ^{⊥}H(P). If this is the case then H^{0}(P) is a set of compact projective generators in H(P).

Proof. If H^{0}(P) is projective and h : H^{0}(P) ⟶ Σ(A) is a morphism with A ∈ H(P), then it is easy to see that for any triangle A ⟶^{f} B ⟶^{g} H^{0}(P) ⟶^{h} Σ(A), the sequence A ⟶^{f} B ⟶^{g} H^{0}(P) is short exact in H(P). Since H^{0}(P) is projective, g splits and then h = 0. It follows that C(H^{0}(P), Σ(A)) = C(ΣLΣ^{−1}R(P), Σ(A)) = C(LΣ^{−1}(P), A) = 0, ∀P ∈ P, ∀A ∈ H(P). Hence LΣ^{−1}(P) ⊆ ^{⊥}H(P). Conversely if this holds, let A ↣ B ↠ C be an extension in H(P). Then there exists a morphism C ⟶ Σ(A) such that A ⟶ B ⟶ C ⟶ Σ(A) is a triangle in C. Since any morphism H^{0}(P) ⟶ Σ(A) is zero, any morphism H^{0}(P) ⟶ C factors through B ⟶ C and this shows that H^{0}(P) is projective in H(P).

Now if the objects H^{0}(P), with P ∈ P, are projective in H(P), the functors H(P)(H^{0}(P), −), P ∈ P, are faithful if they reflect zero objects, and this happens by Lemma 3.1. Hence H^{0}(P) consists of compact projective generators in H(P).

We have the following connection between the generating set P of the torsion theory (X_{P}, Y_{P}) and the full subcategory H^{0}(P) ⊆ H(P), which will be useful for a nice description of the heart.

Lemma 3.3. If C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, then we have the following.

(i) For any T ∈ X_{P} we have an isomorphism: C(P, T) ⟶^{≅} C(P, H^{0}(T)).

(ii) The composite functor P ↪ C ⟶^{H^{0}} H(P) is fully faithful. Hence it induces an equivalence H^{0}|_{P} : P ⟶^{≈} H^{0}(P).

Proof. (i) Consider the glueing triangle for the object Σ^{−1}(T):

RΣ^{−1}(T) ⟶^{f_{Σ^{−1}(T)}} Σ^{−1}(T) ⟶^{g^{Σ^{−1}(T)}} LΣ^{−1}(T) ⟶^{h_{Σ^{−1}(T)}} ΣRΣ^{−1}(T).

Since H^{0}(T) = ΣLΣ^{−1}(T), we have the following triangle in C:

ΣRΣ^{−1}(T) ⟶^{−Σ(f_{Σ^{−1}(T)})} T ⟶^{−Σ(g^{Σ^{−1}(T)})} H^{0}(T) ⟶^{−Σ(h_{Σ^{−1}(T)})} Σ^{2}RΣ^{−1}(T)

Setting μ_{T} := −Σ(g^{Σ^{−1}(T)}) : T ⟶ H^{0}(T), it is not difficult to see that μ : Id_{X_{P}} ⟶ H^{0}i = H^{0}|_{X_{P}} is a natural morphism. By Proposition 2.8, the hypothesis implies that C(P, ΣRΣ^{−1}(T)) = 0 = C(P, Σ^{2}RΣ^{−1}(T)). It follows that C(P, μ_{T}) : C(P, T) ⟶ C(P, H^{0}(T)) is invertible, for any T ∈ X_{P}.

(ii) Let α : P ⟶ Q be a morphism in P. Then we have a morphism of triangles:

ΣRΣ^{−1}(P) ⟶^{−Σ(h_{Σ^{−1}(P)})} P ⟶^{μ_{P}} H^{0}(P) ⟶^{−Σ(h_{Σ^{−1}(P)})} Σ^{2}RΣ^{−1}(P)
ΣRΣ^{−1}(α)↓  ^{α}↓  ^{H^{0}(α)}↓  ^{Σ^{2}RΣ^{−1}(α)}↓
ΣRΣ^{−1}(Q) ⟶^{−Σ(h_{Σ^{−1}(Q)})} Q ⟶^{μ_{Q}} H^{0}(Q) ⟶^{−Σ(h_{Σ^{−1}(Q)})} Σ^{2}RΣ^{−1}(Q)

If H^{0}(α) = 0, then α factors through −Σ(h_{Σ^{−1}(Q)}). Since by Proposition 2.8 we have C(P, ΣRΣ^{−1}(Q)) = 0, it follows that α = 0. We infer that H^{0}|_{P} is faithful. Now let β : H^{0}(P) ⟶ H^{0}(Q) be a morphism, where P and Q lie in P. Since by Proposition 2.8 we have C(P, Σ^{2}RΣ^{−1}(Q))= 0, the composition μ_{P} ∘ β ∘ (−Σ(h_{Σ^{−1}(Q)})) = 0. Hence there exists a morphism α : P ⟶ Q such that α ∘ μ_{Q} = μ_{P} ∘ β. Then μ_{P} ∘ H^{0}(α) = μ_{P} ∘ β, hence H^{0}(α) − β factors through −Σ(h_{Σ^{−1}(P)}). Since C(Σ^{2}RΣ^{−1}(P), H^{0}(Q)) = C(Σ^{2}RΣ^{−1}(P), ΣLΣ^{−1}(Q)) = C(ΣRΣ^{−1}(P), LΣ^{−1}(Q)) = 0, we infer that H^{0}(α) = β. Hence H^{0}|_{P} is full.

Consider the restriction functor

R : C ⟶ Mod(P), R(C) := C(−, C)|_{P}

and let R^{∗} be the restriction of R to the heart H(P):

R^{∗} : H(P) ⟶ Mod(P), R^{∗}(H) := C(−, H)|_{P}

Theorem 3.4. If C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, then the restriction functor

R^{∗} : H(P) ⟶^{≈} Mod(P)

is an equivalence of categories.

Proof. By Lemma 3.1 the heart H(P) is a cocomplete abelian category. We show that the set H^{0}(P) consists of projective objects of the heart. Let P ∈ P, and consider the triangle RΣ^{−1}(P) ⟶ Σ^{−1}(P) ⟶ LΣ^{−1}(P) ⟶ ΣRΣ^{−1}(P)). Let α : LΣ^{−1}(P) ⟶ A be a morphism with A ∈ H(P). Since A ∈ X_{P}, by Proposition 2.8 we have C(Σ^{−1}(P), A) = 0. Hence α factors through LΣ^{−1}(P) ⟶ ΣRΣ^{−1}(P)) via a morphism β ∈ C(ΣRΣ^{−1}(P), A) = C(RΣ^{−1}(P), Σ^{−1}(A)). However the last group is zero since Σ^{−1}(A) ∈ Y_{P}. It follows that C(LΣ^{−1}(P), A) = 0, ∀A ∈ H(P). Then by Lemma 3.2 we infer that H^{0}(P) is projective in H(P). It follows that H^{0}(P) is a set of compact projective generators in H(P). Then by a classical result of Gabriel-Freyd, the restriction functor S : H(P) ⟶ Mod(H^{0}(P)) defined by S(H) = H(P)(−, H)|_{H^{0}(P)}, is an equivalence. By Lemma 3.3 we have an equivalence H^{0}|_{P} : P ⟶^{≈} H^{0}(P), and using this equivalence it is easy to see that the functor S is isomorphic to the functor R^{∗}. We conclude that R^{∗} is an equivalence.

We can get a more pleasant description of the projective objects of the heart H(P) and the equivalence of the Theorem 3.4, if the generating set P of the torsion theory (X_{P}, Y_{P}) lies in the heart. We need the following result whose easy proof is left to the reader.

Lemma 3.5. P ⊆ H(P) iff C(P, Σ^{n}(P)) = 0, ∀n < 0.

Combining Theorem 3.4 and Lemma 3.5, we have the following.

Theorem 3.6. If C(P, Σ^{n}(P)) = 0, ∀n < 0, then the restriction functor

S : H(P) ⟶^{≈} Mod(P), C ⟼ S(C) := C(−, C)|_{P}

is an equivalence of categories iff C(P, Σ^{n}(P)) = 0, ∀n > 0. If this is the case, then Add(P) is the full subcategory of projective objects of H(P).

We recall that, in the language of t-structures, a torsion theory (X , Y) in C is called non-degenerate, if ⋂_{n∈ℤ} Σ^{n}(X) = 0 = ⋂_{n∈ℤ} Σ^{n}(Y) [15]. It is well-known that the non-degenerate property is an important finiteness condition, for instance in this case the torsion and the torsion-free class can be described in terms of the vanishing of the homological functors H^{n} : C ⟶ H where H = X ∩ Σ(Y) is the heart; in addition the functors {H^{n} | n ∈ ℤ} reflect collectively isomorphisms [15].

In this respect the following result provides useful information.

Corollary 3.7. If C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, then we have an inclusion H(P) ⊆ {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≠ 0}, which is an equality iff P generates C. Moreover the following conditions are equivalent:

(i) P generates C and C(P, Σ^{n}(P)) = 0, ∀n ≥ 1.

(ii) (X_{P}, Y_{P}) is non-degenerate and C(P, Σ^{n}(P)) = 0, ∀n ≥ 1.

(iii) H(P) = {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≠ 0}.

Proof. For simplicity we set X_{P}^{≤0} := {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≥ 1}. Assume that C(P, Σ^{n}(P)) = 0, ∀n ≥ 1. Then by Proposition 2.8 we have X_{P} ⊆ X_{P}^{≤0}. Hence H(P) = X_{P} ∩ Σ(Y_{P}) ⊆ X_{P}^{≤0} ∩ Σ(Y_{P}) = {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≠ 0}. If the last inclusion is an equality, then let C be in X_{P}^{≤0}. Consider the triangle holim_{⟶}T_{n} ⟶^{χ_{C}} C ⟶^{g} Ĉ ⟶^{h} Σ(holim_{⟶}T_{n}) constructed in Proposition 2.6. Then

holim_{⟶}T_{n} lies in X_{P} and Ĉ lies in Σ(Y_{P}). It follows that Ĉ lies in X_{P}^{≤0} ∩ Σ(Y_{P}) =

H(P). In particular Ĉ lies in X_{P}. Since X_{P} is closed under extensions, we have that C lies in X_{P}. Hence X_{P} = X_{P}^{≤0} and then by Proposition 2.8 we infer that P generates C. If P generates C, then the equality H(P) = {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ≠ 0} follows from Proposition 2.8. The equivalence (i) ⇔ (iii) follows from Proposition 2.8 and the above arguments. Finally the equivalence (i) ⇔ (ii) follows from the fact that ⋂_{n∈ℤ} Σ^{n}(Y_{P}) = {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ∈ ℤ}.

Remark 3.8. We call a torsion theory (X , Y) in C left, resp. right, nondegenerate if ⋂_{n∈ℤ} Σ^{n}(X) = 0, resp. ⋂_{n∈ℤ} Σ^{n}(Y) = 0. Then (X , Y) is nondegenerate iff it is left and right non-degenerate. If the torsion theory is generated by a set of compact objects P, then since ⋂_{n∈ℤ} Σ^{n}(Y_{P}) = {C ∈ C | C(P, Σ^{n}(C)) = 0, ∀n ∈ ℤ}, it follows that (X_{P}, Y_{P}) is right non-degenerate iff P generates C. If C(P, Σ^{n}(P)) = 0, ∀n ≥ 1, then it is easy to see that ⋂_{n∈ℤ} Σ^{n}(X_{P}) = ⋂_{n∈ℤ} Σ^{n}(Y_{P}). Hence in this case we have that (X_{P}, Y_{P}) is right non-degenerate iff (X_{P}, Y_{P}) is left non-degenerate iff (X_{P}, Y_{P}) is non-degenerate iff P generates C.

Example. Let Ho(Sp) be the stable homotopy category of spectra and let (X_{S^{0}}, Y_{S^{0}}) be the torsion theory in Ho(Sp) compactly generated by the sphere spectrum S^{0}. Since S^{0} generates Ho(Sp), by Corollary 3.7 the torsion theory (X_{S^{0}}, Y_{S^{0}}) is non-degenerate and its heart consists of all spectra C with stable homotopy groups π_{n}(C) = 0, ∀n ≠ 0. Since the endomorphism ring of S^{0} is the ring ℤ of integers, by Theorem 3.4 it follows that the heart is equivalent to the category Ab of abelian groups.

Example. Let (T , F) be a torsion theory in an abelian category A with exact coproducts. Then the induced torsion theory (X (T ), Y(F)) in D(A) is nondegenerate. It is interesting to know under what conditions the torsion theory (X (T ), Y(F)) is compactly generated.

It is useful to know the structure of projective/injective objects of the heart as well as the relationship between the extension functors Ext_{H(P)}^{∗} of H(P) and the graded Hom functors Hom_{C}^{∗} of C which measures the possibility that C can be recovered from the heart. For a description of the structure of the injective objects of H(P) we need the following construction.

If P generates C then by Brown’s representability Theorem [63], for any P ∈ P there exists an object D_{ℚ/ℤ}(P) in C, unique up to isomorphism, and a natural isomorphism of functors

ω : [C(P, −), ℚ/ℤ] ⟶^{≅} C(−, D_{ℚ/ℤ}(P)).

The object D_{ℚ/ℤ}(P) is called the dual object of the compact object P.

Proposition 3.9. Assume that P generates C and C(P, Σ^{n}(P)) = 0, ∀n ≠ 0. The we have the following.

(α) P is a set of projective generators of H(P).

(β) {D_{ℚ/ℤ}(P) | P ∈ P} is a set of cogenerators in C and a set of injective cogen-

erators in H(P). (γ) There exists a natural isomorphism:

Ext_{H(P)}^{n}(−, −) ⟶^{≅} C(−, Σ^{n}(−)) : H(P)^{op} × H(P) ⟶ Ab, ∀n ∈ ℤ.

Proof. (α) By Theorem 3.6 have Proj(H(P)) = Add(P). Hence P is a generating set of projectives in H(P).

(β) If C(Σ^{n}(C), D_{ℚ/ℤ}(P)) = 0, ∀P ∈ P, ∀n ∈ ℤ, then using the natural morphism ω we infer that [C(P, Σ^{n}(C)), ℚ/ℤ] = 0, ∀P ∈ P, ∀n ∈ ℤ. This implies that C = 0 since P generates C and ℚ/ℤ cogenerates Ab. Now C(Σ^{n}(P), D_{ℚ/ℤ}(P)) ≅ [C(P, Σ^{n}(P)), ℚ/ℤ] = 0, ∀n ≠ 0 and ∀P ∈ P. Hence the set D_{ℚ/ℤ}(P) lies in H(P). If A ↣ B ↠ C is an extension in H(P), then there exists a triangle A ⟶ B ⟶ C ⟶ Σ(A) in C. Applying the functor C(−, D_{ℚ/ℤ}(P)) to the extension and using that C(Σ^{−1}(C), D_{ℚ/ℤ}(P)) = [C(P, Σ^{−1}(C)), ℚ/ℤ] = 0, we infer that D_{ℚ/ℤ}(P) is a set of injective objects which obviously cogenerates H(P).

(γ) The sequence F_{n} := C(Σ^{−n}(−), C) : H(P)^{op} ⟶ Ab is obviously an exact connected sequence of functors and F_{0} = Ext_{H(P)}^{0}(−, C) = C(−, C). By a wellknown characterization of Ext^{∗} in an abelian category it suffices to show that F_{n}(Q) = 0, ∀Q ∈ Proj(H(P)), ∀n ≥ 1. Indeed using that Proj(H(P)) = Add(P) it follows that for any projective Q we have C(Σ^{−n}(Q), C) = 0, since Q ∈ Add(P) and C ∈ H(P). Interchanging projectives with injectives, G_{n} := C(A, Σ^{n}(−)) :

H(P) ⟶ Ab constitute an exact connected sequence of functors which kill injectives and G_{0} = Ext_{H(P)}^{0}(A, −) = C(A, −). Hence G_{n} ≅ Ext_{H(P)}^{n}(A, −).

4. Torsion Theories Induced by Tilting Objects

We know from the results recalled in section 3 of Chapter I that if T is a tilting module over an Artin algebra of projective dimension at most one, then the category mod(Γ), where Γ = End_{Λ}(T), can be described as the heart of a torsion theory in D^{b}(mod(Λ)) constructed from T. In this section we use the results of the previous section to show that a similar result holds when we more generally start with a tilting module of finite projective dimension. We also give a new proof of a theorem of Rickard. Throughout this section we assume that the triangulated category C has all small coproducts.

In analogy with Rickard’s definition of a tilting complex in the derived category of a ring, see [69], we make the following definition.

Definition 4.1. An object T ∈ C is called a tilting object if:

(i) T is compact.

(ii) C(T, Σ^{n}(T)) = 0, ∀n ≠ 0.

(iii) {T } generates C. If conditions (i) and (ii) hold, then we call T a partial tilting object.

Example. Let A be an abelian category with exact coproducts. If T^{•} is a bounded complex in A with components compact projective objects, and if D(A)(T^{•}, Σ^{n}(T^{•})) = 0, ∀n ≠ 0, then T^{•} is a partial tilting object in D(A). T^{•} is a tilting complex, if in addition T^{•} is a generator of D(A).

We have the following direct consequence of the results of the previous section.

Corollary 4.2. Let T be a partial tilting object in C. Then T is a tilting object iff the torsion theory (X_{T} , Y_{T}) in C generated by T is non-degenerate iff the heart H(T) = X_{T} ∩ Σ(Y_{T}) is equal to {C ∈ C | C(T, Σ^{n}(C)) = 0, ∀n ≠ 0}. If this is the case, then T is a compact projective generator of the heart H(T) and the functor C(T, −) : H(T) ⟶ Mod(End_{C}(T)) is an equivalence of categories.

If C is compactly generated then by duality we can define an object E ∈ C to be a cotilting object, if: (1) E is pure-injective in C in the sense of [18], (2) C(Σ^{n}(E), E) = 0, ∀n ≠ 0 and (3) {E} cogenerates C, i.e. C(Σ^{n}(C), E) = 0, ∀n ∈ ℤ ⇒ C = 0. By [18] any object of the form D_{ℚ/ℤ}(T) is pure-injective, for any compact object T. Then under the assumptions of the above corollary we have that D_{ℚ/ℤ}(T) is a cotilting object in C and an injective cogenerator of H(T) = Mod(End_{C}(T)).

The following result shows that a tilting object T in the unbounded derived category of a Grothendieck category G induces a triangle equivalence between its bounded derived category and the bounded derived category of the heart of the torsion theory generated by the tilting object T. This presents a torsion-theoretic approach via the heart to the construction of derived equivalences.

Proposition 4.3. Let A be an abelian category with exact coproducts, e.g. a Grothendieck category. If T^{•} is a tilting object in D(A), then there exists a triangle equivalence

D^{b}(A) ⟶^{≈} D^{b}(Mod(End(T^{•}))).

Proof. Since D(A) admits a tilting object, it is compactly generated, and the heart of the torsion theory generated by T^{•} is equivalent to Mod(End(T^{•})). Since the latter has enough projective and injective objects, by [41], [15], there exists an exact functor G : D^{b}(Mod(End(T^{•}))) ⟶ D^{b}(A) extending the identity functor of Mod(End(T^{•}))). By the argument of Theorem 3.3 in [41], G is an equivalence if for any pair of Mod(End(T^{•}))−modules B, B′, the canonical morphism

Ext_{End(T^{•})}^{n}[B, B′] ⟶^{≅} D^{b}(Mod(End(T)))[B, Σ^{n}(B′] ⟶ D^{b}(A)[B, Σ^{n}(B′)]

induced by G is bijective, ∀n ∈ ℤ. However this holds, by Proposition 3.7(γ).

It is well-known that the full subcategory of compact objects in the unbounded derived category D(Mod(Λ)) of a ring Λ coincides, up to equivalence, with the bounded homotopy category H^{b}(P_{Λ}) of the category of finitely generated projective modules. In this case the tilting objects of D(Mod(Λ)) are precisely the tilting complexes of Rickard [69].

The following consequence of Proposition 4.3 gives a torsion theoretic proof of a part of Rickard’s Theorem on the construction of derived equivalences [69].

Corollary 4.4. [69] Let T^{•} ∈ D(Mod(Λ)) be a tilting complex with endomorphism ring Γ := End(T^{•}). Then there exists a triangle equivalence:

D^{b}(Mod(Λ)) ⟶^{≈} D^{b}(Mod(Γ)).

When T is a tilting Λ-module with pd_{Λ}T ≤ 1, we have an associated torsion theory (T , F) in mod(Λ), and when having a torsion theory in mod(Λ) we have an associated torsion theory (X , Y) in D^{b}(mod(Λ)), whose heart is mod(Γ) for Γ = End_{Λ}(T). For a tilting module T of finite projective dimension there is not in general a natural associated torsion theory in mod(Λ) which plays a role in tilting theory. But we can pass directly to D^{b}(mod(Λ)) and consider the pair (X̃_{T} , Ỹ_{T}) in D^{b}(mod(Λ)) generated by T as defined in Section 2 of this chapter, that is: Ỹ_{T} = {C^{•} ∈ D^{b}(mod(Λ)) | (T, Σ^{n}(C^{•})) = 0, ∀n ≤ 0} and X̃_{T} = ^{⊥}Ỹ_{T} where the left orthogonal is formed in D^{b}(mod(Λ)). Note that it is easy to see that when pd_{Λ}T ≤ 1, then T generates the torsion theory (X , Y) in this sense. We want to show that (X̃_{T} , Ỹ_{T}) is a torsion theory also for pd_{Λ}T < ∞ and that mod(Γ) is equivalent to its heart.

In view of the results in Section 3 it is hence an important problem to have conditions ensuring that a given compactly generated torsion theory in the unbounded derived category restricts to a torsion theory in the bounded derived category. Here we deal only with the case that the torsion theory is induced by a (partial) tilting module. So let Λ be a ring (not necessarily an Artin algebra). Recall that a right Λ-module T is called a partial tilting module if Ext_{Λ}^{n}(T, T) = 0, ∀n ≥ 1, and T has a finite exact resolution by finitely generated projective modules.

Let T be a partial tilting right Λ-module, and let Γ = End_{Λ}(T) be its endomorphism ring. We view the module T as a stalk complex concentrated in degree zero in D(Mod(Λ)). Then it is easy to see that T is a compact object in D(Mod(Λ)) with endomorphism ring Γ. It is well-known that we have an adjoint pair ( − ⊗_{Γ}^{𝕃}T, ℝHom_{Λ}(T, −)) of exact functors:

ℝHom_{Λ}(T, −) : D(Mod(Λ)) ⇆ D(Mod(Γ)) : − ⊗_{Γ}^{𝕃} T

Let (X_{T} , Y_{T}) be the torsion theory in D(Mod(Λ)) generated by T. Then

X_{T} ⊆ X_{T}^{≤0} := {C^{•} ∈ D(Mod(Λ)) | (T, Σ^{n}(C^{•})) = 0, ∀n ≥ 1}
Y_{T} = {C^{•} ∈ D(Mod(Λ)) | (T, Σ^{n}(C^{•})) = 0, ∀n ≤ 0}.

By Theorem 3.5 we have that the heart H(T) is equivalent to Mod(Γ). We consider also the natural torsion theory (Z_{Γ}, W_{Γ}) in D(Mod(Γ)) generated by Γ, i.e. Z_{Γ} = {C^{•} ∈ D(Mod(Γ)) | H^{n}(C^{•}) = 0, ∀n > 0} and W_{Γ} = {C^{•} ∈ D(Mod(Γ)) | H^{n}(C^{•}) = 0, ∀n ≤ 0}, with heart Mod(Γ).

To proceed further we need to recall from [15] the concept of t-exactness of exact functors between triangulated categories each endowed with a t-structure. Let F : C ⟶ D be an exact functor between the triangulated categories C, D. We assume that (X , Y) is a torsion theory in C and (Z, W) is a torsion theory in D. Then F is called left t−exact, resp. right t−exact, if F (Y) ⊆ W, resp. F (X) ⊆ Z. Finally the functor F is called t−exact if F is left and right t-exact.

We have the following result.

Proposition 4.5. Let T be a partial tilting Λ-module with endomorphism ring Γ. Then we have the following.

(1) The functor ℝHom_{Λ}(T, −) : D(Mod(Λ)) ⟶ D(Mod(Γ)) is t-exact.

(2) The functor − ⊗_{Γ}^{𝕃} T : D(Mod(Γ)) ⟶ D(Mod(Λ)) is fully faithful t-exact.

Proof. (1) Let X^{•} ∈ X_{T}. By the above observations we have (T, Σ^{n}(X^{•})) = 0, ∀n ≥ 1. Then H^{n}ℝHom_{Λ}(T, X^{•}) = (T, Σ^{n}(X^{•})) = 0 for n ≥ 1. It follows that ℝHom_{Λ}(T, X^{•}) lies in Z_{Γ}, hence ℝHom_{Λ}(T, −) is right t-exact. Now let Y^{•} ∈ Y_{T}, so (T, Σ^{n}(Y^{•})) = 0 for n ≤ 0. Then H^{n}ℝHom_{Λ}(T, Y^{•}) = (T, Σ^{n}(Y^{•})) = 0 for n ≤ 0, hence ℝHom_{Λ}(T, Y^{•}) lies in W_{Γ}, so ℝHom_{Λ}(T, −) is left t-exact. We conclude that ℝHom_{Λ}(T, −) is t-exact.

(2) Let Z^{•} be a complex in Z_{Γ}. By adjointness, for any complex Y^{•} in Y_{T} we have an isomorphism (Z^{•} ⊗_{Γ}^{𝕃} T, Y^{•}) ⟶^{≅} (Z^{•}, ℝHom_{Λ}(T, Y^{•})). Since Y^{•} lies in Y_{T}, by (1) we have ℝHom_{Λ}(T, C^{•}) ∈ W_{Γ}. Since Z^{•} lies in Z_{Γ} and the pair (Z_{Γ}, W_{Γ}) is a torsion theory in D(Mod(Γ)), we infer that (Z^{•} ⊗_{Γ}^{𝕃} T, Y^{•}) = 0, ∀Y^{•} ∈ Y_{T}. Hence Z^{•} ⊗_{Γ}^{𝕃} T lies in X_{T}, for any Z^{•} in Z_{Γ}. In other words − ⊗_{Γ}^{𝕃} T is right t-exact. Next we show that − ⊗_{Γ}^{𝕃} T is fully faithful. First observe that − ⊗_{Γ}^{𝕃} T preserves coproducts and sends Γ to T. Since Ext_{Λ}^{n}(T, T) = 0, ∀n ≠ 0, and Hom_{Λ}(T, T) = Γ, the functor −⊗_{Γ}^{𝕃} T induces bijections (Γ, Σ^{n}(Γ)) ⟶^{≅} (T, Σ^{n}(T)). Then by devissage [52] we have that − ⊗_{Γ}^{𝕃} T is fully faithful.

Now let W^{•} be a complex in W_{Γ}, and consider the standard triangle X_{W^{•}⊗_{Γ}^{𝕃}T} ⟶

W^{•} ⊗_{Γ}^{𝕃} T ⟶ Y^{W^{•}⊗_{Γ}^{𝕃}T} ⟶ Σ(X_{W^{•}⊗_{Γ}^{𝕃}T}) in D(Mod(Λ)) associated to the torsion theory (X_{T} , Y_{T}). Applying the exact functor ℝHom_{Λ}(T, −) to this triangle, we have a triangle ℝHom_{Λ}(T, X_{W^{•}⊗_{Γ}^{𝕃}T}) ⟶ ℝHom_{Λ}(T, W^{•} ⊗_{Γ}^{𝕃} T) ⟶ ℝHom_{Λ}(T, Y^{W^{•}⊗_{Γ}^{𝕃}T}) ⟶ ΣℝHom_{Λ}(T, X_{W^{•}⊗_{Γ}^{𝕃}T}) in D(Mod(Γ)). Since the functor ℝHom_{Λ}(T, −) is t-exact,

we have that ℝHom_{Λ}(T, Y^{W^{•}⊗_{Γ}^{𝕃}T}) ∈ W_{Γ} and ℝHom_{Λ}(T, X_{W^{•}⊗_{Γ}^{𝕃}T}) ∈ Z_{Γ}. Since the functor −⊗_{Γ}^{𝕃}T is fully faithful, the canonical morphism W^{•} ⟶ ℝHom_{Λ}(T, W^{•}⊗_{Γ}^{𝕃}T) is invertible. Since W^{•} lies in W_{Γ} and the latter is left triangulated, it follows that ℝHom_{Λ}(T, X_{W^{•}⊗_{Γ}^{𝕃}T}) lies in Z_{Γ} ∩ W_{Γ} = 0. Then H^{n}ℝHom_{Λ}(T, X_{W^{•}⊗_{Γ}^{𝕃}T}) = (T, Σ^{n}(X_{W^{•}⊗_{Γ}^{𝕃}T})) = 0, ∀n ∈ ℤ. Hence X_{W^{•}⊗_{Γ}^{𝕃}T} lies in X_{T} ∩ Y_{T} = 0. It follows that

W^{•} ⊗_{Γ}^{𝕃} T ⟶^{≅} Y^{W^{•}⊗_{Γ}^{𝕃}T} lies in Y_{T}, hence − ⊗_{Γ}^{𝕃} T is left t-exact.

Corollary 4.6. T is a tilting module iff X_{T} = X_{T}^{≤0}. If T is a tilting module, then the t-exact quasi-inverse triangle equivalences

ℝHom_{Λ}(T, −) : D(Mod(Λ)) ⇆ D(Mod(Γ)) : − ⊗_{Γ}^{𝕃} T

restrict to quasi-inverse equivalences:

X_{T} ⟶^{≈} Z_{Γ} and Y_{T} ⟶^{≈} W_{Γ}.

We assume now that Λ is right coherent and in addition the endomorphism ring Γ = End_{Λ}(T) of T is right coherent. For simplicity we set F = − ⊗_{Γ}^{𝕃} T, G = ℝHom_{Λ}(T, −). Consider the full subcategories:

X_{T}^{b} = X_{T} ∩ D^{b}(mod(Λ)) and Y_{T}^{b} = Y_{T} ∩ D^{b}(mod(Λ)).

Proposition 4.7. If T is a tilting module, then (X_{T}^{b} , X_{T}^{b}) is a torsion theory in D^{b}(mod(Λ)) with heart equivalent to the abelian category mod(Γ).

Proof. By [69] the triangle equivalences G : D(Mod(Λ)) ⇆ D(Mod(Γ)) : F restrict to triangle equivalences G : D^{b}(mod(Λ)) ⇆ D^{b}(mod(Γ)) : F. By [15] it suffices to show that the coreflection functor R : D(Mod(Λ)) ⟶ X_{T} satisfies R(D^{b}(mod(Λ))) ⊆ D^{b}(mod(Λ)). Let C^{•} ∈ D^{b}(mod(Λ)). Then G(C^{•}) lies in D^{b}(mod(Γ)). Obviously the coreflection (= truncation) R′ : D(Mod(Γ)) ⟶ Z_{Γ} satisfies R′(D^{b}(mod(Γ))) ⊆ D^{b}(mod(Γ)). Hence we have the standard triangle R′G(C^{•}) ⟶ G(C^{•}) ⟶ L′G(C^{•}) ⟶ ΣR′G(C^{•}) in D^{b}(mod(Γ)). Applying F, we have a triangle F R′G(C^{•}) ⟶ C^{•} ⟶ F L′G(C^{•}) ⟶ ΣF R′G(C^{•}) in D^{b}(mod(Λ)), where F R′G(C^{•}) ∈ X_{T} and F L′G(C^{•}) ∈ Y_{T}. Hence the triangle is the standard one associated to the torsion theory (X_{T} , Y_{T}), in particular F R′G(C^{•}) ≅ R(C^{•}). Hence R(C^{•}) ∈ D^{b}(mod(Λ)) and the assertion follows.

Since we obviously have Y_{T}^{b} = Ỹ_{T} and by Proposition 4.7 and Proposition 2.8 we have that (X_{T}^{b} , Y_{T}^{b}) is a torsion theory in D^{b}(mod(Λ)) with X̃_{T} = ^{⊥}Ỹ_{T} = X_{T}^{b}, we infer the following consequence.

Corollary 4.8. Let Λ be a right coherent ring and T a tilting Λ-module with endomorphism ring Γ. Then T generates a torsion theory (X̃_{T} , Ỹ_{T}) in D^{b}(mod(Λ)) with heart mod(Γ).

CHAPTER IV Hereditary Torsion Theories in Triangulated Categories

In this chapter we investigate hereditary torsion theories, usually generated by compact objects, in a triangulated category with all small coproducts. We show that compactly generated hereditary torsion theories provide the proper setting for the Morita theory in derived and stable categories, and we give applications to tilting theory by presenting torsion theoretic proofs of some important results of Rickard, Keller and Happel which are central in the Morita theory of derived categories. We apply our results to the study of homological conjectures in the representation theory of Artin algebras by giving alternative formulations inside the unbounded derived category.

1. Hereditary Torsion Theories

In this section we study hereditary torsion theories generated by a set of compact objects, we give necessary conditions for the existence of TTF-theories, and we show that compactly generated hereditary torsion theories provide the proper setting for various important constructions in derived categories. Finally we study the question of when a hereditary torsion theory induces a torsion theory on the full subcategory of compact objects.

Throughout this section we fix a triangulated category C and we assume that C contains all small coproducts. We recall from Section 2 of Chapter I that a torsion theory (X , Y) in C is hereditary iff the torsion class X, or equivalently the torsion-free class Y, is triangulated.

We begin with the following basic result which characterizes the hereditary torsion theories of finite type and gives sufficient conditions for the existence of TTF-theories.

Proposition 1.1. Let (X , Y) be a hereditary torsion theory in C.

(i) If C is compactly generated, then the following are equivalent: (a) (X , Y) is of finite type.

(b) There exists a TTF-theory (X , Y, Z) in C. If one the above conditions holds, then Y is compactly generated.

(ii) If Y is compactly generated and (X , Y) is of finite type, then there exists a TTF-theory (X , Y, Z) in C.

In any of the above cases, there are triangle equivalences ^{⊥}Y = X ⟶^{≈} Z = Y^{⊥}.

Proof. (i) Assume that C is compactly generated. The implication (b) ⇒ (a) is trivial. (a) ⇒ (b) Since (X , Y) is of finite type, Y is closed under coproducts and by Lemma III.1.2 the reflection functor L : C ⟶ Y preserves compact objects. Let T be a generating set of compact objects in C. If Y ∈ Y is such that Y(L(T), Y) = 0 then C(T, Y) = 0, ∀T ∈ T , hence Y = 0. It follows that Y is compactly generated by the set L(T ). Since Y is compactly generated and the inclusion j : Y ↪ C is exact and preserves coproducts, by [63], j admits a right adjoint. By the results of Section I.2, we infer that Y is a TTF-class with TTF-theory (X , Y, Z), where Z = Y^{⊥}.

(ii) The same arguments as above show that if Y is compactly generated and closed under coproducts in C, then there exists a TTF-theory (X , Y, Z) in C even if the latter is not compactly generated.

The last assertion follows from Corollary I.2.9.

Example. Let (𝔗, O) be a ringed space and let U be an open subspace of the space 𝔗 with closed complement V. Then let D(𝔗, O), D(U, O), D(V, O) be the derived categories of sheaves of O−modules over 𝔗, U and V. Then D(V, O) is a TTF-class in D(𝔗, O).

Remark 1.2. It is not difficult to see that if (X , Y) is a torsion theory in C and X is generated by the set T and Y is generated by the set S, then C is generated by the set T ∪ S. Indeed if C is in C and C(T , C) = 0, then by adjointness we have X (T , R(C)) = 0 and then R(C) = 0. Hence C ∈ Y. Now if also C(S, C) = 0, then since C ∈ Y we have C = 0. It follows that C generated by the set T ∪ S.

Let P be a set of compact objects in C and set R := {Σ^{n}(P) | n ∈ ℤ, P ∈ P}. Then as shown in Section III.2, by setting Y_{P} = R^{⊥} and X_{P} = ^{⊥}Y_{P}, we obtain a torsion theory (X_{P}, Y_{P}) of finite type in C which obviously is hereditary. We call (X_{P}, Y_{P}) the hereditary torsion theory generated by P. Since it will be clear from the context, we use the same notation as in the non-hereditary case to denote the hereditary torsion theory generated by P. Note that if (X̂_{P}, Ŷ_{P}) is the nonhereditary torsion theory generated by P, then Y_{P} = ⋂_{n∈ℤ} Σ^{n}(Ŷ_{P}). By Remark III.2.7 the torsion class X_{P} coincides with Loc(P), the smallest thick subcategory of C which is closed under coproducts and contains P. In particular X_{P} is a compactly generated triangulated category with P as a set of compact generators.

Remark 1.3. It is not difficult to see that if P̂ is the thick subcategory of C generated by the set of compact objects P, then P̂ consists of compact objects and the hereditary torsion theory (XP̂, YP̂) generated by P̂ coincides with (X_{P}, Y_{P}).

Observe that if C is compactly generated, then by Proposition 1.1 (X_{P}, Y_{P}) is part of a TTF-theory (X_{P}, Y_{P}, Z_{P}), where Z_{P} = Y_{P}^{⊥} and moreover the TTF-class Y_{P} is compactly generated as a triangulated category. Observe also that Y_{P} = 0 iff P generates C. Hence the size of Y_{P} measures how far P is from being a compact generating set in C.

The following result shows that a partial converse of Proposition 1.1 is true if the torsion theory is generated by compact objects.

Corollary 1.4. Let (X , Y) be a hereditary torsion theory of finite type in C. If X is compactly generated by a set T and Y is compactly generated by a set of objects S which are compact in C, then C is compactly generated by the set T ∪ S.

Proof. By Lemma III.1.2, T is a set of compact objects in C. Since S ⊆ C^{b}, Remark 1.2 implies that T ∪ S is a set of compact generators for C.

Assume now that C is compactly generated and let (X , Y) be a torsion theory of finite type in C. Then by Proposition 1.1 we have a TTF-theory (X , Y, Z) in C.

Recall from Section III.3 that to any compact object P in C we can associate its dual object D_{ℚ/ℤ}(P) which is equipped with a natural isomorphism:

ω : [C(P, −), ℚ/ℤ] ⟶^{≅} C(−, D_{ℚ/ℤ}(P)).

We have the following relationship.

Proposition 1.5. (1) A compact object P lies in X iff its dual object D_{ℚ/ℤ}(P) lies in Z. In particular Y = 0 iff D_{ℚ/ℤ}(P) lies in Z for any compact object P of C.

(2) A compact object P lies in Y iff its dual object D_{ℚ/ℤ}(P) lies in Z^{⊥}. In particular X = 0 (or equivalently Z = 0) iff all compact objects of C lie in Y.

Proof. Using the natural isomorphism ω we have: P ∈ X iff C(P, Y) = 0 iff [C(P, Y), ℚ/ℤ] = 0 iff C(Y, D_{ℚ/ℤ}(P)) = 0 iff D_{ℚ/ℤ}(P) ∈ Y^{⊥} = Z. The last assertion of part (1) is trivial, and part (2) is similar.

The following consequence shows that the dual objects of a generating set of compact objects of X form a cogenerating set of Z.

Corollary 1.6. Let C be a compactly generated triangulated category and let (X_{P}, Y_{P}, Z_{P}) be the TTF-theory generated by a set P of compact objects from C. Then the set {D_{ℚ/ℤ}(P) | P ∈ P} cogenerates Z.

The following three examples illustrate the importance of hereditary torsion theories of finite type and TTF-theories in concrete situations.

A: The Realization of the Derived Category in the Homotopy Category. Let Λ be a ring and let H(Mod(Λ)) be the unbounded homotopy category of all right Λ-modules.

Let (X_{Λ}, Y_{Λ}) be the hereditary torsion theory in H(Mod(Λ)) generated by the stalk complex Λ concentrated in degree zero with stalk Λ. Hence Y_{Λ} = P^{⊥} and X_{Λ} = ^{⊥}Y_{Λ}, where P = {Σ^{n}(Λ) | n ∈ ℤ}. Obviously Λ is a compact object in H(Mod(Λ)), so (X_{Λ}, Y_{Λ}) is of finite type. Since the space H(Mod(Λ))[Σ^{n}(Λ), C^{•}] is identified with the cohomology H^{−n}(C^{•}) of the complex C^{•}, it follows that Y_{Λ} is identified with the unbounded homotopy category H_{Ac}(Mod(Λ)) of acyclic complexes and by Remark III.2.7, X_{Λ} is identified with the localizing subcategory of H(Mod(Λ)) generated by Λ, which we denote by H_{P}(Mod(Λ)). The following wellknown result shows that the unbounded derived category can be realized as a full subcategory of the homotopy category of complexes, hence it has small hom-sets, in particular D(Mod(Λ)) “exists”.

Corollary 1.7. [55], [18], [23] There exists a TTF-theory

(H_{P}(Mod(Λ)), H_{Ac}(Mod(Λ)), H^{I}(Mod(Λ))

in H(Mod(Λ)) which induces triangle equivalences:

H_{P}(Mod(Λ)) ⟵^{≈} D(Mod(Λ)) ⟶^{≈} H^{I}(Mod(Λ)).

Here H^{I}(Mod(Λ)) is the colocalizing subcategory of H(Mod(Λ)) generated by the injective Λ-modules.

Proof. By Proposition I.2.6, we have triangle equivalences D(Mod(Λ)) = H(Mod(Λ))/H_{Ac}(Mod(Λ)) ⟶^{≈} ^{⊥}H_{Ac}(Mod(Λ)) = H_{P}(Mod(Λ)) which show the first part. By [55] the inclusion H_{Ac}(Mod(Λ)) ↪ H(Mod(Λ)) admits a right adjoint and H_{Ac}(Mod(Λ))^{⊥} = H^{I}(Mod(Λ)), so we have the desired TTF-theory and in the same way as above, the right hand side equivalence.

Observe that by the above corollary the torsion class H_{P}(Mod(Λ)) and the torsion-free class H^{I}(Mod(Λ)) are compactly generated. However in general the TTF-class H_{Ac}(Mod(Λ)) is not compactly generated. Indeed if Λ = ℤ is the ring of integers, then by [64] it follows that H(Mod(ℤ)) is not generated by a set, hence it is not compactly generated. Hence by Remark 1.2 the same is true for H_{Ac}(Mod(ℤ)).

B: The Projective Case. The above construction can be performed in the unbounded homotopy category H(P_{Λ}) of the category P_{Λ} of projective right Λmodules which obviously has all small coproducts. This leads to an interesting connection with the stable module category which will be used later.

Consider the hereditary torsion theory of finite type (^{⊥}H_{Ac}(P_{Λ}), H_{Ac}(P_{Λ})) in H(P_{Λ}) generated by the compact object Λ ∈ H(P_{Λ}). Observe that the torsionfree class H_{Ac}(P_{Λ}) is the costabilization of the stable module category \underline{Mod}(Λ) modulo projectives via the exact functor H_{Ac}(P_{Λ}) ⟶ \underline{Mod}(Λ) which sends an acyclic complex of projectives P^{•} to the class of Im(d^{−1}) in the stable category [17]. This means that the above functor is the universal exact functor from a triangulated category to \underline{Mod}(Λ). In particular if Λ is quasi-Frobenius, then the above functor is a triangle equivalence H_{Ac}(P_{Λ}) ⟶^{≈} \underline{Mod}(Λ).

Corollary 1.8. If Λ is left coherent and right perfect, or if Λ has finite right global dimension, then there exists a TTF-theory in H(P_{Λ}):

(^{⊥}H_{Ac}(P_{Λ}), H_{Ac}(P_{Λ}), H_{Ac}(P_{Λ})^{⊥}).

Proof. If Λ is left coherent and right perfect, then by [47] it follows that H_{Ac}(P_{Λ}) is compactly generated. This also follows using that the stable category modulo projectives \underline{Mod}(Λ) is a compactly generated pretriangulated category [19] and H_{Ac}(P_{Λ}) is its costabilization [17]. Then Proposition 1.1 implies the existence of the desired torsion theory.

If r.gl.dimΛ < ∞, then by [76] the homotopy category H(P_{Λ}) is compactly generated. Hence the assertion follows from Proposition 1.1.

Under the assumptions of Corollary 1.8, the torsion class ^{⊥}H_{Ac}(P_{Λ}) in H(P_{Λ}) coincides with the torsion class ^{⊥}H_{Ac}(Mod(Λ)) in H(Mod(Λ)), hence by Corollary 1.7 it is triangle equivalent to the unbounded derived category D(Mod(Λ)). This follows from the fact that ^{⊥}H_{Ac}(Mod(Λ)) is contained in H(P_{Λ}), by the discussion preceding Corollary 1.7.

Remark 1.9. Let Λ be a left coherent and right perfect ring. Since H_{Ac}(P_{Λ}) and ^{⊥}H_{Ac}(P_{Λ}) are compactly generated, it follows by Remark 1.2 that the homotopy category H(P_{Λ}) of projective modules is generated by a set.

We have seen that in general H(Mod(Λ)) is not generated by a set. However H(Mod(Λ)) is generated by a set if Λ is a ring of finite representation type. Indeed, this follows from the above observations since in this case the category Mod(Λ) is equivalent to the category of all projective modules over Γ, where Γ is the Auslander ring of Λ, which is left coherent and right perfect. Recall that the Auslander ring of a representation finite ring Λ is the endomorphism ring of a finitely presented Λ-module X such that mod(Λ) = add(X). More generally one can show that H(Mod(Λ)) is generated by a set if Λ is a right pure semisimple ring.

C: Idempotent Functors. Let Λ be a QF-ring and let \underline{Mod}(Λ) be the stable category modulo projectives. It is triangulated with coproducts, and the stable category \underline{mod}(Λ) of the finitely presented modules is a skeletally small generating thick subcategory which is identified, up to equivalence, with the compact objects of \underline{Mod}(Λ). If P is a set of finitely presented modules, then the stable category \underline{P} is a set of compact objects in \underline{Mod}(Λ). Hence \underline{P} generates a torsion theory of finite type in \underline{Mod}(Λ). For instance we can take Λ = kG, for a finite group G, where k is a field. Then any thick subcategory S ⊆ \underline{mod}(Λ) generates a hereditary torsion theory (E_{S}, F_{S}) in \underline{Mod}(Λ). Hence any module C sits in a triangle E_{S}(\underline{C}) ⟶ \underline{C} ⟶ F_{S}(\underline{C}) ⟶ ΣE_{S}(\underline{C}), with E_{S}(\underline{C}) ∈ E_{S} and F_{S}(\underline{C}) ∈ F_{S}. If Λ = kG is the group algebra over a field k of a finite group G, then the functors E_{S}, F_{S} are the idempotent functors of Rickard [72], associated to the (tensor ideal) thick subcategory S.

If (X , Y) is a hereditary torsion theory of finite type in C, then it is natural to ask if (X , Y) restricts to a torsion theory in the full subcategory of compact objects of C. This question has connections with K-theory and our results will be applied in the next section to give a proof of a result of Happel without using Auslander-Reiten theory.

From now on we assume that C is compactly generated and (X , Y) is a hereditary torsion theory of finite type in C. As usual C^{b} denotes the full subcategory of compact objects of C.

We fix a generating set T of compact objects in C. By Proposition 1.1, we know that there exists a TTF-theory (X , Y, Z) in C where the TTF-class Y is compactly generated as a triangulated category by the set L(T ). Also by Lemma III.1.2 the functor R : C ⟶ X preserves coproducts. Since C is compactly generated, by [63], R admits a right adjoint h : X ⟶ C. Using that R admits the fully faithful left adjoint i, it is not difficult to see that h is fully faithful. Since Y is closed under coproducts in C, by [63] the inclusion j : Y ↪ C admits a right adjoint G : C ⟶ Y.

