Purity and Almost Split Morphisms in Abstract Homotopy Categories

A Unified Approach via Brown Representability^{†}

Apostolos Beligiannis

(abeligia@mail.otenet.gr, abeligia@aegean.gr)

Department of Mathematics, University of the Aegean, 83200 Karlovassi, Samos, Greece.

September 18 2000 Abstract. Our aim in this paper is to develop a theory of purity and to prove in a unified conceptual way the existence of almost split morphisms, almost split sequences and almost split triangles in Abstract Homotopy Categories, a rather omnipresent class of categories of interest in representation theory. Our main tool for doing this is the classical Brown Representability Theorem.

Keywords: Purity, Almost Split Morphisms, Brown Representability, Phantom Maps, Abstract Homotopy Categories, Pure-injectives, Flat Functors, Approximations.

2000 AMS Subj. Class: 16G, 18G, 18E.

1. Introduction

Abstract Homotopy Categories were introduced by E. Brown in the midsixties as the proper framework for the study of the homotopy theory of CW-complexes. In this setting he proved in [18] his celebrated Representability Theorem, a variant of which has recently found important applications in the stable module category of a modular group algebra and more generally in compactly generated triangulated categories, mainly through the work of Rickard [42] and Neeman [40]. Our main purpose in this paper is to develop a theory of purity and a theory of existence of almost split morphisms in an Abstract Homotopy Category, using Brown Representability Theorem as a main catalysing tool.

This aim is justified by the fact that Abstract Homotopy Categories are omnipresent in Representation Theory. They include module categories, locally finitely presented categories with products, compactly generated triangulated categories, categories of projective modules and stable categories modulo projectives over left coherent and right perfect rings, categories of injective modules and stable categories modulo injectives over right Morita rings, (stable) categories of Cohen-Macaulay modules over Gorenstein rings, and many others. Abstract Homotopy Categories permit a unified treatment of many important phenomena occuring in various forms in the above examples, for instance pure homological algebra and its companion existence theory of almost split morphisms, two of the most fundamental concepts of modern

† To appear in Algebras and Representation Theory.

\begin{matrix}c \\ ◯ \end{matrix} 2000 Kluwer Academic Publishers. Printed in the Netherlands.

Representation Theory. Our results which are stated in the setting of an Abstract Homotopy Category, generalize and prove in a unified conceptual way, most of the basic results concerning purity and almost split morphisms which are known to be true in the setting of the above mentioned, more concrete and rather unrelated, examples. Moreover we obtain many new results. We note that our approach was inspired by recent work of H. Krause [37] in the triangulated case. The organization of the article is as follows.

In Section 2 and following [13], we recall some basic facts concerning homotopy colimits, Abstract Homotopy Categories and Brown Representability and we present a large list of examples of algebraic and topological categories which are Abstract Homotopy Categories in a very natural way.

In Section 3, we introduce the class of homological compact objects in an Abstract Homotopy Category C. Roughly speaking these are the objects which behave well with respect to arbitrary coproducts and weak cokernels. Using Brown Representability, we associate to any homological compact object X and any maximal left ideal 𝔪 of End_{C}(X), the 𝔪−dual object 𝔻_{𝔪}(X) of X, which has local endomorphism ring and in a sense has the dual properties of X. This construction turns out to be fundamental, since it is the main tool for the proof of the main result of this section, which asserts that the 𝔪−dual object 𝔻_{𝔪}(X) of any homological compact object X is the source of a left almost split morphism in C. Moreover if C has weak kernels, then any homological compact object with local endomorphism ring is the target of a right almost split morphism in C.

In Section 4 we develop a theory of purity in an Abstract Homotopy Category C, which is based on a class of diagrams in C, called pure-sequences, defined using a fixed subcategory X ⊆ C, called a Whitehead subcategory, which is part of the structure of C. Using these sequences we define pure-projective and pure-injective objects in C. The main tools for the study of purity are the representation categories L(C), D(C) of C. These are Grothendieck categories, and moreover L(C) is a functor category and D(C) is locally coherent, and there are connecting representation functors S: C ⟶ L(C) and T: C ⟶ D(C), which reflect many important pure-theoretic properties of C. Under reasonable conditions we prove that C has enough pure-projective objects and pureinjective envelopes and moreover the functor S induces an equivalence between the pure-projective objects of C and the projective objects of L(C) and the functor T induces an equivalence between the pure-injective objects of C and the injective objects of D(C). In particular it follows that any indecomposable pure-injective object in C has a local endomorphism ring.

The Representation functors S, T are not fully faithful in general. The failure of their fully faithfullness is measured by an ideal of morphisms in C, called the ideal of phantom maps, studied in Section 5. The structure of the ideal of phantom maps is strongly connected with the pure homological behavior of C. Indeed in many cases C is phantomless iff C is pure-semisimple. We characterize the pure-semisimplicity, the finite type property and the naturally defined Ziegler spectrum of C, in terms of properties of the representation categories L(C), D(C), and we present a variant of Auslander’s correspondence [5] in this setting, which gives a bijective correspondence between Morita equivalence classes of representation-finite rings and equivalence classes of special Abstract Homotopy Categories of finite type.

In Section 6 we study another useful version of Brown Representability in an Abstract Homotopy Category and its connection with flat approximations, the latter connection was inspired by a recent result of H. Krause [36] in the triangulated case. In most cases of interest the representation functor S: C ⟶ L(C) has image in the full subcategory Flat(X) of flat contravariant functors from the Whitehead subcategory X mentioned above, to the category of abelian groups. We give necessary (and in many cases sufficient) conditions such that S : C ⟶ Flat(X) is a representation equivalence in the sense of Auslander [3], in terms of projective dimension of flat functors and flat approximations. This has important consequences if C is triangulated, see [12], [21], [39], [41]. Moreover we show that the image ImS of S is functorially finite in the representation category L(C), and the image ImT of T is functorially finite in the representation category D(C).

In Section 7 we apply the theory of the previous sections to the most important examples of Abstract Homotopy Categories, namely those of module categories, locally finitely presented categories with products and compactly generated triangulated categories. In all these examples we recover in a unified way the main results about purity and the existence of almost split morphisms, sequences and triangles on these categories, proved by many authors following the fundamental work of M. Auslander and I. Reiten [7]. This section contains also new results. For instance we prove that a pure-injective object in a compactly generated triangulated category C, occurs as a source of a left almost split morphism in C iff it is a dual object of a compact object. If the compact objects of C form a Krull-Schmidt category, then we prove that a pure-injective object is the source of an Auslander-Reiten triangle in C iff it is the dual object of an indecomposable compact object.

A convention used in the paper is that we compose morphisms f: A ⟶ B, g: B ⟶ C in a given category in the diagrammatic order, i.e. the composition of f, g is denoted by f ∘ g. There two exceptions: we use the usual antidiagrammatic order, when we compose functors and when we apply elements to morphisms in concrete categories. Our additive categories admit finite direct sums.

2. Abstract Homotopy Categories and Brown Representability

2.1. Weak and Homotopy Colimits

Let C be an additive category and let f: A ⟶ B be a morphism in C. We recall that a weak cokernel of f is a morphism h: B ⟶ C such that f ∘ h = 0 and if t: B ⟶ D is a morphism with f ∘ t = 0, then there exists a morphism α: C ⟶ D such that t = h ∘ α. Equivalently the sequence of functors C(C, −) ⟶ C(B, −) ⟶ C(A, −) is exact. The dual notion is weak kernel. We say that a diagram A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} A_{2} ⟶ ⋯ in C is a weak cokernel sequence if f_{i+1} is a weak cokernel of f_{i}, ∀i ≥ 0.

Let I: I ⟶ C be a functor from a small category I. We use the notation I(i) = A_{i} and I(i ⟶ j) := α_{ij}: A_{i} ⟶ A_{j}. 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}, ∀i ∈ I. Hence a weak colimit of I, henceforth denoted by w.lim_{→}A_{i}, is defined as a genuine colimit except for the uniquenness property. In particular a weak colimit is not uniquely determined.

2.2. Abstract Homotopy Categories and Brown Representability

Fix an additive category C with coproducts and weak cokernels. Then it is easy to see that C has weak colimits. We are especially interested for weak colimits of towers of objects, where a tower in C is a diagram of the form A_{0} ⟶^{f_{0}} A_{1} ⟶^{f_{1}} A_{2} ⟶ · · ·. We call a weak colimit of a tower, a homotopy colimit and we denote it by holim_{⟶}A_{i}. A homotopy colimit of the tower above

can be computed as a weak cokernel of the canonical morphism ⊕_{i≥0}A_{i} ⟶^{1−f} ⊕_{i≥0}A_{i}, induced by the morphisms (1_{A_{i}}, −f_{i}): A_{i} ⟶ A_{i} ⊕ A_{i+1} ↪ ⊕_{i≥0}A_{i}. Hence we have a weak cokernel sequence

⨁_{i≥0} A_{i} ⟶^{1−f} ⨁_{i≥0} A_{i} ⟶ holim_{⟶}A_{i}.

Definition 2.1. Let X ⊆ C be a full subcategory of C. A weak colimit w.lim_{→}A_{i} in C is called X −minimal, if for any object X ∈ X the canonical morphism

lim_{→}C(X, A_{i}) ⟶ C(X, w.lim_{→}A_{i})

is an isomorphism. We say that X is a minimal subcategory if every tower A_{0} ⟶ A_{1} ⟶ A_{2} ⟶ ⋯ in C has an X −minimal homotopy colimit.

To proceed further we need the following concept borrowed from topology.

Definition 2.2. A full subcategory X ⊆ C is called a Whitehead subcategory, if X is skeletally small and a morphism f: A → B in C is invertible if C(X, f): C(X, A) → C(X, B) is invertible, ∀X ∈ X.

Example 2.3. (α) Let C be the homotopy category of pointed connected CWcomplexes and let X ⊆ C be the full subcategory consisting of finite complexes. Then by a well-known result of J.H.C. Whitehead, X is Whitehead subcategory of C. This is of course the origin of the terminology [26]. (β) Let C be a locally small additive category with filtered colimits and let X ⊆ C be a skeletally small full subcategory such that any object of C can be expressed as a filtered colimit of objects from X. Then X is a Whitehead subcategory of C [38].

If X is a Whitehead subcategory of C, then it is not difficult to see that any two X −minimal weak colimits are isomorphic (non-canonically). Moreover if the colimit exists in C, then it is isomorphic to the X −minimal weak colimit.

Definition 2.4. [13] An Abstract Homotopy Category is an additive category which admits coproducts, weak cokernels and a minimal Whitehead subcategory.

Note that in general a minimal Whitehead subcategory X ⊆ C is not uniquely determined. In many aspects the theory depends on the choice of X.

Definition 2.5. An additive functor F: C^{op} ⟶ Ab is called half-exact iff F sends a weak cokernel sequence A ⟶ B ⟶ C in C to an exact sequence F (C) ⟶ F (B) ⟶ F (A) in Ab. An additive functor F: C ⟶ Ab is called half-exact, if F sends a weak cokernel sequence as above to an exact sequence F (A) ⟶ F (B) ⟶ F (C) in Ab.

For any object A ∈ C, the representable functor C(−, A): C^{op} ⟶ Ab is halfexact and sends coproducts to products. The following fundamental Brown Representability Theorem for Abstract Homotopy Categories, asserts that the converse is true. This result is of central importance in what follows.

Theorem 2.6. [Brown] [18], [26], [13] Let C be an Abstract Homotopy Category and let F: C^{op} ⟶ Ab be an additive functor. Then the following are equivalent.

(α) F is representable.

(β) F is half-exact and sends coproducts to products.

Brown Representability has many consequences. First we note the following.

Corollary 2.7. [13] An Abstract Homotopy Category has arbitrary products.

We recall that an object X in an additive category C is called compact, if the functor C(X, −): C ⟶ Ab preserves all small coproducts. The full subcategory of C consisting of all compact objects is denoted by C^{b}. A full subcategory X of C is called a compact subcategory if X ⊆ C^{b}. A useful consequence of Brown Representability is the following Adjoint Functor Theorem.

Theorem 2.8. [13] Let C be an Abstract Homotopy Category and let F : C ⟶ D be an additive functor to an additive category D. Then the following are equivalent.

(α) F admits a right adjoint G: D ⟶ C.

(β) F preserves coproducts and weak cokernels.

In case G exists and C admits a compact mimimal Whitehead subcategory X, then: G preserves coproducts iff F preserves compact objects.

Recall that an object X in an additive category C is called finitely presented, if the representable functor C(X, −): C ⟶ Ab commutes with filtered colimits. The full subcategory of finitely presented objects of C is denoted by f.p(C). C is called locally finitely presented [17], if C has filtered colimits, f.p(C) is skeletally small and any object of C is a filtered colimit of finitely presented objects. If C is skeletally small, we denote by Mod(C) the category of additive functors C^{op} ⟶ Ab. There is a well–defined tensor product functor − ⊗_{C} − : Mod(C) × Mod(C^{op}) ⟶ Ab, which satisfies all the usual properties. A functor F in Mod(C) is called flat, if F ⊗_{C} −: Mod(C^{op}) ⟶ Ab is exact. The full subcategory of Mod(C) consisting of all flat functors is denoted by Flat(C). By [17], Flat(C) is locally finitely presented and any locally finitely presented additive category arises in this way.

2.3. Examples

Although the primitive example of an Abstract Homotopy Category in the non-additive setting is the homotopy category of CW-complexes, there is a host of interesting examples in the additive case.

(i) If Λ is a ring, then the category Mod(Λ) of right Λ−modules is an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory mod(Λ) of finitely presented modules, or the full subcategory P_{Λ} of finitely generated projective modules. More generally if C is skeletally small, then the functor category Mod(C) is an Abstract Homotopy Category.

(ii) Let F be locally finitely presented additive category with products. Then by [13], F is an Abstract Homotopy Category with minimal Whitehead subcategory f.p(F). In particular for any skeletally small additive category C with weak cokernels, Flat(C) is an Abstract Homotopy Category.

(iii) Let C be a compactly generated triangulated category [40]. Then C is an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory C^{b} of compact objects [13]. More generally let C be a right triangulated category with coproducts and suspension functor which is a right semi-equivalence in the sense of [2]. If C^{b} is skeletally small and for all compact objects X of C, C(X, A) = 0 implies A = 0, then C is an Abstract Homotopy category. We refer to Section 7 for non-triangulated examples.

(iv) Let Λ be a left coherent and right perfect ring. Then the stable category \underline{Mod}(Λ) of right Λ−modules modulo projectives, is an Abstract Homotopy Category with minimal Whitehead subcategory, the stable category \underline{mod}(Λ) induced by the finitely presented modules. Similarly the category P_{Λ} of projective modules is an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory of finitely generated projective modules [13].

(v) Let Λ be a right Morita ring, i.e. Λ is right Artinian and Mod(Λ) admits a finitely generated injective cogenerator. Then the stable category \overline{Mod}(Λ) of right Λ−modules modulo injectives, is an Abstract Homotopy Category with minimal Whitehead subcategory, the stable category \overline{mod}(Λ) induced by the finitely generated modules. Similarly the category I_{Λ} of injective right Λ−modules is an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory of finitely generated injective modules [13].

One can also add to this list the exactly definable categories and their definable subcategories in the sense of [32]. By (iv) or (v), it follows that the stable category modulo projectives of a quasi-Frobenius ring, is an Abstract Homotopy Category. Note that important examples of quasi-Frobenius rings are the group algebras of finite groups. The example (iii) of a compactly generated triangulated category, shows that we are supplied with many others Abstract Homotopy Categories:

(vi) The Stable Homotopy Category Ho(S) of Spectra [39], is an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory Ho(S)^{b} of finite spectra.

(vii) The unbounded derived category D(Λ) of all right Λ−modules over a ring Λ, is an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory D(Λ)^{b} of perfect complexes.

(viii) Let H be a commutative Hopf algebra over a field. Then the homotopy category of complexes of injective comodules is an Abstract Homotopy Category with minimal Whitehead subcategory the full subcategory induced by the injective resolutions of the simple comodules [27]. If H is finitedimensional, then the stable category of comodules modulo injectives, is an Abstract Homotopy Category with minimal Whitehead subcategory, the stable category induced by the finite dimensional comodules [27].

(ix) The unbounded derived category D(qcX) of the category of quasi-coherent sheaves over a quasi-compact separated scheme X [40].

