THE HOMOLOGICAL THEORY OF CONTRAVARIANTLY FINITE SUBCATEGORIES:

AUSLANDER-BUCHWEITZ CONTEXTS,

GORENSTEIN CATEGORIES AND (CO-)STABILIZATION

Apostolos Beligiannis

Instituto de Matemáticas, U.N.A.M., Mexico 04510, D.F., Mexico. Current Address: Department of Mathematics, University of the Aegean,

83200 Karlovassi, Samos, Greece. Email: abeligia@aegean.gr

1. Introduction

Let C be an abelian or exact category with enough projectives and let P be the full subcategory of projective objects of C. We consider the stable category C/P modulo projectives, as a left triangulated category [14], [36]. Then there is a triangulated category S(C/P) associated to C/P, which is universal in the following sense. There exists an exact functor S : C/P ⟶ S(C/P) such that any exact functor out of C/P to a triangulated category has a unique exact factorization through S. The triangulated category S(C/P) is called the stabilization of C/P and the functor S is called the stabilization functor. There is also the dual construction of the costabilization R(C/P) of C/P, which is a triangulated category equipped with an exact functor R : R(C/P) ⟶ C/P, the costabilization functor, such that any exact functor from a triangulated category to C/P has a unique exact factorization through R. If C has enough injectives we can stabilize and costabilize in the above sense the stable category modulo injectives. These constructions have topological origin and make sense for any stable category C/X, where now C is an additive category and X is a contravariantly or covariantly finite subcategory of C in the sense of Auslander-Smalø [8], assuming that C satisfies some mild condition. The stabilization construction in our setting is due to Heller [33], see also [24], [44], and later was used by Keller-Vossieck in [36]. For the costabilization construction we refer to the work of Grandis [27].

Our purpose in this paper is to investigate when the stabilization S(C/P) or the costabilization R(C/P) can be represented as a full (triangulated) subcategory T of C/P. In the first case we call the abelian or exact category C, P−Gorenstein, and in the second case we call the abelian or exact category C, P−Co-Gorenstein. In both cases T is realized by the stable category A/P of a specific resolving subcategory A of C. In general if A is any resolving subcategory of C, then our results are dealing with the relations between the coordinates of the triple (C, A, P) in the following three levels. First in the exact level, i.e. inside the exact category C, second in the stable level, i.e. inside the stable categories C/P, A/P, C/A and finally in the derived level, i.e. inside the derived categories D^{b}(C), D^{b}(A), D^{b}(P) and their Verdier-quotients D^{b}(C)/D^{b}(P), D^{b}(A)/D^{b}(P), D^{b}(C)/D^{b}(A).

We study P−(Co-)Gorenstein categories with respect to the above three levels, with a close view to applications in the module theory of an associative ring or an Artin algebra. The organization of the article is as follows.

In section 2, we study the relative homological algebra induced by a pair (C, X) consisting of an additive category C and a contravariantly finite subcategory X of C, assuming that any X −epic has kernel in C. Then the stable category C/X is left triangulated and we give necessary and sufficient conditions for C/X to contain a full triangulated subcategory. This suggests to introduce the concept of an X −Gorenstein object of C, which is a natural generalization of a module of zero Gorenstein dimension in the sense of Auslander-Bridger [3]. The full subcategory G_{X} (C) of X −Gorenstein objects of C is of central importance in this paper.

In section 3 we recall the construction of the stabilization and costabilization of a left triangulated category from [33], [36], and we compute them in the case of the left triangulated category C/X induced by the pair (C, X) mentioned above, in terms of complexes of objects of X. The representation of the stabilization of C/X by means of complexes generalizes (and is inspired by) a result of Keller-Vossieck [36]. In this section we introduce the important concept of a (Co-)Gorenstein left triangulated category, which will be used in the next sections.

In section 4 we introduce the concept of an X −Gorenstein category C, where C is an exact category and X is a contravariantly finite subcategory of C such that any X −epic in C is admissible in the sense of Quillen [40]. We prove that if C is X −Gorenstein, then the stabilization of the left triangulated category C/X is realized always by the stable category G_{X} (C)/X of the X −Gorenstein objects of C. The category G_{X} (C) can be thought of as a category of (relative) maximal Cohen-Macaulay objects, and using this category we define the notion of the X −Gorenstein dimension of an object of C. Then we prove that C is X −Gorenstein iff any object of C has finite X −Gorenstein dimension. In this case the category G_{X} (C) is contravariantly finite in C and we give sufficient conditions for the existence of minimal G_{X} (C)−approximations. The results of this section are related to the fundamental work of Auslander-Buchweitz [4] on maximal Cohen-Macaulay approximations in an abelian category. If C is X −Gorenstein, then all the results of Auslander-Buchweitz in [4] are valid in C and conversely if, roughly speaking, the Auslander-Buchweitz theory is true in C, then C is X −Gorenstein. The crucial points of the Auslander-Buchweitz theory have been recently formulated by M. Hashimoto in the concept of an Auslander-Buchweitz context or AB-context for short, which is a triple of full subcategories (A, B, X) of C satisfying certain properties [30]. We define AB-contexts relative to X and we prove that the exact category C is X −Gorenstein iff X is the base of a relative AB-context. We characterize also when the exact category C is X −Co-Gorenstein and we prove that in many cases any X −Gorenstein category is X −Co-Gorenstein.

Inspired by the definition of the stable homotopy groups in Algebraic Topology [24], we introduce in section 5 the concepts of complete X −extension functors and complete X −resolutions of objects of C, for the pair (C, X) mentioned above. The complete X −extension functors can be regarded as generalized Tate-Vogel cohomology functors and the complete X −resolutions as generalized Tate-Vogel resolutions. The main result of section 5 shows that C is X −Gorenstein iff any object has a complete X −resolution, and in this case we can compute the complete X −extension functors via complete X −resolutions. These results generalize the results of Gendrich-Gruenberg [25], Cornick-Kropholler [18], Mislin [39] and Avramov-Buchweitz-Martsinkovsky-Reiten [17] concerning complete projective or injective resolutions and complete extension functors.

The theory developed in sections 4, 5, indicate that the concept of a Gorenstein category which is defined using universal properties of stable categories, unifies the concepts: AB-context, global existence of complete resolutions, global existence of complete extension functors, finiteness of Gorenstein-dimension, and appears to be the natural setting for the study of stable phenomena in module theory.

In section 6 we apply our results to module categories. If Λ is an associative ring, then we denote by Mod(Λ), resp. mod(Λ), the category of all, resp. finitely presented, right Λ−modules and by P_{Λ}, resp, I_{Λ}, the full subcategory of projective, resp. injective, modules. Choosing C = Mod(Λ) and X = P_{Λ}, or X = I_{Λ}, we show that Mod(Λ) is P_{Λ}−Gorenstein iff Mod(Λ) is I_{Λ}−Gorenstein iff any projective right module has finite injective dimension and any injective right module has finite projective dimension. We call these rings right Gorenstein rings. It is easy to see that QF-rings or rings with finite right global dimension are right Gorenstein. It turns out that a Noetherian right Gorenstein ring is left Gorenstein and this class of rings coincides with the class introduced by Iwanaga [35] and studied by Enochs-Jenda et al., in a long series of papers (see for instance [22], [20]). The classical example of a Gorenstein ring is a local Noetherian ring of finite selfinjective dimension [3]. In case of Artin algebras this class of rings coincides with the class of Gorenstein algebras introduced by Auslander-Reiten [5] using tilting theory and studied also by Happel [32] using derived categories. Our theory has as corollaries the corresponding results of these papers, is valid for all modules not only finitely generated, and can be applied also to (Gorenstein) Orders. D. Happel considered in [32] a certain Verdier quotient D_{P} of the bounded derived category D^{b}(Λ) of an Artin algebra Λ, and he computed this quotient in case Λ is Gorenstein. He says that the computation of D_{P} is hard in general. It turns out that D_{P} is the stabilization of the stable category \underline{mod}(Λ) modulo projectives and an easy computation of D_{P} is available only in case Λ is Gorenstein or equivalently if Mod(Λ) is a P_{Λ}−Gorenstein category. We close the paper applying our previous results to the study of some of the homological conjectures for Artin algebras, and we prove some reductions.

Some of the results of this paper were obtained indepedently by L.L Avramov, R.O. Buchweitz, A. Martsinkovsky and I. Reiten [17]. A general convention used in the paper is that we compose morphisms in a category in the diagrammatic order: the composition of f : A ⟶ B and g : B ⟶ C is denoted by f ∘ g.

2. Relative Homology and Stable Categories

Troughout this section we fix a pair (C, X), where C is an additive category and X ⊆ C is a full additive subcategory of C which is closed under direct summands and isomorphisms. First we recall some notions of relative homological algebra extracted from [8], [15], [23]

A morphism f : A ⟶ B in C is called X −epic if the induced morphism C(X , f) : C(X , A) ⟶ C(X , B) in Ab is epic, and f is called X −monic if the induced morphism C(f, X) : C(B, X) ⟶ C(A, X) in Ab is epic. A morphism χ_{A} : X_{A} ⟶ A is a right X −approximation of A [8], if χ_{A} is an X −epic and X_{A} ∈ X. Dually a morphism χ^{A} : A ⟶ X^{A} is a left X −approximation of A if χ^{A} is an X −monic and X^{A} ∈ X. The subcategory X is contravariantly finite (covariantly finite) [8], if any object of C has a right (left) X −approximation. Finally X is functorially finite if X is covariantly and contravariantly finite in C.

Consider a complex A^{•} : ⋯ ⟶ A_{i+1} ⟶ A_{i} ⟶ A_{i−1} ⟶ ⋯ in C. The complex A^{•} is called covariantly X −exact, if the induced complex C(X , A^{•}) : ⋯ ⟶ C(X , A_{i+1}) ⟶ C(X , A_{i}) ⟶ C(X , A_{i−1}) ⟶ ⋯ is exact in Ab. Dually the complex A^{•} is contravariantly X −exact, if the induced complex C(A^{•}, X) : ⋯ ⟶ C(A_{i−1}, X) ⟶ C(A_{i}, X) ⟶ C(A_{i+1}, X) ⟶ ⋯ is exact in Ab. The complex A^{•} is functorially X −exact, if A^{•} is contravariantly X −exact and covariantly X −exact. Using these notions we can define X −resolutions and X −coresolutions of objects of C. If A ∈ C, then an X −resolution of A is a covariantly X −exact complex X_{A}^{•} : ⋯ ⟶ X_{1} ⟶ X_{0} ⟶ A ⟶ 0, where X_{n} ∈ X , ∀n ≥ 0. Then A has finite contravariant X −dimension if there exists an X −resolution of A of the form 0 ⟶ X_{n} ⟶ ⋯ ⟶ X_{0} ⟶ A ⟶ 0. In this case we write X −dimA ≤ n. The least such integer n is the contravariant X −dimension of A and is denoted by X −dimA. The global contravariant X −dimension of C is defined by X −gl.dimC := sup{X −dimA; A ∈ C}. Dually if A ∈ C, then an X −coresolution of A is a contravariantly X −exact complex X_{•}^{A} : 0 ⟶ A ⟶ X^{0} ⟶ X^{1} ⟶ ⋯, where X^{n} ∈ X , ∀n ≥ 0. The object A has finite covariant X −dimension if there exists an X −coresolution of A of the form 0 ⟶ A ⟶ X^{0} ⟶ X^{1} ⟶ ⋯ ⟶ X^{n} ⟶ 0. In this case we write X −co.dimA ≤ n. The covariant X −dimension, X −co.dimA, of A is the least such integer n. The global covariant X −dimension of C is defined by X −gl.co.dimC := sup{X −co.dimA; A ∈ C}.

The following is a direct consequence of the definitions.

Proposition 2.1. (1) X is coreflective in C iff X −gl.dimC = 0.

(2) X is reflective in C iff X −gl.co.dimC = 0.

We denote by C/X the stable category of C with respect to the subcategory X. We recall that the objects of C/X are the objects of C. If A, B are objects of C, then C/X (A, B) is the factor group C(A, B)/C_{X} (A, B), where C_{X} (A, B) is the subgroup of morphisms factorizing through an object of X. If A ∈ C, then we denote by \underline{A} the same object considered as an object of C/X, and if f : A ⟶ B is a morphism in C, then we denote by \underline{f} the residue class of f in C/X (A, B). Setting ϖ(A) = \underline{A} and ϖ(f) = \underline{f}, we obtain the additive projection functor ϖ : C ⟶ C/X.

2.1. Contravariant Finiteness. Suppose that X is contravariant finite and any X −epic has a kernel in C. Then any object in C has an X −resolution. Indeed let χ_{A} : X_{A} ⟶ A be a right X −approximation of A, let k_{A}^{1} : K_{A}^{1} ⟶ X_{A} be the kernel of χ_{A}, and let χ_{A}^{1} : X_{A}^{1} ⟶ K_{A}^{1} be a right X −approximation of K_{A}^{1}. Setting f_{A}^{1} = χ_{A}^{1} ∘ k_{A}^{1} : X_{A}^{1} ⟶ X_{A}^{0} and continuing in this way we obtain a complex

X_{A}^{•} ⋯ ⟶ X_{A}^{i+1} ⟶^{f_{A}^{i+1}} X_{A}^{i} ⟶ ⋯ ⟶ X_{A}^{1} ⟶^{f_{A}^{1}} X_{A}^{0} ⟶^{χ_{A}} A ⟶ 0

which is an X −resolution of A. A deleted X −resolution of A is an X −resolution as above with A deleted. The objects K_{A}^{n} are called the n^{th} − X −sysygy objects of A with respect to the X −resolution X_{A}^{•}.

Now let F : C ⟶ A and G : C^{op} ⟶ A be additive functors with values in an abelian category A. Then as in [15], we can define the left X −derived functor L_{n}^{X} F of F and the right X −derived functor R_{X}^{n} G of G:

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

as follows. If X_{A}^{•} is a deleted X −resolution of A, then L_{n}^{X} F (A) := H_{n}(F (X_{A}^{•})) and R_{X}^{n} G(A) := H^{n}(G(X_{A}^{•})). Then ∀B ∈ C, the contravariant X −extension functors

\underline{Ext}_{X}^{n} (−, B) : C^{op} ⟶ Ab, ∀n ≥ 0

are defined as the right X −derived functor of C(−, B). Similarly for any object C ∈ C, we have the left X −derived functors

L_{n}^{X} (C, −) : C ⟶ Ab, ∀n ≥ 0

defined as the left X −derived functors of C(C, −). In particular there are natural morphisms ϕ_{C,−} : L_{0}^{X} (C, −) ⟶ C(C, −) and ψ_{−,B} : C(−, B) ⟶ \underline{Ext}_{X}^{n} (−, B), ∀B, C ∈ C. Moreover if 0 ⟶ A_{1} ⟶ A_{2} ⟶ A_{3} ⟶ 0 is a contravariantly X −exact complex, then ∀B, C in C, we have long exact sequences:

0 ⟶ \underline{Ext}_{X}^{0} (A_{3}, B) ⟶ \underline{Ext}_{X}^{0} (A_{2}, B) ⟶ \underline{Ext}_{X}^{0} (A_{1}, B) ⟶ \underline{Ext}_{X}^{1} (A_{3}, B) ⟶ ⋯ (1)

⋯ ⟶ L_{1}^{X} (C, A_{3}) ⟶ L_{0}^{X} (C, A_{1}) ⟶ L_{0}^{X} (C, A_{2}) ⟶ L_{0}^{X} (C, A_{3}) ⟶ 0 (2)

Remark 2.2. If X is contravariantly finite and any X −epic has a kernel in C, then the stable category C/X has a natural left triangulated structure (C/X , Ω_{X} , Δ_{X}), where Ω_{X} : C/X ⟶ C/X is the loop functor and Δ_{X} is the triangulation. By construction Ω_{X} (\underline{A}) = \underline{K}_{A}^{1}, where K_{A}^{1} is the kernel of a right X −approximation of A and the triangulation Δ_{X} consists of all diagrams Ω_{X} (\underline{C}) ⟶ \underline{A} ⟶ \underline{B} ⟶ \underline{C} which are induced by X −exact sequences A ⟶^{g} B ⟶^{f} C in C, where g = ker(f). See [14] for details. We consider always the stable category C/X as a left triangulated category with the described left triangulation.

Lemma 2.3. For any object C ∈ C, there are isomorphisms:

L_{n}^{X} (C, −) ⟶^{≅} C/X (\underline{C}, Ω_{X}^{n+1}(−)), ∀n ≥ 1

and an exact sequence, with Im(ϕ_{C,−}) = C_{X} (C, −):

0 ⟶ C/X (\underline{C}, Ω_{X} (−)) ⟶ L_{0}^{X} (C, −) ⟶^{ϕ_{C,−}} C(C, −) ⟶ C/X (\underline{C}, −) ⟶ 0.

Proof. Consider the X −resolution X_{A}^{•} of A as above. Then by definition we have L_{1}^{X} (C, A) = KerC(C, f_{A}^{1})/ImC(C, f_{A}^{2}). Let a : C ⟶ X_{A}^{1} be a morphism with a∘f_{A}^{1} = 0. Then a∘χ_{A}^{1} ∘k_{A}^{1} = 0 ⇒ a∘χ_{A}^{1} = 0 ⇒ ∃!b : C ⟶ K_{A}^{2} such that b∘k_{A}^{2} = a. Define a morphism ρ : L_{1}^{X} (C, A) ⟶ C/X (\underline{C}, Ω_{X}^{2} (\underline{A})) by ρ(a) = \underline{b}. It is easy to see that ρ is a well-defined isomorphism. The general case follows by dimension shifting.

Now from the exact sequence C(C, X_{A}^{1}) ⟶^{C(C,f_{A}^{1})} C(C, X_{A}^{0}) ⟶^{c} L_{0}^{X} (C, A) ⟶ 0, there exists a unique morphism ϕ_{C,A} : L_{0}^{X} (C, A) ⟶ C(C, A) such that c∘ϕ_{C,A} = C(C, χ_{A}). The existence of the desired exact sequence follows by a simple diagram-chasing argument in the following diagram

C(C, X_{A}^{1}) ⟶^{C(C,f_{A}^{1})} C(C, X_{A}^{0}) ⟶^{c} L_{0}^{X} (C, A) ⟶ 0
∃ ! w↓  ^{||}↓  ^{ϕ_{C,A}}↓
0 ⟶ KerC(C, χ_{A}) ⟶ C(C, X_{A}^{0}) ⟶^{C(C,χ_{A})} C(C, A)

2.2. Covariant Finiteness. Suppose that X is covariant finite and any X −monic has a cokernel in C. Then any object in C has an X −coresolution (see [15]). Let F : C ⟶ A and G : C^{op} ⟶ A be additive functors with values in an abelian category A. Then as in subsection 2.1, we can define the X −derived functors of F, G:

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

as follows. If X_{•}^{A} is a deleted X −coresolution of A, then R_{X}^{n} F (A) := H^{n}(F (X_{•}^{A})) and L_{n}^{X} G(A) := H_{n}(G(X_{•}^{A})). Hence ∀B ∈ C, the covariant X −extension functors

\overline{Ext}_{X}^{n} (A, −) : C ⟶ Ab, ∀n ≥ 0

are defined as the right X −derived functor of C(A, −). Similarly for any object C ∈ C, we have the left X −derived functors of C(−, C) denoted by

L_{n}^{X} (−, C) : C^{op} ⟶ Ab, ∀n ≥ 0.

In particular there are natural morphisms ϕ_{−,C} : L_{n}^{X} (−, C) ⟶ C(−, C) and ψ_{A,−} : C(A, −) ⟶ \overline{Ext}_{X}^{n} (A, −), ∀A, C ∈ C. Moreover if 0 ⟶ B^{1} ⟶ B^{2} ⟶ B^{3} ⟶ 0 is a covariantly X −exact complex, then ∀A, C ∈ C, there are long exact sequences:

0 ⟶ \overline{Ext}_{X}^{0} (A, B^{1}) ⟶ \overline{Ext}_{X}^{0} (A, B^{2}) ⟶ \overline{Ext}_{X}^{0} (A, B^{3}) ⟶ \overline{Ext}_{X}^{1} (A, B^{1}) ⟶ ⋯ (3)

⋯ ⟶ L_{1}^{X} (B^{1}, C) ⟶ L_{0}^{X} (B^{3}, C) ⟶ L_{0}^{X} (B^{2}, C) ⟶ L_{0}^{X} (B^{1}, C) ⟶ 0 (4)

By [14], under the above assumptions the stable category C/X has a natural right triangulated structure (C/X , Σ_{X} , ∇_{X}), where Σ_{X} : C/X ⟶ C/X is the loop functor and ∇_{X} is the triangulation, see [14] for details. The dual of Lemma 2.3 also holds:

Lemma 2.4. For any object C ∈ C, there are isomorphisms:

L_{n}^{X} (−, C) ⟶^{≅} C/X (Σ_{X}^{n+1}(−), \underline{C}), ∀n ≥ 1

and an exact sequence, with Im(ϕ_{−,C}) = C_{X} (−, C):

0 ⟶ C/X (Σ_{X} (−), \underline{C}) ⟶ L_{0}^{X} (−, C) ⟶^{ϕ_{−,C}} C(−, C) ⟶ C/X (−, \underline{C}) ⟶ 0.

2.3. Functorial Finiteness. Suppose now that X is a functorially finite subcategory of C, any X −epic has a kernel and any X −monic has a cokernel in C. Then by the above observations, the stable category C/X admits a left traingulated structure (C/X , Ω_{X} , Δ_{X}) and a right triangulated structure (C/X , Σ_{X} , ∇_{X}). Moreover any object of C admits an X −resolution and an X −coresolution, and for any objects A, C ∈ C the derived functors L_{n}^{X} (−, C), L_{n}^{X} (C, −), \underline{Ext}_{X}^{n} (−, A), \overline{Ext}_{X}^{n} (A, −) are defined, ∀n ≥ 0. We note that in general the left and right triangulated structures in the stable category C/X are not the same and C/X is not necessarily triangulated. Indeed if Λ is an Artin algebra, then the full subcategory P_{Λ} of finitely generated projective right Λ−modules, satisfies all the above assumptions. Hence the stable category \underline{mod}(Λ) modulo projectives is left and right triangulated. But \underline{mod}(Λ) is triangulated iff Λ is selfinjective. However we have the following.

Proposition 2.5. The pair (Σ_{X} , Ω_{X}) is an adjoint pair in the stable category C/X. Hence there exist a natural isomorphism:

C/X [Σ_{X} (?), −] ⟶^{≅} C/X [?, Ω_{X} (−)].

Proof. Let A be an arbitrary object of C, and consider the X −exact sequence

0 ⟶ K_{A}^{1} ⟶^{k_{A}^{1}} X_{A}^{0} ⟶^{χ_{A}} A, where χ_{A} is a right X −approximation of A and k_{A}^{1} = ker(χ_{A}). Let χ^{K_{A}^{1}} : K_{A}^{1} ⟶ X^{K_{A}^{1}} be a left X −approximation of A with cokernel l_{1}^{K_{A}^{1}} : X^{K_{A}^{1}} ⟶ L_{1}^{K_{A}^{1}}. Then in C/X we have Ω_{X} (\underline{A}) = \underline{K}_{A}^{1} and Σ_{X} Ω_{X} (\underline{A}) = \underline{L}_{1}^{K_{A}^{1}}. Since χ^{K_{A}^{1}} is a left X −approximation, there exists a morphism m : X^{K_{A}^{1}} ⟶ X_{A}^{0} such that χ^{K_{A}^{1}} ∘ m = k_{A}^{1} and a commutative diagram:

K_{A}^{1} ⟶^{χ^{K{1 \atop A}}} X^{K_{A}^{1}} ⟶^{l_{1}^{K{1 \atop A}}} L_{1}^{K_{A}^{1}} ⟶ 0
||↓  ^{∃m}↓  ^{∃!e}↓
0 ⟶ K_{A}^{1} ⟶^{k_{A}^{1}} X_{A}^{0} ⟶^{χ_{A}} A.

