Relative Homological Algebra and Purity in Triangulated Categories

Apostolos Beligiannis

Fakultät für Mathematik, Universität Bielefeld, D–33501 Bielefeld, Germany. Email address: abelig@mathematik.uni-bielefeld.de, abeligia@mail.otenet.gr.

Contents

1 Introduction 1 2 Proper Classes of Triangles and Phantom Maps 4 3 The Steenrod and Freyd Category of a Triangulated Category 8 4 Projective Objects, Resolutions and Derived Functors 10 5 The Phantom Tower, the Cellular Tower,

Homotopy Colimits and Compact Objects 19 6 Localization and the Relative Derived Category 25 7 The Stable Triangulated Category 29 8 Projectivity, Injectivity and Flatness 30 9 Phantomless Triangulated Categories 35 10 Brown Representation Theorems 40 11 Purity 42 12 Applications to Derived and Stable Categories 52

References 62

1. Introduction

Triangulated categories were introduced by Grothendieck and Verdier in the early sixties as the proper framework for doing homological algebra in an abelian category. Since then triangulated categories have found important applications in Algebraic Geometry, Stable Homotopy Theory and Representation Theory. Our main purpose in this paper is to study a triangulated category, using relative homological algebra which is developed inside the triangulated category.

Relative homological algebra has been formulated by Hochschild in categories of modules and later by Heller and Butler-Horrocks in more general categories with a relative abelian structure. Its main theme consists of a selection of a class of extensions. In triangulated categories there is a natural candidate for extensions, namely the (distinguished) triangles. Let C be a triangulated category with triangulation Δ. We develop a homological algebra in C which parallels the homological algebra in an exact category in the sense of Quillen, by specifying a class of triangles E ⊆ Δ which is closed under translations and satisfies the analogous formal properties of a proper class of short exact sequences. We call such a class of triangles E, a proper class of triangles. Two big differences occur between the homological algebra in a triangulated category and in an exact category. First the absolute theory in a triangulated category, i.e. if we choose Δ as a proper class, is trivial. Second a proper class of triangles is uniquely determined by an ideal Ph_{E}(C) of C, which we call the ideal of E−phantom maps. This ideal is a homological invariant which almost always is non-trivial and in a sense controls the homological behaviour of the triangulated category. However there are non-trivial examples of phantomless categories. To a large extent the relative homological behaviour of C with respect to E depends on the knowledge of the structure of E−phantom maps.

The paper is organized as follows. Section 2 contains the basic definitions about proper classes of triangles and phantom maps in a triangulated category C. We prove that there exists a bijective correspondence between proper classes of triangles and special ideals of C, and we discuss briefly an analogue of Baer’s theory of extensions. In Section 3 we prove that there exists a bijective correspondence between proper classes of triangles in C and Serre subcategories of the category mod(C) of finitely presented additive functors C^{op} ⟶ Ab, which are closed under the suspension functor. Using this correspondence, we associate to any proper class of triangles E in C, an uniquely determined abelian category S_{E}(C), the E−Steenrod category of C. The E−Steenrod category comes equipped with a homological functor S : C ⟶ S_{E}(C), the projectivization functor, which is universal for homological functors out of C which annihilate the ideal of E−phantom maps. The terminology comes from stable homotopy: the Steenrod category of the stable homotopy category of spectra with respect to the proper class of triangles induced from the Eilenberg-MacLane spectrum corresponding to ℤ/(p), is the module category over the mod-p Steenrod algebra. We regard S_{E}(C) as an abelian approximation of C and the idea is to use it as a tool for transfering information between the topological category C and the algebraic category S_{E}(C). This tool is crucial, if the projectivization functor is non-trivial and this happens iff the proper class of triangles E generates C in an appropriate sense.

In Section 4 fixing a proper class of triangles E in a triangulated category C, we introduce E−projective objects, E−projective resolutions, E−projective and E−global dimension and their duals, and we prove the basic tools of homological algebra (Schanuel’s Lemma, Horseshoe Lemma, Comparison Theorem) in this setting. It follows that we can derive additive functors which behave well with respect to triangles in E. We compare the homological invariants of C with respect to E with the homological invariants of its E−Steenrod category, via the projectivization functor. Finally we study briefly the semisimple and hereditary categories.

In Section 5, we associate with any object A in C, the E−phantom and the E−cellular tower of A, which are crucial for the study of the homological invariants of A with respect to E, e.g. the structure of E−phantom maps out of A and the E−projective dimension of A. Using these towers we prove our first main result which asserts that the global dimension of C with respect to E is less than equal to 1 iff the projectivization functor S : C ⟶ S_{E}(C) is full and reflects isomorphisms. The proper setting for the study of the E−phantom and the E−cellular tower is that of a compactly generated triangulated category [55]. In this case homotopy colimits are defined and we prove that under mild assumptions, any object of C is a homotopy colimit of its E−cellular tower and the homotopy colimit of its E−phantom tower is trivial. Finally we study the category of extensions of E−projective objects, the phantom topology and the phantom filtration of C induced by the ideal Ph_{E}(C), and we compute the E−phantom maps which “live for ever”, i.e. maps in ⋂_{n≥1} Ph_{E}^{n}(C), in terms of the E−cellular tower.

In Section 6 we associate with any triangulated category C with enough E−projectives, a new triangulated category D_{E}(C) which we call the E−derived category. Under some mild assumptions D_{E}(C) is realized as a full subcategory of C and is described as the localizing subcategory of C generated by the E−projective objects. Moreover D_{E}(C) is compactly generated and any compactly generated category is of this form. This construction, which generalizes the construction of the usual derived category, allows us to prove the existence of total E−derived functors. Applying these results to the derived category of a Grothendieck category with projectives, we generalize results of Spaltenstein [67], Boekstedt-Neeman [16], Keller [42], Weibel [71], concerning resolutions of unbounded complexes and the structure of the unbounded derived category, proved in the above papers by using essentially a closed model structure on the category of complexes.

Homological Algebra in Δ−Categories 3

In Section 7 we prove that the stable category of C modulo E−projectives, admits a natural left triangulated structure, which in many cases it is useful to study.

The results of the first seven sections refer to general proper classes of triangles in unspecified categories. In Section 8 we study proper classes of triangles E(X) induced by skeletally small subcategories X in a category C with coproducts. Under a reasonable condition on X we show that the Steenrod category of C with respect to the proper class E(X), is the category Mod(X) of additive functors X^{op} ⟶ Ab. We compare the E(X)−homological properties of objects of C with the homological properties of projective, flat and injective functors of the category Mod(X). Our first main result in this section shows that if X compactly generates C, then C has enough E(X)−injective objects in a functorial way and admits E(X)−injective envelopes. This generalizes a result of Krause [50]. Our second main result shows that if X compactly generates C or if any object of C has finite E(X)−projective dimension, then any coproduct preserving homological functor from C to a Grothendieck category, annihilates the ideal of E(X)−phantom maps. This generalizes results obtained independently by Christensen-Strickland [21] and Krause [50].

In Section 9 we characterize the E(X)−semisimple or E(X)−phantomless categories C by a host of conditions, generalizing work of Neeman [54]. Perhaps the most characteristic is the condition that C is a pure-semisimple locally finitely presented category [24], in particular C has filtered colimits. Using this and an old result of Heller [34] (observed also by Keller-Neeman [44]), we give necessary conditions for a skeletally small category D such that the categories of Pro-objects and Ind-objects over D admit a triangulated structure. These conditions are sufficient for the existence of a Puppe-triangulated or “pre-triangulated” structure [57] on these categories, in the sense that the Octahedral Axiom does not necessarily hold.

In Section 10 we study Brown Representation Theorems, via the concept of a representation equivalence functor, i.e. a functor which is full, surjective on objects and reflects isomorphisms. We say that a pair (C, X) consisting of a triangulated category with coproducts C and a full skeletally small triangulated subcategory X ⊆ C, satisfies Brown Representability Theorem or that C is an E(X)−Brown category, if the projectivization functor of C with respect to E(X) induces a representation equivalence between C and the category of cohomological functors over X. Our main result shows that this happens iff X compactly generates C and any cohomological functor over X has projective dimension bounded by 1. This generalizes results of Christensen-Strickland [21] and Neeman [56].

In Section 11 we study the fundamental concept of purity in a compactly generated triangulated category which provides a link between representation-theoretic and homological properties. Motivated by the examples of the stable homotopy category of spectra, the derived category of a ring and the stable module category of a Quasi-Frobenius ring, it is natural to regard a compact object as an analogue of a finitely presented object in a triangulated category. In this spirit the proper class of triangles induced by the compact objects are the pure triangles and the resulting theory is the pure homological algebra of C. In this case the pure Steenrod category of C is the module category Mod(C^{b}), where C^{b} denotes the full subcategory of compact objects of C. We give various formulas for the computation of the pure global dimension and we prove that the pure global dimension bounds the pure global dimension of a smashing or finite localization. Our main result characterizes the pure Brown categories as the pure hereditary ones, and also by a host of equivalent conditions most notable being the fullness of the projectivization functor and the conditions concerning the behaviour of weak colimits. These results are applied directly to the stable homotopy category, which has pure global dimension 1. By the results of Section 9, a compactly generated triangulated category is pure-semisimple iff its pure Steenrod category is locally Noetherian.

Motivated by the pure homological theory of module categories, we define a category to be of finite type if its pure Steenrod category is locally finite. We prove that the phantomless, Brown or finite type property, is preserved under smashing or finite localization, generalizing results of Hovey, Palmieri and Strickland [38].

In Section 12 we apply the results of the previous Sections (mainly about purity), to derived categories of rings and stable categories of Quasi-Frobenius rings. We compute the pure global dimension for many classes of rings and we prove that the derived category D(Λ) of a ring Λ is of finite type iff D(Λ^{op}) is of finite type iff D(Λ) and D(Λ^{op}) are pure semisimple, so finite type is a symmetric condition. However we show that the Brown property is not symmetric in general. Motivated by the pure homological theory of module categories we formulate the Derived Pure Semisimple Conjecture, DPSC for short, which asserts that if D(Λ) is pure semisimple, then D(Λ) is of finite type. DPSC implies the still open Pure Semisimple Conjecture asserting that a right pure semisimple ring is of finite representation type. We prove DPSC for many classes of rings, including Artin algebras. Indeed we characterize the Artin algebras with pure semisimple derived categories as the iterated tilted algebras of Dynkin type. We prove that the analogue of DPSC holds for the stable category of a Quasi-Frobenius ring Λ, showing that Λ is representation-finite iff the stable category of Λ is pure semisimple (of finite type). The analogue of DPSC in a general compactly generated triangulated category fails. We prove this by answering in the negative a seemingly unrelated problem of Roos [64] concerning the structure of Quasi-Frobenius Grothendieck categories. Our results indicate that finite type is the appropriate notion of “representationfiniteness”, at least for stable or derived categories. We close the paper by applying the results of Sections 6, 7 to derived categories. In particular we obtain structure results for the unbounded derived category and we generalize results of Wheeler [72] [73], concerning the structure of the stable derived category.

A general convention used in the paper is that composition of morphisms in a category is meant diagrammatically: if f : A ⟶ B, g : B ⟶ C are morphisms, their composition is denoted by f ∘ g. However we compose functors in the usual (anti-diagrammatic) order. Our additive categories admit finite direct sums.

2. Proper Classes of Triangles and Phantom Maps

2.1. Triangulated Categories. Let C be an additive category and Σ : C ⟶ C be an additive functor. We define the category Diag(C, Σ) as follows. An object of Diag(C, Σ) is a diagram in C of the form A ⟶ B ⟶ C ⟶ Σ(A). A morphism in Diag(C, Σ) between A_{i} ⟶^{f_{i}} B_{i} ⟶^{g_{i}} C_{i} ⟶^{h_{i}} Σ(A_{i}), i = 1, 2, is a triple of morphisms α : A_{1} ⟶ A_{2}, β : B_{1} ⟶ B_{2}, γ : C_{1} ⟶ C_{2}, such that the diagram (†) commutes:

A_{1} ⟶^{f_{1}} B_{1} ⟶^{g_{1}} C_{1} ⟶^{h_{1}} Σ(A_{1})
α↓  ^{β}↓  ^{γ}↓  ^{Σ(α)}↓  (†)
A_{2} ⟶^{f_{2}} B_{2} ⟶^{g_{2}} C_{2} ⟶^{h_{2}} Σ(A_{2}).

A triangulated category [70] is a triple (C, Σ, Δ), where C is an additive category, Σ : C ⟶ C is an autoequivalence of C and Δ is a full subcategory of Diag(C, Σ) which is closed under isomorphisms and satisfies the following axioms:

(T_{1}) For any morphism f : A ⟶ B in C, there exists an object A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) in Δ. For any object A ∈ C, the diagram 0 ⟶ A ⟶^{1_{A}} A ⟶ 0 is in Δ.

(T_{2}) If A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) is in Δ, then B ⟶^{g} C ⟶^{h} Σ(A) ⟶^{−Σ(f)} Σ(B) is in Δ.

(T_{3}) If A_{i} ⟶^{f_{i}} B_{i} ⟶^{g_{i}} C_{i} ⟶^{h_{i}} Σ(A_{i}), i = 1, 2 are in Δ, and if there are morphisms α : A_{1} ⟶ A_{2} and β : B_{1} ⟶ B_{2} such that α ∘ f_{2} = f_{1} ∘ β, then there exists a morphism γ : C_{1} ⟶ C_{2} such that the diagram (†) above is a morphism in Δ.

(T_{4}) The Octahedral Axiom. For its formulation we refer to Proposition 2.1 below.

Proposition 2.1 (see also [51]). Let C be an additive category equipped with an autoequivalence Σ : C ⟶ C and a class of diagrams Δ ⊆ Diag(C, Σ). Suppose that the triple (C, Σ, Δ) satisfies all the axioms of a triangulated category except possibly of the Octahedral axiom. Then the following are equivalent:

(i) Base Change. For any triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ Δ and any morphism ε : E ⟶ C, there exists a commutative diagram

0 ⟶ M ⟶^{=} M ⟶ 0
↓  ^{α}↓  ^{δ}↓  ↓
A ⟶^{f′} G ⟶^{g′} E ⟶^{h′} Σ(A)
∥↓  ^{β}↓  ^{ε}↓  ^{∥}↓
A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A)
↓  ^{γ}↓  ^{ζ}↓  ↓
0 ⟶ Σ(M) ⟶^{=} Σ(M) ⟶ 0

in which all horizontal and vertical diagrams are triangles in Δ. (ii) Cobase Change. For any triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ Δ and any

morphism α : A ⟶ D, there exists a commutative diagram:

0  ⟶ N ⟶^{=} N ⟶ 0
↓  ^{ζ}↓  ^{δ}↓  ↓
Σ^{−1}(C) ⟶^{−Σ^{−1}(h)} A ⟶^{f} B ⟶^{g} C
∥↓  ^{α}↓  ^{β}↓  ^{∥}↓
Σ^{−1}(C) ⟶^{−Σ^{−1}(h′)} D ⟶^{f′} F ⟶^{g′} C
↓  ^{η}↓  ^{ϑ}↓  ↓
0  ⟶ Σ(N) ⟶^{=} Σ(N) ⟶ 0

in which all horizontal and vertical diagrams are triangles in Δ. (iii) Octahedral Axiom. For any two morphisms f_{1} : A ⟶ B, f_{2} : B ⟶ C

there exists a commutative diagram:

A ⟶^{f_{1}} B ⟶^{g_{1}} X ⟶^{h_{1}} Σ(A)
∥↓  ^{f_{2}}↓  ^{α}↓  ^{∥}↓
A ⟶^{f_{1}∘f_{2}} C ⟶^{g_{3}} Y ⟶^{h_{3}} Σ(A)
f_{1}↓  ^{∥}↓  ^{β}↓  ^{Σ(f_{1})}↓
B ⟶^{f_{2}} C ⟶^{g_{2}} Z ⟶^{h_{2}} Σ(B)
↓  ^{0}↓  ^{h_{2}∘Σ(g_{1})}↓  ↓
0 ⟶ Σ(X) ⟶^{=} Σ(X) ⟶ 0

in which all horizontal and the third vertical diagrams are triangles in Δ.

Proof. (i) ⇒ (iii) Let A ⟶^{f_{1}} B ⟶^{f_{2}} C be a diagram in C. By T_{1} there are triangles Σ^{−1}(Z) ⟶^{α} B ⟶^{f_{2}} C ⟶ Z and Σ^{−1}(X) ⟶ A ⟶^{f_{1}} B ⟶ X in Δ. Applying base change for the last triangle along α, it is easy to see that if we arrange properly the resulting diagram, then we obtain an octahedral diagram as in (iii). (iii) ⇒ (i) If M ⟶ E ⟶^{ε} C ⟶^{ζ} Σ(M) is a triangle, then applying the Octahedral Axiom to the composition g ∘ ζ, we obtain a diagram which if we apply Σ^{−1} and we arrange it properly, we have a diagram as in (i). The proof of (ii) ⇔ (iii) is similar.

Hence for any triangulated category we may use the above equivalent forms instead of the Octahedral Axiom, when it is more convenient. Throughout the paper we fix a triangulated category C = (C, Σ, Δ); Σ is the suspension functor, Δ is the triangulation and the elements of Δ are the triangles of C. A triangle (T) : A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ Δ is called split if h = 0. Trivially if (T) is split, then the morphisms f, g induce a direct sum decomposition B ≅ A⊕C. We denote by Δ_{0} the full subcategory of Δ consisting of the split triangles. We call a triangle semi-split, if it is isomorphic to a coproduct of suspensions of triangles of the form A ⟶^{=} A ⟶ 0 ⟶ Σ(A). Any split triangle is semi-split but the converse is not true.

2.2. Proper Classes Of Triangles. A class of triangles E is closed under base change if for any triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ E and any morphism ε : E ⟶ C

as in (i) of Proposition 2.1, the triangle A ⟶^{f′} G ⟶^{g′} E ⟶^{h′} Σ(A) belongs to E. Dually a class of triangles E is closed under cobase change if for any triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ E and any morphism α : A ⟶ D as in (ii) of Proposition

2.1, the triangle D ⟶^{f′} F ⟶^{g′} C ⟶^{h′} Σ(D) belongs to E. A class of triangles E is closed under suspensions if for any triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ E and for

any i ∈ ℤ, the triangle Σ^{i}(A) ⟶^{(−1)^{i}Σ^{i}f} Σ^{i}(B) ⟶^{(−1)^{i}Σ^{i}g} Σ^{i}(C) ⟶^{(−1)^{i}Σ^{i}h} Σ^{i+1}(A) ∈

E. Finally a class of triangles E is called saturated if the following condition holds: if in the situation of Base Change in Proposition 2.1, the third vertical and the second horizontal triangle is in E, then the triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) is in E. An easy consequence of the Octahedral Axiom is that E is saturated iff in the situation of Cobase Change in Proposition 2.1, if the second vertical and the third horizontal triangle is in E, then the triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) is in E.

The following concept is inspired from the definition of an exact category [61].

Definition 2.2. A full subcategory E ⊆ Diag(C, Σ) is called a proper class of triangles if the following conditions are true:

(i) E is closed under isomorphisms, finite coproducts and Δ_{0} ⊆ E ⊆ Δ. (ii) E is closed under suspensions and is saturated. (iii) E is closed under base and cobase change.

Example 2.3. (1) The class of split triangles Δ_{0} and the class of all triangles Δ in C are proper classes of triangles. If E is a proper class of triangles in C then E^{op} is a proper class of triangles in C^{op}. If {E_{i}; i ∈ I} is a family of proper classes of triangles, then ∩_{i∈I}E_{i} is a proper class of triangles. If {E_{i}; i ∈ I} is an increasing chain, then the class of triangles ∪_{i∈I}E_{i} is proper.

(2) Let U : C ⟶ D be an exact functor of triangulated categories and let E be a proper class of triangles in D. Let F := U^{−1}(E) be the class of triangles T in C such that U(T) ∈ E. Then F is a proper class of triangles in C.

(3) Let F : C ⟶ U be a (co-)homological functor from C to an abelian category U. Then we obtain a proper class of triangles E(F) in C as follows:

• A triangle A ⟶ B ⟶ C ⟶ Σ(A) is in E(F) iff ∀i ∈ ℤ, the induced sequence

0 ⟶ F^{i}(A) ⟶ F^{i}(B) ⟶ F^{i}(C) ⟶ 0 is exact in U, where F^{i} = F Σ^{i}.

(4) If X ⊆ C is a class of objects satisfying Σ(X) = X, then we obtain a proper class of triangles E(X), resp. E^{op}(X), in C as follows:

• A triangle A ⟶ B ⟶ C ⟶ Σ(A) is in E(X), resp. E^{op}(X), iff ∀X ∈ X, the induced sequence 0 ⟶ C(X, A) ⟶ C(X, B) ⟶ C(X, C) ⟶ 0, resp. 0 ⟶ C(C, X) ⟶ C(B, X) ⟶ C(A, X) ⟶ 0, is exact in Ab.

The family of proper classes of triangles in C is easily seen to be a (big) poset with 0 (the class Δ_{0}) and 1 (the class Δ), defining E_{1} ⪯ E_{2} ⇔ E_{1} ⊆ E_{2}.

2.3. Phantom Maps. In general it is difficult to distinguish the morphisms occuring in a triangle by a characteristic property. To solve this problem we proceed as follows.

Definition 2.4. Let E be a class of triangles in C (not necessarily proper) and let A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) be a triangle in E.

(i) The morphism f : A ⟶ B is called an E−proper monic. (ii) The morphism g : B ⟶ C is called an E−proper epic. (iii) The morphism h : C ⟶ Σ(A) is called an E−phantom map. The class of E−phantom maps is denoted by Ph_{E}(C).

The concept of phantom maps has homotopy-theoretic origin and the topological terminology is due to A. Heller (see [52]). For a justification of the definition see Section 3 and [52]. Let Ph_{E}(A, B) be the set of all E−phantom maps from A to B and let Ph_{E}(A, B)^{n} be the set of all maps from A to B which can be written as a composition of n E−phantom maps. We set Ph_{E}^{n}(C) = ⋃_{A,B∈C} Ph_{E}^{n}(A, B).

We recall that an ideal of C is an additive subfunctor of C(−, −). An ideal I of C is called Σ−stable if f ∈ I ⇔ Σ^{n}(f) ∈ I, ∀n ∈ ℤ. If I is a Σ−stable ideal of C, then define a class of triangles E_{I} in C as follows. The triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) ∈ E_{I} if h ∈ I(C, Σ(A)). A Σ−stable ideal I in C is called saturated, if the following condition is true: if f is E_{I}−proper epic and f ∘ σ ∈ I then σ ∈ I. By the Octahedral Axiom this is equivalent to the condition that if g is an E_{I}−proper monic and ρ ∘ g ∈ I then ρ ∈ I. Trivially if E is a proper class of triangles, then Ph_{E}^{n}(C) is a Σ−stable saturated ideal of C, ∀n ≥ 1. Conversely if I is a Σ−stable saturated ideal, then the class of triangles E_{I} is proper and we have the relations: Ph_{E_{I}}(C) = I and E_{Ph_{E}(C)} = E. We collect these observations in the following.

Proposition 2.5. The map Φ : I ↦ E_{I} is a poset isomorphism between the poset of Σ−stable saturated ideals of C and the poset of proper classes of triangles in C, with inverse given by Ψ : E ↦ Ph_{E}(C). E.g. Ψ(Δ) = C(−, −) and Ψ(Δ_{0}) = 0.

It follows that E is a proper class of triangles in C iff Ph_{E}(C) is a Σ−stable saturated ideal of C. This indicates an analogy between the homological theory in C based on proper classes of triangles and the formulation by Butler-Horrocks [18] of relative homology in an abelian category, based on subfunctors of Ext.

2.4. Baer’s Theory. Let E be a proper class of triangles in C and let (T) : A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) be a triangle in E. We call h : C ⟶ Σ(A) the characteristic class of (T), and usually we denote it by ch(T) = h. Let A, C be two objects of C and consider the class E^{∗}(C, A) of all triangles A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) in E. We define a relation in E^{∗}(C, A) as follows. If (T_{i}) : A ⟶^{f_{i}} B_{i} ⟶^{g_{i}} C ⟶^{h_{i}} Σ(A), i = 1, 2, are elements of E^{∗}(C, A), then we define (T_{1}) ∼ (T_{2}) if there exists a morphism of triangles

A ⟶^{f_{1}} B_{1} ⟶^{g_{1}} C ⟶^{h_{1}} Σ(A)
∥↓  ^{β}↓  ^{∥}↓  ^{∥}↓
A ⟶^{f_{2}} B_{2} ⟶^{g_{2}} C ⟶^{h_{2}} Σ(A).

Obviously β is an isomorphism and ∼ is an equivalence relation on the class E^{∗}(C, A). Using base and cobase change, it is easy to see that we can define (as in the case of the classical Baer’s theory in an abelian category) a sum in the class E(C, A) := E^{∗}(C, A)/ ∼ in such a way that E(C, A) becomes an abelian group and E(−, −) : C^{op} × C ⟶ Ab an additive bifunctor. Trivially we have the following.

Corollary 2.6. ch is an isomorphism of bifunctors : E(−, −) ⟶ Ph_{E}(−, Σ−).

3. The Freyd and Steenrod Category of a Triangulated Category

3.1. The Freyd Category. Let C be an additive category. We recall that an additive functor F : C^{op} ⟶ Ab is called finitely presented if there exists an exact sequence C(−, A) ⟶ C(−, B) ⟶ F ⟶ 0. We denote by A(C) or mod(C) the category of finitely presented functors and we call it the Freyd category of C. The Yoneda embedding is denoted by 𝕐 : C ↪ A(C). We define also the Freyd category B(C) to be A(C^{op})^{op}, so B(C) = mod(C^{op})^{op} is the dual of the category of finitely presented covariant functors over C. The corresponding Yoneda embedding is denoted by 𝕐^{op} : C ↪ B(C). The category A(C), resp. B(C), has cokernels, resp. kernels, and the functor 𝕐, resp. 𝕐^{op}, has the universal property that any additive functor F : C ⟶ M where M has cokernels, resp. kernels, admits a unique cokernel, resp. kernel, preserving extension through 𝕐, resp. 𝕐^{op}. We recall that a morphism f : A ⟶ B is a weak kernel of g : B ⟶ C, if the sequence of functors

𝕐(A) ⟶^{𝕐(f)} 𝕐(B) ⟶^{𝕐(g)} 𝕐(C) is exact. The notion of weak cokernel is defined dually. By results of Freyd [29], the category A(C), resp. B(C), is abelian iff any morphism in C has a weak kernel, resp. weak cokernel. In this case 𝕐, resp. 𝕐^{op}, embedds C as a class of projective, resp. injective, objects in A(C), resp. B(C), containing enough projective, resp. injective, objects.

We recall that an abelian category A is called Frobenius if A has enough projectives, enough injectives and the projectives coincide with the injectives. By results of Freyd [29], for any triangulated category C, the category A(C) is Frobenius abelian and the Yoneda embedding 𝕐 : C ↪ A(C) realizes C as a class of projective-injective objects in A(C), containing both enough projectives and injectives. Moreover there exists a unique equivalence 𝔻 : A(C) ⟶^{≈} B(C) such that 𝔻𝕐 = 𝕐^{op}. It follows that any object F in A(C) has a description as a kernel, image or cokernel of a morphism between representable functors. The Freyd category is the universal homological category of C in the sence that 𝕐 is a homological functor and if F : C ⟶ K is a homological functor to an abelian category K, then there exists a unique exact functor F^{∗} : A(C) ⟶ K such that F^{∗}𝕐 = F. Note that if idempotents split in C, then 𝕐 induces an equivalence C ≈ Proj(A(C)) = Inj(A(C)).

3.2. The Steenrod Category. Let C be a triangulated category and let I be a class of morphisms in C. We denote by A(I) the full subcategory of A(C) consisting of all functors of the form Im𝕐(h), for h ∈ I. Similarly let B(I) be the full subcategory of B(C) consisting of all functors of the form Im𝕐^{op}(h). Trivially the suspension Σ of C extends to an automorphism of A(C) and B(C), denoted also by Σ. A full subcategory U of A(C) or B(C), is called Σ−stable if U is closed under Σ.

Theorem 3.1. The following statements are equivalent.

(i) I is a Σ−stable saturated ideal of C. (ii) A(I) is a Σ−stable Serre subcategory of A(C). (iii) B(I) is a Σ−stable Serre subcategory of B(C).

If one of the above is true, then there is an equivalence A(I) ⟶^{≈} B(I) which induces an equivalence A(C)/A(I) ⟶^{≈} B(C)/B(I) in the localizations.

Proof. (ii) ⇒ (i) Let A(C)/A(I) be the quotient category and let Q : A(C) ⟶ A(C)/A(I) be the exact quotient functor. Then the composition S_{I} = Q𝕐 is a homological functor and it is easy to see that the kernel ideal kerS_{I} = I. Then obviously I is a Σ−stable saturated ideal of C.

(i) ⇒ (ii) Assume that I is a Σ−stable saturated ideal of C. Then A(I) is a Σ−stable full subcategory of A(C). Let β : G ⟶ H be an epimorphism in A(C) with G ∈ A(I). Then there exists a morphism μ : B ⟶ K in I such that Im𝕐(μ) = G and a morphism ν : D ⟶ L such that Im𝕐(ν) = H. Let 𝕐(μ) = ρ ∘ κ and 𝕐(ν) = σ ∘ λ be the canonical factorizations of 𝕐(μ) and 𝕐(ν). Since β is epic and 𝕐(D) is projective, there exists ω : 𝕐(D) ⟶ G such that ω ∘ β = σ. Similarly since ρ is epic, there exists a morphism 𝕐(ϵ) : 𝕐(D) ⟶ 𝕐(B) such that 𝕐(ϵ) ∘ ρ = ω. Hence 𝕐(ϵ) ∘ ρ ∘ β = σ. On the other hand since 𝕐(L) is injective, there exists a morphism 𝕐(δ) : 𝕐(K) ⟶ 𝕐(L) such that κ ∘ 𝕐(δ) = β ∘ λ. Then 𝕐(ϵ) ∘ 𝕐(μ) ∘ 𝕐(δ) = 𝕐(ϵ) ∘ ρ ∘ β ∘ λ = σ ∘ λ = 𝕐(ν). Hence ϵ ∘ μ ∘ δ = ν. Since μ ∈ I, it follows that ν ∈ I. Hence H ∈ A(I) and A(I) is closed under quotient objects. The proof that A(I) is closed under subobjects is dual and is left to the reader. It remains to prove that A(I) is closed under extensions. Let 0 ⟶ F_{1} ⟶^{α} F_{2} ⟶^{β} F_{3} ⟶ 0 be a short exact sequence in A(C) with F_{1}, F_{3} ∈ A(I). Let A_{i} ⟶^{f_{i}} B_{i} ⟶^{g_{i}} C_{i} ⟶ Σ(A_{i}) be triangles in C such that F_{i} = Im𝕐(g_{i}) and g_{1}, g_{3} ∈ I, i = 1, 3. Let 𝕐(g_{i}) = ρ_{i} ∘ σ_{i} be the canonical factorizations. Since 𝕐(B_{3}) is projective, there exists ξ : 𝕐(B_{3}) ⟶ F_{2} such that ξ ∘ β = ρ_{3}. Since 𝕐(C_{1}) is injective there exists ω : F_{2} ⟶ 𝕐(C_{1}) such that α ∘ ω = σ_{1}. It is easy to see that F_{2}

is the image of the morphism 𝕐(g_{2}), where g_{2} = ( \begin{matrix}g_{1} & 0 \\ τ & g_{3} \end{matrix} ) : B_{1} ⊕ B_{3} ⟶ C_{1} ⊕ C_{3}

and τ : B_{3} ⟶ C_{1} is the unique morphism such that 𝕐(τ) = ξ∘ω. Hence is suffices to show that g_{2} or equivalently τ is in I. Since 𝕐(f_{3})∘ρ_{3} = 𝕐(f_{3})∘ξ∘β = 0, there exists a morphism κ : 𝕐(A_{3}) ⟶ F_{1} such that 𝕐(f_{3}) ∘ ξ = κ ∘ α. Since 𝕐(A_{3}) is projective, there exists λ : 𝕐(A_{3}) ⟶ 𝕐(B_{1}) such that λ ∘ ρ_{1} = κ. Then 𝕐(f_{3}) ∘ ξ = λ ∘ ρ_{1} ∘ α, hence 𝕐(f_{3}) ∘ ξ ∘ ω = λ ∘ ρ_{1} ∘ α ∘ ω = λ ∘ ρ_{1} ∘ σ_{1} = λ ∘ 𝕐(g_{1}). Since λ is of the form 𝕐(μ), it follows that 𝕐(f_{3}) ∘ ξ ∘ ω = 𝕐(f_{3}) ∘ 𝕐(τ) = 𝕐(μ ∘ g_{1}) ⇒ f_{3} ∘ τ = μ ∘ g_{1}. Since g_{1} is in I, it follows that f_{3} ∘ τ ∈ I. But f_{3} is E_{I}−epic. Since I is saturated, we have τ ∈ I. We conclude that F_{2} ∈ A(I) and A(I) is closed under extensions.

The proof of (i) ⇔ (iii) is similar and is left to the reader. Finally it is easy to see that the equivalence 𝔻 : A(C) ⟶ B(C) defined by 𝔻(Coker𝕐(f)) = Ker𝕐^{op}(f), sends A(I) to B(I), so it induces an equivalence A(C)/A(I) ≈ B(C)/B(I).

Corollary 3.2. The map E ↦ A(Ph_{E}(C)) gives a bijection between proper classes of triangles in C and Σ−stable Serre subcategories of A(C), with inverse the map U ↦ E(F_{U}), where F_{U} is the homological functor C ⟶ A(C) ⟶ A(C)/U.