(x) We recall [11] the a ring Λ is called right Gorenstein, if any projective right module has finite injective dimension and any injective right module has finite projective dimension. Important examples of right (and left) Gorenstein rings include all Noetherian rings of finite selfinjective dimension. The category CM(Λ) of Cohen-Macaulay modules over Λ, consists of all

modules A such the A = Ker(f_{0}), where ⋯ ⟶ P^{−1} ⟶ P^{0} ⟶^{f_{0}} P^{1} ⟶ ⋯

is an exact sequence of projective modules which remains exact if we apply to it the functor (−, P), for any projective module P. The dual definition of the category CoCM(Λ) of CoCohen-Macaulay modules using injectives, is left to the reader.

If Λ is a left coherent and right perfect right Gorenstein ring, then the projectively stable category \underline{CM}(Λ) is an Abstract Homotopy Category.

If Λ is a right Morita right Gorenstein ring, then the injectively stable category \overline{CoCM}(Λ) is an Abstract Homotopy Category. We refer to [13],

[11] for details.

Note that any category of the above list admits weak kernels, except possibly of the example (ii) of a locally finitely presented category with products. In what follows when we view a category of the above list as an Abstract Homotopy Category, we will assume tacitly that the choice of a minimal Whitehead subcategory is as described above. We call this choice, the natural choice. If C is an Abstract Homotopy Category with X ⊆ C as a minimal Whitehead subcategory, then without any loss of generality, we shall assume always that X is full additive and closed under direct summands. If C admits cokernels, we may (and will) replace weak cokernels by cokernels and homotopy or filtered weak colimits by genuine colimits.

3. Almost Split Morphisms

Throughout this section we fix an Abstract Homotopy Category C and let X be a fixed minimal Whitehead subcategory of C. In the sequel we follow ideas of M. Auslander [4], W. Crawley-Boevey [17] and H. Krause [37].

3.1. Dual Objects

We fix throughout a compact object X ∈ C. We denote by Λ_{X} := End_{C}(X) the endomorphism ring of X. For any additive functor F: C^{op} ⟶ Ab, define a left Λ_{X}−module structure on the abelian group F (X) as follows. If ρ ∈ Λ_{X} and x ∈ F (X), then ρ ⋆ x := F (ρ)(x). In particular for any object A ∈ C, C(X, A) is a left Λ_{X}−module. Let 𝔪 be a maximal left ideal of Λ_{X}, let S_{𝔪}(X) := Λ_{X}/𝔪 be the corresponding simple left Λ_{X}−module and let I_{𝔪} be its injective envelope. Consider the additive functor

H_{X} := [C(X, −), I_{𝔪}] : C^{op} ⟶ Ab, H_{X}(A) := Hom_{Λ_{X}} (C(X, A), I_{𝔪}).

Since X is compact, the functor H_{X} sends coproducts to products. We want this functor to be representable. For this purpose we introduce the following concept.

Definition 3.1. A compact object X is called homological if the functor C(X, −) : C ⟶ Ab is half-exact. The full subcategory of C consisting of all homological compact objects is denoted by H(C^{b}).

It follows that if X ∈ H(C^{b}), then the functor H_{X} is half-exact and converts coproducts to products. Hence by Brown Representability, there exists an object 𝔻_{𝔪}(X) ∈ C, unique up to isomorphism, and a natural isomorphism:

ω : [C(X, −), I_{𝔪}] ⟶^{≅} C(−, 𝔻_{𝔪}(X)).

From now on we assume that: X ∈ H(C^{b}).

Lemma 3.2. The object 𝔻_{𝔪}(X) has a local endomorphism ring.

Proof. Using the isomorphism ω, we have: End_{C}(𝔻_{𝔪}(X)) = [C(X, 𝔻_{𝔪}(X)), I_{𝔪}] ≅ [[C(X, X), I_{𝔪}], I_{𝔪}] = End_{Λ_{X}} (I_{𝔪}). Since I_{𝔪} is the injective envelope of the simple left Λ_{X}−module S_{𝔪}(X), I_{𝔪}, hence 𝔻_{𝔪}(X), has a local endomorphism ring.

Definition 3.3. The object 𝔻_{𝔪}(X) is called the 𝔪−dual object of X with respect to the maximal left ideal 𝔪 of Λ_{X}.

Plainly the class of m−dual objects of C corresponds bijectively with the disjoint union ⨆{Max(Λ_{X}) | X ∈ Iso(H(C^{b}))}, where Max(Λ_{X}) is the set of maximal left ideals of Λ_{X}. Here Iso(Y) denotes the collection of isoclasses of objects of Y ⊆ C.

Generalizing this construction, let ρ: Γ ⟶ Λ_{X} be a ring morphism and let I be an injective left Γ−module. As above for any additive functor F: C^{op} ⟶ Ab, the abelian group F (X) admits, via the ring morphism ρ, a left Γ−module structure and Brown Representability implies that there exists an object 𝔻_{I}(X) ∈ C, unique up to isomorphism, equipped with a natural isomorphism:

ω : Hom_{Γ}[C(X, −), I] ⟶^{≅} C(−, 𝔻_{I}(X)).

We call 𝔻_{I}(X) the I−dual object of X with respect to the injective Γ−mo– dule I. The notation 𝔻_{I}(X) ∈ C means that a homological compact object X in C is fixed and I is an injective left Γ−module, where ρ: Γ ⟶ Λ_{X} is a fixed ring morphism. The isomorphism ω is a powerful tool for producing new interesting objects.

Example 3.4. (i) Choose Γ = ℤ and I = ℚ/ℤ; in this case we use the notation 𝔻_{ℚ/ℤ}(X). For example if C = Mod(Λ), then for X = Λ and any right Λ−module A, the above isomorphism reduces to: (A_{Λ}, ℚ/ℤ) ≅ _{Λ}(A, 𝔻_{ℚ/ℤ}(Λ)).

(ii) Choose Γ = ℤ, let p be a prime integer and let I be the injective envelope of the simple ℤ−module ℤ/(p), i.e. I is the Prüfer group ℤ_{p^{∞}}. Then the above isomorphism produces the p−dual object 𝔻_{p}(X), with endomorphism ring, the commutative ring End_{ℤ}(ℤ_{p^{∞}}) of p−adic integers. Choosing I = ℚ, we have the rational dual object 𝔻_{ℚ}(X) of X, with endomorphism ring ℚ.

(iii) If Ho(S) is the stable homotopy category of spectra and X is a finite spectrum, then 𝔻_{ℚ/ℤ}(X) is the Brown-Comenetz dual of the Spanier-Whitehead dual of X [19]. The p−dual object 𝔻_{p}(X) is a p−local spectrum [39], for any prime p. If S^{n} is the n^{th}−suspension of the sphere spectrum S^{0}, then let π_{n}(A) := Ho(S)(S^{n}, A) be the stable homotopy groups of A. Then: (π_{n}(A), ℤ_{p^{∞}}) ≅ Ho(S)(A, 𝔻_{p}(S^{n})), (π_{n}(A), ℚ) ≅ Ho(S)(A, 𝔻_{ℚ}(S^{n})) and π_{n}(𝔻_{ℚ/ℤ}(S^{n})) = ℚ/ℤ, ∀n ≥ 0.

We recall that an additive functor F: C^{op} ⟶ Ab is called finitely presented if there exists an exact sequence of functors C(−, A) ⟶ C(−, B) ⟶ F ⟶ 0. Let mod(C) be the category of finitely presented contravariant additive functors. It is well–known that mod(C) is abelian iff C admits weak kernels, see [6].

Lemma 3.5. If 𝔻_{I}(X) is an I−dual object of X, then ∀F ∈ mod(C):

Hom_{Γ}(F (X), I) ⟶^{≅} [F, C(−, 𝔻_{I}(X))].

If C has weak kernels, then C(−, 𝔻_{I}(X)) is an injective object in mod(C).

Proof. Choose a presentation C(−, A) ⟶ C(−, B) ⟶ F ⟶ 0 of F. Then C(X, A) ⟶ C(X, B) ⟶ F (X) ⟶ 0 is exact in Mod(Γ^{op}). Hence the sequence 0 ⟶ [F (X), I] ⟶ [C(X, B), I] ⟶ [C(X, A), I] is exact. Using the isomorphism ω, it follows that [F (X), I] is isomorphic to the kernel of C(B, 𝔻_{I}(X)) ⟶ C(A, 𝔻_{I}(X)) which is isomorphic to [F, C(−, 𝔻_{I}(X))]. If C has weak kernels, then mod(C) is abelian and the above isomorphisms show that C(−, 𝔻_{I}(X)) is an injective object in mod(C).

We denote by S: C ⟶ Mod(X), the restricted Yoneda functor defined by S(A) = C(−, A)|_{X}. In particular S|_{X} is the Yoneda embedding X ↪ Mod(X).

Lemma 3.6. For any F ∈ Mod(X) there exists an isomorphism:

Hom_{Γ}(F (X), I) ⟶^{≅} [F, S(𝔻_{I}(X))].

In particular S(𝔻_{I}(X)) is an injective object in Mod(X).

Proof. First let F be finitely presented, and choose a presentation X (−, Y_{1}) ⟶ X (−, Y_{0}) ⟶ F ⟶ 0. By Yoneda’s Lemma the exact sequence

0 ⟶ [F, S(𝔻_{I}(X))] ⟶ [X (−, Y_{0}), S(𝔻_{I}(X))] ⟶ [X (−, Y_{1}), S(𝔻_{I}(X))]

is isomorphic to the sequence

0 ⟶ [F, S(𝔻_{I}(X))] ⟶ C(Y_{0}, 𝔻_{I}(X)) ⟶ C(Y_{1}, 𝔻_{I}(X)).

As in the proof of Lemma 3.5, the kernel of the last morphism is isomorphic via ω to Hom_{Γ}(F (X), I). Now let F ∈ Mod(X) be arbitrary. It is well known that F is a filtered colimit lim_{→}F_{i} of finitely presented functors. Then

Hom_{Γ}(F (X), I) = Hom_{Γ}(lim_{→}F_{i}(X), I) = lim_{←}Hom_{Γ}(F_{i}(X), I) ≅
≅ lim_{←}[F_{i}, S(𝔻_{I}(X))] ≅ [lim_{→}F_{i}, S(𝔻_{I}(X))] = [F, S(𝔻_{I}(X))].

The injectivity of S(𝔻_{I}(X)) in Mod(X) follows trivially from these isomorphisms.

Corollary 3.7. ∀A ∈ C, the functor S induces a canonical isomorphism

S_{A,𝔻_{I}(X)} : C(A, 𝔻_{I}(X)) ⟶^{≅} [S(A), S(𝔻_{I}(X))], f ↦ S(f).

Proof. Choosing F = S(A) in Lemma 3.6, we infer that we have an isomorphism Hom_{Γ}(C(X, A), I) ≅ [S(A), S(𝔻_{I}(X))]. Plainly the composite isomor-

phism C(A, 𝔻_{I}(X)) ⟶^{ω_{X}^{−1}} Hom_{Γ}(C(X, A), I) ⟶ [S(A), S(𝔻_{I}(X))] is equal to S_{A,𝔻_{I}(X)}.

If Y is a class of objects of C, then Add(Y), resp. add(Y), resp. Prod(Y), denotes the full subcategory of C consisting of all direct summands of arbitrary coproducts, resp. finite coproducts, resp. products, of objects of Y.

Corollary 3.8. If X ⊆ H(C^{b}), then 𝔻_{ℚ/ℤ}(X) = {𝔻_{ℚ/ℤ}(X); X ∈ X } is a “coWhitehead” subcategory of C, i.e. 𝔻_{ℚ/ℤ}(X) is skeletally small and f: A ⟶ B is invertible in C iff C(f, 𝔻_{ℚ/ℤ}(X)) is invertible, ∀X ∈ X. Moreover E := ∏_{X∈Iso(X)} 𝔻_{ℚ/ℤ}(X) is a cogenerator of C and S(E) is an injective cogenerator of Mod(X).

For instance if C = Mod(Λ) and X = P_{Λ}, then we recover the well–known fact that 𝔻_{ℚ/ℤ}(Λ) = Hom_{ℤ}(Λ, ℚ/ℤ) is an injective cogenerator for Mod(Λ).

Assume from now on that the homological compact object X is in X and consider a maximal left ideal 𝔪 of Λ_{X}. It is not difficult to see that defining

X (−, X)_{𝔪}(Y) := {ϕ ∈ X (Y, X) | ∀X ⟶^{θ} Y : θ ∘ ϕ ∈ 𝔪}

we obtain a right ideal in X, i.e. an additive subfunctor X (−, X)_{𝔪} ↪ X (−, X). The corresponding quotient functor is denoted by S_{X,𝔪} := X (−, X)/ X (−, X)_{𝔪}. By [4] the functor S_{X,𝔪} is simple and any simple functor in Mod(X) arises in this way.

Consider the evaluation ω_{X}: [C(X, X), I_{𝔪}] = Hom_{Λ_{X}} (Λ_{X}, I_{𝔪}) ⟶^{≅} C(X, 𝔻_{𝔪}(X)) of the isomorphism ω at X. Let π : Λ_{X} ↠ Λ_{X}/𝔪 be the canonical projection and let μ : Λ_{X}/𝔪 ↣ I_{𝔪} be the canonical inclusion. We denote the image of the composition π ∘ μ under ω_{X} by h_{X}. Hence h_{X}: X ⟶ 𝔻_{𝔪}(X) is the morphism in C defined by h_{X} := ω_{X}(π ∘ μ). Observe that by construction h_{X} ≠ 0.

Theorem 3.9. (i) The functor S(𝔻_{𝔪}(X)) ∈ Mod(X) is the injective envelope of the simple functor S_{X,𝔪}.

(ii) If X ⊆ H(C^{b}) and E ∈ Mod(X) is the injective envelope of a simple functor, then there exists X ∈ X and a maximal left ideal 𝔪 of Λ_{X} such that E = S(𝔻_{𝔪}(X)).

Proof. (i) By Corollary 3.7, the objects 𝔻_{𝔪}(X), S(𝔻_{𝔪}(X)) have isomorphic endomorphism rings. Hence by Lemma 3.2, the ring End(S(𝔻_{𝔪}(X))) is local.

Consider the morphism S(h_{X}): S(X) ⟶ S(𝔻_{𝔪}(X)) in Mod(X) and let κ: F ⟶ S(X) be its kernel and G its image. We claim that G is the simple functor S_{X,𝔪}. Indeed since S(X) = X (−, X), it suffices to show that F = X (−, X)_{𝔪}. Let α: A ⟶ X be any morphism with A ∈ X. Then ω induces a commutative diagram:

\begin{matrix}H_{X}(X) & ⟶^{ω_{X}} & C(X, 𝔻_{𝔪}(X)) \\ ^{H_{X}(α)}↓ & & _{α^{∗}}↓ & (†) \\ H_{X}(A) & ⟶^{ω_{A}} & C(A, 𝔻_{𝔪}(X)) \end{matrix}

where α^{∗} = C(α, 𝔻_{𝔪}(X)). Then α^{∗}ω_{X}(π ∘ μ) = α^{∗}(h_{X}) = α ∘ h_{X} = C(A, h_{X})(α). On the other hand H_{X}(α)(π ∘ μ): C(X, A) ⟶ I_{𝔪} is the map defined by H_{X}(α)(π ∘ μ)(θ) = π ∘ μ(θ ∘ α). Since ω_{A} is invertible, by the commutativity of the above diagram, we have: α ∈ F (A) iff α ∘ h_{X} = 0 iff ∀θ: X ⟶ A, the element π ∘ μ(θ ∘ α) = 0. Obviously this last condition holds iff the composition θ ∘ α ∈ 𝔪. Hence F = X (−, X)_{𝔪} and G = ImS(h_{X}) is the simple functor S_{X,𝔪}. Since End_{C}(S(𝔻_{𝔪}(X))) is local, the inclusion S_{X,𝔪} ↪ S(𝔻_{𝔪}(X)) is an injective envelope.

(ii) Assume that E is the injective envelope of a simple functor S in Mod(X). Then S is of the form S_{X,𝔪} for an object X ∈ X and a maximal left ideal 𝔪 of Λ_{X} = End_{C}(X). Since X ⊆ H(C^{b}), we can perform the construction of the 𝔪−dual object 𝔻_{𝔪}(X) of X with respect to 𝔪 and then obviously E ≅ S(𝔻_{𝔪}(X)).