Lemma 1.10. (1) The essential image of the functor h : X ↪ C is the torsionfree class Z. Moreover the reflection functor C ⟶ Z is isomorphic to hR.

(2) The kernel of the functor G : C ⟶ Y is the torsion-free class Z. Moreover G is the coreflection functor for Y.

Proof. (1) Let X be in X. Then C(Y, h(X)) ≅ X (R(Y), X) = 0. Hence h(X) ∈ Y^{⊥} = Z. Hence h has image in Z. Now if C is an object in C, let A ⟶ C ⟶ hR(C) ⟶ Σ(A) be a triangle in C where C ⟶ hR(C) is the unit of the adjoint pair (R, h) evaluated at C. Applying to this triangle the functor C(X , −) we see easily that C(X , A) = 0, hence A ∈ X^{⊥} = Y. It follows that hR is the coreflection of C in Z. In particular if Z is in Z, then the morphism Z ⟶ hR(Z) is invertible and this shows that the essential image of h coincides with Z.

(2) The proof is similar.

Keeping the above notations we have the following result which gives necessary and sufficient conditions for a hereditary torsion theory in C to induce a torsion theory in the full subcategory C^{b} of compact objects.

Proposition 1.11. Let C be a compactly generated triangulated category and let (X , Y) be a hereditary torsion theory of finite type in C. Then the following conditions are equivalent.

(i) R : C ⟶ X preserves compact objects.

(ii) h : X ↪ C preserves coproducts.

(iii) Z is closed under coproducts, i.e. the torsion theory (Y, Z) is of finite type. (iv) G : C ⟶ Y preserves coproducts.

(v) Any compact object of Y remains compact in C.

(vi) (X^{b}, Y^{b}) is a hereditary torsion theory in C^{b}. (vii) The inclusion X^{b} ↪ C^{b} admits a right adjoint. If one of the above equivalent conditions holds, then the torsion theory (X , Y) is compactly generated by the set of objects R(T ), and X^{b} = X ∩ C^{b}.

Proof. Since h is a right adjoint of R, that (i) is equivalent to (ii) follows from [63]. Since Y is compactly generated by [63], we have that the inclusion j : Y ⟶ C preserves compact objects iff its right adjoint G preserves coproducts, hence (iv) is equivalent to (v). Since G is the coreflection functor for the hereditary torsion theory (Y, Z) by Lemma III.1.2, we have the equivalence of (iii) and (iv). Hence the first five conditions are equivalent. If one of these holds, then let T be a compact object of C and consider the glueing triangle R(T) ⟶ T ⟶ L(T) ⟶ ΣR(T). By (i) we know that R(T) ∈ X^{b} and by Lemma III.1.2 we know that L(T) ∈ Y^{b}. Since obviously C(X^{b}, Y^{b}) = 0, we have that (X^{b}, Y^{b}) is a hereditary torsion theory in C^{b}. If this is true, then obviously R preserves compact objects. Finally it is easy to see that (vi) is equivalent to (vii). Hence all conditions are equivalent.

Finally we claim that if R preserves compactness, then the set R(T ) generates X. Indeed, if X is in X and X (R(T ), X) = 0, then C(T , h(X)) = 0 and then h(X) = 0 since T is a generating set in C. Since h is fully faithful, we have X = 0. Hence R(T ) generates X.

If the torsion theory (X , Y) is generated by a set P of compact objects of C, then X^{b} is the thick subcategory thick(P) of C generated by P. The Neeman-Ravenel Localization Theorem, see [64], asserts that the Verdier-Quotient C^{b}/thick(P) is fully embedded in Y^{b} = (C/X)^{b} and the latter is the closure of the former under direct summands. The following consequence gives a proof of an improved version of the Neeman-Ravenel Theorem under an additional assumption.

Corollary 1.12. Let P be a set of compact objects in C and let (X_{P}, Y_{P}) be the hereditary torsion theory in C generated by P. Then R : C ⟶ X_{P} preserves compact objects iff (X_{P}^{b}, Y_{P}^{b}) is a hereditary torsion theory in C^{b} iff the inclusion thick(P) ↪ C^{b} admits a right adjoint. In this case we have a triangle equivalence C^{b}/thick(P) ⟶^{≈} Y^{b} and an isomorphism K_{0}(C^{b}) ⟶^{≅} K_{0}(thick(P)) ⊕ K_{0}(Y^{b}).

2. Hereditary Torsion Theories and Tilting

Working in a compactly generated triangulated category which is of algebraic origin [56] (for instance derived categories, stable categories) and using the techniques of differential graded algebras developed by Keller [52], we can describe in more explicit terms the torsion class of the hereditary torsion theory generated by a compact object. This description provides simple proofs and slight generalizations of the Morita Theorem for derived categories of Rickard [69], of the Morita Theorem for stable categories of Keller [52] and in particular of Happel’s Theorem [38] on the description of the derived category of an Artin algebra in terms of the stable category of its repetitive algebra. Thus using torsion-theoretic methods we obtain simple proofs and at the same time a unified approach to the above mentioned results. A similar approach was considered independently by Dwyer and Greenless in [31], although all the main ideas are due to Keller [52]. In contrast to the previous chapter we use hereditary torsion theories, so the heart is trivial. In a sense the following results are relative versions of Corollary 1.7.

Rickard’s Theorem. Let Λ be a ring and let T^{•} be a fixed compact object in the unbounded derived category D(Mod(Λ)), i.e. T^{•} is a complex quasi-isomorphic to a bounded complex with components finitely generated projective modules. Let (X_{T^{•}}, Y_{T^{•}}) be the hereditary torsion theory in D(Mod(Λ)) generated by T^{•}. Let Γ = DGHom_{Λ}(T^{•}, T^{•}) be the DG-algebra induced by the complex T^{•} and let D(Γ) be the derived category of Γ in the sense of [52]. By Proposition 1.1 we know that Y_{T^{•}} is a TTF-class, so we have a TTF-theory (X_{T^{•}}, Y_{T^{•}}, Z_{T^{•}}) in D(Mod(Λ)).

The following extends the bounded version of the Morita Theorem proved in Corollary III.4.4 via the heart, and contains the Morita Theorem for derived categories.

Theorem 2.1. [69], [71], [52]

(i) There are triangle equivalences:

D(Mod(Λ)) ↩ X_{T^{•}} ⟵^{≈} D(Γ) ⟶^{≈} Z_{T^{•}}

(ii) The inclusion D(Γ) ↪ D(Mod(Λ)) sends Γ to T^{•} and admits a right adjoint. (iii) T^{•} is a generator of D(Mod(Λ)) iff the inclusion D(Γ) ↪ D(Mod(Λ)) in-

duces a triangle equivalence:

D(Γ) ⟶^{≈} D(Mod(Λ)).

(iv) If Hom_{D(Mod(Λ))}(T^{•}, Σ^{n}(T^{•})) = 0, ∀n ≠ 0, then the DG-algebra Γ is the ordinary ring End_{D(Mod(Λ))}(T^{•}). In particular if T^{•} is a tilting complex, then we have a triangle equivalence D(Mod(Γ)) ⟶^{≈} D(Mod(Λ)) sending Γ to T^{•}.

Proof. (i) As in [52], T^{•} can be considered as Γ-−Λ-bimodule and then the total right derived functor ℝHom_{Λ}(T^{•}, −) : D(Mod(Λ)) ⟶ D(Γ) is defined and admits as a left adjoint the total left derived functor −⊗_{Γ}^{𝕃}T^{•} : D(Γ) ⟶ D(Mod(Λ)). We first show that the image of − ⊗_{Γ}^{𝕃} T^{•} lies in the torsion class X_{T^{•}}. Indeed for any DG-module M and for any torsion-free object Y^{•} ∈ Y_{T^{•}} we have:

Hom_{D(Mod(Λ))}(M ⊗_{Γ}^{𝕃} T^{•}, Y^{•}) ⟶^{≅} Hom_{D(Γ)}(M, ℝHom_{Λ}(T^{•}, Y^{•}))

which is zero by the construction of the torsion-free class Y_{T^{•}}. It follows that we can consider the exact functor −⊗_{Γ}^{𝕃} T^{•} : D(Γ) ⟶ X_{T^{•}} which obviously preserves all small coproducts and moreover we have obviously Γ ⊗_{Γ}^{𝕃} T^{•} = T^{•}. Now the torsion class X_{T^{•}} is a compactly generated triangulated category with T^{•} as a compact generator. By a well-known criterion for a coproduct preserving exact functor to be an equivalence, see Lemma 4.2 in [52], we infer that − ⊗_{Γ}^{𝕃} T^{•} : D(Γ) ⟶ X_{T^{•}} is a triangle equivalence with quasi-inverse the restriction ℝHom_{Λ}(T^{•}, −) : X_{T^{•}} ⟶ D(Γ). This completes the proof of (i) since by Corollary I.2.9, we always have a triangle equivalence X_{T^{•}} ⟶^{≈} Z_{T^{•}}.

(ii) Since X_{T^{•}} is a torsion class, the inclusion X_{T^{•}} ↪ D(Mod(Λ)) admits a right adjoint and the assertion follows from (i).

(iii) If T^{•} is a generator of D(Mod(Λ)) then by construction it follows trivially that the torsion-free class Y_{T^{•}} = 0. Hence the torsion class X_{T^{•}} coincides with the whole derived category D(Mod(Λ)) and then the assertion follows by (i). The converse is trivial, since Γ is a generator of D(Γ).

(iv) If T^{•} is a tilting object, then by (i), (ii), we have a triangle equivalence D(Γ) ⟶^{≈} D(Mod(Λ)). Since Hom_{D(Mod(Λ))}(T^{•}, Σ^{n}(T^{•})) = 0, ∀n ≠ 0, the cohomology of the DG-algebra Γ is concentrated in degree zero. This implies that Γ is (quasi-)isomorphic to End_{D(Mod(Λ))}(T^{•}) and then D(Γ) ⟶^{≈} D(Mod(Γ)).

The above proof shows that the idempotent functor iR : D(Mod(Λ)) ⟶ D(Mod(Λ)) associated to the torsion class X_{T^{•}} is given by ℝHom_{Λ}(T^{•}, −) ⊗_{Γ}^{𝕃} T^{•} : D(Mod(Λ)) ⟶ D(Mod(Λ)).

Keller’s Theorem. Now let F be an exact Frobenius category which admits all small coproducts, for instance Mod(Λ) where Λ is a quasi-Frobenius ring. Then trivially the stable category \underline{F} of F modulo projectives has all small coproducts. We assume that \underline{F} admits a set of compact generators. We fix a compact object T in \underline{F} and let (X_{T} , Y_{T}) be the hereditary torsion theory in \underline{F} generated by T. Then by Proposition 1.1 we have that Y_{T} is a TTF-class so we have a TTF-theory (X_{T} , Y_{T} , Z_{T}) in \underline{F}. By [52], the stable category \underline{F} is triangle equivalent to the homotopy category H_{Ac}(P) of acyclic complexes of projectives of F. Let T^{•} be the object in H_{Ac}(P) which corresponds to T via the above equivalence; this amounts to

choosing a Tate resolution of T, i.e. an unbounded acyclic complex ⋯ ⟶ P^{−1} ⟶^{d^{−1}}

P^{0} ⟶^{d^{0}} P^{1} ⟶ ⋯ of projectives such that Im(d^{−1}) ≅ T in \underline{F}. Let Γ be the DGalgebra DGHom(T^{•}, T^{•}) induced by the complex T^{•}. The following is essentially Keller’s Morita Theorem for stable categories.

Theorem 2.2. [52]

(i) There are triangle equivalences:

\underline{F} ↩ X_{T} ⟵^{≈} D(Γ) ⟶^{≈} Z_{T} .

(ii) The inclusion D(Γ) ↪ \underline{F} admits a right adjoint which sends T to Γ.

(iii) T is a generator of \underline{F} iff the above inclusion D(Γ) ↪ \underline{F} induces a triangle equivalence:

D(Γ) ⟶^{≈} \underline{F}.

(iv) If T is a tilting object, then the DG-algebra Γ is the ordinary ring End\underline{F}(T^{•}) and then we have a triangle equivalence \underline{F} ⟶^{≈} D(Mod(Γ)).

Proof. (i) Identifying \underline{F} with H_{Ac}(P), the object T^{•} induces an exact functor G : \underline{F} ⟶ D(Γ), given by G(P^{•}) = Hom(T^{•}, P^{•}), which preserves coproducts. Then by restriction, G induces an exact functor G : X_{T} ⟶ D(Γ) which preserves coproducts and sends the compact generator T of X_{T} to the compact generator Γ of D(Γ). Then as in the proof of Theorem 2.1, we infer that G induces a triangle equivalence G : X_{T} ⟶^{≈} D(Γ). Parts (ii), (iii), (iv) follow as in Theorem 2.1.

Remark 2.3. In case we work with a set of compact objects, then the DGalgebra Γ above can be replaced by the induced DG-category in the sense of [52].

Happel’s Theorem. For the definition of the repetitive algebra Λ̂ associated to an Artin algebra Λ we refer to [38]. In particular by [38] we have that Λ̂ is selfinjective, hence the stable category \underline{Mod}(Λ̂) is a compactly generated triangulated category and moreover it is easy to see that \underline{Mod}(Λ̂)^{b} = \underline{mod}(Λ̂). It is easy to see that Mod(Λ) is embedded in the stable category \underline{Mod}(Λ̂) and Λ is a compact object in \underline{Mod}(Λ̂). Consider the TTF-theory (X_{Λ}, Y_{Λ}, Z_{Λ}) in \underline{Mod}(Λ̂) generated by Λ.

The following consequence of Theorem 2.2 is due to Happel. Later Keller provided a simpler proof and extended the result to the big module category. We give a torsion theoretic proof of Keller’s version below.

Theorem 2.4. [38], [51] Let Λ be an Artin algebra and let Λ̂ be its repetitive algebra.

(i) There are triangle equivalences

\underline{Mod}(Λ̂) ↩ X_{Λ} ⟵^{≈} D(Mod(Λ)) ⟶^{≈} Z_{Λ}

which induce a full embedding H^{b}(P_{Λ}) ↪ \underline{mod}(Λ̂). (ii) The inclusion H : D(Mod(Λ)) ↪ \underline{Mod}(Λ̂) admits a right adjoint R : \underline{Mod}(Λ̂)

⟶ D(Mod(Λ)) which sends Λ to Λ. (iii) The inclusion H : D(Mod(Λ)) ↪ \underline{Mod}(Λ̂) induces a triangle equivalence

D(Mod(Λ)) ⟶^{≈} \underline{Mod}(Λ̂)

iff Λ generates \underline{Mod}(Λ̂) iff Λ is a tilting object in \underline{Mod}(Λ̂) iff R preserves compact objects iff the inclusion H^{b}(P_{Λ}) ↪ \underline{mod}(Λ̂) admits a right adjoint iff gl dim Λ < ∞.

Proof. All the assertions, except for the last two in part (iii), are consequences of our previous results. If Λ is a tilting object, then H and consequently R are equivalences. Then trivially R preserves compactness. Conversely if the right adjoint R of H preserves compact objects, then by Proposition 1.9 any compact object of Y_{Λ} is a compact object in \underline{Mod}(Λ̂), hence we have an inclusion Y_{Λ}^{b} ⊆ \underline{mod}(Λ̂). By the definition of Y_{Λ}, inside \underline{mod}(Λ̂) we have: Y_{Λ}^{b} ∩ mod(Λ) = 0. Since mod(Λ) generates \underline{mod}(Λ̂), we infer that Y_{Λ}^{b} = 0. Since Y_{Λ} is compactly generated, this implies that Y_{Λ} = 0, hence H and consequently R are equivalences. If this condition holds, then Λ generates \underline{mod}(Λ̂). This implies easily by devissage that Λ has finite global dimension. Conversely if Λ has finite global dimension, then obviously Λ generates \underline{mod}(Λ̂) as a thick subcategory and consequently Λ generates \underline{Mod}(Λ̂) as category with infinite coproducts. Hence the latter coincides with X_{Λ} and consequently H is an equivalence.

Remark 2.5. Happel proved that D^{b}(mod(Λ)) is triangle equivalent to \underline{mod}(Λ̂) iff Λ has finite global dimension, see [38], [39]. For the (⇒) direction he used the Auslander-Reiten theory on the existence of Auslander-Reiten sequences and triangles in the dualizing varieties mod(Λ̂) and \underline{mod}(Λ̂). The above proof avoids this.

Remark 2.6. Using Neeman-Ravenel’s Theorem [64], it follows that the Ver– dier Quotient \underline{mod}(Λ̂)/H^{b}(P_{Λ}) is fully embedded in Y^{b} and the latter is its closure under direct summands: add(\underline{mod}(Λ̂)/H^{b}(P_{Λ}))= Y^{b}. Moreover Y^{b} is the thick subcategory generated by the isoclasses of the class of objects L(\underline{mod}(Λ̂)).

It follows by the proof of Theorem 2.4 that the full exact embedding H : D(Mod(Λ)) ↪ \underline{Mod}(Λ̂) of (ii) above is Happel’s functor constructed inductively in [38]. Hence the above result shows that it admits a right adjoint and computes its image as the full subcategory X_{Λ} or Z_{Λ}. Moreover it computes the triangulated Verdier quotient \underline{Mod}(Λ̂)/D(Mod(Λ)) as the TTF-class Y_{Λ}.

p−groups. The above results have an interesting application to the stable module category of a modular group algebra.

Let G be a finite group and k a field with characteristic p > 0. Then the stable module category \underline{Mod}(kG) is compactly generated and the trivial G−module k is a compact object. Let (X_{k}, Y_{k}, Z_{k}) be the TTF-theory of finite type in \underline{Mod}(kG) generated by k. The following gives a description of the localizing subcategory of \underline{Mod}(kG) generated by the trivial G−module k.

Corollary 2.7. (1) There are triangle equivalences

\underline{Mod}(kG) ↩ X_{k} ⟵^{≈} D(Γ) ⟶^{≈} Z_{k}

where Γ is the DG-algebra DGHom(k, k) of a Tate resolution of k, which is quasiisomorphic to the Tate cohomology ring Ĥ^{∗}(G, k). Hence D(Γ) is triangle equivalent to D(Ĥ^{∗}(G, k)).

(2) If G is a p−group, then the full exact embedding D(Γ) ↪ \underline{Mod}(kG) of (1) induces a triangle equivalence

D(Ĥ^{∗}(G, k))⟶^{≈} \underline{Mod}(kG).

Proof. The first part follows by 2.2((i). Part (2) follows from 2.2((iii), since if G is a p−group, then the trivial G−module k is a generator of \underline{Mod}(kG).

3. Connections with the Homological Conjectures

In this section we use the previous results on hereditary torsion theories to shed new light on some homological conjectures in the representation theory of Artin algebras. We show that an interesting problem in tilting theory, whose solution would imply several homological conjectures, is equivalent to a problem on hereditary torsion theories in derived categories.

Throughout this section we work over an Artin R-algebra Λ where R is a commutative Artin ring. We fix a finitely generated right Λ-module T such that Ext_{Λ}^{n}(T, T) = 0, ∀n ≥ 1. We recall that T is called a Wakamatsu tilting module if the regular right Λ-module Λ admits an exact coresolution

0 ⟶ Λ ⟶ T^{0} ⟶^{f^{0}} T^{1} ⟶ ⋯ ⟶ T^{n} ⟶^{f^{n}} T^{n+1} ⟶ ⋯

where T^{n} lies in add(T) and Ext_{Λ}^{1}(Im(f^{n}), T ) = 0, for any n ≥ 0. Dually T is called Wakamatsu cotilting module if the injective cogenerator D(Λ) admits an exact resolution

⋯ ⟶ T^{−n−1} ⟶^{f_{−n}} T^{−n} ⟶ ⋯ ⟶ T^{−1} ⟶^{f_{0}} T^{0} ⟶ D(Λ) ⟶ 0

where T^{−n} lies in add(T) and Ext_{Λ}^{1}(T, Im(f_{−n})) = 0, for any n ≥ 0. A module is known to be a Wakamatsu tilting module if and only if it is a Wakamatsu cotilting module [80], [27]. An important problem is whether a Wakamatsu (co)tilting module is (co)tilting. In this connection there is the following problem.

• (WT)-Conjecture: If T is a Wakamatsu tilting (cotilting) module of finite projective (injective) dimension, then T is a tilting (cotilting) module. It would be interesting in itself to solve this problem, but even more because of

the following consequence. First we recall that the generalized Nakayama conjecture as proposed by Auslander and Reiten [5] says the following. In the minimal injective resolution 0 ⟶ Λ ⟶ I^{0} ⟶ I^{1} ⟶ ⋯ of Λ each indecomposable injective Λ-module occurs as a summand of some I^{n}. Obviously the generalized Nakayama conjecture implies the classical Nakayama Conjecture which says that if each I^{n} is projective then Λ is self-injective. Then we have the following result, part (i) of which was first observed by A. Buan, using [27].

Proposition 3.1. If the (WT)-Conjecture holds, then we have the following.

(i) The generalized Nakayama Conjecture holds.

(ii) The Gorenstein Symmetry Conjecture holds, that is: if id_{Λ}Λ < ∞, then Λ is Gorenstein, i.e. idΛ_{Λ} < ∞.

Proof. (i) Let 0 ⟶ Λ_{Λ} ⟶ I^{0} ⟶ I^{1} ⟶ ⋯ be the minimal injective resolution of Λ, and let T be the direct sum of the indecomposable injectives occurring as summands of the terms I^{n}, n ≥ 0. Then T is a Wakamatsu (co)tilting module, and hence a cotilting module if the (WT)-conjecture is assumed to hold. Hence the number of non-isomorphic indecomposable summands equals the number of non-isomorphic indecomposable injectives. In other words T contains all indecomposable injectives as summands, hence the generalized Nakayama conjecture holds.

(ii) Since id_{Λ}Λ < ∞, the injective cogenerator D(Λ) has finite projective dimension as a right module. Since D(Λ) is obviously a Wakamatsu tilting module and the (WT)-Conjecture is assumed to hold, D(Λ) is a tilting module. This implies that Λ has a finite exact coresolution by injectives. Hence idΛ_{Λ} < ∞.

Note that the (WT)-Conjecture implies also a related conjecture which says that if the nth term of the minimal injective resolution of Λ has projective dimension ≤ n for each n ≥ 0, then the algebra is Gorenstein, see [9]. Indeed the direct sum of the indecomposable injectives occurring in the resolution form a Wakamatsu tilting module of finite projective dimension. Hence by the (WT)-conjecture this is a tilting module. It follows that Λ_{Λ} has finite injective dimension, and then by [9] Λ is Gorenstein.

We now fix a Wakamatsu tilting module T and assume that T has finite projective dimension.

We view T as an object of the unbounded derived category D(Mod(Λ)) via the full embedding Mod(Λ) ↪ D(Mod(Λ)), which sends a module A to the stalk complex concentrated in degree zero with stalk A. Since T is finitely presented and has finite projective dimension, it follows that T is a compact object in D(Mod(Λ)), that is, T lies in H^{b}(P_{Λ}). Let (X_{T} , Y_{T} , Z_{T}) be the TTF-theory in D(Mod(Λ)) generated by T. Then the TTF-class Y_{T} is given by:

Y_{T} = {C^{•} ∈ D(Mod(Λ)) | (T [n], C^{•}) = 0, ∀n ∈ ℤ}

By Proposition 1.1 we know that the categories X_{T} , Y_{T}, and Z_{T} are compactly generated, and X_{T} and Z_{T} are triangle equivalent. We denote as usual by i : X_{T} ↪ D(Mod(Λ)), j : Y_{T} ↪ D(Mod(Λ)), and k : Z_{T} ↪ D(Mod(Λ)) the inclusion functors, and let (i, R), (L, j), (j, S), (k, T), be the corresponding adjoint pairs.

Let Γ = End(T) be the endomorphism ring of T. Then we have the adjoint pair of functors (F, G), where F := − ⊗_{Γ} T_{Λ} : Mod(Γ) ⟶ Mod(Λ) and G := Hom_{Λ}(_{Γ}T_{Λ}, −) : Mod(Λ) ⟶ Mod(Γ). Note that F preserves products since T is finitely presented. The adjoint pair (F, G) induces an adjoint pair of total derived functors (𝔽, 𝔾) on the level of unbounded derived categories, where 𝔽 := − ⊗_{Γ}^{𝕃} T_{Λ} : D(Mod(Γ)) ⟶ D(Mod(Λ)) and 𝔾 := ℝHom_{Λ}(T, −) : D(Mod(Λ)) ⟶ D(Mod(Γ)).

The following is essentially a consequence of Proposition I.2.11.

Lemma 3.2. The functor 𝔽 is fully faithful and the functor 𝔾 admits a fully faithful right adjoint

ℍ : D(Mod(Γ)) ⟶ D(Mod(Λ)), given by ℍ := ℝHom_{Γ}(_{Λ}ℝHom_{Λ}(T, Λ)_{Γ}, −).

Moreover we have the following identifications:

Im(𝔽) = X_{T} , Ker(𝔾) = Y_{T} , Im(ℍ) = Z_{T} , and iR = 𝔽𝔾, kT = ℍ𝔾.

Proof. By Proposition III.4.5 we have that 𝔽 is fully faithful. Since T is compact, it follows that 𝔾 preserves coproducts. Hence by [63] the functor 𝔽 preserves compact objects. Then by Proposition I.2.11, the functor 𝔾 admits a right adjoint ℍ : D(Mod(Γ)) ⟶ D(Mod(Λ)). By the same Proposition we have identifications Im(𝔽) = X_{T}, Im(ℍ) = Z_{T} and Ker(𝔾) = Y_{T}, and moreover iR = 𝔽𝔾 and kT = ℍ𝔾. Now consider the complex _{Λ}ℝHom_{Λ}(T, Λ)_{Γ} and the canonical morphism ϕ : − ⊗_{Λ}^{𝕃} ℝHom_{Λ}(T, Λ)_{Γ} ⟶ ℝHom_{Λ}(T, −) = 𝔾 of functors D(Mod(Λ)) ⟶ D(Mod(Γ)), where ϕ_{C^{•}} is the evaluation. Since both functors preserve coproducts and ϕ_{Λ} is invertible, by [52] it follows that ϕ is invertible. This implies that the right adjoint ℍ of 𝔾 is given by ℍ = ℝHom_{Γ}(_{Λ}ℝHom_{Λ}(T, Λ)_{Γ}, −).

We have the following direct consequence.

Corollary 3.3. The functor 𝔽 : D(Mod(Γ)) ⟶ D(Mod(Λ)) admits a factorization 𝔽 = i 𝔽^{∗}, where 𝔽^{∗} : D(Mod(Γ)) ⟶^{≈} X_{T} is a triangle equivalence, and the functor ℍ : D(Mod(Γ)) ⟶ D(Mod(Λ)) admits a factorization ℍ = k ℍ^{∗}, where ℍ^{∗} : D(Mod(Γ)) ⟶^{≈} Z_{T} is a triangle equivalence.

It follows from Proposition I.2.11 and the above result that the triangle equivalence Φ : X_{T} ⟶^{≈} Z_{T} and its quasi-inverse Ψ : Z_{T} ⟶^{≈} X_{T} are given as follows:

Ψ := − ⊗_{Λ}^{𝕃} ℝHom_{Λ}(T, Λ) ⊗_{Γ}^{𝕃} T, Φ := ℝHom_{Γ}(ℝHom_{Λ}(T, Λ), ℝHom_{Λ}(T, −))
= ℝHom_{Λ}(ℝHom_{Λ}(T, Λ) ⊗_{Γ}^{𝕃} T, −)

The following result gives necessary and sufficient conditions for the (WT)- Conjecture to be true.

Proposition 3.4. Let T be a Wakamatsu tilting right Λ-module of finite projective dimension with endomorphism ring Γ. Then the following are equivalent.

(i) T is a tilting module.

(ii) The torsion class X_{T} is closed under products in D(Mod(Λ)).

(iii) The total left derived functor 𝔽 := − ⊗_{Γ}^{𝕃} T_{Λ} : D(Mod(Γ)) ⟶ D(Mod(Λ))

preserves products. (iv) Λ ∈ X_{T}.

(v) D(Λ) ∈ Z_{T}.

(vi) X_{T} ⊆ Z_{T}. (vii) Z_{T} ⊆ X_{T}.

(viii) The canonical morphism : ℝHom_{Λ}(T, Λ) ⊗_{Γ}^{𝕃} T ⟶ Λ is invertible.

(ix) The canonical morphism : D(Λ) ⟶ ℝHom_{Λ}(ℝHom_{Λ}(T, Λ) ⊗_{Γ}^{𝕃} T, D(Λ)) is

invertible.

(x) The (non-hereditary) torsion theory in D(Mod(Λ)) generated by T is nondegenerate.

(xi) T has finite projective dimension as a left Γ-module.

Proof. Obviously (i) implies all the other conditions, since if T is a tilting module, then X_{T} = Z_{T} = D(Mod(Λ)), hence Y_{T} = 0, and the functors 𝔽 and 𝔾 are quasi-inverse equivalences. By Lemma 3.2, we have that (iv) is equivalent to (viii) and (v) is equivalent to (ix). Since X_{T} is a localizing subcategory of D(Mod(Λ)), if X_{T} contains the generator Λ, then by infinite devissage we have X_{T} = D(Mod(Λ)). Dually since Z_{T} is a colocalizing subcategory, (v) implies that Z_{T} = D(Mod(Λ)). In both cases we have Y_{T} = 0, hence T generates D(Mod(Λ)) and then T is a tilting module. By Proposition III.4.2 we have that (x) implies (i). If 𝔽 preserves products, then so does the functor 𝔽𝔾 = iR. This implies trivially that X_{T} is closed under products in D(Mod(Λ)). Conversely if this holds, then 𝔽 preserves products since 𝔽 = i 𝔽^{∗} and 𝔽^{∗} is an equivalence. Hence (iii) is equivalent to (ii). By Proposition 3.5 below we have that (iii) is equivalent to (xi). Using the glueing triangles for the torsion theories (X_{T} , Y_{T}) and (Y_{T} , Z_{T}), it is easy to see that (vi) and (vii) are equivalent. It remains to show that (ii) ⇒ (i) and (vi) ⇒ (i).

We first show that the right Λ-module Λ, resp. D(Λ), as a stalk complex concentrated in degree zero is isomorphic in D(Mod(Λ)) to a (countable) homotopy limit, resp. homotopy colimit, of compact objects from X_{T}. Indeed let 0 ⟶ Λ_{Λ} ⟶ T^{0} ⟶ T^{1} ⟶ T^{2} ⟶ ⋯ be an exact coresolution of Λ_{Λ}, where T^{n} is in add(T), ∀n ≥ 0. Then Λ_{Λ} is isomorphic to the complex T^{•} : ⋯ ⟶ 0 ⟶ T^{0} ⟶ T^{1} ⟶ T^{2} ⟶ ⋯ in the unbounded derived category. Filtering the complex T^{•} by using the stupid truncation complexes T_{n}^{•} : ⋯ ⟶ 0 ⟶ T^{0} ⟶ T^{1} ⟶ T^{2} ⟶ ⋯ ⟶ T^{n} ⟶ 0 ⟶ ⋯, we can form in the unbounded derived category the homotopy limit holim_{⟵}T_{n}^{•} of the induced inverse tower ⋯ ⟶ T_{2}^{•} ⟶ T_{1}^{•} ⟶ T_{0}^{•}. In other words we have a triangle

holim_{⟵}T_{n}^{•} ⟶ ∏_{n≥0} T_{n}^{•} ⟶ ∏_{n≥0} T_{n}^{•} ⟶ holim_{⟵}T_{n}^{•}[1]  (T)

in D(Mod(Λ)). By [23] we have an isomorphism T^{•} ⟶^{≅} holim_{⟵}T_{n}^{•}. Hence we infer the following isomorphism in the derived category:

Λ_{Λ} ⟶^{≅} holim_{⟵}T_{n}^{•}  (†)

and by construction all the complexes T_{n}^{•} lie in H^{b}(add(T)) = X_{T}^{b}. Since T is also a Wakamatsu cotilting module, there exists an exact sequence ⋯ ⟶ T^{−3} ⟶ T^{−2} ⟶ T^{−1} ⟶ D(Λ) ⟶ 0, where T^{−n} ∈ add(T), for n ≥ 1. Working as above we infer the following isomorphism in the derived category:

D(Λ)_{Λ} ⟶^{≅} holim_{⟶}T_{−n}^{•}  (††)

where T_{−n}^{•} are bounded complexes with components in add(T). In particular we have that D(Λ) lies in X_{T} since the latter is closed under coproducts, hence under homotopy colimits. This implies that ^{⊥}X_{T} = 0, since (C^{•}, Σ^{n}(D(Λ)))⟶^{≅} H^{n}(C^{•}). Hence if (vi) holds, then Y_{T} = ^{⊥}Z_{T} ⊆ ^{⊥}X_{T} = 0, and then T is a tilting module, i.e. (vi) implies (i). Now if X_{T} is closed under products, then it is closed under homotopy limits. We infer by the isomorphism (†) that X_{T} contains Λ, hence by (iv) we have X_{T} = D(Mod(Λ)). Alternatively if X_{T} is closed under products, then X_{T} is a colocalizing subcategory of D(Mod(Λ)). Since by the isomorphism (††), X_{T} contains D(Λ), it follows by Corollary 1.7 that X_{T} = D(Mod(Λ)). This shows that (ii) implies (i).

The following useful result is due to Bernhard Keller. We thank him for allowing us to include his result here and for some valuable comments on these topics.

Proposition 3.5. [B. Keller] Let _{Γ}T_{Λ} be a Γ-Λ-bimodule which as a left Γmodule is finitely presented. Then the total derived functor −⊗_{Γ}^{𝕃}T_{Λ} : D(Mod(Γ)) ⟶ D(Mod(Λ)) preserves products iff _{Γ}T has finite projective dimension.

Proposition 3.5 suggests the following problem which is of independent interest, since coherent functors play an important role in representation theory, see [50].

Problem. Let F : Mod(Λ) ⟶ Mod(Γ) be a coherent functor, that is, F preserves filtered colimits and products. Under what conditions does the total left derived functor 𝕃F : D(Mod(Λ)) ⟶ D(Mod(Γ)) preserve products?

As an application of the above results we have the following characterization of Gorenstein algebras, where we denote by − ⊗_{Λ}^{𝕃} D(Λ) : D(Mod(Λ)) ⟶ D(Mod(Λ)) the total left derived functor of the Nakayama functor − ⊗_{Λ} D(Λ) : Mod(Λ)) ⟶ Mod(Λ)), and by ℝHom_{Λ}(D(Λ), −) : D(Mod(Λ)) ⟶ D(Mod(Λ)) its right adjoint.

Corollary 3.6. The functor − ⊗_{Λ}^{𝕃} D(Λ) : D(Mod(Λ)) ⟶ D(Mod(Λ)) preserves products iff idΛ_{Λ} < ∞. In particular the algebra Λ is Gorenstein iff the functor − ⊗_{Λ}^{𝕃} D(Λ) preserves products, and the functor ℝHom_{Λ}(D(Λ), −) preserves coproducts iff − ⊗_{Λ}^{𝕃} D(Λ) : D(Mod(Λ)) ⟶^{≈} D(Mod(Λ)) is a triangle equivalence.

Proof. By Proposition 3.5 the functor −⊗_{Λ}^{𝕃}D(Λ) preserves products iff_{Λ}D(Λ) has finite projective dimension, or equivalently Λ_{Λ} has finite injective dimension. On the other hand the functor ℝHom_{Λ}(D(Λ), −) preserves coproducts iff D(Λ)_{Λ} is compact iff D(Λ)_{Λ} has finite projective dimension or equivalently _{Λ}Λ has finite injective dimension. The last equivalence follows from Proposition 3.4.

The following observation shows that the behaviour of the Nakayama functors plays a role in the investigation of when a Wakamatsu tilting module of finite projective dimension is a tilting module.

Lemma 3.7. Y_{T} ⊆ Ker(ℝHom_{Λ}(D(Λ), −)) and Im(− ⊗_{Λ}^{𝕃} D(Λ)) ⊆ X_{T}.

Proof. From the proof of Proposition 3.4 we have that D(Λ) lies in X_{T}. Hence for any complex Y^{•} ∈ Y_{T} we have ℝHom_{Λ}(D(Λ), Y^{•}) = 0, and the first inclusion follows. From the isomorphism (C^{•} ⊗_{Λ}^{𝕃} D(Λ), Y^{•}) ⟶^{≅} (C^{•}, ℝHom_{Λ}(D(Λ), Y^{•})) = 0, it follows that for any complex C^{•} ∈ D(Mod(Λ)), we have that C^{•} ⊗_{Λ}^{𝕃} D(Λ) lies in X_{T}, and the second inclusion follows.

Now let 𝔉 denote the set of isoclasses of indecomposable compact objects of D(Mod(Λ)). Recall that the full subcategory of compact objects of D(Mod(Λ)) can be identified up to equivalence with the bounded homotopy category H^{b}(P_{Λ}), and it is well-known that the latter is a Krull-Schmidt category. By Brown representability, for any object P^{•} ∈ 𝔉, there is a natural isomorphism

((P^{•}, −), k) ⟶^{≅} (−, 𝔻_{k}(P^{•}))

where k denotes the injective envelope of R/radR. As in Proposition III.3.9 it follows that the set 𝔻_{k}(𝔉) := {𝔻_{k}(P^{•}) | P^{•} ∈ 𝔉} is a set of pure-injective cogenerators of D(Mod(Λ)) (see [18] and [49] for the concept of purity in triangulated categories). By results of Happel [38], it follows that for any object P^{•} ∈ 𝔉: 𝔻_{k}(P^{•}) ⟶^{≅} P^{•} ⊗_{Λ}^{𝕃} D(Λ). Hence 𝔻_{k}(𝔉) = 𝔉 ⊗_{Λ}^{𝕃} D(Λ) ⊆ X_{T}.

The following result, which possibly can be proved directly, shows that the (WT)-Conjecture holds if Λ has finite right self-injective dimension.

Proposition 3.8. Let T_{Λ} be a Wakamatsu tilting module of finite projective dimension. If idΛ_{Λ} < ∞, then T is a tilting module.

Proof. Assume that idΛ_{Λ} < ∞ or equivalently that pd_{Λ}D(Λ) < ∞. Let E^{•} be an indecomposable pure-injective complex which lies in Y_{T}. Since the set 𝔻_{k}(𝔉) is a cogenerating set of pure-injectives, it follows that E^{•} is a direct summand of ∏_{P^{•}∈𝔉} 𝔻_{k}(P^{•}) = ∏_{P^{•}∈𝔉}(P^{•} ⊗_{Λ}^{𝕃} D(Λ)). By Proposition 3.5 we infer that

∏_{P^{•}∈𝔉}(P^{•} ⊗_{Λ}^{𝕃} D(Λ))⟶^{≅}( ∏_{P^{•}∈𝔉} P^{•})⊗_{Λ}^{𝕃}D(Λ) and this lies in X_{T} by Lemma 3.7. Hence E^{•} lies in X_{T} as a direct summand of ( ∏_{P^{•}∈𝔉} P^{•})⊗_{Λ}^{𝕃}D(Λ). Since E^{•} ∈ Y_{T}, we have E^{•} = 0. Then by a result of Krause, see Theorem 4.4 in [49], it follows that Λ lies in X_{T}, and then by Proposition 3.4 we have that T is a tilting module.

Corollary 3.9. If Z_{T} is closed under coproducts and ℝHom_{Λ}(D(Λ), Z^{•}) lies in Z_{T} for any complex Z^{•} ∈ Z_{T}, then T is a tilting module.

Proof. By Proposition 1.11 we have that Y_{T}^{b} ⊆ H^{b}(P_{Λ}). Let P^{•} be an indecomposable compact object in Y_{T}. Then 𝔻_{k}(P^{•}) = P^{•} ⊗_{Λ}^{𝕃} D(Λ) lies in X_{T}. For any Z^{•} in Z_{T} we have an isomorphism (P^{•} ⊗_{Λ}^{𝕃} D(Λ), Z^{•})⟶^{≅}(P^{•}, ℝHom_{Λ}(D(Λ), Z^{•})), and the last space is zero by hypothesis. Hence P^{•} ⊗_{Λ}^{𝕃} D(Λ) ∈ ^{⊥}Z_{T} = Y_{T}. Since P^{•} ⊗_{Λ}^{𝕃} D(Λ) lies also in X_{T} we infer that P^{•} ⊗_{Λ}^{𝕃} D(Λ) = 0. By construction this is possible only if P^{•} = 0. It follows that Y_{T}^{b} = 0, hence Y_{T} = 0 since Y_{T} is compactly generated. We infer that T is a tilting module.

The (WT)-Conjecture is related to a problem of Grothendieck groups, which we formulate as the following conjecture.

Conjecture. Let T be a Wakamatsu tilting module of finite projective dimension, and assume that the Grothendieck group K_{0}(Y_{T}^{b}) of the full subcategory of compact objects of Y_{T} is trivial. Then Y_{T} = 0, i.e. T is a tilting module.

More generally the following problem is of interest in connection with the results of this section. In a sense the next problem measures how far a triangulated category is from being algebraic in the sense of Keller [56].

Problem. Let T be a triangulated category with coproducts and a compact generator. Is it true that K_{0}(T^{b}) = 0 implies T^{b} = 0, or equivalently T = 0?

We have the following connection of the above Conjecture with the (WT)- Conjecture, where for a finitely generated module M, s(M) denotes the number of non-isomorphic indecomposable direct summands of M. Note that if T is a tilting module, then s(Λ) = s(T).

Proposition 3.10. Let T_{Λ} be a Wakamatsu tilting module of finite projective dimension and assume that s(Λ) = s(T). If the above Conjecture is true and Z_{T} is closed under coproducts in D(Mod(Λ)), then T is a tilting module.

Proof. Since Z_{T} is closed under coproducts, by Proposition 1.11 we have that R : X_{T} ⟶ D(Mod(Λ)) preserves coproducts. Then by Corollary 1.12 we have an isomorphism K_{0}(H^{b}(P_{Λ})) ⟶^{≅} K_{0}(H^{b}(add(T)))⊕K_{0}(Y_{T}^{b}). Now it is easy to see that K_{0}(H^{b}(P_{Λ})) ⟶^{≅} ℤ^{s(Λ)} and K_{0}(H^{b}(add(T))) ⟶^{≅} ℤ^{s(T)}. Hence the hypothesis s(Λ) = s(T) implies that K_{0}(H^{b}(P_{Λ})) ⟶^{≅} K_{0}(H^{b}(add(T))). We infer that K_{0}(Y_{T}^{b}) = 0, and then Y_{T} = 0 by our assumptions. This implies that T is a tilting module.

It is not known if a partial tilting module with s(T) = s(Λ) is always a tilting module. This is a well-known open problem in representation theory [73]. The above results suggest the following conjecture which lies between the (WT)- Conjecture and the Generalized Nakayama Conjecture.

Conjecture. Let T be a Wakamatsu tilting module of finite projective dimension. If s(T) = s(Λ), then T is a tilting module.

Remark 3.11. Observe that if Z_{T} is closed under coproducts, then the compact generator L(Λ) of Y_{T} is compact in D(Mod(Λ)). Hence by Theorem 2.1, there exists a triangle equivalence Y_{T} ⟶^{≅} D(Mod(Δ)) where Δ is the DG-algebra of endomorphisms of L(Λ). In this case the TTF-theory (X_{T} , Y_{T} , Z_{T}) reduces, up to triangle equivalence, to a TTF-theory (D(Mod(Γ)), D(Mod(Δ)), D(Mod(Γ))), i.e. a recollement of derived categories, in D(Mod(Λ)).

We close this section with the following result which shows that the above conjecture has an affirmative answer if Z_{T} is closed under coproducts.

Corollary 3.12. Let T be a Wakamatsu tilting module of finite projective dimension and assume that s(T) = s(Λ). If Z_{T} is closed under coproducts, then T is a tilting module.

Proof. By Proposition 3.10 we have that K_{0}(Y_{T}) = 0. Then with the notation of the above remark we have K_{0}(D(Mod(Δ))^{b})= 0. However this holds only if the DG-algebra Δ of endomorphisms of the compact object L(Λ) is (quasi-isomorphic to) 0. Hence L(Λ) = 0, or equivalently Λ lies in X_{T}. Then the assertion follows from Proposition 3.4.

We refer to [20] for a discussion of the topics of this section using pure homological algebra and the structure of the Ziegler spectrum, in connection with the Auslander-Reiten theory in triangulated and derived categories.

4. Concluding Remarks and Comments

As noted in Chapter I, torsion theories in a triangulated category are widely known as t-structures. They were introduced by Beilinson-Bernstein-Deligne in [15] as an important tool in the study of cohomology of sheaves over possibly singular spaces. In their language, recollement situations correspond to TTF-theories.

Hereditary torsion theories are known in the recent literature as semi-orthogonal decompositions, a terminology coined by Reiten-Van den Bergh, see [68], to a concept introduced by Bondal and Kapranov, see [25], as a useful tool for the study of coherent sheaves on manifolds.

On the other hand hereditary torsion classes of a torsion theory of finite type in a compactly generated triangulated category are known in the literature as smashing subcategories, generalizing a concept which has its origin in stable homotopy theory. We recall that a Grothendieck category G is called locally coherent if G admits a set of finitely presented generators and the full subcategory f.p(G) of finitely presented objects of G is abelian. It is well-known that hereditary torsion theories of finite type in a locally coherent Grothendieck category G are in bijective correspondence with the Serre subcategories of the abelian category f.p(G) of finitely presented objects of G [50]. This result lies at the heart of the representationtheoretic investigation of G via the theory of the Ziegler spectrum of G. In the triangulated case the substitute of a locally coherent Grothendieck category is a compactly generated triangulated category C and the substitute of the Serre subcategories of f.p(G) are the thick subcategories of the category of compact objects of C.

Hence it is natural to ask if the above correspondence has a triangulated analogue. The validity of the triangulated analogue of the above correspondence is equivalent, in our language, to the following well-known conjecture which is connected with thick subcategories of compact objects. Let C be a compactly generated triangulated category and let C^{b} be the full subcategory of compact objects of C.

Ravenel’s Telescope Conjecture: Any hereditary torsion theory of finite type in C is generated by a set of compact objects, i.e. is of the form (X_{P}, Y_{P}) for a set P ⊆ C^{b} closed under Σ, Σ^{−1}. Equivalently the maps

P ⟼ (X_{P}, Y_{P}) and (X , Y) ⟼ X ∩ C^{b}

are mutually inverse bijections between thick subcategories of C^{b} and hereditary torsion theories of finite type in C.

In this generality the conjecture is false. In our language Keller [53] produced an example of a hereditary torsion theory (X , Y) of finite type in the unbounded derived category D(Mod(Λ)) of a non-Noetherian ring Λ such that the torsion class X contains no non-zero compact object from D(Mod(Λ)), so the torsion theory can not be generated by compact objects. By results of Hopkins-Neeman (see [62]), the conjecture holds in case C is the unbounded derived category of a commutative Noetherian ring. We refer to [49] for a detailed analysis and a partial solution. Another class of categories in which the conjecture is true is the class of compactly generated triangulated categories C which admit filtered colimits, that is: the pure-semisimple categories of [18]. In such a category any object is a coproduct of indecomposable compact objects with local endomorphism ring. This imples trivially that any hereditary torsion theory is generated by compact objects.