Definition 3.3. Let E be a proper class of triangles in C. The Steenrod category, resp. dual Steenrod category, of C with respect to E is defined by:

S_{E}(C) := A(C)/A(Ph_{E}(C)), resp. S_{E}^{op}(C) := B(C)/B(Ph_{E}(C))

where Ph_{E}(C) is the Σ−stable saturated ideal of E−phantom maps.

The canonical functor S : C ⟶ S_{E}(C), resp. T : C ⟶ S_{E}^{op}(C), given by the composition of the Yoneda embedding 𝕐 : C ⟶ A(C), resp. 𝕐^{op} : C ⟶ B(C), and the quotient functor A(C) ⟶ A(C)/A(Ph_{E}(C)), resp. B(C) ⟶ B(C)/B(Ph_{E}(C)), is called the projectivization functor, resp. injectivization functor.

We refer to the next Section for the justification of the above definition. The following result summarizes the above observations and explains the terminology for the phantom maps, since these maps are precisely the maps which are invisible in the Steenrod category.

Theorem 3.4. Let E be a proper class of triangles in C. Then the Steenrod category S_{E}(C) is abelian and the projectivization functor S : C ⟶ S_{E}(C) is a homological functor having the property that S(ϕ) = 0, ∀ϕ ∈ Ph_{E}(C). Moreover the pair (S, S_{E}(C)) has the following universal property.

• If H : C ⟶ M is a homological functor to an abelian category M such that H(ϕ) = 0, ∀ϕ ∈ Ph_{E}(C), then there exists a unique exact functor H^{∗} : S_{E}(C) ⟶ M such that H^{∗}S = H.

Proof. It suffices to prove only the universal property. Let H′ : A(C) ⟶ M be the unique exact functor which extends uniquely H through 𝕐. Since H kills E−phantom maps, H′ kills objects of the Serre subcategory A(Ph_{E}(C)), so there exists a unique exact functor H^{∗} : S_{E}(C) ⟶ M such that H^{∗}Q = H′. Then H^{∗}S = H^{∗}Q𝕐 = H′𝕐 = H and H^{∗} is the unique exact functor with this property.

Observe that S_{Δ}(C) = 0 and S_{Δ_{0}}(C) = A(C). By the above Theorem, there exists an equivalence D : S_{E}(C) ⟶^{≈} S_{E}^{op}(C) such that DS = T. We leave to the reader to formulate the above result for the dual Steenrod category S_{E}^{op}(C) and the injectivization functor T.

We regard the Steenrod category as an abelian approximation of C. So it is useful to know when the projectivization functor S : C ⟶ S_{E}(C) is non-trivial.

Definition 3.5. The proper class of triangles E generates C if the projectivization functor S : C ⟶ S_{E}(C) reflects zero objects, i.e. S(A) = 0 ⇒ A = 0.

We recall that the Jacobson radical J ac(C) of an additive category C is the ideal in C defined by J ac(C)(A, B) = {f : A ⟶ B; ∀g : B ⟶ A, the morphism 1_{A} − f ∘ g is invertible}. If F is an additive functor, we denote by ker(F), resp. Ker(F), the ideal of morphisms, resp. the full subcategory of objects, annihilated by F. If I is a Σ−stable ideal of C, let S_{I} : C ⟶ A(C)/A(I) be the canonical functor, i.e. the composition of 𝕐 : C ↪ A(C) and the quotient functor A(C) ⟶ A(C)/A(I).

Lemma 3.6. ker(S_{I}) = I. Moreover I ⊆ J ac(C) iff Ker(S_{I}) = 0.

Proof. The first assertion is trivial. If I ⊆ J ac(C) and S_{I}(A) = 0, then obviously 1_{A} ∈ I ⊆ J ac(C), hence A = 0. Conversely if Ker(S_{I}) = 0 and f : A ⟶ B is in I, let g : B ⟶ A be any morphism and let α := 1_{A} − f ∘ g. Then trivially S_{I}(α) = S_{I}(1_{A}). If A ⟶^{α} A ⟶ C ⟶ Σ(A) is a triangle in C, then S_{I}(C) = 0, hence C = 0. It follows that α is invertible and this implies that f ∈ J ac(C).

Corollary 3.7. E generates C iff Ph_{E}(C) ⊆ J ac(C).

4. Projective Objects, Resolutions and Derived Functors

We fix throughout a proper class of triangles E in the triangulated category C.

4.1. Projective Objects and Global Dimension.

Definition 4.1. An object P ∈ C, resp. I ∈ C, is called E−projective, resp. E−injective, if for any triangle A ⟶ B ⟶ C ⟶ Σ(A) in E, the induced sequence 0 ⟶ C(P, A) ⟶ C(P, B) ⟶ C(P, C) ⟶ 0, resp. 0 ⟶ C(C, I) ⟶ C(B, I) ⟶ C(A, I) ⟶ 0, is exact in Ab.

We denote by P(E) (I(E)) the full subcategory of E−projective (E−injective) objects of C. As an immediate consequence of the above definition we have that the categories P(E) and I(E) are full, additive, closed under isomorphisms, direct summands and they are Σ−stable, i.e. Σ(P(E)) = P(E) and Σ(I(E)) = I(E).

We say that C has enough E−projectives if for any object A ∈ C there exists a triangle K ⟶ P ⟶ A ⟶^{h} Σ(K) in E with P ∈ P(E). Observe that in this case h is a weakly universal E−phantom map out of A in the sense that any E−phantom map h′ : A ⟶ B factors through h in a not necessarily unique way. Dually one defines when C has enough E−injectives. We study only the case of E−projectives since the study of the E−injectives is dual. However we use freely the dual results when it is necessary. Note that an equivalent formulation of the setup of a proper class of triangles in C such that C has enough E−projectives is that of a projective class of morphisms in C in the sense of Eilenberg-Moore [26], see [19].

We recall that a full subcategory X ⊆ C is called contravariantly finite [6] if for any A ∈ C there exists a morphism f : X ⟶ A with X ∈ X, such that any morphism g : X′ ⟶ A with X′ ∈ X factors through f. The following result is a direct consequence of the definitions and its proof is left to the reader.

Lemma 4.2. (i) ∀P ∈ P(E) : Ph_{E}(P, −) = 0 and ∀I ∈ I(E) : Ph_{E}(−, I) = 0. (ii) If C has enough E−projectives, then E generates C iff A ∈ C and C(P, A) =

0, ∀P ∈ P(E), implies A = 0. (iii) C has enough E−projectives iff P(E) is contravariantly finite. In this case:

E = E(P(E)), i.e. a triangle D ⟶ F ⟶ C ⟶ Σ(D) is in E iff ∀P ∈ P(E) the induced sequence 0 ⟶ C(P, D) ⟶ C(P, F) ⟶ C(P, C) ⟶ 0 is exact. Moreover a morphism f : A ⟶ B is E−phantom iff C(P, f) = 0, ∀P ∈ P(E). Further ∀A ∈ C: A ∈ P(E) iff Ph_{E}(A, −) = 0.

(iv) Let P be a full additive contravariantly finite subcategory of C, closed under direct summands such that Σ(P) = P. Then P = P(E(P)).

Corollary 4.3. The maps P ↦ E(P), E ↦ P(E) are inverse bijections between contravariantly finite Σ−stable additive subcategories of C closed under direct summands and proper classes of triangles E in C such that C has enough E−projectives.

Proposition 4.4. [Schanuel′s Lemma] If K_{i} ⟶^{f_{i}} P_{i} ⟶^{g_{i}} A ⟶^{h_{i}} Σ(K_{i}) are triangles in E with P_{i} ∈ P(E), i = 1, 2, then: K_{1} ⊕ P_{2} ≅ K_{2} ⊕ P_{1}.

Proof. Consider the Octahedral diagram induced from the composition g_{1} ∘ h_{2}:

P_{1} ⟶^{g_{1}} A ⟶^{h_{1}} Σ(K_{1}) ⟶^{−Σ(f_{1})} Σ(P_{1})
∥↓  ^{h_{2}}↓  ↓  ^{∥}↓
P_{1} ⟶^{g_{1}∘h_{2}} Σ(K_{2}) ⟶ X ⟶ Σ(P_{1})
g_{1}↓  ^{∥}↓  ↓  ↓
A ⟶^{h_{2}} Σ(K_{2}) ⟶^{−Σ(f_{2})} Σ(P_{2}) ⟶^{−Σ(g_{2})} Σ(A)
↓  ^{0}↓ ^{−Σ(g_{2})∘Σ(h_{1})}↓  ↓
0 ⟶ Σ^{2}(K_{1}) ⟶^{=} Σ^{2}(K_{1}) ⟶ 0.

Since h_{1}, h_{2} are E−phantoms and P_{1}, Σ(P_{2}) are E−projectives, we have: g_{1} ∘h_{2} = 0 and g_{2} ∘ h_{1} = 0. Hence the second horizontal and the third vertical triangles are split. So X ≅ Σ(K_{2}) ⊕ Σ(P_{1}) ≅ Σ(K_{1}) ⊕ Σ(P_{2}) ⇒ K_{2} ⊕ P_{1} ≅ K_{1} ⊕ P_{2}.

If K ⟶ P ⟶ A ⟶ Σ(K) is in E with P in P(E), then we call the object K, a first E−syzygy of A. An n^{th} E−syzygy of A is defined as usual by induction. By Schanuel’s Lemma any two E−syzygies of A are isomorphic modulo E−projectives.

We define inductively the E−projective dimension E−p.dA of an object A ∈ C as follows. If A ∈ P(E) then E−p.dA = 0. Next if E−p.dA > 0 define E−p.dA ≤ n if there exists a triangle K ⟶ P ⟶ A ⟶ Σ(K) in E with P ∈ P(E) and E−p.dK ≤ n−1. Finally define E−p.dA = n if E−p.dA ≤ n and E−p.dA ≰ n−1. Of course we set E−p.dA = ∞, if E−p.dA ≠ n, ∀n ≥ 0. The E−global dimension E−gl.dimC of C is defined by E−gl.dimC = sup{E−p.dA; A ∈ C}.

Example 4.5. ∀C ≠ 0 : Δ−p.dC = ∞, so if 0 ≠ C : Δ−gl.dimC = ∞. On the other extreme Δ_{0}−gl.dimC = 0.

The following is proved using standard arguments.

Proposition 4.6. (i) ∀A, B ∈ C : E−p.dA ⊕ B = max{E−p.dA, E−p.dB}. (ii) ∀A ∈ C, ∀n ∈ ℤ : E−p.dA = E−p.dΣ^{n}(A). (iii) Let A ⟶ P ⟶ B ⟶ Σ(A) be a triangle in E with P ∈ P(E). Then either

B ∈ P(E) or else E−p.dA = E−p.dB − 1.

4.2. Resolutions and Derived Functors.

Definition 4.7. An E−exact complex X^{•} ⟶ A over A ∈ C is a diagram ⋯ ⟶

X^{n+1} ⟶^{d^{n+1}} X^{n} ⟶ ⋯ ⟶ X^{1} ⟶^{d^{1}} X^{0} ⟶^{d^{0}} A ⟶ 0 in C, such that ∀n ≥ 0:

(i) There are triangles K^{n+1} ⟶^{g^{n}} X^{n} ⟶^{f^{n}} K^{n} ⟶^{h^{n}} Σ(K^{n+1}) in E, where K^{0} := A. (ii) The differential d^{n} = f^{n} ∘ g^{n−1}, ∀n ≥ 1 and d^{0} = f^{0}.

An E−projective resolution of A ∈ C is an E−exact complex P^{•} ⟶ A as above such that P^{n} ∈ P(E), ∀n ≥ 0.

The next result gives a useful characterization of an E−projective resolution.

Proposition 4.8. Assume that C has enough E−projectives and consider a complex

P^{•} : ⋯ ⟶ P^{1} ⟶^{d^{1}} P^{0} ⟶^{d^{0}} A ⟶ 0 in C with P^{n} ∈ P(E), ∀n ≥ 0. Then P^{•} is an

E−projective resolution of A iff the induced complex C(Q, P^{•}) is exact, ∀Q ∈ P(E).

Proof. Assume that the induced complex C(Q, P^{•}) is exact in Ab, ∀Q ∈ P(E). Set

f^{0} := d^{0} and let (T_{0}) : K^{1} ⟶^{g^{0}} P^{0} ⟶^{f^{0}} A ⟶^{h^{0}} Σ(K^{1}) be a triangle in C. Since C(Q, f^{0}) is epic, we have C(Q, h^{0}) = 0, ∀Q ∈ P(E). By Lemma 4.2 it follows that h^{0} is E−phantom, hence (T_{0}) is in E. Since d^{1} ∘ f^{0} = 0, there exists a morphism f^{1} : P^{1} ⟶ K^{1} such that f^{1} ∘ g^{0} = d^{1}. It follows easily that C(Q, f^{1}) is epic. Let

(T_{1}) : K^{2} ⟶^{g^{1}} P^{1} ⟶^{f^{1}} K^{1} ⟶^{h^{1}} Σ(K^{2}) be a triangle in C. As above we have (T_{1}) is in E. Since d^{2} ∘ d^{1} = d^{2} ∘ f^{1} ∘ g^{0} = 0, we have C(P(E), d^{2} ∘ f^{1}) ∘ C(P(E), g^{0}) = 0. Since C(P(E), g^{0}) is monic, we have C(P(E), d^{2} ∘ f^{1}) = 0, hence by Lemma 4.2, d^{2} ∘ f^{1} is E−phantom. Since P^{2} ∈ P(E), by Lemma 4.2, we have d^{2} ∘ f^{1} = 0. So there exists a morphism f^{2} : P^{2} ⟶ K^{2} such that d^{2} = f^{2} ∘ g^{1}. Continuing inductively in this way we see that P^{•} is an E−projective resolution of A. The converse is trivial.

Corollary 4.9. Let 0 ⟶ P_{i}^{n} ⟶ P_{i}^{n−1} ⟶ ⋯ ⟶ P_{i}^{0} ⟶ A_{i} ⟶ 0 be E−projective resolutions of A_{i}, i = 1, 2. If A_{1} ≅ A_{2}, then P_{1}^{0}⊕P_{2}^{1}⊕P_{1}^{2}⊕⋯ ≅ P_{2}^{0}⊕P_{1}^{1}⊕P_{2}^{2}⊕· · ·.

From now on we assume that C has enough E−projectives. For any object A ∈ C we fix an E−projective resolution of A:

P_{A}^{•} ⋯ ⟶ P_{A}^{n+1} ⟶^{ε_{A}^{n+1}} P_{A}^{n} ⟶ ⋯ ⟶ P_{A}^{1} ⟶^{ε_{A}^{1}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶ 0

which by definition, is the “Yoneda composition” of triangles

T_{A}^{n} K_{A}^{n+1} ⟶^{g_{A}^{n}} P_{A}^{n} ⟶^{f_{A}^{n}} K_{A}^{n} ⟶^{h_{A}^{n}} Σ(K_{A}^{n+1}) ∈ E

with P_{A}^{n} ∈ P(E), ∀n ≥ 0 and where K_{A}^{0} := A. Using standard arguments from relative homological algebra, one can prove a version of the Comparison Theorem [68] for E−projective resolutions, the obvious formulation of which is left to the reader. It follows that any two E−projective resolutions of an object are homotopy equivalent. Using Schanuel’s Lemma and the above observations we have the following consequence.

Corollary 4.10. E−p.dA ≤ n iff there exists an E−projective resolution of A of the form: 0 ⟶ P^{n} ⟶ ⋯ ⟶ P^{1} ⟶ P^{0} ⟶ A ⟶ 0.

Proposition 4.11. [Horseshoe Lemma] Let (T) : A ⟶^{α} B ⟶^{β} C ⟶^{γ} Σ(A) be a triangle in E. Then there are E−projective resolutions P^{•}(A), P^{•}(B) and P^{•}(C) of A, B and C respectively, and a commutative diagram

P^{•}(A) ⟶^{p^{•}} P^{•}(B) ⟶^{q^{•}} P^{•}(C) ⟶^{0} Σ(P^{•}(A))
↓  ↓  ↓  ↓
A ⟶^{α} B ⟶^{β} C ⟶^{γ} Σ(A)

such that P^{n}(A) ⟶^{p^{n}} P^{n}(B) ⟶^{q^{n}} P^{n}(C) ⟶^{0} Σ(P^{n}(A)) is a split triangle, ∀n ≥ 0. Such a diagram is called an E−projective resolution of the triangle (T).

Proof. Let f_{A}^{0} : P_{A}^{0} ⟶ A and f_{C}^{0} : P_{C}^{0} ⟶ C be E−proper epics with P_{A}^{0}, P_{C}^{0} ∈ P(E). Since γ is E−phantom, f_{C}^{0} ∘ γ = 0. Using that Σ is an automorphism and a result of Verdier (see p. 378 in [71]), the commutative square on the top left corner below is embedded in a diagram:

P_{C}^{0}  ⟶^{0} Σ(P_{A}^{0}) ⟶^{−Σ(p)} Σ(P_{B}^{0}) ⟶^{Σ(q)} Σ(P_{C}^{0})
f_{C}^{0} ↓  ^{−Σ(f_{A}^{0})}↓  ^{−Σ(f_{B}^{0})}↓  ^{Σ(f_{C}^{0})}↓
C ⟶^{γ} Σ(A) ⟶^{−Σ(α)} Σ(B) ⟶^{−Σ(β)} Σ(C)
h_{C}^{0} ↓  ^{−Σ(h_{A}^{0})}↓  ^{−Σ(h_{B}^{0})}↓  ^{Σ(h_{C}^{0})}↓
Σ(K_{C}^{1}) ⟶^{−Σ(ϕ)} Σ^{2}(K_{A}^{1}) ⟶^{Σ^{2}(ψ)} Σ^{2}(K_{B}^{1}) ⟶^{Σ^{2}(ω)} Σ^{2}(K_{C}^{1})
−Σ(g_{C}^{0})↓  ^{Σ^{2}(g_{A}^{0})}↓  ^{Σ^{2}(g_{B}^{0})}↓  ^{−Σ^{2}(g_{C}^{0})}↓
Σ(P_{C}^{0}) ⟶^{0} Σ^{2}(P_{A}^{0}) ⟶^{−Σ^{2}(p)} Σ^{2}(P_{B}^{0}) ⟶^{Σ^{2}(q)} Σ^{2}(P_{C}^{0})

which is commutative except the lower right square which anticommutes and where the rows and columnns are triangles. Then we have the following commutative diagram in which the first three vertical and horizontal diagrams are triangles:

K_{A}^{1} ⟶^{−ψ} K_{B}^{1} ⟶^{ω} K_{C}^{1} ⟶^{−ϕ} Σ(K_{A}^{1})
g_{A}^{0}↓  ^{g_{B}^{0}} ↓  ^{g_{C}^{0}} ↓  ^{Σ(g_{A}^{0})}↓
P_{A}^{0}  ⟶^{p} P_{B}^{0}  ⟶^{q} P_{C}^{0}  ⟶^{0} Σ(P_{A}^{0})
f_{A}^{0}↓  ^{f_{B}^{0}} ↓  ^{f_{C}^{0}} ↓  ^{Σ(f_{A}^{0})}↓
A ⟶^{α} B ⟶^{β} C ⟶^{γ} Σ(A)
h_{A}^{0}↓  ^{h_{B}^{0}} ↓  ^{h_{C}^{0}} ↓  ^{Σ(h_{A}^{0})}↓
Σ(K_{A}^{1}) ⟶^{−Σ(ψ)} Σ(K_{B}^{1}) ⟶^{Σ(ω)} Σ(K_{C}^{1}) ⟶^{−Σ(ϕ)} Σ(K_{A}^{1}).

Since the second horizontal triangle is split, we have that P_{B}^{0} is in P(E). Applying to the above diagram the homological functor C(P, −), ∀P ∈ P(E), a simple diagram chasing argument shows that 0 ⟶ C(P, K_{A}^{1}) ⟶ C(P, K_{B}^{1}) ⟶ C(P, K_{C}^{1}) ⟶ 0 is exact. By Lemma 4.2, the first horizontal triangle is in E. Similarly the second vertical triangle is in E. Inductively the above procedure completes the proof.

Let F : C ⟶ A and G : C^{op} ⟶ A be additive functors, where A is an abelian category. Then we define the E−derived functors

L_{n}^{E}F : C ⟶ A and R_{E}^{n}G : C^{op} ⟶ A, n ≥ 0

of F and G with respect to E, as follows. Let P^{•} ⟶ C be an E−projective resolution of C ∈ C. Then define L_{n}^{E}F (C) to be the n^{th}−homology of the induced complex F (P^{•}) and R_{E}^{n}G(C) to be the n^{th}−cohomology of the induced complex G(P^{•}). In particular ∀C ∈ C, we define the E−extension functors Ext_{E}^{n}(−, C) by:

Ext_{E}^{n}(−, C) := R_{E}^{n}C(−, C) : C^{op} ⟶ Ab, ∀n ≥ 0.

By the Comparison Theorem the above E−derived functors are well defined.

Corollary 4.12. Let A ⟶ B ⟶ C ⟶ Σ(A) be in E. If F : C ⟶ A and G : C^{op} ⟶ A are additive functors where A is abelian, then we have exact sequences:

⋯ ⟶ L_{1}^{E}F (C) ⟶ L_{0}^{E}F (A) ⟶ L_{0}^{E}F (B) ⟶ L_{0}^{E}F (C) ⟶ 0
0 ⟶ R_{E}^{0}G(C) ⟶ R_{E}^{0}G(B) ⟶ R_{E}^{0}G(A) ⟶ R_{E}^{1}G(C) ⟶ ⋯

In particular for any X ∈ C we have a long exact sequence

0 ⟶ Ext_{E}^{0}(C, X) ⟶ Ext_{E}^{0}(B, X) ⟶ Ext_{E}^{0}(A, X) ⟶ Ext_{E}^{1}(C, X) ⟶ ⋯

Proof. Applying F and G to the E−projective resolution of the triangle as in Proposition 4.11 and taking (co)homology, we get the desired exact sequences.

The easy proof of the following is left to the reader.

Corollary 4.13. Let F : C ⟶ A, resp. G : C^{op} ⟶ A, be a homological, resp. cohomological, functor where A is abelian. Then the natural morphism L_{0}^{E}F ⟶ F, resp. G ⟶ R_{E}^{0}G, is an isomorphism iff F, resp. G, kills E−phantom maps.

4.3. Projectively Generating Proper Classes.

Definition 4.14. A proper class of triangles E in C projectively generates C, if E generates C and C has enough E−projectives. A full subcategory G ⊆ C is called a generating subcategory of C, if G is Σ−stable and ∀A ∈ C: C(G, A) = 0 ⇒ A = 0.

By Lemma 4.2, we have the following.

Corollary 4.15. The map E ⟼ P(E) is a bijection between projectively generating proper classes of triangles in C and contravariantly finite generating subcategories of C closed under direct summands. The inverse is given by P ⟼ E(P).

Remark 4.16. Obviously C has enough Δ−projectives and P(Δ) = 0. Conversely if C has enough E−projectives and P(E) = 0, then E = E(P(E)) = E(0) = Δ. If C ≠ 0, then Δ is not projectively generating. Similarly C has enough Δ_{0}−projectives and P(Δ_{0}) = C. Conversely if C has enough E−projectives and P(E) = C, then E = E(P(E)) = E(C) = Δ_{0}. Δ_{0} is always projectively generating.

The next result, proved independently by J. D. Christensen [19], shows that if E projectively generates C, then the E−projective dimension is controlled by the vanishing of the E−extension functors.

Proposition 4.17. If E projectively generates C, then ∀A ∈ C, ∀n ≥ 0:

E−p.dA ≤ n ⇔ Ext_{E}^{n+1}(A, −) = 0.

Proof. It suffices to prove the (⇐) direction. Consider the triangles T_{A}^{n} and the resolution P_{A}^{•} after Corollary 4.9. Since the cohomology of the short complex C(P_{A}^{n}, −) ⟶ C(P_{A}^{n+1}, −) ⟶ C(P_{A}^{n+2}, −) is Ext_{E}^{n+1}(A, −) = 0, the morphism f_{A}^{n+1} : P_{A}^{n+1} ⟶ K_{A}^{n+1} factors through ε_{A}^{n+1}. Hence ε_{A}^{n+1} ∘ α = f_{A}^{n+1} ∘ g_{A}^{n} ∘ α = f_{A}^{n+1}, for some α : P_{A}^{n} ⟶ K_{A}^{n+1}. Applying the functor C(P, −), ∀P ∈ P(E), to this relation and using that C(P, f_{A}^{n+1}) is epic, we have that C(P, g_{A}^{n}) ∘ C(P, α) = 1_{C(P,K_{A}^{n+1})}, so C(P, (α, f_{A}^{n})) : C(P, P_{A}^{n}) ≅ C(P, K_{A}^{n+1}⊕K_{A}^{n}). Since E generates C, P_{A}^{n} ≅ K_{A}^{n+1}⊕K_{A}^{n}. It follows that K_{A}^{n} ∈ P(E) or equivalently E−p.dA ≤ n.

4.4. The Cartan Morphism. Assume that C has split idempotents and is skeletally small, i.e. the collection of isoclasses of objects of C, denoted henceforth by Iso(C), is a set. The Grothendieck group K_{0}(C, E) of C with respect to E is defined as the quotient of the free abelian group on Iso(C), modulo the subgroup generated by all elements (A) − (B) + (C), where A ⟶ B ⟶ C ⟶ Σ(A) is a triangle in E. If E = Δ, then K_{0}(C, Δ) is the usual group defined by Grothendieck. If E = Δ_{0}, then K_{0}(C, Δ_{0}) is the group K_{0}(C, ⊕) of the monoidal category (C, ⊕). Observe that we have canonical epimorphisms K_{0}(C, Δ_{0}) ⟶ K_{0}(C, E) and K_{0}(C, E) ⟶ K_{0}(C, Δ). In case C has enough E−projectives, then we define the Cartan morphism c_{E} : K_{0}(P(E), ⊕) ⟶ K_{0}(C, E), by c_{E}([P]) = [P].

Proposition 4.18. If ∀C ∈ C : E−p.dC < ∞, then c_{E} is an isomorphism.

Proof. We construct an inverse of c_{E} as follows. Let 0 ⟶ P_{n} ⟶ ⋯ ⟶ P^{0} ⟶ A ⟶ 0 be an E-projective resolution of A ∈ C. Using Schanuel’s and Horseshoe Lemma it follows directly that the assignement [A] ⟶ ∑_{i=0}^{n}(−1)^{i}[P^{i}] is a well defined morphism d_{E} : K_{0}(C, E) ⟶ K_{0}(P(E), ⊕). Clearly d_{E} is the inverse of c_{E}.

4.5. The Steenrod Categories. Let F : C ⟶ K be a homological functor, where K is abelian. Let E(F) be the proper class of triangles in C induced by F. Following R. Street [68], an object P ∈ C is called F −projective if F (P) is projective in K and ∀A ∈ C, the canonical map C(P, A) ⟶ K[F (P), F (A)] is an isomorphism. Let P(F) be the full subcategory of F −projectives. C has enough F −projectives if ∀A ∈ C there exists a triangle K ⟶ P ⟶ A ⟶ Σ(K) ∈ E(F) with P ∈ P(F). It is trivial to see that if C has enough F −projectives, then C has enough E(F)−projectives and P(E(F)) = P(F). The next result shows that if C has enough E−projectives for a proper class of triangles E, then E = E(S) and P(E(S)) = P(S), where S is the projectivization functor. It follows that in this case, our formulation of relative homology in C is equivalent to the theory developed by R. Street in [68].

Proposition 4.19. ∀P ∈ P(E), S(P) ∈ ProjS_{E}(C) and ∀A ∈ C, the canonical map S_{P,A} : C(P, A) ⟶ S_{E}(C)[S(P), S(A)] is an isomorphism. Hence S induces a full embedding P(E) ↪ ProjS_{E}(C). If C has enough E−projectives, then:

(i) S_{E}(C) is equivalent to A(P(E)) and S is isomorphic to the restriction A ⟼ C(−, A)|_{P(E)}. In particular S_{E}(C) has enough projectives and E = E(S).

(ii) If idempotents split in P(E), then S : P(E) ⟶^{≈} ProjS_{E}(C). (iii) A complex P^{•} ⟶ A over A is an E−projective resolution of A iff S(P^{•}) ⟶

S(A) is a projective resolution of S(A) in S_{E}(C).

Proof. Since the kernel ideal of the functor S is Ph_{E}(C), it follows that S_{P,A} is injective, since Ph_{E}(P, A) = 0, ∀P ∈ P(E). Now let α : S(P) ⟶ S(A) be a morphism in S_{E}(C). Since S = Q𝕐, where Q : A(C) ⟶ A(C)/A(Ph_{E}(C)) is the quotient functor, α is represented by a fraction 𝕐(P) ⟵^{s} F ⟶^{f} 𝕐(A), where s has kernel and cokernel in A(Ph_{E}(C)). Let ξ : 𝕐(P) ⟶ G be the cokernel of s. Then G is the image of a morphism 𝕐(h) : 𝕐(C) ⟶ 𝕐(D), where h is an E−phantom map; let 𝕐(h) = ϵ ∘ ι be the canonical factorization. Then there exists a morphism 𝕐(ϕ) : 𝕐(P) ⟶ 𝕐(C) such that 𝕐(ϕ) ∘ ϵ = ξ. Then ξ ∘ ι = 𝕐(ϕ) ∘ ϵ ∘ ι = 𝕐(ϕ ∘ h). Since h is E−phantom and P is E−projective, ϕ ∘ h = 0. Hence ξ = 0 and s is split epic. Let s′ : 𝕐(P) ⟶ F be a morphism such that s′ ∘ s = 1_{𝕐(P)}. Then s′ ∘ f is of the form 𝕐(α′) for α′ : P ⟶ A. Trivially S(α′) = Q(s′ ∘ f) = α. Hence S_{P,A} is surjective. The proof that S(P) is projective is similar and is left to the reader.

(i) One can prove this by using the universal property of S_{E}(C). Here is a quick proof. Since C has enough E−projectives, the subcategory P(E) is contravariantly finite in C. By [13], there exists a short exact sequence of abelian categories 0 ⟶ A(C/P(E)) ⟶ A(C) ⟶^{R} A(P(E)) ⟶ 0, where C/P(E) is the stable category of C modulo the full subcategory P(E) and R is the restriction functor F ↦ F |_{P(E)}. It is easy to see that A(C/P(E)) is equivalent to A(Ph_{E}(C)) so by Definition 3.3, S_{E}(C) is equivalent to A(P(E)). In particular S_{E}(C) has enough projectives. Part (ii) follows from [28] and part (iii) follows from (i).

The above result explains the terminology for the projectivization functor. The next example explains the terminology for the Steenrod category.

Example 4.20. Let C be the stable homotopy category of spectra [51] and consider the full subcategory X = {Σ^{n}(K(ℤ/(p))); n ∈ ℤ}, where K(ℤ/(p)) is the Eilenberg-MacLane spectrum corresponding to ℤ/(p). Then the Steenrod category S_{E^{op}(X)}(C) of C with respect to the proper class of triangles E^{op}(X), is equivalent to the category of modules over the mod-p Steenrod algebra, see [74] for details.

The description of the Steenrod category in Proposition 4.19, allows us to give a formula for the derived functors of an additive functor F : C ⟶ M, where M is abelian. Let G := F |_{P(E)} : P(E) ⟶ M. Since S_{E}(C) = A(P(E)), G has a unique right exact extension G^{∗} : S_{E}(C) ⟶ M through the Steenrod category. The easy proof of the following and its contravariant analogue, is left to the reader.

Corollary 4.21. L_{n}^{E}F ≅ (L_{n}G^{∗})S, ∀n ≥ 0.

Corollary 4.22. (i) For any B ∈ C we have natural isomorphisms

Ext_{E}^{n}(−, B) ≅ Ext_{S_{E}(C)}^{n}[S(−), S(B)], ∀n ≥ 0.

The natural map μ_{−,B} : C(−, B) ⟶ R_{E}^{0}C(−, B) = Ext_{E}^{0}(−, B) coincides with the map S_{−,B} : C(−, B) ⟶ S_{E}(C)[S(−), S(B)]. Hence Kerμ_{−,B} is the right ideal Ph_{E}(−, B).

(ii) ∀A ∈ C : E−p.dA ≥ p.dS(A); if E projectively generates C, then E−p.dA = p.dS(A). In particular E−gl.dimC ≤ gl.dimS_{E}(C).

(iii) Assume that C is skeletally small with split idempotents. If C is E−projectively generated and gl.dimS_{E}(C) < ∞, then the projectivization functor S : C ⟶ S_{E}(C) induces an isomorphism S_{∗} : K_{0}(C, E) ⟶^{≅} K_{0}(S_{E}(C)), and an isomorphism K_{∗}(A(C)) ≅ K_{∗}(S_{E}(C))⊕K_{∗}(A(C/P(E))) in Quillen’s higher K−theory.

Proof. The assertions (i), (ii) follow trivially from Proposition 4.19. The first part of (iii) follows from Proposition 4.18 and the commutativity of the diagram:

K_{0}(P(E), ⊕) ⟶^{≅^{S}} K_{0}(Proj(S_{E}(C)), ⊕)
c_{E} ↓  ^{c_{S_{E} (C)}}↓
K_{0}(C, E) ⟶^{S_{∗}}  K_{0}(S_{E}(C)).

The second part of (iii), follows from a result of Auslander-Reiten [5].

The dual of Proposition 4.19 is also true. We state only the following.

Proposition 4.23. If C has enough E−injectives then the dual Steenrod category of C with respect to E is equivalent to B(I(E)) = mod(I(E)^{op})^{op} and the injectivization functor T : C ⟶ S_{E}^{op}(C) is isomorphic to the restriction A ↦ C(A, −)|_{I(E)}. If I(E) has split idempotents, then T induces an equivalence: I(E) ≈ InjS_{E}^{op}(C).

If X ⊆ C is a class of objects, then add(X) denotes the full subcategory of C consisting of all direct summands of finite coproducts of objects of X.

An E−injective envelope of A ∈ C is an E−proper monic μ : A ⟶ E with E ∈ I(E) such that if μ∘α = μ, then α is an automorphism of E. Without assuming the existence of enough E−injectives we have the following.