3.2. Almost Split Morphisms

We recall that a morphism f: B ⟶ C in a given additive category C is called right almost split, if f is not a split epimorphism and any morphism g: A ⟶ C which is not a split epimorphism factors through f. The morphism f is called right minimal, if any morphism α: B ⟶ B such that α ∘ f = f, is an automorphism. A right minimal right almost split morphism is called a minimal right almost split morphism. The dual notions are left almost split morphism, left minimal morphism and minimal left almost split morphism. We refer to [4] for details and more information concerning these morphisms.

Our main result in this section is the following.

Theorem 3.10. Let 0 ≠ X ∈ H(C^{b}) and let Λ_{X} = End_{C}(X) be its endomorphism ring. Then there exists an object 𝔻_{𝔪}(X) in C with local endomorphism ring and a non-zero morphism h_{X}: X ⟶ 𝔻_{𝔪}(X) satisfying the following properties:

(i) Assume that the endomorphism ring Λ_{X} is local.

(α) If α: A ⟶ X is a non-split epimorphism, then α ∘ h_{X} = 0. Any weak kernel f_{X}: A_{X} ⟶ X of h_{X} in C, is a right almost split morphism.

(β) If C has weak kernels, then the image S_{X} of the morphism C(−, h_{X}) : C(−, X) ⟶ C(−, 𝔻_{𝔪}(X)) in the abelian category mod(C), is a simple functor and the projection C(−, X) ⟶ S_{X} is a projective cover.

(ii) (α) Assume that X ∈ X or Λ_{X} is local and C admits weak kernels. If β : 𝔻_{𝔪}(X) ⟶ B is a non-split monomorphism, then h_{X} ∘ β = 0. Any weak cokernel g^{X}: 𝔻_{𝔪}(X) ⟶ B^{X} of h_{X} in C, is a left almost split morphism.

(β) If C has weak kernels, then the inclusion S_{X} ↪ C(−, 𝔻_{𝔪}(X)) is an injective envelope in mod(C).

Proof. Let 𝔪 be a maximal left ideal of Λ_{X} and consider the non–zero morphism h_{X}: X ⟶ 𝔻_{𝔪}(X) constructed in subsection 3.1. If Λ_{X} is local, it follows that 𝔪 is the unique maximal left ideal of Λ_{X}, i.e. 𝔪 = J ac(Λ_{X}).

(i)(α) Let α: A ⟶ X be a morphism in C which is not a split epimorphism. Consider the commutative diagram (†) in the proof of Theorem 3.9 above, where α^{∗} = C(α, 𝔻_{𝔪}(X)). Then we have α^{∗}ω_{X}(π ∘ μ) = α^{∗}(h_{X}) = α ∘ h_{X}. On the other hand H_{X}(α)(π ∘μ): C(X, A) ⟶ I_{𝔪} is the map defined by H_{X}(α)(π ∘ μ)(β) = π ∘ μ(β ∘ α). The morphism β ∘ α is an element of the local ring Λ_{X}. Since α is non-split epic, it follows that β ∘ α is a non-invertible element of Λ_{X}. Hence β ∘ α ∈ J ac(Λ_{X}), or equivalently π(β ∘ α) = 0. Hence H_{X}(α)(π ∘ μ)(β) = 0. Since β was arbitrary, it follows that H_{X}(α)(π ∘ μ) = 0. By the commutativity of the diagram (†), it follows that α ∘ h_{X} = 0.

If f_{X}: A_{X} ⟶ X is a weak kernel of h_{X} in C, then f_{X} is non-split epic, since h_{X} ≠ 0. Let α: A ⟶ X be a non-split epic. Since α ∘ h_{X} = 0, it follows that α factors through f_{X}. Hence f_{X} is a right almost split morphism.

(i)(β) Let κ: G ↪ C(−, X) be the kernel of C(−, h_{X}): C(−, X) ⟶ C(−, 𝔻_{m}(X)) in mod(C). We claim that G is a proper maximal subfunctor of C(−, X). Indeed let i: F ↪ C(−, X) be a proper subfunctor and let C(−, D) ⟶ C(−, C) ⟶^{ϵ} F ⟶ 0 be a presentation of F. Then by Yoneda the morphism ϵ ∘ i: C(−, C) ⟶ C(−, X) is of the form C(−, α), where α: C ⟶ X. Since F is a proper subfunctor of C(−, X), it follows easily that α is non-split epic. Hence by (i)(α), we have: α ∘ h_{X} = 0. Then C(−, α) ∘ C(−, h_{X}) = 0 and this implies that i ∘ C(−, h_{X}) = 0. Then i factors through κ and this means that G contains F as a subfunctor. Hence G is a maximal proper subfunctor of C(−, X). Consequently S_{X} := ImC(−, h_{X}) is a simple functor in mod(C) and then obviously the projection C(−, X) ⟶ S_{X} is a projective cover.

(ii)(α) If X ∈ X, then by Theorem 3.9, S(𝔻_{𝔪}(X)) is the injective envelope of the simple image S_{X,𝔪} of the morphism S(h_{X}). Let β : 𝔻_{𝔪}(X) ⟶ B be a morphism in C which is not a split monomorphism. Then S(β) is not a monomorphism. Indeed otherwise S(β) will be a split monomorphism, since S(𝔻_{𝔪}(X)) is injective. Using Corollary 3.7, it is easy to see that this implies that β is a split monomorphism and this is not true. Hence S(β) is not a monomorphism. Since S_{X,𝔪} ↪ S(𝔻_{𝔪}(X)) is an injective envelope, this implies that the composition S_{X,𝔪} ↪ S(𝔻_{𝔪}(X)) ⟶ S(B) is zero. Then S(h_{X}) ∘ S(β) = 0. Since X ∈ X, it follows that h_{X} ∘β = 0. If Λ_{X} is local and C admits weak kernels, then the assertion follows directly from the fact that C(−, h_{X}) has simple image and C(−, 𝔻_{𝔪}(X)) is injective in mod(C).

Let g^{X}: 𝔻_{𝔪}(X) ⟶ B^{X} be a weak cokernel of h_{X} in C. Then g^{X} is not a split monomorphism, since h_{X} ≠ 0. If β: 𝔻_{𝔪}(X) ⟶ B is not a split monomorphism, then by the above arguments we have that h_{X} ∘ β = 0, hence β factors through g^{X}. This shows that g^{X} is a left almost split morphism.

(ii)(β) Since 𝔻_{𝔪}(X) has local endomorphism ring, it follows by Yoneda that the functor C(−, 𝔻_{𝔪}(X)) is an indecomposable injective object in mod(C). Hence the inclusion μ: S_{X} ↪ C(−, 𝔻_{𝔪}(X)) is an injective envelope.

Corollary 3.11. Let C be an Abstract Homotopy Category with weak kernels. Then for any object X ∈ H(C^{b}) with a local endomorphism ring, there exists a right almost split morphism f_{X}: A_{X} ⟶ X and a left almost split morphism g^{X}: 𝔻_{𝔪}(X) ⟶ B^{X}.

Corollary 3.12. If C admits kernels and cokernels, then for any homological compact object X with a local endomorphism ring, there exists a minimal right almost split morphism A_{X} ⟶ X and a minimal left almost split morphism 𝔻_{𝔪}(X) ⟶ B^{X}.

Proof. Choose f_{X} to be the kernel and g^{X} to be the cokernel of h_{X}.

The next result relates the existence of almost split morphisms in C with properties of projective or injective functors in Mod(X).

Theorem 3.13. Assume that C has weak kernels and X ⊆ H(C^{b}).

(i) If E is an object of C, then for the following statements:

(α) S(E) is the injective envelope of a simple functor in Mod(X).

(β) E ≅ 𝔻_{𝔪}(X), where X ∈ X and 𝔪 is a maximal left ideal of Λ_{X}.

(γ) The functor S(E) is injective in Mod(X) and there exists a left almost split morphism E ⟶ B in C.

we have (α) ⇔ (β) ⇒ (γ). If in addition any injective object of Mod(X) is in the image of S and if for injective objects S(I), S(J), the canonical morphism C(I, J) ⟶ (S(I), S(J)) is surjective, then we have also (γ) ⇒ (α).

(ii) If P is an object in C, then the following are equivalent:

(α) S(P) is the projective cover of a simple functor in Mod(X).

(β) P ∈ X and P has a local endomorphism ring.

(γ) The functor S(P) is projective in Mod(X) and there exists a right almost split morphism A ⟶ P in C.

Proof. (i) By Theorem 3.9 we have (α) ⇔ (β) and by Theorem 3.10(ii)(α) we have (β) ⇒ (γ). Assume now that the additonal hypotheses are true and let g: E ⟶ B be a left almost split morphism in C, where S(E) is an injective functor. Consider the morphism S(g): S(E) ⟶ S(B) and let μ: S ⟶ S(E) be its kernel. It suffices to show that S is a simple functor. Let α: S ⟶ G be any morphism in Mod(X) and let ν: G ⟶ E(G) be an injective envelope. Since E(G) is injective, there exists a morphism σ: S(E) ⟶ E(G) such that α∘ν = μ∘σ. By hypothesis, there exists an object I ∈ C such that S(I) = E(G) and a morphism σ′: E ⟶ I such that S(σ′) = σ. If σ is a split monomorphism, then obviously α is a monomorphism. If σ is not a split monomorphism, then obviously the same is true for σ′. Since g is left almost split, there exists τ: B ⟶ I such that σ′ = g ∘ τ. Then σ = S(g) ∘ S(τ) and this implies that μ∘σ = α∘ν = 0. Then α = 0, since ν is a monomorphism. Since any morphism α: S ⟶ G is zero or a monomorphism, S is a simple functor.

(ii) We use tacitly the fact that since X is compact, the functor S induces an equivalence between Add(X) and the category of projective objects in Mod(X) [12]. (α) ⇒ (β) Since simple functors are finitely generated, it follows that S(P) is finitely generated, i.e. P ∈ X. Then it is easy to see that End_{C}(P) is local. (β) ⇒ (γ) Follows by Corollary 3.11. (γ) ⇒ (α) Since P is the target of a right almost split morphism, by [10], End_{C}(P) is local. Since S(P) is projective, it is easy to see that S(P) is finitely generated, i.e. P ∈ X. Then the projection X (−, P) ⟶ S_{P,𝔪} is a projective cover, where 𝔪 is the unique maximal ideal of End_{C}(P).

Our last result in this section indicates a useful factorization property.

Corollary 3.14. Let X ∈ H(C^{b}), let ρ: Γ ⟶ End_{C}(X) be a ring morphism and let I be an injective cogenerator of Mod(Γ^{op}). If A ⟶^{f} B ⟶^{g} C ⟶^{h} D is a weak cokernel sequence in C, then: any morphism X ⟶ B factors through f iff any morphism C ⟶ 𝔻_{I}(X) factors through h.

Proof. Since X ∈ H(C^{b}), the sequence C(X, A) ⟶^{f_{∗}} C(X, B) ⟶^{g_{∗}} C(X, C) ⟶^{h_{∗}} C(X, D) is exact in Mod(Γ^{op}), where f_{∗} = C(X, f) and so on. Then any morphism X ⟶ B factors through f iff g_{∗} = 0. Since I is injective, using the isomorphism ω it follows that the sequence C(D, 𝔻_{I}(X)) ⟶^{h^{∗}} C(C, 𝔻_{I}(X)) ⟶^{g^{∗}}

C(B, 𝔻_{I}(X)) ⟶^{f^{∗}} C(A, 𝔻_{I}(X)) is exact in Mod(Γ^{op}), where h^{∗} := C(h, 𝔻_{I}(X)), and so on. Then any morphism C ⟶ 𝔻_{I}(X) factors through h iff g^{∗} = 0. But g^{∗} is isomorphic to the morphism (g_{∗}, I). Since I is a cogenerator, we have g_{∗} = 0 iff g^{∗} = 0.

4. Purity in Abstract Homotopy Categories

Throughout this section we fix an Abstract Homotopy Category C with a fixed minimal Whitehead subcategory X ⊆ C. We assume always that X is compact, i.e. X ⊆ C^{b}, and X has weak cokernels and the inclusion X ↪ C preserves them.

4.1. Purity

A sequence in C is a diagram (E): A ⟶^{g} B ⟶^{f} C such that g ∘ f = 0. A sequence (E) is called pure if for any X ∈ X, the induced sequence 0 ⟶ C(X, A) ⟶ C(X, B) ⟶ C(X, C) ⟶ 0 is exact in Ab. The class of pure sequences is denoted by E. The terminology “pure” will be justified in Section 7.

We call an object P ∈ C pure-projective, if for any pure-sequence A ⟶^{g} B ⟶^{f} C, the induced sequence 0 ⟶ C(P, A) ⟶ C(P, B) ⟶ C(P, C) ⟶ 0 is exact in Ab. The full subcategory of C consisting of all pure-projectives is denoted by P(E). We say that C has enough pure-projectives, if ∀C ∈ C there exists a pure-sequence A ⟶ P ⟶ C, with P ∈ P(E). Such a sequence is called a pure-projective presentation of C. A pure-projective cover of C, is a right minimal morphism f included in a pure-projective presentation A ⟶ P ⟶^{f} C in C. We say that C has pure-projective covers, if any object of C admits a pure-projective cover. We leave to the reader to formulate the dual notions concerning pure-injectivity. The full subcategory of pure-injective objects is denoted by I(E). Finally we call an Abstract Homotopy Category C pure-semisimple, if any pure-sequence A ⟶ B ⟶ C in C splits, i.e. it is

isomorphic to the sequence A ⟶^{(1_{A},0)} A ⊕ C ⟶^{^{t}(0,1_{C})} C.

4.2. Representation Categories

Since by definition X is skeletally small, we can consider the category Mod(X), resp. Mod(X^{op}), of contravariant, resp. covariant, additive functors from X to Ab. We consider also the category mod(X^{op}) of finitely presented covariant functors and we set B(X) := mod(X^{op})^{op}.

Since X has weak cokernels and split idempotents, it follows that B(X) is a skeletally small abelian category with enough injectives and any injective is of the form 𝕐(X), where 𝕐: X ↪ B(X) is the (covariant) Yoneda embedding. Consider the category Flat(B(X)) of flat functors B(X)^{op} ⟶ Ab and set:

L(C) := Mod(X) and D(C) := Flat(B(X)).

In other words D(C) is the conjugate category { \atop Mod(X^{op})} of Mod(X^{op}) in the sense of J.E. Roos [43]. We identify B(X) with the full subcategory of finitely presented objects of D(C), via the embedding M ↦ (−, M), and usually we view the objects of B(X) as finitely presented functors X ⟶ Ab.

We recall that a locally finitely presented Grothendieck category G is called locally coherent, if the full subcategory f.p(G) of finitely presented objects of G is abelian. It follows that D(C) is locally coherent and can be identified with the category of left exact functors B(X)^{op} ⟶ Ab.

As in section 3, let S: C ⟶ L(C), S(A) = C(−, A)|_{X}, be the restricted Yoneda functor. Let i : Flat(B(X)) ↪ Mod(B(X)) be the inclusion functor. By a well–known result of Gabriel [23], i admits an exact left adjoint Θ: Mod(B(X)) ⟶ Flat(B(X)). On the other hand the Yoneda embedding 𝕐: X ↪ B(X) induces a fully faithful right exact functor 𝕐^{!} : Mod(X) ↪ Mod(B(X)) which is a left adjoint (given by the Kan construction) of the induced restriction functor 𝕐_{∗}: Mod(B(X)) ⟶ Mod(X), which is defined by 𝕐_{∗}(F) = F 𝕐. It is well–known that 𝕐^{!} is defined as follows: 𝕐^{!}(F)(M) = F ⊗_{X} B(X)[M, 𝕐(−)]. In particular 𝕐^{!} preserves flatness, so it induces a fully faithful functor 𝕐^{!} : Flat(X) ⟶ D(C). Then we have adjoint pairs (𝕐^{!}, 𝕐_{∗}): L(C) ⟶ Mod(B(X)) and (Θ, i): Mod(B(X)) ⟶ D(C). We denote by T: C ⟶ D(C) the composite functor

T := Θ𝕐^{!}S : C ⟶^{S} L(C) ⟶^{𝕐^{!}} Mod(B(X)) ⟶^{Θ} D(C).