The class of pure-semisimple triangulated categories is very small but not quite trivial. Important examples include the unbounded derived category of a ring which is derived equivalent to a right pure-semisimple right hereditary ring, in particular the unbounded derived category of an iterated tilted algebra of Dynkin type, and the stable module category of representation-finite quasi-Frobenius ring, in particular the stable module category of a self-injective Artin algebra of finite representation type, see [18].

CHAPTER V Torsion Theories in Stable Categories

Let throughout this chapter C be an abelian category with enough projective and enough injective objects. When ω is a functorially finite subcategory of C we know that the stable category C/ω is pretriangulated. A pair (X , Y) of subcategories of C with ω ⊆ X ∩ Y gives rise to a pair (X /ω, Y/ω) of subcategories of C/ω. The aim of this chapter is to obtain descriptions of (X /ω, Y/ω) being a torsion theory in terms of properties of the pair (X , Y). In particular we show that there is a strong relationship with pairs of contravariantly finite and covariantly finite subcategories, orthogonal with respect to Ext^{1}, investigated in [6],[7]. Important examples of such pairs of subcategories in the representation theory of Artin algebras emerge from tilting or cotilting modules.

1. Comparison of Subcategories

Let ω be a functorially finite subcategory of C. Our aim in this section is to compare pairs of subcategories of C and C/ω with respect to being contravariantly or covariantly finite, and with respect to the vanishing of Ext^{1} and Hom respectively (see [3] for a similar type of investigation). We start by recalling some of the main results on pairs (X , Y) investigated in [6]. These are proved in [6] when C is the category of of finitely generated modules over an Artin algebra, but the same proofs work when C is more generally a Krull–Schmidt category.

A subcategory X of C is preresolving if it is closed under extensions and contains the projective objects, and it is resolving if in addition it is closed under kernels of epimorphisms. Dually a subcategory Y of C is precoresolving if it is closed under extensions and contains the injective objects, and it is coresolving if in addition it is closed under cokernels of monomorphisms. Many of the results that follow work more generally, by dropping the assumption that C has enough projectives and injectives, and defining X in C to be preresolving if it is closed under extensions and whenever f : A ⟶ B is a morphism in C such that C(X , f) is an epimorphism, then f is an epimorphism. For a subcategory Z of C we denote by Z^{⊥} the full subcategory of C whose objects are the Y with Ext^{1}(Z, Y) = 0, for all Z ∈ Z, and by ^{⊥}Z the full subcategory of C whose objects are the X with Ext^{1}(X, Z) = 0, for all Z ∈ Z. When we use the symbol “^{⊥} ” in stable categories we mean vanishing with respect to Hom.

Definition 1.1. A pair (X , Y) of full subcategories of C is called a good pair if X is contravariantly finite with X^{⊥} = Y and Y is covariantly finite with X = ^{⊥}Y.

We need the following results from [6].

Theorem 1.2. Let C be a Krull-Schmidt category.

(i) If X is preresolving and contravariantly finite in C, then (X , X^{⊥}) is a good pair.

(ii) If Y is precoresolving and covariantly finite in C, then (^{⊥}Y, Y) is a good pair.

A subcategory ω of X is called a cogenerator for X if for any X ∈ X there exists a short exact sequence 0 ⟶ X ⟶ T ⟶ X′ ⟶ 0 with T ∈ ω and X′ ∈ X. An Ext-injective cogenerator for X is a cogenerator ω ⊆ X such that ω ⊆ X^{⊥}. Dually we have the notions of generators and Ext-projective generators for a subcategory.

Proposition 1.3. Let (X , Y) be a good pair in C with ω = X ∩ Y.

(i) X is resolving iff Y is coresolving.

(ii) If C is a Krull-Schmidt category, then ω is an Ext-injective cogenerator for X and an Ext-projective generator for Y.

We now make some comparisons between parallel properties for (X , Y) and (X /ω, Y/ω). Note that if X is a subcategory of C and Y = X^{⊥}, then automatically Y is closed under extensions and contains the injectives. And if X = ^{⊥}Y for a subcategory Y of C, then X is closed under extensions and contains the projectives.

Lemma 1.4. Let X and Y be subcategories of C with ω ⊆ X ∩ Y, and assume that ω is functorially finite and Ext^{1}(X , Y) = 0.

(i) If (X /ω)^{⊥} = Y/ω, then Y is closed under extensions and cokernels of monics.

(ii) If X /ω = ^{⊥}(Y/ω), then X is closed under extensions and kernels of epics.

Proof. (i) Let 0 ⟶ Y_{1} ⟶ C ⟶ Y_{2} ⟶ 0 be exact in C with Y_{1}, Y_{2} ∈ Y. Since Ext^{1}(X , Y) = 0 and ω ⊆ X, it follows that the sequence 0 ⟶ C(ω, Y_{1}) ⟶ C(ω, C) ⟶ C(ω, Y_{2}) ⟶ 0 is exact. Hence it induces a left triangle Ω(\underline{Y}_{2}) ⟶ \underline{Y}_{1} ⟶ \underline{C} ⟶ \underline{Y}_{2} in the pretriangulated category C/ω. Applying the homological functor (X /ω, −) to the triangle we infer that (X /ω, \underline{C}) = 0, hence \underline{C} ∈ Y/ω. This means that C is in Y so that Y is closed under extensions.

Now let 0 ⟶ Y_{1} ⟶ Y_{2} ⟶^{β} C ⟶ 0 be an exact sequence in C with Y_{1}, Y_{2} ∈ Y.

As above the sequence induces a left triangle Ω(\underline{C}) ⟶ \underline{Y}_{1} ⟶ \underline{Y}_{2} ⟶^{\underline{β}} \underline{C} in C/ω.

It follows that the sequence ⋯ ⟶ (X /ω, \underline{Y}_{1}) ⟶ (X /ω, \underline{Y}_{2}) ⟶^{(X /ω,\underline{β})} (X /ω, \underline{C}) is exact. Now observe that we have an exact commutative diagram

(X , Y_{1}) ⟶ (X , Y_{2}) ⟶^{(X ,β)} (X , C) ⟶ 0
π_{1}↓  ^{π_{2}}↓  ^{π_{3}}↓
(X /ω, \underline{Y}_{1}) ⟶ (X /ω, \underline{Y}_{2}) ⟶^{(X /ω,\underline{β})} (X /ω, \underline{C})

where the π_{i} are the canonical epimorphisms. Since the composition (X , β) ∘ π_{3} is epic, it follows that π_{2} ∘ (X /ω, \underline{β}) is also epic, and this implies that (X /ω, \underline{β}) : (X /ω, \underline{Y}_{2}) ⟶ (X /ω, \underline{C}) is epic. Then (X /ω, \underline{Y}_{2}) = 0, since (X /ω, Y/ω) = 0. Hence (X /ω, \underline{C}) = 0, i.e. \underline{C} ∈ (X /ω)^{⊥} = Y/ω, so that C ∈ Y.

The proof of part (ii) is dual.

We have the following consequence of Lemma 1.4.

Proposition 1.5. Assume that Ext^{1}(X , Y) = 0 and let ω be a functorially finite subcategory of C with ω ⊆ X ∩ Y. If (X /ω, Y/ω) is a torsion theory in C/ω, then X is closed under extensions and kernels of epimorphisms, and Y is closed under extensions and cokernels of monomorphisms.

The following example shows that the two orthogonality conditions are not always related.

Example. Let Λ be a Nakayama algebra with admissible sequence (2, 2, 1), and denote by S_{1}, S_{2}, S_{3} the corresponding simple Λ-modules. We then have the Auslander-Reiten quiver

S_{3} ↗ ( \begin{matrix}S_{2} \\ S_{3} \end{matrix} ) ↘ S_{2} ↗ ( \begin{matrix}S_{1} \\ S_{2} \end{matrix} ) ↘ S_{1}.

Let X = add\{S_{1}, S_{2}, ( \begin{matrix}S_{1} \\ S_{2} \end{matrix} )\} and ω = add\{S_{1}, ( \begin{matrix}S_{1} \\ S_{2} \end{matrix} )\}. Then we have X^{⊥} =

add\{( \begin{matrix}S_{2} \\ S_{3} \end{matrix} ) , ( \begin{matrix}S_{1} \\ S_{2} \end{matrix} ) , S_{1}\} and (X /ω)^{⊥} = add\{S_{3}, ( \begin{matrix}S_{2} \\ S_{3} \end{matrix} ) , S_{1}, ( \begin{matrix}S_{1} \\ S_{2} \end{matrix} )\}/ω.

The orthogonality condition Ext^{1}(X , Y) = 0 is very useful. However if the pair (X /ω, Y/ω) is a torsion theory in C/ω, so that (X /ω, Y/ω) = 0, then in general there is no reason to expect that Ext^{1}(X , Y) = 0 (see Section 2). The following result, which will be useful later, gives a condition for this to happen.

Lemma 1.6. Let X be a contravariantly finite subcategory of C which is closed under kernels of epimorphisms, and let ω be a subcategory of C which consists of Ext-injective objects for X. Then for any subcategory Y of C with the property (X /ω, Y/ω) = 0, we have Ext^{1}(X , Y) = 0.

Proof. Let 0 ⟶ Y ⟶ C ⟶ X ⟶ 0 be an extension with Y ∈ Y and X ∈ X and let f_{C} : X_{C} ⟶ C be a right X -approximation of C. Consider the exact commutative diagram:

0 ⟶ X′ ⟶^{α} X_{C} ⟶^{β} X ⟶ 0
η↓  ^{f_{C}} ↓  ^{||}↓
0 ⟶ Y ⟶^{g} C ⟶^{f} X ⟶ 0

Since X is closed under kernels of epics, X′ ∈ X. Since Y is in Y, the morphism η factors through an object in ω, i.e. there exists a factorization η := η_{1} ∘ η_{2} : X′ ⟶^{η_{1}} T′ ⟶^{η_{2}} Y, with T′ ∈ ω. The push-out of the upper row along η_{1} splits, since the objects of ω are Ext-injective in X. It follows easily from this that η factors through α. This implies that f splits. We infer that Ext^{1}(X , Y) = 0.

We now compare X being contravariantly finite in C with the inclusion X /ω ↪ C/ω having a right adjoint. We start with the following observation.

Lemma 1.7. (1) Assume that ω ⊆ X and ω is contravariantly finite in C. Then X is contravariantly finite in C iff X /ω is contravariantly finite in C/ω.

(2) Assume that ω ⊆ Y and ω is covariantly finite in C. Then Y is covariantly finite in C iff Y/ω is covariantly finite in C/ω.

Proof. (1) Assume that X /ω is contravariantly finite in C/ω and let C ∈ C. Let \underline{α} : \underline{X} ⟶ \underline{C} be a right X /ω-approximation of \underline{C}. Choose an object X_{C} and a morphism f_{C} : X_{C} ⟶ C such that \underline{X}_{C} = \underline{X} and \underline{f}_{C} = \underline{α}. Without loss of generality we may assume that f_{C} is ω-epic (if it is not then f′_{C} := ^{t}(μ, f_{C}) : X′_{C} := T ⊕ X_{C} ⟶ C is ω-epic, where μ : T ⟶ C is a right ω-approximation of C and \underline{X}′_{C} = \underline{X} and \underline{f}′_{C} = \underline{α}). If g : X′ ⟶ C is a morphism with X′ ∈ X, then there exists a morphism \underline{h} : \underline{X}′ ⟶ \underline{X} such that \underline{h} ∘ \underline{α} = \underline{g}. Then g − h ∘ α factors through a right ω-approximation ϕ_{C} : ω_{C} ⟶ C of C, i.e. there exists ρ : X′ ⟶ ω_{C} such that ρ ∘ ϕ_{C} = g − h ∘ f_{C}. Since f_{C} is ω-epic and ω ⊆ X, there exists σ : ω_{C} ⟶ X_{C} such that σ ∘ f_{C} = ϕ_{C}. Then g = (h + ρ ∘ σ) ∘ f_{C}. Hence g factors through f_{C}, and this shows that f_{C} is a right X -approximation of C. Conversely if f_{C} : X_{C} ⟶ C is a right X -approximation of C, then it is easy to see that \underline{f}_{C} : \underline{X}_{C} ⟶ \underline{C} is a right X /ω-approximation of \underline{C} in C/ω. It follows that if X is contravariantly finite in C, then X /ω is contravariantly finite in C/ω. Part (2) is dual.

We have the following connection.

Proposition 1.8. Let ω be a functorially finite subcategory of C.

(α) If ω ⊆ X, then the following are equivalent.

(i) X is closed under cokernels of left ω-approximations and the inclusion X /ω ↪ C/ω admits a right adjoint.

(ii) X is contravariantly finite in C and ∀C ∈ C there exists a right X -approxi– mation 0 ⟶ Y_{C} ⟶ X_{C} ⟶^{f_{C}} C of C, such that the object \underline{Y}_{C} ∈ (X /ω)^{⊥}.

(β) If ω ⊆ Y, then the following are equivalent.

(i) Y is closed under kernels of right ω-approximations and the inclusion Y/ω ↪ C/ω admits a left adjoint.

(ii) Y is covariantly finite in C and ∀C ∈ C and there exists a left Y-appro–

ximation C ⟶^{g^{C}} Y^{C} ⟶ X^{C} ⟶ 0 of C, such that the object \underline{X}^{C} ∈ ^{⊥}(Y/ω).

Proof. (α) (i) ⇒ (ii) Since X is closed under cokernels of left ω-approxima– tions it follows that Σ(X /ω) ⊆ X /ω. Let R : C/ω ⟶ X /ω be the right adjoint of the inclusion i : X /ω ↪ C/ω. Let C ∈ C and let α : R(\underline{C}) ⟶ \underline{C} be the coreflection of \underline{C} in X /ω. If (T) : Ω(\underline{C}) ⟶ \underline{Y}_{C} ⟶ R(\underline{C}) ⟶^{α} \underline{C} is a left triangle in C/ω, then by Lemma II.1.3 we have \underline{Y}_{C} ∈ (X /ω)^{⊥}. As in the above Lemma, α induces a right X -approximation f_{C} : X_{C} ⟶ C of C. Since f_{C} is ω-epic, it induces a left triangle (T′) : Ω(\underline{C}) ⟶ \underline{K}_{C} ⟶ \underline{X}_{C} ⟶ \underline{C} in C/ω, where K_{C} = Ker(f_{C}). By construction the left triangle (T′) coincides with (T), so \underline{K}_{C} ≅ \underline{Y}_{C} ∈ (X /ω)^{⊥}.

(ii) ⇒ (i) Let 0 ⟶ Y^{C} ⟶ X_{C} ⟶^{f_{C}} C be a right X -approximation of C with \underline{Y}^{C} ∈ (X /ω)^{⊥}. Since ω ⊆ X, f_{C} is ω-epic, hence the above sequence induces a left

triangle Ω(\underline{C}) ⟶ \underline{Y}^{C} ⟶ \underline{X}_{C} ⟶^{\underline{f}_{C}} \underline{C} in C/ω. Since \underline{Y}^{C} ∈ (X /ω)^{⊥}, by Lemma II.1.3,

the morphism \underline{f}_{C} : \underline{X}_{C} ⟶^{\underline{f}_{C}} \underline{C} is the coreflection of C/ω in X /ω. It remains to

show that X is closed under left ω-approximations. Let X ⟶^{κ} ω^{X} ⟶^{λ} C ⟶ 0 be a right exact sequence, where κ is a left ω-approximation of X ∈ X. Let 0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C be a left exact sequence, where f_{C} is a right X -approximation of C and \underline{Y}^{C} ∈ (X /ω)^{⊥}. Since ω ⊆ X, there exists β : ω^{X} ⟶ X_{C} such that λ = β ∘ f_{C}. Since λ is epic, so is f_{C} and we have an exact commutative diagram

X ⟶^{κ} ω^{X} ⟶^{λ} C ⟶ 0
∃ α↓  ^{∃ β}↓  ^{||}↓
0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0

Since X ∈ X and \underline{Y}^{C} ∈ (X /ω)^{⊥}, there exists a factorization α = α_{1} ∘ α_{2} : X ⟶^{α_{1}} T ⟶^{α_{2}} Y_{C}, where T ∈ ω. Since κ is a left ω-approximation, there exists ρ : ω^{X} ⟶ T such that α_{1} = κ∘ρ. Setting τ := ρ∘α_{2} it follows that κ∘τ = α. Then by standard arguments, there exists σ : C ⟶ X_{C} such that σ ∘ f_{C} = 1_{C}. Hence C ∈ X as a direct summand of X_{C}. The proof of part (β) is dual.

For ω = 0, we have the following special case, which is also easily seen directly.

Corollary 1.9. (1) X is coreflective in C iff X is contravariantly finite and ∀C ∈ C, there exists a right X -approximation f_{C} : X_{C} ⟶ C of C with Ker(f_{C}) ∈ {C ∈ C | C(X , C) = 0}.

(2) Y is reflective in C iff Y is covariantly finite and ∀C ∈ C, there exists a left Y-approximation f^{C} : C ⟶ Y^{C} of C with Coker(f^{C}) ∈ {C ∈ C | C(C, Y) = 0}.

2. A Description of Torsion Theories

Let X , Y be full subcategories of C such that ω = X ∩ Y is functorially finite in C. We want to get a description of (X /ω, Y/ω) being a torsion theory in C/ω, in terms of properties of X and Y.

For C in C, a torsion theory (X /ω, Y/ω) in C/ω gives a right X -approximation f_{C} : X_{C} ⟶ C and a left Y-approximation g^{C} : C ⟶ Y^{C}. Also we get a left ω-approximation κ : X_{C} ⟶ T with cokernel in X and a left ω-approximation μ : T′ ⟶ Y^{C} with kernel in Y. It turns out that, conversely, if we put this information together in an appropriate commutative diagram in C, we obtain sufficient (and necessary) conditions to have a torsion theory.

Theorem 2.1. If ω is functorially finite in C, then the following are equivalent.

(i) The pair (X /ω, Y/ω) is a torsion theory in C/ω.

(ii) ∀C ∈ C, there exists a bicartesian (i.e. a pull-back and push-out) square

\begin{matrix}X_{C} & ⟶^{f_{C}} & C \\ ^{κ}↓ & _{g^{C}} ↓ & (†) \\ T & ⟶^{μ} & Y^{C} \end{matrix}

with the following properties.

(α) X_{C} ∈ X, Y^{C} ∈ Y and any morphism X ⟶ Y with X ∈ X and Y ∈ Y

factors through an object in ω. (β) κ is a left ω-approximation of X_{C} with cokernel in X. (γ) μ is a right ω-approximation of Y^{C} with kernel in Y.

In this case f_{C} : X_{C} ⟶ C is a right X -approximation of C with kernel in Y and g^{C} : C ⟶ Y^{C} is a left Y-approximation of C with cokernel in X.

Proof. (ii) ⇒ (i) The bicartesian square (†) induces a short exact sequence

0 ⟶ X_{C} ⟶^{(κ,f_{C})} T ⊕ C ⟶^{^{t}(−μ,g^{C})} Y^{C} ⟶ 0  (1)

Since κ is ω-monic, it follows that (κ, f_{C}) is ω-monic. Hence we have a right

triangle ∇(C) : \underline{X}_{C} ⟶^{\underline{f}_{C}} \underline{C} ⟶^{\underline{g}^{C}} \underline{Y}^{C} ⟶^{\underline{h}^{C}} Σ(\underline{X}_{C}) in C/ω. Since μ is ω-epic, it

follows that ^{t}(−μ, g^{C}) is ω-epic. Hence we have a left triangle Δ(C) : Ω(\underline{Y}^{C}) ⟶^{\underline{h}_{C}}

\underline{X}_{C} ⟶^{\underline{f}_{C}} \underline{C} ⟶^{\underline{g}^{C}} \underline{Y}^{C} in C/ω. By condition (α) we have \underline{X}_{C} ∈ X /ω, \underline{Y}^{C} ∈ Y/ω

and (X /ω, Y/ω) = 0. Next we show that g^{C} is ω-monic. Indeed if α : C ⟶ T′ is a morphism with T′ ∈ ω, then by (β), there exists a morphism α′ : T ⟶ T′ such that κ ∘ α′ = α. Since (†) is a push-out diagram, there exists h : Y^{C} ⟶ T′ such that g^{C} ∘ h = α′, hence g^{C} is ω-monic. Dually f_{C} is ω-epic. It remains to show that Σ(X /ω) ⊆ X /ω or equivalently that X is closed under left ω-approximations. Suppose that C ∈ X in the diagram (†). Since Y^{C} ∈ Y, the morphism g^{C} factors through ω, hence \underline{g}^{C} = 0 in C/ω. Then from the right triangle ∇(C) we infer that \underline{Y}^{C} ≅ Σ(\underline{X}_{C}), which by (β) lies in X /ω. Since \underline{Y}^{C} ∈ Y/ω, it follows that \underline{Y}^{C} ∈ X /ω ∩ Y/ω = 0. Hence Y^{C} ∈ ω. Set X^{C} = Coker(g^{C}). Then by the bicartesian square (†) above, X^{C} = Coker(g^{C}) ≅ Coker(κ), so by definition X^{C} ∈ X. Since g^{C} is ω-monic, it follows that Σ(\underline{C}) ≅ \underline{X}^{C} ≅ Σ(\underline{X}_{C}) ∈ X /ω.

(i) ⇒ (ii) Assume now that the pair (X /ω, Y/ω) is a torsion theory in C/ω. Then (X /ω, Y/ω) = 0, X is closed under left ω-approximations because Σ(X /ω) ⊆ X /ω, Y is closed under right ω-approximations because Ω(Y/ω) ⊆ Y/ω and finally for any C ∈ C, there exist triangles:

Ω(\underline{Y}^{C}) ⟶ \underline{X}_{C} ⟶^{\underline{α}} \underline{C} ⟶^{\underline{β}} \underline{Y}^{C} ∈ Δ and \underline{X}_{C} ⟶^{\underline{α}} C ⟶^{\underline{β}} \underline{Y}^{C} ⟶ Σ(\underline{X}_{C}) ∈ ∇

Let g^{C} : C ⟶ Y^{C} be a morphism such that \underline{g̃}^{C} = \underline{β} and let μ : T_{1} ⟶ Y^{C} be a

right ω-approximation of Y^{C}. Consider the left exact sequence 0 ⟶ X̂_{C} ⟶^{(λ,f_{C})}

T_{1} ⊕ C ⟶^{^{t}(−μ,g^{C})} Y^{C}. It is clear that in C/ω we have \underline{X̂}_{C} ≅ \underline{X}_{C} and \underline{f}_{C} = \underline{α}. Let κ : X̂_{C} ⟶ T_{2} be a left ω-approximation of X̂_{C} and consider the right exact

sequence X̂_{C} ⟶^{(κ,λ,f_{C})} T_{2} ⊕ T_{1} ⊕ C ⟶^{^{t}(−ν,−ξ,g^{C})} Ỹ^{C} ⟶ 0. Since (λ, f_{C}) is monic, so

is (κ, λ, f_{C}), hence we have a short exact sequence

0 ⟶ X̂_{C} ⟶^{(κ,λ,f_{C})} T_{2} ⊕ T_{1} ⊕ C ⟶^{^{t}(−ν,−ξ,g^{C})} Ỹ^{C} ⟶  (2)

The above sequence induces a right triangle X̂_{C} ⟶^{\underline{f}_{C}} \underline{C} ⟶^{\underline{g}^{C}} \underline{Ỹ}^{C} ⟶ Σ(X̂_{C}) from which we infer that \underline{Ỹ}^{C} ≅ \underline{Y}^{C} and \underline{g̃}^{C} = \underline{β} = \underline{g}^{C}. In particular Ỹ^{C} ∈ Y. The above construction is included in the following exact commutative diagram:

0 ⟶ X̂_{C} ⟶^{(κ,λ,f_{C})} T_{2} ⊕ T_{1} ⊕ C ⟶^{^{t}(−ν,−ξ,g^{C})} Ỹ^{C} ⟶ 0
||↓  ^{^{t}(0,1_{C})}↓  ^{∃! σ}↓
0 ⟶ X̂_{C} ⟶^{(λ,f_{C})} T_{1} ⊕ C  ⟶^{^{t}(−μ,g^{C})} Y^{C}

Observe that the right square is a pull-back diagram. We claim that the morphism t(−ν, −ξ) : T_{2} ⊕ T_{1} ⟶ Ỹ^{C} is a right ω-approximation. Indeed let ϕ : T ⟶ Ỹ^{C} be a morphism with T ∈ ω. Since by construction −μ : T_{1} ⟶ Y^{C} is a right ω-approximation, there exists a morphism θ : T ⟶ T_{1} such that θ ∘ μ = ϕ ∘ σ. Consider the morphisms (θ, 0) : T ⟶ T_{1} ⊕ C and ϕ : T ⟶ Ỹ^{C}. By the pullback property there exists a unique morphism ζ := (ζ_{1}, ζ_{2}, ζ_{3}) : T ⟶ T_{2} ⊕ T_{1} ⊕ C such that (ζ_{1}, ζ_{2}, ζ_{3}) ∘^{t}(−ν, −ξ, g^{C}) = ϕ and (ζ_{1}, ζ_{2}, ζ_{3}) ∘^{t}(0, 1_{C}) = (θ, 0). It follows trivially that ζ_{3} = 0 and ζ_{2} = θ. Then (ζ_{1}, θ) : T ⟶ T_{2}⊕T_{1} is a morphism such that (ζ_{1}, θ)∘^{t}(−ν, −ξ) = ϕ. Hence ^{t}(−ν, −ξ) : T_{2} ⊕T_{1} ⟶ Ỹ^{C} is a right ω-approximation. Now since κ is a left ω-approximation, so is (κ, λ) : X̂_{C} ⟶ T_{2} ⊕ T_{1}. Hence setting T := T_{2} ⊕ T_{1}, κ̂ := (κ, λ) : X̂_{C} ⟶ T and μ̃ := ^{t}(ν, ξ) : T ⟶ Ỹ^{C}, we obtain a short exact sequence 0 ⟶ X̂_{C} ⟶^{(κ̂,f_{C})} T ⊕ C ⟶^{^{t}(−μ̃,g^{C})} Ỹ^{C} ⟶ 0, which is equivalent to the existence of a bicartesian square with the desired properties.

Assume now that (ii) holds. If τ : C ⟶ Y is a morphism with Y ∈ Y, then by (α), (β), the composition f_{C} ∘ τ factors through κ. Since the square (†) is a pushout, this implies that τ factors through g^{C}. Hence g^{C} is a left Y-approximation of C and then Coker(g^{C}) ≅ Coker(κ) ∈ X. The proof for f_{C} is dual.

By the above theorem, for any torsion theory (X /ω, Y/ω) in C/ω, where ω is functorially finite in C, and any object C in C, there exists an exact sequence

0 ⟶ X_{C} ⟶^{(f_{C},−κ)} C ⊕ T ⟶^{^{t}(g^{C},μ)} Y^{C} ⟶ 0

where: (i) X_{C} ∈ X, Y^{C} ∈ Y and T ∈ ω.

(ii) f_{C} : X_{C} ⟶ C is a right X -approximation of C and g^{C} : C ⟶ Y^{C} is

a left Y-approximation of C. (iii) κ : X_{C} ⟶ T is a left ω-approximation of X_{C} and μ : T ⟶ Y^{C} is a

right ω-approximation of Y^{C}.

Definition 2.2. The short exact sequence above is called the universal exact sequence of C with respect to the torsion theory (X /ω, Y/ω) in C/ω.

Note that for any C in C the universal exact sequence of C remains exact after the application of the contravariant functors C(−Y) and C(−, ω), and the covariant functors C(X , −) and C(ω, −).

Now we deduce a useful consequence of Theorem 2.1.

Corollary 2.3. If ω = X ∩ Y is functorially finite in C and (X /ω, Y/ω) is a torsion theory in C/ω, then we have the following.

(1) X contains the projectives iff ω is a generator of Y.

(2) Y contains the injectives iff ω is a cogenerator of X.

Proof. (1) If X contains the projectives, the morphism f_{C} in the bicartesian diagram (†) is an epimorphism, and then μ is an epimorphism. If C ∈ Y, then the morphism g^{C} is a split monomorphism. Chasing the diagram (†) we see easily that κ is split monomorphism, hence X_{C} is in ω as a direct summand of T. Conversely if ω is a generator of Y then μ is an epimorphism, since it is a right ω-approximation of Y^{C}. This implies that f_{C} is an epimorphism. Since f_{C} is a right X -approximation of C it follows that X contains the projectives. Part (2) is dual.

Theorem 2.1 generalizes the description of a usual torsion theory. Indeed, if (T , F) is a torsion theory in the abelian category C, and if t is the torsion subfunctor, then T ∩ F = 0 is functorially finite and the following square

t(C) ⟶ C
↓  ↓
0 ⟶ C/t(C)

is bicartesian and satisfies the conditions in part (ii) of Theorem 2.1. Moreover the induced short exact sequence (S) : 0 ⟶ t(C) ⟶ C ⟶ C/t(C) ⟶ 0 is the universal exact sequence of C. Conversely if (X , Y) = 0 and we have a bicartesian square as above with t(C) ∈ X, C/t(C) ∈ Y, then the sequence (S) is exact, hence (X , Y) is a torsion theory in C in the usual sense. We infer that (X , Y) is a torsion theory in the usual sense iff (X , Y) is a torsion theory in our sense with X ∩ Y = 0.

Since for a torsion theory (X , Y) in an abelian category C we have in general Ext^{1}(X , Y) ≠ 0, it follows that there exist torsion theories (X /ω, Y/ω) in C/ω where ω = X ∩ Y is functorially finite, such that Ext^{1}(X , Y) ≠ 0. Hence there exist torsion theories in pretriangulated stable categories not arising from good pairs.

3. Good Pairs and Torsion Theories

Assume that ω is a functorially finite subcategory of C and let X and Y be full subcategories of C with ω = X ∩ Y. In this section we investigate the relationship between the pair (X , Y) being good in C and the pair (X /ω, Y/ω) being a torsion theory in C/ω.

We start with the following result, which gives a first connection between torsion theories in C/ω and good pairs in C.

Proposition 3.1. If (X /ω, Y/ω) is a torsion theory in C/ω, then the following are equivalent:

(i) X is closed under kernels of epics, contains the projectives and admits ω as an Ext-injective cogenerator.

(ii) Y is closed under cokernels of monics, contains the injectives and admits ω

as an Ext-projective generator. (iii) (X , Y) is a good pair.

Proof. (i) ⇔ (ii) Assume that (i) holds. Then by Lemma 1.6 we have Ext^{1}(X , Y) = 0. In particular Y ⊆ X^{⊥}. Let C be in X^{⊥}. Since ω is a cogenerator of X by Corollary 2.3, Y contains the injectives. Hence the left Y-approximation g^{C} of C in Theorem 2.1 is a monomorphism. Since its cokernel X^{C} is in X, it follows that g^{C} splits, hence C ∈ Y. We infer that Y = X^{⊥}, in particular the objects of ω are Ext-projectives in Y. Since X contains the projectives, by Corollary 2.3 we have that ω is a generator of Y. It remains to show that Y is closed under cokernels of monics and this follows from Proposition 1.5. Part (ii) ⇒ (i) is dual.

(iii) ⇔ (i) If (i) holds, then by the equivalence of (i) with (ii) and Theorem 2.1 we have that X is contravariantly finite and X^{⊥} = Y, and Y is covariantly finite and ^{⊥}Y = X. Hence (X , Y) is a good pair. The converse follows easily from Proposition 1.5 and Corollary 2.3.

Let (X , Y) be a good pair C, with ω = X ∩Y functorially finite in C. If C is Krull-Schmidt then we know that ω is a cogenerator for X and a generator for Y, but X is not necessarily resolving and Y is not necessarily coresolving. When (X /ω, Y/ω) is a torsion theory in C/ω, the following direct consequence of Proposition 3.1 and Lemmas 1.4 and 1.6, shows that if (X , Y) is a good pair then X is resolving and Y is coresolving, even if C is not Krull-Schmidt. We observe also that in this case, assuming only that ω ⊆ X ∩ Y, we automatically get equality.

Corollary 3.2. If ω ⊆ X ∩ Y is functorially finite in C and (X /ω, Y/ω) is a torsion theory in C/ω, then ω = X ∩ Y and the following are equivalent:

(i) X is resolving and ω is an Ext-injective cogenerator of X.

(ii) Y is coresolving and ω is an Ext-projective generator of Y.

(iii) Ext^{1}(X , Y) = 0.

(iv) (X , Y) is a good pair in C.

We now get further connections between a pair (X , Y) in C being good and the pair (X /ω, Y/ω) being a torsion theory in C/ω, when ω = X ∩Y is functorially finite in C. We also obtain connections with the conditions that the inclusion X /ω ↪ C/ω has a right adjoint, together with Y/ω being orthogonal to X /ω.

Theorem 3.3. Let X and Y be full subcategories of C such that ω := X ∩ Y is functorially finite in C. Assume that Ext^{1}(X , Y) = 0, that X contains the projective objects and Y contains the injective objects, and that ω is a cogenerator for X and a generator for Y. Then the following are equivalent:

(i) The inclusion i : X /ω ↪ C/ω admits a right adjoint and Y/ω = (X /ω)^{⊥}. (ii) The inclusion j : Y/ω ↪ C/ω admits a left adjoint and X /ω = ^{⊥}(Y/ω). (iii) The pair (X /ω, Y/ω) is a torsion theory in C/ω.

If one of the above equivalent conditions holds, then (X , Y) is a good pair in C.

Proof. Obviously (iii) ⇒ (i). We show that (i) ⇒ (iii), (ii). Since ω ⊆ Y it follows that X ⊆ ^{⊥}Y ⊆ ^{⊥}ω, i.e. ω is an Ext-injective cogenerator of X. This implies that X is closed under cokernels of left ω-approximations. Since X contains the projectives, by Proposition 1.8, X is contravariantly finite and for any C ∈ C, there exists a right X -approximation sequence 0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0 of C with Y_{C} ∈ Y. Since ω is a cogenerator of X, there exists an extension 0 ⟶ X_{C} ⟶^{κ} T ⟶^{ν} X^{C} ⟶ 0, where T ∈ ω and X^{C} ∈ X. Consider the exact commutative diagram

0  0
↓  ↓
0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0
||↓  ^{κ}↓  _{g^{C}} ↓
0 ⟶ Y_{C} ⟶^{λ} T ⟶^{μ} Y^{C} ⟶ 0
ν↓  _{f^{C}} ↓
X^{C} ⟶^{=} X^{C}
↓  ↓
0  0

Then obviously the upper right square is bicartesian. Since by Lemma 1.4, Y is closed under extensions and cokernels of monics, it follows that Y^{C} ∈ Y. Since Ext^{1}(X , Y) = 0 and ω ⊆ X ∩ Y, it follows directly that κ is a left ω-approximation of X_{C} and μ is a right ω-approximation of Y^{C}. If C ∈ Y, then since Y is closed under extensions, X_{C} ∈ X ∩ Y = ω. This implies that any morphism X ⟶ Y with X ∈ X factors through X_{Y}, i.e. through an object in ω. By Theorem 2.1, we infer that (X /ω, Y/ω) is a torsion theory in C/ω. In particular the conditions in (ii) hold. It remains to show that (ii) implies (iii); this is proved in a similar way.

Finally if (X /ω, Y/ω) is a torsion theory in C/ω, then the pair (X , Y) is good by Corollary 3.2.

The proof of the above result suggests the following definitions.

Definition 3.4. A good pair (X , Y) in C is called a pretorsion theory if any object of C admits a right X -approximation with kernel in Y and a left Y-approxi– mation with cokernel in X. A pretorsion theory (X , Y) is called hereditary, resp. cohereditary, if X is resolving, resp. Y is coresolving.

Lemma 3.5. (1) If (X , Y) is a good pair in C and any object of C admits a right X -approximation with kernel in Y, then ω is a cogenerator of X.

(2) If (X , Y) is a good pair in C and any object of C admits a left Y-approxima– tion with kernel in X, then ω is a generator of Y.

Proof. We only prove (1) since part (2) is dual. Let X ∈ X and let 0 ⟶ X ⟶ I ⟶ A ⟶ 0 be exact with I injective. Then pulling-back this sequence along the right X -approximation f_{A} : X_{A} ⟶ A of A with Ker(f_{A}) in Y, we obtain extensions 0 ⟶ X ⟶ T ⟶ X_{A} ⟶ 0 and 0 ⟶ Y_{A} ⟶ T ⟶ I ⟶ 0. Since I and Y_{A} are in Y, and X and X_{A} are in X, the extension closure of X and Y implies that T ∈ X ∩ Y = ω. Hence ω is a cogenerator of X.

The following result shows that the conditions in the above definitions can be relaxed.

Proposition 3.6. If (X , Y) is a good pair in C, then the following conditions are equivalent:

(i) Y is coresolving and any object of C admits a right X -approximation with kernel in Y.

(ii) X is resolving and any object of C admits a left Y-approximation with cok-

ernel in X. (iii) (X , Y) is a hereditary pretorsion theory in C. (iv) (X , Y) is a cohereditary pretorsion theory in C.

Proof. By Proposition 1.2, (iii) is equivalent to (iv) and obviously (iv) implies (i) and (iii) implies (ii). It suffices to show that (i) is equivalent to (ii) and this follows from the diagram of the proof of Theorem 3.3, using Lemma 3.5.

It follows that a pretorsion theory is hereditary iff it is cohereditary. The following result characterizes the good pairs (X , Y) which induce torsion theories in C/ω, if the category C is Krull-Schmidt.

Theorem 3.7. If C is a Krull-Schmidt category, then the following are equivalent for a good pair (X , Y) in C with ω = X ∩ Y functorially finite in C.

(i) (X /ω, Y/ω) is a torsion theory in C/ω.

(ii) X is resolving.

(iii) Y is coresolving.

(iv) (X , Y) is a (co)hereditary pretorsion theory in C.

Proof. The equivalence (ii) ⇔ (iii) follows from Proposition 1.3 and the implication (i) ⇒ (ii) follows from Lemma 1.4. The implication (i) ⇒ (iv) follows from Proposition 1.8 and the implication (iv) ⇒ (iii) is trivial. To complete the proof it suffices to show that (ii) ⇒ (i).

(ii) ⇒ (i) By Proposition 1.3 we have that ω is an Ext-injective cogenerator for X and an Ext-projective generator for Y. Let C ∈ C. Since X is contravariantly finite resolving, there exists a short exact sequence 0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0 in C where f_{C} is a right X -approximation of C. Since C is Krull-Schmidt, we can choose f_{C} to be right minimal [10]. This implies by Wakamatsu’s Lemma [7] that Y_{C} lies in X^{⊥} = Y. If C ∈ Y, then X_{C} ∈ ω since by the equivalence (ii) ⇔ (iii), Y is coresolving. Then any morphism X ⟶ C factors through X_{C}. We infer that (X /ω, Y/ω) = 0, hence Y/ω ⊆ (X /ω)^{⊥}. Now let C in C be such that \underline{C} ∈ (X /ω)^{⊥}. Since by Proposition 1.3, ω is an Ext-injective cogenerator for X, there exists an extension 0 ⟶ X_{C} ⟶^{κ} T ⟶^{ν} X^{C} ⟶ 0 where T ∈ ω and X^{C} ∈ X and then κ is a left ω-approximation of X_{C}. Since \underline{C} ∈ (X /ω)^{⊥}, the right X -approximation f_{C} of C factors through κ and we have an exact commutative diagram

0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0
ξ↓  ^{κ}↓  ^{||}↓
0 ⟶ K ⟶^{α} T ⟶^{β} C ⟶ 0

Since Coker(ξ) ≅ Coker(κ) = X^{C} ∈ X and Y_{C} ∈ Y, the morphism ξ splits. This implies trivially that κ splits and then X_{C} ∈ ω as a direct summand of T. Now since ω ⊆ Y and Y is closed under cokernels of monomorphisms, we have that C ∈ Y. It follows that (X /ω)^{⊥} = Y/ω. Then by Proposition 1.8 we infer that the inclusion X /ω ↪ C/ω admits a right adjoint. Then by Theorem 3.2 we have that the pair (X /ω, Y/ω) is a torsion theory in C/ω.

Remark 3.8. The above proof shows that the assumption that C is Krull-Schmidt can be replaced by any one of the following assumptions: any object of C admits a minimal right X -approximation, or any object of C admits a minimal left Y-aproximation. Actually it is not difficult to see that these conditions are equivalent. It is suggestive to call a good pair satisfying any one of these conditions perfect. Then as in the above theorem, if the good pair (X , Y) is perfect, then (X , Y) is a pretorsion theory in C. In this case the pretorsion theory is (co)hereditary iff (X /ω, Y/ω) is a torsion theory in C/ω.

If ω is a functorially finite subcategory of C, then consider the following classes:

(i) P(ω) is the class of hereditary pretorsion theories (X , Y) in C with X ∩Y = ω. (ii) G(ω) is the class of good pairs (X , Y) in C with X ∩ Y = ω and X resolving

or Y coresolving. (iii) T (ω) is the class of torsion theories (X /ω, Y/ω) in C/ω with Ext^{1}(X , Y) = 0.

Our previous considerations allow us to conclude this subsection with the following result, which gives the precise relationship between pretorsion theories in C and torsion theories in C/ω.

Proposition 3.9. If ω is a functorially finite subcategory of C, then the map

P(ω) ⟶ T (ω) : (X , Y) ⟼ (X /ω, Y/ω)

is a bijection. If C is Krull-Schmidt, then the above map gives a bijection

G(ω) ⟶ T (ω) : (X , Y) ⟼ (X /ω, Y/ω).

The following is another example of a torsion theory not induced by a good pair.

Example. Assume that the full subcategory P of projective objects of C is functorially finite, so that the stable category C/P is pretriangulated. Then (C/P, 0) is obviously a torsion theory in C/P but it is not induced by any good pair (X , Y), unless any projective object is injective.

4. Torsion Classes and Cohen-Macaulay Objects

Let as before C be an abelian category with enough projectives and enough injective objects. In this section we are interested in finding when a full subcategory X of C which admits a functorially finite Ext-injective cogenerator ω, induces a torsion class X /ω in the stable category C/ω. And when such a torsion class exists, we are interested in having an explicit description of the corresponding torsion-free class. We also indicate some potential examples of such subcategories X, which arise naturally in practice.

We start by stating some of the results of the previous section as equivalent conditions for the subcategory X of C, leaving the formulation of the dual results concerning Y to the reader.

Proposition 4.1. Let X be a subcategory of C, closed under kernels of epimorphisms and containing the projective objects of C. If X has a functorially finite Ext-injective cogenerator ω, then the following are equivalent.

(i) X /ω is a torsion class in C/ω.

(ii) For any C ∈ C there exists a right X -approximation sequence 0 ⟶ Y_{C} ⟶

X_{C} ⟶ C ⟶ 0 with \underline{Y}_{C} ∈ (X /ω)^{⊥}. (iii) The inclusion X /ω ↪ C/ω has a right adjoint.

Proof. Obviously (i) implies (iii). Since ω is an Ext-injective cogenerator of X, we have that X is closed under cokernels of right ω-approximations. Then the equivalence (ii) ⇔ (iii) follows by Proposition 1.8.

(iii) ⇒ (i) Let Y be the subcategory of C defined by Y/ω = (X /ω)^{⊥}, so that ω = X ∩ Y. It suffices to show that (X , Y) satisfies the assumptions of Theorem 3.3. By Lemma 1.6 we have Ext^{1}(X , Y) = 0. Then by Lemma 1.4 we have that Y is closed under extensions and cokernels of monics. Using that ω is an Ext-injective cogenerator of X we can construct for any C ∈ C, the bicartesian square of the proof of Theorem 3.3, and using this we deduce easily that Y contains the injectives and has ω as an Ext-projective generator. Hence the assumptions of Theorem 3.3 hold for the pair (X , Y), and since the inclusion X /ω ↪ C/ω admits a right adjoint, by Theorem 3.3 we infer that (X /ω, Y/ω) is a torsion theory in C/ω.

In case of Krull-Schmidt subcategories, we have the following characterizations of torsion classes.

Corollary 4.2. Let X be a Krull-Schmidt subcategory of C closed under kernels of epimorphisms and containing the projective objects of C. If X admits a functorially finite Ext-injective cogenerator ω, then the following are equivalent.

(i) X /ω is a torsion class in C/ω.

(ii) X is contravariantly finite in C.

Proof. By Lemma 1.7 we have that X is contravariantly finite in C iff X /ω is contravariantly finite in C/ω. Since ω is an Ext-injective cogenerator for X, we have that X /ω is a right triangulated subcategory of the pretriangulated category C/ω. By Proposition II.2.4 we have that X /ω is contravariantly finite in C/ω iff X /ω is coreflective in C/ω. Then the assertion follows from Proposition 4.1.

Corollary 4.3. Let C be a Krull-Schmidt category and X a preresolving contravariantly finite subcategory of C and assume that ω := X ∩ X^{⊥} is functorially finite in C. Then the following are equivalent.

(i) X is resolving.

(ii) X /ω is a torsion class in C/ω.

In view of the above results it is natural to consider the maximal subcategory of C containing ω as an Ext-injective cogenerator, which from now on we denote by CM(ω). More explicitly if ω is a self-orthogonal subcategory ω ⊆ C, i.e. Ext^{n}(ω, ω) = 0, ∀n ≥ 1, then:

CM(ω) = { C ∈ C | ∃ exact sequence 0 ⟶ C ⟶ T^{0} ⟶^{f^{0}} T^{1} ⟶^{f^{1}} T^{2} ⟶ ⋯,

T^{s} ∈ ω, ∀s ≥ 0 : Ext_{C}^{t}(Ker(f^{n}), ω) = 0, ∀n ≥ 0, ∀t ≥ 1 }.

We call the objects of CM(ω) the Cohen-Macaulay objects of C with respect to ω. If ω = add(T) for a self-orthogonal object T, then we use the notation CM(ω) = CM(T). This category was introduced in [6] (see also [3]), called X_{ω} in [6], in connection with Cohen-Macaulay approximations: if Λ is a commutative Gorenstein ring, then CM(Λ) is the category of (maximal) Cohen-Macaulay modules. Note that by construction and Corollary 4.2 the subcategory CM(ω) contains X for any torsion class X /ω in C/ω arising from a contravariantly finite resolving subcategory X of C. Since by [6], CM(ω) is closed under extensions and kernels of epimorphisms, it is resolving if it contains the projectives. Putting things together we have the following consequence.

Corollary 4.4. Let ω be a functorially finite self-orthogonal subcategory of C. If CM(ω) contains the projectives of C, then the following are equivalent.

(i) CM(ω)/ω is a torsion class in C/ω.

(ii) the inclusion CM(ω)/ω ↪ C/ω admits a right adjoint.

Example. Let ω = P ∩ I be the full subcategory of C consisting of the projective-injective objects. We say that the object C has infinite dominant dimension, and then we write dom.dimC = ∞, if in any injective resolution 0 ⟶ C ⟶ I_{0} ⟶ I_{1} ⟶ ⋯ of C all the injective objects I_{i} are projective. Let Dom(C) be the full subcategory of C consisting of all objects with dom.dimC = ∞. Observe that CM(ω) = Dom(C). If ω is functorially finite in C then by Corollary 4.2 we have that Dom(C)/ω is a torsion class in C/ω iff Dom(C)/ω is coreflective in C/ω. This is equivalent to Dom(C) being contravariantly finite in C, if C is in addition Krull-Schmidt.