Theorem 4.24. The projectivization functor S induces a full embedding

S : I(E) ↪ InjS_{E}(C) and InjS_{E}(C) ⊆ add(ImS).

In particular S : I(E) ⟶^{≈} InjS_{E}(C) if ImS is closed under direct summands. If this is the case, then C has enough E−injectives iff S_{E}(C) has enough injectives. In particular if S_{E}(C) has injective envelopes, then C has E−injective envelopes.

Proof. Let P_{A}^{1} ⟶^{ϵ_{A}^{1}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶ 0 be the start of an E−projective resolution of A ∈ C. If I is E−injective, then obviously the map S_{A,I} : C(A, I) ⟶ S_{E}(C)[S(A), S(I)] is injective. Let μ : S(A) ⟶ S(I) be any morphism. Consider the morphism S(f_{A}^{0}) ∘ μ : S(P_{A}^{0}) ⟶ S(I). Since P_{A}^{0} is E−projective, S(f_{A}^{0}) ∘ μ = S(α), for some morphism α : P_{A}^{0} ⟶ I. Then S(ϵ_{A}^{1})∘S(α) = S(ϵ_{A}^{1})∘S(f_{A}^{0})∘μ = 0, so ϵ_{A}^{1}∘α : P_{A}^{1} ⟶ I is E−phantom. Since P_{A}^{1} is E−projective, ϵ^{1} ∘ α = 0. Since ϵ_{A}^{1} = f_{A}^{1} ∘ g_{A}^{0}, we have that the morphism g_{A}^{0} ∘ α factors through h_{A}^{1} : K_{A}^{1} ⟶ Σ(K_{A}^{2}). So g_{A}^{0} ∘ α = h_{A}^{1} ∘ β, for some morphism β : Σ(K_{A}^{2}) ⟶ I. But since h_{A}^{1} is E−phantom, it follows that g_{A}^{0} ∘ α : K_{A}^{1} ⟶ I is E−phantom. Since I is E−injective, g_{A}^{0} ∘ α = 0. Hence there exists γ : A ⟶ I such that: f_{A}^{0} ∘ γ = β. Then S(f_{A}^{0}) ∘ S(γ) = S(β). Since S(f_{A}^{0}) is epic, μ = S(γ), hence S_{A,I} is an isomorphism. It remains to show that

S(I) ∈ InjS_{E}(C). Let F ∈ S_{E}(C) and let S(P_{1}) ⟶^{S(β)} S(P_{0}) ⟶ F ⟶ 0 be the start of a projective resolution of F. Let (T) : A ⟶^{α} P_{1} ⟶^{β} P_{0} ⟶ Σ(A) be a triangle in C and let K ⟶ P_{2} ⟶^{f} A ⟶ Σ(K) be an E−projective presentation of

A in C. Since the sequence S(A) ⟶^{S(α)} S(P_{1}) ⟶^{S(β)} S(P_{0}) ⟶ F ⟶ 0 is exact, if G = Ker(S(β)), then we have an epimorphism S(f) ∘ coimS(β) : S(P_{2}) ⟶ G.

Then S(P_{2}) ⟶^{S(f∘α)} S(P_{1}) ⟶ S(P_{0}) ⟶ F ⟶ 0 is part of a projective resolution of F. Applying S_{E}(C)[−, S(I)] to this sequence and using that the map S_{C,I} is an isomorphism ∀C ∈ C, it follows easily that the resulting sequence is exact. This means that Ext_{S_{E}(C)}^{1}[F, S(I)] = 0. Hence S(I) is injective.

From the triangle (T) it follows that we have an inclusion ξ : F ↪ S(Σ(A)). If F is injective then F is a direct summand of S(Σ(A)), hence if in addition ImS is closed under direct summands, then F ≅ S(I) for some (obviously E−injective) I. If C has enough E−injectives, let Σ(A) ⟶^{τ} I ⟶ B ⟶ Σ^{2}(A) be a triangle in E with I ∈ I(E). Then ξ ∘ S(τ) : F ⟶ S(I) is monic and S(I) is injective, hence S_{E}(C) has enough injectives. Conversely if S_{E}(C) has enough injectives, ∀C ∈ C, let ζ : S(C) ↪ E be monic with E ∈ InjS_{E}(C). Then E = S(I) for some I ∈ I(E) and ζ = S(ν) for a morphism ν : C ⟶ I in C. Since the triangle C ⟶^{ν} I ⟶ B ⟶ Σ(C) is in E, it follows that C has enough E−injectives. Clearly if S(μ) : S(C) ⟶ S(I) is an injective envelope, then μ : C ⟶ I is an E−injective envelope.

4.6. Semisimple and Hereditary Categories. The following characterization of E−phantomless categories follows easily from our previous results.

Theorem 4.25. The following are equivalent:

(i) E−gl.dimC = 0. (ii) C = P(E). (iii) E = Δ_{0}. (iv) Ph_{E}(C) = 0. (vi) The projectivization functor S : C ⟶ S_{E}(C) is faithful.

(v) The projectivization functor S induces an equivalence S_{E}(C) ⟶^{≈} A(C).

Lemma 4.26. Let G : C ⟶ M be a homological and F : C^{op} ⟶ M be a cohomological functor, where M is abelian. If E−p.dA ≤ 1, there are exact sequences:

0 ⟶ L_{0}^{E}G(A) ⟶ G(A) ⟶ L_{1}^{E}G(Σ(A)) ⟶ 0,
0 ⟶ R_{E}^{1}F (Σ(A)) ⟶ F (A) ⟶ R_{E}^{0}F (A) ⟶ 0.

Proof. Since E−p.dA ≤ 1, an E−projective resolution of A is simply a triangle

P_{A}^{1} ⟶^{g_{A}^{0}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶^{h_{A}^{0}} Σ(P_{A}^{1}) in E, with P_{A}^{1}, P_{A}^{0} ∈ P(E). From the exact sequence:

⋯ ⟶ G(P_{A}^{1}) ⟶^{G(g_{A}^{0})} G(P_{A}^{0}) ⟶ G(A) ⟶ GΣ(P_{A}^{1}) ⟶^{GΣ(g_{A}^{0})} GΣ(P_{A}^{0}) ⟶ ⋯, by

construction we have Coker(G(g_{A}^{0})) = L_{0}^{E}G(A) and Ker(GΣ(g_{A}^{0})) = L_{1}^{E}GΣ(A) and the result follows. The claim for F is similar.

The next result computes the ideal Ph_{E}(C) in case E−gl.dimC ≤ 1 and can be regarded as a Universal Coefficient Theorem.

Theorem 4.27. Let A ∈ C with E−p.dA ≤ 1. Then we have the following. (i) For any B ∈ C there exists a short exact sequence

0 ⟶ Ext_{E}^{1}(Σ(A), B) ⟶ C(A, B) ⟶^{μ_{A,B}} Ext_{E}^{0}(A, B) ⟶ 0.

Hence Ph_{E}(A, B) = Ext_{E}^{1}(Σ(A), B) and A is E−projective iff Ext_{E}^{1}(A, −) = 0. (ii) Ph_{E}(A, −)^{2} = 0. If E−gl.dimC ≤ 1, then the projectivization functor S : C ⟶ S_{E}(C) is full and reflects isomorphisms, E projectively generates C, and the following relations hold:

Ph_{E}(C)^{2} = 0, Ph_{E}(C) ⊆ J ac(C), Ph_{E}(Σ^{−1}−, −) ≅ Ext_{E}^{1}(−, −).

Proof. Part (i) follows from Lemma 4.26 setting F = C(−, B). To prove (ii) let A ⟶^{α} B ⟶^{β} C be E−phantom maps and consider the E−projective resolution of A as in the proof of Lemma 4.26. Since α is E−phantom, the composition f_{A}^{0} ∘ α = 0. So there exists a map y : Σ(P_{A}^{1}) ⟶ B with h_{A}^{0} ∘ y = α. But with P_{A}^{1}, also Σ(P_{A}^{1}) is E−projective. Since β is E−phantom, the composition y ∘ β = 0.

Then α ∘ β = h_{A}^{0} ∘ y ∘ β = 0. Hence Ph_{E}(A, −)^{2} = 0. If E−gl.dimC ≤ 1, then since Ext_{E}^{0}(A, B) ≅ S_{E}(C)[S(A), S(B)], ∀A, B ∈ C, the functor S is full. Since kerS = Ph_{E}(C) is a square zero ideal, it is contained in the Jacobson radical of C. This implies that S reflects isomorphisms. The last claim follows from (i), (ii).

Remark 4.28. If C is skeletally small and A is abelian, let Add(C, A) be the category of additive functors from C to A and let Hom(C, A) be the full subcategory of homological functors. If E−gl.dimC ≤ 1, it is easy to see that the kernel-ideal ker(L_{0}^{E}) of the functor L_{0}^{E} : Hom(C, A) ⟶ Add(C, A), satisfies ker(L_{0}^{E})^{2} = 0. Hence if ϕ : F ⟶ G is a map of homological functors, then ϕ is invertible iff so is L_{0}^{E}(ϕ).

Corollary 4.29. Assume that E−gl.dimC ≤ 1 and let F : C^{op} ⟶ Ab be a cohomological functor. Then the following are equivalent.

(i) F is representable. (ii) There exists Ω ∈ C and a natural isomorphism F |_{P(E)} ⟶^{≅} C(−, Ω)|_{P(E)}.

Proof. (ii) ⇒ (i) Let ω : C(−, Ω)|_{P(E)} ⟶^{≅} F |_{P(E)} be a natural isomorphism. Using ω, it follows easily that the short exact sequence of Lemma 4.26 is isomorphic to:

0 ⟶ Ext_{E}^{1}[Σ−, Ω] ⟶^{β} F ⟶^{α} Ext_{E}^{0}[−, Ω] ⟶ 0.

Let μ_{−,Ω} : C(−, Ω) ⟶ Ext_{E}^{0}[−, Ω] be the canonical morphism. Since α is an epimorphism and C(−, Ω) is a projective functor, there exists a morphism ϕ : C(−, Ω) ⟶ F such that ϕ ∘ α = μ_{−,Ω}. Obviously ϕ_{P} is an isomorphism, ∀P ∈ P(E), since μ_{P,Ω} and α_{P} are. Using the natural morphism ϕ and the projective resolution of A, using that F is cohomological and finally using 5−Lemma, we see directly that ϕ_{A} : C(A, Ω) ⟶ F (A) is an isomorphism. Hence ϕ : C(−, Ω) ⟶^{≅} F.

5. The E−Phantom Tower the E−Cellular Tower Homotopy Colimits and Compact Objects

Throughout this section we assume that E is a proper class of triangles in the triangulated category C, and assume that C has enough E−projectives. Our purpose in this section is to construct for any object A of C, the phantom tower and the cellular tower of A, generalizing topological constructions from [51]. These towers are crucial in the study of the homological behaviour of A with respect to E.

5.1. The Phantom Tower. We fix an object A ∈ C and let

P_{A}^{•} : ⋯ ⟶ P_{A}^{n+1} ⟶^{ε_{A}^{n+1}} P_{A}^{n} ⟶ ⋯ ⟶ P_{A}^{1} ⟶^{ε_{A}^{1}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶ 0

be the E−projective resolution of A as in Section 4. We recall that the resolution P_{A}^{•} is obtained by splicing triangles

T_{A}^{n} : K_{A}^{n+1} ⟶^{g_{A}^{n}} P_{A}^{n} ⟶^{f_{A}^{n}} K_{A}^{n} ⟶^{h_{A}^{n}} Σ(K_{A}^{n+1}) ∈ E

where P_{A}^{n} ∈ P(E), ∀n ≥ 0, K_{A}^{0} := A and ε_{A}^{n} = f_{A}^{n} ∘ g_{A}^{n−1} : P_{A}^{n} ⟶ P_{A}^{n−1}, ∀n ≥ 1.

Consider the morphisms h_{A}^{n−1} : K_{A}^{n−1} ⟶ Σ(K_{A}^{n}), ∀n ≥ 1. By construction all these maps are E−phantoms, i.e. we have h_{A}^{n−1} ∈ Ph_{E}(K_{A}^{n−1}, Σ(K_{A}^{n})). Hence the maps Σ^{n−1}(h_{A}^{n−1}) : Σ^{n−1}(K_{A}^{n−1}) ⟶ Σ^{n}(K_{A}^{n}) are also E−phantom maps. In this way we obtain a tower of objects and E−phantom maps

A ⟶^{h_{A}^{0}} Σ(K_{A}^{1}) ⟶^{Σ(h_{A}^{1})} Σ^{2}(K_{A}^{2}) ⟶ ⋯ ⟶ Σ^{n−1}(K_{A}^{n−1}) ⟶^{Σ^{n−1}(h_{A}^{n−1})} Σ^{n}(K_{A}^{n}) ⟶ ⋯

which we call the E−phantom tower of A, with respect to the proper class E.

Proposition 5.1. For any morphism α : A ⟶ B, the following are equivalent: (i) α ∈ Ph_{E}^{n}(A, B).

(ii) There exists a morphism σ : Σ^{n}(K_{A}^{n}) ⟶ B such that:

(−1)^{n+1}h_{A}^{0} ∘ Σ(h_{A}^{1}) ∘ Σ^{2}(h_{A}^{2}) ∘ ⋯ ∘ Σ^{n−1}(h_{A}^{n−1}) ∘ σ = α.

Proof. (i) ⇒ (ii) If α ∈ Ph_{E}(A, B), then f_{A}^{0} ∘ α = 0, since f_{A}^{0} : P_{A}^{0} ⟶ A and α is E−phantom. Hence there exists σ : Σ(K_{A}^{1}) ⟶ B with h_{A}^{0}∘σ = α. If α ∈ Ph_{E}^{2}(A, B), then α = α_{1} ∘ α_{2} where α_{1} : A ⟶ X and α_{2} : X ⟶ B are E−phantom maps. By the above argument, there exists σ_{1} : Σ(K_{A}^{1}) ⟶ X with h_{A}^{0} ∘ σ_{1} = α_{1}. Since the composition σ_{1} ∘ α_{2} : Σ(K_{A}^{1}) ⟶ B is E−phantom and Σ(P_{A}^{1}) ∈ P(E), we have −Σ(f_{A}^{1}) ∘ σ_{1} ∘ α_{2} = 0. Then as above there exists σ : Σ^{2}(K_{A}^{2}) ⟶ B with −Σ(h_{A}^{1}) ∘ σ = σ_{1} ∘ α_{2}. Then h_{A}^{0} ∘ (−Σ(h_{A}^{1})) ∘ σ = h_{A}^{0} ∘ σ_{1} ∘ α_{2} = α_{1} ∘ α_{2} = α. A simple induction argument completes the proof. The converse is trivial.

5.2. The Cellular Tower. In this subsection we use the phantom tower to construct the cellular tower of an object. We fix an object A in the triangulated category C and consider the E−projective resolution of A as above. For simplicity we set: ω_{A}^{n} := (−1)^{n}h_{A}^{0} ∘ Σ(h_{A}^{1}) ∘ Σ^{2}(h_{A}^{2}) ∘ ⋯ ∘ Σ^{n}(h_{A}^{n}) : A ⟶ Σ^{n+1}(K_{A}^{n+1}), A_{0} := 0

and A_{1} := P_{A}^{0}. Consider the triangle K_{A}^{1} ⟶^{g_{A}^{0}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶^{h_{A}^{0}} Σ(K_{A}^{1}) and form the Cobase change of this triangle along the phantom map −h_{A}^{1} : K_{A}^{1} ⟶ Σ(K_{A}^{2}):

K_{A}^{1} ⟶^{g_{A}^{0}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶^{h_{A}^{0}} Σ(K_{A}^{1})
−h_{A}^{1}↓  ^{α_{1}}↓  ^{∥}↓  ^{−Σ(h_{A}^{1})}↓  (+)
Σ(K_{A}^{2}) ⟶^{β_{2}} A_{2} ⟶^{γ_{2}} A ⟶^{ω_{A}^{1}} Σ^{2}(K_{A}^{2}).

Next consider the third horizontal triangle Σ(K_{A}^{2}) ⟶^{β_{2}} A_{2} ⟶^{γ_{2}} A ⟶^{ω_{A}^{1}} Σ^{2}(K_{A}^{2}) in the above diagram, which by construction is in E, and form the Cobase change of this triangle along the phantom map −Σ(h_{A}^{2}) : Σ(K_{A}^{2}) ⟶ Σ^{2}(K_{A}^{3}):

Σ(K_{A}^{2}) ⟶^{β_{2}} A_{2} ⟶^{γ_{2}} A ⟶^{ω_{A}^{1}} Σ^{2}(K_{A}^{2})
−Σ(h_{A}^{2})↓  ^{α_{2}}↓  ^{∥}↓  ^{−Σ^{2}(h_{A}^{2})}↓
Σ^{2}(K_{A}^{3}) ⟶^{β_{3}} A_{3} ⟶^{γ_{3}} A ⟶^{ω_{A}^{2}} Σ^{3}(K_{A}^{3}).

Inductively the above construction produces a tower

0 = A_{0} ⟶ A_{1} = P_{A}^{0} ⟶^{α_{1}} A_{2} ⟶^{α_{2}} A_{3} ⟶ ⋯ ⟶ A_{n} ⟶^{α_{n}} A_{n+1} ⟶ ⋯

which we call the E−cellular tower of A, and commutative diagrams, ∀n ≥ 1 :

A_{n} ⟶^{γ_{n}} A
α_{n}↓  ^{∥}↓
A_{n+1} ⟶^{γ_{n+1}} A.

The following result is a direct consequence of the above constructions.

Corollary 5.2. For an object A ∈ C, the following are equivalent:

(i) E−p.dA ≤ n. (ii) The morphisms γ_{n+i} : A_{n+i} ⟶ A of the E−cellular tower of A constructed

above are isomorphisms, ∀i ≥ 1. Consequently E−gl.dimC ≤ n iff the cellular tower of any object of C stabilizes

after the (n+1)^{th}−step, i.e. the morphisms α_{m} : A_{m} ⟶ A_{m+1} of the cellular tower are isomorphisms, ∀m ≥ n+1. If this is the case then ∀A ∈ C : A ≅ A_{m}, ∀m ≥ n+1.

Using the E−cellular tower we can prove the main result of this section.

Theorem 5.3. (α) The following statements are equivalent:

(i) The projectivization functor S : C ⟶ S_{E}(C) is full and reflects isomorphisms. (ii) E−gl.dimC ≤ 1.

(iii) E generates C and ImS ⊆ {F ∈ S_{E}(C) : p.dF ≤ 1}.

(β) Assume that idempotents split in P(E). Then {F ∈ S_{E}(C) : p.dF ≤ 1} ⊆ ImS. If E−gl.dimC ≤ 1, then S induces an equivalence S : I(E) ⟶^{≈} InjS_{E}(C); further C has enough E−injectives iff S_{E}(C) has enough injectives.

(γ) If C has enough E−injectives, then the statements in (α) are also equivalent to the following (the inclusion below is an equality if I(E) has split idempotents): (iv) ∀A ∈ C: E−i.dA ≤ 1.

(v) E generates C and ImS ⊆ {F ∈ S_{E}(C) : i.dF ≤ 1}.

Proof. (α) (ii) ⇒ (i) follows from Theorem 4.27. (i) ⇒ (ii) Suppose that (i) is true. Applying to the cobase change diagram (+) the projectivization functor S, we obtain the following commutative diagram of short exact sequences in S_{E}(C):

0 ⟶ S(K_{A}^{1}) ⟶^{S(g_{A}^{0})} S(P_{A}^{0}) ⟶^{S(f_{A}^{0})} S(A) ⟶ 0
−S(h_{A}^{1})↓  ^{S(α_{1})}↓  ^{||}↓
0 ⟶ S(Σ(K_{A}^{2})) ⟶^{S(β_{2})} S(A_{2}) ⟶^{S(γ_{2})} S(A) ⟶ 0

Since h_{A}^{1} is an E−phantom map, S(h_{A}^{1}) = 0. Consider the cokernel ω : S(A_{2}) ⟶ F of S(α_{1}). By the above diagram, there exists a unique isomorphism ζ : S(Σ(K_{A}^{2})) ⟶ F such that S(β_{2}) ∘ ω = ζ. Consider the morphism ω ∘ ζ^{−1} : S(A_{2}) ⟶ S(Σ(K_{A}^{2})). Since S is full, there exists ϕ : A_{2} ⟶ Σ(K_{A}^{2}) such that S(ϕ) = ω∘ζ^{−1}. Next consider the morphism α_{1} ∘ ϕ : P_{A}^{0} ⟶ Σ(K_{A}^{2}). Then S(α_{1} ∘ ϕ) = S(α_{1}) ∘ ω ∘ ζ^{−1} = 0. Hence the morphism α_{1} ∘ ϕ is E−phantom. Then α_{1} ∘ ϕ = 0, since P_{A}^{0} ∈ P(E). From the diagram (+), we have that Σ(g_{A}^{1}) = β_{2} ∘ δ_{1}, where P_{A}^{0} ⟶^{α_{1}} A_{2} ⟶^{δ_{1}} Σ(P_{A}^{1}) ⟶ Σ(P_{A}^{0}) is a triangle in C. It follows that ϕ factors through δ_{1}, i.e. there exists ρ : Σ(P_{A}^{1}) ⟶ Σ(K_{A}^{2}), such that δ_{1} ∘ ρ = ϕ. Then β_{2} ∘ δ_{1} ∘ ρ = β_{2} ∘ ϕ ⇒ Σ(g_{A}^{1}) ∘ ρ = β_{2} ∘ ϕ. But S(β_{2}) ∘ ω = ζ ⇒ S(β_{2}) ∘ ω ∘ ζ^{−1} = 1_{S(Σ(K_{A}^{2}))} ⇒ S(β_{2}) ∘ S(ϕ) = 1_{S(Σ(K_{A}^{2}))} ⇒ S(β_{2}∘ϕ) = 1_{S(Σ(K_{A}^{2}))}. Since S reflects isomorphisms, it follows that β_{2}∘ϕ = Σ(g_{A}^{1})∘ρ is invertible. Then Σ(g_{A}^{1}) is split monic and consequently Σ(f_{A}^{1}) : Σ(P_{A}^{1}) ⟶ Σ(K_{A}^{1}) is split epic. This implies that K_{A}^{1} is E−projective as a direct summand of P_{A}^{1}. Hence E−p.dA ≤ 1. Part (ii) ⇔ (iii) follows from (i) and Corollary 4.22, since if E generates, then E−p.dA = p.dS(A), ∀A ∈ C.

(β) Let F ∈ S_{E}(C) with p.dF ≤ 1. Since P(E) has split idempotents, by Proposition 4.19, S induces an equivalence P(E) ≈ ProjS_{E}(C). Hence F admits a projective

resolution of the form 0 ⟶ S(P_{1}) ⟶^{S(α)} S(P_{0}) ⟶^{f} F ⟶ 0 with P_{1}, P_{0} ∈ P(E). Then the triangle P_{1} ⟶^{α} P_{0} ⟶ A ⟶ Σ(P_{1}) is in E and obviously F ≅ S(A). Hence {F ∈ S_{E}(C) : p.dF ≤ 1} ⊆ ImS. If E−gl.dimC ≤ 1, then by (α) we have that {F ∈ S_{E}(C) : p.dF ≤ 1} = ImS. Hence ImS is closed under direct summands and the remaining assertions are consequences of Theorem 4.24.

Part (γ) is similar and is left to the reader.

5.3. The Category of Extensions and the Phantom Filtration. We recall now a well-known construction from [11]. If X is a class of objects of C, then the category X ⋆ X of extensions of X is defined as follows:

X ⋆ X = add{A ∈ C : ∃ triangle X_{1} ⟶ A ⟶ X_{2} ⟶ Σ(X_{1}) with X_{1}, X_{2} ∈ X }

Inductively we define X^{⋆n} := X ⋆ X ⋆ X ⋆ ⋯ ⋆ X (n−factors). It is shown in [11] that the operation ⋆ is associative, hence the above definition of X^{⋆n} makes sense.

Finally define X^{⋆∞} := ⋃_{n≥1} X^{⋆n}, where we set X^{⋆1} = X. The category X^{⋆n} is called the category of n^{th}−extensions of X, for n = 1, 2, ..., ∞.

Remark 5.4. Let A be an object of C, and consider the E−cellular tower of A:

A_{0} ⟶^{α_{0}} A_{1} ⟶^{α_{1}} A_{2} ⟶^{α_{2}} A_{3} ⟶ ⋯ ⟶ A_{n} ⟶^{α_{n}} A_{n+1} ⟶ ⋯

Then clearly: A_{n} ∈ P(E)^{⋆n}, ∀n ≥ 1.

The next Corollary generalizes results of G.M. Kelly [46] and R. Street [68].

Corollary 5.5. (α) If A ∈ C, then ∀n ≥ 0, consider the following statements: (i) E−p.dA ≤ n. (ii) A ∈ P(E)^{⋆(n+1)}. (iii) Ph_{E}^{n+1}(A, −) = 0. Then (i) ⇒ (ii) ⇒ (iii). In particular if E−p.dA < ∞, then Ph_{E}(A, −) ⊆ J ac(A, −) and the ideal Ph_{E}(A, A) of the ring End_{C}(A) is nilpotent. Hence if an object X has a semiprime endomorphism ring, then either Ph_{E}(X, X) = 0 or else E−p.dX = ∞.

(β) If any object of C has finite E−projective dimension, then C is E−projectively generated and C = P(E)^{⋆∞}.

(γ) If E−gl.dimC ≤ n, or if C = P(E)^{⋆(n+1)}, then: Ph_{E}^{n+1}(C) = 0.

Proof. (α) The implication (i) ⇒ (ii) follows from Corollary 5.2. Let A be in P(E) ⋆ P(E), so there exists a triangle P_{1} ⟶^{f} A ⟶^{g} P_{0} ⟶^{h} Σ(P_{1}) with P_{0}, P_{1} ∈ P(E). If α : A ⟶ B, β : B ⟶ C are E−phantoms, then α = g ∘ ρ and ρ ∘ β = 0, for some ρ : P_{0} ⟶ B. Then α∘β = g ∘ρ∘β = 0. Hence Ph_{E}^{2}(A, −) = 0. Now the implication (ii) ⇒ (iii) follows by induction. Parts (β), (γ) are direct consequences of (α).

Using the proper class E, we define a new class of triangles E^{2} as follows:

• A triangle A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) is in E^{2} if h ∈ Ph_{E}^{2}(C, Σ(A)).

Proposition 5.6. The class of triangles E^{2} in C is proper. If C has enough E−projectives then C has enough E^{2}−projectives and P(E^{2}) = P(E) ⋆ P(E) = P(E)^{⋆2}. Moreover if E generates C then also E^{2} generates C.

Proof. From Proposition 2.5, it follows that E^{2} is a proper class of triangles. From the construction of the E−cellular tower we know that ∀A ∈ C, there exists a

triangle Σ(K_{A}^{2}) ⟶ A_{2} ⟶ A ⟶^{ω_{A}^{1}} Σ^{2}(K_{A}^{2}) such that ω_{A}^{1} ∈ Ph_{E}^{2}(C) and A_{2} ∈ P(E) ⋆ P(E). Hence the above triangle is in E^{2} and it suffices to show that any object in P(E) ⋆ P(E) is E^{2}−projective. So let A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) be a triangle in E^{2} and let α : X ⟶ C be a morphism with X ∈ P(E) ⋆ P(E). Then the composition α ∘ h is in Ph_{E}^{2}(X, Σ(A)). By Corollary 5.5, we have α ∘ h = 0, so α factors through g. Hence X is E^{2}−projective and since P(E) ⋆ P(E) is closed under direct summands, E^{2} has enough E^{2}−projectives and P(E^{2}) = P(E) ⋆ P(E). Since E generates C iff Ph_{E}(C) ⊆ J ac(C) and since Ph_{E^{2}}(C) = Ph_{E}(C)^{2}, the last claim follows trivially.

From now on we set E^{1} := E. We note that Ph_{E^{2}}(C) = Ph_{E^{1}}^{2}(C). The procedure described in the above Proposition produces a filtration of proper classes of triangles

Δ_{0} ⊆ E^{n} ⊆ E^{n−1} ⊆ ⋯ ⊆ E^{2} ⊆ E^{1} = E ⊆ Δ

a filtration of phantom ideals

0 ⊆ Ph_{E^{n}}(C) ⊆ Ph_{E^{n−1}}(C) ⊆ ⋯ ⊆ Ph_{E^{2}}(C) ⊆ Ph_{E^{1}}(C) = Ph_{E}(C) ⊆ C(−, −)

where Ph_{E^{n}}(C) = Ph_{E}^{n}(C), and a filtration of subcategories

0 ⊆ P(E) = P(E^{1}) ⊆ P(E^{2}) ⊆ ⋯ ⊆ P(E^{n−1}) ⊆ P(E^{n}) ⊆ C

where P(E^{n}) = P(E)^{⋆n}. Finally setting E^{∞} = ⋂_{n≥1} E^{n} we obtain a proper class of triangles such that: Ph_{E^{∞}}(C) = ⋂_{n≥1} Ph_{E^{n}}(C) = ⋂_{n≥1} Ph_{E}^{n}(C) and P(E^{∞}) = ⋃_{n≥1} P(E)^{⋆n} = P(E)^{⋆∞}. If C has enough E−projectives then for 1 ≤ n ≤ ∞, C has enough E^{n}−projectives. Moreover if E generates C then the same is true for E^{n}. The above data induce a filtration of Steenrod categories

0 ↪ S_{E^{∞}}(C) ↪ ⋯ ↪ S_{E^{n}}(C) ↪ S_{E^{n−1}}(C) ↪ ⋯ ↪ S_{E^{2}}(C) ↪ S_{E^{1}}(C) ↪ A(C)

and for 1 ≤ n ≤ m ≤ ∞ each inclusion S_{E^{m}}(C) ↪ S_{E^{n}}(C) has an exact right adjoint S_{m,n} : S_{E^{n}}(C) ⟶ S_{E^{m}}(C), with kernel the Serre subcategory A(Ph_{E}^{n}(C))/A(Ph_{E}^{m}(C)) ≈ A(Ph_{E}^{n}(C)/Ph_{E}^{m}(C)). The following is a consequence of our previous results.

Corollary 5.7. For any n with 1 ≤ n ≤ ∞, the following are true.

(i) C is E^{n}−phantomless ⇔ Ph_{E}^{n}(C) = 0 ⇔ C = P(E)^{⋆n} ⇔ E^{n}−gl.dimC = 0. (ii) E^{n}−p.dA ≤ 1 ⇒ ∀B ∈ C: Ph_{E}^{n}(A, B) = Ext_{E^{n}}^{1}(Σ(A), B), Ph_{E}^{2n}(A, B) = 0. (iii) E^{n}−gl.dimC ≤ m ⇒ C = P(E)^{∗n·(m+1)} and Ph_{E}^{n·(m+1)}(C) = 0.

We define the E−phantom dimension E−Ph.dimC of C by inf{n ≥ 0 : Ph_{E}^{n+1}(C) = 0} and the E−extension dimension E−Ext.dimC of C by inf{n ≥ 0 : P(E)^{⋆(n+1)} = C}. Trivially: E−Ph.dimC ≤ E−Ext.dimC ≤ E−gl.dimC. Using Corollary 5.7(i), the Octahedral axiom and the E−cellular tower, it is not difficult to see that E−Ph.dimC = E−Ext.dimC. We thank J.D. Christensen for this nice observation. It is unclear under what conditions the last inequality is an equality.

5.4. Homotopy Colimits and Compact Objects. In this subsection we study the behaviour of the E−cellular tower under (co-)homological functors. We assume throughout that C has countable coproducts.

Let A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} A_{2} ⟶^{f_{2}} ⋯ be a tower of objects in C. The homotopy colimit holim_{⟶}A_{n} of the tower is defined by the following triangle

⨁_{n≥0} A_{n} ⟶^{f} ⨁_{n≥0} A_{n} ⟶^{g} holim_{⟶}A_{n} ⟶^{h} Σ(⨁_{n≥0} A_{n}) (T)

where f is induced by the morphisms A_{n} ⟶^{(1,−f_{n})} A_{n} ⊕ A_{n+1} ↪ ⊕_{n≥0}A_{n}. The homotopy colimit is uniquely determined up to a non-unique isomorphism. We include for completeness the proof of the following well-known result.

Lemma 5.8. Let {A_{n}; f_{n}}_{n≥0} be a tower of objects in C.

(1) If F : C ⟶ Ab is a homological functor which preserves coproducts, then the canonical map lim_{→}F (A_{n}) ⟶ F (holim_{⟶}A_{n}) is invertible.

(2) If F : C^{op} ⟶ Ab is a cohomological functor converting coproducts to products, then the canonical map F (holim_{⟶}A_{n}) ⟶ lim_{←}F (A_{n}) induces a Milnor short exact

sequence: 0 ⟶ lim_{←}^{(1)}F Σ(A_{n}) ⟶ F (holim_{⟶}A_{n}) ⟶ lim_{←}F (A_{n}) ⟶ 0.

Proof. (1) The triangle (T) induces a short exact sequence 0 ⟶ ⊕_{n≥0}F (A_{n}) ⟶ ⊕_{n≥0}F (A_{n}) ⟶ F (holim_{⟶}A_{n}) ⟶ 0 in Ab and the result follows. (2) Applying F to the triangle (T), we obtain the long exact sequence: ⋯ ⟶ F (holim_{⟶}A_{n}) ⟶

∏_{n≥0} F (A_{n}) ⟶ ∏_{n≥0} F (A_{n}) ⟶ F (holim_{⟶}Σ^{−1}(A_{n})) ⟶ ⋯ in Ab, and the result

follows from the construction of lim_{←}^{(i)}, i = 0, 1, of a tower in Ab, see [71].

The following concepts are fundamental.

Definition 5.9. [56] If C is a triangulated category, then an object A ∈ C is called compact if the functor C(A, −) : C ⟶ Ab preserves all small coproducts. The compact part of C is by definition the full subcategory C^{b} of all compact objects.