We recall that an object M in a locally coherent category G is called FPinjective, if Ext_{G}^{1}(F, M) = 0, for any finitely presented object F ∈ G. The full subcategory of FP-injective objects of G is denoted by FPInjG. If A is a skeletally small abelian category, then Ex(A^{op}, Ab) denotes the category of exact functors A^{op} ⟶ Ab.

The following result is basic in what follows.

Lemma 4.1. The image of S lies in the full subcategory FlatL(C) of flat functors of L(C) and the functor 𝕐^{!} induces an equivalence

FlatL(C) ⟶^{≈} Ex(B(X)^{op}, Ab).

The functor T is isomorphic to 𝕐^{!}S and there is an identification

Ex(B(X)^{op}, Ab) = FPInjD(C).

In particular T induces a functor T: C ⟶ FPInjD(C) ↪ D(C).

Proof. Since X has weak cokernels, it is easy to see [14] that a functor F ∈ Mod(X) is flat iff for any weak cokernel sequence X ⟶ Y ⟶ Z in X, the sequence F (Z) ⟶ F (Y) ⟶ F (X) is exact, i.e. F is half-exact. Then ∀A ∈ C, the functor S(A) = C(−, A)|_{X} is flat since it is half-exact and X is closed under weak cokernels in C. For a proof that the functor 𝕐^{!} : FlatL(C) ⟶ Ex(B(X)^{op}, Ab) is an equivalence we refer to [14]. Since ImS consists of flat functors and 𝕐^{!} preserves flatness, from the definition of T it follows directly that T is isomorphic to 𝕐^{!}S. The last assertion is well-known. We include a proof for completeness. If E: B(X)^{op} ⟶ Ab is exact, then there exists a flat functor F : X^{op} ⟶ Ab such that E = 𝕐^{!}(F). Write F = lim_{→}S(X_{i}) as a filtered colimit of representable functors over X. Since 𝕐^{!}

commutes with filtered colimits, E = lim_{→}𝕐^{!}S(X_{i}). By the definition of 𝕐^{!}

it follows that 𝕐^{!}S(X_{i}) = (−, 𝕐(X_{i})), which obviously is an FP-injective object in D(C). Then E is FP-injective, since by [46] a filtered colimit of FP-injective objects in a locally coherent category is again FP-injective. Conversely if E is FP-injective, let 0 ⟶ M_{1} ⟶ M_{2} ⟶ M_{3} ⟶ 0 be an exact sequence in B(X). By the construction of cokernels in D(C), see [23], it follows that 0 ⟶ (−, M_{1}) ⟶ (−, M_{2}) ⟶ (−, M_{3}) ⟶ 0 is exact in D(C). Since E is FP-injective and the (−, M_{i}) are finitely presented, the sequence 0 ⟶ ((−, M_{3}), E) ⟶ ((−, M_{2}), E) ⟶ ((−, M_{1}), E) ⟶ 0 is exact, so its isomorphic copy 0 ⟶ E(M_{3}) ⟶ E(M_{2}) ⟶ E(M_{1}) ⟶ 0 is exact, i.e. E is exact.

The following is a direct consequence of the above Lemma.

Corollary 4.2. If 0 ⟶ F_{1} ⟶ F_{2} ⟶ F_{3} ⟶ 0 is an exact sequence of flat functors in L(C), then the sequence 0 ⟶ 𝕐^{!}(F_{1}) ⟶ 𝕐^{!}(F_{2}) ⟶ 𝕐^{!}(F_{3}) ⟶ 0 is exact in D(C).

From now on we identify the functors T, 𝕐^{!}S : C ⟶ D(C).

Remark 4.3. ∀A ∈ C, ∀X ∈ X, ∀M ∈ B(X):

T(A)(𝕐(X)) = S(A) ⊗_{X} [𝕐(X), 𝕐(−)] = S(A) ⊗_{X} (X, −) ≅ C(X, A).
T(X)(M) = S(X) ⊗_{X} [M, 𝕐(−)] = [M, 𝕐(X)] = M(X).

Remark 4.4. It is easy to see that S preserves products. Using that X consists of compact objects it follows that S preserves coproducts. Since 𝕐^{!} is a left adjoint, it follows that T preserves coproducts. Since a product of FPinjective objects is again FP-injective, it follows directly by Lemma 4.1 that T preserves products.

We call L(C), D(C), the representation categories of C, with respect to the fixed Whitehead subcategory X. The categories L(C), D(C) are Grothendi– eck Abstract Homotopy Categories and together with the connecting representation functors S: C ⟶ L(C), T: C ⟶ D(C), are very useful for the study of purity in C.

Our aim in this Section is, using the Representation Categories, to prove that an Abstract Homotopy Category C satisfying some additional conditions, has enough pure-projectives and pure-injective envelopes. The usefullness of the Representation Categories in the study of purity, emerges from the following result.

Lemma 4.5. Let (E): A ⟶ B ⟶ C be a sequence in C. Then (E) is pure iff the sequence 0 ⟶ S(A) ⟶ S(B) ⟶ S(C) ⟶ 0 is short exact in L(C) iff the sequence 0 ⟶ T(A) ⟶ T(B) ⟶ T(C) ⟶ 0 is short exact in D(C).

Proof. The first assertion is clear. If (E) is pure, then 0 ⟶ S(A) ⟶ S(B) ⟶ S(C) ⟶ 0 is an exact sequence of flat functors in L(C). Since T = 𝕐^{!}S, by Corollary 4.2, the sequence 0 ⟶ T(A) ⟶ T(B) ⟶ T(C) ⟶ 0 is exact in D(C). Conversely if the last sequence is exact, then since its terms are FP-injective objects and ∀X ∈ X, the object T(X) = (−, 𝕐(X)) is finitely presented, the sequence 0 ⟶ (T(X), T(A)) ⟶ (T(X), T(B)) ⟶ (T(X), T(C)) ⟶ 0 is exact. Obviously this sequence is isomorphic to the sequence 0 ⟶ (S(X), S(A)) ⟶ (S(X), S(B)) ⟶ (S(X), S(C)) ⟶ 0, or equivalently to the sequence 0 ⟶ C(X, A) ⟶ C(X, B) ⟶ C(X, C) ⟶ 0. Hence the sequence (E) is pure.

To proceed further in an efficient way we need to impose an axiom. First it is convenient to introduce some terminology.

Definition 4.6. A morphism g: B ⟶ C in C is called a pure-epic, if S(f) is an epimorphism in L(C). A morphism f: A ⟶ B in C is called a pure-monic, if T(f) is a monomorphism in D(C).

Axiom 1. Any pure-epic B ⟶ C, resp. pure-monic A ⟶ B, can be included in a pure-sequence A ⟶ B ⟶ C in C

We assume throughout that Axiom 4.7 holds in C.

4.3. Pure-Projectives

The following result shows the existence of enough pure-projective objects in C and describes their structure.

Proposition 4.7. C has enough pure-projectives and P(E) = Add(X). Moreover ∀P ∈ P(E), ∀A ∈ C, the canonical morphism C(P, A) ⟶ [S(P), S(A)] is an isomorphism and the functor S: C ⟶ L(C) induces an equivalence

S: P(E) ⟶^{≈} ProjL(C).

Proof. Plainly Add(X) ⊆ P(E). Let C be an object in C. Then the set of morphisms I_{C} := {X ⟶ C | X ∈ Iso(X)}, induces a morphism g_{C}: ⊕_{a∈I_{C}} X_{a} ⟶ C. By construction the morphism C(X, g_{C}) is an epimorphism ∀X ∈ X, equivalently S(g_{C}) is an epimorphism and then g_{C} is pure-epic. By Axiom 4.7, there exists a pure-sequence A ⟶ ⊕_{a∈I_{C}} X_{a} ⟶^{g_{C}} C in C. Since ⊕_{a∈I_{C}} X_{a} ∈ P(E), it follows that C has enough pure-projectives. If C ∈ P(E), then the above pure-sequence splits, hence C ∈ Add(X) as a direct summand of ⊕_{a∈I_{C}} X_{a}. We conclude that P(E) = Add(X). Now let A ∈ C and X ∈ X. Then plainly the canonical morphism S_{X,A}: C(X, A) ⟶ [S(X), S(A)] is invertible. Since S preserves coproducts, this isomorphism can be extended to an isomorphism S_{T,A}: C(T, A) ⟶ [S(T), S(A)], for any T being a coproduct of objects of X. From this it follows that S_{P,A} is an isomorphism for any P ∈ P(E) = Add(X). Finally since X is compact, it is trivial to see that the functor S : Add(X) ⟶ ProjL(C) is an equivalence [12] and the last assertion follows.

We recall that a functor category Mod(Y) is called perfect, if any functor has a projective cover, equivalently if any flat functor is projective.

Corollary 4.8. C is pure-semisimple iff any object of C has a pure-projective cover iff the category L(C) is perfect. In this case the functor S induces an equivalence S: C ⟶^{≈} FlatL(C) = ProjL(C).

Proof. If C is pure-semisimple then by Proposition 4.8, C = Add(X). By [13], the functor category L(C) is perfect and S: C ⟶^{≈} FlatL(C) = ProjL(C) is an equivalence. Trivially then C has pure-projective covers. Conversely if C has projective covers then ∀A ∈ C, let B ⟶ P ⟶^{f} A be a sequence with f a pureprojective cover. Using Proposition 4.8, we see easily that S(f): S(P) ⟶ S(A) is a projective cover. It is well-known that the only flat functors admitting projective covers are the projective ones. Since S(A) is flat, S(A) is projective or equivalently A is pure-projective. This implies that C is pure-semisimple. It remains to show that C is pure-semisimple, if L(C) is perfect. But ∀A ∈ C, S(A) is flat, hence projective. It follows that A is pure-projective and then C is pure-semisimple.

4.4. Pure-Injectives

We would like to know under what conditions the Abstract Homotopy Category C admits pure–injective envelopes.

Lemma 4.9. ∀A ∈ C, ∀I ∈ I(E), the canonical morphism S_{A,I}: C(A, I) ⟶ L(C)[S(A), S(I)] is an isomorphism. In particular the functors S : I(E) ⟶ L(C) and T: I(E) ⟶ D(C) are fully faithful.

Proof. Let A, I be in C with I be a pure-injective object. By Proposition 4.8, we can choose pure-projective presentations B ⟶ P_{0} ⟶ A and C ⟶ P_{1} ⟶ B. Then we have an exact sequence S(P_{1}) ⟶ S(P_{0}) ⟶ S(A) ⟶ 0 in L(C). Since I is pure-injective, we have an exact sequence 0 ⟶ C(A, I) ⟶ C(P_{0}, I) ⟶ C(P_{1}, I) and an exact sequence

0 ⟶ [S(A), S(I)] ⟶ [S(P_{0}), S(I)] ⟶ [S(P_{1}), S(I)].

Then we have the following exact commutative diagram:

0 ⟶ C(A, I) ⟶ C(P_{0}, I) ⟶ C(P_{1}, I)
S_{A,I} ↓  ^{S_{P_{0},I}} ↓  ^{S_{P_{1},I}} ↓
0 ⟶ [S(A), S(I)] ⟶ [S(P_{0}), S(I)] ⟶ [S(P_{1}), S(I)]

Since S_{P,A} is invertible, ∀P ∈ P(E), it follows directly that the canonical morphism S_{A,I} is invertible. Hence S: I(E) ⟶ L(C) is fully faithful. Since 𝕐^{!} is fully faithful, it follows that the functor T = 𝕐^{!}S : I(E) ⟶ D(C) is fully faithful.

Viewing FlatL(C) as an Abstract Homotopy Category with minimal Whitehead subcategory 𝕐(X), it follows easily that a sequence F_{1} ⟶ F_{2} ⟶ F_{3} in FlatL(C) is pure iff the sequence 0 ⟶ F_{1} ⟶ F_{2} ⟶ F_{3} ⟶ 0 is exact in L(C).

Proposition 4.10. Any injective object E in D(C) is of the form 𝕐^{!}(F), where F is a pure-injective object in FlatL(C).

Proof. Since E is FP-injective, by Lemma 4.1 there exists a flat functor F ∈ L(C) such that 𝕐^{!}(F) = E. Let F_{1} ↣ F_{2} ↠ F_{3} be a pure–sequence in FlatL(C). By Corollary 4.2, 𝕐^{!}(F_{1}) ↣ 𝕐^{!}(F_{2}) ↠ 𝕐^{!}(F_{3}) is a short exact sequence in D(C). Since 𝕐^{!}(F) is injective, (𝕐^{!}(F_{3}), 𝕐^{!}(F)) ↣ (𝕐^{!}(F_{2}), 𝕐^{!}(F)) ↠ (𝕐^{!}(F_{1}), 𝕐^{!}(F)) is a short exact sequence in Ab. Since 𝕐^{!} is fully faithful, the sequence (F_{3}, F) ↣ (F_{2}, F) ↠ (F_{1}, F) is short exact. It follows that F is a pure-injective object in FlatL(C).

Theorem 4.11. Assume that any pure-injective in FlatL(C) is of the form S(I), for some pure-injective I ∈ C. Then C has pure-injective envelopes and the functor T: C ⟶ D(C) induces an equivalence: T : I(E) ⟶^{≈} InjD(C).

Proof. We show that T(I) is injective in D(C) for any pure-injective I in C. Let μ: T(I) ⟶ E be the injective envelope of T(I) in D(C). By Proposition 4.11, there exists a pure-injective object F in FlatL(C), such that 𝕐^{!}(F) = E. By hypothesis, there exists a pure-injective object J ∈ C such that S(J) = F. Since E = T(J), by Lemma 4.10 the morphism μ is of the form T(α), where α: I ⟶ J. Since T(α) is monic, α is pure-monic. Since I is pure-injective in C, α is split monic. Then obviously T(α) is invertible, so T(I) is injective in D(C). The above argument also shows that if E is an injective object of D(C), then E ≅ T(I), for a pure-injective object I in C. So by Lemma 4.10, T: I(E) ⟶ InjD(C) is an equivalence.

Now let A be in C and let μ: T(A) ⟶ T(I) be an injective envelope in D(C). Since μ is of the form T(f), for a pure-monic f: A ⟶ I, by Axiom 4.7, there exists a pure-sequence A ⟶^{f} I ⟶ C in C, which is a pure-injective copresentation of A. If ρ: I ⟶ I is a morphism in C such that f ∘ ρ = f, then T(f) ∘ T(ρ) = T(f). Since T(f) = μ is an injective envelope, T(ρ) is an automorphism. Since 𝕐^{!} is fully faithful, S(ρ) is an automorphism. Then ρ is an automorphism, since S reflects isomorphisms. Hence C has pure-injective envelopes.

If X is a homological compact object, then choosing Γ = ℤ and I = ℚ/ℤ in subsection 3.1, it follows that the dual object 𝔻_{ℚ/ℤ}(X) of X exists in C and

there exists a natural isomorphism ω: [C(X, −), ℚ/ℤ] ⟶^{≅} C(−, 𝔻_{ℚ/ℤ}(X)).

Theorem 4.12. Assume that X ⊆ H(C^{b}) and that 𝕐^{!} is exact on exact

sequences in L(C) of the form S(A) ⟶^{S(f)} S(B) ⟶^{S(g)} S(C). Then we have the following:

(i) C has pure-injectives envelopes and I(E) = Prod(𝔻_{ℚ/ℤ}(X)).

(ii) T induces an equivalence T : I(E) ⟶^{≈} InjD(C).

Proof. The isomorphism ω above shows directly that any dual object of the form 𝔻_{ℚ/ℤ}(X) with X ∈ X, is pure-injective. In particular Prod(𝔻_{ℚ/ℤ}(X)) ⊆ I(E).