In this way we obtain a morphism ε\underline{A} := \underline{e} : Σ_{X} Ω_{X} (\underline{A}) ⟶ \underline{A} in C/X. Dually we construct a morphism δ\underline{A} : \underline{A} ⟶ Ω_{X} Σ_{X} (\underline{A}) in C/X. We leave to the reader to check that ε is the counit and δ is the unit of an adjoint pair (Σ_{X} , Ω_{X}) in C/X.

Let Ω_{X}^{n} (C), Σ_{X}^{n} (C) be the full additive subcategories of C generated by X and the n^{th}−syzygies, n^{th}−cosyzygies, with respect to X −resolutions, X −coresolutions of objects of C. The next two results are direct consequences of Proposition 2.5.

Corollary 2.6. (1) Ω_{X}^{n} (C/X) is a reflective subcategory of C/X and Σ_{X}^{n} (C/X) is a coreflective subcategory of C/X, ∀n ≥ 0.

(2) Ω_{X}^{n} (C) is a covariantly finite subcategory of C and Σ_{X}^{n} (C/X) is a contravariantly finite subcategory of C/X, ∀n ≥ 0.

Corollary 2.7. ∀A, B ∈ C there are isomorphisms:

L_{n}^{X} (−, B)(A) ⟶^{≅} L_{n}^{X} (A, −)(B), ∀n ≥ 1.

The next result gives some sufficient conditions, for the coincidence of the contravariant and covariant X −extension functors. The proof is the same as in the classical case, using a simple spectral sequence argument.

Proposition 2.8. If \overline{Ext}_{X}^{n} (X , B) = \underline{Ext}_{X}^{n} (A, X) = 0, ∀n ≥ 1 and the canonical morphisms ψ_{A,X} : C(A, X) ⟶ \underline{Ext}_{X}^{0} (A, X), ψ_{X ,B} : C(X , B) ⟶ \overline{Ext}_{X}^{0} (X , B) are invertible, then: \overline{Ext}_{X}^{n} (A, −)(B) ≅ \underline{Ext}_{X}^{n} (−, B)(A), ∀n ≥ 0.

2.4. Stable Triangulated Categories. We close this section studying when a stable category is triangulated. For the theory of (left or right) triangulated categories and exact functors we refer to [14], [31], [36], [45].

Definition 2.9. The category X is called an X −cogenerator of C if for any A ∈ C, there exists an X −epic f : X ⟶ B with X ∈ X such that \underline{A} ≅ \underline{Ker(f)} in C/X.

Dually X is called an X −generator of C if for any B ∈ C, there exists an X −monic f : A ⟶ X with X ∈ X such that \underline{B} ≅ \underline{Coker(f)} in C/X.

If X is contravariantly finite in C and any X −epic has a kernel, the functors \underline{Ext}_{X}^{n} (−, A) are defined, and if X is covariantly finite in C and any X −monic has a cokernel, the functors \overline{Ext}_{X}^{n} (A, −) are defined, for any object A of C. In these cases we define the left X −orthogonal subcategory^{⊥}X and the right X −orthogonal subcategory X^{⊥} of X as follows:

⊥X = \{A ∈ C : \underline{Ext}_{X}^{n} (A, X) = 0, ∀n ≥ 1 and ψ_{A,X} : C(A, X) ⟶^{≅} \underline{Ext}_{X}^{0} (A, X)\},

X^{⊥} = \{B ∈ C : \overline{Ext}_{X}^{n} (X , B) = 0, ∀n ≥ 1 and ψ_{X ,B} : C(X , B) ⟶^{≅} \overline{Ext}_{X}^{0} (X , B)\}.

Definition 2.10. Let A, B, X be full subcategories of C.

(α) If any X −epic has a kernel in C, then A is called X −resolving, if X ⊆ A and A is closed under kernels of X −epics. Moreover if A ⟶^{g} B ⟶^{f} C is a diagram in C, where f is X −epic, g = ker(f) and A, C ∈ A, then B ∈ A.

(β) If any X −monic has a cokernel in C, then B is called X −coresolving, if X ⊆ B and B is closed under cokernels of X −monics. Moreover if A ⟶^{f} B ⟶^{g} C is a diagram in C, where f is X −monic, g = coker(f) and A, C ∈ B, then B ∈ B.

If X is contravariantly finite in C and any X −epic in C has a kernel, then for any X −resolving subcategory A, the stable category A/X is a full left triangulated subcategory of C/X. Dually if X is covariantly finite in C and any X −monic in C has a cokernel, then for any X −coresolving subcategory B, the stable category B/X is a full right triangulated subcategory of C/X [14]. For the notions of an exact category, admissible epic, monic, short exact sequence, we refer to [40].

Theorem 2.11. Let X be a full subcategory of an exact category C.

(1) Suppose that X is contravariantly finite in C and any X −epic is an admissible epic. If A is an X −resolving subcategory of C, the following are equivalent: (α) A/X is a triangulated subcategory of (C/X , Ω_{X} , Δ_{X}). (β) A ⊆ ^{⊥}X and X is an X −cogenerator of A.

(2) Suppose that X is covariantly finite in C and any X −monic is an admissible monic. If B is an X −coresolving subcategory of C, the following are equivalent: (α) B/X is a triangulated subcategory of (C/X , Σ_{X} , ∇_{X}). (β) B ⊆ X^{⊥} and X is an X −generator of B.

(3) If the assumptions in (1), (2) are true for A = B, then the triangulated structures (A/X , Ω_{X} , Δ_{X}) and (A/X , Σ_{X} , ∇_{X}) on A/X coincide: Ω_{X} = Σ_{X}^{−1} and

Ω_{X} (\underline{C}) ⟶^{\underline{h}} \underline{A} ⟶^{\underline{g}} \underline{B} ⟶^{\underline{f}} \underline{C} ∈ Δ_{X} ⇔ \underline{A} ⟶^{\underline{g}} \underline{B} ⟶^{\underline{f}} \underline{C} ⟶^{Σ_{X} (\underline{h})} Σ_{X} (\underline{A}) ∈ ∇_{X}.

Proof. (1) By the above remarks C/X carries a left triangulated structure and A/X is a full left triangulated subcategory of C/X. An easy modification of Theorem 3.3 of [1] in our setting, shows that the loop functor Ω_{X} is fully faithful in A/X iff A ⊆ ^{⊥}X. Trivially Ω_{X} is surjective on objects iff X is an X −cogenerator of A. (2) Follows by duality, and (3) follows from (1), (2) and Proposition 2.5.

Suppose again that X is a full subcategory of C. We define some classes of objects in C, which will play an important role in the next sections. First let P_{X}^{∞}(C) be the full subcategory of C of all objects of finite contravariant X −dimension.

Definition 2.12. (1) C ∈ C is called X −stable if C has a functorially X −exact resolution.

(2) C ∈ C is called X − n−torsion free object, n ≥ 0, if there exists a functorially X −exact sequence 0 ⟶ C ⟶ X^{0} ⟶ ⋯ ⟶ X^{n}, with X^{i} ∈ X, 0 ≤ i ≤ n.

(3) C ∈ C is called X −torsion free, if C is X − n−torsion free, ∀n ≥ 0, i.e. if C has a functorially X −exact coresolution.

(4) C ∈ C is called an X −Gorenstein object if C is X −stable and X −torsion free. Equivalently C has a functorially X −exact resolution and coresolution.

We denote by G_{X} (C) the full subcategory of C consisting of all X −Gorenstein objects. By definition X is a functorially finite subcategory of G_{X} (C). Observe that if X is contravariantly finite in C and any X −epic has a kernel, then C is X −stable iff \underline{Ext}_{X}^{i} (C, X) = 0, ∀i ≥ 1. Hence if any X −epic is epic, then C is X −stable iff C ∈ ^{⊥}X. The category of arbitrary X −syzygy objects is defined by Ω_{X}^{∞}(C) = ⋂_{n≥1} Ω_{X}^{n} (C). Dually if X is covariantly finite in C and any X −monic

has a cokernel, then C is X −torsion free iff \overline{Ext}_{X}^{i} (X , C) = 0, ∀i ≥ 1. Hence if any X −monic is monic, then C is X −torsion free iff C ∈ X^{⊥}. The category of arbitrary X −cosyzygy objects Σ_{X}^{∞}(C) is defined similarly. The final result of this section shows that in many cases, the category G_{X} (C) of X −Gorenstein objects is the largest X −resolving subcategory of C, such that the stable category G_{X} (C)/X is a full triangulated subcategory of C/X. First we recall [33] that an exact category E is called Frobenius, if E has enough projectives and injectives and the projectives coincide with the injectives.

Proposition 2.13. Let C be an exact category and X a full subcategory of C.

(1) If X is contravariantly finite, any X −epic is an admissible epic and any left X −approximation of an X -Gorenstein object is an admissible monic, then G_{X} (C) is an X −resolving Frobenius exact subcategory of C and the stable category G_{X} (C)/X is a full triangulated subcategory of (C/X , Ω_{X} , Δ_{X}).

Moreover if A is a subcategory of C, such that A/X is a triangulated subcategory of C/X, then A is X −resolving and A ⊆ G_{X} (C).

(2) If X is covariantly finite, any X −monic is an admissible monic and any right X −approximation of an X -Gorenstein object is an admissible epic, then G_{X} (C) is an X −coresolving Frobenius exact subcategory of C and the stable category G_{X} (C)/X is a full triangulated subcategory of (C/X , Σ_{X} , ∇_{X}).

Moreover if B is a subcategory of C, such that B/X is a triangulated subcategory of C/X, then B is X −coresolving and B ⊆ G_{X} (C).

Proof. (1) Since any X −epic is admissible epic, it has a kernel in C so C/X is left triangulated. Since any X −epic is epic, we have C(−, B) = \underline{Ext}_{X}^{0} (−, B). If A is X −Gorenstein, then consider a covariantly X −exact X −coresolution 0 ⟶ A ⟶^{α_{0}} X^{0} ⟶^{α_{1}} X^{1} ⟶ ⋯ of A. Since this coresolution is contravariantly X −exact, α_{0} is a left X −approximation, so by hypothesis there exists an admisssible exact sequence 0 ⟶ A ⟶^{α_{0}} X^{0} ⟶^{κ} A^{1} ⟶ 0, and a factorization α_{1} = κ ∘ λ, where λ : A^{1} ⟶ X^{1}. Similarly λ is a left X −approximation of A^{1}, so it is admissible monic. Inductively we see easily that the objects A^{n} are X −Gorenstein and the above X −coresolution of A is a Yoneda composition of admissible and functorially X −exact sequences

0 ⟶ A^{n} ⟶ X^{n} ⟶ A^{n+1} ⟶ 0, A^{0} = A. Suppose now that (†) : A ⟶^{g} B ⟶^{f} C

is a sequence in C with A, C ∈ G_{X} (C), f is an X −epic and g = ker(f), so (†) is admissible and covariantly X −exact. Then we have the long exact sequence: 0 ⟶ C(C, X) ⟶ C(B, X) ⟶ C(A, X) ⟶ \underline{Ext}_{X}^{1} (C, X) ⟶ ⋯ which implies that B ∈ ⊥X and (†) is contravariantly X −exact, since A, C are X −Gorenstein. Moreover using that A, C ∈ G_{X} (C), we can construct by standard arguments a covariantly X −exact X −coresolution of B, from the covariantly X −exact X −coresolutions of A, C. This shows that B ∈ G_{X} (C). Let (†) : A ⟶^{g} B ⟶^{f} C be a sequence in C with B, C ∈ G_{X} (C), f is an X −epic and g = ker(f), so (†) is admissible and covariantly X −exact. As above we see that A ∈ ^{⊥}X and (†) is contravariantly X −exact. Consider the functorially X −exact admissible sequence 0 ⟶ B ⟶^{α} X^{0} ⟶^{β} B^{1} ⟶ 0 which starts a covariantly X −exact X −coresolution of B. Then we have the following exact commutative diagram:

0 ⟶ A ⟶^{g} B ⟶^{f} C ⟶ 0
||↓  ^{α}↓  ^{δ}↓
0 ⟶ A ⟶^{ϕ} X^{0} ⟶^{ψ} D ⟶ 0

and an admissible covariantly X −exact sequence 0 ⟶ C ⟶^{δ} D ⟶^{ϵ} B^{1} ⟶ 0. Since C, B^{1} ∈ G_{X} (C), we have that D ∈ G_{X} (C). Since, as easily seen, the sequence 0 ⟶ A ⟶^{ϕ} X^{0} ⟶^{ψ} D ⟶ 0 is functorially X −exact and D is an X −Gorenstein object, we infer that A is an X −Gorenstein object. Hence G_{X} (C) is an X −resolving subcategory of C. We leave to the reader to show that G_{X} (C) is actually a Frobenius subcategory of C. By Theorem 2.11, G_{X} (C)/X is a full triangulated subcategory of C/X, since by definition X is an X −cogenerator of G_{X} (C) and G_{X} (C) ⊆ ^{⊥}X.

If A ⊆ C is such that A/X is a triangulated subcategory of C/X, then it is not difficult to see that A is X −resolving using that the inclusion A/X ↪ C/X is exact. Then by Theorem 2.11, A ⊆ ^{⊥}X, so any object of A has a contravariantly X −exact resolution. Since X is a cogenerator of A, any object of A has a covariantly X −exact coresolution. We conclude that A ⊆ G_{X} (C). The proof of part (2) is dual.

3. Stable Categories and (Co)Stabilization

In this section we associate to a fixed left triangulated category C two triangulated categories: S(C), the stabilization of C and R(C), the costabilization of C. The category S(C) is the universal triangulated category for exact functors starting at C and R(C) is the universal triangulated category for exact functors ending at C. The existence of S(C) is a result of Heller [33], see also [24], [36], and the existence of R(C) is due to Grandis [27]. Both existence results were inspired by well-known constructions in Algebraic Topology [38], and they are very useful tools for the study of stable categories. We define a looped category to be a pair (A, Ω) where A is an additive category and Ω : A ⟶ A is an additive functor. If (A, Ω), (B, Σ) are looped categories, then a stable functor (A, Ω) ⟶ (B, Σ) is a pair (F, ϕ) where F : A ⟶ B is an additive functor and ϕ : ΣF ⟶^{≅} F Ω is a natural isomorphism.

3.1. Stabilization. Let C = (C, Ω, Δ) be a left triangulated category.

Definition 3.1. The stabilization of C is a pair (S, S(C)), where S(C) is a triangulated category and S : C ⟶ S(C) is an exact functor, the stabilization functor, such that for any exact functor F : C ⟶ D to a triangulated category D, there exists a unique exact functor F^{∗} : S(C) ⟶ D such that: F^{∗}S = F.

We recall the construction of S(C) from [33], which consists of formally inverting the endofunctor Ω. An object of S(C) is a pair (A, n) where A ∈ C and n ∈ ℤ. If n, m ∈ ℤ, then we consider the directed set I_{n,m} = {k ∈ ℤ : k ≥ n, k ≥ m}. The space of morphisms between (A, n), (B, m) ∈ S(C) is defined by

S(C)[(A, n), (B, m)] = lim_{⟶}_{k∈I_{n,m}}C(Ω^{k−n}(A), Ω^{k−m}(B)).

Then S(C) is an additive category and there exists an equivalence Ω̃ : S(C) ⟶ S(C) defined as follows: Ω̃(A, n) = (A, n − 1) and if f̃ : (A, n) ⟶ (B, m) then choose a representative f_{k} : Ω^{k−n}(A) ⟶ Ω^{k−m}(B) where k ∈ I_{n,m} and define Ω̃(f) to be the class of f_{k−1} in S(C)[(A, n − 1), (B, m − 1)]. The inverse of Ω̃ is defined by Ω̃^{−1}(A, n) = (A, n + 1). There exists a natural additive functor S : C ⟶ S(C) defined as follows: S(A) = (A, 0), and if f : A ⟶ B is a morphism in C then S(f) is defined by the zero-representative of f. The functor S is a stable functor, i.e. there exists a natural isomorphism ω : Ω̃S(?) = (?, −1) ≅ (Ω(?), 0) = SΩ(?), and the pair (S, S(C)) has the following universal property. If (D, Σ) is a looped category with Σ a self equivalence of D, and F : C ⟶ D is a stable functor, then there exists a unique stable functor F^{∗} : S(C) ⟶ D such that F^{∗}S = F. Indeed this follows directly by defining F^{∗}(A, n) = Σ^{−n}F (A).

Using the functor S : C ⟶ S(C) and the triangulation Δ of C, we define a triangulation Δ̃ of the pair (S(C), Ω̃) as follows. A diagram Ω̃(C, l) ⟶ (A, n) ⟶ (B, m) ⟶ (C, l) belongs to Δ̃ if there exists k ∈ 2ℤ and a triangle of representatives Ω(Ω^{k−l}(C)) ⟶ Ω^{k−n}(A) ⟶ Ω^{k−m}(B) ⟶ Ω^{k−l}(C) in C. Then the triple (S(C), Ω̃, Δ̃) is a triangulated category and has the required universal property of definition 3.1. In fact the functor S is exact and if F : C ⟶ D is an exact functor to a triangulated category D, then the functor F^{∗} defined above, is the unique exact functor which extends F through the stabilization functor S.

Remark 3.2. In case C has a right triangulated structure (C, Σ, ∇), then the above construction works, producing the stabilization S(C) of C which also is triangulated and satisfies the same universal property. The stabilization has the same objects as before but the space of morphisms is defined by the formula

S(C)[(A, n), (B, m)] = lim_{⟶}_{k∈J_{n,m}}C(Σ^{k+n}(A), Σ^{k+m}(B))

where J_{n,m} = {k ∈ ℤ : k + n, k + m ≥ 0}. In this case we invert the suspension functor Σ to obtain the equivalence Σ̃, where Σ̃(A, n) = (A, n + 1). As above the triangulation ∇ of C induces a triangulation ˜∇ in the stabilization.

The following is a direct consequence of the construction.

Corollary 3.3. (1) C is triangulated iff S : C ⟶^{≈} S(C) is a triangle equivalence.

(2) S(C) = 0 iff ∀A ∈ C, there exists n_{A} ≥ 0, such that: Ω^{n_{A}}(A) = 0.

(3) Consider a morphism f̃ : (A, n) ⟶ (B, m) in S(C). Then f̃ = 0 iff there exists a representative f_{k} : Ω^{k−n}(A) ⟶ Ω^{k−m}(B) of f̃ and l ≥ k, such that: Ω^{l−k}(f_{k}) = 0. Also f̃ is an isomorphism iff there exists a representative f_{k} : Ω^{k−n}(A) ⟶ Ω^{k−m}(B) of f̃ and l ≥ k, such that: Ω^{l−k}(f_{k}) is an isomorphism.

(4) If C, D are triangle equivalent, then S(C), S(D) are triangle equivalent.

The next result is useful studying when an exact functor C ⟶ D to a triangulated category D extends to a triangle equivalence F^{∗} : S(C) ⟶ D. For a proof see [44].

Proposition 3.4. Let F : C ⟶ D be an exact functor to a triangulated category D with translation functor Σ, and let F^{∗} : S(C) ⟶ D be the unique exact functor extending F through the stabilization functor. Then:

(1) F^{∗} is faithful iff for any morphism f : A ⟶ B in C such that F (f) = 0, there exists k ≥ 0 such that: Ω^{k}(f) = 0.

(2) F^{∗} is full iff for any morphism g : F (A) ⟶ F (B) in D, there exists k ≥ 0 and a morphism f : Ω^{k}(A) ⟶ Ω^{k}(B), such that: Σ^{k}(g) = F (f).

(3) F^{∗} is surjective on objects iff for any object D ∈ D, there exists k ≥ 0 and an object A ∈ C, such that: Σ^{k}(D) = F (A).

Corollary 3.5. The functor S : C ⟶ S(C) is faithful (full, resp. dense) iff Ω is faithful (full, resp. dense). In case Ω is fully faithful then S(C) is the smallest triangulated category containing C as a full left triangulated subcategory.

Example 3.6. Let U be an additive category and let K(U) be the (triangulated) homotopy category of complexes over U. Let K^{0]}(U), K^{b,0]}(U) be the full subcategories of negative, negative and bounded below complexes respectively. These categories are right triangulated subcategories of K(U). A simple application of Proposition 3.4 shows that S(K^{0]}(U)) = K^{−}(U) and S(K^{b,0]}(U)) = K^{b}(U). Similar remarks are applied to the derived category, in case U is abelian or exact.

The next result shows that the Grothendieck group [15] is invariant under stabilization.

Proposition 3.7. The stabilization functor S : C ⟶ S(C) induces an isomorphism

K_{0}(S) : K_{0}(C) ⟶^{≅} K_{0}(S(C)).

Proof. Since S is exact, we have the induced morphism K_{0}(S) which is defined by K_{0}(S)[A] = [(A, 0)]. Consider the function ψ : Ob(S(C)) ⟶ K_{0}(C) defined by ψ(A, n) = (−1)^{n}[A]. If (A, n) ≅ (B, m) then there exists k ∈ I_{n,m} and an isomorphism Ω^{k−n}(A) ≅ Ω^{k−m}(B). Then in K_{0}(C) we have (−1)^{k−n}[A] = [Ω^{k−n}(A)] = [Ω^{k−m}(B)] = (−1)^{k−m}[B], hence ψ(A, n) = ψ(B, m). If Ω̃(C, l) ⟶ (A, n) ⟶ (B, m) ⟶ (C, l) is a triangle in S(C), then by definition there exists a triangle Ω(Ω^{k−l}(C)) ⟶ Ω^{k−n}(A) ⟶ Ω^{k−m}(B) ⟶ Ω^{k−l}(C) in C. Then in K_{0}(C) we have (−1)^{k−m}[B] = (−1)^{k−n}[A] + (−1)^{k−l}[C]. This implies that ψ(B, m) = ψ(A, n) + ψ(C, l). Hence there exists a unique group homomorphism ϕ : K_{0}(S(C)) ⟶ K_{0}(C), such that ϕ([A, n]) = ψ([(A, n)]) = (−1)^{n}[A]. If A is an object of C, then ϕK_{0}(S)([A]) = ϕ([(A, 0)]) = [A] and if (A, n) is an object of S(C), then K_{0}(S)ϕ([(A, n)]) = K_{0}(S)((−1)^{n}[A]) = (−1)^{n}[(A, 0)]. But (−1)^{n}[(A, 0)] = (−1)^{−n}[(A, 0)] = [Ω̃^{−n}(A, 0)] = [(A, n)]. So K_{0}(S)ϕ([(A, n)]) = [(A, n)]. This shows that K_{0}(S) is an isomorphism with inverse ϕ.

Consider now an additive category C and a full additive contravariantly finite subcategory X ⊆ C, closed under direct summands and suppose throughout that any X −epic has a kernel in C. Our purpose is to compute the stabilization S(C/X) of the left triangulated category C/X = (C/X , Ω_{X} , Δ_{X}). Let K(X) be the unbounded homotopy category of complexes over X and let K^{−}(X) be the full subcategory consisting of bounded above complexes, where a complex

X^{•} : ⋯ ⟶ X^{n−1} ⟶^{d^{n−1}} X^{n} ⟶^{d^{n}} X^{n+1} ⟶ ⋯ is bounded above if X^{k} = 0 for all

sufficiently large k > 0. We call a complex X^{•} n−acyclic, if the morphism d^{n−1}

admits a factorization X^{n−1} ⟶^{ϵ^{n−1}} Ker(d^{n}) ⟶^{ker(d^{n})} X^{n}, where ϵ^{n−1} is X −epic. The complex X^{•} is called acyclic, if it is n−acyclic, for any n ∈ ℤ. Let K^{−,b}(X) be the full subcategory of K^{−}(X) consisting of all complexes which are acyclic almost everywhere, i.e. except of a finite number of degrees. Then we have exact inclusions of triangulated categories K^{b}(X) ↪ K^{−,b}(X) ↪ K^{−}(X) ↪ K(X), where K^{b}(X) is the bounded homotopy category of complexes over X. Obviously K^{b}(X) is closed under direct summands in K^{−,b}(X), hence the Verdier quotient K^{−,b}(X)/K^{b}(X) is defined and it is a triangulated category.