Dually we denote by CoCM(ω) the maximal subcategory of C which admits ω as an Ext-projective generator. Obviously

CoCM(ω) = { C ∈ C | ∃ exact sequence ⋯ ⟶ T_{2} ⟶^{f_{1}} T_{1} ⟶^{f_{0}} T^{0} ⟶ C ⟶ 0,

T_{s} ∈ ω, ∀s ≥ 0 : Ext_{C}^{t}(ω, Coker(f_{n})) = 0, ∀n ≥ 0, ∀t ≥ 1 }.

If ω = add(T) for T ∈ C, then we use the notation CoCM(ω) = CoCM(T). The duals of the above observations also hold.

Let now (X , Y) be a good pair in the abelian Krull-Schmidt category C, and assume that ω = X ∩ Y is functorially finite in C and X is resolving. Then we have that (X /ω, Y/ω) is a torsion theory in C/ω. In some cases it is possible to get a description of Y (see [3]). For a subcategory Z of C we denote by Ẑ the full subcategory of C whose objects are the C such that there is an exact sequence 0 ⟶ Z_{n} ⟶ ⋯ ⟶ Z_{1} ⟶ Z_{0} ⟶ C ⟶ 0 with the Z_{i} in Z. Dually we denote by Z̃ the full subcategory of C whose objects are the C such that there is an exact sequence 0 ⟶ C ⟶ Z_{0} ⟶ Z_{1} ⟶ ⋯ ⟶ Z_{m} ⟶ 0 with the Z_{i} in Z. For instance if P is the full subcategory of projectives, then { \atop CM(P)} is the full subcategory of objects with finite Gorenstein dimension and P̂ := P^{<∞} is the full subcategory of C consisting of the objects with finite projective dimension. Recall from [4] that the Gorenstein dimension G-dimC of an object C in C is defined inductively as follows. If C is in CM(P), then G-dimC = 0. If t ≥ 1, then G-dimC ≤ t if there exists an exact sequence 0 ⟶ G_{t} ⟶ ⋯ ⟶ G_{1} ⟶ G_{0} ⟶ C ⟶ 0 where G-dimG_{i} = 0, for 0 ≤ i ≤ t. Then G-dimC = t if G-dimC ≤ t and G-dimC ≰ t − 1. Finally if G-dimC ≠ t for any t ≥ 0, then define G-dimC = ∞. It follows that

{ \atop CM(P)} = \{C ∈ C | G-dimC < ∞\}.

Proposition 4.5. With the above notations and assumptions we have: X̂ = C iff Y = ω̂. Dually we have: Ỹ = C iff X = ω̃.

In the not necessarily Krull-Schmidt case we have the following result, which is an easy consequence of our previous results and the results of [3].

Proposition 4.6. If X is a resolving subcategory with a functorially finite Extinjective cogenerator ω, then the following are equivalent:

(i) (X /ω, ω̂/ω) is a torsion theory in C/ω.

(ii) X̂ = C. If this the case, then X = CM(ω).

The above results are true in the more general situation in which C does not necessarily have enough projectives and we don’t necessarily have X̂ = C. Indeed the following remark shows that it is possible to get a good pair in a smaller piece of C and a torsion theory in a smaller piece of C/ω.

Remark 4.7. Let X be a full subcategory of C which is closed under extensions and kernels of epimorphisms, and assume that X admits a cogenerator ω with Ext^{n}(X , ω) = 0, ∀n ≥ 1. It follows by the results of [3] that (X , ω̂) is a hereditary pretorsion theory in X̂ with X ∩ ω̂ = ω, hence a hereditary pretorsion theory in C if X̂ = C. If in addition ω is an Ext-projective generator of X, then it follows easily that the stable category X̂/ω is pretriangulated and then the pair (X /ω, ω̂/ω) is a torsion theory in X̂/ω, hence a torsion theory in C/ω if in addition ω is functorially finite in C and X̂ = C. In particular for any contravariantly finite self-orthogonal subcategory ω of C, the pair (CM(ω), ω̂) is a pretorsion theory in { \atop CM(ω)} and the pair (CM(ω)/ω, ω̂/ω) is a torsion theory in { \atop CM(ω)}/ω. For instance we can take ω = P to be the full subcategory of projective objects of C. Then (CM(P), P^{<∞}) is a good pair in { \atop CM(P)} with CM(P) ∩ P^{<∞} = P and (CM(P)/P, P^{<∞}/P) is a torsion theory in { \atop CM(P)}/P.

Dual observations hold for subcategories admitting an Ext-projective generator and CoCohen-Macaulay objects.

The following is an example from homological group theory which illustrates the above remark.

Example. Let G be a group and k be a commutative ring of coefficients. We denote by B = B(G, k) the set of functions G ⟶ k which take only finitely many different values in k. Then G acts on B by multiplication and B is free as a kmodule. Let X be the full subcategory of Mod(kG) consisting of all kG-modules M such that B ⊗_{k} M is projective. Following [21] we call the modules in X cofibrant modules. It is not difficult to see that X is resolving and X̂ consists of all kGmodules M such that B⊗_{k}M has finite projective dimension. Let ω be the category of projective kG-modules, so ω̂ is the full subcategory of all kG-modules of finite projective dimension. It is not difficult to see that ω is an Ext-injective cogenerator of X, hence by the above remark, (X , ω̂) is pretorsion theory in X̂. We refer to [21] for details and more information. Note that if k has finite global dimension, and the group G is of type F P_{∞}, that is, the trivial G-module k admits an exact resolution by finitely generated projective modules, and belongs to the Kropholler’s class H𝔉 of hierarchically decomposable groups [29], then X̂ = Mod(kG). Hence in this case (X , ω̂) is a pretorsion theory in Mod(kG).

The finiteness condition X̂ = C or Ỹ = C has some interesting consequences on the level of derived categories, as we now explain.

Let (X /ω, Y/ω) be a torsion theory in C arising from a good pair (X , Y) with ω = X ∩ Y functorially finite in C. Then X, Y and ω are closed under extensions in C, hence they are exact subcategories of C in the sense of Quillen; obviously ω carries the split exact structure since it is self-orthogonal. Let D^{b}(X), D^{b}(Y) and D^{b}(ω) be the corresponding bounded derived categories, see [54] for the notion of the derived category of an exact category. Then D^{b}(ω) coincides with the bounded homotopy category H^{b}(ω).

Corollary 4.8. We have a commutative diagram of full exact embeddings:

D^{b}(ω) ⟶ D^{b}(Y)
↓  ↓
D^{b}(X) ⟶ D^{b}(C)

(1) If X̂ = C (equivalently Y ⊆ ω̂), then the above functors induce triangle equivalences D^{b}(X) ⟶^{≈} D^{b}(C) and D^{b}(Y) ⟶^{≈} D^{b}(ω).

(2) If Ỹ = C (equivalently X ⊆ ω̃), then the above functors induce triangle equivalences D^{b}(Y) ⟶^{≈} D^{b}(C) and D^{b}(X) ⟶^{≈} D^{b}(ω).

Proof. We only prove (1) since (2) is dual. By the dual of Theorem 12.1 in [54] the induced functor D^{b}(X) ⟶ D^{b}(C) is fully faithful, if any short exact sequence 0 ⟶ A ⟶ B ⟶ X ⟶ 0 in C with X ∈ X can be embedded in an exact commutative diagram

0 ⟶ X′ ⟶ X_{B} ⟶ X ⟶ 0
↓  ^{f_{B}} ↓  ^{∥}↓
0 ⟶ A ⟶ B ⟶ X ⟶ 0

where X′ and X_{B} are in X. This holds by choosing f_{B} to be a right X -approxima– tion of B; then X′ is in X since X is closed under kernels of epimorphisms. If X̂ = C then the fully faithful functor D^{b}(X) ⟶ D^{b}(C) is an equivalence by devissage. Since X̂ = C implies that Y ⊆ ω̂ again by devissage the full embedding D^{b}(ω) ↪ D^{b}(Y) is an equivalence.

It is not difficult to see that the above full exact embeddings induce short exact sequences of triangulated categories

0 ⟶ D^{b}(ω) ⟶ D^{b}(X) ⟶ T_{r}(X /ω) ⟶ 0
0 ⟶ D^{b}(ω) ⟶ D^{b}(Y) ⟶ T_{l}(Y/ω) ⟶ 0

where T_{r}(X /ω) is the stabilization of the right triangulated torsion class X /ω and T_{l}(Y/ω) is the stabilization of the left triangulated torsion-free class Y/ω.

We close this section by pointing out some interesting examples of good pairs and pretorsion theories in the category Mod(Λ) of all right Λ-modules, where Λ is any ring:

Example. It is not difficult to see that the following are good pairs:

(i) (^{⊥}Y, Y), where Y is a covariantly finite coresolving subcategory of Mod(Λ) consisting of pure-injective modules.

(ii) (X , X^{⊥}), where X is a contravariantly finite resolving subcategory of Mod(Λ)

closed under direct limits. (iii) (^{⊥}T, (^{⊥}T)^{⊥}), where T is a pure-injective module such that any projective

module is a submodule of a product of copies of T, e.g. T = Λ for a right perfect ring Λ.

(iv) (^{⊥}(T^{⊥}), T^{⊥}), where T is a finitely presented module with right perfect endomorphism ring and such that Ext_{Λ}^{1}(T, T) = 0.

Actually the good pairs in (i) and (ii) are pretorsion theories. More generally by [33], if ω ⊆ Mod(Λ) is any skeletally small subcategory, then (^{⊥}(ω^{⊥}), ω^{⊥}) is a good pair in Mod(Λ). Hence if ω is resolving and ω^{⊥} consists of pure-injectives or ⊥(ω^{⊥}) is closed under direct limits, we get a pretorsion theory in Mod(Λ) by (i) or (ii) respectively. By the recent solution of the Flat Cover Conjecture by Bashir, Bican and Enochs [14], it follows that for any ring Λ, the pair (Flat(Λ), Flat(Λ)^{⊥}) is a hereditary pretorsion theory in Mod(Λ). As we now explain this example also shows that there exists a pretorsion theory (X , Y) such that X ∩Y is not functorially finite, hence we cannot speak of a torsion theory in C/X ∩ Y.

Example. Consider the flat pretorsion theory (Flat(Λ), Flat(Λ)^{⊥}) in Mod(Λ) and assume that Λ is a left coherent and right IF-ring, that is any right injective module is flat, which is not right Noetherian. Then it is not difficult to see that the intersection ω = Flat(Λ) ∩ Flat(Λ)^{⊥} is precisely the category of injective right modules, which is contravariantly finite iff Λ is right Noetherian. Note that ω is covariantly finite since Λ is left coherent.

When the article was in the process of writing we learned that the concept of a pretorsion theory was introduced recently by Trlifaj under the name complete cotorsion theories, see [78].

5. Tilting Modules

An important source of examples of good pairs for which the induced torsion or torsion-free class is easily described emerges from cotilting or tilting modules over an Artin algebra Λ. It is well-known that the category mod(Λ) of finitely generated modules is a Krull-Schmidt category.

Let T be a finitely generated Λ-module with Ext_{Λ}^{i}(T, T) = 0, ∀i > 0. Recall that T is a cotilting module if id_{Λ}T < ∞ and the injective cogenerator D(Λ) lies in { \atop add(T)}. Dually T is a tilting module if pd_{Λ}T < ∞ and the projective generator Λ lies in { \atop add(T)}. If ω is a self-orthogonal subcategory then we denote by X_{ω} the subcategory

X_{ω} := { C ∈ mod(Λ) | Ext_{Λ}^{i}(C, ω) = 0, ∀i > 0 }.

The subcategory Y_{ω} is defined dually. If ω = add(T), for a self-orthogonal module T, then we use the notations X_{T} and Y_{T}.

Note. We warn the reader that the above notation is not standard in the literature. The subcategory above is denoted usually by ^{⊥}ω. However in this paper we reserved the last notation for the subcategory {C ∈ mod(Λ) | Ext_{Λ}^{1}(C, ω) = 0}.

If T is a cotilting module, then X_{T} ∩ { \atop add(T)} = add(T), and (X_{T} , { \atop add(T)}) is a good pair with X_{T} resolving. Hence (X_{T} /add(T), { \atop add(T)}/add(T)) is a torsion theory in \underline{mod}_{T} (Λ) := mod(Λ)/add(T) which is pretriangulated, since add(T) is always functorially finite [10]. It is well-known that T is cotilting module iff add(T) is a cogenerator for X_{T} and X̂_{T} = mod(Λ) iff { \atop CM(T)} = mod(Λ), see [6]. We state explicitly the following result which characterizes the cotilting modules in torsion theoretic terms, leaving its dual concerning tilting modules to the reader.

Proposition 5.1. Let Λ be an Artin algebra and let T be a finitely generated Λ-module with Ext_{Λ}^{i}(T, T) = 0, ∀i > 0. Then the following are equivalent.

(i) T is a cotilting module.

(ii) (X_{T} /add(T), { \atop add(T)}/add(T)) is a torsion theory in \underline{mod}_{T} (Λ).

Proof. The implication (i) ⇒ (ii) follows by the above considerations. (ii) ⇒ (i) By induction we see easily that Ext_{Λ}^{1}(X, Y) = 0, ∀X ∈ X_{T} , ∀Y ∈ { \atop add(T)}. Then by Proposition 3.1 and Theorem 3.3 we have that add(T) is an Ext-injective cogenerator for X_{T} and (X_{T} , { \atop add(T)}) is a good pair in mod(Λ). Then by Proposition 4.6 we infer that X̂_{T} = { \atop CM(T)} = mod(Λ), hence T is a cotilting module.

Now let T be a tilting and cotilting module. Then we have the torsion theories:

(i) (X /add(T), Y/add(T)) in \underline{mod}_{T} (Λ), where X = X_{T} and Y = { \atop add(T)}.

(ii) (Z/add(T), W/add(T)) in \underline{mod}_{T} (Λ), where Z = { \atop add(T)} and W = Y_{T}. When Λ has finite global dimension, then by [6] we have X = Z and Y = W, so that the torsion theories coincide. When id_{Λ}T ≤ 1, then X is the torsion-free class of a torsion theory (T , X) in mod(Λ), and when pd_{Λ}T ≤ 1, then Y is the torsion class of a torsion theory (Y, F) in mod(Λ). We have the following close relationship between the torsion theories, a special case of which was observed by Dlab and Ringel in connection with tilting modules associated to quasi-hereditary algebras, see Theorem 4.3 in [30].

Proposition 5.2. With the above notation, if Λ has finite global dimension and T is a classical tilting-cotilting module, then we have equivalences:

X /add(T) ⟶^{≈} F and Y/add(T) ⟶^{≈} T .

Proof. Let t : mod(Λ) ⟶ T be the right adjoint of the inclusion T ↪ mod(Λ) and let f : mod(Λ) ⟶ F be the left adjoint of the inclusion F ↪ mod(Λ). Then for any module C there exists a functorial exact sequence 0 ⟶ t(C) ⟶ C ⟶ f(C) ⟶ 0.

We consider the functor f : X_{T} ⟶ F. Then f(T) = 0, hence there is induced a functor F : X_{T} /add(T) ⟶ F. Let F be a module in F and consider the right X_{T} −approximation sequence 0 ⟶ Y_{F} ⟶ X_{F} ⟶ F ⟶ 0 of F. Then we know that Y_{F} ∈ Y. Hence there exists an extension 0 ⟶ T_{1} ⟶ T_{0} ⟶ Y_{F} ⟶ 0 with T_{i} ∈ add(T). Applying to this sequence the functor (T, −) it follows easily that Ext_{Λ}^{1}(T, Y_{F}) = 0, that is Y_{F} ∈ T . Hence t(X_{F}) = Y_{F} and then by definition F(X_{F}) = F. Hence F is surjective on objects. If α : X_{1} ⟶ X_{2} is a morphism in X_{T} such that f(α) = 0, then obviously α factors through t(X_{2}) ∈ Y. This implies that α factors through add(T), since (X_{T} /add(T), Y/add(T)) = 0. Hence \underline{α} = 0 in X_{T} /add(T). Finally if β : F_{1} ⟶ F_{2} is a morphism in F, then the pull-back of the extension 0 ⟶ Y_{F_{2}} ⟶ X_{F_{2}} ⟶ F_{2} ⟶ 0 along the composition X_{F_{1}} ⟶ F_{1} ⟶ F_{2}, splits since Ext_{Λ}^{1}(X_{F_{1}}, Y_{F_{2}}) = 0. This implies that the composition X_{F_{1}} ⟶ F_{1} ⟶ F_{2} factors through X_{F_{2}} ⟶ F_{2} say via a morphism α : X_{F_{1}} ⟶ X_{F_{2}}. Then by construction F(\underline{α}) = β. Hence F is full. We infer that F : X_{T} /add(T) ⟶ F is an equivalence. The proof of the second equivalence is dual.

The above example shows that it can happen that the torsion or torsion-free class is an abelian category, or even a module category. Indeed, keep the above assumptions and suppose in addition that the projective cover of T is injective. Then as in [30] one can show that the regular module Λ is a projective generator of X /add(T), hence we have an equivalence X /add(T) ≈ mod(\underline{End}_{T} (Λ)). Moreover D(Λ) is an injective cogenerator of W/add(T), hence we have an equivalence (W/add(T))^{op} ⟶^{≈} mod(\underline{End}_{T} (D(Λ)). It is also easy to see that the hearts of all the torsion theories in the above example are trivial.

The triviality of the hearts is a rather typical situation encountered, if the pretriangulated category is not triangulated. However the following example shows that the heart of a torsion theory in a stable non-triangulated category can be a non-zero abelian category.

Example. Let Γ = k[x]/(x^{2}), where k is a field and consider the Artin algebra

Λ = ( \begin{matrix}Γ & 0 \\ Γ & Γ \end{matrix} ). Then Λ is cotilting module of injective dimension 1. The inde-

composable Λ-modules fall into the following three groups, where we represent the Λ-modules via Γ-morphisms, see [12].

(α) X = CM(Λ) = add({(Γ ⟶^{=} Γ), (0 ⟶ Γ), (0 ⟶ k), (k ⟶^{=} k), (k ↪ Γ)}).

(β) Y = P_{Λ}^{<∞} = add({(Γ ⟶^{=} Γ), (0 ⟶ Γ), (Γ ⟶ 0), (Γ ⟶^{f} Γ)}), where f is nonzero and non-invertible. Here P_{Λ}^{<∞} denotes the full subcategory of modules with finite projective dimension.

(γ) U = add({(k ⟶ 0), (Γ ⟶^{g} k)}), where g is non-zero. Denote by H_{l} = X /P_{Λ} ∩ Σ(Y/P_{Λ}) the left heart of the induced torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ). It is not difficult to see that the only indecomposable object in H_{l} is given by (k ↪ Γ) with stable endomorphism ring, the field k. It follows that H_{l} is a non-zero abelian category. Observe that the right heart is trivial. We don’t know if in general the left or right heart of a torsion theory in a stable category is always zero or abelian.

We have seen that the right adjoint R of the inclusion of a torsion class tends to be left exact. However R is not right exact in general. Indeed in the above example, since Hom_{Λ}[(k ⟶ 0), Λ)] = 0, we have a sequence (k ⟶ 0) ⟶ 0 ⟶ 0 ⟶ 0, which induces the right triangle (k ⟶ 0) ⟶ 0 ⟶ 0 ⟶ 0 in \underline{mod}(Λ). The right X -approximation of (k ⟶ 0) is (k ↪ Γ), which is not projective. Since there is no exact sequence (k ↪ Γ) ⟶ P_{1} ⟶ P_{0} ⟶ 0, with P_{1}, P_{0} projectives, by the construction of R, we infer that R is not right exact.

Interesting examples of cotilting modules T occur for quasi-hereditary algebras, where we have that X_{T} is the category of good modules [74]. Other special cases are T = Λ, where Λ is a Gorenstein Artin algebra or a commutative complete local Gorenstein ring, and T = ω, where ω is a dualizing bimodule over a Cohen-Macaulay Artin algebra or a commutative complete local Noetherian Cohen-Macaulay ring. In all these cases X_{T} = CM(T) is the category of (maximal) Cohen-Macaulay modules.

We can also speak more generally of ω being a cotilting subcategory of C, meaning that Ext_{Λ}^{i}(ω, ω) = 0, ∀i > 0, that there is some t ≥ 0 such that id_{Λ}T ≤ t, for each T ∈ ω and each injective object of C lies in ω̂. This gives rise to similar results which generalize easily to the infinite-dimensional case for any ring.

CHAPTER VI Triangulated Torsion(-Free) Classes in Stable Categories

In this chapter we study the natural question of when the torsion, resp. torsionfree, class of a torsion theory in a stable pretriangulated category, is a triangulated subcategory. When this happens the torsion theory is hereditary, resp. cohereditary. An instance of this situation is the case of good triples which, are analogous to TTF-theories and are studied in the last section. Although most of the results of this and the next section are valid in much more general situations, for simplicity and concreteness we work in the category of finitely generated modules over an Artin algebra. In this case the question has strong connections with tilting theory and leads to various torsion-theoretic characterizations of Gorenstein algebras.

1. Triangulated Subcategories

In this brief section we state and prove some general results on triangulated subcategories of a stable category, which will be useful in the later sections. We assume throughout that C is an abelian category.

Lemma 1.1. Let X be a full subcategory of C, closed under kernels of epics, and let ω ⊆ X be a full subcategory.

(α) If ω is an Ext-injective cogenerator of X, then the stable category X /ω is right triangulated.

(β) If ω is contravariantly finite in X and any ω-epic in X is epic, e.g. if ω is an Ext-projective generator of X, then the stable category X /ω is left triangulated.

(γ) If ω is contravariantly finite in X and an Ext-injective cogenerator for X, and if any ω-epic in X is epic, then the stable category X /ω is pretriangulated.

Proof. (α) Obviously ω is covariantly finite in X. To show that X /ω is right triangulated it suffices by [16] to show that C is in X whenever X_{1} ⟶^{f} X_{2} ⟶^{g} C ⟶ 0 is exact in C where X_{1}, X_{2} ∈ X and f is ω-monic. Since X_{1} is embedded in an object in ω, it follows that f is monic. Since ω is a cogenerator for X and X_{2} is in X, there exists an exact sequence 0 ⟶ X_{2} ⟶^{κ} ω_{0} ⟶^{λ} X_{3} ⟶ 0, where ω_{0} ∈ ω and X_{3} ∈ X. Consider the following exact commutative diagram

0 ⟶ X_{1} ⟶^{f} X_{2} ⟶^{g} C ⟶ 0
||↓  ^{κ}↓  ^{α}↓
0 ⟶ X_{1} ⟶^{h} ω_{0} ⟶^{ε} D ⟶ 0

Then κ induces an isomorphism Coker(κ) ≅ Coker(α) ≅ X_{3}. We claim that h is ω-monic. Indeed let ϕ : X_{1} ⟶ T be a morphism with T ∈ ω. Since f is ω-monic, there exists τ : X_{2} ⟶ T such that f ∘ τ = ϕ. Since Coker(κ) ∈ X and ω consists of Ext-injectives in X, it follows that κ is ω-monic. Hence there exists σ : ω_{0} ⟶ T such that κ ∘ σ = τ. Then h ∘ σ = f ∘ κ ∘ σ = f ∘ τ = ϕ. Hence ϕ factors through h and then h is ω-monic, i.e. a left ω-approximation of X_{1}. Since ω is an Ext-injective cogenerator of X, there exists an exact sequence 0 ⟶ X_{1} ⟶^{ζ} ω′_{0} ⟶^{η} X ⟶ 0 with ω′_{0} ∈ ω and X ∈ X. Then obviously ζ is a left ω-approximation of X_{1}. Since h, ζ are both left ω-approximations of X_{1}, standard arguments show that in the stable category C/ω there exists an isomorphism \underline{D} ≅ \underline{X} and then D ∈ X, since X ∈ X. Since Coker(κ) and D are in X and X is closed under kernels of epics, we infer that C ∈ X.

(β) Since any ω-epic in X is epic and X is closed under kernels of epics, we have that any ω-epic in X has a kernel in X. Then by [16], the stable category X /ω is left triangulated. Part (γ) follows easily from (α) and (β).

The following result gives a useful criterion for deciding when a stable category is triangulated.

Proposition 1.2. Let X be a full subcategory of C closed under extensions and kernels of epics, and let ω be an Ext-injective cogenerator for X. Then the stable category X /ω is triangulated iff ω is an Ext-projective generator for X.

Proof. If ω is an Ext-projective generator for X, then obviously ω is contravariantly finite in X and any ω-epic in X is epic. Then by the above lemma, we have that X /ω is pretriangulated. Let 0 ⟶ X ⟶ T ⟶ X′ ⟶ 0 be exact with T ∈ ω and X, X′ ∈ X. Since ω is Ext-injective in X, we have that Σ(\underline{X}) = \underline{X}′ in X /ω. Since ω is Ext-projective in X, we have that ΩΣ(\underline{X}) = Ω(\underline{X}′) = \underline{X}. It follows that the canonical morphism Id_{X /ω} ⟶ ΩΣ is invertible. Dually the canonical morphism ΣΩ ⟶ Id_{X /ω} is invertible. Hence X /ω is triangulated. Conversely if X /ω is a triangulated subcategory of C/ω, then Σ|_{X /ω} is surjective on objects,

so for any object X ∈ X, there exists a right exact sequence X′ ⟶^{β} T ⟶^{α} X ⟶ 0 such that β is ω-monic and T ∈ ω. Since ω is a cogenerator of X, β is monic, so ω is a generator of X. It remains to show that the objects of ω are Ext-projective in X. Let 0 ⟶ X ⟶^{f} A ⟶^{g} T ⟶ 0 be an extension with X ∈ X and T ∈ ω. Then A is in X since the latter is closed under extensions. Since T is Ext-injective in X,

the above extension induces a triangle \underline{X} ⟶^{\underline{f}} \underline{A} ⟶ 0 ⟶ Σ(\underline{X}) in X /ω. Since the latter is triangulated we have that \underline{f} : \underline{X} ⟶ \underline{A} is invertible. Hence there exists a morphism h : A ⟶ X such that the endomorphism 1_{X} − f ∘ h of X factors through an object T′ in ω, i.e. 1_{X} − f ∘ h = κ ∘ λ, where κ : X ⟶ T′ and λ : T′ ⟶ X. Since T′ ∈ ω and the objects of ω are Ext-injective in X, the morphism κ factors through f. Hence there exists a morphism ρ : A ⟶ T′ such that κ = f ∘ ρ. Then 1_{X} = f ∘ h + κ ∘ λ = f ∘ h + f ∘ ρ ∘ λ = f ∘ (h + ρ ∘ λ). We infer that f is split monic and then T is Ext-projective. Hence the objects of ω are Ext-projective in X.

For instance if X is a resolving subcategory of an abelian category C with enough projectives, and the full subcategory ω of projective objects of C form an Ext-injective cogenerator of X, then the stable category X /ω is triangulated.

Example. The full subcategory X of Mod(kG) of cofibrant modules in the example after Remark V.4.7, satisfies the assumptions of the above Proposition, where ω is the full subcategory of projective kG-modules. Hence X /ω is triangulated.

We leave to the reader the dualization of the above result.

2. Triangulated Torsion(-Free) Classes

Throughout this section we fix an Artin algebra Λ. When ω is a functorially finite subcategory of the category mod(Λ) of finitely generated modules, we investigate when a torsion class in the stable pretriangulated category mod(Λ)/ω is triangulated. We deduce also some consequences for Gorenstein algebras.

Let CM(Λ) be the full subcategory of Cohen-Macaulay modules in mod(Λ). Since P_{Λ} is an Ext-projective generator and an Ext-injective cogenerator of CM(Λ), by Proposition 1.2 the stable category \underline{CM}(Λ) modulo projectives is a triangulated subcategory of \underline{mod}(Λ). Since mod(Λ) is a Krull-Schmidt category it follows by Corollary V.4.2 that \underline{CM}(Λ) is a torsion class in \underline{mod}(Λ) if and only if CM(Λ) is contravariantly finite. The next result shows that the example of Cohen-Macaulay modules is universal in the sense that any triangulated torsion class induced by a good pair consists of Cohen-Macaulay modules.

Theorem 2.1. Let (X , Y) be a good pair in mod(Λ) and assume that X is resolving and ω := X ∩ Y is functorially finite. Then the following are equivalent:

(i) The torsion class X /ω is triangulated.

(ii) ω = P_{Λ}. If this is the case, then X ⊆ CM(Λ), P_{Λ}^{<∞} ⊆ Y, the coreflection functor R : mod(Λ)/ω ⟶ X /ω is left exact and the torsion theory (X /ω, Y/ω) is hereditary.

Conversely if (X /P_{Λ}, Y/P_{Λ}) is a torsion theory in \underline{mod}(Λ) with X ⊆ CM(Λ), then (X , Y) is a good pair in mod(Λ) with X ∩ Y = P_{Λ}.

Proof. (ii) ⇒ (i) Since X is resolving we have that X /P_{Λ} is a left triangulated subcategory of \underline{mod}(Λ). By Proposition V.1.3, P_{Λ} is an Ext-injective cogenerator of X. Then Proposition 1.2 implies that X /P_{Λ} is a triangulated subcategory.

(i) ⇒ (ii) Assume that the torsion class X /ω is a triangulated subcategory of \underline{mod}_{ω}(Λ). By Proposition V.1.3, ω is an Ext-injective cogenerator of X and then by Proposition 1.2, ω is an Ext-projective generator of X. Since X contains the projectives, it follows that any projective is in ω, i.e. P_{Λ} ⊆ ω. We show that P_{Λ} is a cogenerator of X. Let X ∈ X. Since ω is a cogenerator of X, there exists a short exact sequence 0 ⟶ X ⟶^{κ} T′ ⟶^{ν} X′ ⟶ 0 with T′ ∈ ω and X′ ∈ X. Let 0 ⟶ K ⟶^{β} P ⟶^{α} T′ ⟶ 0 be a projective cover of T′ and consider the following exact commutative diagram:

0 ⟶ X′′ ⟶^{γ} P ⟶^{δ} X′ ⟶ 0
ϵ↓  ^{α}↓  ^{||}↓
0 ⟶ X ⟶^{κ} T′ ⟶^{ν} X′ ⟶ 0

Then the left square is bicartesian, hence it induces a short exact sequence (E_{1}) : 0 ⟶ X′′ ⟶ X ⊕ P ⟶ T′ ⟶ 0 and a short exact sequence (E_{2}) : 0 ⟶ K ⟶ X′′ ⟶^{ϵ} X ⟶ 0. Since X is resolving and X′, T′ ∈ X, it follows that K, X′′ ∈ X. Applying (−, ω) to the sequence (E_{1}) and using that the subcategory ω is selforthogonal, we see that the resulting sequence is exact. Hence it induces a right triangle \underline{X}′′ ⟶ \underline{X}⊕\underline{P} ⟶ \underline{T}′ ⟶ Σ(\underline{X}′′) in X /ω. Since ω contains the projectives and T′ ∈ ω, it follows that the above right triangle reduces to \underline{X}′′ ⟶^{\underline{ϵ}} \underline{X} ⟶ 0 ⟶ Σ(\underline{X}′′). Since X /ω is triangulated, it follows that \underline{ϵ} : \underline{X}′′ ⟶ \underline{X} is invertible. Since X ∈ X, it follows trivially that the sequence (E_{2}) remains exact after the application of the functor (−, ω), hence it induces a right triangle \underline{K} ⟶ \underline{X}′′ ⟶^{\underline{ϵ}} \underline{X} ⟶ Σ(\underline{K}) in X /ω. Since the latter is triangulated and \underline{ϵ} is invertible, it follows that \underline{K} = 0, hence K ∈ ω. Now using that ω is a self-orthogonal subcategory, it follows that the short exact sequence 0 ⟶ K ⟶ P ⟶^{α} T′ ⟶ 0 splits, hence T′ is projective. Then from the short exact sequence 0 ⟶ X ⟶ T′ ⟶ X′ ⟶ 0, we infer that P_{Λ} is a cogenerator of X. In particular for any object T ∈ ω there exists a short exact sequence 0 ⟶ T ⟶ P ⟶ X ⟶ 0, where P is projective and X ∈ X, which obviously splits since ω is an Ext-injective cogenerator of X. We conclude that T is projective and this implies that ω = P_{Λ}.

Assume now that condition (i) holds. Since X is resolving and P_{Λ} is an Extinjective cogenerator of X, it follows from Section V.4 that X ⊆ CM(Λ). Since Y is coresolving and contains P_{Λ}, it follows easily that P_{Λ}^{<∞} ⊆ Y. It remains to show that the torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) is hereditary. Since X is resolving and ω = P_{Λ} it is easy to check that the inclusion i : X /P_{Λ} ↪ \underline{mod}(Λ) is left exact. By Proposition 3.6 of [6], any exact sequence 0 ⟶ A ⟶ B ⟶ C ⟶ 0 in mod(Λ) induces an exact sequence 0 ⟶ X_{A} ⟶ X_{B} ⟶ X_{C} ⟶ 0 in X where X_{A}, X_{B}, X_{C} are right X -approximations of A, B, C. The last sequence induces a triangle ΩR(\underline{C}) ⟶ R(\underline{A}) ⟶ R(\underline{B}) ⟶ R(\underline{C}) in X /P_{Λ}. If B is projective then the above triangle induces an isomorphism ΩR(\underline{C}) ⟶^{≅} RΩ(\underline{C}) ≅ R(\underline{A}). These facts imply that the coreflection functor R : \underline{mod}(Λ) ⟶ X /P_{Λ} is left exact. Then the idempotent functor iR : \underline{mod}(Λ) ⟶ \underline{mod}(Λ) is left exact, so that the torsion theory (X /P_{Λ}, Y/P_{Λ}) is hereditary. The last assertion follows from Lemma V.1.6 and Proposition V.3.1.

Remark 2.2. The above theorem implies that if Λ has finite global dimension, then Y = mod(Λ) and X = P_{Λ}. Hence for algebras of finite global dimension, the only triangulated torsion class induced by a good pair is the zero subcategory.

The next result describes some important consequences of the existence of a good pair (X , Y) in mod(Λ) with X resolving and X ∩Y consisting of the projectives, which will be useful in Chapter VIII in connection with Cohen-Macaulay modules.

Proposition 2.3. Let (X , Y) be a good pair in mod(Λ) with X resolving and X ∩ Y = P_{Λ}. Then we have the following.

(i) C ∈ X̂ if and only if Y_{C} ∈ P_{Λ}^{<∞}.

(ii) P_{Λ}^{<∞} = Y ∩ X̂.

(iii) X̂ = mod(Λ) iff Y = P_{Λ}^{<∞}.

Proof. (i) Since X contains the projectives it follows that P_{Λ}^{<∞} ⊆ X̂. Assume now that C has a finite exact resolution 0 ⟶ X_{n+1} ⟶ X_{n} ⟶ ⋯ ⟶ X_{1} ⟶ X_{0} ⟶^{α} C ⟶ 0 by modules from X. If n = 0, then consider the pull-back diagram

0 ⟶ Y_{C} ⟶ A ⟶ X_{0} ⟶ 0
∥↓  ↓  ^{α}↓
0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0

of the right X -approximation sequence of C along α. Since X_{0} is in X, α factors through f_{C} and this implies that A ≅ Y_{C} ⊕ X_{0}. But from the above diagram it follows that A is in X since the kernel of α is X_{1} ∈ X and X is closed under extensions. We infer that Y_{C} ∈ X ∩ Y = P_{Λ}. By induction on the length n + 1 of the resolution of C by objects from X, we deduce easily that pd_{Λ}Y_{C} ≤ n. Hence Y_{C} has finite projective dimension.

(ii) If C is in P_{Λ}^{<∞}, then by Theorem 2.1 we have that C is in Y. Since X contains the projectives, we have P̂_{Λ} = P_{Λ}^{<∞} ⊆ X̂. Hence C lies in Y ∩ X̂. Conversely if C lies in Y ∩ X̂, then by (i) we have that Y_{C} ∈ P_{Λ}^{<∞}. However since C is in Y the right X -approximation of C is projective, hence Y_{C} = Ω(C). We infer that C has finite projective dimension.

(iii) Follows from (ii).

We continue with some consequences of Theorem 2.1. First we recall from Section IV.3 that an Artin algebra Λ is called Gorenstein, if Λ has finite injective dimension both as a left and right Λ-module, equivalently Λ is a cotilting module.

The following result characterizes the cotilting projective modules in torsion theoretic terms.

Proposition 2.4. For a Λ-module T the following are equivalent.

(i) T is a cotilting module and the torsion class X_{T} /add(T) is a triangulated subcategory of \underline{mod}_{T} (Λ).

(ii) T is cotilting projective module.

(iii) T is a projective generator and Λ is a Gorenstein algebra.

Proof. (i) ⇒ (ii) By Theorem 2.1 we have that add(T) = P_{Λ}, so T is projective. (ii) ⇒ (iii) Since T is cotilting, T and Λ have the same number of nonisomorphic indecomposable summands. Hence T is a projective generator and then obviously Λ is a cotilting module, and consequently Λ is Gorenstein. (iii) ⇒ (i) The hypothesis implies that Λ is a cotilting module and add(T) = P_{Λ}. Then (i) follows by Theorem 2.1.

The next result describes when the torsion and the torsion-free class are triangulated.

Corollary 2.5. Let (X , Y) be a good pair in mod(Λ) with X resolving and ω = X ∩ Y functorially finite. Then the following are equivalent:

(i) The torsion class X /ω and the torsion-free class Y/ω are triangulated. (ii) Λ is self-injective and ω = P_{Λ}.

Proof. If (i) holds, then by Theorem 2.1 and its dual we have ω = P_{Λ} = I_{Λ}. Hence Λ is self-injective. If (ii) holds, then \underline{mod}(Λ) is triangulated. Since X is resolving, X /P_{Λ} is closed under Ω, and this implies that X /P_{Λ} is a triangulated subcategory of \underline{mod}(Λ). Since Y/P_{Λ} = (X /P_{Λ})^{⊥}, the same is true for Y/P_{Λ}.

Actually in the self-injective case we have a bijection between good pairs and hereditary torsion theories.

Proposition 2.6. Let Λ be a self-injective Artin algebra and let X and Y be full subcategories of mod(Λ). Then the following are equivalent:

(i) (X , Y) is a good pair with X (co-)resolving and X ∩ Y = P_{Λ}.

(ii) (X /P_{Λ}, Y/P_{Λ}) is a (co-)hereditary torsion theory in \underline{mod}(Λ).

(iii) \underline{Hom}_{Λ}(X , Y) = 0, X /P_{Λ} and Y/P_{Λ} are triangulated subcategories of \underline{mod}(Λ)

and the latter is generated by X /P_{Λ} and Y/P_{Λ}. The map (X , Y) ↦ (X /P_{Λ}, Y/P_{Λ}) gives a bijection between good pairs in mod(Λ) with X ∩ Y = P_{Λ} and X resolving, and hereditary torsion theories in \underline{mod}(Λ).

Proof. The implication (i) ⇒ (ii) follows from Corollary 2.5 and the equivalence (ii) ⇒ (iii) follows from Proposition I.1.9. (ii) ⇒ (i) By Proposition V.3.1 it suffices to show that Ext_{Λ}^{1}(X , Y) = 0. Since Λ is self-injective, for any X ∈ X and Y ∈ Y we have Ext_{Λ}^{1}(X, Y) ≅ \underline{Hom}_{Λ}(Ω(X), Y) ≅ \underline{Hom}_{Λ}(X, Σ(Y)). Since X is closed under Ω we infer that Ext_{Λ}^{1}(X , Y) = 0.

If (X , Y) is a good pair with X resolving such that the torsion class X /ω is triangulated, then by Theorem 2.1 we know that ω = P_{Λ} and the coreflection functor R : \underline{mod}(Λ) ⟶ X /P_{Λ} is a left exact functor of pretriangulated categories. It is natural to ask if this functor is the universal left exact functor out of \underline{mod}(Λ) to triangulated categories. In this connection we have the following result, which gives a nice description of the coreflection functor. Its dual is left to the reader.

Proposition 2.7. Let (X , Y) be a good pair in mod(Λ) with ω = X ∩Y functorially finite and X resolving. If the torsion class X /ω in mod(Λ)/ω is triangulated (hence ω = P_{Λ}), then the following are equivalent.

(i) The coreflection functor R : \underline{mod}(Λ) ⟶ X /P_{Λ} is the universal left exact functor out of \underline{mod}(Λ) to triangulated categories, i.e. X /P_{Λ} is the left stabilization of \underline{mod}(Λ).

(ii) Λ is Gorenstein. If this is the case, then X = CM(Λ) and R = Ω^{−d}Ω^{d}, where idΛ = d.

Proof. (i) ⇒ (ii) By Proposition II.5.2 we have that for any module C there exists n ≥ 0 such that Ω^{n}(\underline{C}) ∈ X /P_{Λ}. This implies that X̂ = mod(Λ). Then { \atop CM(Λ)} = mod(Λ) since, by Theorem 2.1 X consists of Cohen-Macaulay modules. By Proposition V.5.1 this implies that Λ is a cotilting module, hence Λ is Gorenstein. (ii) ⇒ (i) Since Λ is a cotilting module, by [6] we have Y ⊆ I_{Λ}^{<∞} = P_{Λ}^{<∞}, and then Y = P_{Λ}^{<∞} by Theorem 2.1. Now (i) follows from Proposition II.5.2. The description of R follows from [17].

3. Good Triples and TTF-Theories

In analogy with the concept of a TTF-theory in an abelian or triangulated category in this section we study the concept of a good triple. We give necessary and sufficient conditions for the existence of a good triple, and we show that any such triple induces two torsion theories in two different stable pretriangulated categories. In addition the torsion class of the first and the torsion-free class of the second are triangulated and they are triangle equivalent.

We continue to work in the category mod(Λ) of finitely generated modules over an Artin algebra Λ.

Definition 3.1. A triple (X , Y, Z) of subcategories of mod(Λ) is called a good triple if (X , Y) and (Y, Z) are good pairs.

The primary motivating example of a good triple is the following.

Example. By Proposition V.5.1 and its dual it follows that for a Gorenstein algebra Λ, P_{Λ}^{<∞} = I_{Λ}^{<∞} is a functorially finite resolving and coresolving subcategory of mod(Λ) and we have good pairs (CM(Λ), P_{Λ}^{<∞}) and (I_{Λ}^{<∞}, CoCM(D(Λ))). Hence

(CM(Λ), P_{Λ}^{<∞} = I_{Λ}^{<∞}, CoCM(D(Λ)))

is a good triple in mod(Λ). Moreover CM(Λ)∩P_{Λ}^{<∞} = P_{Λ} and I_{Λ}^{<∞} ∩CoCM(D(Λ)) = I_{Λ}, we have a torsion theory (CM(Λ)/P_{Λ}, P_{Λ}^{<∞}/P_{Λ}) in \underline{mod}(Λ) with CM(Λ)/P_{Λ} triangulated, and a torsion theory (I_{Λ}^{<∞}/I_{Λ}, CoCM(D(Λ))/I_{Λ}) in \overline{mod}(Λ) with CoCM(D(Λ))/I_{Λ} triangulated. Finally by [8] it follows that the Nakayama functor N^{+} induces a triangle equivalence CM(Λ)/P_{Λ} ⟶^{≈} CoCM(D(Λ))/I_{Λ}.

The next result generalizes the above observations to the non Gorenstein case.

Theorem 3.2. The following are equivalent for a subcategory Y of mod(Λ).

(i) Y is a functorially finite resolving and coresolving subcategory of mod(Λ). (ii) There exists a good triple (X , Y, Z) in mod(Λ) with Y (co)resolving. (iii) There is a torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) with X /P_{Λ} triangulated

and a torsion theory (Y/I_{Λ}, Z/I_{Λ}) in \overline{mod}(Λ) with Z/I_{Λ} triangulated. If one of the above equivalent conditions holds, then there is a triangle equivalence

Φ : X /P_{Λ} ⟶^{≈} Z/I_{Λ}.

Proof. (i) ⇒ (ii) Follows from Theorem V.1.2 and Proposition V.1.3.

(i) ⇒ (iii) Setting X = ^{⊥}Y and Z = Y^{⊥}, we infer by (ii) that (X , Y) and (Y, Z) are good pairs with X resolving and Z coresolving. Obviously P_{Λ} ⊆ X ∩ Y. If T ∈ X ∩Y and 0 ⟶ K ⟶ P ⟶ T ⟶ 0 is exact with P projective, then K ∈ X ∩Y since X , Y are resolving. It follows that the sequence splits since Ext_{Λ}^{1}(X , Y) = 0. Hence T ∈ P_{Λ}. We infer that P_{Λ} = X ∩ Y and dually I_{Λ} = Y ∩ Z. By Theorem V.3.7, we have that (X /P_{Λ}, Y/P_{Λ}) is a torsion theory in \underline{mod}(Λ) and (Y/I_{Λ}, Z/I_{Λ}) is a torsion theory in \overline{mod}(Λ). Then by Proposition 1.2, X /P_{Λ} is a triangulated subcategory of \underline{mod}(Λ) and dually Y/I_{Λ} is a triangulated subcategory of \overline{mod}(Λ).

(iii) ⇒ (i) By Theorem V.2.1, Y is functorially finite. Since Y is a left triangulated subcategory of \underline{mod}(Λ), it follows that P_{Λ} is a generator of Y and since Y is right triangulated subcategory of \overline{mod}(Λ), it follows that I_{Λ} is a cogenerator of Y. Since X /P_{Λ} is a triangulated subcategory of \underline{mod}(Λ), it follows that X is resolving and P_{Λ} is an Ext-injective cogenerator of X. By Proposition V.3.1 we have Ext_{Λ}^{1}(X , Y) = 0, and then by Theorem V.3.3 we infer that X^{⊥} = Y and hence Y is coresolving. Dually we have Ext_{Λ}^{1}(Y, Z) = 0 and by Theorem V.3.3, Z is coresolving having I_{Λ} as an Ext-projective generator and ^{⊥}Z = Y. Hence Y is resolving.

(ii) ⇒ (i) If (X , Y, Z) is a good triple, then obviously Y is functorially finite. If Y coresolving, then X is resolving by Proposition V.1.3 and the first part of the proof of (i) ⇒ (ii) shows that X ∩Y = P_{Λ}, and X /P_{Λ} is a triangulated subcategory of \underline{mod}(Λ). If 0 ⟶ C ⟶ Y_{1} ⟶ Y_{2} ⟶ 0 is exact with Y_{i} ∈ Y, then applying (X, −) to this sequence we infer that Ext^{2}(X, C) = Ext^{1}(Ω(X), C) = 0, ∀X ∈ X. Since X /P_{Λ} is triangulated, any object of X is of the form Ω(X). Hence Ext^{1}(X , C) = 0, i.e. C ∈ Y and Y is coresolving. If Y is coresolving, the dual argument shows that Y is resolving.