Now let A ∈ C and consider the set of morphisms J_{A} := {A ⟶ 𝔻_{ℚ/ℤ}(X) | X ∈ Iso(X)}. Since C as an Abstract Homotopy Category, has products, the set of morphisms J_{A} induces a morphism f_{A}: A ⟶ ∏_{J_{A}} 𝔻_{ℚ/ℤ}(X) in C, with the property that C(f_{A}, 𝔻_{ℚ/ℤ}(X)) is an epimorphism, ∀X ∈ X. We set E := ∏_{J_{A}} 𝔻_{ℚ/ℤ}(X). Then the morphism

[C(X, f_{A}), ℚ/ℤ]: [C(X, E), ℚ/ℤ] ⟶ [C(X, A), ℚ/ℤ]

is isomorphic via ω to the morphism C(f_{A}, 𝔻_{ℚ/ℤ}(X)): C(E, 𝔻_{ℚ/ℤ}(X)) ⟶ C(A, 𝔻_{ℚ/ℤ}(X)). Since C(f_{A}, 𝔻_{ℚ/ℤ}(X)) is an epimorphism and ℚ/ℤ is an injective cogenerator in Ab, it follows that ∀X ∈ X the morphism C(X, f_{A}): C(X, A) ⟶ C(X, E) is a monomorphism. Hence S(f_{A}): S(A) ⟶ S(E) is a monomorphism. By hypothesis, T(f_{A}) is also a monomorphism. Then by Axiom 4.7, there exists a pure-sequence A ⟶ E ⟶ C in C, which is a pureinjective copresentation of A. Hence C has enough pure-injectives. If A is pure-injective, then the morphism f_{A} splits, hence A is a direct summand of E = ∏_{J_{A}} 𝔻_{ℚ/ℤ}(X). It follows that I(E) = Prod(𝔻_{ℚ/ℤ}(X)).

Let E be an injective object in D(C) and consider the functor F : C^{op} ⟶ Ab defined by F (A) = (T(A), E). Since X ⊆ H(C^{b}), S sends a weak cokernel sequence A ⟶ B ⟶ C in C to the exact sequence S(A) ⟶ S(B) ⟶ S(C) in L(C). By hypothesis on 𝕐^{!}, the sequence T(A) ⟶ T(B) ⟶ T(C) is exact in D(C) and since E is injective in D(C), the functor F is half-exact. Plainly F sends coproducts to products, hence by Brown Representability, there exists an isomorphism:

ξ : F = (T(−), E) ⟶^{≅} C(−, I_{E}).

Since T sends pure-sequences in C to short exact sequences in D(C) and since E is injective in D(C), it follows that I_{E} is pure-injective. Then ξ_{|X} : (T(−), E)|_{X} ≅ S(I_{E}) and 𝕐^{!}(ξ_{|X}): 𝕐^{!}(T(−), E)|_{X} ≅ T(I_{E}). It is not difficult to see that the functor 𝕐^{!}(T(−), E)|_{X} is isomorphic to the functor E. Hence T(I_{E}) ≅ E.

Next we show that T(I) is injective in D(C), for any pure-injective object I in C. Let α: T(I) ⟶ E be the injective envelope of T(I) in D(C). By the above argument, there exists a pure-injective object I_{E} such that T(I_{E}) ≅ E. Since T_{A,I_{E}} is invertible, ∀A ∈ C, there exists a morphism f: I ⟶ I_{E} such that T(f) = α. Since T(f) is a monomorphism, there exists a pure-sequence I ⟶ I_{E} ⟶ C in C. Since I is pure-injective, this sequence splits and I is a direct summand of I_{E}. Then T(I) is a direct summand of T(I_{E}) = E, hence T(I) is injective in D(C). Then by Lemma 4.10, T induces an equivalence T : I(E) ⟶^{≈} InjD(C). The existence of pure-injective envelopes in C is proved in the same way as in Theorem 4.12.

Corollary 4.13. Let E be an object in C.

(1) If C satisfies the assumptions of Theorem 4.12 or Theorem 4.13 and E is indecomposable and pure-injective, then E has local endomorphism ring.

(2) If C satisfies the assumptions of Theorem 4.13 then for the statements:

(i) E ≅ 𝔻_{𝔪}(X), where X ∈ X and 𝔪 is a maximal left ideal of End_{C}(X).

(ii) E is pure-injective and there exists a left almost split morphism E ⟶ B in C.

(iii) T(E) is the injective envelope of a simple functor in D(C).

we have (i) ⇒ (ii) ⇒ (iii).

Proof. (1) By hypothesis, T(E) is indecomposable injective in D(C) and End_{C}(E) ≅ End_{D(C)}(T(E)). Then End_{C}(E) is local by [28]. (2) (i) ⇒ (ii) Follows from part (ii) of Theorem 3.10. (ii) ⇒ (iii) The proof is identical with the proof of (β) ⇒ (γ) in part (i) of Theorem 3.13, using that T: I(E) ⟶ InjD(C) is an equivalence.

5. Phantomless Abstract Homotopy Categories, Categories of Finite Type and Ziegler Spectra

Throughout this section we keep the setting, the assumptions and the notations introduced in Section 4. In particular we assume that Axiom 4.7 holds.

5.1. Phantom maps

An important class of morphisms in an Abstract Homotopy Category C is the class of phantom morphisms, i.e. morphisms which are invisible in the representation categories L(C), D(C). In a sense their complexity measures how far is C from being locally finitely presented. More precisely:

Definition 5.1. A morphism f: B ⟶ C is called phantom, if C(X, f) = 0, ∀X ∈ X. C is called phantomless, if there are no no-zero phantom morphisms in C. Equivalently the functor S: C ⟶ L(C) or the functor T: C ⟶ D(C) is faithful.

Let Ph(A, B) be the set of all phantom morphisms from A to B. Plainly Ph(C) = ∪_{A,B∈C}Ph(A, B) concides with the kernel-ideal kerS := {f | S(f) = 0} of S, hence it is an ideal in C, i.e. Ph(−, −) is an additive subfunctor of C(−, −). Since X is a Whitehead subcategory of C, it follows that S reflects isomorphisms. Hence Ph(C) is contained in the Jacobson radical of C. Observe that by Proposition 4.8 and Lemma 4.10, Ph(P, B) = 0 = Ph(A, I), ∀P ∈ P(E), ∀I ∈ I(E). To study phantom morphisms it is useful to impose on C the following axiom.

Axiom 2. If K ⟶^{g} P ⟶^{f} A is a pure-projective presentation in C, then f is a weak cokernel of g.

Note that this Axiom holds in all the examples listed in Section 2. The next lemma shows that the phantomless property of C forces S to be full.

Lemma 5.2. Assume that Axiom 5.2 holds in C. If C is phantomless, then the functor S: C ⟶ L(C) is full and any object of C is a filtered colimit of objects of X.

Proof. Let K ⟶^{g} P ⟶^{f} A be a pure-projective presentation of A in C and let h: A ⟶ C be a weak cokernel of f. Then h is phantom, hence h = 0. It follows that f is an epimorphism. Since f is a weak cokernel of g, for any B ∈ C, we have the following exact commutative diagram:

0 ⟶ C(A, B) ⟶ C(P, B) ⟶ C(K, B)
S_{A,B} ↓  ^{S_{P,B}} ↓  ^{S_{K,B}} ↓
0 ⟶ [S(A), S(B)] ⟶ [S(P), S(B)] ⟶ [S(K), S(B)]

Since C is phantomless, S_{A,B}, S_{K,B} are monomorphisms and since P is pureprojec-tive, S_{P,B} is invertible. A simple diagram chasing shows that S_{A,B} is invertible. Hence S is fully faithful. Now fix an object A ∈ C and consider the category U_{A} with objects, morphisms α: X ⟶ A with X ∈ X. A morphism in U_{A} from α: X_{1} ⟶ A to β: X_{2} ⟶ A is a morphism f: X_{1} ⟶ X_{2} such that f ∘β = α. Since S is fully faithful, the inclusion X ↪ C is dense in the sense of [45]. Then it is well-known that A is a colimit of the functor U_{A} ⟶ C defined by sending α: X ⟶ A to X. Using that X is closed under weak cokernels in C, it is not difficult to see that the category U_{A} is filtered. So any object of C is a filtered colimit of objects of X.

Proposition 5.3. The following are equivalent.

(i) C is a locally finitely presented category and f.p(C) = X.

(ii) The functor S induces an equivalence C ⟶^{≈} FlatL(C).

(iii) The functor T induces an equivalence C ⟶^{≈} FPInjD(C).

(iv) C is phantomless, Axiom 5.2 holds and any functor I: I ⟶ X from a skeletally small filtered category I, has an X -minimal weak colimit in C.

Proof. Plainly (i) ⇒ (ii) and by Lemma 4.1 it follows directly that (ii) ⇔ (iii). (ii) ⇒ (iv) Clearly C is phantomless and Axiom 5.2 holds in C. Moreover any functor I: I ⟶ C admits a filtered colimit, which obviously is an X −minimal weak colimit, since X consists of finitely presented objects. (iv) ⇒ (i) Since C is phantomless by Lemma 5.3, S: C ⟶ L(C) is fully faithful. Let F be a flat functor over X. Then F is a filtered colimit lim_{→}S(X_{i}), where X_{i} = I(i) and I: I ⟶ X is a functor from a skeletally small filtered category I. By hypothesis, there exists in C an X −minimal weak colimit w.lim_{→}X_{i}. Then plainly S(w.lim_{→}X_{i}) ≅ lim_{→}S(X_{i}) = F and S: C ⟶ FlatL(C) is surjective on objects. Hence S is an equivalence.

Remark 5.4. If any morphism in C is a weak kernel, then any epimorphism in C splits. Indeed if f: B ⟶ C is an epimorphism and a weak kernel, then f is a weak kernel of its weak cokernel which is zero. Then 1_{C} factors through f, i.e. f splits.

Lemma 5.5. If any morphism in C is a weak kernel, then: C^{b} = f.p(C).

Proof. Let X be a compact object and let {A_{i}; i ∈ I} be a filtered system in C, with filtered colimit lim_{→}A_{i}, so we have an exact sequence ⊕_{i⟶j}A_{i} ⟶^{g}

⊕_{i∈I}A_{i} ⟶^{f} lim_{→}A_{i} ⟶ 0. Since g is a weak kernel of f and since by Remark

5.5, f is a split epimorphism, it follows that the sequence C(X, ⊕_{i⟶j}A_{i}) ⟶

C(X, ⊕_{i∈I}A_{i}) ⟶ C(X, lim_{→}A_{i}) ⟶ 0 is exact. Since X is compact, it follows

directly that the canonical morphism lim_{→}C(X, A_{i}) ⟶ C(X, lim_{→}A_{i}) is invertible, hence X is finitely presented. The converse is obvious.

Theorem 5.6. If any morphism in C is a weak kernel, then the following statements are equivalent.

(i) C is phantomless.

(ii) C is pure-semisimple.

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

Proof. If C is phantomless, then ∀A ∈ C choose a pure-epic f: P ⟶ A with P be pure-projective. Since any weak cokernel of f is phantom, it follows that f is an epimorphism, which splits by Remark 5.5. Hence A is pureprojective and then C is pure-semisimple. If C is pure-semisimple, then by Corollary 4.8, C = Add(X) is locally finitely presented. It is easy to see that Add(X)^{b} = add(X). Since by our general assumptions X is closed under direct summands, we infer that C^{b} = X. If C is locally finitely presented and C^{b} = X, then by Lemma 5.6, X = f.p(C). Then S: C ⟶ FlatL(C) is an equivalence, in particular C is phantomless.

5.2. Abstract Homotopy Categories of Finite type and Ziegler Spectra

We recall [23] that a Grothendieck category G is called locally Artinian, resp. locally Noetherian, resp. locally finite, if G has a set of Artinian, resp. Noetherian, resp. finite length, generators. Recall also that an additive category A is called a Krull-Schmidt category, if any object of A is a finite coproduct of indecomposable objects and any indecomposable object has a local endomorphism ring. Finally recall that an abelian category is called a length category, if any of its objects has finite length.

Proposition 5.4 and Gabriel’s theory [23], consult also [44], admit the following

Corollary 5.7. C is pure-semisimple iff D(C) is locally Noetherian iff B(X) is a Noetherian abelian category iff the functor T induces an equivalence T: C ⟶^{≈} InjD(C) iff the functor S induces an equivalence S: C ⟶^{≈} ProjL(C). If this is the case, then X is a Krull-Schmidt category and any object of C is in a unique way a coproduct of objects of X having a local endomorphism ring.

Definition 5.8. An Abstract Homotopy Category C is called of finite type iff its representation category D(C) is locally finite.

Proposition 5.9. The following are equivalent.

(i) C is of finite type.

(ii) B(X) is an abelian length category.

(iii) The functor category Mod(X^{op}) is locally finite.

(iv) C is pure semisimple and B(X) is Artinian.

If X is abelian with a finite number of non-isomorphic simple objects, then C is of finite type iff X is a length category with a finite number of non-isomorphic indecomposable objects.

Proof. By Gabriel’s theory [23], D(C) is locally finite iff the category of finitely presented objects of D(C), which is equivalent to B(X), is Artinian and Notherian, hence (i) ⇔ (ii). The equivalence (i) ⇔ (iv) follows from Corollary 5.8. Since B(X) = mod(X^{op})^{op}, the equivalence (ii) ⇔ (iii) follows. The last assertion follows from the above equivalences and well-known results of M. Auslander [5].

Recall from [14] that given a category K, there exists an abelian category A(K), the free abelian category of K, and a functor F: K ⟶ A(K) such that any functor K ⟶ U to an abelian category U, has an, essentially unique, exact factorization through F. By [14], an abelian category A is free iff A is an Auslander category, i.e. A has enough projectives and injectives and has global dimension ≤ 2 and dominant dimension ≥ 2 in the sense that any projective P admits an exact coresolution 0 ⟶ P ⟶ I_{0} ⟶ I_{1}, where I_{0}, I_{1} are projective-injective objects. In this case A is the free abelian category of the full subcategory of projective-injective objects.

Recall that a ring Λ is called representation-finite if Λ is right Artinian and the set of isoclasses of indecomposable finitely presented right Λ−modules is finite. Let F be the collection of Morita equivalence classes of representationfinite rings and let G be the collection of equivalence classes of Abstract Homotopy Categories of finite type with minimal Whitehead subcategory X, such that the abelian category B(X) is free with a finite number of nonisomorphic simple objects. The following consequence of Proposition 5.10, is a variant of Auslander’s correspondence [5].

Corollary 5.10. The map χ: G ⟶ F defined by χ(C) = Λ is a bijection, where Λ = End_{C}(Y), Y = add(Y) and Y is the full subcategory of projectiveinjective objects of B(X). The inverse map is given by ψ(Λ) = Mod(Λ).

Assume now that the hypotheses of Theorem 4.12 or Theorem 4.13 hold. Since the functor T induces an equivalence between the pure-injectives of C and the injectives of D(C), it follows that the isoclasses of indecomposable pure-injective objects of C form a set, since it is well-known [23] that this is true for the isoclasses of indecomposable injective objects in the locally coherent category D(C).

Definition 5.11. The Ziegler Spectrum Zsp(C) of C is defined to be the set of isoclasses of indecomposable pure-injective objects of C.

Following [35] we equip the Ziegler spectrum with a topology in the following way. Let Φ be a class of morphisms in X. An object I ∈ Zsp(C) is called Φ−injective, if C(ϕ, I): C(Y, I) ⟶ C(X, I) is an epimorphism for any ϕ: X ⟶ Y in Φ. We denote by U_{Φ} the subset of Zsp(C) consisting of all Φ−injectives. The next result reduces the study of Zsp(C) to the study of Spectra of more familiar Abstract Homotopy Categories, for which there exists already a rich theory. Its proof is identical with the proof of the corresponding statement for module categories [31], [35].

Corollary 5.12. The subsets {U_{Φ} | Φ is a class of morphisms in X } form the closed sets of a topology in Zsp(C), the Ziegler topology of C, and there is a bijection between the open sets of Zsp(C) and the Serre subcategories of B(X), given by

O ⟼ {M ∈ B(X) | {E ∈ Zsp(C) : D(C)(M, T(E)) ≠ 0} ⊆ O}.

The space Zsp(C) is homeomorphic with the Ziegler spectrum of the Abstract Homotopy Category FlatL(C) and with the spectrum [31] of the category D(C).

6. Representation Equivalences and Flat Approximations

Throughout this section we fix an Abstract Homotopy Category C equipped with a minimal compact Whitehead subcategory X ⊆ C closed under weak cokernels. We assume throughout that C satisfies Axioms 4.7 and 5.2 of Sections 4, 5.