The following result generalizes (and is inspired by) a result of Keller-Vossieck (see [36], where the next result is proved for an exact category with enough injectives objects).

Theorem 3.8. There exists a triangle equivalence

S(C/X) ≅ K^{−,b}(X)/K^{b}(X).

Moreover KerS = P_{X}^{∞}(C)/X and S(C/X) = 0 iff ∀C ∈ C: X −dimC < ∞, i.e. C = P_{X}^{∞}(C). Finally KerS = 0 iff X = P_{X}^{∞}(C) iff sup{X −dimC : C ∈ P_{X}^{∞}(C)} = 0.

Proof. Define a functor F : C ⟶ K^{−,b}(X)/K^{b}(X) as follows. If A is an object of C, let X_{A}^{•} be a deleted X −resolution of A as in section 2. Hence X_{A}^{•} : ⋯ ⟶

X_{A}^{i+1} ⟶^{f_{A}^{i+1}} X_{A}^{i} ⟶ ⋯ ⟶ X_{A}^{1} ⟶^{f_{A}^{1}} X_{A}^{0} ⟶ 0, where X^{i} is in degree −i. We

set F (A) = Q(X_{A}^{•}), where Q : K^{−,b}(X) ⟶ K^{−,b}(X)/K^{b}(X) is the quotient functor. Since any two X −resolutions are homotopy equivalent, the functor F is well-defined. Obviously F (X) = 0. Moreover if C ⟶^{g} B ⟶^{f} A is a contravariantly X −exact sequence in C, then there exists a sequence 0 ⟶ X_{C}^{•} ⟶ X_{B}^{•} ⟶ X_{A}^{•} ⟶ 0 of complexes which is split short exact in each degree. Hence we obtain a triangle X_{A}^{•}[−1] ⟶ X_{C}^{•} ⟶ X_{B}^{•} ⟶ X_{A}^{•} in K^{−,b}(X). Applying the exact quotient functor Q, we see that the contravariantly X −exact sequence C ⟶ B ⟶ A in C induces a triangle F (A)[−1] ⟶ F (C) ⟶ F (B) ⟶ F (A) in K^{−,b}(X)/K^{b}(X). By Theorem 2.2 of [12], there exists a unique exact functor G : C/X ⟶ K^{−,b}(X)/K^{b}(X), such that Gϖ = F, where ϖ : C ⟶ C/X is the projection functor. By the universal property of the stabilization S(C/X), there exists a unique exact functor G^{∗} : S(C/X) ⟶ K^{−,b}(X)/K^{−}(X), such that G^{∗}S = G. We claim that G^{∗} is an equivalence. This is easy to see applying Proposition 3.4 and using the definition of the functor G, and the construction of the quotient K^{−,b}(X)/K^{b}(X) in [45]. The last part is trivial.

Corollary 3.9. Let C be an abelian (or exact) category with enough projectives, and let P be the full subcategory of projective objects of C.

(1) There exists a triangle equivalence

S(C/P) ≈ D^{b}(C)/K^{b}(P).

S(C/P) = 0 iff D^{b}(C) = K^{b}(P) iff any object of C has finite projective dimension.

(2) There exists an isomorphism: K_{0}(C/P) ≅ K_{0}(D^{b}(C)/K^{b}(P)) such that the canonical morphisms K_{0}(P, ⊕) ⟶ K_{0}(K^{b}(P)) and K_{0}(C) ⟶ K_{0}(D^{b}(C)) are embedded in the exact commutative diagram, where c_{C}, c_{C}^{•} are the Cartan morphisms:

Ker(c_{C}) ⟶ K_{0}(P, ⊕) ⟶^{c_{C}} K_{0}(C) ⟶  K_{0}(C/P)  ⟶ 0
≅↓  ≅↓  ≅↓  ≅↓
Ker(c_{C}^{•}) ⟶ K_{0}(K^{b}(P)) ⟶^{c_{C}^{•}} K_{0}(D^{b}(C)) ⟶ K_{0}(D^{b}(C)/K^{b}(P)) ⟶ 0

(3) C is Frobenius iff there exists a triangle equivalence:

C/P ≈ D^{b}(C)/K^{b}(P).

Proof. (1) Follows from Thorem 3.8, since if X = P, then K^{−,b}(X) = D^{b}(C).

(2) By a well known result of Grothendieck, the central square of the above diagram commutes and the middle vertical arrows are invertible. Since the cokernel of c_{C} is the stable Grothendieck of C modulo projectives [15], the result follows from Proposition 3.7, [16] and part (1).

(3) It is well-known that if C is Frobenius, then C/P is triangulated (see [31] or Theorem 2.11). Hence in this case C/P is triangle equivalent to its stabilization, and the result follows from (1). If C/P is triangle equivalent to D^{b}(C)/K^{b}(P), then C/P is triangulated. Then by section 2, we know that C = ^{⊥}P and P is a P−cogenerator of C. The fact that C = ^{⊥}P implies that any projective is injective. Since P is a P−cogenerator of C, for any object of C, there exists a short exact sequence 0 ⟶ C ⟶ P ⟶ D ⟶ 0 with P ∈ P. In particular C has enough injectives and trivially any injective is projective. Hence C is Frobenius.

3.2. Costabilization. Let C = (C, Ω, Δ) be a left triangulated category.

Definition 3.10. The costabilization of C is a pair (R, R(C)), where R(C) is a triangulated category and R : R(C) ⟶ C is an exact functor, the costabilization functor, such that for any exact functor F : D ⟶ C from a triangulated category D, there exists a unique exact functor F^{∗} : D ⟶ R(C) such that: RF^{∗} = F.

We recall the construction of the pair (R, R(C)) from [27], which consists of constructing formal Ω−spectra. An object of R(C) is a family (A_{n}, α_{n}) where n ∈ ℤ and α_{n} : A_{n} ⟶^{≅} Ω(A_{n+1}) is an isomorphism ∀n ∈ ℤ. A morphism f_{•} : (A_{n}, α_{n}) ⟶ (B_{n}, β_{n}) is a family f_{•} = (f_{n}), where f_{n} : A_{n} ⟶ B_{n} is a morphism in C such that the following diagram commutes, ∀n ∈ ℤ:

A_{n}  ⟶^{f_{n}}  B_{n}
α_{n}↓  ^{β_{n}}↓
Ω(A_{n+1}) ⟶^{Ω(f_{n+1})} Ω(B_{n+1}).

Then R(C) is an additive category and defining Ω̂(A_{n}, α_{n}) = (B_{n}, β_{n}) where B_{n} = A_{n−1} and β_{n} = α_{n−1} and for a morphism f_{•} : (A_{n}, α_{n}) ⟶ (B_{n}, β_{n}) in R(C), Ω̂(f_{•}) = g_{•}, where g_{n} = f_{n−1}, we obtain an equivalence Ω̂ : R(C) ⟶ R(C). Hence the pair (R(C), Ω̂) is a looped category. Now define an additive functor R : R(C) ⟶ C by R(A_{n}, α_{n}) = A_{0} and R(f_{•}) = f_{0}. The functor R is stable since RΩ̂(A_{n}, α_{n}) = A_{−1} ≅ Ω(A_{0}) = ΩR(A_{n}, α_{n}), by the isomorphism α_{−1}. The pair (R, R(C)) has the following universal property. If (D, Σ) is a looped category where Σ is a selfequivalence of D, and F : D ⟶ C is a stable functor, then there exists a unique stable functor F^{∗} : D ⟶ R(C) with RF^{∗} = F. Indeed define F^{∗}(D) = (D_{n}, d_{n}) where D_{n} = F Σ^{−n}(D) and d_{n} is the isomorphism D_{n} = F Σ^{−n}(D) = F ΣΣ^{−n−1}(D) ≅ ΩF Σ^{−n−1}(D) = Ω(D_{n+1}). Then RF^{∗}(D) = D_{0} = F (D), and F^{∗} is obviously the unique stable functor lifting F through the costabilization functor R.

Using the functor R : R(C) ⟶ C and the triangulation Δ of C, we define a trian-

gulation Δ̂ in the looped category (R(C), Ω̂) as follows. A diagram Ω̂(C_{n}, γ_{n}) ⟶^{(h_{n})}

(A_{n}, α_{n}) ⟶^{(g_{n})} (B_{n}, β_{n}) ⟶^{(f_{n})} (C_{n}, γ_{n}) belongs to Δ̂ if for any n ∈ ℤ there are tri-

angles Ω(C_{n}) ⟶^{γ_{n−1}^{−1}∘h_{n}} A_{n} ⟶^{g_{n}} B_{n} ⟶^{f_{n}} C_{n} in Δ. Then the triple (R(C), Ω̂, Δ̂) is a triangulated category, the functor R : R(C) ⟶ C is exact and if D is a triangulated category with translation functor Σ, and F : D ⟶ C is an exact functor, then the functor F^{∗} defined above is the unique exact functor lifting F through R. In case C has a right triangulation, then the above construction with the necessary modifications also works, producing the costabilization of C which also is triangulated and satisfies the same universal property. We leave to the reader to state and prove the analogous results of Corollaries 3.3, 3.5 and Proposition 3.4.

Consider now the pair (C, X) where C is an additive category and X ⊆ C is a full additive contravariantly finite subcategory of C, closed under direct summands and suppose that any X −epic has a kernel in C. Our purpose is to compute the costabilization R(C/X) of the left triangulated category C/X = (C/X , Ω_{X} , Δ_{X}). Let K(X) be the unbounded homotopy category over X and let K_{Ac}(X) be the full subcategory of acyclic complexes, as defined in Subsection 3.1. Clearly K_{Ac}(X) is a full triangulated subcategory of K(X).

Theorem 3.11. There exists a triangle equivalence:

K_{Ac}(X) ≈ R(C/X).

Moreover Im(R) = Ω_{X}^{∞}(C)/X. Hence: Im(R) = 0 iff R(C/X) = 0 iff X = Ω_{X}^{∞}(C).

Proof. Define a functor F : K_{Ac}(X) ⟶ R(C/X) as follows. Let X^{•} be an acyclic complex of X −objects. Then by definition, ∀n ∈ ℤ, there are sequences A_{n−1} ⟶^{μ_{n−1}} X^{n} ⟶^{ε_{n}} A_{n}, where ε_{n} is X −epic and μ_{n−1} = ker(ε_{n}), such that the differential d^{n} = ε_{n} ∘μ_{n} : X^{n} ⟶ X^{n+1}. We set F (X^{•}) = (\underline{A}_{n}, \underline{α}_{n}), where \underline{α}_{n} : \underline{A}_{n} ≅ Ω_{X} (\underline{A}_{n+1}) is the natural identification. Let f_{•} : X^{•} ⟶ Y^{•} be a morphism, and let B_{n−1} ⟶^{ν_{n−1}} Y^{n} ⟶^{ζ_{n}} B_{n} be sequences as above, where ζ_{n} is X −epic and ν_{n−1} = ker(ζ_{n}), such that the differential d^{n} = ζ_{n} ∘ ν_{n} : Y^{n} ⟶ Y^{n+1}. Since the morphisms μ_{t−1}, ν_{t−1} are monics, we have that μ_{t−1} : A_{t−1} ⟶ X^{t} is the kernel of d_{X^{•}}^{t} : X^{t} ⟶ X^{t+1} and similarly ν_{t−1} : B_{t−1} ⟶ Y^{t+1} is the kernel of d_{Y^{•}}^{t} : Y^{t} ⟶ Y^{t+1}. Hence there exists a unique morphism ρ_{t−1} : A_{t−1} ⟶ B_{t−1} with ρ_{t−1} ∘ ν_{t−1} = μ_{t−1} ∘ f_{t}. The family of morphisms ρ_{t} : A_{t} ⟶ B_{t} has obviously the property ρ_{t} ∘ν_{t} = μ_{t} ∘f_{t+1} and f_{t} ∘ ζ_{t} = ε_{t} ∘ ρ_{t}, ∀t ∈ ℤ. This means that \underline{α}_{t} ∘ Ω_{X} (\underline{ρ}_{t+1}) = \underline{ρ}_{t} ∘ \underline{β}_{t}, ∀t ∈ ℤ. Hence the family ρ_{•} : (\underline{A}_{t}, \underline{α}_{t}) ⟶ (\underline{B}_{t}, \underline{β}_{t}) is a morphism in R(C/X). We set F (f_{•}) = ρ_{•}. It is easy to see that in this way we obtain an exact functor F : K_{Ac}(X) ⟶ R(C/X). We leave to the reader the easy proof that F is an equivalence. The proof of the last assertion is trivial.

Corollary 3.12. Let C be an exact category with enough projectives, and let P be the full subcategory of projective objects. Then there is a triangle equivalence

R(C/P) ≈ K_{Ac}(P)

and R(C/P) = 0 iff P = Ω^{∞}(C), i.e. the only arbitrary syzygy objects of C are the projectives. In particular C is Frobenius iff there exists a triangle equivalence

C/P ≈ K_{Ac}(P).

We leave to the reader to state and prove the dual results concerning (co-)stabili-zations of right triangulated stable categories C/X, induced by covariantly finite subcategories X in C, such that any X −monic has a cokernel in C. For example stable categories modulo injectives of exact categories with enough injectives.

3.3. (Co-)Gorenstein Left Triangulated Categories. Throughout this subsection C will denote a left triangulated category (C, Ω, Δ). Our purpose here is to examine when the (co-)stabilization of C can be realized as a full subcategory of C. We shall obtain more complete results in the next section when the left triangulated category C is a stable category. We denote as always by S : C ⟶ S(C) the stabilization functor and by R : R(C) ⟶ C the costabilization functor.

Definition 3.13. (1) The left triangulated category C is called Gorenstein if there exists a full left triangulated subcategory V of C, such that the composite functor Si_{V} : V ⟶^{i_{V}} C ⟶^{S} S(C) is a triangle equivalence, where i_{V} : V ↪ C is the inclusion. In this case we say that V realizes the stabilization of C.

(2) The left triangulated category C is called Co-Gorenstein if there exists a full left triangulated subcategory U of C, such that the inclusion i_{U} : U ↪ C is the costabilization functor. In this case we say that U realizes the costabilization of C.

The easy proof of the next Lemma is left to the reader (use Proposition 3.4).

Lemma 3.14. If i_{V} : V ↪ C is a left triangulated subcategory of C, then the loop functor Ω : V ⟶ V is fully faithful ⇔ the functor Si_{V} : V ⟶ S(C) is fully faithful.

Theorem 3.15. The following are equivalent:

(1) C is Gorenstein.

(2) There exists a full triangulated subcategory V of C, such that: ∀C ∈ C and n ∈ ℤ, there exists t ≥ n, 0 with Ω^{t−n}(C) ∈ V.

(3) There exists a full triangulated subcategory V of C, such that: ∀C ∈ C, there exists t ≥ 0 with Ω^{t}(C) ∈ V.

In this case the triangulated category V is uniquely determined up to a triangle equivalence and realizes the stabilization of C.

Proof. (1) ⇒ (2) Suppose that C is Gorenstein, and let i_{V} : V ↪ C be a full triangulated subcategory of C realizing the stabilization. Let C be in C and n ∈ ℤ. Consider the object (C, n) ∈ S(C). Since Si_{V} is dense, there exists an object A ∈ V and an isomorphism f̃ : (C, n) ⟶ Si_{V}(A) = (A, 0), with inverse g̃ : (A, 0) ⟶ (C, n). Choose representatives f_{k} : Ω^{k−n}(C) ⟶ Ω^{k}(A) of f̃ with k ≥ 0, n and g_{l} : Ω^{l}(A) ⟶ Ω^{l−n}(C) of g̃ with l ≥ 0, n. Analyzing the relations g̃ ∘ f̃ = 1_{(C,n)} and f̃∘ g̃ = 1_{(A,0)} and choosing t ≥ k, l, we see that Ω^{t−k}(f_{k}) : Ω^{t−n}(C) ⟶ Ω^{t}(A) is invertible with inverse Ω^{t−l}(g_{l}). Since Ω^{t}(A) ∈ V, the assertion (2) follows. The direction (2) ⇒ (3) is trivial.

(3) ⇒ (1) Since V is a triangulated subcategory, by the above Lemma the exact functor Si_{V} : V ⟶ S(C) is fully faithful. Let (C, n) be an arbitrary object of S(C). Then by hypothesis there exists t ≥ 0 such that Ω^{t}(C) := A ∈ V. Applying the functor S, we have S(Ω^{t}(C)) = S(A) ⇒ S(C) = SΩ^{−t}(A). Since A ∈ V and V is triangulated, we can write Ω^{−t}(A) = Ω^{r}(B) with B ∈ V and r ≥ 0, n. Then (C, n) = Ω̃^{−n}S(C) = Ω̃^{−n}S(Ω^{−t}(A)) = Ω̃^{−n}S(Ω^{r}(B)) = Ω̃^{−n}Ω̃^{r}S(B) = Ω̃^{r−n}S(B) = SΩ^{r−n}(B). Since B is in V, this shows that Si_{V} is dense.

We denote the kernel of the stabilization functor S : C ⟶ S(C), by

P^{∞}(C) = KerS = {C ∈ C : ∃n ≥ 0 : Ω^{n}(C) = 0}.

If C is Gorenstein, we call the uniquely determined triangulated subcategory of C realizing the stabilization, the category of maximal Cohen-Macaulay objects (or maximal CM-objects for short) of C, and we denote it by CM(C).

Proposition 3.16. If the left triangulated category C is Gorenstein, then

(1) The stabilization functor S : C ⟶ CM(C) is given as follows:

∀C ∈ C : S(C) = Ω^{−t}Ω^{t}(C) where t ≥ 0 is such that : Ω^{t}(C) ∈ CM(C).

(2) P^{∞}(C) is a full left triangulated subcategory of C, P^{∞}(C) ⋂ CM(C) = 0 and the inclusions P^{∞}(C) ↪ C and CM(C) ↪ C induce isomorphisms:

K_{0}(P^{∞}(C)) = 0 and K_{0}(CM(C)) ≅ K_{0}(C).

Proof. (1) By Theorem 3.15, ∀C ∈ C there exists t ≥ 0, such that Ω^{t}(C) ∈ CM(C). Suppose also that Ω^{s}(C) ∈ CM(C) with s ≥ 0 and assume without restriction that s = t + r. Then Ω^{−s}Ω^{s}(C) = Ω^{−t−r}Ω^{t+r}(C) = Ω^{−t}Ω^{−r}Ω^{r}Ω^{t}(C) = Ω^{−t}Ω^{t}(C), i.e. the object Ω^{−t}Ω^{t}(C) ∈ CM(C) is uniquely determined. We set S′(C) = Ω^{−t}Ω^{t}(C). If f : C ⟶ D is a morphism in C, define S′(f) = Ω^{−t}Ω^{t}(f). A similar argument as above shows that in this way we obtain a functor S′ : C ⟶ CM(C) which clearly is exact. If F : C ⟶ D is an exact functor to a triangulated category D with translation functor Σ, then define F^{∗} : CM(C) ⟶ D by F^{∗} = F |_{CM(C)}. Trivially F^{∗} is exact and ∀C ∈ C with Ω^{t}(C) ∈ CM(C), we have F^{∗}S′(C) = F^{∗}Ω^{−t}Ω^{t}(C) = Σ^{−t}F^{∗}Ω^{t}(C) = Σ^{−t}F Ω^{t}(C) = Σ^{−t}Σ^{t}F (C) = F (C). It is easy to see that these identifications are natural, so we have F^{∗}S′ = F. If G : CM(C) ⟶ D is another exact functor with GS′ = F, then ∀A ∈ CM(C): G(A) = GS′(A) = F (A) = F^{∗}(A). Hence G = F^{∗}. This shows that S′ = S is the stabilization functor.

(2) Let C ∈ P^{∞}(C) ⋂ CM(C). Since C ∈ P^{∞}(C) we have S(C) = 0 and since C ∈ CM(C) we have S(C) = C, so C = 0. All other assertions are trivial, noting that S(P^{∞}(C)) = 0, since the loop functor Ω in P^{∞}(C) is locally nilpotent.

We turn now our attention to Co-Gorenstein left triangulated categories. We consider the following full subcategory of C:

Ω^{∞}(C) = ⋂_{n≥1} Ω^{n}(C) = {C ∈ C | ∃{C_{n}}_{n≥0} ⊆ C : C_{n} = Ω(C_{n+1}), ∀n ≥ 0, C_{0} = C}.

If C is Co-Gorenstein, we call the uniquely determined triangulated subcategory of C realizing the costabilization, the category of maximal Co-Cohen-Macaulay objects (or maximal Co-CM-objects for short) of C, and we denote it by CoCM(C).

Theorem 3.17. The following are equivalent.

(1) C is Co-Gorenstein.

(2) Ω^{∞}(C) is a triangulated subcategory of C.

(3) There exists a triangulated subcategory U of C with the property: Ω^{∞}(C) ⊆ U. In this case U = CoCM(C) = Ω^{∞}(C), there exists a triangle equivalence R(C) ≈ Ω^{∞}(C) and the inclusion Ω^{∞}(C) ↪ C is the costabilization functor.

Proof. (1) ⇒ (3) If C is Co-Gorenstein, then there exists a triangulated subcategory U of C such that the inclusion i : U ↪ C is the costabilization functor. Since U is triangulated, we have U ⊆ Ω^{∞}(C) and since the strict image of the costabilization functor i is Ω^{∞}(C), we conclude that Ω^{∞}(C) = U. (3) ⇒ (2) Trivial. (2) ⇒ (1) To prove (1) it suffices to show that the costabilization functor R : R(C) ⟶ C is fully faithful, since then R induces an equivalence: R(C) ≈ Ω^{∞}(C). If f_{•} : (A_{n}, α_{n}) ⟶ (B_{n}, β_{n}) is a morphism in R(C) with R(f_{•}) = 0, then Ω^{n}(f_{n}) = 0, ∀n ≥ 0 and f_{n} = 0, ∀n ≤ 0. Since A_{n}, B_{n} ∈ Ω^{∞}(C), and since the latter is triangulated, we have f_{n} = 0, ∀n ∈ ℤ. Hence f_{•} = 0 and this shows that R is faithful. A similar argument shows that R is full, using that ImR = Ω^{∞}(C) is triangulated.

The next result, which is a consequence of Theorems 3.15, 3.17, shows that jn some cases the stabilization and the costabilization of C coincide.

Corollary 3.18. Suppose that C contains a full triangulated subcategory V enjoying the property: ∃d ≥ 0 such that Ω^{d}(C) ⊆ V. Then C is Gorenstein and Co-Gorenstein, there are triangle equivalences

R(C) ⟶^{≈} Ω^{∞}(C) ⟶^{≈} CoCM(C) ⟶^{≈} V ⟶^{≈} CM(C) ⟶^{≈} S(C),

the functor Ω^{−d}Ω^{d} : C ⟶ V is the stabilization functor and the inclusion i_{V} : V ↪ C is the costabilization functor.

We leave to the reader the formulation of the dual concepts and results concerning (Co-)Gorenstein right triangulated categories.

4. Gorenstein Categories and Auslander-Buchweitz Contexts

Throughout this section we assume that C is an exact category and X is a full contravariantly finite additive subcategory of C which is closed under isomorphisms, direct summands, and such that any X −epic is an admissible epic.

4.1. Stabilization and X −Gorenstein Exact categories.

Definition 4.1. The exact category C is called X −Gorenstein if the stable left triangulated category C/X is Gorenstein.

By section 2, we know that if C/X is Gorenstein, then the triangulated subcategory of C/X realizing the stabilization is the stable category of an X −resolving subcategory A of C. So we fix an X −resolving subcategory A of C, and let i_{A} : A/X ↪ C/X be the inclusion functor. Then we know that A/X is left triangulated and the inclusion functor i_{A} is exact. Let Â be the full subcategory of C consisting of all objects having finite X −resolutions by objects of A. Hence an object C ∈ C is in Â if there exists an X −exact sequence 0 ⟶ A_{n} ⟶ ⋯ ⟶ A_{1} ⟶ A_{0} ⟶ C ⟶ 0, where A_{i} ∈ A, ∀i ≥ 0. Let S : C/X ⟶ S(C/X) be the stabilization functor.