The compact part C^{b} of C is a full triangulated subcategory of C. The category C is compactly generated if C has all small coproducts, C^{b} is skeletally small and C(X, A) = 0, ∀X ∈ C^{b} implies that A = 0, i.e. the proper class E(C^{b}) generates C.

By the construction of the E−cellular and the E−phantom tower of an object A, we have a diagram of towers {A_{n}} ⟶ {A} ⟶ {Σ^{n}(K_{A}^{n})} ⟶ {Σ(A_{n})}, such that A_{n} ⟶ A ⟶ Σ^{n}(K_{A}^{n}) ⟶ Σ(A_{n}) is a triangle in C. It is not known if the induced diagram E(A) : holim_{⟶}A_{n} ⟶ A ⟶ holim_{⟶}Σ^{n}(K_{A}^{n}) ⟶ Σ(holim_{⟶}A_{i}) is a triangle in C. This happens if C is a “homotopy” category of a suitable stable closed model category [37], for instance a derived category or a stable module category. The next result follows from the construction of the cellular and phantom tower.

Lemma 5.10. (1) Let T be a compact E−projective object and let holim_{⟶}Σ^{n}(K_{A}^{n}) be a homotopy colimit of the phantom tower of A. Then: C(T, holim_{⟶}Σ^{n}(K_{A}^{n})) = 0.

(2) If Ph_{E}^{t}(A, −) = 0, for some t ≥ 0 or if C is generated by a class X of compact E−projective objects, then: holim_{⟶}Σ^{n}(K_{A}^{n}) = 0.

Proposition 5.11. Let T ∈ C^{b} be a compact object.

(1) Let A ∈ C and let holim_{⟶}A_{n} be a homotopy colimit of the E−cellular tower {A_{n}; n ≥ 0} of A. Then we have an isomorphism: lim_{→}C(T, A_{n}) ≅ C(T, holim_{⟶}A_{n}).

(2) If in addition T ∈ P(E), then: C(T, A) ≅ lim_{→}C(T, A_{n}) ≅ C(T, holim_{⟶}A_{n}).

Proof. Part (1) follows from Lemma 5.8. Part (2) follows from (1), applying C(T, −) to the cobase change diagrams, defining the E−cellular tower of A.

We have the following nice consequence.

Corollary 5.12. If C is generated by a class X of compact E−projective objects, then any object A ∈ C is a homotopy colimit of its E−cellular tower: A ≅ holim_{⟶}A_{n}.

Proof. Since X generates C, the functors C(T, −), T ∈ X collectively reflect isomorphisms and the claim follows from Proposition 5.11.

5.5. The Phantom Topology. Our purpose here is to describe the infinite E− phantom ideal Ph_{E}^{∞}(C) by means of the E−cellular tower. Fix two objects A, B ∈ C.

Definition 5.13. The E−phantom topology of C(A, B) is the topology with open sets ϕ + Ph_{E}^{n}(A, B), n ≥ 0. The topological group C(A, B) is complete (in the E−phantom topology) if it is Hausdorff and any Cauchy sequence has a limit.

Consider the E−phantom tower of A and let ω_{A}^{n} := (−1)^{n}h_{A}^{0} ∘ Σ(h_{A}^{1}) ∘ Σ^{2}(h_{A}^{2}) ∘ ⋯ ∘ Σ^{n}(h_{A}^{n}) : A ⟶ Σ^{n+1}(K_{A}^{n+1}) as in subsection 5.1. Then by Proposition 5.1, we have clearly ImC(ω_{A}^{n}, B) = Ph_{E}^{n+1}(A, B). Now applying the functor C(−, B) to the E−cellular tower of A, we obtain a short exact sequence of inverse systems in Ab:

0 ⟶ {C(A, B)/Ph_{E}^{n}(A, B)} ⟶ {C(A_{n}, B)} ⟶ {ImC(β_{A}^{n}, B)} ⟶ 0.

Hence we have the exact sequence

0 ⟶ lim_{←}C(A, B)/Ph_{E}^{n}(A, B) ⟶ lim_{←}C(A_{n}, B) ⟶ lim_{←}ImC(β_{A}^{n}, B) ⟶

⟶ lim_{←}^{(1)}C(A, B)/Ph_{E}^{n}(A, B) ⟶ lim_{←}^{(1)}C(A_{n}, B) ⟶ lim_{←}^{(1)}ImC(β_{A}^{n}, B) ⟶ 0 (1)

Assume now that C has countable coproducts and is generated by a class of objects X such that Σ(X) = X ⊆ P(E)∩C^{b}. Then since holim_{⟶}A_{n} ≅ A, we have the following exact sequence (2): 0 ⟶ lim_{←}^{(1)}C(Σ(A_{n}), B) ⟶ C(A, B) ⟶ lim_{←}C(A_{n}, B) ⟶ 0. From the short exact sequence of the inverse systems

0 ⟶ {KerC(ω_{A}^{n−1}, B)} ⟶ {C(Σ^{n}(K_{A}^{n}), B)} ⟶ {Ph_{E}^{n}(A, B)} ⟶ 0

using that {KerC(ω_{A}^{n−1}, B)} = {ImC(Σ(β_{A}^{n}), B)}, we obtain the exact sequence

0 ⟶ lim_{←}ImC(Σ(β_{n}), B) ⟶ lim_{←}C(Σ^{n}(K_{A}^{n}), B) ⟶ lim_{←}Ph_{E}^{n}(A, B) ⟶

⟶ lim_{←}^{(1)}ImC(Σ(β^{n}), B) ⟶ lim_{←}^{(1)}C(Σ^{n}(K_{A}^{n}), B) ⟶ lim_{←}^{(1)}Ph_{E}^{n}(A, B) ⟶ 0 (3)

But ∀B ∈ C, the E−phantom tower of A induces a short exact sequence

0 ⟶ lim_{←}^{(1)}C(Σ^{n+1}(K_{A}^{n}), B) ⟶ C(holim_{⟶}Σ^{n}(K_{A}^{n}), B) ⟶ lim_{←}C(Σ^{n}(K_{A}^{n}), B) ⟶ 0.

Since holim_{⟶}Σ^{n}(K_{A}^{n}) = 0, we have lim_{←}^{(1)}C(Σ^{n+1}(K_{A}^{n}), B) = lim_{←}C(Σ^{n}(K_{A}^{n}), B) =

0. Since this holds ∀B ∈ C, by (3): lim_{←}ImC(β_{n}, B) = lim_{←}^{(1)}Ph_{E}^{n}(A, B) = 0,

lim_{←}Ph_{E}^{n}(A, B) ≅ lim_{←}^{(1)}ImC(Σ(β_{n}), B). Since lim_{←}^{(1)}C(A, B)/Ph_{E}^{n}(A, B) = 0, using (1) and collecting all the above information we deduce the following.

Proposition 5.14. (1) lim_{←}C(A_{n}, B) ≅ lim_{←}C(A, B)/Ph_{E}^{n}(A, B).

(2) lim_{←}^{(1)}Ph_{E}^{n}(A, B) = 0.

(3) Ph_{E}^{∞}(A, B) = ⋂_{n≥1} Ph_{E}^{n}(A_{n}, B) = lim_{←}Ph_{E}^{n}(A, B) ≅ lim_{←}^{(1)}C(Σ(A_{n}), B).

(4) C(A, B) is complete in the E−phantom topology iff lim_{←}^{(1)}C(Σ(A_{n}), B) = 0.

It is well known that lim_{←}^{(1)}G_{n} = 0 for an inverse system {G_{n}} of Artinian groups. Hence we have the following consequence.

Corollary 5.15. If E−p.dA < ∞ or A ∈ P(E)^{⋆∞} or C(Σ(A_{n}), B) is Artinian, ∀n ≥ 0, then C(A, B) is complete in the E−phantom topology, i.e. Ph_{E}^{∞}(A, B) = 0.

Note that the above computations are useful for the construction and the study of convergence of the Adams Spectral sequence [68], [19] for the proper class E.

6. Localization and the Relative Derived Category

We consider in this section a triangulated category C equipped with a proper class of triangles E and we assume always that C has enough E−projectives.

We have seen that the class of E−phantom maps Ph_{E}(C) = {f : ∀P ∈ P(E); C(P, f) = 0} is an ideal in C, and the condition Ph_{E}(C) = 0 has nice consequences for the triangulated category C. So it is useful to find a way to “kill” the E−phantom maps. Define a class of morphisms by R := {f | S(f) is invertible}, i.e. ∀A, B ∈ C:

R_{A,B} = {f ∈ C(A, B) | ∀P ∈ P(E); C(P, f) : C(P, A) ⟶^{≅} C(P, B)}.

The easy proof of the following is left to the reader.

Lemma 6.1. The class of morphisms R is a multiplicative system of morphisms in C compatible with its triangulation.

We call the elements of R, E-quasi-isomorphisms. By the above Lemma, the localized category C[R^{−1}] can be calculated by left and right fractions. We recall that C[R^{−1}] is a triangulated category and there exists an exact functor of triangulated categories R : C ⟶ C[R^{−1}], which sends the elements of R to isomorphisms in C[R^{−1}] and has the following universal property. If F : C ⟶ T is an exact functor to a triangulated category T which sends R to isomorphisms, then there exists a unique up to isomorphism exact functor G : C[R^{−1}] ⟶ T with GR = F. From now on we denote by D_{E}(C) := C[R^{−1}] and by R : C ⟶ D_{E}(C) the canonical functor. We call the triangulated category D_{E}(C) the E−derived category of C with respect to E. In general D_{E}(C) lives in a higher universe than C. If A_{E}(C) = KerR = {A ∈ C : R(A) = 0} is the full subcategory of E−acyclic objects, then we have a short exact sequence of triangulated categories:

0 ⟶ A_{E}(C) ⟶^{j} C ⟶^{R} D_{E}(C) ⟶ 0.

Lemma 6.2. The kernel-ideal kerR is contained in the ideal of E−phantom maps.

Proof. If R(f) = 0, then there exists a morphism s ∈ R with f ∘ s = 0. By the definition of R, we have C(P(E), f) = 0. Then by Lemma 4.2, f ∈ Ph_{E}(C).

Lemma 6.3. ∀P ∈ P(E) and ∀B ∈ C, the morphism

R_{P,B} : C(P, B) ⟶ D_{E}(C)[R(P), R(B)] is an isomorphism.

Proof. By Lemma 6.2, R_{P,B} is injective. Let ã : R(P) ⟶ R(B) be a morphism

in D_{E}(C). It is well known that ã can be represented as a diagram P ⟵^{s} X ⟶^{f} B with s ∈ R. Then C(P(E), s) : C(P(E), X) ⟶^{≅} C(P(E), P). Let g ∈ C(P, X) be the unique morphism such that g ∘ s = 1_{P}. By the definition of the morphisms in the category of fractions D_{E}(C), we have R(g ∘ f) = ã, so R_{P,B} is surjective.

By Lemma 6.3 and induction we have the following direct consequence.

Lemma 6.4. ∀P ∈ P(E)^{⋆n}, n = 0, 1, 2, ..., ∞ and ∀B ∈ C, the morphism

R_{P,B} : C(P, B) ⟶ D_{E}(C)[R(P), R(B)] is an isomorphism.

In particular the induced functor R : P(E)^{⋆∞} ↪ D_{E}(C) is fully faithful.

6.1. Relative Derived categories. We seek conditions such that the functor R induces an equivalence between D_{E}(C) and a full subcategory of C. We fix an object A ∈ C, and consider the E−cellular tower of A:

(T) A_{0} ⟶^{α_{0}} A_{1} ⟶^{α_{1}} A_{2} ⟶ ⋯ ⟶ A_{n} ⟶^{α_{n}} A_{n+1} ⟶ ⋯, A_{n} ∈ P(E)^{⋆n}, ∀n ≥ 0.

From now on we assume that the triangulated category C has (countable) coproducts. Consider a homotopy colimit holim_{⟶}A_{n},

⨁_{n≥0} A_{n} ⟶^{ζ} ⨁_{n≥0} A_{n} ⟶^{θ} holim_{⟶}A_{n} ⟶^{η} Σ(⨁_{n≥0} A_{n}) (!)

and the following triangle, where the morphism ψ is naturally induced from (T):

E(A) :  holim_{⟶}A_{n} ⟶^{ψ} A ⟶^{ϕ} Â ⟶^{χ} Σ(holim_{⟶}A_{n})  (!!)

Let { \atop P(E)^{⋆∞}} be the full additive subcategory generated by all objects which are isomorphic to a homotopy colimit of a tower (T) as above with A_{n} ∈ P(E)^{⋆n}, n ≥ 0:

{ \atop P(E)^{⋆∞}} := add\{A ∈ C | A ≅ holim_{⟶}A_{n}, where A_{n} ∈ P(E)^{⋆n}, ∀n ≥ 0\}.

Now we can prove the main result of this Section. First we recall that a localizing subcategory of C is a full triangulated subcategory of C closed under coproducts. If U ⊆ C, then the localizing subcategory Loc<U> generated by U is the smallest localizing subcategory of C which contains U. Finally we set ⊥U := {A ∈ C | C(A, U) = 0, ∀U ∈ U} and Add(U) to be the full subcategory of C consisting of all direct summands of arbitrary coproducts of objects of U.

Theorem 6.5. Assume that there exists a set of compact E−projective objects X in C such that: Σ(X) = X and a morphism f ∈ R iff the morphism C(X , f) is an isomorphism. Then we have the following:

(i) { \atop P(E)^{⋆∞}} = ^{⊥}A_{E}(C) = Loc<P(E)>.

(ii) The functor R induces a triangle equivalence: R : { \atop P(E)^{⋆∞}} ⟶^{≈} D_{E}(C). (iii) The category D_{E}(C) is compactly generated and any compactly generated tri-

angulated category arises in this way.

Proof. (i) Let P ∈ P(E) and L ∈ A_{E}(C). If α : P ⟶ L is a morphism, then R(α) = 0, since L is E−acyclic. Lemma 6.3 implies that α = 0. Hence P(E) ⊆ ^{⊥}A_{E}(C). By induction and using Lemma 6.4 it follows that P(E)^{⋆n} ⊆ ^{⊥}A_{E}(C). Now let holim_{⟶}A_{n} be a homotopy colimit of a tower of objects A_{n} ∈ P(E)^{⋆n}. By Lemma

5.8, we have a short exact sequence 0 ⟶ lim_{←}^{(1)}C(Σ(A_{n}), L) ⟶ C(holim_{⟶}A_{n}, L) ⟶ lim_{←}C(A_{n}, L) ⟶ 0. Since L is an E−acyclic object, we have C(holim_{⟶}A_{n}, L) = 0.

Hence { \atop P(E)^{⋆∞}} ⊆ ^{⊥}A_{E}(C). If A is in ^{⊥}A_{E}(C) and \{A_{n}; n ≥ 0\} is the E−cellular tower of A, then in the triangle (!!) above we have Â ∈ A_{E}(C). Indeed applying C(X , −) to the triangle (!!) and using Proposition 5.11, it follows that C(X , ψ) : C(X , holim_{⟶}A_{n}) ⟶ C(X , A) is an isomorphism. Hence by hypothesis, ψ ∈ R and

this implies that R(Â) = 0, i.e. Â ∈ A_{E}(C). Since A ∈ ^{⊥}A_{E}(C), we have ϕ = 0. It follows that A is a direct summand of holim_{⟶}A_{n}, hence by definition A ∈ { \atop P(E)^{⋆∞}}.

We conclude that { \atop P(E)^{⋆∞}} = ^{⊥}A_{E}(C), so it is a localizing subcategory of C. By construction { \atop P(E)^{⋆∞}} is the localizing subcategory of C generated by P(E).

(ii) It is well-known and easy to see that R induces a full embedding ^{⊥}A_{E}(C) ↪ D_{E}(C) = C/A_{E}(C). Hence R induces a full embedding { \atop P(E)^{⋆∞}} ↪ D_{E}(C). If A is an object in D_{E}(C), then by the above argument the morphism ψ : holim_{⟶}A_{n} ⟶ A

is in R. Since by construction holim_{⟶}A_{n} ∈ { \atop P(E)^{⋆∞}}, it follows that the functor

R : { \atop P(E)^{⋆∞}} ⟶ D_{E}(C) is surjective on objects, so it is a triangle equivalence.

(iii) If A ∈ { \atop P(E)^{⋆∞}}, then from the triangle E(A) we deduce trivially that the morphism ψ : holim_{⟶}A_{n} ⟶ A is an isomorphism. If C(X , A) = 0, then C(X , holim_{⟶}A_{n}) = 0. This implies that the morphism C(X , ζ) in the triangle (!) is an isomorphism. Hence by hypothesis, ζ is in R. It follows that R(holim_{⟶}A_{n}) = 0, so A ≅ holim_{⟶}A_{n} ∈ A_{E}(C). Hence A = 0. Since X consists of compact objects and

it is contained { \atop P(E)^{⋆∞}}, it follows that { \atop P(E)^{⋆∞}} is compactly generated.

It remains to prove that any compactly generated category C arises in this way. Consider the proper class of triangles E = E(C^{b}) in C. Since C^{b} is skeletally small it follows easily (see Section 8) that C has enough E−projectives and P(E) = Add(C^{b}). Since C^{b} generates C, the projectivization functor S reflects isomorphisms, hence R consists of isomorphisms. Choosing X = Iso(C^{b}) to be the set of isoclasses of compact objects above, we have C = C[R^{−1}] = D_{E}(C) = { \atop Add(C^{b})^{⋆∞}}.

In some cases the compactness condition in Theorem 6.5 can be avoided:

Remark 6.6. Let H : C ⟶ M be a coproduct preserving homological functor to an AB4 category M. Assume that H kills E−phantom maps and satisfies the property: H(f) is an isomorphism ⇒ f ∈ R. Then parts (i), (ii) in Theorem 6.5 are true. Indeed it suffices to show that H(ψ) : H(holim_{⟶}A_{n}) ⟶ H(A) is an isomorphism. Observe first that H(holim_{⟶}A_{n}) ≅ lim_{⟶}H(A_{n}) and H(holim_{⟶}Σ^{n}(K_{A}^{n})) ≅ lim_{⟶}H(Σ^{n}(K_{A}^{n})) = 0. Hence applying H to the diagrams defining the E−cellular tower of A, we have H(ψ) : H(holim_{⟶}A_{n}) ⟶^{≅} H(A). Note that under the above assumptions, if f ∈ R, then H(f) is an isomorphism. Indeed since H kills E−phantom maps, H = H^{∗}S where H^{∗} : S_{E}(C) ⟶ M is exact. Hence if f ∈ R, i.e. S(f) is invertible, then so is H(f). It follows that E and H define the same phantom maps.

We call the objects of { \atop P(E)^{⋆∞}}, E−cofibrant. Hence any object of C has an E−cofibrant resolution, i.e. admits an E−quasi-isomorphism to an E−cofibrant object. The following are direct consequences of Theorem 6.5 or Remark 6.6.

Corollary 6.7. Keep the hypothesis of Theorem 6.5 or Remark 6.6. Then the inclusion i : { \atop P(E)^{⋆∞}} ↪ C has a right adjoint i^{!} and the inclusion j : A_{E}(C) ↪ C has a left adjoint j^{∗}. The quotient functor R : C ⟶ D_{E}(C) has a fully faithful left adjoint. For any object A ∈ C, there exists a functorial triangle

i^{!}(A) ⟶ A ⟶ j^{∗}(A) ⟶ Σ(i^{!}(A))

where the involved morphisms are the adjunction morphisms. Any triangle P ⟶ A ⟶ L ⟶ Σ(P) with P ∈ { \atop P(E)^{⋆∞}} and L ∈ A_{E}(C), is isomorphic to the triangle above by a unique isomorphism of triangles which extends the identity of A.

Corollary 6.8. A full triangulated subcategory of D_{E}(C) equals D_{E}(C) if it contains X and is closed under coproducts.

Corollary 6.9. If C is compactly generated then any object A of C is the homotopy colimit holim_{⟶}A_{n}, of a tower \{A_{n}; A_{n} ⟶^{α_{n}} A_{n+1}\}, where A_{n} ∈ { \atop Add(C^{b})^{⋆n}}, ∀n ≥ 0.

6.2. Total Derived Functors. Let C and B be triangulated categories with proper classes of triangles E and F respectively. We assume that the pairs (C, E), (B, F) satisfy the conditions of Theorem 6.5 or Remark 6.6.

Let F : C ⟶ A be an additive functor to an additive category A. The total left E−derived functor of F is an additive functor \underline{\underline{L}}F : D_{E}(C) ⟶ A together with a morphism ζ : \underline{\underline{L}}F R ⟶ F such that for any additive functor G : D_{E}(C) ⟶ A and morphism α : GR ⟶ F, there exists a unique natural morphism ξ : G ⟶ \underline{\underline{L}}F such that ξR ∘ ζ = α. Similarly if F : C ⟶ B is an additive functor, then the total left (E, F)−derived functor of F is an additive functor \underline{\underline{L}}F : D_{E}(C) ⟶ D_{F}(B) which is the total left E−derived functor for the composite R_{B}F : C ⟶ D_{F}(B).

Theorem 6.10. (1) Any additive functor F : C ⟶ A to an additive category A, has a total E−derived functor \underline{\underline{L}}F : D_{E}(C) ⟶ A. If F is homological then so is \underline{\underline{L}}F.

(2) Any additive functor F : C ⟶ B has a total left (E, F)−derived functor \underline{\underline{L}}F : D_{E}(E) ⟶ D_{E}(B). If F is exact then so is \underline{\underline{L}}F.

Proof. In case (2) just set \underline{\underline{L}}F = R_{B}F i : D_{E}(C) ⟶ D_{F}(B). Case (1) is similar.

Remark 6.11. Let U be a Serre subcategory of Ab. Define a class of morphisms R(U) in C as follows: R(U) = ⋃_{A,B∈C} R_{A,B}(U), where a morphism f : A ⟶ B belongs to R_{A,B}(U) if ∀P ∈ P(E), the morphism C(P, f) is an isomorphism in the Gabriel-quotient Ab/U, i.e. kerC(P, f), cokerC(P, f) ∈ U. As above the class R(U) is a multiplicative system of morphisms compatible with the triangulation of C. The class Ph_{E}^{U}(C) of U−, E−phantom maps defined as the class of morphisms f : A ⟶ B such that C(P, f) = 0 in Ab/U, i.e. ImC(P, f) ∈ U, ∀P ∈ P(E), is then an ideal of C. One can develop the theory of the present section to this situation without important changes. The above theory then corresponds to the case U = 0. Note that this applies to the stable homotopy category of spectra, choosing as U suitable Serre subcategories of Ab associated with various sets of primes.

7. The Stable Triangulated Category

Let C be a triangulated category and let E be a proper class of triangles in C. We assume that C has enough E−projectives. We denote by C/P(E) the stable category of C modulo the full subcategory P(E) of E−projective objects. If ϖ : C ⟶ C/P(E) is the canonical functor, then we set: ϖ(A) = \underline{A} and ϖ(f) = \underline{f}.

Our aim is to show that C/P(E) carries in a natural way a left triangulated structure, which in some cases it is useful to study. For the notion of left triangulated categories we refer to [12], or to [43] in which the dual notion is treated.

For each object A ∈ C choose an E−projective presentation of A, i.e. a triangle

K_{A}^{1} ⟶^{g_{A}^{0}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶^{h_{A}^{0}} Σ(K_{A}^{1}) in E with P_{A}^{0} ∈ P(E). By Schanuel’s Lemma, we see that if we choose another E−projective presentation K ⟶ P ⟶ A ⟶ Σ(K) of A, then K_{A}^{1} ⊕ P ≅ K ⊕ P_{A}^{0}. Hence the object K_{A}^{1} is uniquely determined in C/P(E). We denote it by Ω_{E}(\underline{A}) = \underline{K}_{A}^{1}. Let f : A ⟶ B be a morphism in C. Then there are morphisms p_{f}^{0} : P_{A}^{0} ⟶ P_{B}^{0}, k_{f}^{1} : K_{A}^{1} ⟶ K_{B}^{1} and a morphism of triangles

K_{A}^{1} ⟶^{g_{A}^{0}} P_{A}^{0} ⟶^{f_{A}^{0}} A ⟶^{h_{A}^{0}} Σ(K_{A}^{1})
k_{f}^{1} ↓  ^{p_{f}^{0}} ↓  ^{f}↓  ^{Σ(k_{f}^{1} )}↓
K_{B}^{1} ⟶^{g_{B}^{0}} P_{B}^{0} ⟶^{f_{B}^{0}} B ⟶^{h_{B}^{0}} Σ(K_{B}^{1}).

If q_{f}^{0} : P_{A}^{0} ⟶ P_{B}^{0} is another lifting of f which induces a morphism l_{f}^{1} : K_{A}^{1} ⟶ K_{B}^{1} in the above diagram, then h_{A}^{0} ∘ [Σ(k_{f}^{1}) − Σ(l_{f}^{1})] = 0, so there exists a morphism α : Σ(P_{A}^{0}) ⟶ Σ(K_{A}^{1}) such that Σ(k_{f}^{1}) − Σ(l_{f}^{1}) = Σ(g_{A}^{0}) ∘ α. Then the morphism Σ(k_{f}^{1}) − Σ(l_{f}^{1}) or equivalently the morphism k_{f}^{1} − l_{f}^{1} is zero in C/P(E), i.e. \underline{k}_{f}^{1} = \underline{l}_{f}^{1}. Hence setting Ω_{E}(\underline{f}) = \underline{k}_{f}^{1}, we obtain an additive functor Ω_{E} : C/P(E) ⟶ C/P(E),

which we call the E−loop functor of C with respect to E. If T : A ⟶^{α} B ⟶^{β} C ⟶^{γ} Σ(A) is a triangle in E, then T induces a morphism of triangles:

K_{C}^{1} ⟶^{g_{C}^{0}} P_{C}^{0} ⟶^{f_{C}^{0}} C ⟶^{h_{C}^{0}} Σ(K_{C}^{1})
c↓  ^{d}↓  ^{∥}↓  ^{Σ(c)}↓
A ⟶^{α} B ⟶^{β} C ⟶^{γ} Σ(A).

It is easy to see as above that the morphism c : K_{C}^{1} ⟶ A is independent in C/P(E), of the chosen E−projective presentation of C. We call the image of c in C/P(E) the characteristic class of the triangle T and we denote it by \underline{c} := ch(T). Hence the triangle T in E, induces a diagram

\underline{T} : Ω_{E}(\underline{C}) ⟶^{ch(T)} \underline{A} ⟶^{\underline{α}} \underline{B} ⟶^{\underline{β}} \underline{C} in C/P(E).

We define a triangulation Δ_{E} of the pair (C/P(E), Ω_{E}) as follows:

• A diagram Ω_{E}(\underline{C}′) ⟶ \underline{A}′ ⟶ \underline{B}′ ⟶ \underline{C}′ in C/P(E) belongs to Δ_{E} iff it is isomorphic to a diagram of the form \underline{T}, where T is a triangle in E.

Theorem 7.1. The triple (C/P(E), Ω_{E}, Δ_{E}) is a left triangulated category.

Proof. The proof is similar to the proof of the main Theorem of [12]. We prove only how any morphism \underline{f} : \underline{B} ⟶ \underline{C} is embedded in a triangle in C/P(E). Consider the base-change diagram of the E−projective presentation of C along the morphism f:

K_{C}^{1} ⟶^{α} A ⟶^{β} B ⟶^{γ} Σ(K_{C}^{1})
∥↓  ^{δ}↓  ^{f}↓  ^{∥}↓
K_{C}^{1} ⟶^{g_{C}^{0}} P_{C}^{0} ⟶^{f_{C}^{0}} C ⟶^{h_{C}^{0}} Σ(K_{C}^{1}).

Then consider the cobase-change diagram of the E−projective presentation of C along the morphism α : K_{C}^{1} ⟶ A:

K_{C}^{1} ⟶^{g_{C}^{0}} P_{C}^{0} ⟶^{f_{C}^{0}} C ⟶^{h_{C}^{0}} Σ(K_{C}^{1})
α↓  ^{ε}↓  ^{∥}↓  ^{Σ(α)}↓
A ⟶^{ζ} D ⟶^{θ} C ⟶^{η} Σ(A)

We know that the lower triangle in the above diagram is in E. Hence by the definition of Δ_{E} we have a triangle Ω_{E}(\underline{C}) ⟶ \underline{A} ⟶ \underline{D} ⟶ \underline{C} in C/P(E). It is not difficult to see that the above two diagrams produces an isomorphism D ≅ B ⊕ P_{C}^{0}. Hence \underline{D} ≅ \underline{B}, such that the morphism \underline{D} ⟶ \underline{C} is isomorphic to \underline{f}. Hence the

diagram Ω_{E}(\underline{C}) ⟶ \underline{A} ⟶ \underline{B} ⟶^{\underline{f}} \underline{C}, belongs to the triangulation Δ_{E} of C/P(E).

The stable category C/P(E) is not necessarily triangulated. The next result describes when this happens. Its proof is left to the reader (compare [12]).

Theorem 7.2. The stable category C/P(E) is triangulated iff C has enough E−injectives and P(E) = I(E). If C/P(E) is triangulated, then E−gl.dimC = 0 or ∞. Moreover E−gl.dimC = 0 iff E = Δ_{0}.

8. Projectivity, Injectivity and Flatness

Throughout this section we fix a triangulated category C with coproducts and a full skeletally small additive subcategory 0 ≠ X ⊆ C, closed under isomorphisms, Σ, Σ^{−1}, and direct summands. Our aim is to study the homological theory of C based on the proper class of triangles E(X):

• A triangle A ⟶ B ⟶ C ⟶ Σ(A) is in E(X) iff ∀X ∈ X, the induced sequence

0 ⟶ C(X, A) ⟶ C(X, B) ⟶ C(X, C) ⟶ 0 is exact in Ab.

Lemma 8.1. C has enough E(X)−projectives and P(E(X)) = Add(X).

Proof. For any C ∈ C, let I_{C} := {X ⟶ C | X ∈ Iso(X)}. If X_{C} := ⊕_{λ∈I_{C}} X_{λ}, then X_{C} ∈ P(E(X)) since X ⊆ P(E(X)) and P(E(X)) is closed under coproducts. The set I_{C} induces a morphism f : X_{C} ⟶ C, which by construction is an E(X)−proper epic. Then the triangle A ⟶ X_{C} ⟶^{f} C ⟶ Σ(A) ∈ E(X) and C has enough E(X)−projectives. Further Add(X) ⊆ P(E(X)), since P(E(X)) is closed under coproducts, direct summands and contains X. If P ∈ P(E(X)), let A ⟶ X_{P} ⟶^{α} P ⟶ Σ(A) be a triangle in E(X) with X_{P} a coproduct of objects of X. Since P is E(X)−projective, α splits and P is a direct summand of X_{P}, so P ∈ Add(X).

Since C has enough E(X)−projectives, we can apply the results of the previous sections to the pair (C, X). An important example of the above situation occurs in the stable homotopy category of spectra [51], choosing X to be the category of finite spectra. For other examples we refer to Section 12.

Lemma 8.2. (1) f : A ⟶ B is an E(X)−phantom map iff ∀X ∈ X : C(X, f) = 0. (2) The ideal Ph_{E(X)}(C) ⊆ J ac(C) iff ∀X ∈ X : C(X, A) = 0 ⇒ A = 0.

8.1. Projective and Flat Functors. Since X is skeletally small, we can consider the category Mod(X) of right X −modules, i.e. the category of contravariant additive functors from X to the category Ab of abelian groups. Define a functor

S′ : C ⟶ Mod(X), by S′(A) = C(−, A)|_{X} and S′(f) = C(−, f)|_{X}

where C(−, A)|_{X} denotes the restriction of the representable functor to X.

There is a well-defined tensor product functor −⊗_{X} − : Mod(X)×Mod(X^{op}) ⟶ Ab, which satisfies all the usual properties [53]. A functor F in Mod(X) is called flat, if F ⊗_{X} − : Mod(X^{op}) ⟶ Ab is exact. The full subcategory of Mod(X) consisting of flat functors is denoted by Flat(Mod(X)). Also we denote by Proj(Mod(X)), resp. Inj(Mod(X)), the full subcategory of Mod(X) consisting of all projective, resp. injective, objects. Observe that ∀X ∈ X, S′(X) is the representable functor X (−, X), so S′(X) ∈ Proj(Mod(X)).

Remark 8.3. The functor S′ : C ⟶ Mod(X) is homological and a triangle T is in E(X) iff S′(T) is a short exact sequence in Mod(X). Moreover S′ preserves products, and S′ preserves coproducts iff X ⊆ C^{b}. Obviously S′ kills E(X)−phantom maps, so there exists a unique exact functor H : S_{E(X)}(C) ⟶ Mod(X) with HS = S′, where S_{E(X)}(C) is the Steenrod category and S is the projectivization functor.

Proposition 8.4. The following conditions are equivalent:

(i) The functor H : S_{E(X)}(C) ⟶ Mod(X) is an equivalence of categories. (ii) The functor S′ induces an equivalence: P(E(X)) = Add(X) ≈ Proj(Mod(X)). (iii) For any object X ∈ X and any family {X_{i}}_{i∈I} ⊆ X, the natural morphism:

⊕_{i∈I}C(X, X_{i}) ⟶ C(X, ⊕_{i∈I}X_{i}) is an isomorphism. If (i) holds, then identifying S_{E(X)}(C) = Mod(X) and S′ = S, we have that S induces natural isomorphisms:

Ext_{E(X)}^{n}(−, C) ≅ Ext_{Mod(X)}^{n}[S(−), S(C)] : C^{op} ⟶ Ab, ∀C ∈ C, ∀n ≥ 0.

Further the ideal: kerS = Ph_{E(X)}(C), and if X generates C, then ∀A ∈ C : E(X)−p.dA = p.dS(A). In particular: E(X)−gl.dimC ≤ gl.dimMod(X).