Although C is not necessarily equivalent to FlatL(C), there is another kind of equivalence between these categories which is useful in many cases. Recall [3] that a functor F: C ⟶ G is called a representation equivalence, if F is full, surjective on objects and reflects isomorphisms. Since X is closed under weak cokernels in C, by Lemma 4.1, we have ImS ⊆ FlatL(C). We would like to know under what conditions the representation functor S : C ⟶ FlatL(C) is a representation equivalence, so that C will be as close as possible locally finitely presented, if it is not phantomless. Another motivation for the study of this question comes from Algebraic Topology. Since X is closed under weak cokernels in C, by [14], a functor F: X^{op} ⟶ Ab is flat iff F is half-exact. Then S : C ⟶ FlatL(C) is a representation equivalence iff any half-exact functor F: X^{op} ⟶ Ab is of the form S(C) and any morphism F ⟶ G of half-exact functors is of the form S(α). Hence the question above deals with the problem of “extending” half-exact functors X^{op} ⟶ Ab to half-exact (representable) functors C^{op} ⟶ Ab, thus it is related to another version of the classical Adams-Brown Representability [18], [1], a generalization of which in the triangulated case, is treated in detail in [12], [21], [22], [36], [39], [41].

6.1. Flat Approximations

We assume that C has weak kernels and satisfies the assumptions of Theorem 4.12 or Theorem 4.13, so that C admits pure-injective envelopes and T: I(E) ⟶ InjD(C) is an equivalence. Recall that a full subcategory Y of an additive category A is called contravariantly finite [8], if for any object A ∈ A, there exists a morphism f_{A}: Y_{A} ⟶ A with Y_{A} ∈ Y, such that any morphism Y ⟶ A with Y in Y factors through f_{A}. The dual notion is covariantly finite. If Y is both contravariantly and covariantly finite, then it is called functorially finite.

Theorem 6.1. ImS is functorially finite in L(C), i.e. any functor M ∈ L(C) has a right ImS−approximation p_{M}: S(A) ⟶ M and a left ImS−approxima– tion i_{M}: M ⟶ S(B). Moreover ImT is functorially finite in D(C).

Proof. Let M be in L(C) and consider the functor 𝕐^{!}(M) ∈ D(C). Choose a

minimal injective presentation 0 ⟶ 𝕐^{!}(M) ⟶^{μ′} T(I_{0}) ⟶^{f′} T(I_{1}) of 𝕐^{!}(M) in D(C). By our assumptions, f′ is of the form T(f), where f: I_{0} ⟶ I_{1} is a morphism in C. Since 𝕐_{∗} is exact and 𝕐_{∗}𝕐^{!} = Id_{L(C)}, we have an exact

sequence 0 ⟶ M ⟶^{μ} S(I_{0}) ⟶^{S(f)} S(I_{1}), where μ := 𝕐_{∗}(μ′). Let g: A ⟶ I_{0} be a weak kernel of f in C. Since S(g)∘S(f) = 0, there exists a unique morphism p_{M}: S(A) ⟶ M such that p_{M} ∘ μ = S(g). If α: S(B) ⟶ M is a morphism in L(C), then by Lemma 4.10, the morphism 𝕐^{!}(α) ∘ 𝕐^{!}(μ): T(B) ⟶ T(I_{0}) is of the form T(h), where h: B ⟶ I_{0}. Then obviously T(h) ∘ T(f) = 0. Hence by Lemma 4.10, we have h ∘ f = 0. Since g is a weak kernel of f in C, there exists a morphism γ: B ⟶ A such that γ ∘ g = h. Then S(γ) ∘ S(g) = S(h). Since 𝕐^{!}(α) ∘ 𝕐^{!}(μ) = T(h) = 𝕐^{!}S(h) and 𝕐^{!} is faithful, we have α ∘ μ = S(h) = S(γ) ∘ S(g). Then S(γ) ∘ p_{M} ∘ μ = S(γ) ∘ S(g) = α ∘ μ and this implies that S(γ) ∘ p_{M} = α, since μ is a monomorphism. This shows that p_{M} is a right ImS−approximation. Hence ImS is contravariantly finite in L(C). Now

let S(P_{1}) ⟶^{S(f)} S(P_{0}) ⟶^{ϵ} M ⟶ 0 be the start of a projective resolution of M and let P_{1} ⟶^{f} P_{0} ⟶^{g} A be a weak cokernel sequence in C. Then there exists a unique morphism i_{M}: M ⟶ S(A) such that ϵ ∘ i_{M} = S(g). If h: M ⟶ S(B) is a morphism in L(C), then S(f) ∘ ϵ ∘ h = 0. Since P_{0} is pure-projective, the morphism ϵ ∘ h is of the form S(τ), where τ: P_{0} ⟶ B. Hence S(f) ∘ S(τ) = 0 and then f ∘ τ = 0. Hence there exists σ: A ⟶ B such that g ∘ σ = τ. Then S(g) ∘ S(σ) = S(τ) ⇒ ϵ ∘ i_{M} ∘ S(σ) = ϵ ∘ h ⇒ i_{M} ∘ S(σ) = h. Hence i_{M} is a left ImS−approximation.

Using the adjoint pair (𝕐^{!}, 𝕐_{∗}) and the contravariant finiteness of ImS it is easy to see that ImT is contravariantly finite in D(C). Since T preserves products and ImT consists of FP-injective objects, it follows that ImT is closed under products and pure-subobjects, for the purity in D(C) induced by its finitely presented objects. Then it follows by [35], that ImT is covariantly finite in D(C).

Corollary 6.2. If S: C ⟶ FlatL(C), resp. T: C ⟶ FPInjD(C), is surjective on objects, then FlatL(C), resp. FPInjD(C), is functorially finite in L(C), resp. D(C).

Corollary 6.3. Let P be a skeletally small additive category with weak cokernels. Then the category Flat(P) is contravariantly finite in Mod(P) iff Flat(Mod(P)) admits weak kernels. In particular if Λ is a left coherent ring, then the category Flat(Λ) of flat right Λ−modules is contravariantly finite in Mod(Λ) iff Flat(Λ) admits weak kernels.

Proof. If Flat(P) is contravariantly finite, then Flat(P) has weak kernels by [13]. Since P has weak cokernels, the category Flat(P) is an Abstract Homotopy Category and as such satisfies the assumptions of Theorem 4.12. If in addition Flat(P) has weak kernels, then since in our case S is an equivalence, the contravariant finiteness of Flat(P) follows from Corollary 6.2. The last assertion follows from the well–known observation that the category P_{Λ} of finitely generated projective right Λ−modules has weak cokernels iff Λ is left coherent.

6.2. Representation Equivalences

The following result shows a connection between between representation equ– ivalences and flat approximations, first observed by H. Krause [36] in case C is a compactly generated triangulated category.

Let p_{M}: S(A) ⟶ M be the right ImS−approximation and i_{M}: M ⟶ S(B) be the left ImS−approximation of M ∈ L(C), constructed in Theorem 6.1.

Proposition 6.4. If for all functors M ∈ L(C), p_{M} right minimal or i_{M} is left minimal, then the functor S: C ⟶ FlatL(C) is a representation equivalence.

Proof. Let S(P_{1}) ⟶^{S(f)} S(P_{0}) ⟶^{ϵ} M ⟶ 0 be the start of a projective resolution of M, such that i_{M}: M ⟶ S(A) is a minimal left flat approximation of M, where g: P_{0} ⟶ A is a weak cokernel of f and S(g) = ϵ ∘ i_{M}. If M is flat, then obviously i_{M} is invertible. Hence S is surjective on flat functors. If α: S(A) ⟶ S(A′) is a morphism, then by construction and hypothesis, we

can choose projective presentations S(P_{1}) ⟶^{S(f)} S(P_{0}) ⟶^{S(g)} S(A) ⟶ 0 and

S(P′_{1}) ⟶^{S(f′)} S(P′_{0}) ⟶^{S(g′)} S(A′) ⟶ 0 such that g, resp. g′, is a weak cokernel of f, resp. f′. Then α induces morphisms γ: P_{1} ⟶ P′_{1} and β: P_{0} ⟶ P′_{0} such that γ ∘ f′ = f ∘ β and S(β) ∘ S(g′) = S(g) ∘ α. By the first commutativity, there exists δ such that g ∘δ = β ∘g′. Obviously then S(δ) = α, hence S is full. Since S reflects isomorphisms, it is a representation equivalence. The proof for p_{M} is similar.

Remark 6.5. H. Krause [36] proved that if C is a compactly generated triangulated category and X = C^{b}, then the converse holds for p_{M}: S(A) ⟶ M. The corresponding result for the morphism i_{M} is not true in general (even in the setting of [36]). However it is not difficult to see that it is true if any torsionless functor, i.e. a subfunctor of a projective functor, has a projective cover.

We define the pure-projective dimension p.p.dA of an object A ∈ C inductively as follows. If A is pure-projective, then p.p.dA = 0. If p.p.dA > 0, then p.p.dA ≤ n if there exists a pure sequence K ⟶ P ⟶ A in C such that p.p.dK < n and P is pure-projective. Then define p.p.dA = n if p.p.dA ≤ n and p.p.dA ≰ n − 1. If p.p.dA ≠ n, ∀n ≥ 0, then define p.p.dA = ∞. Finally define the pure-global dimension p.gl.dimC of C by p.gl.dimC = sup{p.p.dA | A ∈ C}.

Remark 6.6. Since S reflects isomorphisms it is easy to see (compare [12]) that ∀A ∈ C: p.dS(A) = p.p.dA. Hence: p.gl.dimC ≤ sup{p.dF | F ∈ FlatL(C)}.

We recall [44] that the weight w(X) of X is the cardinality of the disjoint union of the sets C(X_{1}, X_{2}), where X_{1}, X_{2} run over the isoclasses of objects of X. By a result of Simson [44], if w(X) ≤ ℵ_{t} for some t ≥ 0, then sup{p.dF; F ∈ FlatL(C)} ≤ t + 1. Hence in this case p.gl.dimC ≤ t + 1. It follows that if X has a countable skeleton, i.e. if w(X) ≤ ℵ_{0}, then p.gl.dimC ≤ 1.

Lemma 6.7. If p.p.dA ≤ 1, then ∀B ∈ C, the canonical morphism S_{A,B}: C(A, B) ⟶ [S(A), S(B)] is an epimorphism. Hence the functor S is full, if p.gl.dimC ≤ 1.

Proof. Let P_{1} ⟶^{g} P_{0} ⟶^{f} A be a pure sequence in C with P_{1}, P_{0} pure-projectives and let B ∈ C. Since f is a weak cokernel of g, the sequence C(A, B) ⟶ C(P_{0}, B) ⟶ C(P_{1}, B) is exact. Since the sequence 0 ⟶ [S(A), S(B)] ⟶ [S(P_{0}), S(B)] ⟶ [S(P_{1}), S(B)] is exact and the morphisms S_{P_{i},B}: C(P_{i}, B) ⟶ [S(P_{i}), S(B)], i = 0, 1 are invertible, it follows that S_{A,B}: C(A, B) ⟶ [S(A), S(B)] is an epimorphism.

Proposition 6.8. If any flat functor F in L(C) has projective dimension p.dF ≤ 1, then the functor S: C ⟶ FlatL(C) is a representation equivalence.

Proof. Since S reflects isomorphisms, and by Remark 6.6, Lemma 6.7, S is full, it remains to show that S is surjective on flat functors. Let F be a flat functor and let 0 ⟶ R_{1} ⟶ R_{0} ⟶ F ⟶ 0 be a projective resolution of F in L(C). Then the morphism R_{1} ⟶ R_{0} is of the form S(g): S(P_{1}) ⟶ S(P_{0}), for a morphism g: P_{1} ⟶ P_{0} in C. Since 𝕐^{!} is exact on exact sequences of flat functors, it follows that T(g) is a monomorphism, i.e. g is pure-monic. By Axiom 4.7, there exists a pure sequence P_{1} ⟶^{g} P_{0} ⟶ A in C. Then obviously S(A) ≅ F.

Corollary 6.9. S: C ⟶ FlatL(C) is a representation equivalence, if w(X) ≤ ℵ_{0}.

As a consequence we have the following result on flat approximations.

Corollary 6.10. If any flat functor F in L(C) has p.dF ≤ 2 (e.g. if the weight of X is of cardinality w(X) ≤ ℵ_{1}), then FlatL(C) is contravariantly finite in L(C).

Proof. By Corollary 6.2, it suffices to show that S is surjective on flat functors.

Let 0 ⟶ S(P_{2}) ⟶^{S(f_{2})} S(P_{1}) ⟶^{S(f_{1})} S(P_{0}) ⟶^{S(f_{0})} F ⟶ 0 be a projective resolution of the flat functor F in L(C). Then the morphism f_{2} is pure-monic, hence by Axiom 4.7, there exists a pure sequence P_{2} ⟶^{f_{2}} P_{1} ⟶^{g} A in C. Then S(A) ≅ Im(S(f_{1})). Since p.p.dA ≤ 1, by Lemma 6.7 the inclusion morphism S(A) ↪ S(P_{0}) comes from a morphism h: A ⟶ P_{0} and plainly h is pure-monic, since the cokernel of S(h) is flat and 𝕐^{!} is exact on short exact sequences of flat functors. By Axiom 4.7 there exists a pure-sequence A ⟶^{h} P_{0} ⟶ B in C. Then obviously S(B) ≅ F.

The converse of Proposition 6.8 is not true in general. For instance if C is a locally finitely presented category with products, then S: C ⟶ Flat(f.p(C)) is an equivalence but a flat functor over f.p(C) can be of arbitrary large projective dimension. However the converse is true if in C any epimorphism splits and the pure-projective presentations are closed under weak push-outs along phantom morphisms, in the following sense. If K ⟶^{g} P ⟶^{f} A is a pureprojective presentation of A and h: K ⟶ B is a phantom morphism in C, then there exists a commutative diagram:

K ⟶^{g} P ⟶^{f} A
h↓  ^{γ}↓  ^{||}↓
B ⟶^{α} D ⟶^{β} A

such that the lower sequence is pure and the left square is a weak pushout diagram, i.e. the morphism ^{t}(γ, α): P ⊕ B ⟶ D is a weak cokernel of (g, −h): K ⟶ P ⊕ B. Note that these conditions hold trivially if C is a triangulated category.

Theorem 6.11. If any epimorphism in C splits and the pure-projective presentations are closed under weak push-outs along phantom morphisms, then the functor S: C ⟶ FlatL(C) is a representation equivalence iff p.dF ≤ 1, ∀F ∈ FlatL(C).

Proof. (⇒) Since any flat functor F is of the form S(A), by Remark 6.6 it

suffices to show that ∀A ∈ C: p.p.dA ≤ 1. Let K ⟶^{g} P ⟶^{f} A and L ⟶^{g′} Q ⟶^{f′} K be pure-projective presentations and let h: K ⟶ B be a weak cokernel of f′. Consider the weak push-out of h, g, as in the above diagram. Since f′ is pureepic, h is a phantom morphism or equivalently S(h) = 0. Applying the functor S to the diagram above, we have the following exact commutative diagram in L(C):

0 ⟶ S(K) ⟶^{S(g)} S(P) ⟶^{S(f)} S(A) ⟶ 0
0↓  ^{S(γ)}↓  ^{||}↓
0 ⟶ S(B) ⟶^{S(α)} S(D) ⟶^{S(β)} S(A) ⟶ 0

Then there exists a morphism δ′: S(A) ⟶ S(D) such that S(f) ∘ δ′ = S(γ). Since S is full, there exists δ: A ⟶ D such that S(δ) = δ′. It follows easily that S(δ) ∘ S(β) = 1_{S(A)}. Since S reflects isomorphisms, we have that δ ∘ β is invertible. Let ω: S(D) ⟶ F be the cokernel of S(γ). By the above diagram, there exists a unique isomorphism ζ: S(B) ⟶ F such that ζ = S(α) ∘ ω. Consider the morphism ω ∘ ζ^{−1} : S(D) ⟶ S(B). By the fullness of S, there exists a morphism ϕ: D ⟶ B such that S(ϕ) = ω ∘ ζ^{−1}. Then S(α) ∘ S(ϕ) = S(α) ∘ ω ∘ ζ^{−1} = 1_{S(B)}. Since S reflects isomorphisms, we have that α ∘ ϕ is invertible. Now observe that S(γ) ∘ S(ϕ) = S(γ) ∘ ω ∘ ζ^{−1} = 0. Hence γ ∘ ϕ is phantom. Since P is pure-projective it follows that γ ∘ ϕ = 0. Then h = h ∘ α ∘ ϕ = g ∘ γ ∘ ϕ = 0. This implies that f′ is an epimorphism since its weak cokernel is zero. By hypothesis f′ splits. Hence K is pure-projective as a direct summand of Q. This implies that p.p.dA ≤ 1.