Theorem 4.2. (1) The following are equivalent.

(α) The exact functor Si_{A} : A/X ⟶ S(C/X) is fully faithful. (β) The loop functor Ω_{X} : A/X ⟶ A/X is fully faithful. (γ) A ⊆ ^{⊥}X.

(2) The following are equivalent.

(α) The functor Si_{A} : A/X ⟶ S(C/X) is a triangle equivalence.

(β) (i) A ⊆ ^{⊥}X.

(ii) X is an X −cogenerator of A.

(iii) ∀C ∈ C and ∀n ∈ ℤ, ∃t ≥ 0 with t ≥ n, such that: Ω_{X}^{t−n}(\underline{C}) ∈ A/X.

(γ) (i) A ⊆ ^{⊥}X.

(ii) X is an X −cogenerator of A.

(iii) C = Â.

Proof. Part (1) and the direction (α) ⇔ (β) of part (2) are consequences of Lemma 3.14, Theorem 3.15 and the results of section 2. (2)(β) ⇒ (γ) If C ∈ C, then by Theorem 3.15, there exists t ≥ 0 such that Ω_{X}^{t} (\underline{C}) = \underline{A} ∈ A/X. Hence there exists an X −exact sequence 0 ⟶ A ⟶ X_{t−1} ⟶ ⋯ ⟶ X_{0} ⟶ C ⟶ 0 with A ∈ A and X_{i} ∈ X ⊆ A, ∀i = 0, ..., t − 1. This implies that C ∈ Â, and consequently C = Â.

(γ) ⇒ (α) By Theorem 3.15, it suffices to show that for any C ∈ C there exists t ≥ 0 such that Ω_{X}^{t} (\underline{C}) ∈ A/X. Since Â = C, there exists an X −exact sequence 0 ⟶ A_{t} ⟶ A_{t−1} ⟶ ⋯ ⟶ A_{1} ⟶ A_{0} ⟶ C ⟶ 0, where A_{i} ∈ A, ∀i ≥ 0. Consider the X −exact sequence 0 ⟶ A_{t} ⟶ A_{t−1} ⟶ M_{t−1} ⟶ 0, and let Ω_{X} (\underline{M}_{t−1}) ⟶ \underline{A}_{t} ⟶ \underline{A}_{t−1} ⟶ \underline{M}_{t−1} be the induced triangle in C/X. Since A is X −resolving, imbedding the morphism \underline{A}_{t} ⟶ \underline{A}_{t−1} in a triangle in A/X, it follows that Ω_{X} (\underline{M}_{t−1}) ∈ A/X. Continuing in this way we see that Ω_{X}^{k} (\underline{M}_{t−k}) ∈ A/X, and finally that Ω_{X}^{t} (\underline{C}) := \underline{A} ∈ A/X.

Let P_{X}^{∞}(C) of C be the full subcategory of C of all objects with finite contravariant X −dimension. Obviously P_{X}^{∞}(C) is an X −resolving subcategory of C and we know that the induced stable category P_{X}^{∞}(C)/X = Ker(S). So in the notation of Subsection 3.3, P_{X}^{∞}(C)/X = P^{∞}(C/X). If C is X −Gorenstein, then we know that the stabilization of C/X is realized by the full subcategory CM(C/X). Hence there exists an X −resolving subcategory CM(C) of C such that CM(C/X) = CM(C)/X.

Theorem 4.3. If C is X −Gorenstein, then the following are true.

(1) The stabilization functor S : C/X ⟶ CM(C/X) is the coreflection of the category CM(C/X) in C/X and is given as follows:

S(\underline{C}) = Ω_{X}^{−t}Ω_{X}^{t} (\underline{C}), where t ≥ 0 is such that Ω_{X}^{t} (\underline{C}) ∈ CM(C/X).

(2) For any C ∈ C, there exists an X −exact sequence

(A_{C}) 0 ⟶ P_{C} ⟶^{p_{C}} A_{C} ⟶^{α_{C}} C ⟶ 0 with P_{C} ∈ P_{X}^{∞}(C) and A_{C} ∈ CM(C).

CM(C) is a contravariantly finite subcategory of C. The morphism α_{C} : A_{C} ⟶ C gives a right CM(C)−approximation of C.

(3) Let f : A_{C} ⟶ A_{C} be a morphism such that f ∘ α_{C} = α_{C}. Then \underline{f} is an isomorphism. All objects of C have minimal right CM(C)−approximations if CM(C) is a Krull-Schmidt category or if the ideal J_{X} (CM(C)) of morphisms in CM(C) factoring through X, is contained in the Jacobson radical of CM(C).

(4) P_{X}^{∞}(C) ∩ CM(C) = X. Moreover ∀C ∈ C : C ∈ P_{X}^{∞}(C) ⇔ A_{C} ∈ X.

(5) ∀A ∈ CM(C), ∀C ∈ C we have: \underline{Ext}_{X}^{i} (A, C) ≅ A/X (Ω_{X}^{i} (\underline{A}), \underline{A}_{C}), ∀i ≥ 1. Moreover: C/X [CM(C/X), P^{∞}(C/X)] = 0.

(6) CM(C) = ^{⊥}P_{X}^{∞}(C) = ^{⊥}X.

(7) For any C ∈ C, there exists an X −exact sequence

(P^{C}) 0 ⟶ C ⟶^{p^{C}} P^{C} ⟶^{α^{C}} A^{C} ⟶ 0 with P^{C} ∈ P_{X}^{∞}(C) and A^{C} ∈ CM(C).

P_{X}^{∞}(C) = CM(C)^{⊥} is a covariantly finite subcategory of C. The left P_{X}^{∞}(C)− approximation of C is given by the morphism p^{C} : C ⟶ P^{C}.

(8) Let f : P^{C} ⟶ P^{C} be a morphism such that p^{C} ∘ f = p^{C}. Then \underline{f} is a monomorphism and Ω_{X}^{n} (\underline{f}) is an isomorphism ∀n ≥ 1. All objects of C have minimal left P_{X}^{∞}(C)−approximations if P_{X}^{∞}(C) is a Krull-Schmidt category or if the ideal J_{X} (CM(C)) of morphisms in CM(C) factoring through X is contained in the Jacobson radical J ac(CM(C)) of CM(C).

(9) The stable category P_{X}^{∞}(C)/X is a reflective left triangulated subcategory of C/X and the reflection is given by the functor

T : C/X ⟶ P_{X}^{∞}(C)/X , defined by T(\underline{C}) = \underline{P}^{C}.

(10) The category C/X admits a “direct sum decomposition” C/X = P_{X}^{∞}(C)/X ⊕ CM(C/X) in the sense that the sequences of categories and functors

0 ⟶ P_{X}^{∞}(C)/X ⟶^{j} C/X ⟶^{S} CM(C/X) ⟶ 0

0 ⟶ CM(C/X) ⟶^{i} C/X ⟶^{T} P_{X}^{∞}(C)/X ⟶ 0

satisfy the relations: Sj = 0, Ti = 0, Si = Id_{CM(C/X)}, Tj = Id_{P_{X}^{∞}(C)/X}. Moreover there exists an equivalence of categories:

C/X /P_{X}^{∞}(C)/X ≈ CM(C/X)

where the first category indicates localization of the left triangulated category C/X with respect to the (eppaise) subcategory P_{X}^{∞}(C)/X.

(11) The pair [CM(C/X), P^{∞}(C/X)] is a torsion theory in C/X:

CM(C/X)^{⊥} := \{\underline{C} ∈ C/X : C/X (\underline{A}, \underline{C}) = 0, ∀\underline{A} ∈ CM(C/X)\} = P^{∞}(C/X),

⊥P^{∞}(C/X) := \{\underline{C} ∈ C/X : C/X (\underline{C}, \underline{P}) = 0, ∀\underline{P} ∈ P^{∞}(C/X)\} = CM(C/X). In particular there exists a triangle in C/X: Ω_{X} T ⟶ S ⟶ Id_{C/X} ⟶ T.

(12) Consider the relative Grothendieck groups K_{0}(C, X) and K_{0}(CM(C), X) and the stable Grothendieck groups K_{0}(C/X) and K_{0}(CM(C/X)) as defined in [15]. Then there are isomorphisms K_{0}(CM(C), X) ≅ K_{0}(C, X) and K_{0}(CM(C/X)) ≅ K_{0}(C/X) and an exact commutative diagram (†):

K_{0}(X , ⊕) ⟶ K_{0}(CM(C), X) ⟶ K_{0}(CM(C/X)) ⟶ 0
||↓  ≅↓  ≅↓
K_{0}(X , ⊕) ⟶ K_{0}(C, X) ⟶ K_{0}(C/X) ⟶ 0.

Proof. (1) By Proposition 3.16 it suffices to show that the functor S is the coreflection of CM(C/X). If C ∈ C, then by Theorem 4.1, there exists t ≥ 0 such that Ω_{X}^{t} (\underline{C}) := \underline{A} with A ∈ CM(C). Hence there exists an X −resolution (α) : 0 ⟶ A ⟶ X_{C}^{t−1} ⟶ X_{C}^{t−2} ⟶ ⋯ ⟶ X_{C}^{0} ⟶ C ⟶ 0 of C in C. Since A belongs to CM(C) and CM(C/X) is triangulated, \underline{A} is an arbitrary X −syzygy object. Hence there exists an X −exact sequence (β) : 0 ⟶ A ⟶ X^{t−1} ⟶ X^{t−2} ⟶ ⋯ ⟶ X^{0} ⟶ A_{C} ⟶ 0 in CM(C) and \underline{A}_{C} = Ω_{X}^{−t}(\underline{A}) = Ω_{X}^{−t}Ω_{X}^{t} (\underline{C}). Consider the X −exact sequences (γ) : 0 ⟶ A ⟶ X_{C}^{t−1} ⟶ K_{C}^{t−1} ⟶ 0 in C and (δ) : 0 ⟶ A ⟶ X^{t−1} ⟶ A^{t−1} ⟶ 0 in CM(C). Since A^{t−1} by construction is in CM(C) and CM(C) ⊆ ^{⊥}X, the push-out of the above admissible sequence along the morphism A ⟶ X_{C}^{t−1} splits, and this induces a morphism (δ) ⟶ (γ) of short exact sequences. Continuing in this way we obtain finally a morphism of X −resolutions (β) ⟶ (α), in particular we obtain a morphism α_{C} : A_{C} ⟶ C. It is easy to see that \underline{α}_{C} is independent of the above construction and induces a natural morphism \underline{α} : i_{A}S ⟶ Id_{C/X}. Since by construction Ω_{X}^{t} (\underline{α}_{C}) = 1_{Ω_{X}^{t} (\underline{C})}, trivially \underline{α}_{C} gives the coreflection of \underline{C} in CM(C/X). Hence CM(C/X) is a coreflective (triangulated) subcategory of C/X with coreflector the stabilization functor S.

(2) We use the notation of part (1). Adding a right X −approximation X_{C} to A_{C} if necessary and using that any X −epic is admissible, we can assume that α_{C} : A_{C} ⟶ C is an (admissible X −)epic. Hence the X −exact sequence 0 ⟶ P_{C} ⟶ A_{C} ⟶ C ⟶ 0 is defined. This sequence induces a triangle Ω_{X} (\underline{C}) ⟶ \underline{P}_{C} ⟶ \underline{A}_{C} ⟶ \underline{C} in C/X and a triangle Ω̃_{X} S(\underline{C}) ⟶ S(\underline{P}_{C}) ⟶ S(\underline{A}_{C}) ⟶ S(\underline{C}) in S(C/X). But obviously the stabilization functor S induces an isomorphism S(\underline{α}_{C}) : \underline{A}_{C} = S(\underline{A}_{C}) ≅ S(\underline{C}). Hence S(\underline{P}_{C}) = 0 which is equivalent to P_{C} being of finite contravariant X −dimension, i.e. P_{C} ∈ P_{X}^{∞}(C). It remains to show that α_{C} : A_{C} ⟶ C is a right CM(C)−approximation, but this follows directly from (1).

(3) Let f : A_{C} ⟶ A_{C} be a morphism such that f ∘ α_{C} = α_{C}. Then \underline{f} ∘ \underline{α}_{C} = \underline{α}_{C} in C/X. Applying the stabilization functor to this relation we see directly that S(\underline{f}) is an isomorphism. Since \underline{A}_{C} is in CM(C/X), we have that \underline{f} is an isomorphism. If the ideal J_{X} (CM(C)) is contained in J ac(CM(C)) then the projection functor CM(C) ⟶ CM(C/X) reflects isomorphisms. Hence f is an isomorphism and any object of C has a minimal right CM(C)−approximation. If CM(C) is a Krull-Schmidt category, then the proof of Lemma 2.6 of [1] can be applied, showing that any object of C has a minimal right CM(C)−approximation.

(4) That P_{X}^{∞}(C) ∩ CM(C) = X, follows from Proposition 3.16. Suppose now that C ∈ P_{X}^{∞}(C). Then the sequence 0 ⟶ P_{C} ⟶ A_{C} ⟶ C ⟶ 0, induces a triangle Ω_{X} (\underline{C}) ⟶ \underline{P}_{C} ⟶ \underline{A}_{C} ⟶ \underline{C} ⟶ in C/X and then a triangle Ω̃_{X} S(\underline{C}) ⟶ S(\underline{P}_{C}) ⟶ S(\underline{A}_{C}) ⟶ S(\underline{C}) in S(C/X). But since P_{C}, C have finite X −dimension, we have S(\underline{P}_{C}) = S(\underline{C}) = 0. Hence S(\underline{A}_{C}) = 0, so A_{C} is in P_{X}^{∞}(C) ∩ CM(C) = X. Conversely if A_{C} ∈ X, then obviously C ∈ P_{X}^{∞}(C).

(5) Consider objects A ∈ CM(C) and C ∈ C, and let 0 ⟶ Ω_{X} (A) ⟶ X_{A} ⟶ A ⟶ 0 be a right X −approximation of A. Applying the functor C(−, C) to this sequence and using that CM(C) ⊆ ^{⊥}X, it is easy to see that there exists an isomorphism \underline{Ext}_{X}^{1} (A, C) ≅ C/X (Ω_{X} (\underline{A}), \underline{C}). Consider now the triangle Ω_{X} (\underline{C}) ⟶ \underline{P}_{C} ⟶ \underline{A}_{C} ⟶ \underline{C} in C/X induced by the sequence (A_{C}) in (2). Since by (1), CM(C/X) is coreflective in C/X and Ω_{X} (\underline{A}) is in CM(C/X), it follows that we have an isomorphism CM(C/X)(Ω_{X} (\underline{A}), \underline{A}_{C}) ⟶ C/X (Ω_{X} (\underline{A}), \underline{C}). Hence \underline{Ext}_{X}^{1} (A, C) ≅ C/X (Ω_{X} (\underline{A}), \underline{C}) ≅ CM(C/X)(Ω_{X} (\underline{A}), \underline{A}_{C}). The general case follows by dimension shifting since CM(C) is X −resolving.

If \underline{f} : \underline{A} ⟶ \underline{P} is a morphism in C/X with A ∈ CM(C) and P ∈ P_{X}^{∞}(C), then from the right CM(C)−approximation sequence (A_{P}): 0 ⟶ P_{P} ⟶ A_{P} ⟶ P ⟶ 0, we see that f factors through A_{P}. But by (4), A_{P} ∈ X. Hence \underline{f} = 0 in C/X.

(6) By hypothesis on CM(C) and parts (3), (4) we have CM(C) ⊆ ^{⊥}P_{X}^{∞}(C) ⊆ ^{⊥}X. By part (2), if C ∈ ^{⊥}P_{X}^{∞}(C), then C ∈ CM(C) as a direct summand of A_{C}. Hence CM(C) = ^{⊥}P_{X}^{∞}(C) and it remains to show that ^{⊥}X ⊆ ^{⊥}P_{X}^{∞}(C). So let C ∈ ^{⊥}X. We first prove that any morphism f : C ⟶ P with P ∈ P_{X}^{∞}(C) factors through X. If P ∈ X, this is trivial. If X −dimP = 1 and 0 ⟶ X_{1} ⟶ X_{0} ⟶ P ⟶ 0 is a X −resolution, then the pull-back sequence along the morphism f splits since C ∈ ^{⊥}X. An easy induction argument shows our claim. Now as in the proof of (4), we have \underline{Ext}_{X}^{1} (C, P) ≅ C/X (Ω_{X} (\underline{C}), \underline{P}) = 0, since P ∈ P_{X}^{∞}(C) and Ω_{X} (C) ∈ ^{⊥}X. By dimension shifting we conclude that C ∈ ^{⊥}P_{X}^{∞}(C).

(7) Consider the X −exact sequence (A_{C}) of (2) and let 0 ⟶ A_{C} ⟶^{κ} X ⟶^{λ} A^{C} ⟶ 0 be an X −exact sequence in CM(C) with X ∈ X, which exists since A_{C} ∈ CM(C). Since the morphisms p_{C} : P_{C} ⟶ A_{C} and κ : A_{C} ⟶ X are admissible monics, their composition p_{C} ∘ κ : P_{C} ⟶ X is admissible monic. Let 0 ⟶ P_{C} ⟶ X ⟶ P^{C} ⟶ 0 be the induced admissible short exact sequence in C. Then we have a pull-back diagram, which defines the object P^{C}, and a push-out diagram:

\begin{matrix}P_{C} & ⟶^{p_{C}} & A_{C} & ⟶^{α_{C}} & C \\ ^{||}↓ & & ^{κ}↓ & _{p^{C}} ↓ \\ P_{C} & ⟶^{p_{C}∘κ} & X & ⟶^{ν} & P^{C} \end{matrix}  \begin{matrix}A^{C} & ⟶^{κ} & X & ⟶^{λ} & A^{C} \\ ^{α_{C}} ↓ & & ^{ν}↓ & & ^{||}↓ \\ C & ⟶^{p^{C}} & P^{C} & ⟶^{α^{C}} & A^{C}. \end{matrix}

Since the admissible sequence 0 ⟶ A_{C} ⟶ X ⟶ A^{C} ⟶ 0 is X −exact, from the above diagram it follows directly that the admissible sequence (P^{C}) : 0 ⟶

C ⟶^{p^{C}} P^{C} ⟶^{α^{C}} A^{C} ⟶ 0 is X −exact, hence induces a triangle Ω̃S(\underline{A}^{C}) ⟶ S(\underline{C}) ⟶ S(\underline{P}^{C}) ⟶ S(\underline{A}^{C}) in CM(C/X). But from (2) we have \underline{A}^{C} = Ω_{X}^{−1}(\underline{A}_{C}) = Ω_{X}^{−t−1}Ω_{X}^{t} (\underline{C}). Hence Ω̃S(\underline{A}^{C}) = Ω̃S(Ω_{X}^{−t−1}Ω_{X}^{t} (\underline{C})) = Ω̃Ω̃_{X}^{−t−1}S(Ω_{X}^{t} (\underline{C})) = S(\underline{C}). Obviously the morphism Ω̃S(\underline{A}^{C}) ⟶ S(\underline{C}) in the above triangle is the identity, hence S(\underline{P}^{C}) = 0 and then P^{C} belongs to P_{X}^{∞}(C). Now let f : C ⟶ Q be a morphism with Q ∈ P_{X}^{∞}(C) and consider the push-out sequence 0 ⟶ Q ⟶ M ⟶ A^{C} ⟶ 0 of (P^{C}) along the morphism f. Trivially the push-out sequence is also admissible and X −exact, hence is split by (6). It follows that the morphism p^{C} : C ⟶ P^{C} is the left P_{X}^{∞}(C)−approximation of C, since then f factors through p^{C}.

Finally by (6), P_{X}^{∞}(C) ⊆ CM(C)^{⊥}. If C ∈ CM(C)^{⊥}, then the X −exact sequence (P^{C}) splits, and C ∈ P_{X}^{∞}(C) as a direct summand of P^{C}. So P_{X}^{∞}(C) = CM(C)^{⊥}.

(8) Let f : P^{C} ⟶ P^{C} be a morphism such that p^{C} ∘ f = p^{C}. Then from the sequence (P^{C}) of (7), there exists a unique morphism g : A^{C} ⟶ A^{C} such that α^{C} ∘ g = f ∘ α^{C}. Since A^{C} ∈ CM(C), as in (3) we have that \underline{g} is an isomorphism. Then the triple of morphisms (1\underline{C}, \underline{f}, \underline{g}) is an endomorphism of the induced triangle in C/X of the sequence (P^{C}). Applying Yoneda’s lemma we conclude directly that Ω_{X} (\underline{f}) is an isomorphism and \underline{f} is a monomorphism. If the ideal J_{X} (CM(C)) is contained in J ac(CM(C)) then the projection functor CM(C) ⟶ CM(C/X) reflects isomorphisms. Hence g is an isomorphism and this implies that f is an isomorphism. Hence any object of C has a minimal left P_{X}^{∞}(C)−approximation. If P_{X}^{∞}(C) is Krull-Schmidt, then as in (3) we can apply the method of proof of Lemma 2.6 of [1], to conclude that any object of C has a minimal left P_{X}^{∞}(C)−approximation.

(9) It suffices to show that P_{X}^{∞}(C) is reflective in C/X. For any C ∈ C, we claim

that the object P^{C} of part (7) is uniquely determined in C/X. Indeed if 0 ⟶ C ⟶^{p^{C}}

P^{C} ⟶^{α^{C}} A^{C} ⟶ 0 and 0 ⟶ C ⟶^{q^{C}} Q^{C} ⟶^{β^{C}} B^{C} ⟶ 0 are left P_{X}^{∞}(C)−approximations of C with A^{C}, B^{C} ∈ CM(C), then we have the following commutative diagrams

\begin{matrix}C & ⟶^{p^{C}} & P^{C} & ⟶^{α^{C}} & A^{C} \\ ^{||}↓ & & ^{f}↓ & & ^{∃ρ}↓ \\ C & ⟶^{q^{C}} & Q^{C} & ⟶^{β^{C}} & B^{C} \end{matrix}  \begin{matrix}C & ⟶^{q^{C}} & Q^{C} & ⟶^{β^{C}} & B^{C} \\ ^{||}↓ & & ^{g}↓ & & ^{∃σ}↓ \\ C & ⟶^{p^{C}} & P^{C} & ⟶^{α^{C}} & A^{C}. \end{matrix}

It is easy to see then that there are morphisms k : A^{C} ⟶ P^{C} and m : B^{C} ⟶ Q^{C} such that: 1_{P^{C}} − f ∘ g = α^{C} ∘ k and 1_{Q^{C}} − g ∘ f = β^{C} ∘ m. But from part (5) we have that \underline{k} = 0 and \underline{m} = 0 in C/X. Hence \underline{f} and \underline{g} are isomorphisms in C/X. We set T(\underline{C}) := \underline{P}^{C}. If Q ∈ P_{X}^{∞}(C) and \underline{f} : \underline{C} ⟶ \underline{Q} is a morphism, then by part (7) there exists a morphism g : P^{C} ⟶ Q with p^{C} ∘ g = f. Then \underline{p}^{C} ∘ \underline{g} = \underline{f}. If \underline{h} : \underline{P}^{C} ⟶ \underline{Q} is another morphism with \underline{p}^{C} ∘ \underline{h} = \underline{f}, then p^{C} ∘ (g − h) = k ∘ m where k : C ⟶ X, m : X ⟶ Q and X ∈ X. Let t : P^{C} ⟶ X be a morphism with p^{C} ∘t = k. Then p^{C} ∘(g −h) = p^{C} ∘t∘m, hence the morphism g −h−t∘m = α^{C} ∘z for a unique morphism z : A^{C} ⟶ Q. Since Q has finite X −dimension, \underline{z} = 0. Hence \underline{g} − \underline{h} = \underline{t} ∘ \underline{m} = 0 and \underline{g} = \underline{h}. So \underline{p}^{C} : \underline{C} ⟶ T(\underline{C}) is the reflection of \underline{C} in P_{X}^{∞}(C)/X.