Now if (i) holds then the last assertion follows from [8]. For completeness we include the argument for the existence of an equivalence Φ : X /P_{Λ} ⟶ Z/I_{Λ} and we show in addition that Φ is exact. If X ∈ X, let 0 ⟶ Z_{X} ⟶^{g_{X}} Y_{X} ⟶^{f_{X}} X ⟶ 0 be a minimal right Y-approximation sequence of X. By Wakamatsu’s Lemma it follows that Z_{X} ∈ Y^{⊥} = Z. We set Φ(\underline{X}) = \underline{Z}_{X}. Using that (Y/I_{Λ}, Z/I_{Λ}) = 0, it is easy to see that Φ is a well defined additive functor. Dually for any Z ∈ Z, let 0 ⟶ Z ⟶^{g^{Z}} Y^{Z} ⟶^{f^{Z}} X^{Z} ⟶ 0 be a minimal left Y-approximation sequence of Z. As above we have that X^{Z} ∈ ^{⊥}Y = X. Setting Ψ(\underline{Z}) = \underline{X}^{Z} and using that (X /P_{Λ}, Y/P_{Λ}) = 0, it is easy to see that we obtain a well defined additive functor Ψ : Z/I_{Λ} ⟶ X /P_{Λ}. Since Ext^{1}(X , Y) = 0, it follows that 0 ⟶ Z_{X} ⟶^{g_{X}} Y_{X} ⟶^{f_{X}} X ⟶ 0 is a left Y-approximation sequence of Z_{X}. This implies easily that

if 0 ⟶ Z_{X} ⟶^{g^{Z_{X}}} Y^{Z_{X}} ⟶^{f^{Z_{X}}} X^{Z_{X}} ⟶ 0 is a minimal left Y-approximation sequence of Z_{X}, then X ≅ X^{Z_{X}} ⊕ Y, where Y ∈ Y. Then Y ∈ X ∩ Y = P_{Λ}, hence \underline{X} ≅ \underline{X}^{Z_{X}} in X /P_{Λ}, i.e. \underline{X} ≅ ΨΦ(\underline{X}). Dually \underline{Z} ≅ ΦΨ(\underline{Z}), ∀Z ∈ Z. Hence Φ, Ψ are mutually inverse equivalences.

Finally we show that Φ is exact. Let X be in X and let 0 ⟶ Ω(X) ⟶ P ⟶ X ⟶ 0 be an exact sequence with P projective. Let 0 ⟶ Z_{Ω(X)} ⟶ Y_{Ω(X)} ⟶ Ω(X) ⟶ 0 be the minimal left Y-approximation of Ω(X). Then in Z/I_{Λ} we have ΦΩ(\underline{X}) = \underline{Z}_{Ω(X)}. Let 0 ⟶ Z_{Ω(X)} ⟶ I ⟶ Σ(Z_{Ω(X)}) ⟶ 0 be an exact sequence with I injective. The above exact sequences can be embedded in the following exact commutative diagram of short exact sequences:

0 ⟶ Z_{Ω(X)} ⟶ Y_{Ω_{X}} ⟶ Ω(X) ⟶ 0
↓  ↓  ↓
0 ⟶  I  ⟶ I ⊕ P ⟶ P ⟶ 0
↓  ↓  ↓
0 ⟶ Σ(Z_{Ω(X)}) ⟶ Ỹ ⟶ X ⟶ 0

Since Y is coresolving and contains the projectives, we have that Ỹ is in Y. Since Σ(Z_{Ω(X)}) is in Z and Ext^{1}(Y, Z) = 0, it follows that the lower exact sequence is a right Y-approximation sequence of X. If 0 ⟶ Z_{X} ⟶ Y_{X} ⟶ X ⟶ 0 is the minimal right Y-approximation sequence of X, then standard arguments show that Σ(Z_{Ω(X)}) ≅ Z_{X} ⊕ Y with Y in Y. Then Y ∈ Y ∩ Z = I_{Λ}. It follows that in Z/I_{Λ} we have Σ(\underline{Z}_{Ω(X)}) ≅ \underline{Z}_{X}, and then ΣΦΩ(\underline{X}) ≅ Φ(\underline{X}) or equivalently ΦΩ(\underline{X}) ≅ Σ^{−1}Φ(\underline{X}). It is easy to see that this induces a natural isomorphism ζ : ΦΩ ⟶^{≅} Σ^{−1}Φ, so that Φ commutes with the loop functors Ω, Σ^{−1}. It remains to show that Φ sends triangles in X /P_{Λ} to triangles in Z/I_{Λ}. Let 0 ⟶ X_{1} ⟶ X_{2} ⟶ X_{3} ⟶ 0 be an exact sequence in mod(Λ) with the X_{i} in X. Then by [6], there exists a short exact sequence 0 ⟶ Z_{X_{1}} ⟶ Z ⟶ Z_{X_{3}} ⟶ 0 where Z ∈ Z is such that \underline{Z} ≅ \underline{Z}_{X_{2}} in Z/I_{Λ}. Hence Φ(\underline{X}_{2}) ≅ \underline{Z} and then using the isomorphism ζ we have a triangle Σ^{−1}Φ(\underline{X}_{3}) ⟶ Φ(\underline{X}_{1}) ⟶ Φ(\underline{X}_{2}) ⟶ Φ(\underline{X}_{3}) in Z/I_{Λ}. Hence Φ is exact.

For the concept of an Auslander-Reiten triangle we refer to [38], [67].

Corollary 3.3. Let (X , Y, Z) be a good triple in mod(Λ). Then the triangulated categories X /P_{Λ} and Z/I_{Λ} admit Auslander-Reiten triangles from both sides. In particular X /P_{Λ} and Z/I_{Λ} admit a Serre functor (see [67]).

Proof. Since X is contravariantly finite, X has right almost split morphisms: if X is indecomposable in X and ρ : C ⟶ X is a right almost split morphism in mod(Λ), then the composition β : X_{C} ⟶^{f_{C}} C ⟶^{ρ} X is a right almost split morphism in X, where f_{C} is a right X -approximation of C. If X is non-projective, then it is easy to see that \underline{β} : \underline{X}_{C} ⟶ \underline{X} is a right almost split morphism in X /P_{Λ}. Since X /P_{Λ} is Krull-Schmidt, it follows that for any indecomposable object \underline{X} there exists a minimal right almost split morphism \underline{α} : \underline{X}′ ⟶ \underline{X} in X /P_{Λ}. Since X /P_{Λ} is triangulated, the triangle Ω(\underline{X}′′) ⟶ \underline{X}′ ⟶^{\underline{α}} \underline{X} ⟶ \underline{X}′′ is an Auslander-Reiten triangle, by [1], [38]. Hence X /P_{Λ} has Auslander-Reiten triangles from the right.

By duality we have that Z/I_{Λ} has Auslander-Reiten triangles from the left. Since X /P_{Λ} and Z/I_{Λ} are triangle equivalent, both of them have Auslander-Reiten triangles from left and right, in particular they admit a Serre functor by [67].

Remark 3.4. In general the triangle equivalence Φ : X /P_{Λ} ⟶^{≈} Z/I_{Λ} of Theorem 3.2 may not induced by a functor defined on \underline{mod}(Λ). However there is a triangle equivalence between X /P_{Λ} and Z/I_{Λ} which enjoys this property, and which has interesting connections with the equivalence DTr : \underline{mod}(Λ) ⟶^{≈} \overline{mod}(Λ) [12].

Let Y be a functorially finite resolving and coresolving subcategory of mod(Λ). Then DTr restricts to a triangle equivalence DTr : X /P_{Λ} ⟶^{≈} Z/I_{Λ} with quasiinverse induced by TrD, where X = ^{⊥}Y and Z = Y^{⊥}. This was observed in [8] in case Λ is a Gorenstein algebra and Y = P_{Λ}^{<∞}. Here we present a torsion-theoretic proof in the general case.

Recall that if X is a finitely generated Λ-module, then there is an isomorphism of functors Ext_{Λ}^{1}(−, DTr(X)) ⟶ D\underline{Hom}_{Λ}(\underline{X}, −) (see [12]). If X is in X, then evaluating the above isomorphism at Y we have Ext_{Λ}^{1}(Y, DTr(X)) ≅ D\underline{Hom}_{Λ}(X, Y) = 0. Hence DTr(X) lies in Y^{⊥} = Z. It follows that DTr restricts to a functor DTr : X /P_{Λ} ⟶ Z/I_{Λ} which obviously is an equivalence with quasi-inverse induced by TrD. Moreover the equivalence DTr is exact. Indeed, if 0 ⟶ X_{1} ⟶ X_{2} ⟶ X_{3} ⟶ 0 is an exact sequence with the X_{i} in X, then by a result of Auslander-Bridger [4] there is an induced exact sequence 0 ⟶ d(X_{3}) ⟶ d(X_{2}) ⟶ d(X_{1}) ⟶^{ϑ} Tr(X_{3}) ⟶ Tr(X_{2}) ⟶ Tr(X_{3}) ⟶ 0 in mod(Λ^{op}), where d = Hom_{Λ}(−, Λ) and Tr is the transpose. However since the projectives are Ext-injective in X, the connecting morphism ϑ vanishes. Hence we have an exact sequence 0 ⟶ Tr(X_{3}) ⟶ Tr(X_{2}) ⟶ Tr(X_{1}) ⟶ 0, which induces an exact sequence 0 ⟶ DTr(X_{1}) ⟶ DTr(X_{2}) ⟶ DTr(X_{3}) ⟶ 0 in mod(Λ). By the above argument the last sequence lies in Z. This implies easily that DTr : X /P_{Λ} ⟶^{≈} Z/I_{Λ} is an exact equivalence of triangulated categories. It is not difficult to see that there is a natural morphism φ : Φ ⟶ DTr between the equivalences Φ, DTr : X /P_{Λ} ⟶ Z/I_{Λ}. In general φ is not invertible. A necessary and sufficient condition for φ\underline{X} to be invertible is that the middle part of an almost split sequence starting at any indecomposable non-projective direct summand of X ∈ X, lies in Y.

Remark 3.5. Let us denote by τ the Auslander-Reiten translate and by τ^{−} its inverse, in both the categories X /P_{Λ} and Z/I_{Λ}. If the object R(\underline{DTr(X)}) is indecomposable, then it is easy to see that τ(\underline{X}) ≅ R(\underline{DTr(X)}), since then the Auslander-Reiten triangle starting at \underline{X} is the image of the triangle Ω(\underline{X}) ⟶ \underline{DTr(X)} ⟶ \underline{A} ⟶ \underline{X} under the functor R, where 0 ⟶ DTr(X) ⟶ A ⟶ X ⟶ 0 is an almost split sequence. In particular if for any indecomposable non-injective Z in Z its minimal right X -approximation X_{Z} is indecomposable up to projective summands, then τ(\underline{X}) = R(\underline{DTr(X)}) in X /P_{Λ} and τ(\underline{Z}) = \underline{DTrR(Z)} in Z/I_{Λ}. Dually if for any indecomposable non-projective X in X, its minimal left Z-approximation Z^{X} is indecomposable up to injective summands, then τ^{−}(\underline{X}) = \underline{TrDS(X)} in X /P_{Λ} and τ^{−}(\underline{Z}) = S(\underline{TrD(Z)}) in Z/I_{Λ}, where S denotes the reflection \overline{mod}(Λ) ⟶ Z/I_{Λ}. Note that by the results of [8], all the above remarks can be applied to the good triple (CM(Λ), P_{Λ}^{<∞} = I_{Λ}^{<∞}, CoCM(D(Λ))), where Λ is Gorenstein algebra.

If (X , Y) is a good pair then in general we don’t know if X is covariantly finite or Y is contravariantly finite. This happens for instance if the good pair is induced by a tilting or cotilting module, see [6]. However the following remark and its corollary shows that if (X , Y, Z) is a good triple, then all the categories X , Y, Z are functorially finite.

Remark 3.6. Let (X , Y, Z) be a good triple in mod(Λ). We know that X /P_{Λ} = ⊥(Y/P_{Λ}) and Z/I_{Λ} = (Y/I_{Λ})^{⊥} and (X /P_{Λ})^{⊥} = Y/P_{Λ} and ^{⊥}(Z/I_{Λ}) = Y/I_{Λ}. The orthogonal subcategories ^{⊥}(X /P_{Λ}) and (Z/I_{Λ})^{⊥} also have a computable description. If C ∈ \overline{mod}(Λ), then C ∈ (Z/I_{Λ})^{⊥} iff \overline{Hom}_{Λ}(Z, C) = 0. Since D\overline{Hom}_{Λ}(Z, C) ≅ Ext^{1}(TrD(C), Z), we have C ∈ (Z/I_{Λ})^{⊥} iff TrD(C) ∈ Y/P_{Λ}. It follows that the functor TrD : (Z/I_{Λ})^{⊥} ⟶^{≈} Y/P_{Λ} is an equivalence and similarly the functor DTr : ^{⊥}(X /P_{Λ}) ⟶^{≈} Y/I_{Λ} is an equivalence.

Corollary 3.7. The subcategory (Z/I_{Λ})^{⊥} is coreflective in \overline{mod}(Λ) and the subcategory ^{⊥}(X /P_{Λ}) is reflective in \underline{mod}(Λ). In particular the categories X and Z are functorially finite in mod(Λ).

Proof. Let \underline{C} be in \overline{mod}(Λ), and let \underline{Y}_{TrD(\underline{C})} ⟶ TrD(\underline{C}) be the coreflection of TrD(\underline{C}) in Y/P_{Λ}. Then DTr(\underline{Y}_{TrD(\underline{C})}) ⟶ \underline{C} is the coreflection of \underline{C} in (Z/I_{Λ})^{⊥}. Similarly for any \underline{C} in \underline{mod}(Λ), the morphism \underline{C} ⟶ TrD(\underline{Y}^{DTr(\underline{C})}) is the reflection of \underline{C} in ^{⊥}(X /P_{Λ}), where DTr(\underline{C}) ⟶ \underline{Y}^{DTr(\underline{C})} is the reflection of DTr(\underline{C}) in Y/I_{Λ}. The last assertion follows from Lemma V.1.7.

We have seen that for a good triple (X , Y, Z), the categories X /P_{Λ} and Z/I_{Λ} are always triangle equivalent. The next result shows that if X = Z, then the algebra Λ is self-injective.

Corollary 3.8. The subcategories X and Z coincide iff Λ is self-injective and the TTF-theory (X /P_{Λ}, Y/P_{Λ}, Z/P_{Λ}) in \underline{mod}(Λ) splits. In this case also Y/P_{Λ} is triangulated and all the above subcategories are closed under DTr and TrD, and the stable category admits a decomposition \underline{mod}(Λ) = X /P_{Λ} × Y/P_{Λ} = Y/I_{Λ} × Z/I_{Λ}.

Proof. If X = Z, then P_{Λ} = X ∩Y = Y∩Z = I_{Λ}. Hence Λ is self-injective and then obviously Y/P_{Λ} is triangulated. In particular the triple (X /P_{Λ}, Y/P_{Λ}, Z/P_{Λ}) is a TTF-theory in \underline{mod}(Λ). By the discussion after Proposition II.2.3, we have that the TTF-theory (X /P_{Λ}, Y/P_{Λ}, Z/P_{Λ}) in \underline{mod}(Λ) splits. Working as in Remark 3.4 we infer easily that X , Y, Z are closed under DTr, TrD. The converse and the last assertion are easy and are left to the reader.

If Λ is self-injective, then the following consequence of Theorem 3.2 describes the TTF-theories of \underline{mod}(Λ) in terms of subcategories of mod(Λ).

Corollary 3.9. Let Λ be a self-injective algebra and let Y be a full subcategory of mod(Λ). Then the following are equivalent.

(i) Y is a functorially finite resolving and coresolving subcategory of mod(Λ). (ii) There exists a TTF-theory (X /P_{Λ}, Y/P_{Λ}, Z/P_{Λ}) in \underline{mod}(Λ).

In the self-injective case the following corollary shows that any good triple in mod(Λ), equivalently any TTF-theory in \underline{mod}(Λ), produces an infinite number of good triples in the module category, equivalently an infinite number of TTF-theories in the stable category.

Corollary 3.10. Let Λ be a self-injective algebra and let (X , Y, Z) be a good triple in mod(Λ). Then (^{⊥}X , X , Y) and (Y, Z, Z^{⊥}) are good triples. Iterating this process ∀n ≥ 0 we get good triples (^{(n+2)⊥}Y,^{(n+1)⊥}Y,^{(n)⊥}Y) in mod(Λ), where (n)⊥Y is the n−th iterated left orthogonal of Y and ^{(0)⊥}Y := Y and good triples (Y^{(n)⊥}, Y^{(n+1)⊥}, Y^{(n+2)⊥}) in mod(Λ), where Y^{(n)⊥} is the n−th iterated right orthogonal of Y and Y^{(0)⊥} := Y.

The above good triples induce an infinite number of TTF-theories (T_{n}, F_{n}, R_{n}) in \underline{mod}(Λ), where F_{n} = ^{(n)⊥}(Y/P_{Λ}) for n ≤ 0 and F_{n} = (Y/P_{Λ})^{(n)⊥} for n ≥ 0.

If Y is a functorially finite resolving and coresolving subcategory of mod(Λ), then by Theorem 3.2, Y/P_{Λ} is a left triangulated subcategory of \underline{mod}(Λ) and Y/I_{Λ} is a right triangulated subcategory of \overline{mod}(Λ). Hence we can form the left triangulated Verdier quotient \underline{mod}(Λ)⧸Y/P_{Λ} and the right triangulated Verdier quotient \overline{mod}(Λ)⧸Y/I_{Λ}. We recall that \underline{mod}(Λ)⧸Y/P_{Λ} is the localization of \underline{mod}(Λ) at the class 𝔏_{Y} of morphisms \underline{f} : \underline{A} ⟶ \underline{B} for which there exists a left triangle

Ω(\underline{B}) ⟶ \underline{Y} ⟶ \underline{A} ⟶^{\underline{f}} \underline{B} with Y in Y, that is \underline{mod}(Λ)⧸Y/P_{Λ} is obtained by formally inverting the class of morphisms 𝔏_{Y} [35]. Dually \overline{mod}(Λ)⧸Y/I_{Λ} is a localization of \overline{mod}(Λ) at the class ℜ_{Y} of morphisms \underline{f} : \underline{A} ⟶ \underline{B} for which there exists a right

triangle \underline{A} ⟶^{\underline{f}} \underline{B} ⟶ \underline{Y} ⟶ Σ(\underline{A}) with Y in Y. The following result describes the above Verdier quotients and can be regarded as an analogue of Corollary I.2.9.

Theorem 3.11. Let Y be a functorially finite resolving and coresolving subcategory of mod(Λ) and consider the torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) and the torsion theory (Y/I_{Λ}, Z/I_{Λ}) in \overline{mod}(Λ). Then there are triangle equivalences:

\underline{mod}(Λ)⧸Y/P_{Λ} ⟶^{≈} X /P_{Λ} and \overline{mod}(Λ)⧸Y/I_{Λ} ⟶^{≈} Z/I_{Λ}.

In particular the localization categories \underline{mod}(Λ)⧸Y/P_{Λ} and \overline{mod}(Λ)⧸Y/I_{Λ} are triangulated and there is a triangle equivalence \underline{mod}(Λ)⧸Y/P_{Λ} ⟶^{≈} \overline{mod}(Λ)⧸Y/I_{Λ}.

Proof. By Theorem 3.2 we have that X /P_{Λ} is triangulated, in particular Σ(X /P_{Λ}) = X /P_{Λ} where Σ is the left adjoint of Ω in \underline{mod}(Λ), and Z/I_{Λ} is triangulated, in particular Ω(Z/I_{Λ}) = Z/I_{Λ} where Ω is the right adjoint of Σ in \overline{mod}(Λ). Then the result is a consequence of Proposition II.4.1 and Theorem 3.2.

We close this chapter with an application to the stable Grothendieck group K_{0}(\underline{mod}(Λ)), which is one of the few known invariants under stable equivalences of algebras. Under the assumptions of Theorem 3.11, the torsion theory (X /P_{Λ}, Y/P_{Λ}) is hereditary and the torsion theory (Y/I_{Λ}, Z/I_{Λ}) is cohereditary. Then we have the following consequence of Corollaries II.5.6 and II.5.7.

Corollary 3.12. We have isomorphisms:

K_{0}(\underline{mod}(Λ)) ≅ K_{0}(X /P_{Λ}) ⊕ K_{0}(Y/P_{Λ}),
K_{0}(\overline{mod}(Λ)) ≅ K_{0}(Y/I_{Λ}) ⊕ K_{0}(Z/I_{Λ})

3. GOOD TRIPLES AND TTF-THEORIES 99

If Y ⊆ P_{Λ}^{<∞} or Y ⊆ I_{Λ}^{<∞}, they reduce to isomorphisms:

K_{0}(Y/P_{Λ}) = 0 = K_{0}(Y/I_{Λ})

K_{0}(\underline{mod}(Λ)) ⟶^{≅} K_{0}(X /P_{Λ}) ⟶^{≅} K_{0}(Z/I_{Λ}) ⟵^{≅} K_{0}(\overline{mod}(Λ)).

If the algebra Λ is Gorenstein, then we have the following consequence. Corollary 3.13. If Λ is Gorenstein, there are triangle equivalences: \underline{CM}(Λ) ⟶^{≈} \underline{mod}(Λ)⧸P_{Λ}^{<∞}/P_{Λ} ⟶^{≈} \overline{mod}(Λ)⧸I_{Λ}^{<∞}/I_{Λ} ⟶^{≈} \overline{CoCM}(D(Λ))

and isomorphisms K_{0}(\underline{mod}(Λ)) ⟶^{≅} K_{0}(\underline{CM}(Λ)) ⟶^{≅} K_{0}(\underline{CoCM}(D(Λ))) ⟵^{≅} K_{0}(\overline{mod}(Λ)).

CHAPTER VII Torsion Theories and Closed Model Structures

Our aim in this chapter is to present and investigate an interesting connection between torsion theories and good pairs on one hand, and closed model structures in the sense of Quillen [66] on the other hand. A similar approach in a different context was developed independently by Hovey [46]. Since torsion theories can be regarded as a generalized form of tilting theory, the above connection shows that tilting theory admits a natural interpretation in the more general context of closed model categories, which, as is well known, is the proper framework for the study of homotopical/homological algebra. As a byproduct we obtain new classes of torsion theories, since closed model structures are omnipresent in algebra and topology. Our main results in this chapter give a classification of pretorsion theories in an abelian category and a classification of torsion theories in the stable category of an abelian category, in terms of closed model structures.

1. Preliminaries on Closed Model Categories

In this section we collect some preliminary concepts and results concerning closed model categories, a class of categories introduced by Quillen in the late sixties as the proper conceptual framework for doing homotopy theory in more general categories than the category of topological spaces.

Let C be an additive category. Recall that a closed model structure on C in the sense of Quillen is a triple (ℭ, 𝔉, 𝔚) of classes of morphisms, satisfying the following axioms [Mi], i = 2, ..., 5. The morphisms in 𝔉 are the fibrations, the morphisms in ℭ are the cofibrations, and the morphisms in 𝔚 are the weak equivalences. Similarly we denote by 𝔗𝔉 = 𝔉 ∩ 𝔚 the class of trivial fibrations and by 𝔗ℭ = ℭ ∩ 𝔚 the class of trivial cofibrations.

[M2] [Two out of three axiom] If f, g are composable morphisms in C and two of f, g and f ∘ g are weak equivalences, then so is the third.

[M3] [Retract axiom] If f and g are morphisms in C and f is a retract of g in the category of morphisms C^{2}, and if g is a weak equivalence, fibration, or a cofibration, then so is f.

[M4] [Lifting axiom] Consider the following commutative diagram in C:

\begin{matrix}A & ⟶^{α} & C \\ ^{i}↓ & & ^{p}↓ \\ B & ⟶^{β} & D \end{matrix}

If either i is a cofibration and p is a trivial fibration, or i is a trivial cofibration and p is a fibration, then there exists a morphism λ : B ⟶ C such that i ∘ λ = α and λ ∘ p = β.

[M5] [Factorization axiom] Any morphism α in C admits a factorization α = i ∘ g, where i is a cofibration and g is a trivial fibration, and a factorization α = f ∘ p, where f is a trivial cofibration and p is a fibration.

A closed model category is an additive category C equipped with a closed model structure (ℭ, 𝔉, 𝔚), which satisfies the following axiom:

[M1] C has kernels and cokernels.

Note that, contrary to the recent treatments of closed model categories [45], [43], we don’t require the existence of all small limits and colimits or functoriallity of the factorizations in the axioms [M1] and [M5] respectively as have been formulated in [45], [43]. This approach has the advantage that our results are applicable to a wider class of categories which are of interest in representation theory. As an important example we mention the category of finitely presented modules over an Artin algebra.

Consider an additive closed model category C with closed model structure (ℭ, 𝔉, 𝔚). We denote by Cof the category of cofibrant objects, that is the objects C such that 0 ⟶ C is a cofibration, and by Fib the category of fibrant objects, that is the objects F such that F ⟶ 0 is a fibration. Of special importance to us are the following subcategories:

TCof = {X ∈ C | 0 ⟶ X is a trivial cofibration} ⊆ Cof

TFib = {Y ∈ C | Y ⟶ 0 is a trivial fibration} ⊆ Fib

ω_{c} = Cof ∩ TFib and ω_{f} = TCof ∩ Fib.

We call the objects in TCof trivially cofibrant and the objects in TFib trivially fibrant. Note that ω_{c}, ω_{f} ⊆ Cof ∩ Fib, that is the objects of the subcategories ω_{c}, ω_{f} are fibrant and cofibrant. By the Retract axiom [M2] it follows easily that all the categories Cof, Fib, TCof, TFib, ω_{c}, ω_{f} are closed under direct summands.

We need the following easy result. For a proof we refer to [45], [43].

Lemma 1.1. (i) If p : B ⟶ C is a trivial fibration, then any morphism

γ : X ⟶ C from a cofibrant object X to C factors through p. (ii) If i : A ⟶ B is a trivial cofibration, then any morphism α : A ⟶ Y from A

to a fibrant object Y factors through i. (iii) Let 0 ⟶ F ⟶ A ⟶^{p} B be an exact sequence in C. If p is a fibration then

F ∈ Fib, and if p is a trivial fibration, then F ∈ TFib. (iv) Let A ⟶^{i} B ⟶ C ⟶ 0 be an exact sequence in C. If i is a cofibration then

C ∈ Cof, and if i is a trivial cofibration, then C ∈ TCof.

A very useful property of model categories is that any one of the classes of morphisms which occur in the pairs {ℭ, 𝔗𝔉}, {𝔗ℭ, 𝔉}, characterizes the other via a lifting property of an appropriate commutative square. To make this precise we need the following definition.

Consider a commutative diagram in C:

\begin{matrix}A & ⟶^{α} & C \\ ^{i}↓ & & ^{p}↓ \\ B & ⟶^{β} & D \end{matrix}

Definition 1.2. A lifting of the above square is a morphism λ : B ⟶ C such that i ∘ λ = α and λ ∘ p = β. If such a lifting exists, then the morphism i is said to have the left lifting property with respect to p and p is said to have the right lifting property with respect to i.

For the easy proof of the following we refer to [45], [43].

Lemma 1.3. (i) A morphism is a cofibration iff it has the left lifting prop-

erty with respect to all trivial fibrations. (ii) A morphism is a trivial cofibration iff it has the left lifting property with

respect to all fibrations. (iii) A morphism is a fibration iff it has the right lifting property with respect to

all trivial cofibrations. (iv) A morphism is a trivial fibration iff it has the right lifting property with

respect to all cofibrations.

For all unexplained concepts and results we use in this chapter, we refer to the standard references [66], [45], [43].

2. Approximation Sequences and Good Pairs

In this section we study the natural question of when a closed model structure on an additive category is determined by objects, more precisely by the subcategories of (trivial) fibrant or cofibrant objects. We shall show that this question is connected with the existence of certain (pre)torsion theories and good pairs. We fix throughout an additive closed model category C with closed model structure (ℭ, 𝔉, 𝔚).

We begin with the following basic result which shows that there exists a remarkable similarity between certain exact sequences induced by a closed model category structure and the approximation sequences induced by a good pair.

Proposition 2.1. Let C be a closed model category. Then we have the following.

(i) The category Cof of cofibrant objects is contravariantly finite in C, and for any A in C there exists an exact sequence 0 ⟶ Y_{A} ⟶^{g_{A}} C_{A} ⟶^{f_{A}} A, where f_{A} is a right cofibrant approximation of A (and a trivial fibration) and the object Y_{A} ∈ TFib.

(ii) The category Fib of fibrant objects is covariantly finite in C and for any A

in C there exists an exact sequence A ⟶^{g^{A}} F^{A} ⟶^{f^{A}} X^{A} ⟶ 0, where g^{A} is a left fibrant approximation of A (and a trivial cofibration) and the object X^{A} ∈ TCof.

(iii) The category TCof is contravariantly finite in C, and for any A in C there exists an exact sequence 0 ⟶ F_{A} ⟶^{ψ_{A}} X_{A} ⟶^{ϕ_{A}} A, where ϕ_{A} is a right TCof−approximation of A (and a fibration) and the object F_{A} ∈ Fib.

(iv) The category TFib is covariantly finite in C and for any object A in C there

exists an exact sequence A ⟶^{ψ^{A}} Y^{A} ⟶^{ϕ^{A}} C^{A} ⟶ 0, where ψ^{A} is a left TFib−approximation of A (and a cofibration) and the object C^{A} ∈ Cof.

(v) (Cof/ω_{c}, TFib/ω_{c}) = 0 and (TCof/ω_{f}, Fib/ω_{f}) = 0, i.e. any map C ⟶ Y with C ∈ Cof and Y ∈ TFib, factors through an object in ω_{c}, and any map X ⟶ F with F ∈ Fib and X ∈ TCof, factors through an object in ω_{f}.

Proof. (i), (ii) Let A ∈ C and let 0 ⟶ C_{A} ⟶^{f_{A}} A be a factorization of 0 ⟶ A into a cofibration followed by a trivial fibration. Then by definition C_{A} is cofibrant. Consider the exact sequence 0 ⟶ Y_{A} ⟶^{g_{A}} C_{A} ⟶^{f_{A}} A in C. Since f_{A} is a trivial fibration, by Lemma 1.1 we have that Y_{A} is in TFib. Also since f_{A} is a trivial fibration, the above lemma implies that any morphism C ⟶ A with C cofibrant, factors through f_{A}. We infer that f_{A} is a right Cof-approximation of A and by construction f_{A} is a trivial fibration. Part (ii) is dual.

(iii), (iv) Let A be in C. Then by (ii) there exists a trivial cofibration f^{A} : A ⟶ F^{A} with F^{A} fibrant which is a left fibrant approximation of A. Let F^{A} ⟶^{c} Y^{A} ⟶ 0 be a factorization of F^{A} ⟶ 0 into a cofibration and a trivial fibration. Then Y^{A} is in TFib and we claim that ψ^{A} := f^{A} ∘ c : A ⟶ Y^{A} is a left TFib-approximation of A. Indeed since cofibrations are closed under composition, ψ^{A} is a cofibration. Let α : A ⟶ Y be a morphism with Y ∈ TFib. Consider the diagram

A ⟶^{α} Y
ψ^{A}↓  ↓
Y^{A} ⟶ 0

Since Y ⟶ 0 is a trivial fibration, by the Lifting axiom, α factors through ψ^{A}. Hence ψ^{A} is a left TFib-approximation of A and a cofibration, and then C^{A} := Coker(ψ^{A}) is cofibrant as a cokernel of a cofibration. Part (iii) is dual.

(v) Let Y be in TFib and consider the right cofibrant approximation sequence 0 ⟶ F_{Y} ⟶ C_{Y} ⟶^{f_{Y}} Y of (i). Since any morphism C ⟶ Y with C cofibrant factors through C_{Y} it suffices to show that C_{Y} is in ω_{c}. Since C_{Y} is already cofibrant it suffices to show that C_{Y} ⟶ 0 is a trivial fibration. Consider the following commutative diagram

A ⟶^{α} C_{Y}
i↓  ↓
B ⟶ 0

where i is a cofibration. Since Y is in TFib, the morphism Y ⟶ 0 is a trivial fibration, hence by the Lifting axiom the morphism α∘f_{Y} : A ⟶ Y factors through i, i.e. there exists a morphism β : B ⟶ Y such that i ∘ β = α ∘ f_{Y}. Hence we have the commutative square

A ⟶^{α} C_{Y}
i↓  ^{f_{Y}} ↓
B ⟶^{β} Y

where i is a cofibration and f_{Y} is a trivial fibration. Then by the Lifting axiom [M5], α factors through i. Hence C_{Y} ⟶ 0 is a trivial fibration since it has the right lifting property with respect to cofibrations. We infer that C_{F} ∈ ω_{c}. Hence (Cof/ω_{c}, TFib/ω_{c}) = 0. The proof that (TCof/ω_{f}, Fib/ω_{f}) = 0 is dual.

The above result allows us to obtain a homological connection between the full subcategories of (trivially) fibrant or cofibrant objects and the classes of (trivial) fibrations or cofibrations.

For a class 𝔉 of morphisms in a category C, we denote by P(𝔉) the full subcategory of C consisting of all objects P with the property that for any morphism f : A ⟶ B in 𝔉, any morphism P ⟶ B factors through f. Dually for a class ℭ of morphisms in C, we denote by I(ℭ) the full subcategory of C consisting of all objects I with the property that for any morphism g : A ⟶ B in ℭ, any morphism A ⟶ I factors through g. We call the objects in P(𝔉) relative 𝔉-projectives and the objects in I(ℭ) relative ℭ-injectives.

Proposition 2.2. For a closed model structure (ℭ, 𝔉, 𝔚) in C we have:

Cof = P(𝔗𝔉), Fib = I(𝔗ℭ), TCof = P(𝔉), TFib = I(ℭ).

Proof. If C is in Cof, and p : A ⟶ B is a trivial fibration, then by Lemma 1.2((i) any morphism C ⟶ B factors through p. Hence C is 𝔗𝔉-projective. Conversely if P is in P(𝔗𝔉), then by Proposition 2.1, there exists a cofibrant approximation f_{P} : C_{P} ⟶ P of P which is a trivial fibration. Hence f_{P} splits since P is 𝔗𝔉-projective. Since Cof is closed under direct summands and C_{P} is cofibrant, it follows that P is cofibrant. Hence we proved that Cof = P(𝔗𝔉).

Now let X be in TCof and let p : A ⟶ B be a fibration. If α : X ⟶ B is a morphism, then by Lemma 1.3(ii) the square

\begin{matrix}0 & ⟶ & A \\ ↓ & & ^{p}↓ \\ X & ⟶^{α} & B \end{matrix}

admits a lifting since 0 ⟶ X is a trivial cofibration. Hence X is in P(𝔉). Conversely if P is 𝔉-projective then by Proposition 2.1((iii) there exists a right TCof-approxi– mation ϕ_{P} : X_{P} ⟶ P and the morphism ϕ_{P} is a fibration. Hence ϕ_{P} splits, and this implies that P is in TCof since the latter is closed under direct summands. This proves that TCof = P(𝔉). The remaining equalities are proved in a similar way.

In particular we see that the classes of morphisms 𝔉 and 𝔗𝔉 are projective classes in C and the classes of morphisms ℭ and 𝔗ℭ are injective classes in C, in the sense of relative homological algebra introduced and developed by Eilenberg-Moore [32].

The above result suggests the following problem:

Problem. Under what conditions is a closed model structure (ℭ, 𝔉, 𝔚) in C determined by its subcategories Cof, Fib, TCof, TFib ?

The above problem splits into two parts:

(i) Let X , Y, X_{t}, Y_{t} be full additive subcategories of C. Define ℭ′ to be the class of X_{t}-epics, 𝔉′ to be the class of Y_{t}-monics, and 𝔚′ to be the class of morphisms which can be written as a composition of a Y-monic followed by an X -epic. Under what conditions is the triple (ℭ′, 𝔉′, 𝔚′) a closed model structure in C ?

(ii) If we choose above X_{t} = TCof, X = Cof, Y_{t} = TFib, Y = Fib, for a closed model structure (ℭ, 𝔉, 𝔚) in C, then is it true that (ℭ, 𝔉, 𝔚) coincides with the closed model structure (ℭ′, 𝔉′, 𝔚′) ?

In the next sections we’ll give fairly general answers to the above problem which is connected rather surprisingly with pretorsion theories.

3. Good Pairs and Closed Model Structures

From now on we assume that C is an abelian category. In this section we give a first answer to the problem of when a closed model structure is determined by objects, by examining more closely the connections between closed model structures and good pairs.

We begin with the following useful result which connects the vanishing of the first extension functor with the lifting property of commutative squares defined in Section 1, and provides a tool for checking whether various subcategories in a closed model category are orthogonal with respect to the vanishing of Ext^{1}.

Let X and Y be in C and consider the following exact commutative diagram: Y

c↓
A ⟶^{α} C
i↓  ^{p}↓
B ⟶^{β} D
d↓
X

where the sequences 0 ⟶ A ⟶^{i} B ⟶^{d} X ⟶ 0 and 0 ⟶ Y ⟶^{c} C ⟶^{p} D ⟶ 0 are exact.

Extension-Lifting Lemma 3.1. The following are equivalent:

(i) Ext^{1}(X, Y) = 0.

(ii) Any commutative diagram as above with fixed X and Y admits a lifting: there exists a morphism λ : B ⟶ C such that i ∘ λ = α and λ ∘ p = β.

Proof. (i) ⇒ (ii) Consider the pull-back diagram

0 ⟶ Y ⟶^{ϵ} K ⟶^{ζ} B ⟶ 0
∥↓  ^{γ}↓  ^{β}↓
0 ⟶ Y ⟶^{c} C ⟶^{p} D ⟶ 0

Since i ∘ β = α ∘ p, there exists a unique morphism ϕ : A ⟶ K such that ϕ ∘ ζ = i and ϕ ∘ γ = α. Then we have the following exact commutative diagram

0 ⟶ A ⟶^{ϕ} K ⟶^{ξ} L ⟶ 0
∥↓  ^{ζ}↓  ^{η}↓
0 ⟶ A ⟶^{i} B ⟶^{d} X ⟶ 0

Since the kernel of η is isomorphic to Y, the hypothesis implies that η is split epic, and its kernel ker(η) : Y ⟶ L is split monic. By the above diagram ker(η) admits a factorization ϵ ∘ ξ. It follows that ϵ is split monic or equivalently ζ : K ⟶ B is split epic. Then by the first diagram above there exists a morphism θ : B ⟶ C such that θ ∘p = β. Then i∘θ ∘p = i∘β = α∘p ⇒ (α−i∘θ)∘p = 0. Hence there exists a unique morphism μ : A ⟶ Y such that μ ∘ c = α − i ∘ θ = 0. Since Ext^{1}(X, Y) = 0, the push-out of the extension 0 ⟶ A ⟶^{i} B ⟶^{d} X ⟶ 0 along μ, splits. Hence there exists a morphism ν : B ⟶ Y such that i ∘ ν = μ. Then i ∘ ν ∘ c + i ∘ θ = α. Setting λ := θ + ν ∘ c we have α = i ∘ λ and λ ∘ p = θ ∘ p + ν ∘ c ∘ p = θ ∘ p = β. Hence λ : B ⟶ C has the desired property.

(ii) ⇒ (i) Let 0 ⟶ Y ⟶^{c} C ⟶^{p} X ⟶ 0 be an extension. Then we have the exact commutative diagram preceding the lemma, if we set A = C, B = A ⊕ X, α = 1_{C}, β = ^{t}(p, 1_{X}), i = (1_{A}, 0), d = ^{t}(0, 1_{X}). Then the hypothesis implies that there exists a morphism ^{t}(κ, λ) : C ⊕ X ⟶ C such that ^{t}(κ, λ) ∘ p = 1_{X}. From this it follows that λ ∘ p = 1_{X}. Hence p splits, that is Ext^{1}(X, Y) = 0.

By part (v) of Proposition 2.1 the subcategories Cof and TFib are orthogonal with respect to Hom in the stable category C/Cof ∩ TFib, and the categories TCof and Fib are orthogonal with respect to Hom in the stable category C/TCof∩Fib. The following result gives sufficient conditions for the pairs of subcategories (Cof, TFib) and (TCof, Fib) to be orthogonal in C with respect to Ext^{1}.

Lemma 3.2. (i) If any trivial fibration is epic, then ^{⊥}TFib ⊆ Cof.

(ii) If any fibration is epic, then ^{⊥}Fib ⊆ TCof.

(iii) If any trivial cofibration is monic, then TCof^{⊥} ⊆ Fib.

(iv) If any cofibration is monic, then Cof^{⊥} ⊆ TFib.

(v) If any monic with cokernel in TCof is a trivial cofibration, then:

Ext^{1}(TCof, Fib) = 0.

(vi) If any epic with kernel in TFib is a trivial fibration, then:

Ext^{1}(Cof, TFib) = 0.

Proof. We prove only (i), (ii), (v), since the proof of (iii), (iv), (vi) is dual.

(i) Let A ∈ ^{⊥}TFib and consider the right cofibrant approximation 0 ⟶ Y_{A} ⟶ C_{A} ⟶^{f_{A}} A of A. Since f_{A} is a trivial fibration, f_{A} is epic. Since Y_{A} ∈ TFib and A ∈ ^{⊥}TFib, the sequence splits. Hence A is cofibrant since it is a direct summand of C_{A}. We infer that ^{⊥}TFib ⊆ Cof.

(ii) Let A ∈ ^{⊥}Fib and consider the right TCof-approximation 0 ⟶ F_{A} ⟶ X_{A} ⟶^{ϕ_{A}} A of A. Then F_{A} is fibrant and ϕ_{A} is epic as a fibration. Since A ∈ ^{⊥}Fib, the sequence splits, and we infer that A is in TCof since it is a direct summand of X_{A}.

(v) Let 0 ⟶ F ⟶^{i} A ⟶ X ⟶ 0 be an extension with F fibrant and X in TCof. Then by hypothesis i is a trivial cofibration. Since F is fibrant, i splits. Hence Ext^{1}(X, F) = 0.

The next result gives sufficient conditions for the pairs (Cof, TFib), (TCof, Fib) to be good in C. For examples satisfying the conditions below we refer to the next section.

Proposition 3.3. (1) (Cof, TFib) is a good pair in C, provided that:

(i) Any cofibration is monic.

(ii) Any trivial fibration is epic and any epic with kernel in TFib is a trivial

fibration.

(2) (TCof, Fib) is a good pair in C, provided that:

(i) Any fibration is epic.

(ii) Any trivial cofibration is monic and any monic with cokernel in TCof is a trivial cofibration.

Proof. We prove only part (1) since the proof of (2) is dual. By Proposition 2.1 we know that Cof is contravariantly finite and TFib is covariantly finite. By parts (i), (iv), (vi) of Lemma 3.2 we have Cof^{⊥} = TFib and ^{⊥}TFib = Cof. Hence (Cof, TFib) is a good pair in C.

Putting together the conditions in (1) and (2) of Proposition 3.3, we get a characterization of the closed model structures (ℭ, 𝔉, 𝔚) in C such that both pairs (Cof, TFib) and (TCof, Fib) are good in C.

Theorem 3.4. Let C be an abelian closed model category with enough projective and injective objects. Then the following conditions are equivalent.

(i) (a) Any fibration is epic and any epic with kernel in TFib is a trivial

fibration.

(b) Any cofibration is monic and any monic with cokernel in TCof is a

trivial cofibration.

(ii) (Cof, TFib) and (TCof, Fib) are good pairs in C. If this is the case and TFib = TCof, then (Cof, TFib = TCof, Fib) is a good triple in C and the assignments C ↦ F_{C} and F ↦ C^{F} induce inverse equivalences:

Φ : Cof/ω_{c} ⟶^{≈} Fib/ω_{f} and Ψ : Fib/ω_{f} ⟶^{≈} Cof/ω_{c}.

Proof. (i) ⇒ (ii) Follows by Proposition 3.3.

(ii) ⇒ (i) Let p : B ⟶ A be a fibration. Since (TCof, Fib) is a good pair, TCof contains the projectives. Hence the right TCof-approximation ϕ_{A} : X_{A} ⟶ A of A is an epimorphism. Since X_{A} is in TCof, the morphism 0 ⟶ X_{A} is a trivial cofibration. By the Lifting axiom [M4] the square

0 ⟶ B
↓  ^{p}↓
X_{A} ⟶^{ϕ_{A}} A

admits a lifting. Hence ϕ_{A} factors through p, and this implies that p is an epimorphism. By duality any cofibration is a monomorphism. Now let 0 ⟶ Y ⟶ B ⟶^{p} A ⟶ 0 be an extension with Y in TFib. By the Lifting axiom, to show that p is a trivial fibration, it suffices to show that for any commutative square

\begin{matrix}K & ⟶^{α} & B \\ ^{i}↓ & & ^{p}↓ \\ L & ⟶^{β} & A \end{matrix}

where i is a cofibration, there exists a morphism λ : L ⟶ B which factorizes α through i and β through p. Since i is a cofibration, we have an exact sequence 0 ⟶ K ⟶^{i} L ⟶ C ⟶ 0 and C is cofibrant. Since Ext^{1}(C, Y) = 0, Lemma 3.1 shows that the factorizing morphism λ exists. Hence any epimorphism with kernel in TFib is a trivial fibration, and by duality any monomorphism with cokernel in TCof is a trivial cofibration.

Finally if TCof = TFib, then the existence of the equivalences Φ and Ψ is proved exactly as in Theorem 3.2 of Chapter VI.

Keeping the assumptions of Theorem 3.4, the results obtained so far combined with the results of Chapters V and VI, imply the following.

Corollary 3.5. (i) If ω_{c} is functorially finite in C, then Cof is closed under under kernels of epics iff (Cof/ω_{c}, TFib/ω_{c}) is a torsion theory in the pretriangulated category C/ω_{c}.

(ii) If ω_{f} is functorially finite in C, then Fib is closed under cokernels of monics iff (TCof/ω_{f}, Fib/ω_{f}) is a torsion theory in the pretriangulated category C/ω_{f}.

(iii) If TCof = TFib and Cof is closed under kernels of epics and Fib is closed under cokernels of monics, then C is a Frobenius category with all objects fibrant and cofibrant, and moreover ω_{c} = TFib = ω_{f} = TCof is the category of projective-injective objects.

The following gives a partial answer to the problem posed in Section 2.

Remark 3.6. Under the assumptions of Theorem 3.4, the closed model structure (ℭ, 𝔉, 𝔚) is completely determined by the subcategories Cof, Fib, TCof, TFib. Indeed, we have the following descriptions of the (trivial) fibrations, (trivial) cofibrations and weak equivalences in terms of the above subcategories.

(i) 𝔉 is the class of TCof-epics with fibrant kernel.

(ii) ℭ is the class of TFib-monics with cofibrant cokernel.

(iii) 𝔗𝔉 is the class of Cof-epics with trivially fibrant kernel.

(iv) 𝔗ℭ is the class of Fib-monics with trivially cofibrant cokernel.

(v) 𝔚 is the class of morphisms which can be written as a composition of a Fibmonic with trivially cofibrant cokernel followed by a Cof-epic with trivially fibrant kernel.

Let us show (i). For a subcategory X ⊆ C, we denote by E(X) the class of X - epics. By Proposition 2.2 we have P(𝔉) = TCof. Hence E(P(𝔉)) = E(TCof). Since obviously 𝔉 ⊆ E(P(𝔉)), we have 𝔉 ⊆ E(TCof). Since TCof is contravariantly finite and C has enough projectives, E(TCof) consists of epics. Since by Lemma 1.2 the kernel of a fibration is fibrant, it suffices to show that any TCof-epic p with fibrant kernel is a fibration. By Lemma 1.3 it suffices to show that p has the right lifting property with respect to trivial cofibrations. This follows from the Extension-Lifting Lemma 3.1 and condition (ii) of Theorem 3.4.

For a more complete answer to the problem of which closed model structures are determined by objects we refer to Theorems 4.6 and 4.7 below.

4. Closed Model Structures and (Pre)Torsion Theories

In this section we describe all pretorsion theories in an abelian closed model category in terms of closed model structures. This description will be used in the next section for the classification of all torsion theories of a stable category.

Although all the results of this section are valid (with the necessary modifications) in any abelian category, as in Chapter V, for simplicity we assume throughout this section that C is an abelian category with enough projective and injective objects. We fix a pair (X , Y) of full additive subcategories in C, which are closed under direct summands and isomorphisms, and let ω = X ∩ Y.