Theorem 6.11, together with Corollary 6.9, generalizes the well–known result of Brown [18] and Adams [1], see also [39], on the homotopy category of pointed CW-complexes to the non–stable case and answers a question of Heller [26]. Since the assumptions of Theorem 6.11 hold in triangulated categories, we recover the generalizations of Adams-Brown Representability Theorem to triangulated categories studied in [12], [21], [41].

7. Applications

7.1. Locally Finitely Presented Categories

Let F be a locally finitely presented additive category with products. We view F as an Abstract Homotopy Category with minimal Whitehead subcategory f.p(F). Clearly f.p(F) is compact and closed under (weak) cokernels in F. Moreover it is not difficult to see that F satisfies Axiom 4.7 and the condition imposed on Theorem 4.12. Then F is phantomless and the theory of Section 4 applied to F recovers the theory of purity developed by Crawley-Boevey [17], and by Gruson-Jensen [24], Simson [44] and others (see [28]), if in addition F is abelian. In particular, F has enough pure-projective objects and admits pureinjective envelopes. Concerning almost split morphisms, we have the following direct consequence of Theorem 3.10.

Theorem 7.1. Let X be a homological finitely presented object in F. Then the 𝔪−dual object 𝔻_{𝔪}(X) is pure-injective and there exists a left almost split morphism g_{X}: 𝔻_{𝔪}(X) ⟶ F^{X} in F. If F has weak kernels and X has a local endomorphism ring, then there exists a right almost split morphism f_{X}: F_{X} ⟶ X in F.

Theorem 7.1 and the results of sections 3, 4 can be applied to the locally finitely presented category F of flat quasi-coherent sheaves for a non-singular irreducible curve/surface X over a field, where f.p(F) is the category of vector bundles over X [17]. Similar remarks are hold for categories of rational G−modules with good filtrations over an algebraic group G [17]. Details are left to the interested reader.

7.2. Right Triangulated Categories with a Right Semi-Equivalence Suspension Functor

Let C be a right triangulated [2] (or suspended [30]) category with suspension functor Σ: C ⟶ C. Recall that Σ is called a right semi-equivalence [2], if Σ is fully faithful and ImΣ is closed under extensions in the following sense: if Σ(A) ⟶ X ⟶ Σ(C) ⟶ Σ^{2}(A) is a triangle in C, then X = Σ(B) for some B in C. The following are non-trivial examples of right triangulated categories with a right semi-equivalence suspension functor, which are not necessarily triangulated.

Example 7.2. (1) Let (C^{≤0}, C^{≥0}) be a t−structure in a triangulated category C. Then C^{≤0} is right triangulated with a right semi-equivalence suspension functor.

(2) Let C be an abelian category with enough injectives and let I be the full subcategory of injective objects of C. By [2], a full subcategory X /I of C/I is right triangulated with right semi-equivalence suspension functor iff X is closed under extensions, cokernels of monomorphisms and I ⊆ X ⊆ I^{⊥}, where I^{⊥} := {A ∈ C | Ext^{n}(X, A) = 0, ∀X ∈ I, ∀n ≥ 1}. For instance we can take X = I^{⊥}.

Throughout we fix a right triangulated category C with a right semiequivalence suspension functor Σ. We assume that C has coproducts and Σ preserves coproducts. By [2], the functor C(A, −): C ⟶ Ab is half-exact, ∀A ∈ C. Hence H(C^{b}) = C^{b} and then by Lemma 3.10 of [13], we have the following consequence.

Lemma 7.3. Any compact subcategory of C is minimal.

The above Lemma and the Adjoint Functor Theorem 2.8 admit the following.

Corollary 7.4. If C contains a compact Whitehead subcategory, then C is an Abstract Homotopy Category and the suspension functor Σ admits a right adjoint Ω.

From now on we assume that C contains a compact Whitehead subcategory X and we view C as an Abstract Homotopy Category. The following important class of triangles was introduced by Happel [25], generalizing the fundamental notion of an almost split sequence in an abelian category, due to Auslander-Reiten [7].

Definition 7.5. A triangle A ⟶^{g} B ⟶^{f} C ⟶ Σ(A) in C is called an Auslan– der-Reiten triangle, if g is left almost split and f is right almost split.

Theorem 7.6. If X is a compact object in C with local endomorphism ring, then there exists an Auslander-Reiten triangle:

𝔻_{𝔪}(X) ⟶^{g^{X}} A ⟶^{f_{X}} Σ(X) ⟶^{−Σ(h_{X})} Σ𝔻_{𝔪}(X)

and an Auslander-Reiten triangle E ⟶ B ⟶ X ⟶^{h_{X}} 𝔻_{𝔪}(X), if 𝔻_{𝔪}(X) ∈ ImΣ.

Proof. Consider the morphism h_{X}: X ⟶ 𝔻_{𝔪}(X) constructed in Theorem 3.10

and let X ⟶^{h_{X}} 𝔻_{𝔪}(X) ⟶^{g^{X}} A ⟶^{f_{X}} Σ(X) be a triangle in C. By Theorem 3.10, g^{X} is left almost split. Since Σ is fully faithful, End_{C}(Σ(X)) ≅ End_{C}(X) is

local. Then by [2], f_{X} is right almost split. Hence the triangle (∗) : 𝔻_{𝔪}(X) ⟶^{g^{X}}

A ⟶^{f_{X}} Σ(X) ⟶^{−Σ(h_{X})} Σ𝔻_{𝔪}(X) is an Auslander-Reiten triangle. If 𝔻_{𝔪}(X) ∈ ImΣ, let E ∈ C be such that Σ(E) = 𝔻_{𝔪}(X). Since ImΣ is closed under extensions, A = Σ(B), for some B ∈ C. Moreover End_{C}(E) ≅ End_{C}(Σ(E)) ≅ End_{C}(𝔻_{𝔪}(X)) is local. Then by [2], there exists a triangle (†) : E ⟶^{α} B ⟶^{β} X ⟶^{h_{X}} Σ(E) in C and the morphism β is a weak kernel of h_{X}. Hence by Theorem 3.10, β is a right almost split morphism. Then by [2], it follows that (†) is an Auslander-Reiten triangle in C.

Since any compact object in C is homological, Corollary 3.14 implies the following generalization of a result of H. Krause [37].

Corollary 7.7. Let X be a compact object in C, let ρ : Γ ⟶ Λ_{X} := End_{C}(X) be a ring morphism and let I be an injective left Γ−module. If A ⟶^{f} B ⟶^{g} C ⟶^{h} Σ(A) is a triangle in C, then there exist isomorphisms:

Hom_{Λ_{X}} [CokerC(X, Σ(f)), I] ≅ CokerC(f, Ω𝔻_{I}(X)) ≅ KerC(g, 𝔻_{I}(X)).

If I is a cogenerator, then any morphism X ⟶ C factors through g iff any morphism A ⟶ Ω𝔻_{I}(X) factors through f. In particular X ∈ X iff 𝔻_{I}(X) is pure-injective.

It is not difficult to see that if Σ preserves pure monics, then a sequence A ⟶ B ⟶ C in C is pure iff there exists a triangle A ⟶ B ⟶ C ⟶^{h} Σ(A) in C such that 0 ⟶ C(X, A) ⟶ C(X, B) ⟶ C(X, C) ⟶ 0 is exact, ∀X ∈ X.

7.3. Triangulated Categories

Let C be a triangulated category which admits all small coproducts. We assume that C is compactly generated [40]. This means that C admits a set of compact objects S such that if C(Σ^{n}(S), A) = 0, ∀S ∈ S, ∀n ∈ ℤ, then A = 0. By [13], this is equivalent to say that the full subcategory C^{b} of compact objects is a (minimal) Whitehead subcategory of C, i.e. C is an Abstract Homotopy Category. The purity in C based on the Whitehead subcategory C^{b} is denoted by E and coincides with (and recovers) the theory of purity and phantom maps developed in [12], [20], [34], [39].

It is not difficult to see [14] that C enjoys the very useful property that the Representation Categories of C coincide: D(C) = Mod(C^{b}) = L(C). Hence Theorem 7.6 admits the following consequence, in which the existence of Auslander-Reiten triangles and pure-injective envelopes, is due to H. Krause.

Theorem 7.8. [37], [12], [34] Let X be a compact object in C with local endomorphism ring. Then there exists an Auslander-Reiten triangle

Σ^{−1}𝔻_{𝔪}(X) ⟶ A ⟶ X ⟶^{h_{X}} 𝔻_{𝔪}(X).

Moreover C has enough pure-projectives, pure-injective envelopes and there exist equivalences P(E) ≈ Add(C^{b}) and I(E) ≈ Prod(𝔻_{ℚ/ℤ}(C^{b})). C is puresemisimple iff C has pure-projective covers iff C is phantomless iff C is locally finitely presented iff Mod(C^{b}) is locally Noeherian iff Mod(C^{b}) is perfect.

Let S: C ⟶ L(C) = Mod(C^{b}) be the restricted Yoneda functor, defined by S(C) = C(−, C)|_{C^{b}}. The following result characterizes the pure-injective objects of C occuring as a source of a left almost split morphism in C and, under the assumption that C^{b} is a Krull-Scmidt category, gives necessary and sufficient conditions for the existence of Auslander-Reiten triangles in C starting at a pure-injective object.

Theorem 7.9. For an object E in C, the following are equivalent:

(i) E is pure-injective and there exists a left almost split morphism E ⟶ B in C.

(ii) S(E) is the injective envelope of a simple functor.

(iii) E ≅ 𝔻_{𝔪}(X), where X is compact and 𝔪 is a maximal left ideal of End_{C}(X).

If C^{b} is a Krull-Schmidt category, then the above are also equivalent to:

(iv) E is pure-injective and there exists an Auslander-Reiten triangle E ⟶ B ⟶ P ⟶ Σ(E) in C.

(v) E ≅ 𝔻_{𝔪}(X), for an indecomposable compact object X and a maximal left ideal 𝔪 of End_{C}(X).

Proof. That the first three statements are equivalent is a direct consequence of Theorems 3.10, 3.13 and Corollary 4.13, since the hypotheses imposed on these results hold trivially in C. Assume that C^{b} is a Krull-Schmidt category. (i) ⇒ (iv) By part (iii), S(E) is the injective envelope of a simple functor, which by Section 3, is of the form S_{X,𝔪}, where X ∈ C^{b}. Since C^{b} is Krull-Schmidt, we can choose X to be indecomposable. By Theorem 3.9, the injective envelope of S_{X,𝔪} is isomorphic to S(𝔻_{𝔪}(X)). Hence S(𝔻_{𝔪}(X)) ≅ S(E) and this implies that E ≅ 𝔻_{𝔪}(X). If Σ^{−1}(𝔻_{𝔪}(X)) ⟶ A ⟶ X ⟶ E is the Auslander-Reiten triangle of Theorem 7.8 ending at X, then E ⟶ Σ(A) ⟶ Σ(X) ⟶ Σ(E) is the desired Auslander-Reiten triangle starting at E. (iv) ⇒ (v) Let E ⟶^{g} B ⟶^{f} P ⟶^{h} Σ(E) be an Auslander-Reiten triangle in C. If P is not compact, then any morphism α: X ⟶ P with X ∈ C^{b} is not a split epimorphism, hence α factors through f. Then the triangle is pure and since E is pure-injective, g is split monic and this is not true. Hence P ∈ C^{b} and then by using (iii), we have an isomorphism E ≅ 𝔻_{𝔪}(X), where X = Σ(P), since the end terms of an Auslander-Reiten triangle are uniquely determined up to isomorphism [25]. The direction (v) ⇒ (iii) is trivial.

We refer to [15] for necessary and sufficient conditions for the existence of Auslander-Reiten triangles in C^{b}.

7.4. Spectra and Complexes

The results of subsection 7.3 are directly applicable to the Stable Homotopy Category Ho(S) of spectra [39], where Ho(S)^{b} is the full subcategory of finite spectra and to the unbounded derived category D(Λ) of a ring Λ, where D(Λ)^{b} is, up to equivalence, the homotopy category H^{b}(P_{Λ}) of perfect complexes, i.e. bounded complexes with components finitely generated projective modules. For instance we have the following.

Corollary 7.10. (1) For any finite spectrum X with local endomorphism ring, there exists an Auslander-Reiten triangle Σ^{−1}𝔻_{𝔪}(X) ⟶ A ⟶ X ⟶ 𝔻_{𝔪}(X) in Ho(S). Moreover for any finite spectrum X, the 𝔪−dual spectrum 𝔻_{𝔪}(X) is pure-injective and occurs as the source of an almost split morphism in Ho(S).

(2) For any perfect complex X^{•} in D(Λ) with local endomorphism ring, there exists an Auslander-Reiten triangle 𝔻_{𝔪}(X^{•})[−1] ⟶ A^{•} ⟶ X^{•} ⟶ 𝔻_{𝔪}(X^{•}) in D(Λ). Moreover for any perfect complex X^{•}, the 𝔪−dual complex 𝔻_{𝔪}(X^{•}) is pure-injective and occurs as the source of an almost split morphism in D(Λ).

(3) If H^{b}(P_{Λ}) is a Krull-Schmidt category, then for a pure-injective complex E^{•} ∈ D(Λ), there exists an Auslander-Reiten triangle E^{•} ⟶ A^{•} ⟶ B^{•} ⟶ E^{•}[1] in D(Λ) iff there exists an indecomposable perfect complex X^{•}, such that E^{•} = 𝔻_{𝔪}(X^{•}), in which case B^{•} = X^{•}[1].

It is easy to see H^{b}(P_{Λ}) is Krull-Schmidt, if Λ is an Artin R−algebra. If I is the injective envelope of R/J ac(R) and D = Hom_{R}(−, I) is the usual duality of Artin algebras, then for any perfect complex X^{•} and any complex A^{•} ∈ D(Λ), the isomorphism ω of subsection 3.1, reduces to the rather well–known isomorphism: D(X^{•}, A^{•}) ≅ (A^{•}, 𝔻_{I}(X^{•})), see [25]. It follows that 𝔻_{I}(X^{•}) ≅ X^{•} ⊗_{Λ}^{L} D(Λ), where −⊗_{Λ}^{L} D(Λ): D(Λ) ⟶ D(Λ) is the total left derived functor of the Nakayama functor − ⊗_{Λ} D(Λ) : Mod(Λ) ⟶ Mod(Λ).

Since the endomorphism ring ℤ of the sphere spectrum S^{0} ∈ Ho(S)^{b} is not semiperfect, it follows that Ho(S)^{b} is not a Krull-Schmidt category. However the full subcategory Ho(S)_{p}^{b} of p−local finite spectra [39], where p is a prime, is a Krull-Schmidt subcategory of the category Ho(S)_{p} of p−local spectra, so one can characterize the p−local pure-injective spectra occuring as a source of an Auslander-Reiten triangle in the p−local category, along the lines of Theorem 7.9.

7.5. Module Categories

If Γ is an associative ring, then we view the category Mod(Γ) of right Γ−mo– dules as an Abstract Homotopy Category with minimal Whitehead subcategory, the full subcategory mod(Γ) of finitely presented modules. Obviously Mod(Γ) satisfies the conditions imposed on Theorem 4.12. The Representation Categories L(Γ) := L(Mod(Γ)) = Mod(mod(Γ)) and D(Γ) := D(Mod(Γ)) = Mod(mod(Γ^{op})^{op}) have been used for long time as an indispensable tool for the study of purity of modules. The description of the functor S is clear and it is easy to see that T is isomorphic to the functor M ↦ M ⊗_{Γ} −: mod(Γ^{op}) ⟶ Ab. The theory of purity developed in Section 4, applied to Mod(Γ) coincides and recovers the well-known and extensively studied purity in the sense of Cohn. In particular the Ziegler spectrum as defined in Section 5, coincides with the Ziegler spectrum defined by Ziegler [47] and studied extensively by Krause [35] and others. Theorem 4.12 and Corollary 5.8 recover the well– known result that Mod(Γ) has pure-injective envelopes and that Γ is right pure-semisimple iff Mod(Γ) has pure-projective covers iff L(Γ) is perfect iff D(Γ) is locally Noetherian [44], [28]. Proposition 5.10 recovers the well-known result that Γ is representation-finite iff D(Γ) is locally finite [5].