Proof. (ii) ⇒ (iii) Let {X_{i}}_{i∈I} be any family of objects in X. Since ⊕_{i∈I}X_{i} is in P(E(X)), by hypothesis we have ⊕_{i∈I}S′(X_{i}) ≅ S′(⊕_{i∈I}X_{i}). It follows that for any X ∈ X: ⊕_{i∈I}C(X, X_{i}) = ⊕_{i∈I}S′(X_{i})(X) ≅ S′(⊕_{i∈I}X_{i})(X) = C(X, ⊕_{i∈I}X_{i}).

(iii) ⇒ (ii) Since S′ restricted to X is the Yoneda embedding, it follows that S′|_{X} : X ↪ P is fully faithful where P is the category of finitely generated projective functors in Mod(X). Since Proj(Mod(X)) = Add(P), the hypothesis implies easily that S′|_{X} can be extended to a full embedding P(E(X)) = Add(X) ↪ Proj(Mod(X)). Let F be a projective functor. Then there exists a family {X_{i}}_{i∈I} ⊆ X such that F ⊕ G = ⊕_{i∈I}S′(X_{i}) = S′(⊕_{i∈I}X_{i}), for some functor G. This decomposition gives us an idempotent f : S′(⊕_{i∈I}X_{i}) ⟶ S′(⊕_{i∈I}X_{i}) with Im(f) = F. Then there exists an idempotent e : ⊕_{i∈I}X_{i} ⟶ ⊕_{i∈I}X_{i} such that S′(e) = f. Since C has coproducts, by [28] the idempotent e splits producing a direct sum decomposition ⊕_{i∈I}X_{i} = K ⊕ L with Im(e) = K. Then obviously S′(K) ≅ F and consequently S′ : P(E(X)) ⟶ Proj(Mod(X)) is an equivalence.

(i) ⇔ (ii) If condition (i) holds, then H restricts to an equivalence between ProjS_{E(X)}(C) and Proj(Mod(X)). Since P(E(X)) = Add(X) has split idempotents, by Proposition 4.19 we have an equivalence P(E(X)) ≈ ProjS_{E(X)}(C), induced by S. It follows that S′ = HS induces an equivalence P(E(X)) ≈ Proj(Mod(X)). Conversely if (ii) holds, then the functor S′ induces the desired equivalence H : S_{E(X)}(C) = A(P(E(X))) ⟶^{≈} A(Proj(Mod(X))) = Mod(X).

The last assertions follow from Section 4.

We call X self-compact if X satisfies condition (iii) of Proposition 8.4. From now on we write S = S′, so that if X is self-compact, S′ is the projectivization functor and S_{E(X)}(C) = Mod(X). We would like to know when S has its image in the subcategory Flat(Mod(X)) of flat functors. The motivation for this is that in case X is triangulated, the category Coh(X^{op}, Ab), resp. Hom(X , Ab), of cohomological, resp. homological, functors over X and the category Flat(Mod(X)), resp. Flat(Mod(X^{op})), of flat right, resp. left, X −modules, coincide [13]. It is natural to compare C with the cohomological functors over X, via the homological functor S.

Proposition 8.5. The following conditions are equivalent:

(i) ImS ⊆ Flat(Mod(X)). (ii) X has weak cokernels and the inclusion i : X ↪ C preserves them. (iii) X has weak cokernels and any morphism in X is a weak kernel. (iv) X has weak cokernels and any injective right X −module is flat.

Proof. (ii) ⇒ (i) Since X has weak cokernels, it follows from [13] that a functor F ∈ Mod(X) is flat iff F sends diagrams of the form (∗) : X ⟶^{f} Y ⟶^{g} Z, where g is a weak cokernel of f in X, to exact sequences in Ab. Hence we must show that if (∗)

is a diagram as above, then S(A)(Z) ⟶^{S(A)(g)} S(A)(Y) ⟶^{S(A)(f)} S(A)(X) is exact,

∀A ∈ C. Obviously this diagram is isomorphic to the complex (†) : C(Z, A) ⟶^{C(g,A)}

C(Y, A) ⟶^{C(f,A)} C(X, A). Let X ⟶^{f} Y ⟶^{g′} A′ ⟶^{h} Σ(X) be a triangle in C. Since g is a weak cokernel of f in X and the inclusion i : X ↪ C preserves weak cokernels, there are morphisms τ : Z ⟶ A′ and ρ : A′ ⟶ Z such that: g′ = g∘τ and g = g′∘ρ. Then let ω ∈ KerC(f, A) be such that f ∘ ω = 0. Since g′ is a weak cokernel of f, there exists a morphism ϕ : A′ ⟶ A with g′ ∘ϕ = ω. Then g ∘τ ∘ϕ = ω ⇒ ω ∈ ImC(g, A). Hence (†) is exact in Ab, and consequently S(A) is a flat functor in Mod(X).

(i) ⇒ (ii) Let f : X ⟶ Y be a morphism in X and let X ⟶^{f} Y ⟶^{g} A ⟶^{h} Σ(X) be a triangle in C. Since S is homological, we have an exact sequence in Mod(X):

S(X) ⟶^{S(f)} S(Y) ⟶^{S(g)} S(A) ⟶^{S(h)} S(Σ(X)). Let κ : S(Y) ⟶ F be the cokernel of S(f), and let λ : F ↪ S(A) be the canonical inclusion, such that S(g) = κ ∘ λ. Since S(X), S(Y) are representable, F is finitely presented. By hypothesis S(A) is a flat functor. Hence the morphism λ factors through some representable functor X (−, Z) = S(Z) with Z ∈ X. So there are morphisms m : F ⟶ S(Z) and ν : S(Z) ⟶ S(A) such that: λ = m ∘ ν. Since Z ∈ X, and S|_{X} is fully faithful, there exists a unique morphism γ : Z ⟶ A such that ν = S(γ). Similarly for the morphism κ ∘ m : S(Y) ⟶ S(Z), there exists a unique morphism α : Y ⟶ Z with: κ ∘ m = α. Then S(g) = κ ∘ λ = κ ∘ m ∘ ν = S(α) ∘ S(γ) = S(α ∘ γ). Since S|_{X} is fully faithful, g = α ∘ γ. We show that α : Y ⟶ Z is a weak cokernel of f in X. First S(f) ∘ S(g) = 0 ⇒ S(f) ∘ κ ∘ λ = 0 ⇒ S(f) ∘ κ = 0 ⇒ S(f) ∘ κ ∘ m = 0 ⇒ S(f) ∘ S(α) = 0 ⇒ f ∘ α = 0. Now let ρ : Y ⟶ W be a morphism in X such that f ∘ ρ = 0. Since g is a weak cokernel of f in C, there exists a morphism τ : A ⟶ W with ρ = g ∘ τ. But then ρ factors through α as ρ = α ∘ γ ∘ τ. Hence X has weak cokernels. To show that the inclusion i : X ↪ C preserves them, it suffices to show that if ω : Y ⟶ B is a morphism in C with f ∘ ω = 0, then ω factors through α. If f ∘ ω = 0, there exists ϕ : A ⟶ B with g ∘ ϕ = ω. Then α ∘ γ ∘ ϕ = ω.

(iii) ⇔ (iv) Follows from [13]. (ii) ⇔ (iii) The easy proof is left to the reader.

8.2. Injective Objects and Functors. Assume, besides the default assumptions concerning the pair (C, X), that X generates C and consists of compact objects.

Let F ∈ Flat(Mod(X^{op})), i.e. F : X ⟶ Ab is a flat functor. Consider the functor

𝔻[S(−) ⊗_{X} F] : C^{op} ⟶ Ab (∗)

where 𝔻 = Hom_{ℤ}(−, ℚ/ℤ) and 𝔻[S(−) ⊗_{X} F](A) = Hom_{ℤ}[S(A) ⊗_{X} F, ℚ/ℤ]. Since F is a flat, the above functor is cohomological and converts coproducts to products. Since C is compactly generated by X, by Brown Representability Theorem [55] there exists an object D(F) ∈ C, unique up to isomorphism, and a natural isomorphism

ω : 𝔻[S(−) ⊗_{X} F] ⟶^{≅} C[−, D(F)] (+)

Since S sends triangles in E(X) to short exact sequences in Mod(X), it follows that 𝔻[S(−) ⊗_{X} F], hence C[−, D(F)], sends triangles in E(X) to short exact sequences in Ab. We infer that D(F) is an E(X)−injective object in C.

Choose F = X (X, −) in (∗), where X ∈ X. Then 𝔻[S(−) ⊗_{X} X (X, −)] = 𝔻C(X, −). In this case we denote the object D(X (X, −)) by 𝔻(X). Observe that if C(C, 𝔻(X)) = 0, ∀X ∈ X, then 𝔻C(X, C) = 0 ⇒ C(X, C) = 0, ∀X ∈ X. Since X generates C, C = 0. Hence the set {𝔻(X); X ∈ Iso(X)} cogenerates C.

For a class of objects Y ⊆ C, let Prod(Y) be the full subcategory of retracts of products of objects of Y. The following generalizes results of [50], [21].

Theorem 8.6. C has E(X)−injective envelopes and I(E(X)) = Prod(𝔻(X)). The functor S induces an equivalence S : I(E(X)) ⟶^{≈} Inj(Mod(X)). Finally for any E(X)−injective object E, the ring Λ := End_{C}(E) is F −semiperfect, i.e. Λ/J ac(Λ) is Von Neumann regular and idempotents can be lifted modulo J ac(Λ). In particular an indecomposable E(X)−injective object in C has a local endomorphism ring.

Proof. For C ∈ C, let J_{C} := {C ⟶ 𝔻(X) | X ∈ Iso(X)}. Set E_{C} = ∏_{μ∈J_{C}} 𝔻(X_{μ}). The set J_{C} induces a morphism g_{C} : C ⟶ E_{C}. Let (T) : C ⟶^{g_{C}} E_{C} ⟶ A ⟶ Σ(C) be a triangle in C. Applying ∀X ∈ X, the functor C(−, 𝔻(X)) to the triangle it follows by construction that 0 ⟶ C(A, 𝔻(X)) ⟶ C(E_{C}, 𝔻(X)) ⟶ C(C, 𝔻(X)) ⟶ 0 is exact. But this sequence is isomorphic to the sequence 0 ⟶ 𝔻C(X, C) ⟶ 𝔻C(X, E_{C}) ⟶ 𝔻C(X, A) ⟶ 0. Since ℚ/ℤ cogenerates Ab, the sequence 0 ⟶ C(X, C) ⟶ C(X, E_{C}) ⟶ C(X, A) ⟶ 0 is exact, i.e. (T) ∈ E(X). Since E_{C} is E(X)−injective, we have that C has enough E(X)−injectives and the characterization of E(X)−injective objects follows. By Theorem 4.24, S : I(E(X)) ↪ Inj(Mod(X)) is fully faithful. Let F be an injective functor in Mod(X). Applying Brown Representability to the functor C(S(−), F), we have an isomorphism ω : C(S(−), F) ⟶ C(−, E_{F}) and plainly E_{F} is in I(E(X)). Then ω|_{X} : C(S(−), F)|_{X} = F ≅ S(E_{F}), so S is surjective. By Theorem 4.24, C has E(X)−injective envelopes. The last assertions are true since they are true in Mod(X), see [39].

We close this subsection showing that any object of C admits a functorial “embedding” into an E(X)−injective object. We assume from now on that X is a triangulated subcategory of C. By Proposition 8.5, for any object A ∈ C, S(A) is flat right X −module. It is well-known that 𝔻S(A) is an injective left X −module. Since X is a triangulated subcategory of C, by the dual of Proposition 8.5, we infer that 𝔻S(A) is a flat left X −module. Hence choosing F = 𝔻S(A) in (∗), we obtain an E(X)−injective object D(𝔻S(A)) in C, which we denote by D(A), equipped with the natural isomorphism ω : 𝔻[S(−) ⊗_{X} 𝔻S(A)] ⟶^{≅} C[−, D(A)].

Theorem 8.7. The assignement A ↦ D(A) defines an additive functor D : C ⟶ C and ∀A ∈ C, the object D(A) is E(X)−injective. Further there exists a natural morphism δ : Id_{C} ⟶ D, such that δ_{A} : A ⟶ D(A) is an E(X)−proper monic.

Proof. It is well known [65] that there is a canonical morphism F ⊗_{X} 𝔻(G) ⟶ 𝔻(F, G) which is invertible if F is finitely presented. Hence we have a natural morphism ϕ : S(−) ⊗_{X} 𝔻S(A) ⟶ 𝔻[S(−), S(A)] such that ϕ|_{X} is invertible. We denote by ψ : C(−, A) ⟶ C[−, D(A)] the composition

C(−, A) ⟶^{S_{−,A}} [S(−), S(A)] ⟶^{μ} 𝔻^{2}[S(−), S(A)] ⟶^{𝔻(ϕ)}

⟶^{𝔻(ϕ)} 𝔻[S(−) ⊗_{X} 𝔻S(A)] ⟶^{ω} C[−, D(A)]

where μ is the canonical monomorphism. Observe that ψ|_{X} : S(A) ⟶ S(D(A)) is a monomorphism. By Yoneda’s Lemma there exists a unique morphism δ_{A} : A ⟶ D(A) such that ψ = C(−, δ_{A}). Then ψ|_{X} = C(−, δ_{A})|_{X} = S(δ_{A}) and since ψ|_{X} is a monomorphism, the same is true for S(δ_{A}), i.e. δ_{A} is an E(X)−proper monic. We leave to the reader to prove that A ↦ D(A) defines an additive functor D : C ⟶ C and {δ_{A}}_{A∈C} are the components of a natural morphism δ : Id_{C} ⟶ D.

Note that we have the formula: SD ≅ 𝔻^{2}S, so D can be regarded as a “double dual” functor in C. The next result is a direct consequence of the above Theorems.

Corollary 8.8. ∀A, B ∈ C, ∀X ∈ X: Ph_{E(X)}(A, 𝔻(X)) = 0 = Ph_{E(X)}(A, D(B)). Moreover f ∈ Ph_{E(X)}(C) iff D(f) = 0 iff C(f, 𝔻(X)) = 0, ∀X ∈ X.

8.3. Homological Functors Preserving Coproducts. Assume from now on that X is self-compact and satisfies the following condition (††) X has weak kernels and the inclusion X ↪ C preserves them, which is dual to the condition (ii) of Proposition 8.5.

Fix a coproduct preserving homological functor H : C ⟶ Ab. Set H′ := H|_{X} : X ⟶ Ab and let H^{∗} := − ⊗_{X} H′ : Mod(X) ⟶ Ab be the unique colimit preserving functor extending H′, through the Yoneda embeding X ↪ Mod(X). Using Corollary 4.21, it is not difficult to see that H^{∗}S ≅ L_{0}^{E(X)}H. Let ω : L_{0}^{E(X)}H ⟶ H be the canonical morphism, in particular the restriction ω|_{P(E(X))} is invertible. Since H is homological and X satisfies condition (††), by [13] it follows easily that H′ is flat, so H^{∗} is exact (see also [50]). We infer that H^{∗}S = L_{0}^{E(X)}H is homological.

Lemma 8.9. Assume that (α) : any object of C has finite E(X)−projective dimension or (β): X ⊆ C^{b} and X generates C. Then ω : L_{0}^{E(X)}H ⟶^{≅} H.

Proof. (α) Assume first that E(X)−p.dA ≤ 1 and let P_{1} ⟶ P_{0} ⟶ A ⟶ Σ(P_{1}) be an E(X)−projective resolution of A. Since ω|_{P(E(X))} is invertible, applying the homological functors L_{0}^{E(X)}H, H to the above triangle and using 5-Lemma, it follows that ω_{A} : L_{0}^{E(X)}H(A) ⟶ H(A) is invertible. Since any object of C has finite E(X)−projective dimension, by induction we have that ω is an isomorphism.

(β) Set U := {A ∈ C | ω_{A} : L_{0}^{E(X)}H(A) ⟶^{≅} H(A)}. Since L_{0}^{E(X)}H, H are

coproduct preserving homological functors, U is a full triangulated subcategory of C, closed under coproducts and contains the compact generating subcategory X of C. It follows that U = C and consequently ω is invertible.

Theorem 8.10. Under the assumptions of Lemma 8.9, we have the following.

(i) There exists an exact colimit preserving functor H^{∗} : Mod(X) ⟶ Ab, unique up to isomorphism, such that: H^{∗}S = H. The functor H^{∗} admits a right adjoint which preserves injectives and we can identify S(−) ⊗_{X} H|_{X} = H.

(ii) H(f) = 0, ∀f ∈ Ph_{E(X)}(C). (iii) If Flat^{b}(Mod(C^{op})) is the category of coproduct preserving homological func-

tors C ⟶ Ab, then the assignment F ⟼ F^{∗}S induces an equivalence: Flat(Mod(X^{op})) ⟶^{≈} Flat^{b}(Mod(C^{op})), with inverse given by H ⟼ H|_{X}.

Proof. (i) Follows from the above analysis and Lemma 8.9. (ii) Follows from (i), since S kills E(X)−phantom maps. (iii) The easy proof is left to the reader.

Remark 8.11. In Theorem 8.10, Ab can be replaced by any abelian AB5 category.

8.4. The Roos Spectral Sequence. Let {F_{i}} be a filtered system of functors in Mod(X) and let G be another functor. Then by [65] we have a Roos’s spectral sequence: E_{2}^{p,q} = lim_{←}^{(p)}Ext^{q}[F_{i}, G] ⇒ Ext^{n}[lim_{→}F_{i}, G]. Assume now that X compactly generates C and satisfies condition (ii) of Proposition 8.5. Let A, B ∈ C. Since S(A) is flat, there exists a functor I : I ⟶ X from a small filtered category I such that S(A) = lim_{→}S(X_{i}), where X_{i} = I(i). Trivially the Roos spectral sequence for the functors S(A), S(B) collapses giving isomorphisms

lim_{←}_{I}^{(n)}C(X_{i}, B) ≅ Ext^{n}[S(A), S(B)] ≅ Ext_{E(X)}^{n}[A, B], ∀n ≥ 0.

Corollary 8.12. E(X)−i.dB ≤ n iff lim_{←}_{I}^{(t)}C(X_{i}, B) = 0, for all t ≥ n + 1 and for

all filtered direct systems {X_{i}} of objects of X.

Let A_{0} ⟶ A_{1} ⟶ ⋯ be a tower of objects of C. Since X is compact, the functor S preserves coproducts. Hence by Lemma 5.8 we have: S(holim_{⟶}A_{i}) ≅ lim_{→}S(A_{i}).

Since for a tower {G_{n}} in Ab, lim_{←}^{(p)}G_{n} vanishes ∀p ≥ 2, the next result follows from Roos’s spectral sequence by standard arguments.

Corollary 8.13. ∀B ∈ C and ∀n ≥ 0, there exists a short exact sequence

0 ⟶ lim_{←}^{(1)}Ext_{E(X)}^{n−1}[A_{i}, B] → Ext_{E(X)}^{n}[holim_{⟶}A_{i}, B] ⟶ lim_{←}Ext_{E(X)}^{n}[A_{i}, B] ⟶ 0.

If the tower above is the E−cellular tower of A, then since A ≅ holim_{⟶}A_{i}, the above short exact sequence gives a method to compute the extension functors of A by means of the extension functors of its cells A_{i}.

9. Phantomless Triangulated Categories

Let C be a triangulated category with coproducts and let X ⊆ C be a skeletally small full additive subcategory, closed under Σ, Σ^{−1}, direct summands and isomorphisms. Our main result in this section characterizes the E(X)−phantomless triangulated categories and generalizes results of Neeman [54]. First we need a simple Lemma and some definitions.

Lemma 9.1. If C has products and X ⊆ C^{b}, then for any set of objects {A_{i}}_{i∈I} ⊆ C, the canonically defined triangle ⊕_{i∈I}A_{i} ⟶ ∏_{i∈I} A_{i} → A → Σ(⊕_{i∈I}A_{i}) is in E(X).

Proof. Since S preserves products and also coproducts (since X ⊆ C^{b}), the exact sequences ⋯ ⟶ S(⊕_{i∈I}A_{i}) ⟶ S(∏_{i∈I} A_{i}) ⟶ S(A) ⟶ ⋯ and ⋯ ⟶ ⊕_{i∈I}S(A_{i}) ⟶ ∏_{i∈I} S(A_{i}) ⟶ S(A) ⟶ ⋯ are isomorphic. Since in module categories the morphism from a coproduct to the product is a monomorphism, we see that 0 ⟶ S(⊕_{i∈I}A_{i}) ⟶ S(∏_{i∈I} A_{i}) ⟶ S(A) ⟶ 0 is exact and the assertion follows.

We recall [13] that an additive category C is called weak abelian if C has weak kernels, weak cokernels and any morphism is a weak kernel and a weak cokernel. Trivially triangulated categories are weak abelian. We recall [30] that a Grothendieck category G is called locally Noetherian, resp. locally Artinian, resp. locally finite, if G has a set of generators consisting of Noetherian, resp. Artinian, resp. finite length, objects. A functor category is called perfect [39] if any flat functor is projective. We need the following result from [13].

Proposition 9.2. [13] Let C be a skeletally small additive category with split idempotents. Then the following are equivalent:

(i) Mod(C) is a Frobenius category.

(ii) C is weak abelian and Mod(C) is locally Noetherian. (iii) C is weak abelian and Mod(C) is perfect. (iv) C is weak abelian and Proj(Mod(C)), resp. Inj(Mod(C)), is closed under prod-

ucts, resp. coproducts. In this case the projective, injective and flat functors coincide. Moreover the module category Mod(C) is Frobenius and locally finite iff Mod(C^{op}) is Frobenius and locally finite iff C is weak abelian and Mod(C), Mod(C^{op}) are locally Noetherian (perfect).

Let F be an additive category. We recall that an object A ∈ F is called finitely presented if the functor F(A, −) commutes with filtered colimits. Following [24] we say that F is locally finitely presented if F has filtered colimits, any object of F is a filtered colimit of finitely presented objects and the full subcategory of finitely presented objects of F is skeletally small. Finally we recall that an additive category is a Krull-Schmidt category, if any of its objects is a finite coproduct of indecomposable objects and any indecomposable object has a local endomorphism ring. In case C is compactly generated and X = C^{b}, then parts (1), (12), (14), (15) of the next result have been observed independently by H. Krause [50].

Theorem 9.3. The following statements are equivalent:

(1) The functor S : C ⟶ Mod(X) induces an equivalence: C ≈ Flat(Mod(X)).

(2) X ⊆ C^{b} and E(X)−gl.dimC = 0.

(3) X ⊆ C^{b} and Ph_{E(X)}(C) = 0.

(4) X ⊆ C^{b} and C = Add(X), resp. X ⊆ C^{b} generates C and C = Prod(𝔻(X)).

(5) X generates C, X ⊆ C^{b} and the module category Mod(X) is Frobenius.

(6) X generates C, X ⊆ C^{b} is weak abelian and Add(X), resp. I(E(X)), is closed under products, resp. coproducts.

(7) X generates C, X ⊆ C^{b} is weak abelian and C is E(X)−Frobenius, i.e. C has enough E(X)−injectives and P(E(X)) = I(E(X)).

(8) The unique (exact) extension S^{∗} : A(C) ⟶ Mod(X) of S through the Freyd category A(C) of C, is an equivalence of categories.

(9) For any abelian category M and for any homological functor H : C ⟶ M, there exists a unique exact functor H^{∗} : Mod(X) ⟶ M with H^{∗}S = H.

(10) X generates C, X ⊆ C^{b} is weak abelian (resp. and C has products) and the following conditions are true: (α) For any family {X_{i}}_{i∈I} ⊆ X the canonical (mono-)morphism

μ : ⨁_{i∈I} C(−, X_{i}) ⟶ ∏_{i∈I} C(−, X_{i}), (resp. and ν : ⨁_{i∈I} X_{i} ⟶ ∏_{i∈I} X_{i}) splits.

(β) For any set I and directed family {X_{i}}_{i∈I} ⊆ X and for any Y ∈ X :

lim_{←}_{I}^{(n)}C(X_{i}, Y) = 0, ∀n ≥ 1.

(11) X generates C, X ⊆ C^{b} is weak abelian and if {X_{i}; i ∈ I} and {Y_{i}; j ∈ J} are filtered direct systems of objects of X, then:

lim_{←}_{I}^{(n)}lim_{→}_{J}C(X_{i}, Y_{j}) = 0, ∀n ≥ 1.

(12) X ⊆ C^{b} is a Krull Schmidt weak abelian category which generates C and for any sequence X_{1} ⟶^{f_{1}} X_{2} ⟶^{f_{2}} X_{3} ⟶ ⋯ of non-isomorphisms between indecomposable objects of X, there exists N ∈ ℕ, such that f_{1}∘f_{2}∘⋯∘f_{N} = 0.

(13) X ⊆ C^{b} generates C and E(X) = lim_{→}Δ_{0}, i.e. a triangle is in E(X) iff it is a filtered colimit of split triangles.

(14) X ⊆ C^{b} and if A ∈ C then A = ⨁_{i∈I} X_{i}, where X_{i} ∈ X and End_{C}(X_{i}) is a local ring. Moreover any two such decompositions are isomorphic.

(15) X = C^{b} and C is a locally finitely presented category.

If one of the above conditions is true, then C is a pure semisimple locally finitely presented category with products with X = C^{b} as its full subcategory of finitely presented objects. Further the projectivization functor S induces equivalences

C = Add(C^{b}) ≈ Proj(Mod(C^{b})) = Flat(Mod(C^{b})) = Inj(Mod(C^{b}))

and a triangle equivalence: Δ/Δ_{0} ≈ \underline{Mod}(C^{b}), where Δ/Δ_{0} is the stable triangulated category of all triangles modulo the split triangles [13] and \underline{Mod}(C^{b}) is the stable triangulated category of the module category Mod(C^{b}) modulo projectives.

Proof. In the first part of the proof we show that the first nine statements are equivalent by the scheme: (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (1), (5) ⇔ (6), (7), (8) and (8) ⇔ (9). In the second part we show that (5) ⇔ (10), (11), (12), that (3) ⇔ (13), that (4) ⇔ (14) and finally that (1) ⇔ (15).

If (1) is true, then since S is faithful, we have kerS = Ph_{E(X)}(C) = 0. Trivially X ⊆ C^{b} generates C. Hence (1) ⇒ (2). Trivially (2) ⇒ (3), (3) ⇒ (4) and by Theorem 4.25 we have (4) ⇒ (5). If (5) is true then since X ⊆ C^{b}, by Proposition 8.4, the functor S : P(E(X)) ⟶ Proj(Mod(X) = Flat(Mod(X) is an equivalence. Since by Proposition 9.2, X is weak abelian, it follows by Proposition 8.5 that ImS consists of flat = projective modules. Now since X generates C , ∀A ∈ C: E(X)−p.dA = p.dS(A) = 0. This implies that E(X)−gl.dimC = 0, hence C = P(E(X)) ≈ Flat(Mod(X)), so (1) is true. (5) ⇔ (6) Since Add(X) ≈ Proj(Mod(X)) and I(E(X)) ≈ Inj(Mod(X)), the category Add(X), resp. I(E(X)), is closed under products, resp. coproducts, iff Proj(Mod(X)), resp. Inj(Mod(X)), is closed under products, resp. coproducts. Then the assertion follows from Proposition 9.2. (5) ⇔ (7) If X compactly generates C and X is weak abelian, then trivially Mod(X) is Frobenius iff C is E(X)−Frobenius and the claim follows. (5) ⇒ (8) Since (5) ⇒ (2) and Mod(X) is the Steenrod category of C, (8) follows from Theorem 4.25. That (8) ⇒ (5) is a consequence of the fact that A(C) is Frobenius. Finally (8) ⇔ (9) follows from the fact that A(C) is the universal homological category of C.

So far we proved that the conditions (1), ..., (9) are equivalent. Observe that the conditions (1), (4), imply that X = C^{b}. In particular X is a triangulated subcategory of C and C is compactly generated.

(5) ⇔ (10) If (5) is true then ⊕_{i∈I}C(−, X_{i}) is injective, since Mod(X) is Frobenius. Hence the canonical monomorphism μ splits. Let {X_{i}}_{i∈I} ⊆ X, with I a filtered set and let Y ∈ X. Consider the (flat) functor F = lim_{→}_{I}X (−, X_{i}). From

the Roos spectral sequence of subsection 8.4 for the functors F, X (−, Y), it follows that lim_{←}_{I}^{(n)}C(X_{i}, Y) ≅ Ext^{n}[F, X (−, Y)], ∀n ≥ 1. Since any flat is projective, we

have Ext^{n}(F, X (−, Y)) = 0, ∀n ≥ 1, hence the condition (10)(β) holds. For the parenthetical case observe that since Flat(Mod(X)) has products, by (1) we have that C has products, hence the canonical morphism ν is defined. By Lemma 9.1, ν is an E(X)−proper monic, which splits since Ph_{E(X)}(C) = 0 by (2). Conversely if (10) is true, then by (β) we have Ext^{n}[F, X (−, Y)] = 0, ∀n ≥ 1, for any flat functor F and any object Y ∈ X, again from the Roos’s spectral sequence. Since X is weak abelian, X has weak cokernels so by [40], we have that X (−, Y) is pure injective. Then by condition (α), any free functor is pure injective. It follows that any projective is pure injective and then by [40], the category Mod(X) is perfect, i.e. any flat functor is projective. By Proposition 9.2, Mod(X) is Frobenius. For the parenthetical case observe that μ is isomorphic to S(ν), since X ⊆ C^{b}.

(5) ⇔ (11) As in the proof of (5) ⇔ (10), using the Roos’s spectral sequence.

(5) ⇔ (12) Suppose that (5) is true. Then Mod(X) is perfect and the assertion follows from the characterization of perfect functor categories [39]. If (12) is true, by [39] we have that Mod(X) is perfect, hence Frobenius since X is weak abelian.

(4) ⇔ (14) If (4) is true then by (5), Mod(X) is perfect and C is identified with Proj(Mod(X)). Then the assertion follows from [39]. The converse is trivial.

(3) ⇔ (13) If any triangle in E(X) is a filtered colimit of split triangles, then obviously C is E(X)−phantomless. Conversely if C is E(X)−phantomless, then the assertion follows from the equivalence of (3) with the condition (1).

(1) ⇔ (15) If (1) is true, then C = Flat(Mod(X)) which obviously is locally finitely presented. Moreover by (4), the full subcategory of finitely presented objects of C coincides with X = C^{b}. By [24] and condition (11), we have that C is puresemisimple. Conversely if (15) is true, and if Y is the category of finitely presented objects of C, then obviously Y ⊆ C^{b} and we know by [24] that the functor S : C ⟶ Flat(Mod(Y)) is an equivalence. By condition (1) with X replaced by Y it follows that C is E(Y)−phantomless. This implies as above that Y = C^{b}. Since by hypothesis C^{b} = X, it follows that X = Y and (15) implies (1).

The last part follows from the above proof, noting that by [13], we have always an equivalence Δ/Δ_{0} ≈ \underline{A(C)}, where the latter is the stable category of the Frobenius category A(C) modulo projectives.

Example 9.4. Any Semisimple algebraic stable homotopy category in the sense of Hovey, Palmieri and Strickland [38], satisfies any one of the conditions of Theorem 9.3. For other examples we refer to Section 12.

We recall that an additive category D is called (von Neumann) regular if for any morphism f : X ⟶ Y in D, there exists g : Y ⟶ X such that: f ∘ g ∘ f = f. It is not difficult to see that a triangulated category D is abelian iff D is semisimple abelian iff D is regular with split idempotents. It is well known that if D is regular and skeletally small, then all left or right modules over D are flat.

Corollary 9.5. If C is compactly generated, then the following are equivalent: (i) All triangles in C are semi-split. (ii) C is E(C^{b})−phantomless and C^{b} is von Neumann regular. (iii) C is E(C^{b})−phantomless and C^{b} is (semisimple) abelian. (iv) The Steenrod category Mod(C^{b}) is semisimple abelian.

(v) The projectivization functor S : C ⟶ Mod(C^{b}) is an equivalence.

9.1. Puppe Triangulations. Let F be an abelian category with enough injectives. By [43], the stable category \underline{F} modulo injectives, is right triangulated with suspension functor Ω^{−1} : \underline{F} ⟶ \underline{F}.

Definition 9.6. A stable structure on F is a pair (Σ, δ), where Σ : F ⟶ F is an automorphism of F and δ : Σ ⟶ Ω^{−3} is a natural isomorphism in \underline{F}, such that δΩ^{−1} + Ω^{−1}δ = 0. The last condition makes sense since we can identify ΣΩ^{−1} ≅ Ω^{−1}Σ. F is called Σ−stable if F admits a stable structure (Σ, δ).

Observe that if F is Σ−stable, then F is Frobenius. Now let P be an additive category equipped with an automorphism Σ : P ⟶ P. A class P of diagrams A ⟶ B ⟶ C ⟶ Σ(A) in P is called a Puppe-triangulation of the pair (P, Σ), if P satisfies all the axioms of a triangulation, except possibly of the Octahedral Axiom. In this case P is called Puppe-triangulated, see [57] where the terminology pretriangulated is used. However the last term has been used by many authors to mean different things and we shall avoid it. Our terminology is justified by the fact that according to A. Heller [34], D. Puppe was the first who introduced the Axioms T_{1}, T_{2}, T_{3} of Section 2, see for instance [60].

Example 9.7. Let (T , Σ, Δ) be a Puppe-triangulated category. Then the Freyd category A(T ) is Frobenius and the suspension Σ of T induces an auto-equivalence

Σ_{∗} : A(T ) ⟶ A(T ), defined by Σ_{∗}(F) = CokerT (−, Σ(f)), where T (−, A) ⟶^{T (−,f)} T (−, B) ⟶ F ⟶ 0 is a finite presentation of F ∈ A(T ). We leave to the reader to check that in fact A(T ) is Σ_{∗}−stable.

The following is due to Heller and was observed independently by Keller-Neeman.