Consider the following classes of morphisms in C:

(i) ℭ_{ω} is the class of monics with cokernel in X.

(ii) 𝔗ℭ_{ω} is the class of split monics with cokernel in ω.

(iii) 𝔉_{ω} is the class of ω-epics.

(iv) 𝔗𝔉_{ω} is the class of epics with kernel in Y.

(v) 𝔚_{ω} is the class of morphisms α which admit a factorization α = α_{1} ∘ α_{2} where α_{1} ∈ 𝔗ℭ_{ω} and α_{2} ∈ 𝔗𝔉_{ω}.

We call the morphisms in 𝔉_{ω}, resp. 𝔗𝔉_{ω}, contravariant ω-fibrations, resp. contravariant ω-trivial fibrations. We call the morphisms in ℭ_{ω}, resp. 𝔗ℭ_{ω}, contravariant ω-cofibrations, resp. contravariant ω-trivial cofibrations. Finally we call the morphisms in 𝔚_{ω}, contravariant ω-weak equivalences. Our aim is to investigate when the triple (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) is a closed model structure in C.

We begin with the following useful result which gives, under some reasonable conditions, a handy description of the contravariant ω-weak equivalences.

Lemma 4.1. Assume that ω is contravariantly finite, Ext^{1}(X , Y) = 0 and (X /ω)^{⊥} = Y/ω. If X contains the projectives and admits ω as a cogenerator, then the following are equivalent for a morphism α : A ⟶ B in C.

(i) α : A ⟶ B is in 𝔚_{ω}.

(ii) (\underline{X}, \underline{α}) : (\underline{X}, \underline{A}) ⟶ (\underline{X}, \underline{B}) is invertible in C/ω, for any X ∈ X.

Proof. We observe first that the contravariant finitenness of ω implies that C/ω is left triangulated. Assume that α is in 𝔚_{ω}. Then there exists a factorization

A ⟶^{(1_{A},0)} A ⊕ T ⟶^{^{t}(α,κ)} B, where T is in ω and ^{t}(α, κ) is epic with kernel in Y. Since Ext^{1}(X , Y) = 0, the morphism ^{t}(α, κ) is ω-epic, hence we have a left triangle Ω(\underline{B}) ⟶ \underline{Y} ⟶ \underline{A} ⟶^{\underline{α}} \underline{B} in C/ω. Applying (\underline{X}, −) to this triangle, we have the following exact commutative diagram:

\begin{matrix}C(X, A ⊕ T) & ⟶^{(X ,^{t}(α,κ))} & C(X, B) \\ ^{π_{1}}↓ & & ^{π_{2}}↓ & (†) \\ C/ω(\underline{X}, \underline{A}) & ⟶^{(\underline{X},\underline{α})} & C/ω(\underline{X}, \underline{A}) \end{matrix}

Since Ext^{1}(X , Y) = 0, the morphism C(X,^{t}(α, κ)) is epic for any X ∈ X. By the above diagram this implies that (\underline{X}, \underline{α}) is epic. However since the cone of \underline{α} is in Y/ω it follows that (\underline{X}, \underline{α}) is monic. Hence (\underline{X}, \underline{α}) is invertible for any X in X. Conversely assume that this condition holds. First we show that if α : A ⟶ B is an ω-epic, such that (\underline{X}, \underline{α}) is invertible for any X in X, then α is epic with kernel in Y. Consider the diagram

\begin{matrix}C(X, A) & ⟶^{(X ,α)} & C(X, B) \\ ^{π_{1}}↓ & & ^{π_{2}}↓ & (††) \\ C/ω(\underline{X}, \underline{A}) & ⟶^{(\underline{X},\underline{α})} & C/ω(\underline{X}, \underline{A}) \end{matrix}

Let X be in X and let f : X ⟶ B be a morphism. Then by the above diagram there exists a unique morphism \underline{g} : \underline{X} ⟶ \underline{A}, such that \underline{g} ∘ \underline{α} = \underline{f}. Hence f − g ∘ α factors through a right ω-approximation τ_{B} : T ⟶ B, say as: f − g ∘ α = λ ∘ τ_{B}, where λ : T_{B} ⟶ B. Since α is an ω-epic, there exists a morphism μ : T_{B} ⟶ A such that μ ∘ α = τ_{B}. Then f − g ∘ α = λ ∘ μ ∘ α. Hence f = (g + λ ∘ μ) ∘ α. It follows that any morphism from an object of X to B factors through α. Since X contains the projectives, this implies that α is an epimorphism. It remains to show that the kernel Y of α is in Y. Since ω is an (Ext-injective) cogenerator of X we have that ω is covariantly finite in X and the cokernel of a left ω-approximation of any object of X lies in X. In particular the suspension functor Σ : X /ω ⟶ X /ω is defined and we have a natural isomorphism (Σ(\underline{X}), \underline{A}) ⟶^{≅} (\underline{X}, Ω(\underline{A})) in C/ω, for any X ∈ X and any object A in C. Now since α is an ω-epic, we have a left triangle Ω(\underline{B}) ⟶ \underline{Y} ⟶ \underline{A} ⟶^{\underline{α}} \underline{B} in C/ω. Since (\underline{X}, \underline{α}) is invertible for any X ∈ X, applying to the above diagram the functor (X /ω, −) and using the above adjunction isomorphism, we infer that \underline{Y} ∈ (X /ω)^{⊥} = Y/ω. Hence the kernel Y of α lies in Y.

Now let α : A ⟶ B be such that (\underline{X}, \underline{α}) is invertible for any X in X. Consider the ω-epic ^{t}(α, τ_{B}) : A ⊕ T_{B} ⟶ B, where τ_{B} : T_{B} ⟶ B is a right ω-approximation of B. This morphism is obviously isomorphic to α in C/ω. Hence it enjoys the same property as \underline{α}. Then by the above argument we infer that ^{t}(α, τ_{B}) is epic

with kernel in Y. Then α admits a factorization A ⟶^{(1_{A},0)} A⊕T_{B} ⟶^{^{t}(α,τ_{B})} B, where T_{B} is in ω and ^{t}(α, τ_{B}) is in 𝔗𝔉_{ω}. Hence α is in 𝔚_{ω}.

The following main result of this section gives necessary and sufficient conditions for the existence of closed model structures in C, arising from good pairs.

First we recall that the homotopy category Ho(C) of a closed model category structure (ℭ, 𝔉, 𝔚) is defined as the category of fractions C[𝔚^{−1}], obtained by formally inverting the weak equivalences, see [66]. A more convenient description of the homotopy category is that Ho(C) is the factor category Cof ∩ Fib/ ∼ of the category of fibrant and cofibrant objects modulo a suitable equivalence relation ∼, called the homotopy relation, defined on the class of morphisms between fibrant and cofibrant objects.

Theorem 4.2. The following conditions are equivalent:

(i) The triple (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) defines a closed model structure in C such that: ℭ_{ω} is the class of cofibrations, 𝔉_{ω} is the class of fibrations, and 𝔚_{ω} is the class of weak equivalences.

(ii) (X , Y) is a pretorsion theory and ω is contravariantly finite in C. If this is the case, then Cof_{ω} = X and TCof_{ω} = ω, and TFib_{ω} = Y and Fib_{ω} = C. Moreover the associated Quillen homotopy category Ho(C) is equivalent to the stable right triangulated category X /ω.

Proof. (i) ⇒ (ii) If the triple (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) defines a closed model structure in C, then obviously we have Cof_{ω} = X, TCof_{ω} = ω, TFib_{ω} = Y and Fib_{ω} = C. Then the assertion follows by Propositions 2.1 and 3.3(1).

(ii) ⇒ (i) We check the axioms [Mi], i = 1, ⋯, 5, of a closed model category. First note that [M1] holds since C as an abelian category has kernels and cokernels.

[M4] Consider a commutative diagram

\begin{matrix}A & ⟶^{α} & C \\ ^{i}↓ & & ^{p}↓ \\ B & ⟶^{β} & D \end{matrix}

Assume first that i is in ℭ_{ω} and p is in 𝔗𝔉_{ω}. Then i is monic with cokernel in X and p is epic with kernel in Y. Since Ext^{1}(X , Y) = 0, there exists a lifting by the lifting-extension Lemma 3.1. Assume now that i is in 𝔗ℭ_{ω} and p is in 𝔉_{ω}. Since i is

a split monic with cokernel in ω we can assume that i is the inclusion A ⟶^{(1,0)} A⊕T where T ∈ ω, and then the morphism β is of the form ^{t}(β_{1}, β_{2}) : A ⊕ T ⟶ D. Since p is ω-epic and T is in ω, there exists a morphism t : T ⟶ C such that t ∘ p = β_{2}.

Then the morphism ^{t}(α, t) : A ⊕ T ⟶ C is a lifting of the above square. Hence axiom [M4] holds.

[M5] Let α : A ⟶ B be a morphism in C. Let τ_{B} : T_{B} ⟶ B be a right ω-

approximation of B. Then α can be factorized as follows A ⟶^{(1_{A},0)} A⊕T_{B} ⟶^{^{t}(α,τ)} B where the first morphism is in 𝔗ℭ_{ω} and the second is in 𝔉_{ω}. Hence α can be written as a composition of a morphism in 𝔗ℭ_{ω} followed by a morphism in 𝔉_{ω}. To prove that α can be written as a composition of a morphism in ℭ_{ω} followed by a morphism in 𝔗𝔉_{ω}, we show first that any monic μ and any epic ϵ has this property. So let μ : A ⟶ B be monic, and consider the following exact commutative diagram

0 ⟶ A ⟶^{i} K ⟶^{ξ} X_{C} ⟶ 0
∥↓  ^{p}↓  ^{f_{C}} ↓
0 ⟶ A ⟶^{μ} B ⟶^{β} C ⟶ 0

where the second row is induced by pulling-back the first row along the right X - approximation of C. Then by definition i is in ℭ_{ω}. Since X is contravariantly finite and contains the projectives, the morphism f_{C}, hence the morphism p, is epic. Since (X , Y) is a pretorsion theory, the kernel of f_{C}, hence the kernel of p, lies in Y. It follows that p is in 𝔗𝔉_{ω}. Hence any monic is a composition of a morphism in ℭ_{ω} followed by a morphism in 𝔗𝔉_{ω}. Next let β : B ⟶ C be epic. Then consider the following exact commutative diagram:

0 ⟶ K ⟶ B ⟶^{ϵ} C ⟶ 0
g^{K} ↓  ^{i}↓  ^{∥}↓
0 ⟶ Y^{K} ⟶ L ⟶^{p} C ⟶ 0

where the second row is induced by pushing-out the upper row along the left Yap–proximation of K. Then by definition p lies in 𝔗𝔉_{ω}. Since Y contains the injectives, g^{K} is monic. Since (X , Y) is a pretorsion theory, the cokernel of g^{K}, hence the cokernel of i, lies in X. Hence i is in ℭ_{ω} and this shows that any epic is a composition of a morphism in ℭ_{ω} followed by a morphism in 𝔗𝔉_{ω}.

Assume now that α is arbitrary and consider the factorization A ⟶^{(1_{A},0)} A ⊕

B ⟶^{^{t}(α,1_{B})} B of α. Since (1_{A}, 0) is monic, there exists a factorization (1_{A}, 0) : A ⟶^{i_{1}} D ⟶^{p_{1}} A ⊕ B, where i_{1} ∈ ℭ_{ω} and p_{1} ∈ 𝔗𝔉_{ω}. Then the morphism p_{1} ∘ ^{t}(α, 1_{B}) : D ⟶ B is epic, hence it admits a factorization p_{1} ∘ ^{t}(α, 1_{B}) : D ⟶^{i_{2}} E ⟶^{p_{2}} B where i_{2} ∈ ℭ_{ω} and p_{2} ∈ 𝔗𝔉_{ω}. It suffices to show that the monic i_{1} ∘ i_{2} : A ⟶ E has cokernel in X. Since i_{1} and i_{2} are monics, there exists a short exact sequence 0 ⟶ Coker(i_{1}) ⟶ Coker(i_{1} ∘ i_{2}) ⟶ Coker(i_{2}) ⟶ 0. Since Coker(i_{1}), Coker(i_{2}) ∈ X and X is closed under extensions, it follows that Coker(i_{1} ∘ i_{2}) ∈ X. Hence i := i_{1} ∘ i_{2} ∈ ℭ_{ω}, and consequently α admits a factorization α = i ∘ p where i ∈ ℭ_{ω} and p := p_{2} ∈ 𝔗𝔉_{ω}.

Before we prove that the remaining axioms [M3] and [M2] hold, we recall from Chapter V, Section 3, that the good pair (X , Y) satisfies all the properties of Lemma 4.1. Hence a morphism α : A ⟶ B is in 𝔚_{ω} iff (\underline{X}, \underline{α}) : (\underline{X}, \underline{A}) ⟶ (\underline{X}, \underline{B}) is invertible for any X ∈ X.

[M3] Let α : A ⟶ B be a retract of a morphism β : C ⟶ D in the category of morphisms. Choosing direct sum decompositions C ⟶^{≅} A ⊕ A′ and D ⟶^{≅} B ⊕ B′ it is easy to see that we have a diagram

C ⟶^{≅} A ⊕ A′
β↓  ↓( \begin{matrix}α & 0 \\ 0 & α′ \end{matrix} )
D ⟶^{≅} B ⊕ B′

If β ∈ ℭ_{ω}, then β is monic with cokernel in X. The above diagram shows that α is monic with cokernel in X as a direct summand of Coker(β). If β ∈ 𝔉_{ω}, i.e. β is ω-epic, then trivially so is α. Finally if β ∈ 𝔚_{ω}, then obviously the same is true for α, using the description of morphisms in 𝔉_{ω}.

[M2] We have to show that if two out of the three morphisms α, β, α ∘ β are in 𝔚_{ω}, then so is the third. This follows from the description of morphisms of 𝔚_{ω} in Lemma 4.1.

Assume now that (ii) holds and consider the closed model structure of (i). It is not difficult to see that two morphisms α, β : A ⟶ B in C are left homotopic iff α − β factors through an object in Y and they are right homotopic iff α − β factors through an object in ω. Since C ∩X = X is the full subcategory of cofibrant-fibrant objects, and a morphism between objects of X factors through an object in Y iff it factors through an object in ω, we infer that Ho(C) is equivalent to the stable category X /ω and the canonical functor from C to the homotopy category Ho(C) is isomorphic to the functor C ⟶ X /ω, A ↦ R(\underline{A}), where as usual R : C/ω ⟶ X /ω is the right adjoint of the inclusion X /ω ↪ C/ω.

Let (X , Y) be a pretorsion theory in C with ω = X ∩ Y contravariantly finite.

Definition 4.3. The closed model structure (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) of Theorem 4.2 is called the contravariant ω−closed model structure associated to the pretorsion theory (X , Y).

Example. There are pretorsion theories (X , Y) not defining a closed model structure. Indeed, in general the flat pretorsion theory (Flat(Λ), Flat(Λ)^{⊥}) in Mod(Λ) has ω = Flat(Λ) ∩ Flat(Λ)^{⊥} not contravariantly finite. Hence in general there cannot exist a contravariant ω-closed model structure in Mod(Λ) induced by the flat pretorsion theory.

Example. Let G be a group and k be a commutative ring of coefficients, and let B be the set of functions G ⟶ k which take only finitely many different values in k. If X = {X ∈ Mod(kG) | B ⊗_{k} X is projective} and ω is the full subcategory of projective kG-modules, then by the example after Remark V.4.7, we have a pretorsion theory (X , ω̂) in X̂ = {X ∈ Mod(kG) | pd_{kG}B ⊗_{k} X < ∞}. Since the subcategory ω of projective kG-modules is contravariantly finite and all closed model theoretic constructions of Theorem 4.2 can be performed in X̂, Theorem 4.2 produces a closed model structure in X̂. For this closed model structure the modules in X are the cofibrant objects and the modules of finite projective dimension are the trivially fibrant objects. The associated Quillen homotopy category is the stable category X /ω, which is triangulated by Proposition VI.1.2. Note that the existence of this closed model structure was first observed by D. Benson, see Theorem 10.2 in [21]. By the example after Remark V.4.7, the above closed model structure extends to a closed model structure in the whole module category Mod(kG), if k has finite global dimension and the group G is of type F P_{∞} and belongs to Kropholler’s class of groups H𝔉.

We recall from Chapter II that the homotopy category of an abelian closed model category C is pretriangulated. To state our next result we need to recall a definition from [45].

Definition 4.4. A closed model structure in C is called stable, if the associated pretriangulated homotopy category is triangulated.

The following characterizes the stable closed model structures arising from pretorsion theories.

Corollary 4.5. The triple (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) is a stable closed model structure iff (X , Y) is a pretorsion theory in C, and any contravariant ω-fibration is an epimorphism iff (X , Y) is a pretorsion theory in C with X ∩ Y = P.

Proof. By Theorem 4.2 we know that the homotopy category of the contravariant ω-closed model structure is equivalent to X /ω. As in Theorem VI.2.1 we have that the latter is triangulated iff P = ω. If 𝔉_{ω} consists of epimorphisms, then by Proposition 3.3(2) we have a good pair (TCof_{ω}, Fib_{ω}). But by the above theorem we have ω = TCof_{ω} and Fib_{ω} = C. We infer that ω = P. Conversely if ω = P, then obviously 𝔉_{ω} consists of epimorphisms.

Using the above theorem we can get the following classification of closed model structures arising from pretorsion theories in C.

Theorem 4.6. There exists a bijective correspondence between pretorsion theories (X , Y) in C with X ∩Y contravariantly finite in C, and closed model structures (ℭ, 𝔉, 𝔚) in C such that:

(i) Any cofibration is monic and any object of C is fibrant.

(ii) Trivial fibrations are epics and any epic with kernel in TFib is a trivial

fibration. (iii) TCof = Cof ∩ TFib.

The bijection is given as follows:

Φ : (X , Y) ⟼ (ℭ_{X ∩Y}, 𝔉_{X ∩Y}, 𝔚_{X ∩Y})
Ψ : (ℭ, 𝔉, 𝔚) ⟼ (Cof, TFib).

The above bijection induces a bijection between pretorsion theories (X , Y) in C with X ∩ Y = P and stable closed model structures (ℭ, 𝔉, 𝔚) satisfying (i), (ii), (iii) and (iv), where:

(iv) 𝔉 consists of epimorphisms.

Proof. If (X , Y) is a pretorsion theory in C with ω = X ∩ Y contravariantly finite in C, then by Theorem 4.2, Φ(X , Y) = (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) is a closed model structure in C which satisfies the conditions (i) and (ii). Moreover we have ω = TCof_{ω} = X ∩ Y = Cof_{ω} ∩ TFib_{ω} which is contravariantly finite by assumption. Hence the resulting closed model structure also satisfies (iii). Finally observe that by Theorem 4.2 we have ΨΦ(X , Y) = Ψ(ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) = (X , Y).

Conversely let (ℭ, 𝔉, 𝔚) be a closed model structure in C satisfying the conditions (i), (ii) and (iii). Then by Theorem 3.4 we have that Ψ(ℭ, 𝔉, 𝔚) = (Cof, TFib) is a good pair, and by Proposition 2.1 we have that the intersection Cof ∩ TFib is contravariantly finite, since it is equal to TCof. Finally Proposition 2.1 implies that the good pair (Cof, TFib) is a pretorsion theory in C.

It remains to show that ΦΨ(ℭ, 𝔉, 𝔚) = Φ(Cof, TFib) = (ℭ, 𝔉, 𝔚). In other words we have to show that the closed model structures

(ℭ_{Cof∩TFib}, 𝔉_{Cof∩TFib}, 𝔚_{Cof∩TFib}) and (ℭ, 𝔉, 𝔚)

coincide. We set ω := Cof ∩ TFib which is equal to TCof by hypothesis. We first prove that ℭ = ℭ_{ω}. If i is in ℭ, then by (i) we have that i is monic. Since the cokernel of a cofibration is cofibrant we have that Coker(i) lies in Cof. Hence i is a contravariant ω-cofibration and this shows that ℭ ⊆ ℭ_{ω}. Now let i : A ⟶ B be a contravariant ω-cofibration. Then we have a short exact sequence 0 ⟶ A ⟶^{i} B ⟶ C ⟶ 0 where C is cofibrant. To show that i is a contravariant ω-cofibration, by Lemma 1.3 it suffices to show i has the left lifting property with respect to all contravariant ω-trivial fibrations, which by definition are exactly the epimorphisms with kernel in TFib. This follows by the Extension-Lifting Lemma 3.1, since C is cofibrant and Ext^{1}(C, TFib) = 0. Next we show that 𝔉 = 𝔉_{ω}. Let p : C ⟶ D be a contravariant ω-fibration, that is p is an ω-epic. To show that p is a fibration it suffices to show that p has the right lifting property with respect to all trivial cofibrations. Let i : A ⟶ B be a trivial cofibration; then Coker(i) is in TCof = ω. By Lemma 1.1, any morphism out of A to a fibrant object factors through i. Since all objects are fibrant, it follows that i is split monic. Hence i is a split monic with cokernel in ω, and then without loss of generality we can assume that i is given by

A ⟶^{(1_{A},0)} A ⊕ T where T ∈ ω. Consider a commutative diagram

A ⟶^{α} C
(1_{A},0)↓  ^{p}↓
A ⊕ T ⟶^{^{t}(κ,λ)} D

Since T is in ω and p is an ω-epic, there exists a morphism τ : T ⟶ C such that τ ∘ p = λ. Then the morphism ^{t}(α, τ) : A ⊕ T ⟶ C is a lifting of the above square. We infer that 𝔉_{ω} ⊆ 𝔉. Conversely let p : C ⟶ D be a fibration. Let T be in ω and let α : T ⟶ D be a morphism. Since ω = TCof we have that 0 ⟶ T is a trivial cofibration. Since any fibration has the right lifting property with respect to trivial cofibrations, the following commutative diagram

\begin{matrix}0 & ⟶ & C \\ ↓ & & ^{p}↓ \\ T & ⟶^{α} & D \end{matrix}

admits a lifting, which means that α factors through p. Hence p is an ω-epic, that is p ∈ 𝔉_{ω}, and this shows that 𝔉 ⊆ 𝔉_{ω}. We infer that 𝔉 = 𝔉_{ω}. It remains to show that 𝔚 = 𝔚_{ω}. This follows easily from the fact that in a closed model structure the fibrations together with the cofibrations determine the weak equivalences, see [43]. This completes the proof that the functions Φ and Ψ are mutually inverse.

What we proved so far in this section can be dualized. For the convenience of the reader we state the dual results, leaving their proof to the reader.

Keeping the set-up of this section we continue to assume that X and Y are full additive subcategories of the abelian category C which are closed under direct summands and isomorphisms, and we let ω = X ∩ Y.

In a dual manner we consider the following classes of morphisms in C:

(i) ℭ^{ω} is the class of ω-monics.

(ii) 𝔗ℭ^{ω} is the class of monics with cokernel in X.

(iii) 𝔉^{ω} is the class of epics with kernel in Y.

(iv) 𝔗𝔉^{ω} is the class of split epics with kernel in ω.

(v) 𝔚^{ω} is the class of morphisms α which admit a factorization α = α_{1} ∘ α_{2} where α_{1} ∈ 𝔗ℭ^{ω} and α_{2} ∈ 𝔗𝔉^{ω}.

We call the morphisms in 𝔉^{ω}, resp. 𝔗𝔉^{ω}, covariant ω-fibrations, resp. covariant ω-trivial fibrations. We call the morphisms in ℭ^{ω}, resp. 𝔗ℭ^{ω}, covariant ω-cofibrations, resp. covariant ω-trivial cofibrations. Finally we call the morphisms in 𝔚^{ω}, covariant ω-weak equivalences.

The following result, which we state without proof, gives the dual version of Theorems 4.2 and 4.6.

Theorem 4.7. (1) The following are equivalent:

(i) The triple (ℭ^{ω}, 𝔉^{ω}, 𝔚^{ω}) defines a closed model structure on C such that: ℭ^{ω} is the class of cofibrations, 𝔉^{ω} is the class of fibrations, and 𝔚^{ω} is the class of weak equivalences.

(ii) (X , Y) is a pretorsion theory and ω is covariantly finite in C. If this is the case, then Cof^{ω} = C and TFib^{ω} = ω, and Fib^{ω} = Y and TCof^{ω} = X. Moreover the associated Quillen homotopy category Ho(C) is equivalent to the stable left triangulated catetory Y/ω.

(2) The triple (ℭ^{ω}, 𝔉^{ω}, 𝔚^{ω}) is a stable closed model structure in C iff (X , Y) is a pretorsion theory in C and any covariant ω-cofibration is a monomorphism iff (X , Y) is a pretorsion theory in C with X ∩ Y = I.

(3) There exists a bijective correspondence between pretorsion theories (X , Y) in C with X ∩ Y covariantly finite in C and closed model structures (ℭ, 𝔉, 𝔚) in C such that:

(i) Any fibration is epic and any object of C is cofibrant.

(ii) Trivial cofibrations are monics and any monic with kernel in TCof is a trivial

cofibration. (iii) TFib = TCof ∩ Fib.

The bijective correspondence is given as follows:

Φ̂ : (X , Y) ⟼ (ℭ^{X ∩Y}, 𝔉^{X ∩Y}, 𝔚^{X ∩Y})
Ψ̂ : (ℭ, 𝔉, 𝔚) ⟼ (TCof, Fib).

The above bijection induces a bijection between pretorsion theories (X , Y) in C with X ∩ Y = I and stable closed model structures (ℭ, 𝔉, 𝔚) satisfying (i), (ii), (iii) and (iv), where:

(iv) ℭ consists of monomorphisms.

Let (X , Y) be a pretorsion theory in C with ω = X ∩ Y covariantly finite.

Definition 4.8. The closed model structure (ℭ^{ω}, 𝔉^{ω}, 𝔚^{ω}) of Theorem 4.7 is called the covariant ω−closed model structure associated to the pair (X , Y).

Example. Let Λ be a left coherent ring and let X = Flat(Λ) be the category of flat right Λ-modules. Then ω = X ∩X^{⊥} which, by Section V.4, is the category of flat and pure-injective modules, is covariantly finite. Since (X , X^{⊥}) is a pretorsion theory, by Theorem 4.7 we get a closed model structure in Mod(Λ) where the flat modules are the trivially cofibrant objects, and the modules in X^{⊥}, widely known as the cotorsion modules, are the fibrant objects.

Combining Theorems 4.6 and 4.7 we have the following consequence.

Corollary 4.9. Assume that C has enough projectives and enough injectives. Then C is Frobenius iff (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) and (ℭ^{ω}, 𝔉^{ω}, 𝔚^{ω}) are stable closed model structures in C.

Let (X , Y) be a pretorsion theory in C with X ∩ Y := ω. If ω is contravariantly finite, then we can define in C the contravariant ω-closed structure, and when ω is covariantly finite, we can define in C the covariant ω-closed structure. If ω is functorially finite, so that both closed model structures are defined, then there is an additional closed model structure defined in C which is induced by ω. Indeed, by [19] any functorially finite subcategory ω defines a closed model structure (ℭ(ω), 𝔉(ω), 𝔚(ω)) in C, with cofibrations the ω-monics, fibrations the ω-epics and weak equivalences the morphisms which are isomorphisms in C/ω. This suggests the following definition which will be useful in the next section.

Definition 4.10. If ω is a functorially finite subcategory of C, then the closed model structure (ℭ(ω), 𝔉(ω), 𝔚(ω)) is called the functorial ω−closed model structure.

Note that by [19], the Quillen homotopy category of the functorial ω-closed model structure (ℭ(ω), 𝔉(ω), 𝔚(ω)) is the stable pretriangulated category C/ω.

Remark 4.11. Observe that between the triples (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}), (ℭ^{ω}, 𝔉^{ω}, 𝔚^{ω}) we have the following relations:

(i) 𝔉^{ω} = 𝔗𝔉_{ω} which are epics and ℭ_{ω} = 𝔗ℭ^{ω} which are monics.

(ii) TCof_{ω} = Cof_{ω} ∩ TFib_{ω} and TFib^{ω} = Fib^{ω} ∩ TCof^{ω}.

(iii) TCof_{ω} = ω = TFib^{ω} and Fib_{ω} = C = Cof^{ω}.

5. A Classification of (Pre)Torsion Theories

In this section we classify in terms of closed model structures all pretorsion theories (X , Y) in C with the property that X ∩ Y is functorially finite in C. This classification will be used for the classification of torsion theories in the stable category of C modulo a functorially finite subcategory.

The last remark of the previous section suggests the following definition, which can be considered as “glueing” together closed model structures.

Definition 5.1. An (ordered) pair of closed model structures (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) and (ℭ_{2}, 𝔉_{2}, 𝔚_{2}) in C are called compatible if:

(i) 𝔉_{2} = 𝔗𝔉_{1} which are epics and ℭ_{1} = 𝔗ℭ_{2} which are monics.

(ii) TCof_{1} = Cof_{1} ∩ TFib_{1} = TFib_{2} = Fib_{2} ∩ TCof_{2}.

(iii) Fib_{1} = C = Cof_{2}.

Let (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) and (ℭ_{2}, 𝔉_{2}, 𝔚_{2}) be a pair of compatible closed model structures in C. It follows easily from the definition and Lemma 1.3 that 𝔗𝔉_{2} ⊆ 𝔉_{1} and 𝔗ℭ_{1} ⊆ ℭ_{2}. Then by Proposition 2.2 we have the following relations:

TCof_{2} = Cof_{1}, TFib_{1} = Fib_{2}, TCof_{1} ⊆ Cof_{2}, TFib_{2} ⊆ Fib_{1}.

Moreover by Proposition 2.1, TCof_{1} = TFib_{2} := ω is a functorially finite subcategory of C. Hence the functorial ω-closed model structure (ℭ(ω), 𝔉(ω), 𝔚(ω)) is defined in C. This motivates us to make the following definition.

Definition 5.2. Let (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) and (ℭ_{2}, 𝔉_{2}, 𝔚_{2}) be a pair of compatible closed model structures in C. Their intersection (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) ∩ (ℭ_{2}, 𝔉_{2}, 𝔚_{2}) is defined to be the functorial ω-closed model structure (ℭ(ω), 𝔉(ω), 𝔚(ω)), where TCof_{1} = TFib_{2} := ω.

We have the following consequence of the above results.

Theorem 5.3. If the abelian category C has enough projectives and enough injectives, then there exists a bijective correspondence between pretorsion theories (X , Y) in C with X ∩Y functorially finite in C and compatible closed model structures (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) and (ℭ_{2}, 𝔉_{2}, 𝔚_{2}) in C. The bijection is given as follows:

(X , Y) ⟼ {(ℭ_{X ∩Y}, 𝔉_{X ∩Y}, 𝔚_{X ∩Y}), (ℭ^{X ∩Y}, 𝔉^{X ∩Y}, 𝔚^{X ∩Y})}
{(ℭ_{1}, 𝔉_{1}, 𝔚_{1}), (ℭ_{2}, 𝔉_{2}, 𝔚_{2})} ⟼ (Cof_{1}, Fib_{2})

The following consequence of Theorem 5.3 shows that if we have a pair of compatible closed model structures in C, then we obtain a torsion theory in the homotopy category of the closed model structure of their intersection; moreover any torsion theory in the homotopy category is obtained in this way, provided that the torsion and the torsion free subcategories are orthogonal with respect to Ext^{1}.

Theorem 5.4. Assume that the abelian category C has enough projectives and enough injectives and let ω be a functorially finite subcategory of C. Then the correspondence of Theorem 5.3 induces a bijective correspondence

{ torsion theories (X /ω, Y/ω) in C/ω with Ext^{1}(X , Y) = 0 } ⟼

{ compatible closed model structures (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) and (ℭ_{2}, 𝔉_{2}, 𝔚_{2}) in C such that : Cof_{1} is closed under kernels of epics, or equivalently Fib_{2} is closed under cokernels of monics, and : Cof_{1} ∩ Fib_{2} = ω }

Under this correspondence:

(i) The stable pretriangulated category C/ω is the homotopy category of the functorial ω-closed model structure and corresponds to the homotopy category of the intersection (ℭ_{1}, 𝔉_{1}, 𝔚_{1}) ∩ (ℭ_{2}, 𝔉_{2}, 𝔚_{2}).

(ii) The torsion class X /ω is the homotopy category of the contravariant ω-clo– sed model structure and corresponds to the homotopy category of the closed model structure (ℭ_{1}, 𝔉_{1}, 𝔚_{1}).

(iii) The torsion-free class Y/ω is the homotopy category of the covariant ω-clo– sed model structure and corresponds to the homotopy category of the closed model structure (ℭ_{2}, 𝔉_{2}, 𝔚_{2}).

It follows by the above result that torsion theories appear as “glueing” together homotopy categories of compatible closed model structures inside the homotopy category of the closed model structure of their intersection. This gives a conceptual homotopy-theoretic interpretation of the theory developed in Chapters V and VI, and puts it in the proper framework.

The correspondences given by the above theorems combined with the correspondence of Proposition VI.2.6 allow us to deduce the following.

Corollary 5.5. Let C be a Frobenius abelian category. Then there exist bijective correspondences between:

(i) Compatible stable closed model structures in C.

(ii) Hereditary pretorsion theories (X , Y) in C with X ∩ Y the projectives P. (iii) Hereditary torsion theories in C/P.

We can define closed model structures in C starting from a subcategory instead of X , Y and ω. Indeed, let X be a full preresolving subcategory of C. Define ℭ_{X} to be the class of monomorphisms with cokernel in X, 𝔉_{X} to be the class of all epics, 𝔗ℭ_{X} to be the class of all monics with projective cokernel, 𝔗𝔉_{X} to be the class of epics with kernel in X^{⊥}, and finally 𝔚_{X} to be the class of morphisms which can be written as a composition of a morphism in 𝔗ℭ_{X} followed by a morphism in 𝔗𝔉_{X}.

Dually for a precoresolving subcategory Y of C, define ℭ^{Y} to be the class of all monics, 𝔉^{Y} to be the class of all epics with kernel in Y, 𝔗ℭ^{Y} to be the class of all monics with cokernel in ^{⊥}Y, 𝔗𝔉^{Y} to be the class of epics with injective kernel, and finally 𝔚^{Y} to be the class of morphisms which can be written as a composition of a morphism in 𝔗ℭ^{Y} followed by a morphism in 𝔗𝔉^{Y}.

Theorem 5.6. (1) Assume that C is Krull-Schmidt or else C has filtered colimits and X is closed under filtered colimits. If X ∩ X^{⊥} = P, then the triple (ℭ_{X} , 𝔉_{X} , 𝔚_{X}) is a (stable) closed model structure in C iff X is contravariantly finite.

(2) Assume that C is Krull-Schmidt or else C is Grothendieck and 𝔗ℭ^{Y} is closed under filtered colimits. If ^{⊥}Y ∩ Y = I, then the triple (ℭ^{Y}, 𝔉^{Y}, 𝔚^{Y}) is a (stable) closed model structure in C iff Y is covariantly finite.

Proof. (1) If the preresolving subcategory X is contravariantly finite, then either of the imposed assumptions on C and/or X implies that (X , X^{⊥}) is a pretorsion theory in C, see the example at the end of Section V.4. Since X ∩ X^{⊥} = P is contravariantly finite, by Theorem 4.2 we have a closed model structure (ℭ_{X} , 𝔉_{X} , 𝔚_{X}) and obviously Cof_{X} = X, Fib_{Y} = C, TCof_{X} = P, and TFib_{X} = X^{⊥}. The converse follows from Proposition 2.1, since Cof_{X} = X. The proof of (2) is dual.

We have seen that for any tilting or cotilting module T over an Artin algebra Λ, there is associated in a natural way a pretorsion theory. Since ω = add(T) is functorially finite, Theorems 4.6 and 4.7 can be applied. It follows that any tilting or cotilting module defines natural closed model structures in mod(Λ). We leave to the reader to write down explicitly the associated closed model structures. We note only that the well-known equivalences and dualities induced by a tilting or cotilting module can be recovered as Quillen equivalences between the above closed model structures.

Good Triples. We close this section by discussing briefly the connections between closed model structures and good triples.

Let (X , Y, Z) be a good triple in the abelian category C and assume that the pairs (X , Y) and (Y, Z) are pretorsion theories in C. As in the proof of Theorem VI.3.2 we have that Y is closed under kernels of epics iff Y is closed under cokernels of monics. In other words (X , Y) is a hereditary pretorsion theory iff (Y, Z) is a hereditary pretorsion theory. Assume in the sequel that one of these conditions holds. Then X is closed under kernels of epics and Z is closed under cokernels of monics, and moreover X ∩ Y = P and Y ∩ Z = I. Moreover, if P is covariantly finite, then (X /P, Y/P) is a torsion theory in C/P with X /P triangulated, and if I is contravariantly finite, then (Y/I, Z/I) is a torsion theory in C/I with Z/I triangulated. By Theorems 4.6 and 4.7, the pretorsion theories (X , Y) and (Y, Z) induce four closed model structures on C:

(ℭ_{P}, 𝔉_{P}, 𝔚_{P}), (ℭ^{P}, 𝔉^{P}, 𝔚^{P}), (ℭ_{I}, 𝔉_{I}, 𝔚_{I}), (ℭ^{I}, 𝔉^{I}, 𝔚^{I}).

It is not difficult to see that the adjoint pair (Id_{C}, Id_{C}) of identity functors is a Quillen equivalence, in the sense of [45], between (ℭ_{P}, 𝔉_{P}, 𝔚_{P}) and (ℭ^{I}, 𝔉^{I}, 𝔚^{I}). Hence the identity functor Id_{C} induces a (triangle) equivalence between the homotopy categories, in other words a (triangle) equivalence X /P ⟶^{≈} Z/I. This is a closed model theoretic proof of Theorem VI.3.2.

A special case of the above situation is the case of the good triple (CM(Λ), P_{Λ}^{<∞} = I_{Λ}^{<∞}, CoCM(D(Λ))) where Λ is a Gorenstein algebra. We leave to the reader to write down the four closed model structures. For an arbitrary Artin algebra Λ, working with all (not necessarily finitely generated) modules, we will show in the next chapter that there exists a pair of compatible closed model structures in Mod(Λ), hence four in all, and two of them are stable. For the first stable closed model structure the Cohen-Macaulay modules appear as cofibrant objects and for the second, the CoCohen-Macaulay modules appear as fibrant objects.

A nice illustration of the above results is the following example which shows that derived categories are homotopy categories of stable closed model structures induced by good triples in the category of complexes.

Let Λ be a ring and let C(Mod(Λ)) be the category of complexes of right Λ-modules. Let C_{0}(Mod(Λ)) be the full subcategory of contractible complexes. It is easy to see that the latter is functorially finite, hence by [19], the stable category C(Mod(Λ))/C_{0}(Mod(Λ)) is identified with the homotopy category H(Mod(Λ)). We denote by C_{Ac}(Mod(Λ)) the full subcategory of acyclic complexes and by C_{P}(Mod(Λ)), resp. C^{I}(Mod(Λ)), the full subcategory of homotopically projective, resp. injective, complexes [55]. It is not difficult to see that C_{P}(Mod(Λ)) ∩ C_{Ac}(Mod(Λ)) = C_{0}(Mod(Λ)) = C_{Ac}(Mod(Λ)) ∩ C^{I}(Mod(Λ)). We denote as in Chapter III by H_{P}(Mod(Λ)), H^{I}(Mod(Λ)), H_{Ac}(Mod(Λ)), the induced homotopy categories. By Corollary IV.1.7 we have a TTF-theory

(H_{P}(Mod(Λ)), H_{Ac}(Mod(Λ))), H^{I}(Mod(Λ)))

in H(Mod(Λ)), and it is not difficult to see that Ext^{1}(C_{P}(Mod(Λ)), C_{Ac}(Mod(Λ))) = Ext^{1}(C_{Ac}(Mod(Λ)), C^{I}(Mod(Λ))) = 0. Hence by Theorem 4.2 we have a closed model structure (ℭ_{P}, 𝔉_{P}, 𝔚_{P}) in C(Mod(Λ) with corresponding Quillen homotopy category H_{P}(Mod(Λ)) and a closed model structure (ℭ^{I}, 𝔉^{I}, 𝔚^{I}) in C(Mod(Λ)) with corresponding Quillen homotopy category H^{I}(Mod(Λ)). By Corollary IV.1.7 both Quillen homotopy categories are triangle equivalent to the unbounded derived category D(Mod(Λ)).

It follows that Theorems 4.2 and 4.7 produce the projective and injective closed model structures on the category of complexes investigated in [45]. In our terminology the projective model structure of [45] is the contravariant ω-closed model structure (ℭ_{ω}, 𝔉_{ω}, 𝔚_{ω}) associated with the good pair (C_{P}(Mod(Λ)), C_{Ac}(Mod(Λ))), where ω = C_{P}(Mod(Λ)) ∩ C_{Ac}(Mod(Λ)) is the full subcategory of contractible complexes. For instance for the projective closed model structure, the cofibrations are the monomorphisms with cokernel a homotopically projective complex, the fibrations are the epimorphisms, and the weak equivalences are the morphisms which are compositions α ∘ β where α is split monic with contractible cokernel and β is an epimorphism with acyclic kernel. In particular the weak equivalences are quasi-isomorphisms. The injective closed model structure has a similar description.

Note that the above analysis with the necessary modifications works if we replace the module category with a Grothendieck category (with projectives).

For the history of the development of these closed model structures we refer to

CHAPTER VIII Generalized Tate-Vogel Cohomology

Let C be an abelian category and let ω be a functorially finite subcategory of C. Our aim in this chapter is to construct and investigate universal (co)homology theories in C which are induced by a (co)hereditary torsion theory defined in the pretriangulated stable category C/ω. We study also the connections and the interplay between relative homology and these new (co)homology theories and we give applications to Cohen-Macaulay modules and modules of finite projective dimension over an Artin algebra.

1. Hereditary Torsion Theories and Homological Functors

In this section, using the left or right stabilization of a pretriangulated category C, we define universal (co)homology theories on C with respect to a (co)hereditary torsion theory.

We fix throughout a torsion theory (X , Y) in the pretriangulated category C, and as usual let R : C ⟶ X be the right adjoint of the inclusion i : X ↪ C and L : C ⟶ Y the left adjoint of the inclusion j : Y ↪ C. We denote by P : C ⟶ T_{l}(C), resp. Q : C ⟶ T_{r}(C), the stabilization functor of C, when the latter is a considered as a left, resp. right, triangulated category.

Recall from section II.5 that if the torsion theory (X , Y) is hereditary, then X is pretriangulated, and the inclusion functor i : X ↪ C and the coreflection functor R : C ⟶ X are left exact. In this case there exist unique exact functors i^{∗} : T_{l}(X) ⟶ T_{l}(C) and R^{∗} : T_{l}(C) ⟶ T_{l}(X), such that the following diagrams commute:

\begin{matrix}X & ⟶^{i} & C \\ ^{P}↓ & & ^{P}↓ \\ T_{l}(X) & ⟶^{i^{∗}} & T_{l}(C) \end{matrix}  \begin{matrix}C & ⟶^{R} & X \\ ^{P}↓ & & ^{P}↓ \\ T_{l}(C) & ⟶^{R^{∗}} & T_{l}(X) \end{matrix}

where P : X ⟶ T_{l}(X) is the stabilization functor of X when the latter is considered as a left triangulated category. Moreover R^{∗} is a right adjoint of i^{∗}, and (T_{l}(X), T_{l}(Y)) is a hereditary torsion theory in T_{l}(C). Hence we have a short exact sequence of triangulated categories:

0 ⟶ T_{l}(Y) ⟶^{j^{∗}} T_{l}(C) ⟶^{R^{∗}} T_{l}(X) ⟶ 0

Dually if the torsion theory (X , Y) is cohereditary, then Y is pretriangulated, the inclusion functor j : Y ↪ C and the reflection functor L : C ⟶ Y are right exact, and there exist unique exact functors j^{∗} : T_{r}(Y) ⟶ T_{r}(C) and L^{∗} : T_{r}(C) ⟶ T_{r}(Y), such that the following diagrams commute:

\begin{matrix}Y & ⟶^{j} & C \\ ^{Q}↓ & & ^{Q}↓ \\ T_{r}(X) & ⟶^{j^{∗}} & T_{r}(C) \end{matrix}  \begin{matrix}C & ⟶^{L} & Y \\ ^{Q}↓ & & ^{Q}↓ \\ T_{r}(C) & ⟶^{L^{∗}} & T_{r}(Y) \end{matrix}

where Q : Y ⟶ T_{r}(Y) is the stabilization functor of Y when the latter is considered as a right triangulated category. Moreover L^{∗} is a left adjoint of j^{∗}, and (T_{r}(X), T_{r}(Y)) is a hereditary torsion theory in T_{r}(C). Hence we have a short exact sequence of triangulated categories:

0 ⟶ T_{r}(X) ⟶^{i^{∗}} T_{r}(C) ⟶^{L^{∗}} T_{r}(Y) ⟶ 0

The above considerations suggest the following.

Definition 1.1. Let (X , Y) be a hereditary torsion theory in C. Then the n−th projective extension bifunctor

Êxt_{(X ,Y)}^{n}(−, −) : C^{op} × C ⟶ Ab, n ∈ ℤ

of C with respect to (X , Y) is defined as follows:

Êxt_{(X ,Y)}^{n}(A, B) := T_{l}(X)[Ω^{n}R^{∗}P(A), R^{∗}P(B)] ⟶^{≅} T_{l}(X)[Ω^{n}PR(A), PR(B)] ⟶^{≅}

lim_{⟶}_{k,k+n≥0}X [Ω^{k+n}R(A), Ω^{k}R(B)] ≅ lim_{⟶}_{k,k+n≥0}C[Ω^{k+n}R(A), Ω^{k}(B)].

Let (X , Y) be a cohereditary torsion theory in C. Then the n−th injective extension bifunctor

Ẽxt_{(X ,Y)}^{n}(−, −) : C^{op} × C ⟶ Ab, n ∈ ℤ

of C with respect to (X , Y) is defined as follows:

Ẽxt_{(X ,Y)}^{n}(A, B) := T_{r}(Y)[L^{∗}Q(A), Σ^{n}L^{∗}Q(B)] ⟶^{≅} T_{r}(Y)[QL(A), Σ^{n}QL(B)] ⟶^{≅}

lim_{⟶}_{k,k+n≥0}Y[Σ^{k}L(A), Σ^{k+n}L(B)] ≅ lim_{⟶}_{k,k+n≥0}C[Σ^{k}(A), Σ^{k+n}L(B)].

Example. Let C be triangulated and let (X , Y) be a hereditary torsion theory in C. Then (X , Y) is also cohereditary, and according to Chapter I, X and Y are thick subcategories of C; in particular all the involved stabilization functors are triangle equivalences in this case. Then Êxt_{(X ,Y)}^{n}(A, B) = X (Ω^{n}R(A), R(B)) ≅

C(Ω^{n}R(A), B) and Ẽxt_{(X ,Y)}^{n}(A, B) = Y(L(A), Σ^{n}L(B)) ≅ C(A, Σ^{n}L(B)).

Example. The smallest torsion theory (0, C) and the largest torsion theory (C, 0) in C are obviously hereditary and cohereditary. In this case we have

Ẽxt_{(C,0)}^{n}(−, −) = 0 = Êxt_{(0,C)}^{n}(−, −) and Êxt_{(C,0)}^{n}(−, −) = Ẽxt_{(0,C)}^{n}(−, −).

The above example suggests the following definition. The terminology will be justified in the next section.