We recall the fundamental concept of an almost split sequence.

Definition 7.11. [7] An exact sequence 0 ⟶ A ⟶^{g} B ⟶^{f} C ⟶ 0 is called an almost split sequence, if f is right almost split and g is left almost split.

We recall that the transpose Tr(X) of a finitely presented right Γ−module X is defined by the exact sequence Hom_{Γ}(P_{0}, Γ) ⟶ Hom_{Γ}(P_{1}, Γ) ⟶ Tr(X) ⟶ 0 in mod(Γ^{op}), where P_{1} ⟶ P_{0} ⟶ X ⟶ 0 is a projective presentation of X in mod(Γ). The following consequence of Theorem 3.10 is the original basic existence result of almost split morphisms and almost split sequences due to M. Auslander and I. Reiten [4], [7]. We include a proof as a sample application of Brown Representability and the machinery of Abstract Homotopy Categories. For part (β) it is essential to work not in Mod(Γ) but in a more suitable Abstract Homotopy Category.

Theorem 7.12. [4] (α) Let P be a projective Γ−module with local endomorphism ring. Then there exists a non-zero map h_{P} : P ⟶ 𝔻_{𝔪}(P) in Mod(Γ), such that 𝔻_{𝔪}(P) is an indecomposable injective module, Im(h_{P}) = S_{𝔪,P} is a simple module, the inclusion Rad(P) = Ker(h_{P}) → P is minimal right almost split and the projection 𝔻_{𝔪}(P) ⟶ Coker(h_{P}) = 𝔻_{𝔪}(P)/Soc(𝔻_{𝔪}(P)) is minimal left almost split.

(β) Let X be a non-projective finitely presented module with local endomorphism ring. Then there exists a functor 𝔻_{𝔪}(−, \underline{X}) : \underline{mod}(Γ)^{op} ⟶ Ab, an isomorphism of functors 𝔻_{𝔪}(−, \underline{X})(?) ≅ Ext_{Γ}^{1}[?, Hom_{Λ_{X}} (Tr(X), I_{𝔪})], and an almost split sequence:

0 ⟶ Hom_{Λ_{X}} [Tr(X), I_{𝔪}] ⟶ A ⟶ X ⟶ 0

where Λ_{X} = End(\underline{X}) and 𝔪 is a maximal ideal of Λ_{X}. Further Hom_{Λ_{X}} [Tr(X), I] is pure-injective in Mod(Γ). If Γ is an Artin algebra, then Hom_{Λ_{X}} [Tr(X), I_{𝔪}] ≅ DTr(X) and there exists an isomorphism of functors

𝔻_{𝔪}(−, \underline{X})(?) ≅ D\underline{Hom}_{Γ}[X,?] ≅ Ext_{Γ}^{1}[?, DTr(X)] : \underline{mod}(Γ)^{op} ⟶ Ab

where DTr is the usual “dual of the transpose” [7].

Proof. (α) Since P has local endomorphism ring, it follows that P is finitely generated, in particular P is compact. Viewing Mod(Γ) as an abelian Abstract Homotopy Category, with minimal Whitehead subcategory the full subcategory P_{Γ} of finitely generated projective modules, all the assertions are consequences of Theorem 3.10.

(β) Let \underline{mod}(Γ) be the stable category of finitely presented modules modulo projectives. We view the category Mod(\underline{mod}(Γ)) of contravariant additive functors \underline{mod}(Γ)^{op} ⟶ Ab, as an Abstract Homotopy Category in which the functor (−, \underline{X}) is a homological compact projective object. Since Λ_{X} := End(\underline{X}) is a factor ring of End(X), it is a local ring with unique maximal ideal 𝔪. By Brown Representability, there exists a functor 𝔻_{𝔪}(−, \underline{X}) in Mod(\underline{mod}(Γ)) and a natural isomorphism

ω: Hom_{Λ_{X}} [((−, \underline{X}),?), I_{𝔪}] ⟶^{≅} [?, 𝔻_{𝔪}(−, \underline{X})].

Then for any finitely presented Γ−module C, ω_{(−,\underline{C})} induces isomorphisms:

Hom_{Λ_{X}} [((−, \underline{X}), (−, \underline{C})), I_{𝔪}] ≅ Hom_{Λ_{X}} [(\underline{X}, \underline{C}), I_{𝔪}] ≅
≅ [(−, \underline{C}), 𝔻_{𝔪}(−, \underline{X})] ≅ 𝔻_{𝔪}(−, \underline{X})(\underline{C}).

But it is well known [7] that we have isomorphisms (2):

Hom_{Λ_{X}} [(\underline{X}, \underline{C}), I_{𝔪}] ≅ Hom_{Λ_{X}} [T or_{1}^{Γ}(Tr(X), C), I_{𝔪}] ≅
≅ Ext_{Γ}^{1}[C, Hom_{Λ_{X}} (Tr(X), I_{𝔪})].

It follows that we have an isomorphism of functors

𝔻_{𝔪}(−, \underline{X})(?) ≅ Ext_{Γ}^{1}[?, Hom_{Λ_{X}} (Tr(X), I_{𝔪})] (3)

Now by Yoneda’s Lemma the morphism h_{(−,\underline{X})} : (−, \underline{X}) ⟶ 𝔻_{𝔪}(−, \underline{X}) constructed in Theorem 3.10 corresponds to an element Δ_{1} of 𝔻_{𝔪}(−, \underline{X})(\underline{X}). Under the isomorphism (3), Δ_{1} corresponds to an element Δ_{2} of Ext_{Γ}^{1}[X, Hom_{Λ_{X}} (Tr(X), I_{𝔪})]. Using the properties of h_{(−,\underline{X})} in Theorem 3.10, it is trivial to check that Δ_{2} represents an almost split sequence 0 ⟶ Hom_{Λ_{X}} [Tr(X), I_{𝔪}] ⟶ A ⟶ X ⟶ 0 in Mod(Γ).

Now let (E): 0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0 be a pure-exact sequence in Mod(Γ). Then (X, g): (X, B) ⟶ (X, C) is an epimorphism. This implies that (\underline{X}, \underline{g}): (\underline{X}, \underline{B}) ⟶ (\underline{X}, \underline{C}) is an epimorphism, hence [(\underline{X}, \underline{g}), I_{𝔪}] : [(\underline{X}, \underline{C}), I_{𝔪}] ⟶ [(\underline{X}, \underline{B}), I_{𝔪}] is a monomorphism. The pure-exact sequence (E) induces a long exact sequence

0 ⟶ (C, Hom_{Λ_{X}} [Tr(X), I_{𝔪}]) ⟶ (B, Hom_{Λ_{X}} [Tr(X), I_{𝔪}])
⟶^{f^{∗}} (A, Hom_{Λ_{X}} [Tr(X), I_{𝔪}]) ⟶ Ext_{Γ}^{1}(C, Hom_{Λ_{X}} [Tr(X), I_{𝔪}])

⟶^{g^{∗}} Ext_{Γ}^{1}(B, Hom_{Λ_{X}} [Tr(X), I_{𝔪}]) ⟶ ⋯

Using isomorphism (2), we have that g^{∗} is isomorphic to the morphism [(\underline{X}, \underline{g}), I_{𝔪}] which is a monomorphism; so f^{∗} is an epimorphism. This shows that any morphism A ⟶ Hom_{Λ_{X}} [Tr(X), I_{𝔪}] factors through f, i.e. Hom_{Λ_{X}} [Tr(X), I_{𝔪}] is pure-injective.

If Γ is an Artin algebra over a commutative Artin ring R, then we can choose as I_{𝔪} to be the injective envelope I of R/J ac(R). Then the functor D = Hom_{R}(−, I) is the usual duality of Artin algebras. It follows directly from the construction that there exists an isomorphism 𝔻_{I}(−, \underline{X})(?) ≅ D\underline{Hom}_{Γ}[X,?] ≅ Ext_{Γ}^{1}[?, DTr(X)].

Using Corollary 3.14 and the isomorphisms of the proof of Theorem 7.12, we have the following well-known basic result of M. Auslander.

Corollary 7.13. [4] Let X ∈ mod(Γ), let ρ: Λ ⟶ End_{Γ}(X) be a ring morphism and let I be an injective Λ^{op}−module. If 0 ⟶ A ⟶^{f} B ⟶^{g} C ⟶ 0 be a short exact sequence in Mod(Γ), then there exists an isomorphism

Hom_{Λ}[Coker(X, g), I] ≅ Coker[f, Hom_{Λ}(Tr(X), I)]

which is functorial with respect to short exact sequences. In particular, if the Λ^{op}−module I is an injective cogenerator, then: every morphism X ⟶ C factors through g iff every morphism A ⟶ Hom_{Λ}(Tr(X), I) factors through f.

Working stably as in Theorem 7.12, one can characterize along the lines of Theorem 7.9 the pure-injective modules occuring as a source of a left split morphism, see [16]. If mod(Γ) is a Krull-Schmidt category, one can characterize the pure-injective non-injective modules occuring as a source of an almost split sequence, see [33].

We leave to the reader to apply the theory of Sections 3, 4, 5 to the other examples of Abstract Homotopy Categories listed in Section 2, for instance to stable module categories (examples (iv), (v) of section 2). See also the recent paper by P. Jørgensen [29] for a discussion of phantom maps in the setting of stable categories. Finally note that one can obtain the existence of almost split morphisms and (relative) almost split sequences in a dualizing variety [9], and the existence of Serre-Grothendieck duality for derived categories of quasi-coherent sheaves over a smooth projective algebraic variety, using Brown Representability and the machinery of Abstract Homotopy Categories. Details are left to the reader.

Acknowledgements. The main part of this work was done during a stay at the University of Bielefeld in the summer of 1998. This stay was made possible by a post-doctoral scholarship from the TMR network “Algebraic Lie Representations”. The author expresses his gratitude to the members of the Representation Theory Group at Bielefeld, and especially to Prof. Dr. C. M. Ringel, for the warm hospitality and to Prof. H. Krause for the inspiring discussions. Finally the author thanks the referee for the useful comments.

References

1. J.F. Adams, A Variant of Brown’s Representability Theorem, Topology 10 (1971), 185–198.

2. I. Assem, A. Beligiannis and N. Marmaridis, Right Triangulated Categories with Right Semi-equivalebce, in: Algebras and Modules II (Geiranger 1996), CMS Conf. Proc. 24 Amer. Math. Soc. Providence, RI, (1998), 17–37.

3. M. Auslander, Representation Dimension of Artin Algberas, Queen Mary College Mathematical Notes, London, (1971).

4. M. Auslander, Functors and Morphisms Determined by Objects, in: Representation Theory of Algebras, Ed. R. Gordon, Dekker, New York (1978), 1–244.

5. M. Auslander, Representation Theory of Artin Algebras II, Comm. Algebra 2 (1974), 269–310.

6. M. Auslander and I. Reiten, Stable Equivalence of Dualizing R−Varieties, Advances in Math. (1974), 306–366.

7. M. Auslander, I. Reiten, Representation Theory of Artin Algebras III: Almost Split Sequences, Comm. Algebra 3 (1974), 239–294.

8. M. Auslander and S. Smalø, Preprojective Partitions over Artin Algebras, J. Algebra 66 (1980), 61–122.

9. M. Auslander and S. Smalø, Almost Split Sequences in Subcategories, J. Algebra 69 (1981), 426–454.

10. M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics 36, (1995).

11. A. Beligiannis, The Homological Theory of Contravariantly Finite Subcategories, Comm. Algebra 28 Issue 10, (2000).

12. A. Beligiannis, Relative Homological Algebra and Purity in Triangulated Categories, J. Algebra 227 (2000), 268–361.

13. A. Beligiannis, Homotopy Theory of Modules and Gorenstein Rings, Preprint, University of Bielefeld, (1998). To appear in Math. Scandinavica.

14. A. Beligiannis, On the Freyd Categories of an Additive Category, Preprint, University of Ioannina. To appear.

15. A. Beligiannis, Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings, Preprint, University of the Aegean, (2000).

16. W. Crawley- Boevey, Modules of Finite Length over their Endomorphism Ring, London Math. Soc. Lec. Note Series 168 (1992), 127–184.

17. W. Crawley- Boevey, Locally Finitely Presented Additive Categories, Comm. Algebra 22 (1994), 1644–1674.

18. E.H. Brown, Abstract Homotopy Theory, Trans. Amer. Math. Soc. 119 (1965), 79–85.

19. E.H. Brown and M. Comenetz, Pontrjagin Duality for Generalized Homology and Cohomology Theories, Amer. J. Math. 98 (1976), 1–27.

20. J.D. Christensen, Ideals in Triangulated Categories: Phantoms, Ghosts and Skeleta, Advances in Math. 136 (1998), 209–339.

21. J.D. Christensen and N. Strickland, Phantom Maps and Homology Theories, Topology 37 (1998), 339–364.

22. J.D. Christensen, B. Keller and A. Neeman, Failure of Brown Representability in Derived Categories, To appear in Topology.

23. P. Gabriel, Des Catégories Abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.

24. L. Gruson and C.U. Jensen, Dimensions Cohomologiques Relieés aux Foncteurs lim_{←}^{(i)}, in Lecture Notes in Math. 867 (1981), 234–294.

25. D. Happel, Triangulated Categories in the Representation Theory of Finite-Di– mensional Algebras, LMS Lecture Notes, vol 119, Cambridge University Press, 1988.

26. A. Heller, Completions in Abstract Homotopy Theory, Trans. A.M.S. 147 (1970), 573–602.

27. M. Hovey, J. Palmieri and N. Strickland, Axiomatic Stable Homotopy Theory, Memoirs Amer. Math. Soc., vol. 610 (1997).

28. C.U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach, New York, (1989).

29. P. Jørgensen, Phantom Maps of Modules, Preprint, Matematisk Institute Køben– havns Universitet, (1999).

30. B. Keller and D. Vossieck, Sous les Catégories Dérivées, C. R. Acad. Sci. Paris 305 Série I (1987), 225–228.

31. H. Krause, The Spectrum of a Locally Coherent Category, J. Pure Appl. Algebra 114 (1997), 259–271.

32. H. Krause, Exactly Definable Categories, J. Algebra 201 (1998), 456–492 .

33. H. Krause, Generic Modules over Artin Algebras, Proc. London Math. Soc. 76 (1998), 276–306.

34. H. Krause, Smashing Subcategories and the Telescope Conjecture-An Algebraic Approach, Invent. Math. 139 (2000), 99–133.

35. H. Krause, The Spectrum of a Module Category, to appear in Memoirs of AMS.

36. H. Krause, Brown Representability and Flat Covers, To appear in J. Pure and Appl. Algebra.

37. H. Krause, Auslander-Reiten Theory via Brown Representability, To appear in K-Theory.

38. M. Makkai and A.M. Pitts, Some Results on Locally Finitely Presentable Categories, Trans. Amer. Math. Soc. 299 (1987), 473–496.

39. H.R. Margolis, Spectra and the Steenrod Algebra, North Holland Math. Library 29 (1983).

40. A. Neeman, The Grothendieck Duality Theorem via Bousfield’s Techniques and Brown Representability, J. Amer. Math. Soc. 9 (1996), 205–236.

41. A. Neeman, On a Theorem of Brown and Adams, Topology 36 (1997), 619–645.

42. J. Rickard, Idempotent Modules in the Stable Category, J. London Math. Soc. 56 (1997), 149–170.

43. J.E. Roos, Locally Noetherian Categories and Strictly Linearly Compact Rings, in Lecture Notes in Math. 92, Berlin, Heidelberg, New York (1969), 197–277.

44. D. Simson, On Pure Global Dimension of Locally Finitely Presented Grothendieck categories, Fund. Math. 96 (1977), 91–116.

45. H. Schubert, Categories, Springer-Verlag, Berlin, Heidelberg, New York, (1972).

46. B. Stenstrom, Coherent Rings and FP-injective Modules, J. London. Math. Soc. 2 (1970), 323–329.

47. M. Ziegler, Model Theory of Modules, Ann. Pure and Appl. Logic 26 (1984), 149–213.

pureart.tex; 24/09/2000; 21:26; p.42