Theorem 9.8. [34], [44] Let F be a Frobenius category and Σ : F ⟶ F be an automorphism of F. If P is the full subcategory of projective-injective objects of F, then there exists a bijective correspondence between Puppe-triangulations P of the pair (P, Σ) and Σ−stable structures (Σ, δ) on F.

In the next two paragraphs and for reasons of comparison, we rename triangulated categories, and we call them Verdier-triangulated categories. Hence a Verdier-triangulated category is a Puppe-triangulated category which satisfies the Octahedral Axiom. Theorem 9.8 suggests the following conjecture, which we believe is due to Bernhard Keller and Amnon Neeman.

Conjecture 9.9. There exists a Σ−stable Frobenius category C, such that the Puppe-triangulated category (P, Σ, P) of Theorem 9.8 is not Verdier-triangulated.

The following provides support for the above conjecture to be true.

Remark 9.10. By example 9.7, a Puppe-triangulated category T is embedded in a Σ−stable Frobenius category as a full subcategory of projective-injective objects. It follows that the above Conjecture fails iff any Puppe-triangulated category is Verdier-triangulated. If the latter holds, then it would follow that for any Verdiertriangulated category T , the Octahedral Axiom T_{4} of Verdier is a formal consequence of the Axioms T_{1}, T_{2}, T_{3}. This is very unlikely to be true, but it is not known if there exists a counterexample.

9.2. Triangulations and Categories of Pro-objects and Ind-objects. Let X be a skeletally small triangulated category. It is very natural to ask if the categories Proj(Mod(X)), Flat(Mod(X)) admit a triangulated structure in such a way that the inclusions X ↪ Proj(Mod(X)), X ↪ Flat(Mod(X)) are exact. We devote this subsection to a discussion of this question.

By Theorem 9.3, if a triangulated category C is E(X)−phantomless with X as in Theorem 9.3, then X = C^{b} is triangulated, the Steenrod category Mod(X) is Frobenius and its category of projectives (equivalently the category of cohomological functors over X), has a triangulated structure which extends the triangulated structure of X. The following result provides a partial converse.

Theorem 9.11. Let D be a skeletally small additive category with split idempotents. Then for the following statements:

(i) The category Flat(Mod(D)) is triangulated. (ii) Mod(D) is Σ−stable. we have (i) ⇒ (ii) and (ii) implies that Flat(Mod(D)) is Puppe-triangulated.

If (i) holds, then D is triangulated, the inclusion i : D ↪ Flat(Mod(D)) is exact, Mod(D) is Frobenius and there are identifications:

Add(D) = Proj(Mod(D)) = Flat(Mod(D)) = Inj(Mod(D)).

Proof. If C := Flat(Mod(D)) is triangulated with suspension Σ, then C has coproducts and C^{b} is a full triangulated subcategory of C. By [24], the category of finitely presented objects of Flat(Mod(D)) can be identified with D. Trivially D ⊆ C^{b} generates C. Let α : F ⟶ G be an E(C^{b})−phantom map in C, i.e. (C^{b}, α) = 0. Then (D, α) = 0, which trivially by Yoneda implies that α = 0. By Theorem 9.3, it follows that D = C^{b}, hence D is a triangulated subcategory of C and Mod(D) is Frobenius, which trivially is Σ−stable. Hence (i) ⇒ (ii). Conversely by Theorem 9.8, (ii) implies that Mod(D) is Frobenius and Flat(Mod(D)) is Puppe-triangulated.

For a skeletally small additive category D, let Pro(D), Ind(D) be the induced Pro-, Ind-categories, see [31]. In [13], we gave necessary and sufficient conditions for Pro(D), Ind(D) to be abelian. It is useful to know when the Pro-, Ind-categories can be endowed with a triangulated structure, since these categories are usually the codomain of derived functors defined on triangulated categories, see [25].

Corollary 9.12. (a) For the following statements:

(i) The category Ind(D), resp. Pro(D), is triangulated. (ii) Mod(D), resp. Mod(D^{op}), is a Σ−stable category. we have (i) ⇒ (ii) and (ii) implies that Ind(D), resp. Pro(D), is Puppe-triangulated.

(b) For the following statements:

(i) The categories Pro(D), Ind(D) are triangulated. (ii) Mod(D), or equivalently Mod(D^{op}), is a locally finite Σ−stable category. we have (i) ⇒ (ii) and (ii) implies that Ind(D) and Pro(D) are Puppe-triangulated. If (i) holds, then Mod(D), Mod(D^{op}) are locally finite Frobenius categories, D is triangulated and the inclusions D ↪ Ind(D), D ↪ Pro(D) are exact.

Proof. It is well known that we can identify Ind(D) = Flat(Mod(D)) and Pro(D) = Flat(Mod(D^{op}))^{op}, and we have an equivalence: Pro(D)^{op} ≈ Ind(D^{op}). Then the assertions are consequences of Theorem 9.11 and Proposition 9.2.

Note that if Conjecture 9.9 is not true, then the map C ↦ Mod(C^{b}) gives a bijective correspondence between E(C^{b})−phantomless compactly generated triangulated categories and Σ−stable Frobenius module categories over skeletally small triangulated categories. The inverse map is given by Mod(D) ↦ Ind(D).

10. Brown Representation Theorems

Throughout this section we assume that the triangulated category C has coproducts. We fix a skeletally small full additive subcategory X ⊆ C which is closed under isomorphisms, direct summands and Σ, Σ^{−1}. Consider the homological functor

S′ : C ⟶ Mod(X), S′(A) = C(−, A)|_{X} and S′(f) = C(−, f)|_{X}

where C(−, A)|_{X} denotes the restriction of the representable functor to X. We recall that an additive functor F : A ⟶ B is called complete [45] or a representation equivalence [1], if F is full, surjective on objects and reflects isomorphisms.

Definition 10.1. The pair (C, X) satisfies the Brown Representability Theorem (BRT for short) if the canonical functor S′ : C ⟶ Mod(X) induces a representation equivalence between C and the flat functors Flat(Mod(X)) over X.

Our aim is to characterize when the pair (C, X) satisfies BRT. By Section 8, the functor S′ satisfies ImS′ ⊆ Flat(Mod(X)) iff the following condition is true: (†) X has weak cokernels and the inclusion functor X ↪ C preserves them. Further S′ induces an equivalence between P(E(X)) and Proj(Mod(X)) iff X is selfcompact iff Mod(X) is the Steenrod category of C with respect to E(X) and S′ = S is the projectivization functor. Our main result in this section is the following:

Theorem 10.2. The following statements are equivalent:

(i) The pair (C, X) satisfies BRT. (ii) (α) X is self-compact, generates C and satisfies condition (†).

(β) Flat(Mod(X)) = {F ∈ Mod(X) | p.dF ≤ 1}. (iii) (α) X is self-compact and satisfies condition (†).

(β) E(X)−gl.dimC ≤ 1.

(γ) Any F ∈ Flat(Mod(X)) has finite projective dimension.

Proof. (i) ⇒ (ii) By BRT, ImS′ = Flat(Mod(X)), hence condition (†) is true. Since S′ : C/Ph_{E(X)}(C) ≈ Flat(Mod(X)), it follows directly that X is self-compact. Since idempotents split in P(E(X)) = Add(X), by Theorem 5.3, Flat(Mod(X)) = ImS′ = {F ∈ Mod(X) | p.dF ≤ 1}. Finally Theorem 5.3 ensures that X generates C and E(X)−gl.dimC ≤ 1. Part (ii) ⇒ (iii) is trivial since if X generates C and condition (†) is true, then: E(X)−gl.dimC ≤ sup{p.dF | F ∈ Flat(Mod(X))}.

(iii) ⇒ (i) Since E(X)−gl.dimC ≤ 1, by Theorem 5.3 the functor S′ = S : C ⟶ Flat(Mod(X)) is full and reflects isomorphisms. It remains to show that S is surjective on objects. Let F be a flat functor, let 0 ⟶ S(P_{n}) ⟶^{a} S(P_{n−1}) ⟶^{b} S(P_{n−2}) ⟶ ⋯ ⟶ S(P_{0}) ⟶ F ⟶ 0 be a projective resolution and let b = d∘e be the canonical factorization, where d : S(P_{n−1}) ⟶ G and e : G ↪ S(P_{n−2}). Then there exists f : P_{n} ⟶ P_{n−1} such that S(f) = a, since S : P(E(X)) ≈ Proj(Mod(X)). Since S(f) is monic, the triangle P_{n} ⟶^{f} P_{n−1} ⟶ L_{n−1} ⟶ Σ(P_{n}) is in E(X), and then S(L_{n−1}) ≅ G. Since S is full, there exists a morphism g : L_{n−1} ⟶ P_{n−2} with S(g) = e. Since S(g) is monic, the triangle L_{n−1} ⟶^{g} P_{n−2} ⟶ L_{n−2} ⟶ Σ(L_{n−1}) is in E(X), and then S(L_{n−2}) ≅ Im(P_{n−2} ⟶ P_{n−3}). Continuing in this way, it follows that F is in ImS, so S is surjective on objects.

If (C, X) satisfies BRT, then we have the classical version of Brown Representability Theorem [55], [51].

Corollary 10.3. If the pair (C, X) satisfies BRT, and if F : C^{op} ⟶ Ab is a cohomological functor converting coproducts to products, then F is representable.

Proof. Since the restriction functor F̃ := F |_{X} ∈ Mod(X) is flat, by BRT there exists A ∈ C, such that α : S(A) = C(−, A)|_{X} ⟶^{≅} F |_{X} = F̃. Since P(E(X)) = Add(X) and F sends coproducts to products, α can be extended to an isomorphism β : F |_{P(E(X))} ≅ C(−, A)|_{P(E(X))}. Then the result follows from Corollary 4.29.

The author is indebted to the referee for the following nice remark.

Remark 10.4. If the pair (C, X) satisfies BRT, then C is compactly generated. Indeed it suffices to show that X ⊆ C^{b}. Let {A_{i}; i ∈ I} be a set of objects in C and choose an E(X)−projective resolution P_{i1} ⟶ P_{i0} ⟶ A_{i} ⟶ Σ(P_{i1}) of A_{i}, ∀i ∈ I. Then the sum ⊕P_{i1} ⟶ ⊕P_{i0} ⟶ ⊕A_{i} ⟶ Σ(⊕P_{i1}) of these resolutions is a triangle which is an E(X)−projective resolution of ⊕A_{i}. If X ∈ X, then applying C(X, −) to the triangle and using 5−Lemma and the self-compactness of X, it follows easily that the canonical map ⊕C(X, A_{i}) ⟶ C(X, ⊕A_{i}) is invertible. Hence X is compact.

A direct consequence of the above representability result is the following.

Corollary 10.5. If the pair (C, X) satisfies BRT, then C has products.

The following is a consequence of Theorems 5.3, 4.24.

Corollary 10.6. If BRT holds for the pair (C, X), then C has E(X)−injective envelopes and the functor S induces an equivalence S : I(E(X)) ⟶^{≈} Inj(Mod(X)).

Corollary 10.7. Assume that X is self-compact, generates C and satisfies condition (†). If (α) any flat right X −module is a direct union of countably presented pure submodules or (β) if any flat right X −module is countably presented or (γ) if the module category Mod(X) satisfies one of the following conditions: p.gl.dimMod(X) ≤ 1 or KdimMod(X) ≤ 1, where p.gl.dimMod(X) denotes pure global dimension and KdimMod(X) denotes Krull dimension [30] of the Grothendieck category Mod(X), then the pair (C, X) satisfies BRT.

Proof. By [65], [39], we have sup{p.dF; F ∈ Flat(Mod(X))} ≤ p.gl.dimMod(X) ≤ KdimMod(X). So the result follows from Theorem 10.2 in case (γ). In the first two cases it follows from [40], [65], since then for any flat F: p.dF ≤ 1.

Corollary 10.8. Let C/Add(X), C/I(E(X)) be the stable categories of C modulo Add(X), I(E(X)) and let \underline{Flat}(Mod(X)), \overline{Flat}(Mod(X)) be the stable categories of flat modules modulo projectives, injectives. If the pair (C, X) satisfies BRT, then S induces equivalences C/Add(X) ≅ \underline{Flat}(Mod(X)) and C/I(E(X)) ≅ \overline{Flat}(Mod(X)). C/Add(X) has kernels and coproducts and C/I(E(X)) has cokernels and products.

Proof. By BRT the functor S induces a full and surjective on objects functor S : C/Add(X) ⟶ \underline{Flat}(Mod(X)). Let f : A ⟶ B be a morphism in C such that S(f) factors through a projective G = S(P), P ∈ Add(X), via the morphisms α = S(g) : S(A) ⟶ S(P) and β = S(h) : S(P) ⟶ S(B). Then f − g ∘ h : A ⟶ B is a phantom map. Let P_{1} ⟶^{σ} P_{0} ⟶^{ρ} A ⟶^{τ} Σ(P_{1}) be an E(X)−projective resolution of A. Since f − g ∘ h is phantom, we have ρ ∘ (f − g ∘ h) = 0, hence there exists ω : Σ(P_{1}) ⟶ B such that τ ∘ ω = f − g ∘ h or f = τ ∘ ω + g ∘ h. Passing to the stable category we have f = 0 in C/Add(X). This means that S|_{C/Add(X)} is faithful, hence an equivalence. Finally the category \underline{Flat}(Mod(X)) is a left triangulated category [12], and since for any flat F we have p.dF ≤ 1, it is trivial to see that \underline{Flat}(Mod(X)) has kernels. Since Flat(Mod(X)) has coproducts it is easy to see that the same is true for \underline{Flat}(Mod(X)). The case of injectives is treated similarly.

There are pairs for which BRT fails (see Section 12). Let C^{ℵ_{0}} be the full subcategory of C consisting of objects A for which there exists a triangle P_{1} ⟶ P_{0} ⟶ A ⟶ Σ(P_{1}) in E(X) with P_{1}, P_{0} direct summands of countable coproducts of objects of X. Let Ph_{E(X)}(C^{ℵ_{0}}) be the restriction of Ph_{E(X)}(C) to C^{ℵ_{0}} and let Flat^{ℵ_{0}}(Mod(X)) be the full subcategory of countably presented flat functors [39], [65].

Proposition 10.9. The functor S induces a representation equivalence S : C^{ℵ_{0}} ⟶ Flat^{ℵ_{0}}(Mod(X)). Hence: C^{ℵ_{0}}/Ph_{E(X)}(C^{ℵ_{0}}) ≈ Flat^{ℵ_{0}}(Mod(X)).

Proof. By construction we have that S has image in Flat^{ℵ_{0}}(Mod(X)), and any object A ∈ C^{ℵ_{0}} has E(X)−p.dA ≤ 1. Since [65] any countably presented flat functor F has p.dF ≤ 1, we see that the functor S is surjective on objects.

Recall from [65] that the weight w(X) of a skeletally small category X is defined by w(X) := card{⨆ X (X, Y) : X, Y ∈ Iso(X)}.

Proposition 10.10. [65] If X is a skeletally small additive category with weight w(X) ≤ ℵ_{t}, for some t ≥ 0, then any flat module X −module F has p.dF ≤ t + 1.

The following consequence is a generalization of the main result of [56].

Corollary 10.11. If for the pair (C, X), the subcategory X has weight w(X) ≤ ℵ_{0} and satisfies condition (†), then the following are equivalent.

(i) The pair (C, X) satisfies BRT. (ii) X is a compact generating subcategory of C.

For examples of pairs (C, X) satisfying BRT we refer to Sections 11, 12.

11. Purity

Throughout this Section we fix a compactly generated triangulated category C. Our aim here is to apply the theory of Section 8, in case X = C^{b} is the full subcategory of compact objects. The homological algebra in C based on the proper class of triangles E(C^{b}) is called the pure homological algebra of C. This kind of purity was also considered independently by H. Krause [50].

Terminology 11.1. A pure triangle in C is a triangle in E(C^{b}). The pure projective, resp. pure injective, objects PProj(C), resp. PInj(C), are the E(C^{b})−projective, resp. E(C^{b})−injective, objects. The pure projective dimension p.p.dA of A ∈ C is defined as E(C^{b})−p.dA. The pure global dimension p.gl.dimC is E(C^{b})−gl.dimC and the pure extension functors PExt^{∗} are the functors Ext_{E(C^{b})}^{∗}. The ideal Ph(C) of pure phantom maps is the ideal Ph_{E(C^{b})}(C), and so on.

By Section 8 we know that C has enough pure-injectives and enough pureprojectives with PProj(C) = Add(C^{b}) and PInj(C) = Prod(𝔻(C^{b})). The pure Steenrod category of C is the module category Mod(C^{b}) and the projectivization functor is the restricted Yoneda functor S : C ⟶ Mod(C^{b}), which induces an equivalence between PProj(C) and Proj(Mod(C^{b})) and between PInj(C) and Inj(Mod(C^{b})). The motivating source of the above definitions and terminology is explained below. Its main theme is to try to describe objects of a larger category by filtered (homotopy) colimits of objects of a smaller and better behaved full subcategory.

11.1. Motivation. It is well-known that the proper framework for the study of purity in a module category is that of a locally finitely presented Grothendieck category [65]. If G is a locally finitely presented Grothendieck category, and f.p(G) is the full subcategory of finitely presented objects, then the restricted Yoneda functor H : G ⟶ Mod(f.p(G)) is fully faithful and identifies G with the full subcategory Flat(Mod(f.p(G))) of flat functors and the pure-projective objects of G with the projective objects of Mod(f.p(G)). A short exact sequence (E) in G is pure iff H(E) is exact. The pure global dimension of the Grothendieck category G is computed as p.gl.dimG = sup{p.dF | F ∈ Flat(Mod(f.p(G)))}. In particular G is puresemisimple iff the module category Mod(f.p(G)) is perfect, i.e. any flat functor is projective [65]. Hence the pure homological theory of G is strongly connected with the behaviour of Mod(f.p(G)) via the functor H.

Now if C is a triangulated category with coproducts such that C^{b} is skeletally small, it is natural to consider C^{b} as the full subcategory of “finitely presented” objects, since ∀X ∈ C^{b}, the functor C(X, −) preserves coproducts. This analogy is justified by the fact that in the stable homotopy category the compact objects are the finite spectra and also in the derived category where the compact objects are the perfect complexes which behave by [69], like finitely presented objects. See also Section 12, where the example of the stable module category provides further justification. If C has a set of compact generators, i.e. C is compactly generated, then C can be considered as the triangulated analogue of a locally finitely presented Grothendieck category. Then the Steenrod category Mod(C^{b}) is the analogue of Mod(f.p(G)) and the functor S : C ⟶ Mod(C^{b}) is the analogue of the functor H above. A triangle (T) is in E(C^{b}) iff S(T) is short exact. Hence the triangles in E(C^{b}) can be considered as the “pure” triangles. The functor S is not fully faithful in general, due to the presence of phantom maps and this is the major difference between the two theories. Indeed by Theorem 9.3, S is fully faithful iff C is phantomless iff S induces an equivalence between C and Flat(Mod(C^{b})). Also Theorem 9.3 characterizes the phantomless categories as the categories C such that Mod(C^{b}) is perfect or equivalently all pure triangles are split. Hence the phantomless categories are the pure-semisimple categories and the global dimension of C with respect to the pure triangles is the pure-global dimension of C.

We recall that a right C^{b}−module G is called FP-injective if Ext^{1}(F, G) = 0 for any finitely presented module F. Since C^{b} is triangulated, by [13] or [50], the category Flat(Mod(C^{b})) of flat right C^{b}−modules, the category Coh((C^{b})^{op}, Ab) of cohomological functors over C^{b} and the category FPInj(Mod(C^{b})) of FP-injective right C^{b}−modules are identical.

Proposition 11.2. p.gl.dimC = sup{p.p.dA; A ∈ C | p.p.dA < ∞} =

= sup{p.dF ; F ∈ Flat(Mod(C^{b})) | p.dF < ∞} =

= sup{p.dF ; F ∈ Flat(Mod(C^{b}))} = sup{i.dF ; F ∈ FPInj(Mod(C^{b}))} =

= sup{i.dF ; F ∈ Flat(Mod(C^{b}))} = sup{p.dF ; F ∈ FPInj(Mod(C^{b}))}.

Proof. Since ∀A ∈ C, p.p.dA = p.dS(A), and since ImS ⊆ Flat(Mod(C^{b})), we have p.gl.dimC ≤ sup{p.dF; F ∈ Flat(Mod(C^{b}))}. By a result of Jensen [65], sup{p.dF; F ∈ Flat(Mod(C^{b}))} = sup{p.dF; F ∈ Flat(Mod(C^{b})) | p.dF < ∞}. We show that if F is a flat right C^{b}−module of finite projective dimension, then there exists A ∈ C such that p.p.dA = p.dF. If p.dF ≤ 1, then we know that there exists A ∈ C such that F = S(A). Suppose that ∞ > p.dF ≥ 2. There exists a small filtered category I and a functor I : I ⟶ C^{b} such that F = lim_{→}_{I}S(X_{i}). Consider the triangle ⊕_{I}X_{i} ⟶ ⊕_{I}X_{i} ⟶ A ⟶ Σ(⊕_{I}X_{i}) in C.

Then we have a long exact sequence ⋯ ⟶ ⊕_{I}S(Σ^{−1}(X_{i})) ⟶^{α} S(Σ^{−1}(A)) ⟶^{β} ⊕_{I}S(X_{i}) ⟶^{γ} ⊕_{I}S(X_{i}) ⟶^{δ} S(A) ⟶ · · ·. The equivalence Σ induces an autoequivalence Σ_{∗} of Flat(Mod(C^{b})) given by Σ_{∗}(F) = F Σ with inverse Σ_{∗}^{−1} defined by Σ_{∗}^{−1}(F) = F Σ^{−1}. Then F = Im(δ) and Im(α) = F Σ^{−1}. Let H = Im(β) and G = Im(γ). Obviously p.dΣ_{∗}(F) = p.dF = p.dΣ_{∗}^{−1}(F) and we have an exact sequence 0 ⟶ Σ_{∗}^{−1}(F) ⟶ S(Σ^{−1}(A)) ⟶^{β} ⊕_{I}S(X_{i}) ⟶^{γ} ⊕_{I}S(X_{i}) ⟶ F ⟶ 0. From this exact sequence, since p.dF ≥ 2 and ⊕_{I}S(X_{i}) is a projective module, we have p.dH = p.dF − 2 = p.dΣ_{∗}^{−1}(F) − 2. But then from the short exact sequence 0 ⟶ Σ_{∗}^{−1}(F) ⟶ S(Σ^{−1}(A)) ⟶ H ⟶ 0, and the fact that F, hence Σ_{∗}^{−1}(F), has finite projective dimension, we deduce that p.dF = p.dΣ_{∗}^{−1}(F) = p.dS(Σ^{−1}(A)) = p.dS(A) = p.p.dA. Hence for any flat right C^{b}−module F with p.dF < ∞, there exists an object A ∈ C, such that p.p.dA = p.dF. This implies trivially that sup{p.dF; F ∈ Flat(Mod(C^{b})) | p.dF < ∞} ≤ p.gl.dimC. As a consequence we have that the first three equalities are true. Using that the flat and the FP-injective modules coincide, the proof of the remaining equalities is left to the reader.

By Proposition 10.10, we have the following consequence.

Corollary 11.3. If w(C^{b}) ≤ ℵ_{t}, t ≥ 0, then: p.gl.dimC ≤ t + 1.

If C is the stable homotopy category of spectra, then it is well known that C is compactly generated and the full subcategory C^{b} of finite spectra has a countable skeleton. Hence by Corollary 11.3, p.gl.dimC ≤ 1.

Corollary 11.4. If C is the stable homotopy category, then p.gl.dimC = 1.

Proof. If p.gl.dimC = 0, then by Theorem 9.3, Mod(C^{b}) is perfect. It is easy to see [65] that this implies that any compact object X has a perfect endomorphism ring. In C this is not true, e.g. choose X to be the sphere spectrum.

Remark 11.5. The pure global dimension is invariant under triangle equivalence. Indeed any triangle equivalence F : C ⟶^{≈} D restricts to an equivalence F^{b} : C^{b} ⟶^{≈} D^{b}. Hence p.gl.dimC = p.gl.dimD, by Proposition 11.2.

We denote the full subcategories of Mod(C^{b}) consisting of the modules of finite projective, resp. injective, dimension by Proj^{<∞}(Mod(C^{b})), resp. Inj^{<∞}(Mod(C^{b})). Since the flat and the FP-injective modules coincide, it is trivial to see that:

Proj^{<∞}(Mod(C^{b})) ⊆ Flat(Mod(C^{b})) = FPInj(Mod(C^{b})) ⊇ Inj^{<∞}(Mod(C^{b})).

Finally the finitistic projective dimension of Mod(C^{b}) is defined as f.p.d Mod(C^{b}) = sup{p.dF; F ∈ Proj^{<∞}(Mod(C^{b}))}. The definition of the finitistic injective dimension f.i.d Mod(C^{b}) of Mod(C^{b}) is similar. Note that since C^{b} is triangulated, by [13] the flat or FP-injective dimension of any left or right C^{b}−module is 0 or ∞.

11.2. Pure Hereditary and Brown Categories. Let I : I ⟶ C be a functor from a small filtered category I. We use the notations: A_{i} = I(i) for i ∈ I and α_{ij} = I(i ⟶ j) : A_{i} ⟶ A_{j} for an arrow i ⟶ j in I. A weak colimit of the functor I is an object A in C together with morphisms f_{i} : A_{i} ⟶ A, which are compatible with the system {A_{i}, α_{ij}} in the sense that for any arrow i ⟶ j in I we have f_{i} = α_{ij} ∘ f_{j}, and if g_{i} : A_{i} ⟶ B is another compatible family, then there exists a (not necessarily unique) morphism ω : A ⟶ B such that f_{i} ∘ ω = g_{i}, for any object i ∈ I. For example a homotopy colimit of a tower is a weak colimit. A weak colimit of the functor I is denoted by w.lim_{⟶}_{I}A_{i}. By [51] a weak colimit of the

functor I can be constructed as a weak cokernel θ_{I} : ⨁_{i∈I} A_{i} ⟶ w.lim_{⟶}_{I}A_{i} of the

canonical morphism ζ_{I} : ⨁_{α_{ij}∈I^{2}} A_{i} ⟶ ⨁_{i∈I} A_{i}. Hence it can be computed from

the triangle ⨁_{α_{ij}∈I^{2}} A_{i} ⟶^{ζ_{I}} ⨁_{i∈I} A_{i} ⟶^{θ_{I}} w.lim_{⟶}_{I}A_{i} ⟶ Σ(⨁_{α_{ij}∈I^{2}} A_{i}).

Definition 11.6. [51] A weak colimit w.lim_{⟶}_{I}A_{i} of a functor I : I ⟶ C is called minimal if the canonical morphism

lim_{→}_{I}S(A_{i}) ⟶ S(w.lim_{⟶}_{I}A_{i})

is an isomorphism. We denote a minimal weak colimit by m.w.lim_{⟶}_{I}A_{i}.

By Lemma 5.8 the homotopy colimit of a tower is a minimal weak colimit.

Definition 11.7. A triangulated category C is called a Brown category, if C has coproducts, C^{b} is skeletally small and the pair (C, C^{b}) satisfies BRT, i.e. the projectivization functor S : C ⟶ Flat(Mod(C^{b})) is a representation equivalence.

Part (1) ⇔ (4) in the next result is due to Neeman [56], (see also [19], [21]) and parts (1) ⇔ (9) ⇒ (11) were proved independently by Hovey, Palmieri and Strickland [38], see also [51]. We believe that our proofs are simpler. Note that the characterizations (4), (5) below are identical with the characterizations of flat modules over the Steenrod algebra, see [51].

Theorem 11.8. Let C be a compactly generated triangulated category. Then the following are equivalent:

(1) C is a Brown category.

(2) The projectivization functor S : C ⟶ Mod(C^{b}) is full.

(3) p.gl.dimC ≤ 1.

(4) Let F ∈ Mod(C^{b}). Then F ∈ Flat(Mod(C^{b})) iff p.dF ≤ 1.

(5) Let F ∈ Mod(C^{b}). Then F ∈ Flat(Mod(C^{b})) iff i.dF ≤ 1.

(6) f.p.d Mod(C^{b}) ≤ 1.

(7) f.i.d Mod(C^{b}) ≤ 1.

(8) If I : I ⟶ C^{b} and J : J ⟶ C^{b} are functors from small filtered categories, with I(i) = X_{i}, J (j) = Y_{j}, then:

lim_{←}_{I}^{(n)}lim_{→}_{J} C(X_{i}, Y_{j}) = 0, ∀n ≥ 2.

(9) Any functor I : I ⟶ C^{b} from a small filtered category I has a minimal weak colimit m.w.lim_{⟶}_{I}X_{i} in C.

(10) Any functor I : I ⟶ C^{b} from a small filtered category I has a weak colimit w.lim_{⟶}_{I}X_{i} in C with the following property. For any coproduct preserving homological functor F : C ⟶ Ab, the canonical map ϕ below is invertible:

ϕ : lim_{→}_{I}F (X_{i}) ⟶^{≅} F (w.lim_{⟶}_{I}X_{i}).

(11) Any object of C is a minimal weak colimit of a functor I : I ⟶ C^{b} from a small filtered category.

If one of the above equivalent conditions is true, then any two mimimal weak colimits are isomorphic and for any weak colimit w.lim_{⟶}_{I}X_{i} with X_{i} ∈ C^{b}, we have:

w.lim_{⟶}_{I}X_{i} ≅ m.w.lim_{⟶}_{I}X_{i} ⊕ P, where P ∈ Add(C^{b}).

Proof. Using Theorems 5.3, 10.2 and Proposition 11.2 it follows trivially that the first seven conditions are equivalent. We include only a proof that (2) ⇒ (1). If S is full, then by Theorem 5.3 we have that p.gl.dimC ≤ 1. Then Proposition 11.2 ensures that any flat functor F ∈ Mod(C^{b}) has p.dF ≤ 1. Finally by Theorem 10.2 it follows that C is Brown category.

(1) ⇔ (8). Let F, G be flat functors. Then F is a filtered colimit of representables S(X_{i}) = C^{b}(−, X_{i}) and G is a filtered colimit of representables S(Y_{j}) = C^{b}(−, Y_{j}), where I, J are small filtered categories and I : I ⟶ C^{b}, J : J ⟶ C^{b} are functors with I(i) = X_{i} and J (j) = Y_{j}. By the Roos’s spectral sequence of subsection 8.4 for

the functors F, G, we have: lim_{←}_{I}^{(n)}C(X_{i}, G) ≅ lim_{←}_{I}^{(n)}lim_{→}_{J} C(X_{i}, Y_{j}) ≅ Ext^{n}(F, G), ∀n

≥ 0. If (1) holds, then the above isomorphism implies (8), by (3). Conversely if (8) holds, then Ext^{n}(F, G) = 0, ∀n ≥ 2, for any two flat functors F, G. By [40], this implies that any flat G has pure injective dimension bounded by one. Since flat and FP-injective functors coincide and the pure injective dimension coincide with the injective dimension for FP-injective functors, (3) follows.

(9) ⇒ (1) Let F = lim_{→}_{I}S(X_{i}) be a flat functor, where I : I ⟶ C^{b} is a functor and I is a small filtered category. Consider a minimal weak colimit m.w.lim_{⟶}_{I}X_{i}

of I in C. By hypothesis: F = lim_{→}_{I}S(X_{i}) ≅ S(m.w.lim_{⟶}_{I}X_{i}), so S is essentially surjective. Since C is compactly generated, S reflects isomorphisms and it remains to prove that S is full. It suffices to show that if I : I ⟶ C^{b} and J : J ⟶ C^{b} are functors from small filtered categories, then any map α′ : S(m.w.lim_{⟶}_{I}X_{i}) ⟶

S(m.w.lim_{⟶}_{J} Y_{j}) is of the form S(α). By construction we have projective presen-

tations: S(⊕X_{i}) ⟶^{S(ζ_{I})} S(⊕X_{i}) ⟶^{ξ_{I}} S(m.w.lim_{⟶}_{I}X_{i}) ⟶ 0 and S(⊕Y_{j}) ⟶^{S(ζ_{J})}

S(⊕Y_{j}) ⟶^{ξ_{J}} S(m.w.lim_{⟶}_{I}Y_{j}) ⟶ 0, and there exists a commutative diagram:

S(⊕X_{i}) ⟶^{S(ζ_{I})} S(⊕X_{i}) ⟶^{ξ_{I}} S(m.w.lim_{⟶}_{I}X_{i}) ⟶ 0
∃β′↓  ^{∃γ′}↓  _{α′}↓
S(⊕Y_{j}) ⟶^{S(ζ_{J})} S(⊕Y_{j}) ⟶^{ξ_{J}} S(m.w.lim_{⟶}_{J} Y_{j}) ⟶ 0

Since S induces an equivalence between Add(C^{b}) and Proj(Mod(C^{b})), there exist morphisms β : ⊕X_{i} ⟶ ⊕Y_{j}, γ : ⊕X_{i} ⟶ ⊕Y_{j} such that S(β) = β′, S(γ) = γ′ and β ∘ ζ_{J} = ζ_{I} ∘ γ. Since m.w.lim_{⟶}_{I}X_{i} and m.w.lim_{⟶}_{J} Y_{j} are weak cokernels of ζ_{I} and

ζ_{J} respectively, there exists a morphism α : m.w.lim_{⟶}_{I}X_{i} ⟶ m.w.lim_{⟶}_{J} Y_{j} such that

γ ∘ θ_{J} = θ_{I} ∘ α. But then obviously S(α) = α′. Hence S is full.