Definition 1.2. The n−th projective extension bifunctor with respect to the hereditary torsion theory (C, 0) is called the n−th Tate-Vogel projective extension bifunctor of C and is denoted by Êxt_{C}^{n}(−, −). In other words

Êxt_{C}^{n}(A, B) := T_{l}(C)[Ω^{n}P(A), P(B)].

The n−th injective extension bifunctor with respect to the cohereditary torsion theory (0, C) is called the n−th Tate-Vogel injective extension bifunctor of C and is denoted by Ẽxt_{C}^{n}(−, −). In other words

Ẽxt_{C}^{n}(A, B) := T_{r}(C)[Q(A), Σ^{n}Q(B)].

From now on we assume that the torsion theory (X , Y) is hereditary. The corresponding results concerning cohereditary torsion theories follow by duality.

The following result presents some properties of the bifunctor Êxt_{(X ,Y)}^{n}(−, −).

Proposition 1.3. (1) If A or B is in Y then Êxt_{(X ,Y)}^{n}(A, B) = 0, ∀n ∈ ℤ.

(2) Êxt_{(X ,Y)}^{0}(A, A) = 0 iff R(A) ∈ P^{<∞}(C) iff Ω^{n}(A) ∈ Y for some n ≥ 0.

(3) For any left triangle Ω(C) ⟶ A ⟶ B ⟶ C in C and for any D ∈ C there exists an infinite long exact sequence:

⋯ ⟶ Êxt_{(X ,Y)}^{−1}(A, D) ⟶ Êxt_{(X ,Y)}^{0}(C, D) ⟶ Êxt_{(X ,Y)}^{0}(B, D) ⟶
⟶ Êxt_{(X ,Y)}^{0}(A, D) ⟶ Êxt_{(X ,Y)}^{1}(C, D) ⟶ Êxt_{(X ,Y)}^{1}(B, D) ⟶ ⋯

and an infinite long exact sequence:

⋯ ⟶ Êxt_{(X ,Y)}^{−1}(D, C) ⟶ Êxt_{(X ,Y)}^{0}(D, A) ⟶ Êxt_{(X ,Y)}^{0}(D, B) ⟶
⟶ Êxt_{(X ,Y)}^{0}(D, C) ⟶ Êxt_{(X ,Y)}^{1}(D, A) ⟶ Êxt_{(X ,Y)}^{1}(D, B) ⟶ ⋯

Proof. Parts (1) and (2) follow directly from the definitions. Part (3) follows from the fact that the stabilization T_{l}(X) is a triangulated category.

Observe that the functor R^{∗} : T_{l}(C) ⟶ T_{l}(X) induces a natural morphism Êxt_{C}^{n}(−, −) ⟶ Êxt_{(X ,Y)}^{n}(−, −), ∀n ∈ ℤ, which is induced by the lifting of the adjunction iR ⟶ Id_{C}. Using Proposition 1.3, the glueing triangle ΩL(A) ⟶ R(A) ⟶ A ⟶ L(A), ∀A ∈ C, and Corollary II.5.5, we have the following.

Corollary 1.4. Let A, B be objects in C. Then there exists a natural in A and B infinite long exact sequence:

⋯ ⟶ Êxt_{C}^{n}(L(A), B) ⟶ Êxt_{C}^{n}(A, B) ⟶ Êxt_{(X ,Y)}^{n}(A, B) ⟶ ⋯

In particular the natural map Êxt_{C}^{n}(A, −) ⟶ Êxt_{(X ,Y)}^{n}(A, −) is invertible for all n ∈ ℤ iff L(A) ∈ P^{<∞}(C), that is: Ω^{n}L(A) = 0 for some n ≥ 0.

Finally the natural map Êxt_{C}^{n}(−, −) ⟶ Êxt_{(X ,Y)}^{n}(−, −) is invertible for all n ∈ ℤ iff Y ⊆ P^{<∞}(C) iff X̂ = C.

Part (3) of Proposition 1.3 suggests the following definition, which extends the usual definition of Grothendieck of homology theories (or homological functors) defined on an abelian category. Of course if we view an abelian category as pretriangulated with zero loop and suspension functor and with left and right triangles to be the exact sequences, then the two definitions agree.

Definition 1.5. A homology theory on C is a sequence of additive functors F_{∗} := {F_{n}}_{n∈ℤ}, F_{n} : C ⟶ Ab, such that for any left triangle Ω(C) ⟶ A ⟶ B ⟶ C in C there exists a long exact sequence

⋯ ⟶ F_{−1}(C) ⟶ F_{0}(A) ⟶ F_{0}(B) ⟶ F_{0}(C) ⟶ F_{1}(A) ⟶ ⋯

which is natural with respect to morphisms of left triangles.

In the sequel we shall need the following which extends a well-known concept of completeness of homological functors defined on an abelian category, see [61].

Definition 1.6. Let Y be a full subcategory of C. A homology theory {F_{n}}_{n∈ℤ} is called Y−complete if F_{n}(Y) = 0, ∀n ∈ ℤ.

The following are examples of homology theories on C.

Example. (i) For any object C ∈ C the sequence of functors Êxt_{(X ,Y)}^{∗}(C, −)

is a Y-complete homology theory on C. (ii) For any C ∈ C we define a sequence of functors π_{n}^{C} : C ⟶ Ab as fol-

lows. For n ≥ 0 we set π_{n}^{C} = C(Ω^{n}(C), −) and π_{−n}^{C}(C) = C(C, Ω^{n}(−)) ⟶^{≅} C(Σ^{n}(C), −). It is not difficult to see that if Σ is fully faithful, then the sequence of functors π_{∗}^{C} is a homology theory in C which is Y-complete iff C ∈ X.

Let F_{•} = {F_{n}} be a homology theory on C. Consider the category Ab^{ℤ} as a graded category equipped with the shift automorphism S : Ab^{ℤ} ⟶ Ab^{ℤ} which sends the graded object {A_{n}} to {S(A_{n})} which in degree n has the object A_{n−1}. Then a homology theory is a homological functor F_{•} : C ⟶ Ab^{ℤ} such that F_{•}Ω ≅ SF_{•}. From now on we identify homology theories C ⟶ Ab and homological functors C ⟶ Ab^{ℤ} which commute with Ω and S.

The following gives a connection between homology theories in the left triangulated category C and homological functors over its stabilization T_{l}(C).

Lemma 1.7. For any homology theory F_{•} : C ⟶ Ab^{ℤ}, there exists a homological functor F_{•}^{∗} : T_{l}(C) ⟶ Ab^{ℤ}, unique up to isomorphism, such that F_{•}^{∗}P ≅ F_{•}. The functor F_{•} ↦ F_{0}^{∗} establishes an equivalence between the category of homology theories on C and the category of homological functors T_{l}(C) ⟶ Ab.

Proof. Defining F_{•}^{∗}(C, n) = S^{−n}F_{•}(C) it is easy to see that F_{•}^{∗} is a homological functor on T_{l}(C). Moreover F_{•}^{∗}P(C) = F_{•}^{∗}(C, 0) = F_{•}(C), hence F_{•}^{∗}P ≅ F_{•}. Trivially F_{•}^{∗} is the unique homological functor on T_{l}(C) which extends F_{•} via the stabilization functor. This shows that the functor F_{•} ↦ F_{•}^{∗} is an equivalence between homology theories on C and homology theories on T_{l}(C). Since the latter is triangulated, the map F_{•}^{∗} ↦ F_{0}^{∗} gives an equivalence between the category of homology theories on T_{l}(C) and the category of homological functors T_{l}(C) ⟶ Ab.

2. Torsion Theories and Generalized Tate-Vogel (Co-)Homology

In this section we study the relationship between (co)hereditary torsion theories defined in the stable category of an abelian category and generalized Tate-Vogel (co)homology functors defined in the abelian category. Our main result shows that the extension functor Ext^{∗} admits a completion, which we call the generalized Tate-Vogel extension functor, with respect to a hereditary or cohereditary torsion theory defined on a suitable stable pretriangulated category, thus generalizing work of Mislin [61]. Moreover we show that the completion can be described as an appropriate Eckmann-Hilton homotopy functor.

Although all the main results of this section are valid for any abelian category C with enough projectives or injectives having a functorially finite subcategory ω, for concreteness and simplicity we work in the category mod(Λ) of finitely generated modules over an Artin algebra Λ, and for the same reasons we choose our functorially finite subcategory ω to be full subcategory P_{Λ} of projectives, or dually the full subcategory ω = I_{Λ} of injectives.

We start with a reasonable pair of full subcategories of mod(Λ) which induce a hereditary torsion theory in the stable category \underline{mod}(Λ) modulo projectives. Throughout this section our working setup is the following.

• [Setup] (X , Y) is a fixed pair (not necessarily good) of subcategories of

mod(Λ) such that: (i) X is resolving. (ii) P_{Λ} ⊆ X ∩ Y. (iii) The pair (X /P_{Λ}, Y/P_{Λ}) is a torsion theory in \underline{mod}(Λ). In particular we don’t assume that Ext_{Λ}^{1}(X , Y) = 0.

Example. The main examples of the above set-up are the following.

(i) X = mod(Λ) and Y = P_{Λ}. Then the pair (mod(Λ), P_{Λ}) is not necessarily good, but (\underline{mod}(Λ), 0) is a hereditary torsion theory in \underline{mod}(Λ).

(ii) (X , Y) is a good pair with X resolving and X ∩ Y = P_{Λ}. For instance we can take X = P_{Λ} and Y = mod(Λ). Then (X /P_{Λ}, Y/P_{Λ}) is a hereditary torsion theory in \underline{mod}(Λ) with X /P_{Λ} triangulated.

Under the above assumptions it is easy to see that X ∩Y = P_{Λ}. As usual we denote by R : \underline{mod}(Λ) ⟶ X /P_{Λ} the right adjoint of the inclusion i : X /P_{Λ} ↪ \underline{mod}(Λ) and by L : \underline{mod}(Λ) ⟶ Y/P_{Λ} the left adjoint of the inclusion j : Y/P_{Λ} ↪ \underline{mod}(Λ). Since X is resolving, we have that X /P_{Λ} is a pretriangulated subcategory of \underline{mod}(Λ) and the functors i and R are left exact, i.e. the torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) is hereditary. We denote by P the stabilization functor of any of the categories X /P_{Λ}, Y/P_{Λ}, \underline{mod}(Λ) with respect to their left triangulation. By Proposition II.5.3, we know that the hereditary torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) induces a hereditary torsion theory (T_{l}(X /P_{Λ}), T_{l}(Y/P_{Λ})) in the stabilization T_{l}(\underline{mod}(Λ)) of \underline{mod}(Λ) with respect to its left triangulation. In other words we have a short exact sequence of triangulated categories

0 ⟶ T_{l}(Y/P_{Λ}) ⟶^{j^{∗}} T_{l}(\underline{mod}(Λ)) ⟶^{R^{∗}} T_{l}(X /P_{Λ}) ⟶ 0

and the functor R^{∗} admits a fully faithful left adjoint i^{∗}, and the functor j^{∗} admits a left adjoint. Now we can use the results from Section 1 to transfer the concept of a projective extension bifunctor with respect to a hereditary torsion theory in the stable pretriangulated category, to the module category.

Definition 2.1. The n−th complete projective extension bifunctor of mod(Λ) with respect to the pair (X , Y):

Êxt_{(X ,Y)}^{n}(−, −) : mod(Λ)^{op} × mod(Λ) ⟶ Ab, n ∈ ℤ

is defined to be the complete extension bifunctor Êxt_{(X /P_{Λ},Y/P_{Λ})}^{n}(−, −) of \underline{mod}(Λ) with respect to the hereditary torsion theory (X /P_{Λ}, Y/P_{Λ}). In other words

Êxt_{(X ,Y)}^{n}(A, B) := T_{l}(X /P_{Λ})[Ω^{n}R^{∗}P(\underline{A}), R^{∗}P(\underline{B})]
⟶^{≅} lim_{⟶}_{k,k+n≥0}\underline{Hom}_{Λ}[Ω^{k+n}R(A), Ω^{k}(B)].

Example-Definition. Let Êxt_{(mod(Λ),P_{Λ})}^{∗}(−, −) be the complete projective extension bifunctor with respect to the largest hereditary torsion theory (\underline{mod}(Λ), 0) in \underline{mod}(Λ). Since in this case R = Id_{\underline{mod}(Λ)}, we have:

Êxt_{(mod(Λ),P_{Λ})}^{∗}(A, B) = Hom_{T_{l}(\underline{mod}(Λ))}[Ω^{n}P(\underline{A}), P(\underline{B})], n ∈ ℤ

In other words the bifunctor Êxt_{(mod(Λ),P_{Λ})}^{∗}(−, −) is what is known in the literature as the projective Tate-Vogel extension bifunctor and from now on we denote it by Êxt_{T V}^{n} (−, −).

Note that the Tate-Vogel bifunctor was introduced in the 90’s by Vogel (see [37]), Benson-Carlson (see [22]) and Mislin (see [61]), using different methods. See also [17] for another approach which is closer to the present construction. More generally using the short exact sequences of Proposition 1.3 and the fact that T_{l}(P_{Λ}^{<∞}/P_{Λ}) = 0, it follows that the projective Tate-Vogel extension bifunctors are the complete projective extension bifunctors with respect to any hereditary torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) such that the torsion-free class Y consists of modules of finite projective dimension.

Example. (i) For any module C the sequence of functors {Êxt_{(X ,Y)}^{n}(C, −)}_{n∈ℤ} is a homological functor, which is Y-complete by construction, that is we have Êxt_{(X ,Y)}^{∗}(C, Y) = 0.

(ii) Any module C defines a homological functor {Ext_{Λ}^{n}(C, −)}_{n∈ℤ}, where of course we set Ext_{Λ}^{n}(C, −) = 0, ∀n < 0.

(iii) Any module C defines a homological functor \{\underline{Π}_{C}^{n}(−)\}_{n∈ℤ}, as follows:

\underline{Π}_{C}^{n}(−) =: \{ \begin{matrix}Ext_{Λ}^{n}(C, −) & n ≥ 1 \\ \underline{Hom}_{Λ}(C, Ω^{−n}(−)) & n ≤ 0 \end{matrix}

In other words \underline{Π}_{C}^{n}(−) is the n−th projective homotopy functor of C introduced by Eckmann-Hilton, see [42]. From now on we call \underline{Π}_{C}^{n}(−) the n-th Eckmann-Hilton projective homotopy functor of C.

Observe that the negative parts of the homological functors in (ii), (iii), are P_{Λ}−complete. However \underline{Π}_{C}^{∗}(−) is P_{Λ}−complete iff Ext_{Λ}^{n}(C, Λ) = 0, ∀n ≥ 1, and Ext_{Λ}^{∗}(C, −) is P_{Λ}−complete iff Ext_{Λ}^{n}(C, Λ) = 0, ∀n ≥ 0.

We are going to show that for any module C, the Y-complete projective extension functor Êxt_{(X ,Y)}^{∗}(C, −) is the Y-completion of the usual extension functor Ext_{Λ}^{∗}(C, −), in the sense of the following definition.

Definition 2.2. The Y−completion of the homological functor {F_{n}}_{n∈ℤ} : mod(Λ) ⟶ Ab is a Y-complete homological functor {F̂_{n}}_{n∈ℤ} : mod(Λ) ⟶ Ab together with a morphism of homological functors ϕ^{∗} : {F_{n}}_{n∈ℤ} ⟶ {F̂_{n}}_{n∈ℤ}, which satisfies the following universal property: any morphism α^{∗} : {F_{n}}_{n∈ℤ} ⟶ {G_{n}}_{n∈ℤ} to a Y-complete homological functor {G_{n}}_{n∈ℤ} factors uniquely through ϕ^{∗}. In this case ϕ^{∗} is called the Y−completion morphism.

We give first some preliminary results on Y-complete homological functors defined on mod(Λ). So we fix for a moment a Y-complete homological functor {F_{n}}_{n∈ℤ}. Since P_{Λ} ⊆ Y, it follows that F_{n}(P) = 0, ∀P ∈ P_{Λ}, ∀n ∈ ℤ. Hence each F_{n} induces a functor F_{n} : \underline{mod}(Λ) ⟶ Ab.

Lemma 2.3. There are natural isomorphisms

(i) F_{−n} ⟶^{≅} F_{0}Ω^{n}, ∀n ≥ 0.

(ii) F_{k} ⟶^{≅} F_{k+1}Ω and F_{k−n} ⟶^{≅} F_{k}Ω^{n}, ∀k ∈ ℤ, ∀n ≥ 0.

(iii) ∀C ∈ mod(Λ), ∀n ≥ 0, ∀k ∈ ℤ:

(Ext_{Λ}^{n}(C, −), F_{k}) ⟶^{≅} F_{k}(Ω^{n}(C)) ⟵^{≅} (\underline{Π}_{C}^{n}, F_{k}).

(iv) ∀C ∈ mod(Λ), ∀n < 0, ∀k ∈ ℤ: (\underline{Π}_{C}^{n}, F_{k}) = F_{k}TrΩ^{−n}Tr(C).

(v) ∀C ∈ mod(Λ), the right X -approximation sequence 0 ⟶ Y_{C} ⟶ X_{C} ⟶^{f_{C}} C ⟶ 0 induces a commutative diagram of isomorphisms, ∀n ≥ 0, ∀k ∈ ℤ:

(Ext_{Λ}^{n}(X_{C}, −), F_{k}) ⟶^{(f_{C}^{n},F_{k})} (Ext_{Λ}^{n}(C, −), F_{k})
≅↓  ≅↓
F_{k}(Ω^{n}(X_{C})) ⟶^{F_{k}(Ω^{n}(f_{C}))} F_{k}(Ω^{n}(C))
≅↓  ≅↓
F_{k−n}(X_{C})  ⟶^{F_{k−n}(f_{C})}  F_{k−n}(C)

where f_{C}^{n} = Ext_{Λ}^{n}(f_{C}, −) and the vertical morphisms are the isomorphisms in (iii) and (ii).

Proof. The assertions in (i), (ii) follow easily form the fact that each F_{k} is half-exact and kills projective modules. If Ω(C) ↣ P (C) ↠ C is a short exact sequence with P (C) projective, then applying (−, F_{k}) to the exact sequence of functors (C, −) ↣ (P (C), −) ⟶ (Ω(C), −) ↠ Ext_{Λ}^{1}(C, −) and using Yoneda’s Lemma we deduce an isomorphism: (Ext_{Λ}^{1}(C, −), F_{k}) ⟶^{≅} F_{k}(Ω(C)). Then part (iii) follows by induction. Part (iv) follows from the isomorphisms \underline{Hom}_{Λ}(C, Ω^{n}(−)), F_{k}) ⟶^{≅} \underline{Hom}_{Λ}(Σ^{n}(C), −), F_{k}) ⟶^{≅} F_{k}(Σ^{n}(C)) and the fact that, according to Section I.5, Σ^{n} = TrΩ^{n}Tr is a left adjoint of Ω^{n}. Finally part (v) follows from the isomorphisms of the previous parts and the fact that each F_{k} kills the objects of Y.

Since the homological functor {Êxt_{(X ,Y)}^{m}(A, −)}_{m∈ℤ} is Y-complete, we have the following direct consequence.

Lemma 2.4. If A, C ∈ mod(Λ), then for any n ≥ 0, m ∈ ℤ:

(Ext_{Λ}^{n}(C, −), Êxt_{(X ,Y)}^{m}(A, −)) ⟶^{≅} lim_{⟶}_{k+m,k+n≥0}\underline{Hom}_{Λ}[Ω^{k+m}R(\underline{C}), Ω^{k+n}(\underline{A})].

For any module C we now define a specific morphism φ_{C,−}^{∗} : Ext_{Λ}^{∗}(C, −) ⟶

Êxt_{(X ,Y)}^{∗}(C, −) as follows. For n < 0 we set φ_{C,−}^{n} = 0. For n ≥ 0, let f_{C} be the (minimal) right X -approximation of C. Then we have an induced morphism Ω^{n}(f_{C}) : Ω^{n}(X_{C}) ⟶ Ω^{n}(C), which in turn serves as a representative of a unique morphism

P(\underline{f}_{C}) : lim_{⟶}_{k+n≥0}\underline{Hom}_{Λ}[Ω^{k+n}R(\underline{C}), Ω^{k+n}(\underline{C})] ⟶^{≅} T_{l}(\underline{mod}(Λ)[(Ω^{n}PR(\underline{C}), Ω^{n}P(\underline{C})]

and then by Lemma 2.4 this morphism represents a unique morphism

φ_{C,−}^{n} : Ext_{Λ}^{n}(C, −) ⟶ Êxt_{(X ,Y)}^{n}(C, −).

Theorem 2.5. The morphism φ_{C,−}^{∗} : Ext_{Λ}^{∗}(C, −) ⟶ Êxt_{(X ,Y)}^{∗}(C, −) is the Ycompletion of the Ext-homological functor Ext_{Λ}^{∗}(C, −).

Proof. Let F_{∗} : mod(Λ) ⟶ Ab be a Y-complete homological functor and let ξ_{∗} : Ext_{Λ}^{∗}(C, −) ⟶ F_{∗} be a morphism of homological functors. Then we have ξ_{n} = 0, for n < 0 and by Lemma 2.4, we have that ξ_{n} is represented by a unique element ζ_{n} ∈ F_{n}(Ω^{n}(C)). We define a morphism ξ̂_{∗} : Êxt_{(X ,Y)}^{∗}(C, −) ⟶ F_{∗} as follows. First

let n ≥ 0, let A be a module and let α ∈ Êxt_{(X ,Y)}^{∗}(C, A). Choose a representative

\underline{α}_{k} : Ω^{n+k}R(\underline{C}) ⟶ Ω^{k}(\underline{A}) in Êxt_{(X ,Y)}^{∗}(C, A), where k ≥ 0. Then F_{n+k}(\underline{α}_{k}) : F_{n+k}Ω^{n+k}R(\underline{C}) ⟶ F_{n+k}Ω^{k}(\underline{A}). By Lemma 2.3 this morphism is isomorphic to the morphism F_{n}(\underline{α}_{k}) : F_{n}Ω^{n}R(\underline{C}) ⟶ F_{n}(\underline{A}). It is clear that the construction is independent of the choice of the representative \underline{α}_{k} of α. By Lemma 2.3 the morphism F_{n}(Ω^{n}(f_{C})) : F_{n}Ω^{n}R(\underline{C}) ⟶ F_{n}(Ω^{n}(\underline{C})) is invertible. Consider the composition

F_{n}(Ω^{n}(\underline{C})) ⟶^{F_{n}(Ω^{n}(f_{C}))^{−1}} F_{n}Ω^{n}R(\underline{C}) ⟶^{F_{n}(\underline{α}_{k})} F_{n}(\underline{A})

and define ξ̂_{n}(α) := F_{n}(\underline{α}_{n})((F_{n}Ω^{n}(f_{C}))^{−1}(ζ_{n})). It is easy to see that the above construction is functorial, hence ξ̂_{n} : Êxt_{(X ,Y)}^{n}(C, −) ⟶ F_{n} is a natural morphism for n ≥ 0. Now we define the morphism ξ̂_{−n} for n > 0. Consider the morphism ξ̂_{0} : Êxt_{(X ,Y)}^{0}(C, −) ⟶ F_{0} and define ξ̂_{−n} = ξ̂_{0}Ω^{n} : Êxt_{(X ,Y)}^{0}(C, Ω^{n}(−)) ⟶ F_{0}Ω^{n} ≅

F_{−n}, where the last isomorphism follows by Lemma 2.4 and Êxt_{(X ,Y)}^{0}(C, Ω^{n}(−)) =

T_{l}(\underline{mod}(Λ))[PR(\underline{C}), PΩ^{n}(−)] = T_{l}(\underline{mod}(Λ))[Ω^{−n}PR(\underline{C}), P(−)] = Êxt_{(X ,Y)}^{−n}(C, −).

We claim that ξ̂_{∗} ∘ φ_{∗}^{C} = ξ_{∗}. For n < 0 this is clear since then φ_{C}^{n} = ξ_{n} = 0. Assume now that n ≥ 0. Since φ_{C}^{n} is represented by Ω^{n}(f_{C}) using the isomorphisms of Lemmas 2.3 and 2.4 it suffices to show that ξ̂_{n}(Ω^{n}(f_{C})) = ξ_{n}. Using the definition of ξ̂_{n} we have ξ̂_{n}(Ω^{n}(f_{C})) = F_{n}(Ω^{n}(f_{C}))((F_{n}Ω^{n}(f_{C}))^{−1}(ζ_{n})) = ζ_{n} which represents the morphism ξ_{n}. Since the construction is independent of the choices for representatives, we have ξ̂_{n}(Ω^{n}(f_{C})) = ξ_{n}. We infer that ξ̂_{∗} ∘φ_{C}^{∗} = ξ_{∗}. The easy proof that ξ̂_{∗} is the unique morphism of homological functors with this property is left to the reader.

Remark 2.6. Observe that for any n ∈ ℤ, Êxt_{(X ,Y)}^{n}(−, −) is a bifunctor, but enjoys the universal property of the above theorem only in the first variable. The following corollary suggests that the functor Êxt_{(X ,Y)}^{n}(−, −) can be considered as a generalized (projective) Tate-Vogel bifunctor.

Applying Theorem 2.5 to the largest hereditary torsion theory (\underline{mod}(Λ), 0) in \underline{mod}(Λ) we get the following result of Mislin [61].

Corollary 2.7. For any module C, the Tate-Vogel projective extension functor Êxt_{T V}^{∗} (C, −) is the P_{Λ}−completion of the homological functor Ext_{Λ}^{∗}(C, −).

If X ∈ X satisfies Ext_{Λ}^{1}(X, Y) = 0, then the Eckmann-Hilton projective homotopy functor \underline{Π}_{X}^{∗} is obviously a Y-complete homological functor. It is natural to ask if in this case the canonical morphism p_{X}^{∗} : Ext_{Λ}^{∗}(X, −) ⟶ \underline{Π}_{X}^{∗}(−) is the Y-completion of Ext_{Λ}^{∗}(X, −), where p_{X}^{n} = 0 for n < 0, p_{X}^{0} is the canonical map Hom_{Λ}(X, −) ⟶ \underline{Hom}_{Λ}(X, −), and p_{X}^{n} is the canonical map Ext_{Λ}^{∗}(X, −) ⟶ \underline{Hom}_{Λ}(Ω^{n}(X), −) for n ≥ 1. This is not true in general. However we have the following.

Corollary 2.8. For any module X ∈ X ∩^{⊥}Y, the following are equivalent:

3. RELATIVE HOMOLOGY AND GENERALIZED TATE-VOGEL (CO)HOMOLOGY 129

(i) X ∈ ^{⊥}P_{Λ}.

(ii) The map p_{X}^{∗} : Ext_{Λ}^{∗}(X, −) ⟶ \underline{Π}_{X}^{∗}(−) is the Y-completion of Ext_{Λ}^{∗}(X, −).

Proof. (i) ⇒ (ii) By Lemma 5.12 of [17] we have that the canonical maps \underline{Hom}_{Λ}(\underline{X}, \underline{B}) ⟶ T_{l}(\underline{mod}(Λ)[P(\underline{X}), P(\underline{B})] and Ext_{Λ}^{n}(X, B) ⟶ \underline{Hom}_{Λ}(Ω^{n}(\underline{X}), \underline{B}) are invertible, ∀n ≥ 1, ∀X ∈ ^{⊥}P_{Λ}, and any module B. It follows from this that Êxt_{(X ,Y)}^{∗}(X, −) ⟶^{≅} \underline{Hom}_{Λ}(Ω^{∗}(\underline{X}), −) = \underline{Π}_{X}^{∗}(−), and then p_{X}^{∗} is isomorphic to φ_{X}^{∗}.

(ii) ⇒ (i) The hypothesis implies that the canonical morphism p_{X}^{∗} : \underline{Π}_{X}^{∗} ⟶ Êxt_{(X ,Y)}^{∗}(X, −) is invertible. In particular the canonical map \underline{Hom}_{Λ}(\underline{X}, −) ⟶ T_{l}(\underline{Hom}(Λ))[P(\underline{X}), P(−)] is invertible. Then by Theorem 4.2 of [17], X ∈ ^{⊥}P_{Λ}.

The following result shows that the Y-completion of the usual homological functor Ext_{Λ}^{∗}(C, −) with respect to the hereditary torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) can be realized as the classical Eckmann-Hilton projective homotopy functor of the right X -approximation of C.

Corollary 2.9. Let (X , Y) be a good pair in mod(Λ) with X resolving and such that X ∩ Y = P_{Λ}. Then the Y-completion of the functor Ext_{Λ}^{∗}(C, −) is the Eckmann-Hilton projective homotopy functor \underline{Π}_{X_{C}}^{∗} (−) of the (minimal) right X - approximation X_{C} of C.

Proof. By Theorem VI.2.1 we have X ⊆ ^{⊥}P_{Λ}. Then for any module C, using the above Corollary and the fact that R(\underline{C}) = \underline{X}_{C}, we have:

Êxt_{(X ,Y)}^{∗}(C, −) ⟶^{≅} T_{l}(\underline{mod}(Λ))[Ω^{∗}PR(\underline{C}), P(−)] ⟶^{≅} \underline{Π}_{X_{C}}^{∗} (−).

Remark 2.10. All the results of this section can be dualized working with pairs (Y, Z) with Z coresolving and Y ∩ Z = I_{Λ}, and assuming that (Y/I_{Λ}, Z/I_{Λ}) is a cohereditary torsion theory in \overline{mod}(Λ). In this case for any module C the usual cohomological functor Ext_{Λ}^{∗}(−, C) admits the Y-completion Ẽxt_{(Y,Z)}^{∗}(−, C). We leave to the reader to state the dual results.

3. Relative Homology and Generalized Tate-Vogel (Co)Homology

In this section we concentrate on complete projective or injective extension bifunctors with respect to torsion theories in \underline{mod}(Λ) or in \overline{mod}(Λ), such that the torsion or the torsion-free class is triangulated, and we study the connections between generalized Tate-Vogel cohomology and relative homology. In this case there are defined two relative homological theories in mod(Λ). The first one is induced by the torsion or torsion-free subcategory and the other one is the Generalized Tate-Vogel (Co)Homology. We compare these with the absolute homological theory represented by Ext_{Λ}^{∗} by providing comparison morphisms which fit in an infinite long exact sequence, thus generalizing recent results of Avramov-Martsinkovsky [13].

Let throughout this section (X , Y) be a fixed good pair in mod(Λ) with ω functorially finite and X resolving, and assume that ω := X ∩ Y = P_{Λ}. Then we know that the torsion class X /P_{Λ} is triangulated, so it coincides with its left stabilization. Then for any two modules A and B, the Y-complete extension bifunctor with respect to the torsion theory (X /P_{Λ}, Y/P_{Λ}) in \underline{mod}(Λ) is given by

Êxt_{(X ,Y)}^{n}(A, B) = \underline{Hom}_{Λ}[Ω^{n}R(\underline{A}), \underline{B}] ≅ \underline{Hom}_{X /P_{Λ}}[Ω^{n}R(\underline{A}), R(\underline{B})] ⟶^{≅} \underline{Π}_{X_{A}}^{n}(B) where X_{A} is the (minimal) right X -approximation of A. Note that by Theorem VI.2.1, we have Êxt_{(X ,Y)}^{∗}(A, B) = 0 if A or B has finite projective dimension.

Consider now the completion morphism

φ_{−,−}^{n} : Ext_{Λ}^{n}(−, −) ⟶ Êxt_{(X ,Y)}^{n}(−, −), ∀n ∈ ℤ

with respect to Y-complete homological functors out of Ext_{Λ}^{n}(−, −). The following collects some basic properties of the bifunctor Êxt_{(X ,Y)}^{∗}(−, −) and the completion morphism φ_{−,−}^{∗}. Part (iii) below was observed by A. Martsinkovsky in a different setting.

Proposition 3.1. (i) The morphism φ_{X,−}^{n} is invertible ∀n ≥ 1 iff X ∈ X. (ii) The kernel of ϕ_{−,−}^{0} is the ideal of mod(Λ) consisting of all morphisms fac-

torizing through an object of Y. (iii) The kernel of ϕ_{−,−}^{1} is the subfunctor F_{X} ⊆ Ext_{Λ}^{1}(−, −) consisting of all ex-

tensions 0 ⟶ A ⟶ B ⟶ C ⟶ 0 such that the sequence 0 ⟶ Hom_{Λ}(X , A) ⟶ Hom_{Λ}(X , B) ⟶ Hom_{Λ}(X , C) ⟶ 0 is exact.

Proof. (i) Since X /P_{Λ} is triangulated it is easy to see that the canonical morphism Ext_{Λ}^{n}(X, B) ⟶ \underline{Hom}_{Λ}[Ω^{n}(\underline{X}), \underline{B}] is invertible, for any X ∈ X, for any B ∈ mod(Λ) and for any n ≥ 1. The assertion follows easily from this.

(ii) Let α : A ⟶ B be a morphism in the kernel of ϕ_{A,B}^{0}, i.e. R(\underline{α}) = 0. Then f\underline{A} ∘ \underline{α} = 0, hence \underline{α} factors through L(\underline{A}), i.e. there exists a morphism \underline{ρ} : L(\underline{A}) ⟶ \underline{B} such that g\underline{A} ∘ \underline{ρ} = \underline{α}. This means that g^{A} ∘ ρ − α factors through the left projective approximation P^{A} of A, i.e. there is a morphism ν : P^{A} ⟶ B such that g^{A} ∘ ρ − α = p^{A} ∘ ν. Now the push-out of the extension 0 ⟶ A ⟶ Y^{A} ⟶ X^{A} ⟶ 0 along p^{A} splits since X^{A} is in X and P^{A} is projective. Hence there is a morphism μ : Y^{A} ⟶ P^{A} such that p^{A} = g^{A} ∘ μ. Then g^{A} ∘ ρ − α = g^{A} ∘ μ ∘ ν and this implies that α factors through g^{A}. Conversely if α factors through an object in Y, then obviously ϕ_{A,B}^{0}(\underline{α}) = R(\underline{α}) = 0.

(iii) Let (E) : 0 ⟶ A ⟶ B ⟶^{ϵ} C ⟶ 0 be a short exact sequence in mod(Λ). We assume that (E) as an element of Ext_{Λ}^{1}(C, A), it is represented by a morphism ω : Ω(C) ⟶ A, that is there exists a push-out diagram:

0 ⟶ Ω(C) ⟶ P (C) ⟶ C ⟶ 0
ω↓  ↓  ^{||}↓
0 ⟶ A ⟶ B ⟶^{ϵ} C ⟶ 0

Then (E) lies in the kernel of φ_{C,A}^{1} iff the morphism R(\underline{ω}) : RΩ(\underline{C}) ⟶ R(\underline{A}) is zero. For any X in X we have an exact commutative diagram

Hom_{Λ}(X, C) ⟶ Ext^{1}_{Λ}(X, Ω(C))
||↓  _{ω^{∗}}↓
Hom_{Λ}(X, C) ⟶ Ext_{Λ}^{1}(X, A)

Since X ∈ X, we have isomorphisms Ext_{Λ}^{1}(X, Ω(C)) = \underline{Hom}_{Λ}(Ω(\underline{X}), Ω(\underline{C})) and Ext_{Λ}^{1}(X, A) = \underline{Hom}_{Λ}(Ω(\underline{X}), \underline{A}), and then ω^{∗} is the morphism which sends \underline{α} : Ω(\underline{X}) ⟶ Ω(\underline{C}) to the composition \underline{α} ∘ \underline{ω} : Ω(\underline{X}) ⟶ \underline{A}. However R(\underline{ω}) = 0 is equivalent to ω_{X}^{∗} = 0, for any X ∈ X. Hence (E) lies in the kernel of φ_{C,A}^{1} iff the connecting morphism Hom_{Λ}(X, C) ⟶ Ext_{Λ}^{1}(X, A) is zero for any X ∈ X. The last condition is equivalent to the fact that the sequence 0 ⟶ Hom_{Λ}(X, A) ⟶ Hom_{Λ}(X, B) ⟶ Hom_{Λ}(X, C) ⟶ 0 is exact for any X ∈ X, that is (E) ∈ F_{X}.

Since X is contravariantly finite, the additive subfunctor F_{X} has enough projectives. We denote by Ext_{X}^{∗} (C, A) the relative derived functors, see for instance [11]. They can be computed using exact resolutions of A by objects from X which have the property that they remain exact after the application of (X, −) for any X ∈ X. Contravariant finiteness of X ensures that such resolutions exist in C. The following result, which generalizes a recent result of Avramov-Martsinkovsky [13], gives a nice connection between the three involved homological theories represented by the bifunctors Ext_{X}^{∗} (−, −), Ext_{Λ}^{∗}(−, −) and Êxt_{(X ,Y)}^{∗}(−, −), by providing comparison morphisms.

Theorem 3.2. There exists a long exact sequence:

0 ⟶ Ext_{X}^{1} (−, −) ⟶ Ext_{Λ}^{1}(−, −) ⟶^{ϕ_{−,−}^{1}} Êxt_{(X ,Y)}^{1}(−, −) ⟶ ⋯

⋯ ⟶ Ext_{X}^{n} (−, −) ⟶ Ext_{Λ}^{n}(−, −) ⟶^{ϕ_{−,−}^{n}} Êxt_{(X ,Y)}^{n}(−, −) ⟶ ⋯

Proof. Let C be a module and let 0 ⟶ Y_{C} ⟶^{g_{C}} X_{C} ⟶^{f_{C}} C ⟶ 0 be the right X -approximation sequence of C with Y_{C} in Y. If C is in Y, then X_{C} is projective. This implies that for any Y ∈ Y we have Ω_{X} (Y) ≅ Ω(Y), where Ω_{X} (Y) is the first relative X -syzygy, and an isomorphism Ext_{X}^{1} (Y, −) ⟶^{≅} Ext_{Λ}^{1}(Y, −). Since Y is closed under syzygies, this implies that the canonical map Ext_{X}^{n} (Y, −) ⟶ Ext_{Λ}^{n}(Y, −) is invertible, for any Y in Y and n ≥ 0. Now the X -approximation sequence of C induces the usual long exact sequence

⋯ ⟶ Ext_{Λ}^{n}(Y_{C}, −) ⟶ Ext_{Λ}^{n+1}(C, −) ⟶ Ext_{Λ}^{n+1}(X_{C}, −) ⟶ ⋯

By the above observation we have an isomorphism Ext_{Λ}^{n}(Y_{C}, −) ⟶ Ext_{X}^{n} (Y_{C}, −). Since Y_{C} is the first relative X -syzygy of C, the last extension space is isomorphic to Ext_{X}^{n+1}(C, −). On the other hand by Corollary 2.9 the extension space

Ext_{Λ}^{n+1}(X_{C}, −) identifies with Êxt_{X}^{n+1}(C, −), via the completion morphism ϕ_{X_{C},−}^{n+1}. It is easy to see that the above isomorphisms make the next diagram commutative:

⋯ ⟶ Ext_{Λ}^{n}(Y_{C}, −) ⟶ Ext_{Λ}^{n+1}(C, −) ⟶ Ext_{Λ}^{n+1}(X_{C}, −) ⟶ ⋯
≅↓  ^{∥}↓  ≅↓
⋯ ⟶ Ext_{X}^{n+1}(C, −) ⟶^{θ_{C,−}^{n+1}} Ext_{Λ}^{n+1}(C, −) ⟶^{ϕ_{C,−}^{n+1}} Êxt_{(X ,Y)}^{n+1}(C, −) ⟶ ⋯

where ϕ_{C,−}^{n+1} is the Y-completion morphism and θ_{C,−}^{n+1} is the canonical map induced by the inclusion Ext_{X}^{1} (−, −) ↪ Ext_{Λ}^{1}(−, −). Since we have an identification Ext_{X}^{1} (C, −) = Coker((X_{C}, −) ⟶ (Y_{C}, −)), and by Proposition 3.1 we have an ex-

act sequence 0 ⟶ Ext_{X}^{1} (−, −) ⟶ Ext_{Λ}^{1}(−, −) ⟶ Êxt_{Λ}^{1}(−, −), the result follows.

As a consequence of the above result we can compare dimensions. First observe that for any module C we have X −pdC ≤ pdC, with equality if the latter is finite. Also from the proof of the above Theorem we have sup{pdY | Y ∈ Y} = sup{X −pdY | Y ∈ Y} ≤ X −gl.dim Λ. Since by Theorem VI.2.1, Y ⊆ P_{Λ}^{<∞}, we have the following inequalities:

fin.p.dim Λ ≤ sup{pdY | Y ∈ Y} ≤ X −gl.dimΛ ≤ gl.dimΛ

where fin.p.dimΛ is the finitistic projective dimension of Λ. The last inequality follows from the fact that finiteness of gl.dimΛ implies by Remark VI.2.2 that X = P_{Λ}. Of course in this case the last inequality becomes an equality. Obviously the finiteness of any of the last three dimensions forces Λ to be Gorenstein. Indeed if the second is finite, then Y ⊆ P_{Λ}^{<∞}. This implies that X = CM(Λ) and consequently Λ is Gorenstein since, for instance by Corollary 1.4, we have { \atop CM(Λ)} = mod(Λ). In general using that Y contains the injectives and any injective resolution of Λ remains exact after the application of Hom_{Λ}(X, −) for any X ∈ X, it is not difficult to see that the following holds:

X −gl.dim Λ < ∞ ⇔ idΛ < ∞ and Y ⊆ P_{Λ}^{<∞}

in which case we have X −gl.dimΛ = idΛ = fin.p.dimΛ and Λ is Gorenstein.

Except for the three homological theories represented by the homological bifunctors Ext_{X}^{∗} (−, −), Ext_{Λ}^{∗}(−, −) and Êxt_{(X ,Y)}^{∗}(−, −), there is a fourth one, namely

the Tate-Vogel extension bifunctor Êxt_{T V}^{∗} (−, −). We close this section by comparing the bifunctors Êxt_{T V}^{∗} (−, −) and Êxt_{(X ,Y)}^{∗}(−, −).

Let C be a module and consider the completion morphism ζ_{C,−}^{∗} : Ext_{Λ}^{∗}(C, −)

⟶ Êxt_{T V}^{∗} (C, −) of the extension functor Ext_{Λ}^{∗}(C, −) with respect to the subcategory P_{Λ}, and the completion morphism ϕ_{C,−}^{∗} : Ext_{Λ}^{∗}(C, −) ⟶ Êxt_{(X ,Y)}^{∗}(C, −) of the extension functor Ext_{Λ}^{∗}(C, −) with respect to the subcategory Y. Since P_{Λ} ⊆ Y, by the universal property of Êxt_{T V}^{∗} (C, −), there exists a unique morphism ρ_{C,−}^{∗} of P_{Λ}−homological functors making the following diagram commutative:

Ext_{Λ}^{∗}(C, −) ⟶^{ζ_{C,−}^{∗}} Êxt_{T V}^{∗} (C, −)
∥↓  ^{∃! ρ_{C,−}^{∗}}↓
Ext_{Λ}^{∗}(C, −) ⟶^{ϕ_{C,−}^{∗}} Êxt_{(X ,Y)}^{∗}(C, −)

It is not difficult to see that ρ_{C,−}^{∗} coincides with the natural morphism constructed in Corollary 1.4, in particular ρ_{C,−}^{∗} extends to a morphism of bifunctors

ρ_{−,−}^{∗} : Êxt_{T V}^{∗} (−, −) ⟶ Êxt_{(X ,Y)}^{∗}(−, −)

which we call the comparison morphism. The following result shows that the comparison morphism measures how far Λ is from being Gorenstein.

Proposition 3.3. The natural morphism ρ_{C,−}^{∗} is invertible iff C ∈ X̂. In particular the comparison morphism ρ_{−,−}^{∗} is invertible iff Λ is Gorenstein.

Proof. Let ρ_{C,−}^{∗} be invertible. Then by Corollary 1.4, Y_{C} lies in P_{Λ}^{<∞}, and this implies that C lies in X̂. Conversely if C ∈ X̂, then by Proposition VI.2.3, Y_{C} has finite projective dimension, and then ρ_{C,−}^{∗} is invertible by Corollary 1.4.

Using Remark V.4.7 it follows that the results obtained so far in this section hold in the following more general setting. Consider the full subcategory CM(Λ) of Cohen-Macaulay modules and the full subcategory P_{Λ}^{<∞} of modules of finite projective dimension. Then (CM(Λ), P_{Λ}^{<∞}) is a good pair in the subcategory { \atop CM(Λ)} of modules with finite Gorenstein projective dimension, and moreover CM(Λ) ∩ P_{Λ}^{<∞} = P_{Λ}. In addition the stable category { \atop CM(Λ)}/P_{Λ} is pretriangulated, and the pair (CM(Λ)/P_{Λ}, P_{Λ}^{<∞}/P_{Λ}) is a torsion theory in { \atop CM(Λ)}/P_{Λ} with CM(Λ)/P_{Λ} triangulated.

Then all our previous results are valid if we consider only modules in the full exact subcategory { \atop CM(Λ)} of mod(Λ). In particular the exact sequence of Theorem 3.2 generalizes Theorem 7.1 of [13], where the sequence of Theorem 3.2 was established for any module of finite Gorenstein dimension in the first variable, using (complete) resolutions.

Remark 3.4. We leave to the reader to state and prove the dual results concerning complete injective extension bifunctors arising from injective good pairs, that is good pairs (Y, Z) with Y ∩ Z = I_{Λ}, and such that the torsion-free class Z/I_{Λ} is triangulated.

4. Good Triples

In this section we study the complete extension bifunctors arising from a good triple. We show that the two (co)homology theories which arise from the two good pairs involved in a good triple are isomorphic, and we show that the same is true for the relative homological theories defined by the torsion and the torsion-free subcategory.

We continue to work in the category mod(Λ) of finitely generated right Λmodules over an Artin algebra Λ, and we fix a good triple (X , Y, Z) in mod(Λ) with Y (co)resolving. Then X ∩ Y = P_{Λ} and Y ∩ Z = I_{Λ}, the torsion class X /P_{Λ} in \underline{mod}(Λ) is triangulated, and the torsion-free class Z/I_{Λ} in \overline{mod}(Λ) is triangulated. Hence the results of the previous section can be applied to both good pairs (X , Y) and (Y, Z). We shall show that the complete extension bifunctors Êxt_{(X ,Y)}^{n}(−, −), Ẽxt_{(Y,Z)}^{n}(−, −) : mod(Λ)^{op} × mod(Λ) ⟶ Ab are isomorphic, ∀n ∈ ℤ. We recall that by Theorem VI.3.2 there exists a triangle equivalence Φ : X /P_{Λ} ⟶^{≈} Z/I_{Λ} with quasi-inverse Ψ.

First we need the following result which shows that the two reasonable possible ways to go from mod(Λ) to X /P_{Λ} and to Z/I_{Λ} are isomorphic.

Lemma 4.1. There exist commutative diagrams of functors

mod(Λ) ⟶^{π} \overline{mod}(Λ)
π↓  _{Σ^{−1}S}↓
\underline{mod}(Λ) ⟶^{ΦR} Z/I_{Λ}

mod(Λ) ⟶^{π} \underline{mod}(Λ)
π↓  _{Ω^{−1}R}↓
\overline{mod}(Λ) ⟶^{ΨS} X /P_{Λ}

that is there exist natural isomorphisms of functors

φ : Σ^{−1}Sπ ⟶^{≅} ΦRπ and ψ : ΨSπ ⟶^{≅} Ω^{−1}Rπ.