(10) The first part is a consequence of (9) and (1). Let T be the class of mor-

phisms \underline{f} : \underline{C} ⟶ \underline{D} in C/X, such that in a triangle Ω_{X} (\underline{D}) ⟶ \underline{P} ⟶ \underline{C} ⟶^{\underline{f}} \underline{D}, the object \underline{P} ∈ P_{X}^{∞}(C)/X. Obviously the functor S : C/X ⟶ CM(C/X) sends the class T to isomorphisms. If F : C/X ⟶ D is a functor with the same property, then since the morphism \underline{α}_{C} : \underline{A}_{C} ⟶ \underline{C} of the sequence (A_{C}) belongs to T, F (\underline{α}_{C}) is an isomorphism. Define a functor F^{∗} : CM(C/X) ⟶ D by F^{∗} = F |_{CM(C/X)}. Then ∀\underline{C} ∈ C/X we have F^{∗}S(\underline{C}) = F^{∗}(\underline{A}_{C}) = F (\underline{A}_{C}) ≅ F (\underline{C}). Hence F^{∗}S = F. Trivially F^{∗} is the unique functor with this property. This implies that CM(C/X) is equivalent to the localization C/X /P_{X}^{∞}(C)/X = C/X [T^{−1}].

(11) Follows easily from part (5) and parts (2), (9).

(12) The diagram (†) is obviously commutative. By the results of section 3, the morphism K_{0}(CM(C/X)) ⟶ K_{0}(C/X) is an isomorphism, since CM(C/X) is the stabilization of C/X. Hence the middle morphism of (†) is an epimorphism, which by [40] is actually an isomorphism, since { \atop CM(C)} = C.

Many of the consequences of the above Theorem are identical with the theory developed by Auslander-Buchweitz in [4], in case C is abelian. The crucial points of the Auslander-Buchweitz theory have been formulated by Hashimoto (see [30] or Subsection 4.3 below), in the concept of an Auslander-Buchweitz context in an abelian category. In our relative setting this concept can be formulated as follows.

Definition 4.4. Let C be an exact category and consider full additive subcategories A, B, X of C. The triple (A, B, X) is called an X −Auslander-Buchweitz context (or X −AB-context), if the following conditions are true: (α) A is X −resolving. (β) B is closed under extensions of X −exact admissible sequences, direct sum-

mands and cokernels of X −monics. (γ) A ∩ B = X is an X −cogenerator of A, and any X −exact admissible short

exact sequence 0 ⟶ X ⟶ C ⟶ A ⟶ 0 with X ∈ X and A ∈ A, splits. (δ) Â = C.

Next we characterize the exact categories which admit an X −AB-context.

Theorem 4.5. The following are equivalent.

(1) The subcategory X is part of an X −Auslander-Buchweitz context (A, B, X).

(2) The category C is X −Gorenstein. In this case: A = G_{X} (C) = ^{⊥}X = ^{⊥}P_{X}^{∞}(C) = CM(C) and B = P_{X}^{∞}(C).

Proof. (2) ⇒ (1) is the content of Theorem 4.3. Conversely let (A, B, X) be an X −Auslander-Buchweitz context. Then Â = C, and X is a cogenerator of A. Since any X −exact admissible sequence of the form 0 ⟶ X ⟶ C ⟶ A ⟶ 0 with X ∈ X and A ∈ A, splits, and since A is X −resolving, we have directly that A ⊆ ^{⊥}X. By Theorem 4.2, the subcategory A realizes the stabilization, so C is X −Gorenstein.

It remains to show that A = G_{X} (C) and B = P_{X}^{∞}(C). Let A ∈ A. Then the X −resolution of A is by definition contravariantly X −exact and it is covariantly X −exact since A = ^{⊥}X. Since X is a cogenerator of A, there exists an admissible contravariantly X −exact sequence 0 ⟶ A ⟶ X^{0} ⟶ B ⟶ 0 with B ∈ A. Since B ∈ A and A = ^{⊥}X, this sequence is also contravariantly X −exact. Continuing in this way we obtain a contravariantly X −exact X −coresolution of A. Hence A has a covariantly X −exact X −resolution and a contravariantly X −exact X −coresolution, i.e. A ∈ G_{X} (C). Conversely if C ∈ G_{X} (C), then since the X −resolution of C is contravariantly X −exact, we have C ∈ ^{⊥}X = A. Hence A = G_{X} (C). Since X ⊆ B, by property (β) of definition 4.4 it follows easily that P_{X}^{∞}(C) ⊆ B. Let C ∈ B and let 0 ⟶ C ⟶ P^{C} ⟶ A^{C} ⟶ 0 be the left P_{X}^{∞}(C)−approximation of C. By (β), (γ) of definition 4.4, we have A^{C} ∈ B ∩ A = X, so the above sequence splits since it is X −exact. Then C ∈ P_{X}^{∞}(C), since B is closed under direct summands.

Hence for any X −Gorenstein category C, we always have: G_{X} (C) = CM(C).

Definition 4.6. Let A be an X −resolving subcategory of C and C ∈ C. Then C has finite A−resolution dimension iff there exists an X −exact sequence 0 ⟶ A_{n} ⟶ ⋯ ⟶ A_{1} ⟶ A_{0} ⟶ C ⟶ 0 with A_{k} ∈ A, ∀k = 0, 1, ..., n. In this case the least such integer n is called the A−resolution dimension of C and is denoted by A−res.dimC. Otherwise we define A−res.dimC = ∞. The global A−resolution dimension of C is defined by A−gl.res.dimC = sup{A−res.dimC; C ∈ C}.

It is not difficult to see, using that A is X −resolving, that A−res.dimC is welldefined. Obviously Â = C iff any object of C has finite A−resolution dimension.

Corollary 4.7. Suppose that any left X −approximation of an X −Gorenstein object of C is admissible monic. Then the following are equivalent:

(1) The category C is X −Gorenstein.

(2) Any object of C has finite G_{X} (C)−resolution dimension, i.e. C = { \atop G_{X} (C)}.

Proof. (1) ⇒ (2) Follows from Theorems 4.3 and 4.5. By hypothesis and Proposition 2.13, G_{X} (C) is X −resolving and the stable category G_{X} (C)/X is triangulated. Then condition (2) implies that C is X −Gorenstein by Theorem 4.2.

Corollary 4.8. Suppose that X is an X −cogenerator of ^{⊥}X and for any object C ∈ C, there exists d ≥ 1 such that \underline{Ext}_{X}^{i} (C, X) = 0, ∀i ≥ d.

Then the triple (^{⊥}X , P_{X}^{∞}(C), X) is an X −Auslander-Buchweitz context.

Proof. The last assumption implies that for any C ∈ C there exists d ≥ 0 such that Ω_{X}^{d} (C) ∈ ^{⊥}X. Hence C = ^{ ˆ⊥}X and then the triple (^{⊥}X , P_{X}^{∞}(C), X) is an X −Auslander-Buchweitz context by Theorem 4.2.

4.2. Costabilization and X −Co-Gorenstein Exact categories. We turn now our attention to the representation of the costabilization R(C/X) of C/X as a full subcategory of C/X. Let again A be an X −resolving subcategory of C, and consider the costabilization functor R : R(C/X) ⟶ C/X. By the description of R(C/X) in Theorem 3.11, it follows that the essential image Im(R) of R is the full subcategory Ω_{X}^{∞}(C/X) = Ω_{X}^{∞}(C)/X of C/X induced by all objects which are arbitrary X −syzygies. Hence A/X ⊆ Im(R) iff A ⊆ Ω_{X}^{∞}(C).

Definition 4.9. The exact category C is called X −Co-Gorenstein if the left triangulated category C/X is Co-Gorenstein.

Theorem 4.10. The following are equivalent.

(1) C is X −Co-Gorenstein.

(2) Ω_{X}^{∞}(C) ⊆ ^{⊥}X.

(3) There exist an X −resolving subcategory A of C satisfying the following: (α) A ⊆ ^{⊥}X. (β) X is an X −cogenerator of A. (γ) Ω_{X}^{∞}(C) ⊆ A.

In one of the above equivalent statements is true, then A = Ω_{X}^{∞}(C) and if the left X −approximation of any X −Gorenstein object is admissible monic, then A = Ω_{X}^{∞}(C) = G_{X} (C) and A/X = CoCM(C/X).

Proof. The proof is a direct consequence of Theorem 3.17 and section 2. If C is X −Co-Gorenstein, then by (3) it follows trivially that A ⊆ G_{X} (C). If A ∈ G_{X} (C), then by hypothesis and section 2, G_{X} (C)/X is a triangulated subcategory of C/X. Hence the exact inclusion G_{X} (C)/X ↪ C/X, factors uniquely through the inclusion A/X ↪ C/X. But then obviously G_{X} (C)/X ⊆ A/X. Hence A = G_{X} (C).

Our next result which is a consequence of Corollary 3.18, shows that in some cases any X −Gorenstein exact category is X −Co-Gorenstein. For instance this is true if C is X −Gorenstein with G_{X} (C)−gl.res.dimC < ∞.

Corollary 4.11. Suppose that X is an X −cogenerator of A and A ⊆ ^{⊥}X. If Ω_{X}^{d} (C/X) ⊆ A/X for some d ≥ 0, then C is X −Gorenstein and X −Co-Gorenstein and there are triangle equivalences

R(C/X) ⟶^{≈} A/X ⟶^{≈} S(C/X),

the functor Ω_{X}^{−d}Ω_{X}^{d} : C/X ⟶ A/X is the stabilization functor and the inclusion i_{A} : A/X ↪ C/X is the costabilization functor.

Remark 4.12. All the above results are true in case we deal with the dual situation of relative injectives, i.e. when X is a covariantly finite subcategory of the exact category C and any X −monic is an admissible monic. Then the stable category C/X is right triangulated and its stabilization S(C/X) (costabilization R(C/X)) is defined. Then C is called X −(Co-)Gorenstein if the category C/X is (Co-)Gorenstein as a right triangulated category. We entrust to the reader the definition of an X −Auslander-Buchweitz context in this case, which however we call an dual X −Auslander-Buchweitz context.

4.3. Gorenstein and Co-Gorenstein Abelian Categories. We assume in this subsection that C is an abelian category. Then a resolving subcategory of C is a full additive subcategory of C which is closed under extensions, kernels of epics and contains the projective objects. Dually a coresolving subcategory of C is a full additive subcategory of C which is closed under extensions, cokernels of monics and contains the injective objects. We denote always by P (I) the full subcategories of projective (injective) objects and by P^{∞} (I^{∞}) the full subcategory of objects with finite projective (injective) dimension. We call P−cogenerators simply cogenerators.

An Auslander-Buchweitz context as defined in [30], is a triple (A, B, X) of full subcategories of C, such that the following conditions are true:

(i) A is closed under extensions, direct summands and kernels of epics. B is closed under extensions, direct summands and cokernels of monics.

(ii) X = A ∩ B is a cogenerator of A with the property

X ⊆ A^{⊥} = {C ∈ C : Ext^{i}(A, C) = 0, ∀i ≥ 1}.

(iii) C = Â, i.e. any object of C has a finite resolution by objects of A.

We refer to [4], [30] for examples of Auslander-Buchweitz contexts, in Commutative Algebra, Algebraic Geometry and Ring Theory. Given an Auslander-Buchweitz context (A, B, X) in the abelian category C, we are interested in describing the stable category C/X, when the latter is left triangulated. This happens if X is in addition contravariantly finite. So the results of Subsection 4.1 give a clear picture of this situation. Now we restrict ourselves in the case when the abelian category C has enough projectives (resp. injectives), studying when P (resp. I) is part of an Auslander-Buchweitz context (A, B, P) (resp. dual Auslander-Buchweitz context (A, B, I)) in C. We shall see that in this case our results are more complete. So from now on suppose that C has enough projectives. In case C has enough injectives we entrust to the reader to formulate the dual definitions of an I−Gorenstein category, B−gl.cores.dimC for a coresolving subcategory of B of C, and of a dual Auslander-Buchweitz context. The dual results using injectives are also true.

Corollary 4.13. The following are equivalent.

(1) The subcategory P is part of an Auslander-Buchweitz context (A, B, P).

(2) The triple (^{⊥}P, P^{∞}, P) is an Auslander-Buchweitz context.

(3) The category C is P−Gorenstein.

(4) The natural functor ^{⊥}P/P ⟶ D^{b}(C)/K^{b}(P) is a triangle equivalence.

(5) Any object of C has finite P−Gorenstein resolution dimension.

(6) The natural functor D^{b}(G_{P}(C)) ⟶ D^{b}(C) is a triangle equivalence. In this case A = ^{⊥}P = ^{⊥}P^{∞} = G_{P}(C) and B = P^{∞}.

Proof. By our previous results, the statements (1) to (5) are equivalent. So it suffice to show that (3) is equivalent to (6). We view G_{P}(C) as an exact subcategory of C with enough projectives. If one of the statements (1) to (5) is true, then we know from Corollary 4.7 that { \atop G_{P}(C)} = C. Then by [28] we have that the natural functor D^{b}(G_{P}(C)) ⟶ D^{b}(C) is a triangle equivalence. Conversely if the above functor is a triangle equivalence, then the Verdier-quotients D^{b}(G_{P}(C))/K^{b}(P) and D^{b}(C)/K^{b}(P) = S(C/P) are triangle equivalent. Since G_{P}(C)/P is triangulated, by section 3, D^{b}(G_{P}(C))/K^{b}(P) = G_{P}(C)/P. Hence the canonical functor G_{P}(C)/P ⟶ S(C/P) is a triangle equivalence. By Theorem 4.2, C is P−Gorenstein.

Corollary 4.14. Let A, B be P−Gorenstein abelian (or exact) categories with enough projectives. If there exists an equivalence G_{P}(A) ≈ G_{P}(B), then there exists a triangle equivalence: D^{b}(A) ≈ D^{b}(B).

Corollary 4.15. The following are equivalent.

(1) There exists an Auslander-Buchweitz context (C, B, P).

(2) The category C is Frobenius.

(3) The natural functor C/P ⟶ D^{b}(C)/K^{b}(P) is a triangle equivalence.

(4) The natural functor K_{Ac}(P) ⟶ C/P is a triangle equivalence.

(5) Any object of C is P−Gorenstein. In this case B = P.

Theorem 4.16. Let C be an abelian category with exact products and coproducts and with enough projectives and injectives. Then the following are equivalent.

(1) C is P−Gorenstein.

(2) C is I−Gorenstein.

(3) (^{⊥}P, P^{∞}, P) is an Auslander-Buchweitz context.

(4) (I^{⊥}, I^{∞}, I) is a dual Auslander-Buchweitz context.

(5) d := sup{p.d.I : I ∈ I} = sup{i.d.P : P ∈ P} < ∞.

(6) P^{∞} = I^{∞}.

(7) K^{b}(P) = K^{b}(I) as full subcategories of D^{b}(C).

(8) The natural functor ^{⊥}P/P ⟶ D^{b}(C)/K^{b}(P) is a triangle equivalence.

(9) The natural functor I^{⊥}/I ⟶ D^{b}(C)/K^{b}(I) is a triangle equivalence.

(10) Any object of C has finite P−Gorenstein resolution dimension.

(11) Any object of C has finite I−Gorenstein resolution dimension.

(12) The natural functor D^{b}(G_{P}(C)) ⟶ D^{b}(C) is a triangle equivalence.

(13) The natural functor D^{b}(G_{I}(C)) ⟶ D^{b}(C) is a triangle equivalence. If one of the above equivalent conditions is true, then we have the following: (α) C is P−Co-Gorenstein and I−Co-Gorenstein and

G_{P}(C)−gl.res.dimC = G_{I}(C)−gl.cores.dimC = d.

(β) P^{∞} = I^{∞} is functorially finite in C, G_{P}(C) = Ω^{d}(C) is contravariantly finite in C, G_{I}(C) = Σ^{d}(C) is covariantly finite in C, G_{P}(C)/P is coreflective in C/P

and G_{I}(C)/I is reflective in C/I.

(γ) If P is covariantly finite in C, then G_{P}(C) is functorially finite in C and G_{P}(C)/P is reflective in C/P. If I is contravariantly finite in C, then G_{I}(C) is functorially finite in C and G_{I}(C)/I is coreflective in C/I.

(δ) There exist triangle equivalences:

K_{Ac}(P) ≈ G_{P}(C)/P ≈ ^{⊥}P/P ≈ ^{⊥}(P^{∞})/P ≈ D^{b}(C)/K^{b}(P) ≈
≈ D^{b}(C)/K^{b}(I) ≈ (I^{∞})^{⊥}/I ≈ I^{⊥}/I ≈ G_{I}(C)/I ≈ K_{Ac}(I).

(ϵ) The costabilization functors are the inclusions

G_{P}(C)/P ↪ C/P and G_{I}(C)/I ↪ C/I

and the stabilization functors are given by

Ω^{−d}Ω^{d} : C/P ⟶ G_{P}(C)/P and Σ^{−d}Σ^{d} : C/I ⟶ G_{I}(C)/I.

(ζ) C is Frobenius iff G_{P}(C) = G_{I}(C).

Proof. By our previous results we have that (1) is equivalent to (3), (8), (10) and (12). By duality (2) is equivalent to (4), (9), (11) and (13). By the results of [25],

[18] we have (5) ⇔ (6). Also it is easy to see that (7) is equivalent to (6). So it suffices to show the equivalence (1) ⇔ (5). Then (2) ⇔ (5) will follow by duality. We set d = sup{p.d.I : I ∈ I} and δ = sup{i.d.P : P ∈ P}.

(1) ⇒ (5) Suppose that C is P−Gorenstein. By Theorem 4.2 we have that ∀C ∈ C there exists r_{C} ≥ 0 such that Ω^{r_{C}} (C) ∈ ^{⊥}P. Hence ∀C ∈ C : Ext^{i}(Ω^{r_{C}} (C), P) = 0, ∀i ≥ 1 or equivalently Ext^{i+r_{C}} (C, P) = 0, ∀i ≥ 1. This implies that any projective object has finite injective dimension, i.e. P ⊆ I^{∞}. If I is an injective object, then the left P^{∞}−approximation (P^{I}) of Theorem 4.3 splits, hence I has finite pojective dimension as a direct summand of P^{I}, so I ⊆ P^{∞}. By the arguments of [25], [18] it follows that d = δ < ∞.

(5) ⇒ (1) Since δ = d < ∞, for any C ∈ C we have Ω^{d}(C) ∈ ^{⊥}P, hence Ω^{d}(C) = ^{⊥}P. Let A ∈ ^{⊥}P and consider an injective resolution 0 ⟶ A ⟶ I^{0} ⟶ I^{1} ⟶ ⋯ of A, with corresponding cosyzygies B^{n} = Σ^{n}(A), ∀n ≥ 1. The exact sequence 0 ⟶ A ⟶ I^{0} ⟶ B^{1} ⟶ 0 induces a triangle Ω(\underline{B}^{1}) ⟶ \underline{A} ⟶ \underline{I}^{0} ⟶ \underline{B}^{1} in C/P. Applying the stabilization functor S to this triangle and using that I ⊆ P^{∞}, we have S(\underline{A}) ≅ SΩ(\underline{B}^{1}) = SΩ(\underline{Σ(A)}). Inductively we have S(\underline{A}) ≅ SΩ^{k}(\underline{B}^{k}) = SΩ^{k}(\underline{Σ^{k}(A)}), ∀k ≥ 1, in particular S(\underline{A}) ≅ SΩ^{d+1}(\underline{B}^{d+1}) = SΩ^{d+1}(\underline{Σ^{d+1}(A)}). Since Ω^{d+1}(\underline{Σ^{d+1}(A)}) ∈ ^{⊥}P/P and since the stabilization functor S is fully faithful restricted to ^{⊥}P/P, we have \underline{A} ≅ Ω^{d+1}(\underline{Σ^{d+1}(A)}). Setting A′ = Ω^{d}Σ^{d+1}(A) ∈ ^{⊥}P we see that A′ ∈ ^{⊥}P and Ω(\underline{A}′) = \underline{A}. Hence P is a cogenerator of ^{⊥}P. Since Ω^{d}(C) = ^{⊥}P, by Corollary 4.11 we have that C is P−Gorenstein, P−Co-Gorenstein and G_{P}(C)−gl.res.dimC = d < ∞.

If one of the equivalent conditions (1), (13) is true, then as the above proof shows, we have Ω^{d}(C) = ^{⊥}P and Σ^{d}(C) = I^{⊥}. Hence by Corollary 4.11 and its dual, C is P−Co-Gorenstein and I−Co-Gorenstein. Also the above proof shows that P ⊆ I^{∞} and I ⊆ P^{∞}, hence trivially P^{∞} = I^{∞} and by the Theorem 4.3 and its dual we see that these categories are functorially finite. Since C is P−Gorenstein, by Theorem 4.2 the category G_{P}(C) is contravariantly finite in C and G_{P}(C)/P is coreflective in C/P. Dually G_{I}(C) is covariantly finite in C and G_{I}(C)/I is reflective in C/I. If P is covariantly finite in C, then by section 2, we have that Ω^{d}(C) is covariantly finite. Since G_{P}(C) = ^{⊥}P = Ω^{d}(C), we conclude that G_{P}(C) is functorially finite. Dually if I is contravariantly finite, then G_{I}(C) is functorially finite. The other assertions in parts (α), (β), (γ), (δ), (ϵ) are consequences of our previous results.

Finally if G_{P}(C) = G_{I}(C), and if I is an injective object, the I is P−Gorenstein. But since I also has finite projective dimension it is projective by Theorem 4.3. Hence any projective is injective. By duality any injective is projective and C is Frobenius. Conversely if C is Frobenius, then obviously G_{P}(C) = C = G_{I}(C).

The following are consequences of Theorem 4.10 and Corollary 4.8.

Corollary 4.17. The abelian category C with enough projectives (injectives) is P−Co-Gorenstein ( I−Co-Gorenstein ) iff Ω^{∞}(C) ⊆ ^{⊥}P ( Σ^{∞}(C) ⊆ I^{⊥} ).

Corollary 4.18. Let C be an abelian category with enough projectives and enough injectives and let A, resp. B, be a resolving, resp. coresolving, subcategory of C.

(1) Suppose that P is a cogenerator of A and A ⊆ ^{⊥}P. If there exists d ≥ 0, such that Ω^{d}(C) ⊆ A, then there are triangle equivalences

K_{Ac}(P) ≈ A/P ≈ D^{b}(C)/K^{b}(P),

the functor Ω^{−d}Ω^{d} : C/P ⟶ A/P is the stabilization functor and the inclusion i_{A} : A/P ↪ C/P is the costabilization functor.

(2) Suppose that I is generator of B and B ⊆ I^{⊥}. If there exists d ≥ 0, such that Σ^{d}(C) ⊆ B, then there are triangle equivalences

K_{Ac}(I) ≈ B/I ≈ D^{b}(C)/K^{b}(I),

the functor Σ^{−d}Σ^{d} : C/I ⟶ B/I is the stabilization functor and the inclusion i_{B} : B/I ↪ C/I is the costabilization functor.

(3) Suppose that C in addition has exact products and coproducts and A satisfies the conditions in (1) or B satisfies the conditions in (2). Then C is P−Gorenstein, P−Co-Gorenstein, I−Gorenstein, I−Co-Gorenstein.

Remark 4.19. If in the definition 4.4 of an X −Auslander-Buchweitz context we remove condition (δ) that C = Â and we add the condition B ⊆ Â, then the triple (A, B, X) is called a weak X −Auslander-Buchweitz context, see [30]. All the results of this section are true for weak X −Auslander-Buchweitz contexts in C, but now C has to be replaced everywhere by Â.

5. Complete Resolutions and Complete (Co-)Homological Functors

5.1. Complete Resolutions and Complete Extension Functors. Troughout this section we fix a pair (C, X), where C is an additive category and X ⊆ C is a full contravariantly finite additive subcategory of C which is closed under direct summands, such that any X −epic has a kernel. Then C/X is left triangulated.

Definition 5.1. The complete X −extension bifunctors of C are defined by

Êxt_{X}^{n} (−, −) : C^{op} × C ⟶ Ab,
Êxt_{X}^{n} (A, B) = Hom_{S(C/X)}[(\underline{A}, −n), (\underline{B}, 0)], ∀n ∈ ℤ.