(1) ⇒ (9) By the equivalence of (1) with (3), for any flat right C^{b}−module F, we have p.dF ≤ 1. Let I : I ⟶ C^{b} be a functor from a small filtered category I and consider the flat modules F = lim_{→}_{I}S(X_{i}) and G = S(w.lim_{⟶}_{I}X_{i}). Then identifying

S(⊕X_{i}) with ⊕S(X_{i}), we have the following exact commutative diagram

⊕S(X_{i}) ⟶^{S(ζ_{I})} ⊕S(X_{i}) ⟶^{α} F ⟶ 0
∥↓  ^{∥}↓  ^{∃! ω}↓
S(⊕X_{i}) ⟶^{S(ζ_{I})} S(⊕X_{i}) ⟶^{S(θ_{I})} G

By BRT there exists A ∈ C and a morphism f : A ⟶ w.lim_{⟶}_{I}X_{i}, such that S(A) = F

and S(f) = ω. Similarly there exists a (unique) morphism μ : ⊕X_{i} ⟶ A, such that S(μ) = α. Let S(ζ_{I}) = β ∘ γ be the canonical factorization of S(ζ_{I}) in Mod(C^{b}), where β : S(⊕X_{i}) ⟶ H and γ : H ⟶ S(⊕X_{i}). By BRT there exists Q ∈ C and (unique) morphisms κ : ⊕X_{i} ⟶ Q and λ : Q ⟶ ⊕X_{i} such that S(κ) = β and S(λ) = γ. Since S(μ), is epic the triangle Q ⟶^{λ} ⊕X_{i} ⟶^{μ} A ⟶^{ν} Σ(Q) is pure. Since p.dF ≤ 1, the functor H = S(Q) is projective, hence the morphism S(κ) is split epic. Since S reflects isomorphisms, κ is split epic so there exists ρ : Q ⟶ ⊕X_{i} with ρ ∘ κ = 1_{Q} inducing a direct sum decomposition ⊕X_{i} ≅ Q ⊕ P. Then we have the following morphisms of triangles:

Q ⟶^{λ} ⊕X_{i} ⟶^{μ}  A  ⟶^{ν} Σ(Q)
ρ↓  ^{∥}↓  ^{f}↓  ^{Σ(ρ)}↓
⊕X_{i} ⟶^{ζ_{I}} ⊕X_{i} ⟶^{θ_{I}} w.lim_{⟶}_{I}X_{i} ⟶^{η_{I}} Σ(⊕X_{i})
κ↓  ^{∥}↓  ^{∃g}↓  ^{Σ(κ)}↓
Q ⟶^{λ} ⊕X_{i} ⟶^{μ}  A  ⟶^{ν} Σ(⊕X_{i}).

Since ρ ∘ κ = 1_{Q}, the morphism f ∘ g is invertible, in particular f is split monic. It is easy to see that Coker(f) = Σ(P), hence w.lim_{⟶}_{I}X_{i} ≅ A ⊕ Σ(P). Observe that ζ_{I} ∘ μ = κ ∘ λ ∘ μ = 0. Let ϕ : ⊕X_{i} ⟶ E be a morphism such that ζ_{I} ∘ ϕ = 0. Since by construction θ_{I} is a weak cokernel of ζ_{I}, we have ϕ = θ_{I} ∘ ψ for a morphism ψ : w.lim_{⟶}_{I}X_{i} ⟶ E. Hence ϕ = μ ∘ f ∘ ψ and this implies that the morphism μ : ⊕X_{i} ⟶ A is a weak cokernel of ζ_{I}. Hence A is a weak colimit of the functor I : I ⟶ C^{b}. Since by construction S(A) = lim_{→}_{I}S(X_{i}), we conclude that A is a

minimal weak colimit of the functor I. If A′ is another minimal weak colimit, then S(A) ≅ S(A′) and by BRT we have A ≅ A′. The above argument shows that if Â is any other weak colimit, then Â ≅ A ⊕ P where P ∈ Add(C^{b}).

(9) ⇔ (10) Note that by Theorem 8.10, a homological functor F : C ⟶ Ab preserves coproducts iff F ≅ G^{∗}S, where G^{∗} is exact and preserves colimits. From this it follows directly that (9) implies (10). Conversely part (9) follows from (10), choosing F = C(X, −) : C ⟶ Ab, for any X ∈ C^{b}.

(1) ⇔ (11) If C is Brown and A ∈ C, then S(A) = lim_{→}_{I}S(X_{i}), where I : I ⟶ C^{b}

is a functor from a small filtered category I. By (9), the functor I has a minimal weak colimit A′ = m.w.lim_{⟶}_{I}X_{i}. Then S(A) ≅ S(A′) and since by (1), S is a

representation equivalence, we have A ≅ A′. Conversely if (11) holds, let A ∈ C. Then as above A = m.w.lim_{⟶}_{I}X_{i}. By construction of the weak colimit of I, there

exists a triangle ⊕X_{j} ⟶^{α} ⊕X_{i} ⟶^{β} A ⟶^{γ} Σ(⊕X_{j}). By the construction of the colimit of the functor SI : I ⟶ Mod(C^{b}), it follows directly that γ is a pure phantom map, so the above triangle is pure. This implies that p.p.dA ≤ 1.

Corollary 11.9. Suppose that p.gl.dimC ≤ 1. If A = m.w.lim_{⟶}_{I}X_{i} with X_{i} ∈ C^{b}, then for any B ∈ C there exists a short exact sequence:

0 ⟶ lim_{←}_{I}^{(1)}C(Σ(X_{i}), B) ⟶ C(A, B) ⟶ lim_{←}_{I}C(X_{i}, B) ⟶ 0.

Hence PExt^{1}(Σ(A), B) = Ph(A, B) = lim_{←}_{I}^{(1)}C(Σ(X_{i}), B) and Ph^{2}(C) = 0. If A = m.w.lim_{⟶}_{I}X_{i}, B = m.w.lim_{⟶}_{J} Y_{j} are representations of A, B as mini-

mal weak colimits of functors I : I ⟶ C^{b} and J : J ⟶ C^{b}, then Ph(A, B) = lim_{←}_{I}^{(1)}lim_{→}_{J} C(Σ(X_{i}), Y_{j}) and there exists a short exact sequence

0 ⟶ lim_{←}_{I}^{(1)}lim_{→}_{J} C(Σ(X_{i}), Y_{j}) ⟶ C(A, B) ⟶ lim_{←}_{I} lim_{→}_{J} C(X_{i}, Y_{j}) ⟶ 0

Proof. By Theorem 4.27, there exists a short exact sequence 0 ⟶ PExt^{1}(Σ(A), B) ⟶ C(A, B) ⟶ PExt^{0}(A, B) ⟶ 0. Since Ext^{n}[S(A), S(B)] ≅ PExt^{n}(A, B), ∀n ≥ 0, we have Ext^{1}[S(Σ(A)), S(B)] ≅ Ext^{1}[lim_{→}_{I}S(Σ(X_{i})), S(B)] ≅ PExt^{1}(Σ(A), B). Using the Roos’s spectral sequence as in the proof of Theorem 11.8 we see directly that Ext^{1}[lim_{→}_{I}S(Σ(X_{i})), S(B)] ≅ lim_{←}_{I}^{(1)}[S(Σ(X_{i}), S(B)] ≅ lim_{←}_{I}^{(1)}C(Σ(X_{i}), B). In

the same way PExt^{0}(A, B) ≅ lim_{←}_{I}(X_{i}, B) and the assertion follows.

Corollary 11.10. If C, D are Brown categories, then:

C^{b} ≈ D^{b} ⇔ C/Ph(C) ≈ D/Ph(D) ⇔ PInj(C) ≈ PInj(D).

Proof. Trivial since C/Ph(C) ≈ Flat(Mod(C^{b})), D/Ph(D) ≈ Flat(Mod(D^{b})).

Remark 11.11. [21] If p.gl.dimC ≤ 1, then we have a +-linear extension of categories

0 ⟶ Ph(C) ⟶ C ⟶^{S} Flat(Mod(C^{b})) ⟶ 0 [E]

in the sense of [9], which represents an element of the second Hochschild-Mitchell-Baues-Wirsching cohomology group H^{2}[Flat(Mod(C^{b})), Ph(C)]. This element is zero if the above extension splits and this happens if there exists a functor T : Flat(Mod(C^{b})) ⟶ C such that ST = Id. If this is true, then ∀B ∈ C, the short exact sequence 0 ⟶ PExt^{1}[Σ−, B] ⟶ C(−, B) ⟶ PExt^{0}[−, B] ⟶ 0 splits. This implies that PExt^{1}[−, B] is a cohomological functor as a direct summand of C(−, Σ(B)). Hence if P_{1} ⟶ P_{0} ⟶ A ⟶ Σ(P_{1}) is a pure triangle with P_{1}, P_{0} ∈ PProj(C), then we have a long exact sequence ⋯ ⟶ PExt^{1}[Σ^{2}(P_{1}), B] ⟶ PExt^{1}[Σ(A), B] ⟶ PExt^{1}[Σ(P_{0}), B] ⟶ ⋯ and this shows that PExt^{1}[Σ(A), B] = 0. Hence B is pureinjective. It follows that p.gl.dimC = 0. Hence if C is pure-hereditary, then C is pure-semisimple iff the extension [E] splits. Since the stable homotopy category of spectra C has p.gl.dimC = 1, we have H^{2}[Flat(Mod(C^{b})), Ph(C)] ≠ 0. In this case all the above results are applied to this situation. In particular we recover the various (topological) versions of Brown Representability Theorem.

Remark 11.12. If p.gl.dimC ≤ 2, then the projectivization functor S : C ⟶

Flat(Mod(C^{b})) is surjective on objects. Indeed let 0 ⟶ S(P_{2}) ⟶^{S(f)} S(P_{1}) ⟶^{S(g)} S(P_{0}) ⟶^{ϵ} F ⟶ 0 be a projective resolution of the flat functor F. If P_{2} ⟶^{f} P_{1} ⟶^{β} B ⟶ Σ(P_{2}) is a triangle in C, then obviously ImS(g) ≅ S(B) and coimS(g) = S(β). Hence imS(g) : S(B) ↪ S(P_{0}). Since p.p.dB ≤ 1, by Theorem 4.27 it follows that imS(g) = S(α) for some morphism α : B ⟶ P_{0}. If B ⟶^{α} P_{0} ⟶ A ⟶ Σ(B) is a triangle in C, then obviously F ≅ S(A). In case C is the unbounded derived category D(Λ), where Λ is a right hereditary ring, then by [22] the converse is true.

The interpretation of the pure global dimension in terms of properties of the projectivization functor S shows that we have, in general non-reversible, implications:

S is faithful ⇒ S is full ⇒ S is surjective on objects.

11.3. Homological Functors. If p.gl.dimC = 0, then by Theorem 9.3, any homological functor H : C ⟶ G to an abelian category G, has a unique exact extension H^{∗} : Mod(C^{b}) ⟶ U, such that H^{∗}S = H. If the homological functor H preserves coproducts and G is Grothendieck, then without any restriction on p.gl.dimC we have the following consequence of Theorem 8.10, part (i) of which was observed also independently by Christensen-Strickland [21] in the Brown case, and by Krause [50] in general. Part (ii) is due to Krause [50] with a different proof. Note that A. Neeman proved in [57] that any product preserving homological functor C ⟶ Ab is representable. However the proof of this general result is much more difficult.

Corollary 11.13. Let H : C ⟶ G be a homological functor which preserves coproducts to a Grothendieck category G. Then:

(i) H(ϕ) = 0, for any pure-phantom map ϕ and there exists a unique coproduct preserving functor H^{∗} : Mod(C^{b}) ⟶ G, such that H^{∗}S = H. Hence S is the universal coproduct preserving homological functor out of C to Grothendieck categories. The functor H^{∗} is exact, admits a right adjoint which preserves injectives and we can identify H^{∗} = − ⊗_{C^{b}} H|_{C^{b}} and H = S(−) ⊗_{C^{b}} H|_{C^{b}}.

(ii) If G = Mod(Λ) for a ring Λ, then H preserves products iff H ≅ C(T, −), where T ∈ C^{b}. In this case H^{∗} has a left adjoint which preserves projectives.

Proof. Part (i) follows from Theorem 8.10. If H : C ⟶ Mod(Λ) preserves products, then H^{∗} preserves limits, so it is representable: H^{∗} ≅ (M, −). Since by (i), H^{∗} is exact and preserves colimits, M is a finitely generated projective functor, hence of the form M = S(T), for T ∈ C^{b}. Then H ≅ H^{∗}S ≅ (S(T), S(−)) ≅ C(T, −).

If H : C ⟶ Ab is a homological functor, then Ĥ := S(−) ⊗_{C^{b}} H|_{C^{b}} : C ⟶ Ab is a homological functor preserving coproducts and there is a natural morphism ϕ : Ĥ ⟶ H. It follows by the above result that ϕ is invertible iff H preserves coproducts. Consider now the dual situation. Let F : C^{op} ⟶ Ab be a cohomological functor, and consider the functor F̃ := (S(−), F |_{C^{b}}) : C^{op} ⟶ Ab. It is not difficult to see that F̃ sends coproducts to products and there exists a natural morphism ψ : F ⟶ F̃. However F̃ is not cohomological in general. Indeed F̃ is cohomological iff ψ is invertible iff F = C(−, E), where E is pure-injective. If for A ∈ C, D(A) denotes the category of morphisms X_{a} ⟶ A with X_{a} ∈ C^{b}, then Ĥ(A) = lim_{→}_{D(A)}H(X_{a})

and F̃(A) = lim_{←}_{D(A)}F (X_{a}) are the functors considered by Margolis [51]. If BRT

holds, then the next Corollary, which gives a trivial proof to a result of Adams [51], shows that F̃ satisfies a weak exact condition.

Corollary 11.14. If p.gl.dimC ≤ 1 and P ⟶^{f} A ⟶^{g} B ⟶ Σ(P) is a triangle in C, where P is pure-projective, then the sequence F̃(B) ⟶ F̃(A) ⟶ F̃(P) is exact.

Proof. By BRT, F |_{C^{b}} = S(C). Hence F̃ = [S(−), S(C)] = PExt^{0}(−, C). Consider the complex (∗) : [S(B), S(C)] ⟶ [S(A), S(C)] ⟶ [S(P), S(C)]. Let α : S(A) ⟶ S(C) be such that S(f) ∘ α = 0. By BRT, α = S(β). Then f ∘ β is pure-phantom. Since P is pure-projective, f ∘ β = 0. Hence β = g ∘ γ, where γ : B ⟶ C. Then α factors through (S(g), S(C)) and this means that the complex (∗) is exact.

Remark 11.15. Suppose C is monogenic [38], i.e. C admits a compact object T such that C coincides with the localizing subcategory generated by T. For example a tilting complex [62] in the derived category of a ring or the sphere spectrum [51] in the stable homotopy category. The relative homological algebra based on the proper class E(X), where X = add{Σ^{n}(T); n ∈ ℤ}, equivalently on the proper class E(H) where H = C(T, −), can be considered as the “absolute homological algebra” of C and is studied in [14], see also subsections 12.4, 12.5. The pure and the absolute theory are related by a Butler-Horrocks spectral sequence [18], see [14].

Remark 11.16. Assume that C has all small coproducts, and let X be an α−localizing subcategory of C in the sense of [57], where α is an infinite cardinal. Then the Steenrod category of C with respect to E(X) is the category Ex(X^{op}, Ab) of all functors X^{op} ⟶ Ab converting coproducts of fewer than α objects of X to products in Ab. See [57] for a detailed analysis of this situation which can be regarded as a higher analogue of the present theory, which in turn corresponds to the case α = ℵ_{0}.

11.4. Pure Global Dimension under Smashing and Finite Localization. Fix a compactly generated triangulated category C. We recall that a localizing subcategory L of C is called smashing if the inclusion functor i : L ↪ C has a right adjoint which preserves coproducts, equivalently the quotient functor π : C ⟶ C/L has a (fully faithful) right adjoint τ : C/L ⟶ C which preserves coproducts. It is easy to see [50] that if L ⊆ C is smashing, then the Verdier quotient C/L is compactly generated and the functor π sends C^{b} to (C/L)^{b}, so π induces an exact functor π^{b} : C^{b} ⟶ (C/L)^{b}. Since π is exact and preserves coproducts, by Corollary 11.13 or [50], there exists a unique exact colimit preserving functor π_{b}^{∗} : Mod(C^{b}) ⟶ Mod((C/L)^{b}) such that π_{b}^{∗}S_{C} = S_{C/L}π, where S_{C}, S_{C/L} are the projectivization functors. Observe that by the above relation, π_{b}^{∗} preserves flat functors and moreover projective functors, since π preserves compact objects. Similarly since τ is exact and preserves coproducts, there exists a unique exact colimit preserving functor τ_{b}^{∗} : Mod((C/L)^{b}) ⟶ Mod(C^{b}) such that S_{C}τ = τ_{b}^{∗}S_{C/L}. Since πτ = Id_{C/L}, the above properties imply that π_{b}^{∗}τ_{b}^{∗} ≅ Id_{Mod((C/L)^{b})}. If lim_{→}S_{C/L}(Y_{j}) is a flat functor over (C/L)^{b}, then since τ_{b}^{∗} preserves filtered colimits, τ_{b}^{∗}(lim_{→}S_{C/L}(Y_{j})) = lim_{→}τ_{b}^{∗}S_{C/L}(Y_{j}) ≅ lim_{→}S_{C}τ(Y_{j}). Since S_{C}τ(Y_{j}) are flat functors and a filtered colimit of flat functors is flat, it follows that τ_{b}^{∗} preserves flatness.

Theorem 11.17. (1) If L is a smashing subcategory of C, then:

p.gl.dimC/L ≤ p.gl.dimC.

(2) If L is a localizing subcategory of C which is generated by a set of compact objects from C, so L is compactly generated and smashing, then:

max{p.gl.dimL, p.gl.dimC/L} ≤ p.gl.dimC.

Proof. (1) Let F be a flat functor over (C/L)^{b}. Let P^{•} ⟶ τ_{b}^{∗}(F) be a projective resolution of the flat functor τ_{b}^{∗}(F) in Mod(C^{b}). Since π_{b}^{∗} is exact and preserves projectives, π_{b}^{∗}(P^{•}) ⟶ π_{b}^{∗}τ_{b}^{∗}(F) ≅ F is a projective resolution of F in Mod((C/L)^{b}) and the assertion follows. The second part is proved in a similar way.

The following consequence is due to Hovey, Palmieri and Strickland [38].

Corollary 11.18. Smashing or finite localizations of Brown, resp. phantomless, categories are Brown, resp. phantomless.

11.5. Compactly Generated Triangulated Categories of Finite Type. By Theorem 9.3 a compactly generated triangulated category C is pure-semisimple iff its pure Steenrod category Mod(C^{b}) is locally Noetherian or perfect. The pure homological theory of module categories [65], [39], suggests the following definition.

Definition 11.19. A compactly generated triangulated category C is said to be of finite type if its pure Steenrod category Mod(C^{b}) is locally finite.

Categories of finite type can be regarded as the “representation-finite” triangulated categories. For a justification of this statement we refer to the next Section.

Corollary 11.20. If L is a smashing subcategory of C and C is of finite type, then so is C/L. If L is a localizing subcategory of C generated by a set of compact objects from C and C is of finite type, then so are L, C/L.

Proof. If L is smashing then by [50], Mod((C/L)^{b}) is a localized quotient of Mod(C^{b}). Since C is of finite type, Mod(C^{b}) is locally finite, hence by [59], Mod((C/L)^{b}) is locally finite, i.e. C/L is of finite type. The second part is left to the reader.

The pure homological theory of module categories suggests the following

Question 11.21. Let C be a pure-semisimple compactly generated triangulated category. Is it true that C is of finite type?

A positive answer to Question 11.21 would follow from an affirmative answer to the following problem about Grothendieck categories posed by J.E. Roos [64] almost thirty years ago.

Roos’s Problem 11.22. Let G be a locally Noetherian Grothendieck category in which any injective object is projective. Is it true that G is locally finite?

Indeed if C is pure-semisimple, then the module category Mod(C^{b}) is a locally Noetherian Frobenius Grothendieck category. If Roos’s Problem has an affirmative answer, then Mod(C^{b}) is locally finite, so C is of finite type. In Section 12 we shall see that in general both Question 11.21 and Roos’s problem have a negative answer.

11.6. Krull-Gabriel Dimension and Dual Pure Global Dimension. We close this section by presenting a characterization of categories of finite type, using two new dimensions which are invariant under triangle equivalence.

Define the dual pure-global dimension p.gl.dimC^{op} of C, as the supremum of the projective dimensions in Mod((C^{b})^{op}) of the homological functors C^{b} ⟶ Ab:

p.gl.dimC^{op} := sup{p.dF | F ∈ Flat(Mod((C^{b})^{op}))}.

Note that p.gl.dimC^{op} is not the pure-global dimension of C^{op}, since C^{op} is never compactly generated [57]. However the above definition is reasonable, since if D(Λ) is the unbounded derived category of a ring Λ, then from the results of the next section it follows that p.gl.dimD(Λ)^{op} = p.gl.dimD(Λ^{op}). Similarly if \underline{Mod}(D) is the stable category of a locally finite Frobenius module category Mod(D), then p.gl.dim\underline{Mod}(D)^{op} = p.gl.dim\underline{Mod}(D^{op}). Finally if C is the stable homotopy category, then using the Spanier-Whitehead duality functor D : C^{b} ⟶^{≈} (C^{b})^{op} [51], it follows that p.gl.dimC^{op} = p.gl.dimC = 1. More generally if T , S are compactly generated and there exists an equivalence (T^{b})^{op} ≈ S^{b}, then p.gl.dimT^{op} = p.gl.dimS and p.gl.dimS^{op} = p.gl.dimT .

Finally let KGdimC^{b} be the Krull-Gabriel dimension [32], [49], of the abelian category A(C^{b}) = mod(C^{b}). Note that this dimension is the finitely presented version of the Krull dimension KdimMod(C^{b}) of Mod(C^{b}) defined by Gabriel [30]. By [49], KdimMod(C^{b}) ≤ KGdimC^{b} with equality if C is pure-semisimple. It seems that the Krull-Gabriel dimension is more stable than the pure global dimensions, and deserves further study. For instance it is symmetric: KGdimC^{b} = KGdim(C^{b})^{op}, see [32]. We denote this common value by KGdimC and by abuse of language we call it the Krull-Gabriel dimension of C. Note that by [32] it follows easily that max{p.gl.dimC^{op}, p.gl.dimC} ≤ KGdimC.

The next result shows that Question 11.21 has an affirmative answer if the implication p.gl.dimC = 0 ⇒ p.gl.dimC^{op} = 0 is true.

Proposition 11.23. (1) p.gl.dimC^{op} = 0 iff the Steenrod category Mod(C^{b}) is locally Artinian. More generally assume that for any compact object X, any decreasing filtered family of subobjects of C^{b}(−, X) in A(C^{b}), contains a cofinal subfamily of cardinality ℵ_{t}, t ≥ −1. Then p.gl.dimC^{op} ≤ t + 1.

(2) The following statements are equivalent:

(i) C is of finite type. (ii) p.gl.dimC^{op} = 0 = p.gl.dimC. (iii) KGdimC = 0.

Proof. Part (1) follows easily from results of Gruson-Jensen [32] and Simson [65]. Part (2) follows from part (α) and Proposition 9.2, using the well-known duality mod(C^{b})^{op} ≈ mod((C^{b})^{op}) given by CokerC^{b}(−, f) ⟼ KerC^{b}(f, −).

Question 11.24. What is the Krull-Gabriel dimension of a Brown category? In particular what is KGdimC, if C is the stable homotopy category? Certainly KGdimC ≥ 1 if C is Brown. It seems reasonable to conjecture that KGdimC = 1.

12. Applications to Derived and Stable Categories

In this Section we apply our previous results (mainly about purity) to derived categories of rings and to stable module categories of Quasi-Frobenius rings.

12.1. Derived Categories. Let Λ be an associative ring (with 1 always). We denote by D(Λ) the unbounded derived category of the abelian category Mod(Λ) of all right Λ−modules. We denote also by Proj(Λ) the full subcategory of all projective modules and by P_{Λ} the full subcategory of finitely generated projective modules in Mod(Λ). Finally we denote by H^{b}(P_{Λ}) the bounded homotopy category of P_{Λ}, and by D^{b}(mod(Λ)) the bounded derived category of finitely presented modules. It is well known [62] that D(Λ) is a compactly generated triangulated category and H^{b}(P_{Λ}) is, up to equivalence, the full subcategory of compact objects of D(Λ), i.e. D(Λ)^{b} ≈ H^{b}(P_{Λ}). For the notion of pure-semisimplicity in module categories, we refer to [65], [39]. Our first result is a consequence of Corollary 11.3.

Proposition 12.1. If |Λ| ≤ ℵ_{t}, for some t ≥ 0, then: p.gl.dimD(Λ) ≤ t + 1.

Example 12.2. Let D(Qcℙ^{n}(k)) be the derived category of quasi-coherent sheaves on the projective n−space over the field k. By Beilinson’s description [10] of the category D(Qcℙ^{n}(k)), it follows that if |k| ≤ ℵ_{t}, then p.gl.dimD(Qcℙ^{n}(k)) ≤ t + 1.

Proposition 12.3. (1) If p.gl.dimD(Λ) = 0, then r.p.gl.dimΛ = 0, i.e. Λ is a right pure-semisimple ring. Moreover r.gl.dimΛ < ∞.

(2) If Λ is derived equivalent to a right pure-semisimple right hereditary ring, then D(Λ) is pure-semisimple.

(3) Λ is a semisimple ring iff D(Λ) is (semisimple) abelian iff all triangles in D(Λ) are semi-split iff there exists a (uniquely determined) set of division rings {Γ_{i}; i ∈ I} and an equivalence: D(Λ) ≈ ∏_{i∈I} Mod(Γ_{i}).

Proof. (1) By Theorem 9.3, the module category Mod(H^{b}(P_{Λ})) is Frobenius, in particular it is locally Noetherian and perfect. Consider the inclusion i : P_{Λ} ↪ H^{b}(P_{Λ}), and let i^{∗} : Mod(H^{b}(P_{Λ})) ⟶ Mod(P_{Λ}) = Mod(Λ) be the restriction functor. It is well-known [59] that i^{∗} has a fully faithful right adjoint i^{!} given by the Kan construction. Hence Mod(Λ) is a localized quotient of Mod(H^{b}(P_{Λ})) and since the latter is locally Noetherian, we have that Mod(Λ) is also locally Noetherian, i.e. Λ is right Noetherian. Let A ∈ Mod(Λ) and let A^{•}[0] ∈ D(Λ) be the module A considered as stalk complex concentrated in degree zero. Since D(Λ) is pure-semisimple, we have D(Λ) = Add(H^{b}(P_{Λ})). Hence there exists a family {X_{i}^{•} : i ∈ I} ⊆ H^{b}(P_{Λ}) such that A^{•}[0] ⊕ Y^{•} = ⊕_{i∈I}X_{i}^{•}, for some complex Y^{•}. Applying the cohomology functor H^{0}, we have: A ⊕ H^{0}(Y^{•}) = ⊕_{i∈I}H^{0}(X_{i}^{•}). Since Λ is right Noetherian, H^{0}(X_{i}^{•}) is a finitely presented module, ∀i ∈ I. Hence A is a direct summand of a coproduct of finitely presented modules, so A is pureprojective. Hence any right module is pure-projective and Λ is right pure semisimple ring. It remains to show that r.gl.dimΛ < ∞. If A ∈ Mod(Λ) is an indecomposable module, then by Theorem 9.3, A^{•}[0] = ⊕_{i∈I}P_{i}^{•} where each P_{i}^{•} is compact with local endomorphism ring. Since A^{•}[0] is indecomposable, we have that A^{•}[0] is isomorphic to some P_{i}^{•} ∈ H^{b}(P_{Λ}). Of course this implies that A has finite projective dimension. Since A was an arbitrary indecomposable, and since Λ is right Artinian (as a right pure semisimple ring), we have that Λ has finite (right) global dimension.

(2) Let Λ be a right pure-semisimple right hereditary ring. Since r.gl.dimΛ ≤ 1, for any complex X^{•} ∈ D(Λ) we have X^{•} ≅ ⊕_{i∈ℤ}Σ^{−i}(H^{i}(X^{•})), see [54]. Since Λ is right hereditary, ∀i ∈ ℤ, the module H^{−i}(Σ^{i}(X^{•})) is isomorphic to an object in H^{b}(Proj(Λ)). Since Λ is right pure semisimple, H^{−i}(Σ^{i}(X^{•})) is a direct summand of a coproduct of finitely presented modules, and since Λ is right coherent as a right hereditary ring, any finitely presented module is isomorphic to an object in H^{b}(P_{Λ}). Putting these things together we have that X^{•} ∈ Add(H^{b}(P_{Λ})). This shows that D(Λ) = Add(H^{b}(P_{Λ})). Then by Theorem 9.3, D(Λ) is pure semisimple. This completes the proof since by Remark 11.5, the pure global dimension of a derived category is invariant under derived equivalence.

(3) Using Corollary 9.5, the easy proof is left to the reader.

Corollary 12.4. Let Λ be a countable ring. If Λ is not right Artinian or right perfect, or r.gl.dimΛ = ∞, then p.gl.dimD(Λ) = 1. In particular p.gl.dimD(ℤ) = 1.

We recall that a ring Λ is called representation-finite, if Λ is right Artinian and the set of isoclasses of indecomposable finitely presented right modules is finite. It is well known that the representation-finite commutative rings are exactly the Artinian principal ideal rings. Proposition 12.3 implies easily the following.

Corollary 12.5. Let Λ be a commutative ring. Then p.gl.dimD(Λ) = 0 iff Λ is a finite product of fields.

Example 12.6. Let H be the category of commutative cocommutative connected graded Hopf algebras over a field k with char(k) = p > 0, generated by elements of degrees 2p^{i}, i ≥ 0. By [66], H is a Grothendieck (functor) category which is hereditary and pure-semisimple. As in Proposition 12.3, D(H) is pure-semisimple.

Remark 12.7. If p.gl.dimD(Λ) = 0, then it is not true that Λ is right hereditary.

Indeed let Γ be a representation-finite piecewise hereditary algebra over an algebraically closed field, with gl.dimΓ ≥ 2 [33]. Then there exists a representation finite hereditary algebra Λ and a derived equivalence D(Λ) ≈ D(Γ). By Proposition 12.3, Γ is right pure-semisimple but not hereditary.

The author is indebted to the referee for the nice proof of the following result.

Proposition 12.8. If Λ is a right hereditary ring, then:

r.p.gl.dimΛ = p.gl.dimD(Λ).

Proof. By a result of A. Neeman [54], any complex A^{•} in D(Λ) is isomorphic to the coproduct of its cohomology groups: A^{•} = ⊕_{n∈ℤ}Σ^{−n}(H^{n}(A^{•})). Hence it suffices to show that p.p.dA = p.p.dA^{•}[0], ∀A ∈ Mod(Λ).

Let Y : Mod(Λ) ⟶ Mod(H^{b}(P_{Λ})) be the composition of the embedding Mod(Λ) ↪ D(Λ) given by A ↦ A^{•}[0], with the projectivization functor S : D(Λ) ⟶ Mod(H^{b}(P_{Λ})). The cohomology functor H^{0} : D(Λ) ⟶ Mod(Λ), preserves coproducts and kills pure-phantom maps. Hence it can be extended uniquely to an exact colimit preserving functor H̃^{0} : Mod(H^{b}(P_{Λ})) ⟶ Mod(Λ), which is given by F ↦ F (Λ) and satisfies H̃^{0}Y = Id_{Mod(Λ)}. It is not difficult to see that Y commutes with filtered colimits. Using the well-known fact that a short exact sequence in a module category is pure iff it is a filtered colimit of split short exact sequences, it follows that Y preserves pure-exactness. Since Λ is right hereditary, hence right coherent, it follows trivially that Y takes pure-projective modules to projective functors. Hence the image under Y of a pure-projective resolution of A, is a projective resolution of Y(A) in Mod(H^{b}(P_{Λ})). The latter can be lifted to a pure-projective resolution of A^{•}[0] in D(Λ). Conversely H̃^{0} preserves pure-exactness, since it is exact and colimit preserving. Since Λ is right coherent as a right hereditary ring, it follows that the cohomology functor H^{0} induces a functor H^{0} : H^{b}(P_{Λ}) ⟶ mod(Λ). Since H̃^{0} is exact, preserves coproducts and H̃^{0}S = H^{0}, it follows easily that H̃^{0} preserves pure-projectivity. Now any pure-projective resolution of A^{•}[0] in D(Λ) induces via S, a (pure-)projective resolution of Y(A) in Mod(H^{b}(P_{Λ})), which in turn induces via H̃^{0}, a pure-projective resolution of A in Mod(Λ).

The above observations about pure-projective resolutions in Mod(Λ) and in D(Λ), imply trivially that p.p.dA = p.p.dA^{•}[0].

Proposition 12.9. (see also [56]) Let Λ be a right coherent ring and assume that D(Λ) is a Brown category. Then for any flat right Λ−module F, we have: p.dF ≤ 1. If moreover Λ has finite weak global dimension, then r.p.gl.dimΛ ≤ 1.

Proof. Let A be a right Λ−module, and consider the stalk complex A^{•}[0]. By

BRT there exists a pure triangle (†) : P_{1}^{•} ⟶^{f^{•}} P_{0}^{•} ⟶^{g^{•}} A^{•}[0] ⟶^{h^{•}} Σ(P_{1}^{•}), such that P_{1}^{•}, P_{0}^{•} are in Add(H^{b}(P_{Λ})). Applying the cohomology functor H^{0}, we deduce a short exact sequence (††) : 0 ⟶ K ⟶ L ⟶ A ⟶ 0, where K, L are direct summands of coproducts of cohomology modules of complexes in H^{b}(P_{Λ}). Since Λ is right coherent, these cohomologies are sums of finitely presented modules. This implies that K, L are pure-projective modules. If A is flat, then (††) is pure, hence p.p.dA ≤ 1. But then p.dA ≤ 1, since p.dA = p.p.dA for A flat.