Proof. Let A be a module and consider the minimal right X -approximation sequence 0 ⟶ Y_{A} ⟶ X_{A} ⟶^{f_{A}} A ⟶ 0 of A and the minimal right Y-approximation sequence 0 ⟶ Z_{X_{A}} ⟶ Y_{X_{A}} ⟶^{g_{X_{A}}} X_{A} ⟶ 0 of X_{A}. Then by definition the object \underline{Z}_{X_{A}} in \overline{mod}(Λ) is the object ΦRπ(A). The above exact sequences are included in the following exact commutative diagram

0 ⟶ U ⟶ Y_{X_{A}} ⟶ A ⟶ 0
↓  ^{g_{X_{A}}} ↓  ^{||}↓
0 ⟶ Y_{A} ⟶ X_{A} ⟶^{f_{A}} A ⟶ 0

which induces an exact sequence 0 ⟶ Z_{X_{A}} ⟶ U ⟶ Y_{A} ⟶ 0. Since Z_{X_{A}} is in Z and Y_{A} is in Y, the sequence splits, hence U ≅ Z_{X_{A}} ⊕ Y_{A}. Since the left square in the above diagram is bicartesian, it induces an exact sequence 0 ⟶ Z_{X_{A}} ⊕ Y_{A} ⟶ Y_{A} ⊕ Y_{X_{A}} ⟶ X_{A} ⟶ 0. Then we have a right triangle \underline{Z}_{X_{A}} ⊕ \underline{Y}_{A} ⟶ \underline{Y}_{A} ⊕ \underline{Y}_{X_{A}} ⟶ \underline{X}_{A} ⟶ Σ(\underline{Z}_{X_{A}} ⊕ \underline{Y}_{A}) in \overline{mod}(Λ). Applying to this triangle the right exact reflection functor S : \overline{mod}(Λ) ⟶ Z/I_{Λ} and using that S(Y/I_{Λ}) = 0, we have an isomorphism S(\underline{X}_{A}) ⟶^{≅} Σ(\underline{Z}_{X_{A}}) = ΣΦRπ(A). Applying S to the right triangle \underline{Y}_{A} ⟶ \underline{X}_{A} ⟶ \underline{A} ⟶ Σ(\underline{Y}_{A}) in \overline{mod}(Λ), we have an isomorphism S(\underline{X}_{A}) ⟶^{≅} S(\underline{A}). Combining the above isomorphisms we infer that we have an isomorphism Sπ(A) = S(\underline{A}) ⟶^{≅} ΣΦRπ(A) or equivalently an isomorphism φ : Σ^{−1}Sπ(A) ⟶^{≅} ΦRπ(A). We leave to the reader the easy proof that φ is functorial as well as the proof of the existence of the isomorphism ψ.

Theorem 4.2. The complete extension bifunctors

Êxt_{(X ,Y)}^{n}(−, −) and Ẽxt_{(Y,Z)}^{n}(−, −) : mod(Λ)^{op} × mod(Λ) ⟶ Ab

are isomorphic, for any n ∈ ℤ.

Proof. Using the isomorphisms of Lemma 4.1 and the exactness of Φ, we have for any n ∈ ℤ:

Êxt_{(X ,Y)}^{n}(A, B) = \underline{Hom}_{X /P_{Λ}}[Ω^{n}R(\underline{A}), R(\underline{B})]
≅ \overline{Hom}_{Z/I_{Λ}}[ΦΩ^{n}R(\underline{A}), ΦR(\underline{B})]
≅ \overline{Hom}_{Z/I_{Λ}}[Σ^{−n}ΦR(\underline{A}), Σ^{−1}S(\underline{B})]
≅ \overline{Hom}_{Z/I_{Λ}}[Σ^{−1}S(\underline{A}), Σ^{−1}Σ^{n}S(\underline{B})]
≅ \overline{Hom}_{Z/I_{Λ}}[S(\underline{A}), Σ^{n}S(\underline{B})]
= Ẽxt_{(Y,Z)}^{n}(A, B).

Corollary 4.3. Let (X , Y, Z) be a good triple in mod(Λ) and assume that Y is (co)resolving. Then the Y-completion of the homological functor Ext_{Λ}^{∗}(A, −) evaluated at B is isomorphic to the Y-completion of the cohomological functor Ext_{Λ}^{∗}(−, B) evaluated at A.

The following consequence gives a simple proof of a result of [17] which was obtained independently by Nucinkis [65], using satellites.

Corollary 4.4. The following statements are equivalent:

(i) Λ is Gorenstein.

(ii) The projective Tate-Vogel homology bifunctor is isomorphic to the injective Tate-Vogel homology bifunctor:

Êxt_{T V}^{∗} (−, −) ⟶^{≅} Ẽxt_{T V}^{∗} (−, −).

Proof. (i) ⇒ (ii) If Λ is Gorenstein, then we have a good triple (X , Y, Z) with Y = P_{Λ}^{<∞} = I_{Λ}^{<∞}. Hence T_{l}(Y/P_{Λ}) = 0 = T_{r}(Y/I_{Λ}) and then T_{l}(\underline{mod}(Λ)) = X /P_{Λ} and T_{r}(\overline{mod}(Λ)) = Z/I_{Λ}. It follows that we have isomorphisms

Êxt_{T V}^{∗} (−, −) ⟶^{≅} Êxt_{(X ,Y)}^{∗}(−, −) ⟶^{≅} Ẽxt_{(Y,Z)}^{∗}(−, −) ⟵^{≅} Ẽxt_{T V}^{∗} (−, −).

(ii) ⇒ (i) Let I be a module of finite injective dimension. Then we have 0 =

Ẽxt_{(Y,Z)}^{∗}(I, I) ⟶^{≅} Êxt_{(X ,Y)}^{∗}(I, I), hence Êxt_{(X ,Y)}^{0}(I, I) = T_{l}(\underline{mod}(Λ))[P(I), P(I)] = 0. However this holds iff I has finite projective dimension. Dually if P has finite projective dimension, the P has finite injective dimension. We conclude that P_{Λ}^{<∞} = I_{Λ}^{<∞}, so Λ is Gorenstein.

The fact that the complete extension bifunctors are isomorphic is reflected in the following result which shows that the additive subfunctors F_{X} and F^{Z} of Ext_{Λ}^{1}(−, −) coincide.

Corollary 4.5. The subfunctors F_{X} and F^{Z} coincide. In particular there exists an isomorphism of bifunctors

Ext_{X}^{∗} (−, −) ⟶^{≅} Ext_{Z}^{∗}(−, −).

Proof. By [10] we have F_{X} = F^{DTr(X)}. Since by Remark VI.3.4, DTr : X /P_{Λ} ⟶ Z/I_{Λ} is a triangle equivalence, we have F_{X} = F^{Z}, and this implies the existence of the isomorphism between the X -, Z-relative extension functors.

Remark 4.6. As an illustration of the fact that all the main results of this chapter are valid in any abelian category with enough projectives and/or injectives, we show that F_{X} = F^{Z} without using the (specific nature of) functor DTr.

Let (E) : 0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0 be an extension in mod(Λ). Then for any Z ∈ Z we have the following commutative diagram of exact sequences:

0 ⟶ Hom_{Λ}(C, Z) ⟶^{(β,Z)} Hom_{Λ}(B, Z) ⟶^{(α,Z)} Hom_{Λ}(A, Z) ⟶^{ϑ} Ext_{Λ}^{1}(C, Z)
π↓  ^{π}↓  ^{π}↓  ^{ω}↓
⟶ \overline{Hom}_{Λ}(C, Z) ⟶^{(\overline{α},Z)} \overline{Hom}(B, Z) ⟶^{(\overline{β},Z)} \overline{Hom}_{Λ}(A, Z) ⟶^{φ} \overline{Hom}(C, Σ(Z))

where the vertical arrows are the canonical epimorphisms and the composition ϑ∘ω acts as follows. Consider the exact commutative diagram (†):

0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0
ρ↓  ^{σ}↓  ^{τ}↓
0 ⟶ Z ⟶^{i_{Z}} I(Z) ⟶^{ϵ_{Z}} Σ(Z) ⟶ 0

where i_{Z} is the injective envelope of Z. Then ω(ϑ(ρ)) = \overline{τ}. Let (E) be in F_{X}.

We first show that φ = 0. Let ρ : A ⟶ Z with Z ∈ Z. and consider the diagram (†) above. Then consider the right X -approximation sequence 0 ⟶ Y_{Σ(Z)} ⟶

X_{Σ(Z)} ⟶^{f_{Σ(Z)}} Σ(Z) ⟶ 0 of Σ(Z). Since X_{C} lies in X, the morphism f_{C} ∘ τ factors through f_{Σ(Z)}. Hence there exists a morphism ξ : X_{C} ⟶ X_{Σ(Z)} such that ξ ∘ f_{Σ(Z)} = f_{C} ∘ τ. Since the extension (E) is in F_{X}, the morphism f_{C} factors through β, say as μ ∘ β = f_{C}. Then ξ ∘ f_{Σ(Z)} = μ ∘ β ∘ τ = μ ∘ σ ∘ ϵ. Hence ξ ∘ f_{Σ(Z)} factors through an injective. Since Y contains the injectives, it follows that \underline{ξ} ∘ \underline{f}_{Σ(Z)} factors through an object of Y/P_{Λ}. Since \underline{X}_{C} ∈ X /P_{Λ} we have that \underline{ξ} ∘ \underline{f}_{Σ(Z)} = \underline{f}_{C} ∘ \underline{τ} = 0. Then from the right triangle \underline{X}_{C} ⟶^{\underline{f}_{C}} \underline{C} ⟶^{\underline{g}^{C}} \underline{Y}^{C} ⟶ Σ(\underline{X}_{C})

in \underline{mod}(Λ) we have that \underline{τ} factors through \underline{g}^{C} : \underline{C} ⟶ \underline{Y}^{C} say as \underline{τ} = \underline{g}^{C} ∘ \underline{ζ}, where ζ : Y^{C} ⟶ Σ(Z). This implies that τ − g^{C} ∘ ζ factors through a projective P, say as τ − g^{C} ∘ ζ = κ ∘ λ, where λ : P ⟶ Σ(Z). Since P ∈ Y, we have that λ factors through the right Y-approximation of Σ(Z). Since Z lies in Z, the sequence 0 ⟶ Z ⟶ I(Z) ⟶ Σ(Z) ⟶ 0 is a right Y-approximation of Σ(Z). Hence λ factors through ϵ_{Z}. It follows that \overline{τ} = \overline{g}^{C} ∘ \overline{ζ} in \overline{mod}(Λ). But ζ : Y^{C} ⟶ Σ(Z) where Y^{C} ∈ Y and Σ(Z) ∈ Z. Since \overline{Hom}_{Λ}(Y/I_{Λ}, Z/I_{Λ})= 0, it follows that \overline{τ} = 0. Then from the first diagram above we have ω(ϑ(ρ)) = 0. Since this happens for any ρ ∈ Hom_{Λ}(A, Z), we have that ϑ ∘ ω = 0. Then π ∘ φ = 0. Since π is an epimorphism, we have that φ = 0.

Now since φ = 0, it follows that for any morphism ρ : A ⟶ Z, with Z in Z, there exists a morphism h : B ⟶ Z such that \overline{α} ∘ \overline{h} = \overline{ρ}. Hence α ∘ h − ρ factors through an injective envelope i_{A} : A ⟶ I(A), say as α ∘ h − ρ = i_{A} ∘ γ. But since I(A) is injective, i_{A} factors through α, say as α ∘ δ = i_{A}. Then α ∘ h − ρ = α ∘ δ ∘ γ. Hence ρ = α ∘ (h − δ ∘ γ), i.e. ρ factors through α. This implies that (α, Z) is an epimorphism, i.e. the extension (E) belongs to F^{Z}. Hence F_{X} ⊆ F^{Z}.

Dual arguments show that F^{Z} ⊆ F_{X}. Hence F_{X} = F^{Z}.

As usual any one of the subfunctors F_{X} and F^{Z} defines a relative homological theory in mod(Λ), see [10]. The corresponding global dimensions are denoted by X -gl.dimΛ and Z-gl.dimΛ. We have the following direct consequence, whose proof we leave to the reader.

Corollary 4.7. X -gl.dimΛ = Z-gl.dimΛ, and this common value is finite iff Λ is Gorenstein.

Remark 4.8. By our previous results, for a good pair (X , Y) with X ∩ Y = P_{Λ}, we have the following: T_{l}(Y/P_{Λ}) = 0 ⇔ R^{∗} : T_{l}(\underline{mod}(Λ)) ⟶ X /P_{Λ} is a triangle equivalence ⇔ Y consists of modules of finite projective dimension ⇔ Λ is Gorenstein. Dually if we have a good pair (Y, Z) with Y ∩ Z = I_{Λ}, then the following are equivalent: T_{r}(Y/I_{Λ}) = 0 ⇔ L^{∗} : T_{r}(\overline{mod}(Λ)) ⟶ Z/I_{Λ} is a triangle equivalence ⇔ Y consists of modules of finite injective dimension ⇔ Λ is Gorenstein. Hence the existence of such good pairs (X , Y) and/or (Y, Z) and more generally of good triples (X , Y, Z), provides a generalization of the Gorenstein case.

Any Artin algebra Λ admits the good pair (P_{Λ}, mod(Λ)) with P_{Λ} ∩ mod(Λ) = P_{Λ} and the good pair (mod(Λ), I_{Λ}) with mod(Λ) ∩ I_{Λ} = I_{Λ}. We call these pairs trivial. A good triple (X , Y, Z) is trivial iff X = P_{Λ} or equivalently Z = I_{Λ}.

In view of the results of this and the previous sections it is natural to pose the following problems:

Problem. (i) Characterize the Artin algebras Λ such that mod(Λ) admits

a non trivial good pair (X , Y) with X ∩ Y = P_{Λ}. (ii) Characterize the Artin algebras Λ such that mod(Λ) admits a non-trivial

good triple (X , Y, Z).

Certainly the class of Gorenstein algebras belongs to all the three classes above, but we don’t know if there are additional algebras. In the next section we shall show that any Artin algebra belongs to any of the above classes, provided we work with the category of all modules.

5. Big Cohen-Macaulay Modules

In this last section of the present chapter we concentrate on Cohen-Macaulay modules over an Artin algebra Λ. Although these modules are not necessarily finitely generated, usually they behave better than the finitely generated ones. Using recent results of P. Jørgensen and applying our previous results of this chapter, we construct (co)hereditary torsion theories in the stable module category of Λ modulo projectives or injectives and universal (co)homology theories in the module category. We also construct natural closed model structures in the module category. For these torsion theories, resp. closed model structures, the big Cohen Macaulay modules appear as torsion modules, resp. cofibrant objects, and the modules of virtually finite projective dimension as torsion-free modules, resp. fibrant objects.

Throughout this section we fix an Artin algebra Λ, and let Mod(Λ) be the category of all right Λ-modules. Since Λ is left coherent and right perfect, by Section II.1.5 the category P_{Λ} of projective modules is functorially finite. Hence the stable category \underline{Mod}(Λ) modulo projectives is pretriangulated. Dually since Λ is right Noetherian, the category I_{Λ} of injective modules is functorially finite. Hence the stable category \overline{Mod}(Λ) modulo injectives is pretriangulated.

We denote by N^{+} : Mod(Λ) ⟶ Mod(Λ) the Nakayama functor defined by N^{+}(A) = A ⊗_{Λ} D(Λ) and by N^{−} : Mod(Λ) ⟶ Mod(Λ) its right adjoint defined by N^{−}(A) = Hom_{Λ}(D(Λ), A). It is not difficult to see that the adjoint pair (N^{+}, N^{−}) induces an equivalence N^{+} : P_{Λ} ⟶^{≈} I_{Λ} with quasi-inverse N^{−} : I_{Λ} ⟶^{≈} P_{Λ}. The Nakayama functors N^{+}, N^{−} have been successfully used by H. Krause in his work on the invariance of the representation type under stable equivalence. For us these functors are useful since they induce an equivalence between the full subcategory of Cohen-Macaulay modules and the full subcategory of Co-Cohen Macaulay modules. We recall the definitions of these subcategories from Chapter V.

Let CM(P_{Λ}), resp. CoCM(I_{Λ}), be the category of Cohen-Macaulay, resp. Co-Cohen-Macaulay, objects as defined in Section V.4:

CM(P_{Λ}) = {C ∈ Mod(Λ) | ∃ exact sequence 0 ⟶ C ⟶ T^{0} ⟶^{f^{0}} T^{1} ⟶^{f^{1}} T^{2} ⟶ ⋯,

T^{s} ∈ P_{Λ}, ∀s ≥ 0 : Ext_{Λ}^{t}(Ker(f^{n}), P_{Λ}) = 0, ∀n ≥ 0, ∀t ≥ 1 },

CoCM(I_{Λ}) = { C ∈ Mod(Λ) | ∃ exact sequence ⋯ ⟶ T_{2} ⟶^{f_{1}} T_{1} ⟶^{f_{0}} T_{0} ⟶ C ⟶ 0, T_{s} ∈ I_{Λ}, ∀s ≥ 0 : Ext_{Λ}^{t}(I_{Λ}, Coker(f_{n})) = 0, ∀n ≥ 0, ∀t ≥ 1 }.

Definition 5.1. The modules in CM(P_{Λ}) are called big Cohen-Macaulay modules and the modules in CoCM(I_{Λ}) are called big Co-Cohen-Macaulay modules.

By the results of Chapter V we know that CM(P_{Λ}) is a resolving subcategory of Mod(Λ) with \underline{CM}(P_{Λ}) triangulated, and CoCM(I_{Λ}) is a coresolving subcategory of Mod(Λ) with \overline{CoCM}(I_{Λ}) triangulated.

Proposition 5.2. The adjoint pair (N^{+}, N^{−}) induces an equivalence

N^{+} : CM(P_{Λ}) ⟶^{≈} CoCM(I_{Λ})

with quasi-inverse N^{−} : CoCM(I_{Λ}) ⟶^{≈} CM(P_{Λ}) and an exact equivalence of triangulated categories

N^{+} : \underline{CM}(P_{Λ}) ⟶^{≈} \overline{CoCM}(I_{Λ})

with quasi-inverse N^{−} : \overline{CoCM}(I_{Λ}) ⟶^{≈} \underline{CM}(P_{Λ}).

Proof. First we show that the left derived functor L_{n}N^{+}(X) = Tor_{n}^{Λ}(X, D(Λ)) = 0, ∀X ∈ CM(P_{Λ}) and the right derived functor R^{n}N^{−}(Z) = Ext_{n}^{Λ}(D(Λ), Z) = 0, ∀Z ∈ CoCM(I_{Λ}). For this we recall the following well-known formula of Cartan-Eilenberg:

Ext_{Λ}^{n}(A, D(B)) ⟶^{≅} DTor_{n}^{Λ}(A, B), ∀n ≥ 0, ∀A ∈ Mod(Λ), ∀B ∈ Mod(Λ^{op}).

Setting B = _{Λ}D(Λ) it follows that DL_{n}N^{+}(X) = DTor_{n}^{Λ}(X, D(Λ)) ⟶^{≅} Ext_{Λ}^{n}(X, Λ). Since any big Cohen-Macaulay module X lies in^{⊥}P_{Λ} we infer that DL_{n}N^{+}(X) = 0, ∀n ≥ 1. Hence L_{n}N^{+}(X) = 0, ∀n ≥ 1, and dually R^{n}N^{−}(Z) = 0, ∀n ≥ 1, ∀Z ∈ CoCM(I_{Λ}). Hence the Nakayama functor N^{+} is exact on exact sequences with right end term a big Cohen-Macaulay module and the Nakayama functor N^{−} is exact on exact sequences with left end term a big CoCohen-Macaulay module.

Now let X be a big Cohen-Macaulay module. Then there exists an exact coresolution 0 ⟶ X ⟶ P^{0} ⟶ P^{1} ⟶ ⋯ of X by projective modules. Since the Nakayama functor N^{+} is exact on Cohen-Macaulay modules, we have that 0 ⟶ N^{+}(X) ⟶ N^{+}(P^{0}) ⟶ N^{+}(P^{1}) ⟶ ⋯ is an injective resolution of N^{+}(X). Applying the left exact functor N^{−} to this sequence we have an exact sequence 0 ⟶ N^{−}N^{+}(X) ⟶ N^{−}N^{+}(P^{0}) ⟶ N^{−}N^{+}(P^{1}). Since the adjunction morphisms P^{i} ⟶ N^{−}N^{+}(P^{i}) are invertible, we infer that the adjunction morphism X ⟶ N^{−}N^{+}(X) is invertible. Now let ⋯ ⟶ P_{−2} ⟶ P_{−1} ⟶ X ⟶ 0 be a projective resolution of X. Since the Nakayama functor N^{+} is exact on Cohen-Macaulay modules, we have an exact resolution ⋯ ⟶ N^{+}(P_{−2}) ⟶ N^{+}(P_{−1}) ⟶ N^{+}(X) ⟶ 0 of N^{+}(X) by injectives. Then to show that N^{+}(X) is a CoCohen-Macaulay module it suffices to show that if we apply Hom_{Λ}(I, −) for any injective module I to the above coresolution, then we still get an exact sequence. Since D(Λ) is an injective cogenerator, it suffices to show that the above resolution remains exact after the application of N^{−}. However this follows trivially since the resulting sequence is isomorphic to the original one. We infer that for any Cohen-Macaulay module X, N^{+}(X) is a CoCohen-Macaulay module and in addition the adjunction morphism X ⟶ N^{−}N^{+}(X) is invertible. Hence we have a fully faithful functor N^{+} : CM(P_{Λ}) ⟶ CoCM(I_{Λ}). By duality, for any CoCohen-Macaulay module Z, N^{−}(Z) is a Cohen-Macaulay module and in addition the adjunction morphism N^{+}N^{−}(Z) ⟶ Z is invertible. This shows that N^{+} : CM(P_{Λ}) ⟶^{≈} CoCM(I_{Λ}) with quasi-inverse N^{−} : CoCM(I_{Λ}) ⟶^{≈} CM(P_{Λ}). Since the functor N^{+} is exact on Cohen-Macaulay modules and sends projectives to injectives, it follows that the induced functor N^{+} : \underline{CM}(P_{Λ}) ⟶^{≈} \overline{CoCM}(I_{Λ}) is a triangle equivalence with quasi-inverse N^{+}.

In particular from the above result we have natural isomorphisms

N^{+}Ω ⟶^{≅} Σ^{−1}N^{+} : \underline{CM}(P_{Λ}) ⟶ \overline{CoCM}(I_{Λ})
N^{−}Σ ⟶^{≅} Ω^{−1}N^{−} : \overline{CoCM}(I_{Λ}) ⟶ \underline{CM}(P_{Λ})

Now let A and B be arbitrary modules. Let 0 ⟶ Ω^{2}(A) ⟶ P_{1} ⟶ P_{0} ⟶ A ⟶ 0 be the start of a projective resolution of A, and let 0 ⟶ B ⟶ I_{0} ⟶ I_{1} ⟶ Σ^{2}(B) ⟶ 0 be the start of an injective resolution of B. We define the modules DTr(A) and TrD(B) by the exact sequences:

0 ⟶ DTr(A) ⟶ N^{+}(P_{1}) ⟶ N^{+}(P_{0}) ⟶ N^{+}(A) ⟶ 0
0 ⟶ N^{−}(B) ⟶ N^{−}(I^{0}) ⟶ N^{−}(I^{1}) ⟶ TrD(B) ⟶ 0

Note that DTr is not the composition of the operators D and Tr as in the finitely presented case. However DTr coincides with the usual operator in case we work with finitely generated modules. The following well-known result, first observed by H. Krause in [48], follows easily from the fact that the Nakayama functors restrict to an equivalence P_{Λ} ⟶^{≈} I_{Λ}.

Lemma 5.3. The assignements A ↦ DTr(A) and B ↦ TrD(B) induce well defined additive functors

DTr : \underline{Mod}(Λ) ⟶ \overline{Mod}(Λ) and TrD : \overline{Mod}(Λ) ⟶ \underline{Mod}(Λ)

which are mutually inverse equivalences.

We have the following direct consequence of Lemma 5.3 and Proposition 5.2, which shows that the stable equivalences induced by the Nakayama functors N^{+}, N^{−} and the operators DTr and TrD are compatible.

Lemma 5.4. The equivalences DTr and TrD restrict to triangle quasi-inverse equivalences

DTr : \underline{CM}(P_{Λ}) ⟶^{≈} \overline{CoCM}(I_{Λ}) and TrD : \overline{CoCM}(I_{Λ}) ⟶^{≈} \underline{CM}(P_{Λ})

such that the following diagrams are commutative:

\underline{CM}(P_{Λ}) ⟶^{N^{+}} \overline{CoCM}(I_{Λ})
i↓  _{Σ^{−2}}↓
\underline{Mod}(Λ) ⟶^{DTr} \overline{Mod}(Λ)

\overline{CoCM}(I_{Λ}) ⟶^{N^{−}} \underline{CM}(P_{Λ})
k↓  _{Ω^{−2}}↓
\overline{Mod}(Λ) ⟶^{TrD} \underline{Mod}(Λ)

where i and k are the inclusion functors.

Let H(P_{Λ}) be the (unbounded) homotopy category of P_{Λ} and let H(I_{Λ}) be the (unbounded) homotopy category of I_{Λ}. We denote by H_{Ac}(P_{Λ}) the full subcategory of H(P_{Λ}) consisting of the acyclic complexes of projectives and by H_{Ac}(I_{Λ}) the full subcategory of H(I_{Λ}) consisting of the acyclic complexes of injectives. Obviously H_{Ac}(P_{Λ}) is a thick subcategory of H(P_{Λ}) and H_{Ac}(I_{Λ}) is a thick subcategory of H(I_{Λ}). We denote by E_{P} the full subcategory of H_{Ac}(P_{Λ}) consisting of the acyclic complexes of projectives P^{•} such that the complex (P^{•}, P) is acyclic for all projective modules P. Dually we denote by E_{I} the full subcategory of H_{Ac}(I_{Λ}) consisting of the acyclic complexes of injectives I^{•} such that the complex (I, I^{•}) is acyclic for all injective modules I. It is easy to see that the functor P^{•} ↦ Ker(d^{0}) induces an equivalence E_{P} ⟶ \underline{CM}(P_{Λ}) and the functor I^{•} ↦ Ker(d^{0}) induces an equivalence E_{I} ⟶ \overline{CoCM}(P_{Λ}). In particular E_{P} and E_{I} are triangulated. The following result is an easy consequence of Proposition 5.2 and the definitions.

Proposition 5.5. The adjoint pair (N^{+}, N^{−}) induces quasi-inverse triangle equivalences

N^{+} : H(P_{Λ}) ⟶^{≈} H(I_{Λ}), N^{−} : H(I_{Λ}) ⟶^{≈} H(P_{Λ})

which induce quasi-inverse triangle equivalences

N^{+} : E_{P} ⟶^{≈} E_{I}, N^{−} : E_{I} ⟶^{≈} E_{P}.

Remark 5.6. In general the triangle equivalences N^{+} : H(P_{Λ}) ⟶^{≈} H(I_{Λ}) and N^{−} : H(I_{Λ}) ⟶^{≈} H(P_{Λ}) do not restrict to equivalences between H_{Ac}(P_{Λ}) and H_{Ac}(I_{Λ}). A necessary and sufficient condition for this to happen is that Λ must be Co-Gorenstein in the sense of [17]. Since by [17] any Gorenstein algebra is Co-Gorenstein, this holds for Gorenstein algebras. Note that for Gorenstein algebras we have H_{Ac}(P_{Λ}) = E_{P} and H_{Ac}(I_{Λ}) = E_{I}.

Now we can state the main result of this section.

Theorem 5.7. (i) The inclusion \underline{CM}(P_{Λ}) ↪ \underline{Mod}(Λ) admits a right adjoint and the inclusion \overline{CoCM}(I_{Λ}) ↪ \overline{Mod}(Λ) admits a left adjoint.

(ii) There is a hereditary pretorsion theory (CM(P_{Λ}), Y) in Mod(Λ).The full subcategory CM(P_{Λ}) is functorially finite and moreover: CM(P_{Λ})∩Y = P_{Λ} and P_{Λ}^{<∞} = Y ∩ { \atop CM(P_{Λ})}.

(iii) There is a cohereditary pretorsion theory (W, CoCM(I_{Λ})) in Mod(Λ). The full subcategory CoCM(I_{Λ}) is functorially finite and moreover: CoCM(I_{Λ}) ∩ W = I_{Λ} and I_{Λ}^{<∞} = { \atop CoCM(I_{Λ})} ∩ W.

(iv) The pair (\underline{CM}(P_{Λ}), Y/P_{Λ}) is a hereditary torsion theory in \underline{Mod}(Λ), and

the pair (W/I_{Λ}, \overline{CoCM}(I_{Λ})) is a cohereditary torsion theory in \overline{Mod}(Λ).

Proof. (i) That the inclusion \underline{CM}(P_{Λ}) ↪ \underline{Mod}(Λ) admits a right adjoint is a result of [47]. We show that the inclusion \overline{CoCM}(I_{Λ}) ↪ \overline{Mod}(Λ) admits a left adjoint using the results of [47] and results of Neeman. Indeed consider the full subcategory E_{I} ↪ H_{Ac}(I_{Λ}). The functor H^{0}N^{−} : H_{Ac}(I_{Λ}) ⟶ Ab is a homological functor which preserves coproducts, where H^{0} is the 0−th cohomology functor, and it is easy to see that E_{I} = {I^{•} ∈ H_{Ac}(I_{Λ}) | H^{0}N^{−}Σ^{n}(I^{•}) = 0, ∀n ∈ ℤ}. Since the homological functor H^{0}N^{−} preserves coproducts, by [47], the Verdier quotient H_{Ac}(I_{Λ})/E_{I} has small hom-sets. Since E_{I} is closed under products and H^{0}N^{−} preserves products, it is easy to see that the quotient functor q : H_{Ac}(I_{Λ}) ⟶ H_{Ac}(I_{Λ})/E_{I} preserves products. By a result of Neeman [64], a homological product preserving functor from a compactly generated triangulated category to a Grothendieck category is representable. This implies that a product preserving exact functor from a compactly generated triangulated category to a triangulated category admits a left adjoint. Since by the dual results of [47], H_{Ac}(I_{Λ}) is compactly generated and q preserves products, we have that q admits a left adjoint. Then by a well-known result of Verdier [79], the inclusion E_{I} ↪ H_{Ac}(I_{Λ}) admits a left adjoint. Since E_{I} is triangle equivalent to \overline{CoCM}(I_{Λ}), it follows that the inclusion \overline{CoCM}(I_{Λ}) ↪ \overline{Mod}(Λ) admits a left adjoint.

(ii) By Corollary V.4.4 it follows that \underline{CM}(P_{Λ}) is a torsion class in \underline{Mod}(Λ). If Y/P_{Λ} is the corresponding torsion-free class, then by Proposition V.3.1 we have that (CM(P_{Λ}), Y) is a good pair in Mod(Λ). Since CM(P_{Λ}) is triangulated, by Theorem VI.2.1 we have that CM(P_{Λ}) ∩ Y = P_{Λ} and P_{Λ}^{<∞} ⊆ Y. Finally by Proposition VI.2.3 we have P_{Λ}^{<∞} = Y ∩ { \atop CM(P_{Λ})}. Finally we show that CM(P_{Λ}) is covariantly finite. Let A be any module and let g : N^{+}(A) ⟶ Z^{N^{+}(A)} be a

left CoCohen-Macaulay approximation. The composition A ⟶^{δ_{A}} N^{−}N^{+}(A) ⟶^{N^{−}(g)} N^{−}(Z^{N^{+}(A)}), where δ_{A} is the unit, is then a left Cohen-Macaulay approximation of A. Indeed we know that N^{−}(Z^{N^{+}(A)}) is Cohen-Macaulay. Let α : A ⟶ X be a morphism where X is Cohen-Macaulay. Then N^{+}(X) is CoCohen-Macaulay, hence there exists a morphism ρ : Z^{N^{+}(A)} ⟶ N^{+}(X) such that g ∘ρ = N^{+}(α). Then δ_{A}∘N^{−}(g)∘N^{−}(ρ) = δ_{A}∘N^{−}N^{+}(α). But δ_{A}∘N^{−}N^{+}(α) = α∘δ_{X} and δ_{X} is invertible since X is Cohen-Macaulay. Hence α factors through N^{−}(Z^{N^{+}(A)}) via the morphism δ_{A} ∘ N^{−}(g) ∘ N^{−}(ρ) ∘ δ_{X}^{−1}. Hence the morphism δ_{A} ∘ N^{−}(g) : A ⟶ N^{−}(Z^{N^{+}(A)}) is a left Cohen-Macaulay approximation of A, i.e. CM(P_{Λ}) is covariantly finite.

Part (iii) is dual to (ii) and part (iv) follows from our previous results.

Definition 5.8. We call the modules in P_{Λ}^{<∝} := CM(P_{Λ})^{⊥} the modules with virtually finite projective dimension, and the modules in I_{Λ}^{<∝} := ^{⊥}CoCM(I_{Λ}) the modules with virtually finite injective dimension.

In this notation we have a hereditary pretorsion theory (CM(P_{Λ}), P_{Λ}^{<∝}) and a cohereditary pretorsion theory (I_{Λ}^{<∝}, CoCM(I_{Λ})) in Mod(Λ). Moreover we have:

P_{Λ}^{<∞} = P_{Λ}^{<∝} ∩ { \atop CM(P_{Λ})} and CM(P_{Λ}) ∩ P_{Λ}^{<∝} = P_{Λ}, and dually I_{Λ}^{<∞} = I_{Λ}^{<∝} ∩

{ \atop CoCM(I_{Λ})} and CoCM(I_{Λ}) ∩ I_{Λ}^{<∝} = I_{Λ}.

The above result shows that we can define new cohomology theories for any Artin algebra Λ, as the generalized Tate-Vogel Cohomology theories with respect to the torsion class induced by the big Cohen-Macaulay objects and the torsion-free class induced by the big CoCohen-Macaulay objects. We say that a module has finite Gorenstein projective, resp. injective, dimension if it admits a finite exact resolution, resp. coresolution, by big Cohen-Macaulay, resp. CoCohen-Macaulay, modules.

For the convenience of the reader we summarize in the following theorem the basic properties of these new cohomology theories, which possibly will have future applications.

Theorem 5.9. For any n ∈ ℤ, there are bifunctors

Êxt_{CM}^{n}(−, −) : Mod(Λ)^{op} × Mod(Λ) ⟶ Ab

the projective Cohen-Macaulay bifunctor, and

Ẽxt_{CM}^{n}(−, −) : Mod(Λ)^{op} × Mod(Λ) ⟶ Ab

the injective Cohen-Macaulay bifunctor. These are defined by

Êxt_{CM}^{n}(A, B) := \underline{Hom}_{Λ}[Ω^{n}R(\underline{A}), \underline{B}] ⟶^{≅} \underline{Hom}_{\underline{CM}(P_{Λ})}[Ω^{n}R(\underline{A}), R(\underline{B})]
Ẽxt_{CM}^{n}(A, B) := \overline{Hom}_{Λ}[\underline{A}, Σ^{n}S(\underline{B})] ⟶^{≅} \overline{Hom}_{\overline{CoCM}(I_{Λ})}[S(\underline{A}), Σ^{n}S(\underline{B})]

and they satisfy the following properties.

(i) The functors Êxt_{CM}^{∗}(−, −) and Ẽxt_{CM}^{∗}(−, −) are homological in each variable.

(ii) Êxt_{CM}^{∗}(A, B) vanishes if A or B is a module of (virtually) finite projective dimension and Ẽxt_{CM}^{∗}(A, B) vanishes if A or B is a module of (virtually) finite injective dimension.

(iii) For any module C, the functor Êxt_{CM}^{∗}(C, −) ⟶^{≅} \underline{Π}_{X_{C}}^{∗} is the completion of Ext_{Λ}^{∗}(C, −) with respect to the full subcategory of modules with virtually finite projective dimension. Here \underline{Π}_{X_{C}}^{∗} is the Eckmann-Hilton projective homotopy functor of the right Cohen-Macaulay approximation of C.

(iv) For any module A, the functor Ẽxt_{CM}^{∗}(−, A) ⟶^{≅} \overline{Π}_{Z^{A}}^{∗} is the completion of Ext_{Λ}^{∗}(−, A) with respect to the full subcategory of modules with virtually finite injective dimension. Here \overline{Π}_{Z^{A}}^{∗} is the Eckmann-Hilton injective homotopy functor of the left CoCohen-Macaulay approximation of A.

(v) For any module C, the P_{Λ}^{<∝}-completion morphism ζ_{C,−}^{∗} : Ext_{Λ}^{∗}(C, −) ⟶

Êxt_{CM}^{∗}(C, −), factors uniquely through the P_{Λ}-completion morphism χ_{C,−}^{∗} :

Ext_{Λ}^{∗}(C, −) ⟶ Êxt_{T V}^{∗} (C, −). The resulting comparison morphism of homological functors ρ_{C,−}^{∗} : Êxt_{T V}^{∗} (C, −) ⟶ Êxt_{CM}^{∗}(C, −) is invertible iff C has finite Gorenstein projective dimension.

(vi) For any module A, the I_{Λ}^{<∝}-completion morphism η_{C,−}^{∗} : Ext_{Λ}^{∗}(−, A) ⟶

Ẽxt_{CM}^{∗}(−, A), factors uniquely through the I_{Λ}-completion morphism ψ_{−,A}^{∗} :

Ext_{Λ}^{∗}(−, A) ⟶ Ẽxt_{T V}^{∗} (−, A). The resulting comparison morphism of homological functors τ_{−,A}^{∗} : Ẽxt_{T V}^{∗} (−, A) ⟶ Ẽxt_{CM}^{∗}(−, A) is invertible iff A has finite Gorenstein injective dimension.

(vii) Λ is Gorenstein iff the comparison morphism

ρ_{−,−}^{∗} : Êxt_{T V}^{∗} (−, −) ⟶ Êxt_{CM}^{∗}(−, −)

is invertible iff the comparison morphism

τ_{−,−}^{∗} : Ẽxt_{T V}^{∗} (−, −) ⟶ Ẽxt_{CM}^{∗}(−, −)

is invertible. (viii) There exists a long exact sequence

0 ⟶ Ext_{CM(P_{Λ})}^{1}(−, −) ⟶ Ext_{Λ}^{1}(−, −) ⟶ Êxt_{CM}^{1}(−, −) ⟶ ⋯

⋯ ⟶ Ext_{CM(P_{Λ})}^{n}(−, −) ⟶ Ext_{Λ}^{n}(−, −) ⟶ Êxt_{CM}^{n}(−, −) ⟶ ⋯

(ix) There exists a long exact sequence

0 ⟶ Ext_{CoCM(I_{Λ})}^{1}(−, −) ⟶ Ext_{Λ}^{1}(−, −) ⟶ Ẽxt_{CM}^{1}(−, −) ⟶ ⋯

⋯ ⟶ Ext_{CoCM(I_{Λ})}^{n}(−, −) ⟶ Ext_{Λ}^{n}(−, −) ⟶ Ẽxt_{CM}^{n}(−, −) ⟶ ⋯

If r.Fin.p.dimΛ, resp. r.Fin.i.dimΛ, are the big Finitistic projective, resp. injective, dimension, then our previous results imply the following useful bounds on the big finitistic projective and injective dimensions in terms of the relative homological dimensions arising from the big (Co)Cohen-Macaulay modules and the modules of virtually finite projective or injective dimension.

Corollary 5.10. For any Artin algebra Λ we have the following inequalities:

r.Fin.p.dimΛ ≤ sup{pdY | Y ∈ P_{Λ}^{<∝}} ≤ CM(P_{Λ})−gl.dimΛ ≤ gl.dimΛ

r.Fin.i.dimΛ ≤ sup{idY | Y ∈ I_{Λ}^{<∝}} ≤ CoCM(I_{Λ})−gl.dim Λ ≤ gl.dimΛ

and finiteness of any of these dimensions, except for the big finitistic projective or injective dimension, forces Λ to be Gorenstein.

If Λ is Gorenstein then we know that P_{Λ}^{<∞} = I_{Λ}^{<∞}. It is not difficult to see that this extends to the big module category: Λ is Gorenstein iff P_{Λ}^{<∞} = I_{Λ}^{<∞}. This analogy with the Gorenstein situation suggest the following definition.

Definition 5.11. An Artin algebra Λ is called virtually Gorenstein if we have an equality: P_{Λ}^{<∝} = I_{Λ}^{<∝}.

If Λ is Gorenstein, then by Corollary 5.10 we have: P_{Λ}^{<∝} := CM(P_{Λ})^{⊥} = P_{Λ}^{<∞} = I_{Λ}^{<∞} = ^{⊥}CoCM(I_{Λ}) := I_{Λ}^{<∝}, hence Λ is virtually Gorenstein. We don’t know if the converse holds.

In analogy with the Gorenstein case we have the following characterization of virtually Gorenstein algebras.

Theorem 5.12. For an Artin algebra Λ the following are equivalent.

(i) Λ is virtually Gorenstein.

(ii) The projective and injective Cohen-Macaulay complete extension bifunctors are isomorphic:

Êxt_{CM}^{∗}(−, −) ⟶^{≅} Ẽxt_{CM}^{∗}(−, −).

(iii) There exists a good triple (CM(P_{Λ}), P_{Λ}^{<∝} = I_{Λ}^{<∝}, CoCM(I_{Λ})) in Mod(Λ).

(iv) F_{CM(P_{Λ})} = F^{CoCM(I_{Λ})}. That is for any short exact sequence 0 ⟶ A ⟶^{g}

B ⟶^{f} C ⟶ 0 in Mod(Λ), any morphism X ⟶ C with X ∈ CM(P_{Λ}) factors f iff any morphism A ⟶ Z with Z ∈ CoCM(I_{Λ}) factors g.

(v) The relative extension functors with respect to the subcategories CM(P_{Λ})

and CoCM(I_{Λ}), are isomorphic: Ext_{CM(P_{Λ})}^{∗}(−, −) ⟶^{≅} Ext_{CoCM(I_{Λ})}^{∗}(−, −). If this is the case then CM(P_{Λ})−gl.dimΛ = CoCM(I_{Λ})−gl.dimΛ and this common dimension is finite iff Λ is Gorenstein.

Proof. Obviously (i) is equivalent to (iii). If Λ is virtually Gorenstein, then the isomorphism in (ii) follows from Theorem 4.2. Conversely if (ii) holds, then working as in the proof of Corollary 4.3 we infer that P_{Λ}^{<∝} = I_{Λ}^{<∝}, hence Λ is virtually Gorenstein. Now obviously (iv) is equivalent to (v), and (i) implies (v) by Corollary 4.5. Finally if (v) holds then the isomorphism in part (ii) follows from the exact sequences of Theorem 3.2.

Since the categories P_{Λ} and I_{Λ} are functorially finite, the results of Chapter VII can be applied for the pretorsion theories produced by Theorem 5.7. It follows that they are induced four compatible model structures on Mod(Λ). For possible future use and for the convenience of the reader we write down explicitly the closed model structures:

\underline{(ℭ_{P}, 𝔉_{P}, 𝔚_{P})} [The contravariant projective CM-closed model structure] Here ℭ_{P} is the class of monics with cokernel a big Cohen-Macaulay module. 𝔗ℭ_{P} is the class of split monics with cokernel a projective module. 𝔉_{P} is the class of epimorphisms. 𝔗𝔉_{P} is the class of epimorphisms with kernel a module of virtually finite projective dimension. Finally 𝔚_{P} is the class of morphisms A ⟶ B which induce an isomorphism modulo projectives between the Cohen-Macaulay approximations of A and B.

For the closed model structure (ℭ_{P}, 𝔉_{P}, 𝔚_{P}), CM(P_{Λ}) is the class of cofibrant objects, all modules are fibrant, the trivially cofibrant objects are the projective modules, and the modules of virtually finite projective dimension are the trivially fibrant objects. The associated Quillen homotopy category of (ℭ_{P}, 𝔉_{P}, 𝔚_{P}) is the stable triangulated category \underline{CM}(P_{Λ}).

\underline{(ℭ^{P}, 𝔉^{P}, 𝔚^{P})} [The covariant projective CM-closed model structure] Here ℭ^{P} is the class of P_{Λ}-monics. 𝔗ℭ^{P} is the class of monics with cokernel a big Cohen-Macaulay module. 𝔉^{P} is the class of epimorphisms with kernel a module with virtually finite projective dimension. 𝔗𝔉^{P} is the class of split epimorphisms with kernel a projective module. Finally 𝔚^{P} is the class of morphisms A ⟶ B which induce an isomorphism modulo projectives, between the left P_{Λ}^{<∝}- approximations of A and B.

For the closed model structure (ℭ^{P}, 𝔉^{P}, 𝔚^{P}), all modules are cofibrant, CM(P_{Λ}) is the class of trivially cofibrant objects, the trivially fibrant objects are the projective modules, and the fibrant objects are the modules of virtually finite projective dimension. The associated Quillen homotopy category of (ℭ^{P}, 𝔉^{P}, 𝔚^{P}) is the stable pretriangulated category P_{Λ}^{<∝}/P_{Λ}.

\underline{(ℭ_{I}, 𝔉_{I}, 𝔚_{I})} [The contravariant injective CM-closed model structure] Here ℭ_{I} is the class of monics with cokernel a module of virtually finite injective dimension. 𝔗ℭ_{I} is the class of split monics with injective cokernel. 𝔉_{I} is the class of I_{Λ}- epics. 𝔗𝔉_{I} is the class of epics with kernel a big CoCohen-Macaulay module. Finally 𝔚_{I} is the class of morphisms A ⟶ B which induce an isomorphism modulo injectives, between the left I_{Λ}^{<∝}-approximations of A and B.

For the closed model structure (ℭ_{I}, 𝔉_{I}, 𝔚_{I}), the cofibrant objects are the modules of virtually finite injective dimension, all modules are fibrant, the trivially cofibrant objects are the injectives, and the trivially fibrant objects are the big CoCohen-Macaulay modules. The associated Quillen homotopy category of (ℭ_{I}, 𝔉_{I}, 𝔚_{I}) is the stable pretriangulated category I_{Λ}^{<∝}/I_{Λ}.

\underline{(ℭ^{I}, 𝔉^{I}, 𝔚^{I})} [The covariant injective CM-closed model structure] Here ℭ^{I} is the class of monics. 𝔗ℭ^{I} is the class of monics with cokernel a module of virtually finite injective dimension. 𝔉^{I} is the class of epimorphisms with kernel a big CoCohen-Macaulay module. 𝔗𝔉^{I} is the class of split epimorphisms with injective kernel. Finally 𝔚^{I} is the class of morphisms A ⟶ B which induce an isomorphism modulo injectives, between the left big CoCohen-Macaulay approximations.

For the closed model structure (ℭ^{I}, 𝔉^{I}, 𝔚^{I}), all modules are cofibrant, CoCM(I_{Λ}) is the class of fibrant objects, the trivially fibrant objects are the injective modules, and the trivially cofibrant objects are the modules of virtually finite injective dimension. The associated Quillen homotopy category of (ℭ^{I}, 𝔉^{I}, 𝔚^{I}) is the stable triangulated category \overline{CoCM}(I_{Λ}).

It is not difficult to see that the adjoint pair (N^{+}, N^{−}) of Nakayama functors on Mod(Λ) is a Quillen equivalence between the closed model structures (ℭ_{P}, 𝔉_{P}, 𝔚_{P}) and (ℭ^{I}, 𝔉^{I}, 𝔚^{I}), that is: it induces an equivalence \underline{CM}(P_{Λ}) ⟶^{≈} \overline{CoCM}(I_{Λ}) between their homotopy categories. Hence we obtain a model theoretic proof of Proposition 5.2.

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