If S : C/X ⟶ S(C/X) is the stabilization functor, then since (\underline{A}, −n) = Ω̃_{X}^{n} (\underline{A}, 0) = Ω̃_{X}^{n} S(\underline{A}), it follows that

Êxt_{X}^{n} (A, B) = Hom_{S(C/X)}[Ω̃_{X}^{n} S(\underline{A}), S(\underline{B})] = Hom_{S(C/X)}[S(\underline{A}), Ω̃_{X}^{−n}S(\underline{B})] ≅

≅ lim_{⟶}_{k,k+n≥0}C/X [Ω_{X}^{k+n}(\underline{A}), Ω_{X}^{k} (\underline{B})].

The above definition is inspired by the definition of the stable homotopy groups of spheres and CW-complexes in algebraic topology, see for instance [24], [38]. We shall see in the next section that the complete X −Extension Bifunctors can be regarded as generalized Tate-Vogel cohomology functors.

Remark 5.2. In case X is a covariantly finite subcategory of C and any X −monic has a cokernel in C, so the stable category C/X is right triangulated, then we can define the complete X −Extension Bifunctors as follows:

Êxt_{n}^{X} (−, −) : C^{op} × C ⟶ Ab,
Êxt_{n}^{X} (A, B) = Hom_{S(C/X)}[(\underline{A}, 0), (\underline{B}, n)] = S(C/X)[S(\underline{A}), Σ̃_{X}^{n} S(\underline{B})] =

= lim_{⟶}_{k,k+n≥0}C/X (Σ_{X}^{k} (\underline{A}), Σ_{X}^{k+n}(\underline{B})), ∀n ∈ ℤ,

where S(C/X) is the stabilization of C/X as a right triangulated category, i.e. inverting the suspension functor Σ_{X} to an automorphism Σ̃_{X} of S(C/X). Observe that in case X is functorially finite, any X −monic has a cokernel and any X −epic has a kernel in C, then the above complete X −extension bifunctors are different, since in general the stabilizations of the stable category, first as a left triangulated category and second as a right triangulated category are not equivalent.

Remark 5.3. (1) From the above description we see that we can define Yoneda products. Indeed if α̃ ∈ Êxt_{X}^{n} (A, B) and β̃ ∈ Êxt_{X}^{m}(B, C), then α̃ : Ω̃_{X}^{n} S(\underline{A}) ⟶ S(\underline{B}) and β̃ : Ω̃_{X}^{m}S(\underline{B}) ⟶ S(\underline{C}). Then define the Yoneda product α̃ ⊙ β̃ as the

composition Ω̃^{m}(α̃) ∘ β̃ ∈ Êxt_{X}^{n+m}(A, C). In this way for any object A ∈ C, we obtain a ℤ−graded ring Êxt_{X}^{∗} (A, A).

(2) Êxt_{X}^{n} (Ω_{X}^{r} (A), Ω_{X}^{s} (B)) = Êxt_{X}^{n+r−s}(A, B), ∀n ∈ ℤ, ∀r, s ≥ 0.

(3) If C/X is triangulated, then since C/X is triangle equivalent to S(C/X):

Êxt_{X}^{n} (A, B) = Hom_{C/X} [Ω^{n}(\underline{A}), \underline{B}] ≅ Hom_{C/X} [\underline{A}, Ω^{−n}(\underline{B})], ∀n ∈ ℤ.

The next result shows in particular that the complete X −cohomology of C is non-trivial only if X −gl.dimC = ∞.

Proposition 5.4. (1) If A or B ∈ P_{X}^{∞}(C), then Êxt_{X}^{∗} (A, B) = 0. In particular if X −gl.dimC < ∞, then Êxt_{X}^{∗} (−, −) = 0.

(2) Let A ⟶^{g} B ⟶^{f} C be a sequence in C with f an X −epic and g = ker(f). Then for any D ∈ C, there are long exact sequences:

⋯ ⟶ Êxt_{X}^{1} (A, D) ⟶ Êxt_{X}^{0} (C, D) ⟶ Êxt_{X}^{0} (B, D) ⟶ Êxt_{X}^{0} (A, D) ⟶

⟶ Êxt_{X}^{−1}(C, D) ⟶ Êxt_{X}^{−1}(B, D) ⟶ Êxt_{X}^{−1}(A, D) ⟶ Êxt_{X}^{−2}(C, D) ⟶ ⋯ (α)

⋯ ⟶ Êxt_{X}^{−1}(D, C) ⟶ Êxt_{X}^{0} (D, A) ⟶ Êxt_{X}^{0} (D, B) ⟶ Êxt_{X}^{0} (D, C)

⟶ Êxt_{X}^{1} (D, A) ⟶ Êxt_{X}^{1} (D, B) ⟶ Êxt_{X}^{1} (D, C) ⟶ Êxt_{X}^{2} (D, A) ⟶ ⋯ (β)

Proof. (2) The sequence A ⟶ B ⟶ C induces a triangle Ω_{X} (\underline{C}) ⟶ \underline{A} ⟶ \underline{B} ⟶ \underline{C} in C/X by the definition of the triangulation Δ_{X} in [14]. Applying the stabilization functor S : C/X ⟶ S(C/X) we have a triangle (\underline{C}, −1) ⟶ (\underline{A}, 0) ⟶ (\underline{B}, 0) ⟶ (\underline{C}, 0) in S(C/X). Since S(C/X) is triangulated, applying to this triangle the cohomological functor Hom_{S(C/X)}[?, (\underline{D}, 0)] we get the long exact sequence (α) and applying the homological functor Hom_{S(C/X)}[(\underline{D}, 0),?] we get the long exact sequence (β). Part (1) follows directly from the definition.

Our purpose here is to compute the complete X −extension bifunctors by using suitable resolutions. To simplify things, we suppose throughout that C is an exact category, any X −epic is an admissible epic and any left X −approximation (which always exist) of an X −Gorenstein object, is an admissible monic.

Definition 5.5. A complete X −resolution of A ∈ C is a functorially X −exact complex

X_{c}^{•}(A) ⋯ ⟶ X^{−n} ⟶ ⋯ ⟶ X^{−1} ⟶ X^{0} ⟶ X^{1} ⟶ ⋯ ⟶ X^{n} ⟶ ⋯

for which there exists t := t_{A} ∈ ℤ, such that the complex ⋯ ⟶ X^{t−2} ⟶ X^{t−1} ⟶ X^{t} ⟶ 0 coincides with a part ⋯ ⟶ X_{A}^{s} ⟶ X_{A}^{s−1} ⟶ X_{A}^{s} of an X −resolution ⋯ ⟶ X_{A}^{s} ⟶ X_{A}^{s−1} ⟶ ⋯ ⟶ X_{A}^{1} ⟶ X_{A}^{0} ⟶ A ⟶ 0 of A.

The following is a direct consequence of the definition.

Lemma 5.6. An object A has a complete X −resolution iff there exists t ≥ 0 such that Ω_{X}^{t} (A) is an X −Gorenstein object or equivalently if A has finite X −Gorenstein resolution dimension, i.e. A ∈ { \atop G_{X} (C)}.

In particular any X −Gorenstein object A has a complete X −resolution (choose t = t_{A} = 0 in definition 5.5). For simplicity we set G(C/X) := G_{X} (C)/X.

Theorem 5.7. The following are equivalent:

(1) The category C is X −Gorenstein.

(2) Any object of C has a complete X −resolution.

Proof. (1) ⇒ (2) If C is X −Gorenstein, then by section 4, we know that there exists d ≥ 0, such that Ω_{X}^{d} (A) is an X −Gorenstein object. Since Ω_{X}^{d} (A) is an arbitrary X −syzygy object, there exists a functorially X − exact complex 0 ⟶ Ω_{X}^{d} (A) ⟶ X^{0} ⟶ X^{1} ⟶ · · ·. Composing the complex 0 ⟶ X^{0} ⟶ X^{1} ⟶ ⋯ with a deleted X −resolution ⋯ ⟶ X^{−2} ⟶ X^{−1} ⟶ 0 of Ω_{X}^{d} (A), since Ω_{X}^{d} (A) ∈ ^{⊥}X, we obtain a functorially X −exact complex, which obviously is a complete X −resolution of A.

(2) ⇒ (1) Suppose that any object C of C has a complete X −resolution X_{c}^{•}(C). Then by the above Lemma, there exists d = d_{C} ≥ 0, such that Ω_{X}^{d} (C) is an X −Gorenstein object. By Corollary 4.7, we have that C is X −Gorenstein.

Corollary 5.8. Suppose that C is X −Gorenstein. Then ∀A, B ∈ C:

Êxt_{X}^{n} (A, B) ≅ G(C/X)[Ω_{X}^{n−t+r}Ω_{X}^{t} (\underline{A}), Ω_{X}^{r} (\underline{B})], ∀n ∈ ℤ,

where t, r ≥ 0 are such that: Ω_{X}^{t} (\underline{A}), Ω_{X}^{r} (\underline{B}) ∈ G(C/X).

Proof. This follows from the description of the stabilization of the X −Gorenstein category C as the stable category G(C/X) in section 4.

Corollary 5.9. Suppose that there exists d ≥ 0, such that Ω_{X}^{d} (C) ⊆ G_{X} (C). Then the complete X −extension functors are given by:

Êxt_{X}^{n} (−, −) ≅ G(C/X)[Ω_{X}^{n} Ω_{X}^{d} (−), Ω_{X}^{d} (−)], ∀n ∈ ℤ.

5.2. Complete (Co-)Homological Functors. Our purpose in this subsection is to define under some assumptions, another sequence of (co-)homological functors, using resolutions of objects. Consider an object A in C, let

X_{A}^{•} ⋯ ⟶ X_{A}^{i+1} ⟶^{f_{A}^{i+1}} X_{A}^{i} ⟶ ⋯ ⟶ X_{A}^{1} ⟶^{f_{A}^{1}} X_{A}^{0} ⟶^{χ_{A}} A ⟶ 0

be an X −resolution of A and let

X_{•}^{A} 0 ⟶ A ⟶^{χ^{A}} X_{0}^{A} ⟶^{f_{1}^{A}} X_{1}^{A} ⟶ ⋯ ⟶ X_{i}^{A} ⟶^{f_{i+1}^{A}} X_{i+1}^{A} ⟶ ⋯

be an X −coresolution of A. The complex

X^{•}(A) ⋯ ⟶ X_{A}^{1} ⟶^{f_{A}^{1}} X_{A}^{0} ⟶^{ϑ_{A}} X_{0}^{A} ⟶^{f_{1}^{A}} X_{1}^{A} ⟶ ⋯

where ϑ_{A} := χ_{A} ∘ χ^{A}, is called an X −biresolution of A. Since we consider all our complexes as cohomological, X_{0}^{A} is in degree 0 and X_{A}^{0} is in degree −1.

Let A, B be two objects in C having X −biresolutions. We define X −homology groups Ĥ_{n}^{X} (A, B) and X −cohomology groups Ĥ_{X}^{n} (A, B), as follows:

Ĥ_{n}^{X} (A, B) = H^{n}(A, X^{•}(B)) and Ĥ_{X}^{n} (A, B) = H^{n}(X^{•}(A), B).

Let 0 ⟶ A ⟶ B ⟶ C ⟶ 0 be a functorially X −exact sequence in C of objects having X −biresolutions. By section 2, ∀D ∈ C, the following sequences are exact:

⋯ ⟶ Ĥ_{n}^{X} (D, A) ⟶ Ĥ_{n}^{X} (D, B) ⟶ Ĥ_{n}^{X} (D, C) ⟶ Ĥ_{n−1}^{X}(D, A) ⟶ ⋯
⋯ ⟶ Ĥ_{X}^{n} (C, D) ⟶ Ĥ_{X}^{n} (B, D) ⟶ Ĥ_{X}^{n} (A, D) ⟶ Ĥ_{X}^{n+1}(C, D) ⟶ ⋯

Our aim is to study the relationship between the complete X −(co)homology functors Ĥ_{n}^{X} (A, B), Ĥ_{X}^{n} (A, B) and the complete X −extensions functors Êxt_{X}^{n} (A, B),

and Êxt_{n}^{X} (A, B), n ∈ ℤ, when all these functors make sense.

So let X be functorially finite in C, any X −epic has a kernel and any X −monic has a cokernel in C. As always Σ_{X} is the left adjoint of Ω_{X} in C/X as in section 2.

Proposition 5.10. For any A, B ∈ C, we have the following

(1) Ĥ_{−n}^{X}(A, B) = Ĥ_{X}^{−n}(A, B) = C/X (\underline{A}, Ω^{n}(\underline{B})) = C/X (Σ^{n}(\underline{A}), \underline{B}), ∀n ≥ 2. If the right X −approximation of A is epic and the left X −approximation of B is monic, then the above identifications are true, ∀n ≥ 0.

(2) If \underline{Ext}_{X}^{n} (A, X) = \overline{Ext}_{X}^{n} (X , B) = 0, ∀n ≥ 1, then Ĥ_{n}^{X} (A, B) = Ĥ_{X}^{n} (A, B) = \underline{Ext}_{X}^{n} (A, B) = \overline{Ext}_{X}^{n} (A, B), ∀n ≥ 1.

(3) If C(A, B) = \underline{Ext}_{X}^{0} (A, B) = \overline{Ext}_{X}^{0} (A, B) and \underline{Ext}_{X}^{n} (A, X) = \overline{Ext}_{X}^{n} (X , B) = 0, ∀n ≥ 1, then Ĥ_{X}^{n} (A, −)(B) ≅ Ĥ_{n}^{X} (−, B)(A), ∀n ∈ ℤ.

Proof. Parts (1), (2) are consequences of Proposition 2.5 and Proposition 2.8. Part (3) follows from (1), (2) and the definitions.

The following result is a direct consequence of the definitions.

Proposition 5.11. The following are equivalent.

(1) A ∈ G_{X} (C), i.e. A is an X −Gorenstein object.

(2) Any X −coresolution of A is covariantly X −exact and any X −resolution of A is contravariantly X −exact.

(3) Ĥ_{n}^{X} (X , A) = Ĥ_{X}^{n} (A, X) = 0, ∀n ∈ ℤ.

(4) A ∈ ^{⊥}X ∩ X^{⊥}.

Let A, B be objects of C and consider the complex C(X^{•}(A), B). By diagram chasing, there exists the following commutative diagram with exact rows:

C/X (Σ_{X} (\underline{A}), \underline{B}) ⟶ L_{0}^{X} (A, B) ⟶ C(A, B) ⟶ C/X (\underline{A}, \underline{B})
μ↓  ^{||}↓  ↓  ^{ν}↓
Ĥ_{X}^{−1}(A, B) ⟶ L_{0}^{X} (A, B) ⟶ \underline{Ext}_{X}^{0} (A, B) ⟶ Ĥ_{X}^{0} (A, B).

From the above diagram we see that that there exist natural morphisms

μ : C/X (Σ_{X} (−), \underline{B}) ≅ C/X (−, Ω_{X} (\underline{B})) ⟶ Ĥ_{X}^{−1}(−, B), ν : C/X (−, \underline{B}) ⟶ Ĥ_{X}^{0} (−, B)

with μ monic. Hence the morphisms μ_{A} : C/X (\underline{A}, Ω_{X} (\underline{B})) ⟶ Ĥ_{X}^{−1}(A, B) and ν_{A} : C/X (\underline{A}, \underline{B}) ⟶ Ĥ_{X}^{0} (A, B) are isomorphisms iff C(A, B) ≅ \underline{Ext}_{X}^{0} (A, B), which obviously happens if the right X −approximation of A is epic. Suppose that the right X −approximations of A and any of its X −syzygies K_{A}^{n} are epics. From the exact sequence 0 ⟶ C(A, B) ⟶ C(X_{A}^{0}, B) ⟶ C(K_{A}^{1}, B) ⟶ \underline{Ext}_{X}^{1} (A, B) ⟶ 0, we see directly that there exists an epic \underline{Ext}_{X}^{1} (A, B) ⟶ C/X (Ω_{X} (\underline{A}), \underline{B}). By dimension shifting we obtain in this way epics ε_{A,B}^{n} : \underline{Ext}_{X}^{n} (A, B) ⟶ C/X (Ω_{X}^{n} (\underline{A}), \underline{B}), ∀n ≥ 1.

Dually consider the complex C(A, X^{•}(B)). Then we have the following commutative diagram, with exact rows

C/X (\underline{A}, Ω_{X} (\underline{B})) ⟶ L_{0}^{X} (A, B) ⟶ C(A, B) ⟶ C/X (\underline{A}, \underline{B})
ξ↓  ^{||}↓  ↓  ^{ζ}↓
Ĥ_{−1}^{X}(A, B) ⟶ L_{0}^{X} (A, B) ⟶ \overline{Ext}_{X}^{0} (A, B) ⟶ Ĥ_{0}^{X} (A, B)

and natural morphisms ξ : C/X (\underline{A}, Ω_{X} (−)) ⟶ Ĥ_{−1}^{X}(A, −) and ζ : C/X (\underline{A}, −) ⟶ Ĥ_{0}^{X} (A, −), with ξ monic. Hence the morphisms ξ_{B} : C/X (\underline{A}, Ω_{X} (\underline{B})) ⟶ Ĥ_{−1}^{X}(A, B) and ζ_{B} : C/X (\underline{A}, Ω_{X} (\underline{B})) ⟶ Ĥ_{0}^{X} (A, B) are isomorphisms iff C(A, B) ≅ \overline{Ext}_{X}^{0} (A, B), which obviously happens if the left X −approximation of B is monic. If the left X −approximations of B and any of its X −cosyzygies L_{n}^{B} are monics, then we have epimorphisms \overline{Ext}_{X}^{n} (A, B) ⟶ C/X (\underline{A}, Σ_{X}^{n} (\underline{B})), ∀n ≥ 1.

Suppose that any X −epic is epic. Then there is a natural morphism

σ_{−,B}^{n} : Ĥ_{X}^{n} (−, B) ⟶ Êxt_{X}^{n} (−, B), ∀B ∈ C, ∀n ∈ ℤ.

Indeed in this case Ĥ_{X}^{−n}(A, B) = C/X (Σ_{X}^{n} (\underline{A}), \underline{B}) ≅ C/X (\underline{A}, Ω^{n}(\underline{B})), ∀n ≥ 0, ∀A ∈ C. Hence ∀n ≥ 0 we have the natural morphism:

σ_{−,B}^{−n} : Ĥ_{X}^{−n}(−, B) ⟶^{≅} C/X (−, Ω^{n}(\underline{B})) ⟶^{S} S(C/X)[S(−), Ω̃_{X}^{n} S(\underline{B})] = Êxt_{X}^{−n}(−, B),

and for n ≥ 1, we have the morphisms:

σ_{−,B}^{n} : Ĥ_{X}^{n} (−, B) = \underline{Ext}_{X}^{n} (−, B) ⟶^{ε_{−,B}^{n}} C/X (Ω_{X}^{n} (−), \underline{B}) ⟶^{S}

⟶^{S} S(C/X)[Ω̃_{X}^{n} S(−), S(\underline{B})] = Êxt_{X}^{n} (−, B).

Lemma 5.12. Let C be an exact category and X be a contravariantly finite subcategory such that any X −epic is admissible epic. Then the morphism

S_{A,B} : C/X (\underline{A}, \underline{B}) ⟶ S(C/X)[S(\underline{A}), S(\underline{B})]

is an isomorphism, ∀A ∈ ^{⊥}X and ∀B ∈ C.

Proof. Let \underline{f} : \underline{A} ⟶ \underline{B} be a morphism with S(\underline{f}) = 0. Then there exists t ≥ 0 such that Ω_{X}^{t} (\underline{f}) = 0. Choose X −resolutions X_{A}^{•} and X_{B}^{•} of A and B, with corresponding X −syzygies K_{A}^{∗}, K_{B}^{∗}. We denote by k_{f}^{n} : K_{A}^{n} ⟶ K_{B}^{n} the induced by f morphisms, such that \underline{k}_{f}^{n} = Ω_{X}^{n} (\underline{f}). Since Ω_{X}^{t} (\underline{f}) = 0, the morphism k_{f}^{t} factors through X_{B}^{t}. But since A ∈ ^{⊥}X, taking the push-out of the X −exact admissible sequence 0 ⟶ K_{A}^{t} ⟶ X_{A}^{t−1} ⟶ K_{A}^{t−1} ⟶ 0 along the morphism K_{A}^{t} ⟶ X_{B}^{t} we see that there exists a morphism X_{A}^{t−1} ⟶ X_{B}^{t} such that composing this morphism with K_{A}^{t} ⟶ X_{A}^{t−1} we obtain the morphism K_{A}^{t} ⟶ X_{B}^{t}. This implies that in the diagram

0 ⟶ K_{A}^{t} ⟶ X_{A}^{t−1} ⟶ K_{A}^{t−1} ⟶ 0
k_{f}^{t} ↓  ^{x_{f}^{t−1}}↓  ^{k_{f}^{t−1}}↓
0 ⟶ K_{B}^{t} ⟶ X_{B}^{t−1} ⟶ K_{B}^{t−1} ⟶ 0

the morphism k_{f}^{t} factors through X_{A}^{t−1}. But then the morphism k_{f}^{t−1} factors through X_{B}^{t−1}. Hence Ω_{X}^{t−1}(\underline{f}) = \underline{k}_{f}^{t−1} = 0. Continuing in this way we have finally that \underline{f} = 0. Hence S_{A,B} is a monomorphism. Suppose now that f̃: S(\underline{A}) ⟶ S(\underline{B}) is a morphism in S(C/X) and choose a representative \underline{f}_{s} : Ω_{X}^{s} (\underline{A}) ⟶ Ω_{X}^{s} (\underline{B}) of f̃, where s ≥ 0. Let g_{s} : K_{A}^{s} ⟶ K_{B}^{s} be a morphism such that \underline{g}_{s} = \underline{f}_{s}. Arguing as above and using that A ∈ ^{⊥}X, we see that there are morphisms \underline{g}_{t} : K_{A}^{t} ⟶ K_{B}^{t}, ∀t ≤ s making the above diagram commutative. We set g = g_{0} : A ⟶ B. Using Corollary 3.3 it follows that S(\underline{g}) = f̃. Hence S_{A,B} is an epimorphism.

The next result shows that the complete X −Extensions functors Êxt_{X}^{∗} (A, B) defined using the contravariant finiteness of X and the stabilization functor S : C/X ⟶ S(C/X), can be computed via resolutions if A has a complete X −resolution or equivalently if A has finite X −Gorenstein resolution dimension.

Theorem 5.13. Let C be an exact category and X be a contravariantly finite subcategory of C, such that any X −epic is an admissible epic and the left X −approximation of any X −Gorenstein object of C is admissible monic. Then for any B ∈ C the following natural morphism is defined and is an isomorphism:

σ_{−,B}^{∗} : Ĥ_{X}^{∗} (−, B) ⟶ Êxt_{X}^{∗} (−, B) : { \atop G_{X} (C)}^{op} ⟶ Ab.

Proof. Since any object of finite X −Gorenstein resolution dimension has a complete X −resolution, the morphism σ_{A,B}^{∗} is defined, ∀B ∈ C. Suppose first that A is X −Gorenstein. Then we have seen in section 4 that the morphisms ϵ_{A,B} above are isomorphisms. Hence to prove that the morphisms σ_{A,B}^{∗} are isomorphisms for A an X −Gorenstein object, it suffices to prove that the morphisms S_{A,B} : C/X (\underline{A}, \underline{B}) ⟶ S(C/X)[S(\underline{A}), S(\underline{B})] are isomorphisms, for all B ∈ C. But this follows from the above Lemma, since G_{X} (C) ⊆ ^{⊥}X. Now if A ∈ { \atop G_{X} (C)} has a complete X −resolution then Ω_{X}^{t} (A) is in G_{X} (C) for some t ≥ 0. By dimension shifting we conclude that the morphism σ_{A,B}^{∗} is an isomorphism, since the functors Êxt_{X}^{∗} (−, B) and Ĥ_{X}^{∗} (−, B) are both cohomological with respect to X −exact admissible short exact sequences.