If w.gl.dimΛ < ∞, then it suffices to prove that (††) is pure. Let M be a finitely presented right Λ−module and α : M ⟶ A be a morphism. Since Λ is right coherent, w.gl.dimΛ < ∞ and M is finitely presented, it follows that M^{•}[0] is

compact. Since h^{•} is pure-phantom, the composition M^{•}[0] ⟶^{α^{•}[0]} A^{•}[0] ⟶^{h^{•}} Σ(P_{1}^{•}) is zero. So there exists a map β^{•} : M^{•}[0] ⟶ P_{0}^{•} such that β^{•} ∘g^{•} = α^{•}[0]. Applying H^{0}, we have H^{0}(β^{•}) ∘ H^{0}(g^{•}) = α. Hence α factors through L, so (††) is pure.

Remark 12.10. It is not difficult to see that if Λ is a right coherent ring with w.gl.dimΛ < ∞, then: r.p.gl.dimΛ ≤ p.gl.dimD(Λ). We refer to the work of Christensen, Keller and Neeman [22] for details and a thorough discussion of related topics. In particular in [22] it is proved that the above inequality can be strict.

Since the pure global dimension coincides with the global dimension for a von Neumann regular ring (obviously coherent) [39], we have the following consequence.

Corollary 12.11. Let Λ be a von Neumann regular ring. Then p.gl.dimD(Λ) ≤ 1 iff Λ is right hereditary. Moreover p.gl.dimD(Λ) = 0 iff Λ is semisimple Artinian.

By Theorem 3.2 of [7], we have the following.

Corollary 12.12. Let Λ = k<X> be the free algebra over a set of variables X, with |X| ≤ ℵ_{0}. If |Λ| = max{ℵ_{0}, |k|} = ℵ_{t}, then: p.gl.dimD(Λ) ≤ 1 ⇔ t = 0. In case X = {x}, then for the polynomial ring k[x]: p.gl.dimD(k[x]) = 1.

Other examples of rings Λ with a Brown derived category are: Dedekind domains and free Boolean rings on countably many generators, see [39].

Example 12.13. There exists a ring Λ such that p.gl.dimD(Λ) ≥ 2 or equivalently D(Λ) is not a Brown category. Indeed take Λ to be a von Neumann regular ring with r.gl.dimΛ ≥ 2 (for instance ∏_{1}^{∞} ℂ or a von Neumann regular ring with |Λ| = ℵ_{n}, n ≥ 2, see [39] ). Then by Corollary 12.11, D(Λ) is not Brown.

That BRT does not hold in general was observed first by B. Keller (see [56]) and is a consequence of the following results of B. Osofsky [58]. Let Λ be a ring with cardinality ℵ_{n} and r.gl.dimΛ = κ. Suppose that Λ is a regular local ring with |Λ| = |Λ/J ac(Λ)| or a complete regular local ring or Λ = k[x_{1}, ..., x_{κ}] and k is a field. Then if Q is the quotient field, we have p.d_{Λ}Q = min(κ, n + 1). In particular assuming the Continuum Hypothesis 2^{ℵ_{0}} = ℵ_{1}, we have p.d_{ℝ[x,y,z]}ℝ(x, y, z) = 2. Hence there are coherent rings Λ having flat modules with arbitrary large finite projective dimension. For example let k be a field and let t be determined by the equation ℵ_{t} = max(ℵ_{0}, |k|). Then the flat module k(x_{1}, x_{2}, ..., x_{n}) over the coherent ring k[x_{1}, x_{2}, ..., x_{n}] has projective dimension t + 1. By Proposition 12.9 for these rings, D(Λ) is not Brown, if t ≥ 1. By Remark 12.10, it follows that for any n ∈ ℕ ∪ {∞}, there exists a (von Neumann regular) ring Λ with p.gl.dimD(Λ) = n.

The next example shows that the left and right pure global dimensions of the derived category of a ring can be different, if they are positive. In particular it shows that the Brown property is not left-right hand symmetric.

Example 12.14. There exists a ring Λ with the following properties:

(i) p.gl.dimD(Λ) ≠ p.gl.dimD(Λ^{op}). (ii) D(Λ) is a Brown category, but D(Λ^{op}) is not.

Indeed Kaplansky in [41] constructed a von Neumann regular ring Λ of cardinality |Λ| = 2^{ℵ_{0}} with r.gl.dimΛ = 1 and l.gl.dimΛ ≥ 2. By Corollary 12.11, p.gl.dimD(Λ) = 1 and by Remark 12.10, p.gl.dimD(Λ^{op}) ≥ 2. Hence by Theorem 11.8, D(Λ) is Brown and D(Λ^{op}) is not. Assuming Continuum Hypothesis 2^{ℵ_{0}} = ℵ_{1}, it follows from Proposition 12.1 that p.gl.dimD(Λ^{op}) = 2.

12.2. The Derived Pure Semisimple Conjecture, Categories of Finite Type and Roos’s Problem. The next result shows that finite type is a symmetric condition for derived categories.

Proposition 12.15. The following are equivalent:

(i) D(Λ) is of finite type. (ii) D(Λ^{op}) is of finite type. (iii) D(Λ) and D(Λ^{op}) are pure-semisimple. (iv) KGdimD(Λ) = 0.

Proof. The natural duality P_{Λ}^{op} ≈ P_{Λ^{op}} extends to a duality H^{b}(P_{Λ})^{op} ≈ H^{b}(P_{Λ^{op}}), so (D(Λ)^{b})^{op} ≈ D(Λ^{op})^{b}. Using the notation of Section 11, we have p.gl.dimD(Λ)^{op} = p.gl.dimD(Λ^{op}) and the assertion follows from Proposition 11.23.

Corollary 12.16. (1) If D(Λ) is of finite type, then Λ is representation-finite of

finite global dimension.

(2) If Λ is (derived equivalent to) a representation-finite right hereditary ring, then D(Λ) is of finite type.

(3) Let Λ be a PI-ring or a ring with a Morita duality, for instance an Artin algebra or a Quasi-Frobenius ring. If D(Λ) is pure semisimple, then Λ is of finite representation type and of finite global dimension.

(4) If Λ is representation-infinite Artin algebra of finite global dimension, then KGdimD(Λ) ≠ 1. If Λ is derived equivalent to a hereditary algebra, then KGdimD(Λ) ≠ 1, independently of the representation type.

Proof. By a well-known result of Auslander [39], a ring is representation-finite iff it is left and right pure-semisimple. Then (1), (2) follow from Propositions 12.3, 12.15.

(3) follows from results of Simson, Herzog [35] claiming that right pure semisimple rings with polynomial identity or with Morita duality, are representation-finite.

(4) Since Λ has finite global dimension, the canonical full embedding mod(Λ) ↪ D(Λ) has its image in H^{b}(P_{Λ}). As in the proof of Proposition 12.3, it follows that Mod(mod(Λ)) is a localized quotient of Mod(H^{b}(P_{Λ})). It follows from results of Gabriel [30] that KdimMod(mod(Λ)) ≤ KdimMod(H^{b}(P_{Λ})) ≤ GKdimD(Λ). Assume that GKdimD(Λ) = 1, so KdimMod(mod(Λ)) ≤ 1. Since KdimMod(mod(Λ)) ≠ 1 for any Artin algebra (see [47], [36]), it follows that KdimMod(mod(Λ)) = 0. Then by a result of M. Auslander [2], it follows that Λ is representation finite and this is not the case. Hence GKdimD(Λ) ≠ 1. If Λ is derived equivalent to a hereditary algebra Γ and GKdimD(Λ) = 1, then GKdimD(Γ) = 1 and as above we deduce that Γ is representation finite. Hence by (2), D(Λ) is of finite type. Then by Proposition 12.15, GKdimD(Λ) = 0 and this is not true. Hence GKdimD(Λ) ≠ 1.

Part (4) of the above Corollary is a partial analogue of a module-theoretic result of Krause [47] and Herzog [36]. We don’t know if for any Artin algebra Λ it holds that GKdimD(Λ) ≠ 1.

We recall that the still open Pure Semisimple Conjecture, asserts that a right pure-semisimple ring Λ is left pure-semisimple, equivalently Λ is representationfinite. We formulate an analogue of this Conjecture in the derived category.

Derived Pure Semisimple Conjecture 12.17. Pure semisimplicity in D(Λ) is left-right hand symmetric: p.gl.dimD(Λ) = 0 ⇔ p.gl.dimD(Λ^{op}) = 0. Equivalently if D(Λ) is pure semisimple then D(Λ) is of finite type.

Remark 12.18. The truth of Derived Pure Semisimple Conjecture (DPSC) implies the truth of the Pure Semisimple Conjecture (PSC). Indeed if DPSC is true and PSC is not, then by [35], there exists a right hereditary right pure-semisimple ring which is not left pure-semisimple. By Proposition 12.3, D(Λ) is pure-semisimple, hence also D(Λ^{op}) is pure-semisimple. By Proposition 12.3 again, Λ is left puresemisimple, which is not true. Hence PSC is true. Note that by Propositon 12.3, for (left and right) hereditary rings, DPSC is equivalent to PSC.

Example 12.19. The following example provides a negative answer to Roos’s Problem 11.22, to Question 11.21 and to the Derived Pure Semisimple Conjecture for “rings with several objects” in the sense of Mitchell [53].

Consider ℵ_{t}, t ≥ 0 as a totally ordered set, let k be a field of cardinality ℵ_{0} and let [ℵ_{t}, Mod(k)] be the category of k−linear representations of ℵ_{t}. Let k(ℵ_{t}) be the ringoid which yields an equivalence [ℵ_{t}, Mod(k)] ≈ Mod(k(ℵ_{t})^{op}). By a result of Brune [17], Mod(k(ℵ_{t})^{op}) is hereditary and pure-semisimple and p.gl.dimMod(k(ℵ_{t})) = t + 1 if t < ℵ_{0} and ∞ if t ≥ ℵ_{0}. As in Proposition 12.3, it follows that the (compactly generated) derived category D(Mod(k(ℵ_{t})^{op})) is puresemisimple but the derived category D(Mod(k(ℵ_{t}))) is not. Using Proposition 12.15 we infer that D(Mod(k(ℵ_{t})^{op})) is pure-semisimple but not of finite type.

On the other hand since D(Mod(k(ℵ_{t})^{op})) is pure semisimple, we have that the Grothendieck category Mod(D(Mod(k(ℵ_{t})^{op}))^{b}) is Frobenius but not locally finite since otherwise, D(Mod(k(ℵ_{t})^{op})) would be of finite type. Note that by the Faith-Walker Theorem [27], the answer to Roos’s Problem is affirmative for rings.

For the representation-theoretic notions and arguments used in the following result (which shows that for Artin algebras, DPSC is true), we refer to [33].

Theorem 12.20. For an Artin algebra Λ the following are equivalent: (i) D(Λ) is of finite type. (ii) D(Λ) is pure-semisimple. (iii) Λ is an iterated tilted algebra of Dynkin type. (iv) Λ is derived equivalent to a representation finite hereditary algebra. In this case, the set of isoclasses of indecomposable compact objects of D(Λ) up to shift, is finite.

Proof. By Proposition 12.15, (i) ⇒ (ii) and by [33], (iii) ⇒ (iv). By Corollary 12.16, (iv) ⇒ (i). So it remains to prove that (ii) implies (iii).

(ii) ⇒ (iii) By Corollary 12.16, Λ is a representation finite algebra of finite global dimension. Let Λ̂ be the repetitive algebra of Λ. By [33] we have a triangle equivalence H^{b}(P_{Λ}) = D^{b}(mod(Λ)) ≅ \underline{mod}(Λ̂). We shall show that Λ̂ is locally representation finite. Let P_{Λ} be the category of projective objects in mod(Λ̂). Since mod(Λ̂) is a Frobenius category, by Proposition 9.2, P_{Λ} is a weak abelian category. Since mod(Λ̂) is a length category (any object has finite length), the module category Mod(P_{Λ}) is locally Noetherian, hence it is perfect according to Proposition 9.2. Since H^{b}(P_{Λ}) ≅ \underline{mod}(Λ̂), we have that Mod(\underline{mod}(Λ̂))) is also perfect. By a result of Simson [66], since Mod(\underline{mod}(Λ̂))) and Mod(P_{Λ}) are perfect, the module category Mod(mod(Λ̂))) is also perfect or equivalently locally Noetherian. Let Γ ⊆ Λ̂ be a finite full convex subcategory. Then we have an inclusion mod(Γ) ⊆ mod(Λ̂). This inclusion implies that Mod(mod(Γ)) is a localizing subcategory of Mod(mod(Λ̂)). Since the latter is a locally Noetherian category, so is Mod(mod(Γ)). But since Γ is a Artin algebra, by a well-known result of Auslander [39], we have that Γ is representation finite. Hence Λ̂ is locally representation finite. By [33], Λ is an iterated tilted algebra of Dynkin type. The last claim is trivial and is left to the reader.

The next Corollaries follow from [7], [8] and the above results.

Corollary 12.21. Let Λ be a finite dimensional hereditary k−algebra over an algebraically closed field k of cardinality ℵ_{t}. Then we have the following:

(i) If t = 0, then p.gl.dimD(Λ) = 0 or 1. ( 0 occurs iff Λ is representation-finite). (ii) If t > 0, then:

(α) If Λ is representation-finite, then p.gl.dimD(Λ) = 0.

(β) If Λ is tame, then p.gl.dimD(Λ) = 2. (γ) If Λ is wild, then p.gl.dimD(Λ) = t + 1.

Corollary 12.22. Let Q be a quiver with or without oriented cycles and let Λ = k[Q] be its quiver-algebra over an algebraically closed field k of cardinality ℵ_{t}. Then:

(i) If t = 0, then p.gl.dimD(Λ) = 0 or 1, the value 0 occurs iff Q is Dynkin. (ii) If t > 0, then:

(α) If Q is a Dynkin diagram, then p.gl.dimD(Λ) = 0. (β) If Q is an oriented cycle, then p.gl.dimD(Λ) = 1. (γ) If Q is extended Dynkin not an oriented cycle, then p.gl.dimD(Λ) = 2. (δ) In all the remaining cases, Q is a wild quiver and p.gl.dimD(Λ) = t + 1.

Corollary 12.23. Let Λ be a finite dimensional local or radical squared zero k−algebra over an algebraically closed field of uncountable cardinality ℵ_{t}. If D(Λ) is Brown, then Λ is representation-finite.

12.3. Stable Module Categories. Let C be a skeletally small additive category such that the module category Mod(C) is Frobenius. It is not difficult to see that the stable category \underline{Mod}(C) is compactly generated and \underline{Mod}(C)^{b} = \underline{mod}(C). If Mod(C^{op}) is also Frobenius, then the Auslander-Bridger transpose duality functor Tr : \underline{mod}(C)^{op} ⟶^{≈} \underline{mod}(C^{op}), see [3], shows that (\underline{Mod}(C)^{b})^{op} ≈ \underline{Mod}(C^{op})^{b}.

Proposition 12.24. p.gl.dimMod(C) = p.gl.dim\underline{Mod}(C). In particular if w(C) ≤ ℵ_{t}, for some t ≥ 0, then: p.gl.dim\underline{Mod}(C) ≤ t + 1.

Proof. Since any projective right C−module is pure projective, it follows trivially that \underline{M} ∈ \underline{Mod}(C) is pure-projective iff M is pure-projective. Similarly it is not difficult to see that a short exact sequence 0 ⟶ K ⟶ L ⟶ M ⟶ 0 in Mod(C) is pure iff the induced triangle \underline{K} ⟶ \underline{L} ⟶ \underline{M} ⟶ Σ(\underline{K}) is pure in \underline{Mod}(C); further any pure triangle in \underline{Mod}(C) arises in this way. It follows that a pure-projective resolution of M induces a pure-projective resolution of \underline{M} and any pure-projective resolution of \underline{M} in the stable category arises in this way. This implies that p.p.dM = p.p.d\underline{M}, ∀M ∈ Mod(C). Hence we have p.gl.dimMod(C) = p.gl.dim\underline{Mod}(C).

Theorem 12.25. If Mod(C) is Frobenius then the following are equivalent: (i) \underline{Mod}(C) is pure semisimple (of finite type). (ii) Mod(C) is pure semisimple (and locally Artinian). (iii) Mod(\underline{mod}(C)) is locally Noetherian or perfect (and locally Artinian). If \underline{Mod}(C) is of finite type, then Mod(C^{op}) is Frobenius and \underline{Mod}(C^{op}) is of finite type. Finally if one of the above equivalent conditions is true, then:

\underline{Mod}(C) ≅ Proj(Mod(\underline{mod}(C))) = Flat(Mod(\underline{mod}(C))) = Inj(Mod(\underline{mod}(C))).

Proof. By Theorem 9.3, (i) ⇔ (iii) and by Proposition 12.24, (i) ⇔ (ii). The easy proof of the remaining assertions and the parenthetical case is left to the reader.

If Λ is a QF-ring, then Mod(Λ) is a Frobenius. Hence \underline{Mod}(Λ) is a compactly generated triangulated category and \underline{Mod}(Λ)^{b} = \underline{mod}(Λ). The following consequence of Theorem 12.25 generalizes a result of [48], [13].

Corollary 12.26. Let Λ be a QF-ring. Then the following are equivalent: (i) \underline{Mod}(Λ), equivalently \underline{Mod}(Λ^{op}), is pure-semisimple (of finite type). (ii) The set of isoclasses of indecomposable compact objects of \underline{Mod}(Λ) is finite. (iii) Λ is representation finite.

If Λ is an Artin algebra, we denote by T(Λ) = Λ ⋉ D(Λ) the trivial extension of Λ by the minimal injective cogenerator [33]. The following consequence of Theorems 12.20 and 12.25, shows that the finite type property coincides for the three triangulated categories D(Λ), \underline{Mod}(T(Λ)), \underline{Mod}(Λ̂), associated to Λ.

Corollary 12.27. For an Artin algebra Λ the following are equivalent. (i) The derived category D(Λ) of Λ is of finite type. (ii) The stable module category \underline{Mod}(T(Λ)) is of finite type. (iii) The stable module category \underline{Mod}(Λ̂) of its repetitive algebra Λ̂, is of finite type. (iv) Λ is an iterated tilted algebra of Dynkin type.

Corollary 12.28. Let Λ be a finite dimensional self-injective local k−algebra, over an algebraically closed field k. Then the following statements are equivalent:

(i) \underline{Mod}(Λ) is a Brown category, i.e. p.gl.dim\underline{Mod}(Λ) ≤ 1. (ii) Either Λ is representation finite or else k is a countable field.

Proof. That (ii) implies (i) follows from Proposition 12.24. Suppose that \underline{Mod}(Λ) is Brown. If p.gl.dim\underline{Mod}(Λ) = 0, then Λ is representation-finite. If p.gl.dim\underline{Mod}(Λ) = 1, then by Proposition 12.24, r.p.gl.dimΛ = 1. Then by [8], k is countable.

Using induction on the p−Sylow subgroups, we have the following consequence first observed by D. Benson and G. Gnacadja.

Corollary 12.29. [15] Let Λ = kG be the group algebra of a finite group G over an algebraically closed field k with char(k) = p. Then the following are equivalent:

(i) \underline{Mod}(Λ) is a Brown category, i.e. p.gl.dim\underline{Mod}(Λ) ≤ 1. (ii) Either G has cyclic p−Sylow subgroups or else k is a countable field.

If Λ and Γ are derived equivalent rings, then p.gl.dimD(Λ) = p.gl.dimD(Γ). If in addition Λ, Γ are self-injective Artin algebras, then by [63], there exists a triangle equivalence \underline{Mod}(Λ) ≈ \underline{Mod}(Γ). Hence p.gl.dim\underline{Mod}(Λ) = p.gl.dim\underline{Mod}(Γ). Finally if Λ is an Artin algebra of finite global dimension, then by Happel’s theorem [33], there exists a triangle equivalence D(Λ) ≈ \underline{Mod}(Λ̂), where Λ̂ is the repetitive algebra. Since Mod(Λ̂) is Frobenius, in case gl.dimΛ < ∞ we have p.gl.dimD(Λ) = p.gl.dim\underline{Mod}(Λ̂). It is interesting to have characterizations in representation-theoretic terms of Artin algebras, resp. self-injective Artin algebras, with Brown derived categories, resp. stable module categories.

If the derived (resp. stable module) category of a ring (resp. QF-ring) is a Brown category, then the results of Sections 10, 11 are true for the triangulated category D(Λ) (resp. \underline{Mod}(Λ)). We leave the formulations of these results to the reader.

Remark 12.30. The results of this Section indicate that finite type is the appropriate notion of “representation-finiteness” for derived and stable categories. It is unclear what are the appropriate notions for tameness and wildness in these categories in analogy with the tame-wild dichotomy of finite dimensional algebras.

12.4. Localization and the Derived Category. Let Λ be an associative ring. We denote by H(Λ), the unbounded homotopy category of all right Λ−modules and by H(Proj(Λ)), the homotopy category of all projective right Λ−modules.

Definition 12.31. A complex P^{•} is called a Cartan-Eilenberg projective complex (CE-projective complex, for short), if P^{•} is homotopy equivalent to a complex having projective components and zero differential.

The full subcategory of H(Λ) of all CE-projective complexes is denoted by P_{CE}. Setting E := E(P_{CE}), we obtain a proper class of triangles in H(Λ), and H(Λ) has enough projectives and P(E) = P_{CE}, (see for instance [23], [26]). Choosing C = H(Λ) and X = {Σ^{n}(Λ) | n ∈ ℤ} in Theorem 6.5 and keeping the terminology and notation of Section 6, we have the following result proved in [16], [67], [42], [71], by using essentially the natural closed model structure of the category of complexes.

Theorem 12.32. (1) { \atop P(E)^{⋆∞}} is the full subcategory H_{P}(Λ) of H(Λ) consisting of all complexes A^{•} which are homotopy equivalent to complexes P^{•} with the following property. There exists a tower of complexes

0 ⊆ P_{0}^{•} ⊆ P_{1}^{•} ⊆ ⋯ ⊆ P_{n}^{•} ⊆ P_{n+1}^{•} ⊆ ⋯ ⊆ P^{•}, such that :

(†) P^{•} = ⋃_{n≥0} P_{n}^{•}, each inclusion P_{n}^{•} ⊆ P_{n+1}^{•} splits in each component, and

each quotient complex P_{n+1}^{•}/P_{n}^{•} is a CE−projective complex.

(2) D_{E}(H(Λ)) = D(Λ), the quotient functor H(Λ) ⟶ D(Λ) admits a fully faithful left adjoint and A_{E}(H(Λ)) is the subcategory H_{Ac}(Λ) of acyclic complexes.

(3) The canonical functor H_{P}(Λ) ⟶ D(Λ) is a triangle equivalence.

(4) The inclusion H_{Ac}(Λ) ↪ H(Λ) has a left adjoint a, the inclusion H_{P}(Λ) ↪ H(Λ) has a right adjoint p and any complex A^{•} is contained in a triangle

p(A^{•}) ⟶ A^{•} ⟶ a(A^{•}) ⟶ Σ(p(A^{•})),
where p(A^{•}) ∈ H_{P}(Λ) and a(A^{•}) ∈ H_{Ac}(Λ).

(5) Any additive functor F : Mod(Λ) ⟶ Mod(Γ) has a total left derived functor

\underline{\underline{L}}F : D(Λ) ⟶ D(Γ).

It is clear from Section 6 that the above Theorem holds in much more general situations. For instance in the Stable Homotopy Category, and in the derived category of a Grothendieck category with projectives, where as noted in [16], [67], [42], [71], the theory implies the existence of total left derived functors defined on unbounded complexes. The dual result concerning injectives and total right derived functors also holds, and can be applied for instance to the derived category of any Grothendieck category, e.g. to categories of sheaves, since the theory of the previous Sections can be dualized (compactness can be avoided using the cohomology functor as in Remark 6.6). The main Theorem of Section 6 also covers the construction of the various derived categories associated with projective classes in [20].

12.5. The Stable Derived Category. For simplicity we assume that Λ is an Artin algebra. We denote by D^{b}(Λ) the bounded derived category D^{b}(mod(Λ)). It is well known that D^{b}(Λ) = H^{−,b}(P_{Λ}) = H^{+,b}(I_{Λ}), where I_{Λ} is the full subcategory of mod(Λ) consisting of injective modules and H^{−,b}(P_{Λ}) (H^{+,b}(I_{Λ})) is the full subcategory of H^{−}(P_{Λ}) (H^{+}(I_{Λ})), consisting of complexes with bounded cohomology. As in subsection 12.4, the Cartan-Eilenberg projectives P_{CE} in D^{b}(Λ) form the full subcategory of E−projectives of a proper class of triangles E, and moreover using the above identifications, D^{b}(Λ) has enough E−injectives, namely the full subcategory I_{CE} of the Cartan-Eilenberg injective complexes in D^{b}(Λ).

The projectively stable (bounded) derived category \underline{D}^{b}(Λ) of Λ, is defined to be the stable category D^{b}(Λ)/P_{CE}, and similarly the injectively stable (bounded) derived category \overline{D}^{b}(Λ) of Λ, is defined to be the stable category D^{b}(Λ)/I_{CE}. By Section 7, we have that \underline{D}^{b}(Λ) is left triangulated and \overline{D}^{b}(Λ) is right triangulated.

Proposition 12.33. Λ is self-injective iff \underline{D}^{b}(Λ) is triangulated iff \overline{D}^{b}(Λ) is triangulated. If Λ is self-injective, then: \underline{D}^{b}(Λ) = \overline{D}^{b}(Λ).

Proof. Let 0 ⟶ A ⟶ B ⟶ C ⟶ 0 be an exact sequence in mod(Λ). It induces a triangle A[0] ⟶ B[0] ⟶ C[0] ⟶ A[1] ∈ E in D^{b}(Λ). If \underline{D}^{b}(Λ) is triangulated, then Λ[0] ∈ P_{CE} ⊆ I_{CE}. Hence the sequence 0 ⟶ (C[0], Λ[0]) ⟶ (B[0], Λ[0]) ⟶ (A[0], Λ[0]) ⟶ 0, equivalently the sequence 0 ⟶ (C, Λ) ⟶ (B, Λ) ⟶ (A, Λ) ⟶ 0, is exact and Λ is self-injective. Conversely if Λ is self-injective, then P_{Λ} = I_{Λ}. Then obviously P_{CE} = I_{CE} and by Theorem 7.2, \underline{D}^{b}(Λ) is triangulated.

Our aim is to show that the stable derived category \underline{D}^{b}(Λ) is connected with the stable module category \underline{mod}(Λ), which is also left triangulated, in the same way as D^{b}(Λ) is connected with mod(Λ). Consider the projective resolution functor ϱ : mod(Λ) ⟶ D^{b}(Λ), which assigns a deleted projective resolution to a Λ−module. Trivially ϱ sends projective modules to CE-projective complexes, hence it induces a functor ℜ : \underline{mod}(Λ) ⟶ \underline{D}^{b}(Λ), making the following diagram commutative:

P_{Λ} ⟶ mod(Λ) ⟶ \underline{mod}(Λ)
↓  ^{ϱ}↓  ^{ℜ}↓
P_{CE} ⟶ D^{b}(Λ) ⟶ \underline{D}^{b}(Λ)

The cohomology functors H^{n} : D^{b}(Λ) ⟶ mod(Λ), n ∈ ℤ, send the class of triangles E in D^{b}(Λ) to short exact sequences in mod(Λ) and they send P_{CE} to P_{Λ}, hence there are induced functors \underline{H}^{n} : \underline{D}^{b}(Λ) ⟶ \underline{mod}(Λ). We denote by H := ⊕_{n∈ℤ}H^{n} and \underline{H} := ⊕_{n∈ℤ}\underline{H}^{n}. Using the definition of the functor ℜ and the construction of the triangulation of \underline{D}^{b}(Λ) as in Section 7, it is easy to deduce the following.

Proposition 12.34. The functors ℜ, \underline{H} are exact, ℜ is fully faithful and \underline{H}ℜ = Id_{\underline{mod}(Λ)}. Further: E−gl.dimD^{b}(Λ) = gl.dimΛ and the functor ℜ induces an isomorphism K_{0}(\underline{mod}(Λ)) ≅ K_{0}(\underline{D}^{b}(Λ)) between the Grothendieck groups of the left triangulated categories \underline{mod}(Λ) and \underline{D}^{b}(Λ).

Lemma 12.35. A morphism f^{•} : A^{•} ⟶ B^{•} in D^{b}(Λ) is an isomorphism in \underline{D}^{b}(Λ) iff ∀n ∈ ℤ, \underline{H^{n}(f^{•})} : \underline{H^{n}(A^{•})} ⟶ \underline{H^{n}(B^{•})} are isomorphisms in \underline{mod}(Λ).

Proof. If \underline{f}^{•} is an isomorphism in \underline{D}^{b}(Λ), then there exists g^{•} : B^{•} ⟶ A^{•} such that 1_{A^{•}} − f^{•} ∘ g^{•}, 1_{B^{•}} − g^{•} ∘ f^{•} factor through some CE-projective complex. Then 1_{H^{n}(A^{•})} − H^{n}(f^{•}) ∘ H^{n}(g^{•}) and 1_{H^{n}(B^{•})} − H^{n}(g^{•}) ∘ H^{n}(f^{•}) factor through some projective module. Hence \underline{H^{n}(f^{•})} are isomorphisms in \underline{mod}(Λ), ∀n ∈ ℤ. Conversely since D^{b}(Λ) is a Krull-Schmidt category, we can assume that A^{•}, B^{•} do not have summands with projective homology. If \underline{H^{n}(f^{•})} are isomorphisms in \underline{mod}(Λ), then so are H^{n}(f^{•}). Hence f^{•} is invertible in D^{b}(Λ), hence in \underline{D}^{b}(Λ).

In the following result we denote by mod(Λ)^{b} the full subcategory of the ℤ−graded category mod(Λ)^{ℤ} consisting of all the bounded objects.

Corollary 12.36. (1) The homology functor H : D^{b}(Λ) ⟶ mod(Λ)^{b} is a representation equivalence iff the functor ℜ is an equivalence iff gl.dimΛ ≤ 1.

(2) If Λ is indecomposable, then \underline{D}^{b}(Λ) is abelian iff Λ is hereditary 1-Gorenstein.

Proof. Part (1) follows as in Theorem 10.2, using Proposition 12.34 and the fact that Λ is hereditary iff any indecomposable complex in D^{b}(Λ) is a stalk complex in some degree [33]. If Λ is indecomposable, then by [4], Λ is hereditary 1−Gorenstein iff \underline{mod}(Λ) is abelian. Part (2) follows from this, using (1) and Proposition 12.34.

The following generalizes results of Wheeler [72], in which Lemma 12.35 and the next result are proved in the case of a self-injective algebra. We note that by Proposition 12.33, in case Λ is self-injective, the stable derived category is triangulated. This is also a result of Wheeler [73], proved by different methods.

Theorem 12.37. Let Λ, Γ be Artin algebras, and let (F, G) be an adjoint pair of exact functors, F : mod(Λ) ⇆ mod(Γ) : G. Then the following are equivalent:

(i) The functors F, G preserve projectives and the induced functors \underline{F} : \underline{mod}(Λ) ⟶ \underline{mod}(Γ) and \underline{G} : \underline{mod}(Γ) ⟶ \underline{mod}(Λ) are inverse triangle equivelences.

(ii) The induced functors F^{•} : D^{b}(Λ) ⟶ D^{b}(Γ) and G^{•} : D^{b}(Γ) ⟶ D^{b}(Λ) preserve CE−projectives and the induced functors \underline{F}^{•} : \underline{D}^{b}(Λ) ⟶ \underline{D}^{b}(Γ) and \underline{G}^{•} : \underline{D}^{b}(Γ) ⟶ \underline{D}^{b}(Λ) are inverse triangle equivalences.

Proof. Fix counit ε : F G ⟶ Id_{mod(Γ)} and unit δ : Id_{mod(Λ)} ⟶ GF. Let F^{•}, G^{•} be the induced functors on the derived categories. Then ε, δ induce morphisms ε^{•}, δ^{•} which are counit and unit respectively of an adjoint pair (F^{•}, G^{•}). If condition (i) holds, then since F, G preserve projectives, the functors F^{•}, G^{•} preserve CEprojectives, so we have induced functors \underline{F}^{•}, \underline{G}^{•} on the stable derived categories. It is easy to see that (\underline{F}^{•}, \underline{G}^{•}) is also an adjoint pair, with counit \underline{ε}^{•} and unit \underline{δ}^{•}. By Lemma 12.35, it suffices to show that \underline{H^{n}(ε^{•})}, \underline{H^{n}(δ^{•})} are isomorphisms. This is immediate, from the fact that the induced morphisms \underline{ε}, \underline{δ} are isomorphisms in the stable module categories by hypothesis. The converse follows similarly.

Since by [63], derived equivalent self-injective algebras are stably equivalent by an equivalence satisfying condition (i) of the above Theorem, we have the following.

Corollary 12.38. Let Λ, Γ be derived equivalent self-injective Artin algebras. Then the triangulated categories \underline{D}^{b}(Λ) and \underline{D}^{b}(Γ) are triangle equivalent.

Remark 12.39. The results of this section generalize to the homotopy category of complexes of injective comodules, resp. stable category of comodules modulo injectives, over commutative, resp. finite dimensional, Hopf algebras over a field. Also most of the results concerning the stable derived category, generalize to the unbounded case and are valid for any ring. Details are left to the reader.

Acknowledgements. Most of the results of this paper were obtained and presented in lectures, between January and April 1997 during a one year stay at the Instituto De Matemáticas, U.N.A.M., Mexico, by a postdoctoral fellowship. The final version of this paper was written during a stay at the University of Bielefeld, thanks to a fellowship from the TMR-network “Algebraic Lie Representations”. The author thanks all the members of Representation Theory group at both universities, and especially Prof. J. A. de la Peña and Prof. C.M. Ringel, for their warm hospitality. The author thanks H. Krause for the very enjoyable and interesting discussions and Prof. D.J. Benson, Dr. G. Gnacadja for their useful comments. Finally the author is grateful to Prof. J.D. Christensen, to Prof. B. Keller, and especially to the referee, for their very valuable comments, suggestions and corrections which resulted in a significant improvement of the paper.

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