Suppose that any X −monic is monic. Then dually there is a natural morphism

τ_{n}^{A,−} : Ĥ_{n}^{X} (A, −) ⟶ Êxt_{n}^{X} (A, −), ∀A ∈ C, ∀n ∈ ℤ

but here the stabilization used in the definition of the complete X −cohomology refers to the stabilization of C/X as a right triangulated category, i.e. inverting the functor Σ_{X}. The next result, which we state without proof, shows that the complete X −Extensions functors Êxt_{∗}^{X} (A, B) defined by the covariant finiteness of X, can be computed via resolutions if B has a complete X −resolution or equivalently A has finite X −Gorenstein coresolution dimension. The full subcategory of C consisting of all objects with finite X −exact coresolutions by objects of G_{X} (C), i.e. the dual of { \atop G_{X} (C)}, is denoted by { \atop G_{X} (C)}.

Theorem 5.14. Let C be an exact category and X be a covariantly finite subcategory of C, such that any X −monic is admissible monic and the right X −approximation of any X −Gorenstein object of C is admissible epic. Then ∀A ∈ C, the following natural morphism is defined and is an isomorphism:

τ_{A,−}^{∗} : Ĥ_{∗}^{X} (A, −) ⟶ Êxt_{∗}^{X} (A, −) : { \atop G_{X} (C)} ⟶ Ab.

6. (Co-)Gorenstein Rings: Applications to Ring Theory

In this section we apply the results of the previous sections to the familiar setting of module categories. We fix throughout an associative ring Λ. We denote by Mod(Λ) the category of right Λ−modules and by P_{Λ}, resp. I_{Λ}, the full subcategory of projective, resp. injective, modules. The category of finitely presented right Λ−modules is denoted by mod(Λ), and its full subcategory of projective, resp. injective, modules is denoted by P_{Λ}, resp. I_{Λ}. The induced stable categories are denoted by: Mod(Λ)/P_{Λ} = \underline{Mod}(Λ), Mod(Λ)/I_{Λ} = \overline{Mod}(Λ), mod(Λ)/P_{Λ} = \underline{mod}(Λ) and mod(Λ)/I_{Λ} = \overline{mod}(Λ).

Throughout we choose C to be the module category Mod(Λ), resp. mod−Λ, and X to be one of the categories P_{Λ}, I_{Λ}, resp. P_{Λ}, I_{Λ}. We use the terminology and notations of the previous sections applied to the above choices.

6.1. Homologically finite Subcategories.

Proposition 6.1. Let Λ be an arbitrary ring.

(1) P_{Λ} is covariantly finite ⇔ Λ is left coherent and right perfect. In this case the loop functor Ω : \underline{Mod}(Λ) ⟶ \underline{Mod}(Λ) admits a left adjoint Σ_{P}.

(2) I_{Λ} is contravariantly finite ⇔ Λ is right Noetherian. In this case the suspension functor Σ : \overline{Mod}(Λ) ⟶ \overline{Mod}(Λ) admits a right adjoint Ω_{I}.

(3) _{Λ}P is covariantly finite and I_{Λ} is contravariantly finite ⇔ Λ is right Artinian.

(4) P_{Λ} is covariantly finite ⇔ Λ is left coherent. If Λ is coherent then we have an adjoint pair (Ω, Σ_{P}) defined on the left triangulated category \underline{mod}(Λ), and an adjoint pair (Ω, Σ_{P}) defined on the left triangulated category \underline{mod}(Λ^{op}).

Proof. It is well known [12], [19] that Λ is left coherent and right perfect iff P_{Λ} is covariantly finite, that Λ is left coherent iff P_{Λ} is covariantly finite and finally that Λ right Noetherian iff I_{Λ} is contravariantly finite. Since Λ is right Artinian iff Λ is right Noetherian and left perfect, (3) is a consequence of (1), (2). If Λ is coherent, then mod(Λ), mod(Λ^{op}) are abelian with projectives, and P_{Λ}, _{Λ}P are covariantly finite in them. The remaining assertions follow from section 2.

By the above result, if Λ is an Artinian ring then all the subcategories P_{Λ}, I_{Λ}, ΛP and I_{Λ} are functorially finite. For n ≥ 1, let Ω^{n}(Mod(Λ)) be the full additive subcategory of Mod(Λ) generated by P_{Λ} and the n^{th}−syzygy modules, and let Ω^{∞}(Mod(Λ)) = ⋂_{n≥1} Ω^{n}(Mod(Λ)) be the full subcategory of arbitrary syzygy modules. The categories Σ^{n}(Mod(Λ)) and Σ^{∞}(Mod(Λ)) are defined similarly.

Proposition 6.2. (1) If Λ is left coherent and right perfect then Ω^{n}(Mod(Λ)) is covariantly finite in Mod(Λ) and Ω^{n}(\underline{Mod}(Λ)) is reflective in \underline{Mod}(Λ), ∀n ≥ 1.

The category Ω^{∞}(Mod(Λ)) is a covariantly finite subcategory of Mod(Λ) and the category Ω^{∞}(\underline{Mod}(Λ)) is a reflective subcategory of \underline{Mod}(Λ).

(2) If Λ is coherent then the categories Ω^{n}(mod(Λ)), Ω^{n}(mod(Λ^{op})) are covariantly finite in mod(Λ), mod(Λ^{op}) and the categories Ω^{n}(\underline{mod}(Λ)), Ω^{n}(\underline{mod}(Λ^{op})) are reflective in \underline{mod}(Λ), \underline{mod}(Λ^{op}) respectively, ∀n ≥ 1.

Proof. Let Σ_{P} be the left adjoint of Ω in the stable category, which exists by Proposition 6.1. Let A be a right Λ−module, and let

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

be a P_{Λ}−coresolution, which is a result of the composition of the exact sequences

A ⟶^{π_{0}^{A}} P_{0}^{A} ⟶ L_{1}^{A}, L_{1}^{A} ⟶^{π_{1}^{A}} P_{1}^{A} ⟶ L_{2}^{A} and so on, where π_{n}^{A} are left P_{Λ}−approximations and L_{n}^{A} = Coker(π_{n−1}^{A}). Then in \underline{Mod}(Λ), we have Σ_{P}^{n}(\underline{A}) = \underline{L}_{n}^{A} and ΩΣ_{P}^{n}(\underline{A}) = \underline{Ker(π_{n−1}^{A})}. We set for simplicity Σ_{P}^{n}(A) = L_{n}^{A}. Then ΩΣ_{P}^{n}(A) = Ker(π_{n−1}^{A}), and we have the following exact commutative diagrams, ∀n ≥ 0:

Σ_{P}^{n}(A) ⟶ P_{n}^{A} ⟶ Σ_{P}^{n+1}(A) ⟶ 0
∃μ_{n}↓  ^{||}↓  ^{||}↓
0 ⟶ ΩΣ_{P}^{n+1}(A) ⟶ P_{n}^{A} ⟶ Σ_{P}^{n+1}(A)

Hence we obtain morphisms \underline{μ}_{n} : Σ_{P}^{n}(\underline{A}) ⟶ ΩΣ_{P}^{n+1}(\underline{A}), ∀n ≥ 0. This system of morphisms induces a tower

\underline{A} ⟶^{\underline{μ}_{0}} ΩΣ_{P}(\underline{A}) ⟶^{Ω(\underline{μ}_{1})} Ω^{2}Σ_{P}^{2}(\underline{A}) ⟶^{Ω^{2}(\underline{μ}_{2})} Ω^{3}Σ_{P}^{3}(\underline{A}) ⟶ ⋯

in \underline{Mod}(Λ), and we have projective presentations of increasing length:

(A_{1}) : 0 ⟶ ΩΣ_{P}(A) ⟶ P_{0}^{A} ⟶ Σ_{P}(A) ⟶ 0,
(A_{2}) : 0 ⟶ Ω^{2}Σ_{P}^{2}(A) ⟶ Q^{1,1} ⟶ P_{1}^{A} ⟶ Σ_{P}^{2}(A) ⟶ 0,
(A_{3}) : 0 ⟶ Ω^{3}Σ_{P}^{3}(A) ⟶ Q^{2,2} ⟶ Q^{2,1} ⟶ P_{2}^{A} ⟶ Σ_{P}^{3}(A) ⟶ 0, ⋯

such that the tower A ⟶ ΩΣ_{P}(A) ⟶ Ω^{2}Σ_{P}^{2}(A) ⟶ Ω^{3}Σ_{P}^{3}(A) ⟶ ⋯ induces morphisms between the presentations. Hence we obtain a direct system of projective presentations A ⟶ (A_{1}) ⟶ (A_{2}) ⟶ · · ·. Taking direct limits in this direct system, we obtain an exact sequence

0 ⟶ lim_{⟶}Ω^{n}Σ_{P}^{n} ⟶ lim_{⟶}Q^{n,n} ⟶ lim_{⟶}Q^{n+1,n} ⟶ lim_{⟶}Q^{n+2,n} ⟶ ⋯

Since Λ is left coherent and right perfect, the above exact sequence is a coresolution of lim_{⟶}Ω^{n}Σ_{P}^{n}(A) by projectives. Hence lim_{⟶}Ω^{n}Σ_{P}^{n}(A) ∈ Ω^{∞}(Mod(Λ)), and there

exists a canonical map d_{∞}^{2}(A) := lim_{⟶}d_{n}^{2}(A) : A ⟶ lim_{⟶}Ω^{n}Σ_{P}^{n}(A), where d_{n}^{2}(A) =

μ_{0} ∘ Ω(μ_{1}) ∘ ⋯ ∘ Ω^{n−1}(μ_{n−1}) : A ⟶ Ω^{n}Σ_{P}^{n}(A). Observe that the morphisms d_{n}^{2}(A) : A ⟶ Ω^{n}Σ_{P}^{n}(A) by construction are left Ω^{n}(Mod(Λ))−approximations of A. We claim that the map d_{∞}^{2}(A) is a left Ω^{∞}(Mod(Λ))−approximation of A. Indeed let α : A ⟶ B with B ∈ Ω^{∞}(Mod(Λ)). Since B ∈ Ω^{n}(Mod(Λ)), ∀n ≥ 1, and since d_{n}^{2}(A) are left Ω^{n}(Mod(Λ))−approximations of A, there are morphisms g_{n} : Ω^{n}Σ_{P}^{n}(A) ⟶ B with d_{n}^{2}(A)∘g_{n} = α. Taking direct limits, we have that d_{∞}^{2}(A)∘ lim_{⟶}g_{n} = α. Hence Ω^{∞}(Mod(Λ)) is covariantly finite. Let Ω^{∞}(\underline{A}) be the image of

lim_{⟶}Ω^{n}Σ_{P}^{n}(A) in \underline{Mod}(Λ). Then obviously the morphism \underline{d}_{∞}^{2}(\underline{A}) : \underline{A} ⟶ Ω^{∞}(\underline{A}) is

the reflection of \underline{A} in Ω^{∞}(\underline{Mod}(Λ)) and the morphism d_{n}^{2}(\underline{A}) : \underline{A} ⟶ Ω^{n}Σ_{P}^{n}(\underline{A}) is the reflection of \underline{A} in Ω^{n}(\underline{Mod}(Λ)). Part (2) is left to the reader.

If Λ is right Noetherian, then there is a similar result for the categories of cosyzygy modules. We leave its formulation to the reader. Now let Λ be left coherent and right perfect. Consider the reflection Ω^{∞} : \underline{Mod}(Λ) ⟶ Ω^{∞}(\underline{Mod}(Λ)) of \underline{Mod}(Λ) in Ω^{∞}(\underline{Mod}(Λ)) constructed in the Proposition 6.2, and let R : R(\underline{Mod}(Λ)) ⟶ \underline{Mod}(Λ) be the costabilization functor. By section 3, R(\underline{Mod}(Λ)) = K_{Ac}(P_{Λ}) is the homotopy category of acyclic complexes of projectives.

Corollary 6.3. Let Λ be left coherent and right perfect. If Mod(Λ) is P_{Λ}−Co-Gorenstein, then the costabilization functor R : R(\underline{Mod}(Λ)) ⟶ \underline{Mod}(Λ) admits a left adjoint Ω̃^{∞} : \underline{Mod}(Λ) ⟶ R(\underline{Mod}(Λ)).

Proof. If Mod(Λ) is P_{Λ}−Co-Gorenstein, then from section 4 we have an identification R(\underline{Mod}(Λ)) = Ω^{∞}(\underline{Mod}(Λ)) and the assertion follows by Proposition 6.2.

Remark 6.4. (1) Our standard assumptions in this paper refer to pairs (C, X), where C is an additive category and any X −epic has a kernel, and then we usually require that any left X −approximation of an X −Gorenstein object is an admissible monic. The most natural example is the pair (Mod(Λ), P_{Λ}). Dually the most natural example of a pair satisfying the dual assumptions is the pair (Mod(Λ), I_{Λ}).

(2) One can define X −Gorenstein objects in Mod(Λ), choosing X to be the full subcategory of flat modules or the FP-injective modules or any other interesting subcategory of modules and to apply the theory. We leave the details to the reader.

Suppose now that Λ is left coherent and right perfect, so that P_{Λ} is functorially finite. Then as in section 2, the functors \underline{Ext}_{P_{Λ}}^{n}(A, B) and \overline{Ext}_{P_{Λ}}^{n}(A, B) are defined. Obviously \underline{Ext}_{P_{Λ}}^{n}(−, −) are the usual extension functors. The subcategory P_{Λ}^{⊥} = \{A ∈ Mod(Λ) : \overline{Ext}_{P_{Λ}}^{n}(P_{Λ}, A) = 0, ∀n ≥ 1 and \overline{Ext}_{P_{Λ}}^{0}(P_{Λ}, A) = Hom_{Λ}(P_{Λ}, A)\} consists of of all modules A such that there exists an exact sequence 0 ⟶ A ⟶ P^{0} ⟶ P^{1} ⟶ ⋯, where P^{i} are projective modules, such that the sequence ⋯ ⟶ Hom_{Λ}(P^{1}, P_{Λ}) ⟶ Hom_{Λ}(P^{0}, P_{Λ}) ⟶ Hom_{Λ}(A, P_{Λ}) ⟶ 0 is exact. Observe that the subcategory P_{Λ}^{⊥} is defined always, although the derived functors \overline{Ext}_{P_{Λ}}^{n} may not exist globally. Then for an arbitrary ring Λ, the subcategory P_{Λ}^{⊥} denotes the full subcategory of all modules with the property described above. For an arbitrary ring Λ the definitions of section 2, take the following form.

Definition 6.5. (1) A module A is called (projectively) stable if A ∈ ^{⊥}P_{Λ}. (2) A module A is called n−torsion-free if

\overline{Ext}^{i}(P_{Λ}, A) = 0, 1 ≤ i ≤ n and \overline{Ext}_{P_{Λ}}^{0}(P_{Λ}, A) = Hom_{Λ}(P_{Λ}, A).

(3) A is called torsion-free if A ∈ P_{Λ}^{⊥}, or equivalently A is n−torsion-free, ∀n ≥ 1.

We leave to the reader the formulation of the above definitions using the subcategory P_{Λ} and the subcategories I_{Λ}, I_{Λ}. The above terminology is natural since if Λ is a two-sided Noetherian (or more generally two-sided coherent) ring , then the modules in P_{Λ}^{⊥} are exactly the n−torsion free modules ∀n ≥ 1, in the sense of Auslander-Bridger [3].

Definition 6.6. [20], [3] A module A is called Gorenstein-projective if A is stable and torsionfree, i.e. A ∈ ^{⊥}P_{Λ} ∩P_{Λ}^{⊥}. Equivalently there exists an exact sequence ⋯ ⟶ P^{−1} ⟶ P^{0} ⟶ P^{1} ⟶ ⋯ of projective modules with Im(P^{−1} ⟶ P^{0}) = A and the sequence remains exact applying Hom_{Λ}(−, P_{Λ}). The full subcategory of all Gorenstein-projective modules is denoted by G_{P}(Mod(Λ)), and the induced stable category modulo projectives is denoted by G_{P}(\underline{Mod}(Λ)). The full subcategory G_{I}(Mod(Λ)) of Gorenstein-injective modules is defined dually and the induced stable category modulo injectives is denoted by G_{I}(\overline{Mod}(Λ)).

By the results of sections 2, 4, G_{P}(Mod(Λ)) is the largest resolving subcategory of Mod(Λ), such that the stable category G_{P}(\underline{Mod}(Λ)), is full triangulated subcategory of \underline{Mod}(Λ) and dually G_{I}(Mod(Λ)) is the largest coresolving subcategory of Mod(Λ), such that the stable category G_{I}(\overline{Mod}(Λ)), is a full triangulated subcategory of \overline{Mod}(Λ). If Λ is a two-sided Noetherian (or coherent) ring, the finitely presented Gorenstein-projective modules G_{P}(mod(Λ)), are exactly the modules in mod(Λ) with zero G-dimension in the sense of [3]. We note that the categories of Gorensteinprojective and Gorenstein-injective modules were introduced also in [20].

In case Λ is left and right coherent, then it is not difficult to see that the left adjoint Σ_{P} of the loop functor Ω : \underline{mod}(Λ) ⟶ \underline{mod}(Λ) is given by Σ_{P} = TrΩTr, where Tr is the Auslander-Bridger duality [3], and there exists an exact sequence 0 ⟶ Ext^{1}(TrA, Λ) ⟶ A ⟶ P^{A} ⟶ Σ_{P}(A) ⟶ 0, where A ⟶ P^{A} is the left P_{Λ}−approximation. This follows from the easily established fact that a left P_{Λ}− approximation of A can be obtained as follows. We denote by d = (−)^{∗} both the Λ−dual functors, d = (−)^{∗} = Hom_{Λ}(−, Λ). Let f : Q ⟶ A^{∗} be an epimorphism

with Q ∈ _{Λ}P. Then the composition A ⟶ A^{∗∗} ⟶^{f^{∗}} Q^{∗} is a left P_{Λ}−approximation of A, where A ⟶ A^{∗∗} is the canonical morphism. Moreover, setting D_{k}^{2} := Ω^{k}Σ_{P}^{k}, the unit d2 ; \underline{A} : \underline{A} ⟶ D_{k}^{2}(\underline{A}) of the adjoint pair (Σ_{P}^{n}, Ω^{n}) is the natural morphism introduced in [3], in case Λ is Noetherian. Similarly setting J_{k}^{2} = Σ_{P}^{k}Ω^{k}, the counit of the adjoint pair (Σ_{P}^{k}, Ω^{k}) is the natural morphism introduced in [3]. Note that the functors J_{k}^{2}, D_{k}^{2} play a fundamental role in [3]. One can develop the theory of [3] for arbitrary modules in a right Noetherian ring Λ, using that in the stable category \overline{Mod}(Λ) modulo injectives, the suspension functor Σ : \overline{Mod}(Λ) ⟶ \overline{Mod}(Λ) has a right adjoint Ω_{I} and similarly in a left coherent and right perfect ring using that in the stable category \underline{Mod}(Λ) modulo projectives, the loop functor Ω : \underline{Mod}(Λ) ⟶ \underline{Mod}(Λ) has a left adjoint Σ_{P}. Hence in each case one can define the corresponding functors D̃_{k}^{2} = Ω_{I}^{k}Σ^{k}, J̃_{k}^{2} = Σ^{k}Ω_{I}^{k}, and D̂_{k}^{2} = Ω^{k}Σ_{P}^{k}, Ĵ_{k}^{2} = Σ_{P}^{k}Ω^{k} and also the corresponding (adjunction) morphisms.

6.2. Complete Extension Functors, Complete Resolutions and Gorenstein Rings. A complete projective (injective) resolution of a module is a P_{Λ}− complete (I_{Λ}−complete) resolution in the sense of section 5. In this case the notions of complete projective or injective resolutions coincide with the notions introduced in [18]. First we note the following Corollary of section 5. The last part follows from the description of the stabilization of the stable module category in section 3 and the Morita theorem of Rickard [41].

Corollary 6.7. (1) A right Λ−module A has a complete projective resolution iff A has finite Gorenstein-projective resolution dimension.

(2) A right Λ−module A has a complete injective resolution iff A has finite Gorenstein-injective resolution dimension.

(3) ∀B ∈ Mod(Λ), the complete projective extension functors are computed as:

Ĥ_{P}^{∗}(−, B) ≅ Êxt_{P}^{∗}(−, B) : { \atop G_{P}(Mod(Λ))}^{op} ⟶ Ab.

(4) ∀A ∈ Mod(Λ), the complete injective extension functors are computed as:

Ĥ_{∗}^{I}(A, −) ≅ Êxt_{∗}^{I}(A, −) : { \atop G_{I}(Mod(Λ))} ⟶ Ab.

(5) If Λ and Γ are derived equivalent rings (and Λ, Γ are right coherent), then the projective stabilizations of their modules categories are triangle equivalent:

S(\underline{Mod}(Λ)) ≈ S(\underline{Mod}(Γ)) (and S(\underline{mod}(Λ)) ≈ S(\underline{mod}(Γ))).

Definition 6.8. An arbitrary ring Λ is called a right Gorenstein ring if any projective right module has finite injective dimension and any injective right module has finite projective dimension.

The following result presents various characterizations and properties of right Gorenstein rings. Its proof is a direct consequence of the results of the previous sections and of Propositions 6.1, 6.2.

Theorem 6.9. Let Λ be an arbitrary ring. Then the following are equivalent.

(1) Λ is a right Gorenstein ring.

(2) Mod(Λ) is P_{Λ}−Gorenstein category.

(3) Mod(Λ) is I_{Λ}−Gorenstein category.

(4) (^{⊥}P_{Λ}, P_{Λ}^{∞}, P_{Λ}) is an Auslander-Buchweitz context.

(5) (I_{Λ}^{⊥}, I_{Λ}^{∞}, I_{Λ}) is a dual Auslander-Buchweitz context.

(6) P_{Λ}^{∞} = I_{Λ}^{∞}

(7) d := sup{p.d.I : I ∈ I_{Λ}} = sup{i.d.P : P ∈ P_{Λ}} < ∞.

(8) The functor ^{⊥}P_{Λ}/P_{Λ} ⟶ D^{b}(Mod(Λ))/K^{b}(P_{Λ}) is a triangle equivalence.

(9) The functor I_{Λ}^{⊥}/I_{Λ} ⟶ D^{b}(Mod(Λ))/K^{b}(I_{Λ}) is a triangle equivalence.

(10) K^{b}(I_{Λ}) = K^{b}(P_{Λ}) as full subcategories of D^{b}(Mod(Λ)).

(11) Any right Λ−module has a complete projective resolution.

(12) Any right Λ−module has a complete injective resolution.

(13) Any right Λ−module has finite Gorenstein-projective resolution dimension.

(14) Any right Λ−module has finite Gorenstein-injective resolution dimension.

(15) The natural functor D^{b}(G_{P}(Mod(Λ))) ⟶ D^{b}(Mod(Λ)) is a triangle equivalence.

(16) The natural functor D^{b}(G_{I}(Mod(Λ))) ⟶ D^{b}(Mod(Λ)) is a triangle equivalence.

If one of the above equivalent statements is true, then we have the following: (α) Mod(Λ) is P_{Λ}−Co-Gorenstein and I_{Λ}−Co-Gorenstein. Moreover:

G_{P}(Mod(Λ))−gl.res.dim(Mod(Λ)) = G_{I}(Mod(Λ))−gl.cores.dim(Mod(Λ)) = d.

(β) The categories P_{Λ}^{∞} = I_{Λ}^{∞} are functorially finite, the category G_{P}(Mod(Λ)) is contravariantly finite and the category G_{I}(Mod(Λ)) is covariantly finite.

(γ) If Λ is left coherent and right perfect, then G_{P}(Mod(Λ)) is functorially finite. If Λ is right Noetherian, then G_{I}(Mod(Λ)) is functorially finite.

(δ) We have: Ω^{∞}(Mod(Λ)) = Ω^{d}(Mod(Λ)) = G_{P}(Mod(Λ)) and Σ^{∞}(Mod(Λ)) = Σ^{d}(Mod(Λ)) = G_{I}(Mod(Λ)). Hence P_{Λ}^{∞} = P_{Λ}^{≤d} and I_{Λ}^{∞} = I_{Λ}^{≤d}, where P_{Λ}^{≤d}, resp. I_{Λ}^{≤d}, is the full subcategory of all modules having projective, resp. injective, dimension bounded by d. The right finitistic projective dimension FPD(Λ) and the right finitistic injective dimension FID(Λ) of Λ are finite: FPD(Λ) = FID(Λ) = d < ∞.

(ϵ) There are triangle equivalences:

K_{Ac}(P_{Λ}) ≈ G_{P}(\underline{Mod}(Λ)) ≈ ^{⊥}P_{Λ}/P_{Λ} ≈ ^{⊥}(P_{Λ}^{∞})/P_{Λ} ≈ D^{b}(Mod(Λ))/K^{b}(P_{Λ}) ≈

≈ D^{b}(Mod(Λ))/K^{b}(I_{Λ}) ≈ (I_{Λ}^{∞})^{⊥}/I_{Λ} ≈ I_{Λ}^{⊥}/I_{Λ} ≈ G_{I}(\overline{Mod}(Λ)) ≈ K_{Ac}(I_{Λ}). The costabilization functors are the inclusions

G_{P}(\underline{Mod}(Λ)) ↪ \underline{Mod}(Λ), G_{I}(\overline{Mod}(Λ)) ↪ \overline{Mod}(Λ)

and the stabilization functors are given by

Ω^{−d}Ω^{d} : \underline{Mod}(Λ) ⟶ G_{P}(\underline{Mod}(Λ)), Σ^{−d}Σ^{d} : \overline{Mod}(Λ) ⟶ G_{I}(\overline{Mod}(Λ)).

(ζ) The complete projective extension functors are given ∀B ∈ Mod(Λ), by

Ext_{P}^{n}(−, B) = \underline{Mod}(Λ)[Ω^{n}Ω^{d}(−), Ω^{d}(\underline{B})], ∀n ∈ ℤ.

The complete injective extension functors are given ∀A ∈ Mod(Λ), by

Ext_{n}^{I}(A, −) = \overline{Mod}(Λ)[Σ^{d}(\underline{A}), Σ^{n}Σ^{d}(−)], ∀n ∈ ℤ.

The triangle equivalence G_{P}(\underline{Mod}(Λ)) ≈ G_{I}(\overline{Mod}(Λ)) in (ϵ), induces isomorphisms

Ext_{P}^{∗}(−, B)(A) ≅ Ext_{∗}^{I}(A, −)(B).

Hence the complete extension bifunctor is defined:

Ext^{∗}(−, −) : Mod(Λ)^{op} × Mod(Λ) ⟶ Ab.

(η) If Λ and Γ are derived equivalent right Gorenstein rings then there are triangle equivalences:

G_{P}(\underline{Mod}(Λ)) ≈ G_{P}(\underline{Mod}(Γ)) and G_{I}(\overline{Mod}(Λ)) ≈ G_{I}(\overline{Mod}(Γ)).

(θ) Λ is a QF-ring ⇔ K_{Ac}(P_{Λ}) ≈ \underline{Mod}(Λ) ≈ D^{b}(Mod(Λ))/K^{b}(P_{Λ}) ⇔ K_{Ac}(I_{Λ}) ≈ \overline{Mod}(Λ) ≈ D^{b}(Mod(Λ))/K^{b}(I_{Λ}) ⇔ G_{P}(Mod(Λ)) = G_{I}(Mod(Λ)) ⇔ Λ is a right Gorenstein ring of dimension zero ⇔ the above equivalences are true for Λ^{op}.

(ι) Λ is (homologically) regular, i.e. any right Λ−module has finite projective dimension ⇔ G_{P}(Mod(Λ)) = P_{Λ} ⇔ G_{I}(Mod(Λ)) = I_{Λ} ⇔ the stabilization of \underline{Mod}(Λ) is trivial ⇔ the stabilization of \overline{Mod}(Λ) is trivial. Similarly for left Λ−modules.

It follows by the above Theorem and also by the results of sections 4, 5 that the complete projective extension functors coincide with the Tate-Vogel cohomology functors, see [17], [18], [25], [26], [39]. Our results generalize the corresponding results of the above papers in much more general situations. In particular the above Theorem shows that for module categories, the Gorenstein property of Mod(Λ) with respect to the projectives and the injectives coincides.

From the above Theorem if Λ is a QF-ring or if r.gl.dimΛ < ∞ (more generally if any right module has finite projective dimension), then Λ is right Gorenstein. Observe however that in these cases the theory is trivial.

Remark 6.10. The general setting of sections 3, 4, 5 can be applied directly to the study of lattices over (Gorenstein) orders in the sense of Auslander [2], with analogous results. We leave the details to the reader.

Corollary 6.11. For a Noetherian ring Λ the following are equivalent.

(1) Λ is a right Gorenstein ring.

(2) Λ is a left Gorenstein ring.

(3) i.d_{Λ}Λ < ∞ and i.dΛ_{Λ} < ∞.

Proof. (1) ⇔ (3) If Λ is a right Gorenstein ring, then i.dΛ_{Λ} < ∞. By a result of Iwanaga [35], i.d_{Λ}Λ = sup{flat.dimE_{Λ}; E_{Λ} ∈ I_{Λ}}. Since any right injective has finite projective dimension bounded by d, we have i.d_{Λ}Λ < ∞. The converse follows from the results of [35]. The equivalence (2) ⇔ (3) follows similarly.

So the Gorenstein property is symmetric for Noetherian rings. In this case a Noetherian left (or right) Gorenstein ring is called simply a Gorenstein ring. By the above Corollary it follows that our definition of a Gorenstein ring agrees in the Noetherian case, with the definition introduced by Iwanaga [35] and used extensively by Enochs-Jenda, et al, in a large list of papers, see for instance [20], [22]. Hence our theory covers, presents new features, and generalizes the corresponding theory developed in [20], [22]. In particular we recover the results of [5] which were obtained using tilting theory, since an Artin algebra is called Gorenstein in the sense of Auslander-Reiten iff i.d_{Λ}Λ < ∞ and i.dΛ_{Λ} < ∞.

Example 6.12. Let Λ be a Noetherian ring.

(1) Trivially Λ is Gorenstein in case Λ is QF or of finite global dimension.

(2) If Λ is Gorenstein and G a finite group, then the group ring ΛG is Gorenstein. Any Quasi-Frobenius extension of a Gorenstein ring is Gorenstein [35]. It is not difficult to see that the ring of the lower triangular matrices of any size over a Gorenstein ring is Gorenstein.

(3) If Λ is an Artin algebra and G a finite group of automorphisms of Λ, then the skew group ring ΛG is Gorenstein iff Λ is Gorenstein [5]. If Λ, Γ are finitedimensional algebras over a field k, then Λ⊗_{k} Γ is Gorenstein iff Λ, Γ are Gorenstein [5]. Also it is not difficult to see [13] that if Λ is a Cohen-Macaulay Artin algebra [5] with dualizing bimodule ω, then the trivial extension Λ ⋉ω is Gorenstein.

(4) Let Λ be an F−Gorenstein Artin algebra, were F is an additive subfunctor of Ext_{Λ}^{1}(−, −) with enough projectives and injectives, in the sense of Auslander-Sølberg [9]. Then mod(Λ) is P(F)−Gorenstein and I(F)−Gorenstein category in the sense of section 4, where P(F) and I(F) are the categories of F−projective and F−injective modules respectively.

(5) If Λ is a local Artin algebra with J ac(Λ)^{2} = 0, then: Λ is Gorenstein iff Λ is representation-finite.

(6) There is a extensive literature concerning commutative (local) Noetherian Gorenstein rings. For more information we refer to [3], [11].

Definition 6.13. A ring Λ is called right Co-Gorenstein if any arbitrary syzygy module is P_{Λ}−torsion-free or equivalently if Mod(Λ) is P_{Λ}−Co-Gorenstein.

We have seen that if Λ is Gorenstein then Λ is left (and right) Co-Gorenstein. We don’t know if the converse is true. We don’t know also if for a (Noetherian) ring Λ, being left Co-Gorenstein is equivalent to being right Co-Gorenstein.

6.3. Artin Algebras. For the remaining of this section, we assume that Λ is an Artin algebra [10]. We denote by d = Hom_{Λ}(−, Λ) or by (−)^{∗} both the Λ−dual functors and by D the usual duality of Artin algebras. All the results of this and the previous sections are true for the module categories Mod(Λ) or mod(Λ) of a (Gorenstein) Artin Algebra Λ. In particular we have the following.

Corollary 6.14. (i) The following are equivalent.

(1) Λ is Gorenstein.

(2) mod(Λ) or equivalently Mod(Λ) is a Gorenstein category.

(3) P_{Λ}^{∞} = I_{Λ}^{∞}.

(4) i.d_{Λ}Λ < ∞ and i.dΛ_{Λ} < ∞, in which case i.d_{Λ}Λ = i.dΛ_{Λ}.

(5) There exists a triangle equivalence G_{P}(\underline{mod}(Λ)) ≈ D^{b}(mod(Λ))/K^{b}(P_{Λ}) or equivalently a triangle equivalence G_{P}(\underline{Mod}(Λ)) ≈ D^{b}(Mod(Λ))/K^{b}(P_{Λ}).

(6) There exists a triangle equivalence G_{I}(\overline{mod}(Λ)) ≈ D^{b}(mod(Λ))/K^{b}(I_{Λ}) or equivalently a triangle equivalence G_{I}(\overline{Mod}(Λ)) ≈ D^{b}(Mod(Λ))/K^{b}(I_{Λ}).

(7) The left-hand side analogues of (2), (3), (5) and (6). If this is the case, the categories P_{Λ}^{∞}, I_{Λ}^{∞}, G_{P}(mod(Λ)), G_{I}(mod(Λ)) are functorially finite in mod(Λ) and any finitely presented right Λ−module has a minimal left and right X −approximation, where X = P_{Λ}^{∞}, I_{Λ}^{∞}, G_{P}(mod(Λ)), or G_{I}(mod(Λ)). The categories G_{P}(mod(Λ)), G_{I}(mod(Λ)), P_{Λ}^{∞}, I_{Λ}^{∞} have Auslander-Reiten sequences, and the triangulated categories G_{P}(\underline{mod}(Λ)), G_{I}(\overline{mod}(Λ)) are triangle equivalent and they have Auslander-Reiten triangles.

(ii) (α) If Λ is Gorenstein, then there are isomorphisms

K_{0}(mod(Λ)) ≅ K_{0}(G_{P}(mod(Λ))) and K_{0}(\underline{mod}(Λ)) ≅ K_{0}(G_{P}(\underline{mod}(Λ))).

(β) If Λ, Γ are derived equivalent Gorenstein algebras, then

G_{P}(\underline{mod}(Λ)) ≈ G_{P}(\underline{mod}(Γ)), |Det(c_{Λ})| = |Det(c_{Γ})|, Ker(c_{Λ}) = Ker(c_{Γ}).

Moreover the triangle equivalence G_{P}(\underline{mod}(Λ)) ≈ G_{P}(\underline{mod}(Γ)) lifts to a triangle equivalence \underline{mod}(Λ) ≈ \underline{mod}(Γ) iff Λ, Γ are selfinjective.

(γ) If two Gorenstein Artin algebras have equivalent categories of Gorenstein -projective or Gorenstein-injective modules, then they are derived equivalent.

(δ) Λ is selfinjective ⇔ K_{Ac}(P_{Λ}) ≈ \underline{mod}(Λ) ≈ D^{b}(mod(Λ))/K^{b}(P_{Λ}) ⇔ K_{Ac}(I_{Λ}) ≈ \overline{mod}(Λ) ≈ D^{b}(mod(Λ))/K^{b}(I_{Λ}) ⇔ the above equivalences are true for Λ^{op}.

Remark 6.15. The equivalence (i)(1) ⇔ (2) has been proved first by Hoshino [34] and by Auslander-Reiten [5] and in case Λ is a commutative local Noetherian ring it has been proved by Auslander-Bridger [3]. The last part of (i) has been proved first by Auslander-Reiten [5], using tilting theory. The direction (1) ⇒ (4) in (i) of the above Corollary was proved first by Rickard [42] in the selfinjective case and then by Happel [32] in the Gorenstein case. The generalized form of the Happel-Rickard Theorem can be stated as follows (this is also a consequence of a general result due to Keller-Vossieck, see [36]):

• For any right coherent ring Λ the stabilization of \underline{mod}(Λ) is triangle equivalent to D^{b}(mod(Λ))/K^{b}(P_{Λ}). Further if two right coherent rings Λ and Γ are derived equivalent, then their stable module categories \underline{mod}(Λ), \underline{mod}(Γ) have triangle equivalent stabilizations.

An Artin algebra Λ is called right P_{Λ}-Co-Gorenstein if any finitely presented arbitrary syzygy right module is torsion-free or equivalently if mod(Λ) is P_{Λ}−Co-Gorenstein category. Similarly we can define, using the costabilization of \overline{mod}(Λ), when Λ is right I_{Λ}−Co-Gorenstein. Since the duality D induces an exact duality D : R(\underline{mod}(Λ)) ⟶ R(\overline{mod}(Λ^{op})) and a duality D : Ω^{∞}(mod(Λ)) ⟶ Σ^{∞}(mod(Λ^{op})), we have that Λ is right P_{Λ}−Co-Gorenstein iff Λ is left _{Λ}I−Co-Gorenstein. From now on we call a P_{Λ}−Co-Gorenstein algebra, simply right Co-Gorenstein.

Corollary 6.16. (1) Suppose that f.p.dΛ < ∞. Then for the costabilization functor R : R(\underline{mod}(Λ)) ⟶ \underline{mod}(Λ) we have: KerR = 0.

(2) If the costabilization functor R : R(\underline{mod}(Λ)) ⟶ \underline{mod}(Λ) satisfies KerR = 0 ( in particular if Λ is right Co-Gorenstein ), then Λ satisfies the so-called Nunke condition for finitely presented left modules: if A is a finitely generated left Λ−module satisfying Ext_{Λ}^{n}(A, Λ) = 0, ∀n ≥ 0, then A = 0. In particular if Λ is right Co-Gorenstein, then Λ satisfies the Generalized Nakayama Conjecture [10].

Proof. (1) If Λ satisfies the finitistic dimension conjecture with f.p.dΛ = d < ∞ and P^{•} is a non contractible complex in KerR, then we have a non contractible complex of projectives 0 ⟶ P^{0} ⟶^{f_{0}} P^{1} ⟶^{f_{1}} ⋯ in mod(Λ). Hence ∃t ≥ 0 such that Im(f_{t}) is not projective. Since p.d.Im(f_{t+d}) < ∞, we have p.d.Im(f_{t+d}) ≤ d. Hence Im(f_{t}) is projective and this is not the case. So KerR = 0.

(2) Let P^{•} ⟶ A be a projective resolution of A. Then we have an acyclic complex of projectives P^{•∗} in mod(Λ). Viewing P^{•∗} as an object of the costabilization R(\underline{mod}(Λ)), we have that R(P^{•∗}) = 0. Since KerR = 0, P^{•∗} = 0 in R(\underline{mod}(Λ)), i.e. P^{•∗} is contractible. But then P^{•} = P^{•∗∗} is contractible, and then A = 0.

Proposition 6.17. (1) Λ is left Co-Gorenstein iff Ω^{∞}(mod(Λ^{op})) ⊆ ⊥ ΛP. In this case we have: Ω^{∞}(mod(Λ^{op})) = G_{P}(mod(Λ^{op})) ⊆ ⊥ ΛP and ^{⊥}P_{Λ} = G_{P}(mod(Λ)) ⊆ Ω^{∞}(mod(Λ)).

(2) Λ is right Co-Gorenstein iff Ω^{∞}(mod(Λ)) ⊆ ^{⊥}P_{Λ}. In this case we have: Ω^{∞}(mod(Λ)) = G_{P}(mod(Λ)) ⊆ ^{⊥}P_{Λ} and ⊥ ΛP = G_{P}(mod(Λ^{op})) ⊆ Ω^{∞}(mod(Λ^{op})).

Proof. (1) By our previous results it suffices to show that if Λ is left Co-Gorenstein then ^{⊥}P_{Λ} ⊆ G_{P}(mod(Λ)). Let A in ^{⊥}P_{Λ}. Then Tr(A) ∈ Ω^{∞}(mod(Λ^{op})). Since Λ is left Co-Gorenstein, Ω^{∞}(mod(Λ^{op})) ⊆ ⊥ ΛP. Hence Tr(A) ∈ ⊥ ΛP and A is Gorensteinprojective [3]. Part (2) is dual.

The following is a direct consequence of the above Proposition.

Corollary 6.18. (1) Λ is right Co-Gorenstein iff the Λ−dual functor d induces a duality d : ⊥ ΛP ⟶ Ω^{∞}(mod(Λ)) or equivalently an exact duality d : G_{P}(\underline{mod}(Λ^{op})) ⟶ R(\underline{mod}(Λ)).

(2) Λ is left Co-Gorenstein iff the Λ−dual functor d induces a duality d : ^{⊥}P_{Λ} ⟶ Ω^{∞}(mod(Λ^{op})) or equivalently an exact duality d : G_{P}(\underline{mod}(Λ)) ⟶ R(\underline{mod}(Λ^{op})).

(3) Λ is left and right Co-Gorenstein iff the Λ−dual functor d induces an exact duality d : R(\underline{mod}(Λ)) ⟶ R(\underline{mod}(Λ^{op})). In this case we have:

Ω^{∞}(mod(Λ)) = G_{P}(mod(Λ)) = ^{⊥}P_{Λ} and ⊥ ΛP = G_{P}(mod(Λ^{op})) = Ω^{∞}(mod(Λ^{op})).

Lemma 6.19. (1) The following are equivalent: (α) i.dΛ_{Λ} < ∞. (β) P_{Λ}^{∞} ⊆ I_{Λ}^{∞}. (γ) ∃d ≥ 0 : Ω^{d}(mod(Λ)) ⊆ ^{⊥}P_{Λ}. If this is the case, then: ^{⊥}P_{Λ}^{∞} = ^{⊥}P_{Λ} and ⊥ ΛP = G_{P}(mod(Λ^{op})).

(2) If i.dΛ_{Λ} < ∞, then Λ is right Co-Gorenstein.

(3) If the category Ω^{d}(mod(Λ)) is closed under extensions, ∀d ≥ 1, then Λ is right Co-Gorenstein.

Proof. Part (1) is easy and the proof is left to the reader. Suppose i.dΛ_{Λ} = d < ∞. Then by (1) we have Ω^{d}(mod(Λ)) ⊆ ^{⊥}P_{Λ}. But then Ω^{∞}(mod(Λ)) ⊆ ^{⊥}P_{Λ}. Then Λ is right Co-Gorenstein by Proposition 6.17. Part (3) follows from [7].

Proposition 6.20. The following are equivalent. (α) Λ is Gorenstein. (β) Λ is left Co-Gorenstein and i.dΛ_{Λ} < ∞. (γ) Λ is right Co-Gorenstein and i.d_{Λ}Λ < ∞.

Proof. We prove only that (α) is equivalent to (β), since the proof of the other parts is similar. By our previous results condition (α) implies (β). Suppose that (β) is true. Since i.dΛ_{Λ} = d < ∞, we have by Lemma 6.19 that Ω^{d}(mod(Λ)) ⊆ ^{⊥}P_{Λ}. Hence by Proposition 6.17(1), we have Ω^{d}(mod(Λ)) ⊆ ^{⊥}P_{Λ} = G_{P}(mod(Λ)) ⊆ Ω^{∞}(mod(Λ)) ⊆ Ω^{d}(mod(Λ)). This implies that G_{P}(mod(Λ)) = Ω^{d}(mod(Λ)). Then by Corollaries 4.11, 6.14 we have that Λ is Gorenstein.

The next result includes simple proofs of some results due to Auslander-Reiten [7], see also [29]. For the notion of a k−Gorenstein algebra we refer to [7].

Corollary 6.21. (1) Suppose that Λ is k−Gorenstein ∀k. Then Λ is left and right Co-Gorenstein and Λ satisfies the Generalized Nakayama Conjecture.

(2) The following are equivalent: (α) Λ is Gorenstein. (β) Λ is k−Gorenstein ∀k and i.dΛ_{Λ} < ∞ (or i.d_{Λ}Λ < ∞ ). (γ) Λ is k−Gorenstein ∀k and P_{Λ}^{∞} (or P_{Λ^{op}}^{∞} ) is contravariantly finite. (δ) Λ is k−Gorenstein ∀k and f.p.dΛ < ∞ (or f.i.dΛ < ∞ . (ϵ) Λ is right Co-Gorenstein and f.p.dΛ < ∞ (or f.i.dΛ < ∞ ). (ζ) Λ is right Co-Gorenstein and { \atop Ω^{∞}(mod(Λ))} = mod(Λ).

Proof. (1) Suppose that Λ is k−Gorenstein for all k. Then by [3] we have that ∀d ≥ 0, any d−syzygy module is d−torsionfree. Hence the category Ω^{∞}(mod(Λ)) coincides with the category of torsionfree modules. Then by definition Λ is right Co-Gorenstein. Since the notion of a k−Gorenstein algebra is left-right symmetric, we have also that Λ is left Co-Gorenstein.

(2) (α) ⇔ (β) Follows directly from (1) and the above Proposition. (α) ⇒ (γ) follows from Corollary 6.14. (γ) ⇒ (δ) holds for any Artin algebra (see [6]) and (δ) ⇒ (ϵ) follows from (1). (ϵ) ⇒ (α) Let fin.p.dimΛ = d < ∞. Then P^{≤d}(mod(Λ)) = P^{∞}(mod(Λ)), where P^{≤d}(mod(Λ)) is the full subcategory of all modules with projective dimension bounded by d. By [7], we have that Ω^{d}(mod(Λ)) = Ω^{d+t}(mod(Λ)), ∀t ≥ 0. Hence Ω^{∞}(mod(Λ)) = Ω^{d}(mod(Λ)). Since Λ is right Co-Gorenstein, by Theorem 4.10 we have Ω^{∞}(mod(Λ)) = G_{P}(mod(Λ)). Hence G_{P}(mod(Λ)) = Ω^{d}(mod(Λ)), and by Corollary 6.14, Λ is Gorenstein. The equivalence (α) ⇔ (ζ) follows from our previous results. The parenthetical cases are treated similarly.

By the above results it is reasonable to conjecture:

• Λ is right Co-Gorenstein ⇔ Λ is left Co-Gorenstein.

If the conjecture is true, by Proposition 6.18 and Lemma 6.17(2), the Auslander-Reiten Conjecture, that any Artin algebra is Gorenstein if i.dΛ_{Λ} < ∞, is true. In any case we have the inclusions Gor ⊆ ∀k−Gor ⊆ Co − Gor between Gorenstein algebras, k−Gorenstein algebras ∀k and (left and right) Co-Gorenstein algebras. The above inclusions also show that if any (left and right) Co-Gorenstein algebra is Gorenstein, then another Conjecture due to Auslander-Reiten is true, namely that any k−Gorenstein algebra ∀k is Gorenstein.

The next Corollary follows directly from the above results and its proof is left to the reader. We note only that if dom.dimΛ = ∞ then Ω^{∞}(mod(Λ)) = Dom(Λ), is the full subcategory of modules of infinite dominant dimension [37].

Corollary 6.22. The following are equivalent.

(1) Λ is selfinjective.

(2) Λ is Gorenstein and dom.dimΛ = ∞.

(3) Λ is ( left or right ) Co-Gorenstein and dom.dimΛ = ∞.

Clearly the Λ−dual functors induce a duality d : G_{P}(mod(Λ)) ⟶ G_{P}(mod(Λ^{op})) and an exact duality \underline{d} : G_{P}(\underline{mod}(Λ)) ⟶ G_{P}(\underline{mod}(Λ^{op})), and the Nakayama functor N^{+} induces an equivalence N^{+} : G_{P}(mod(Λ)) ⟶ G_{I}(mod(Λ)) and a triangle equivalence \underline{N}^{+} : G_{P}(\underline{mod}(Λ)) ⟶ G_{I}(\overline{mod}(Λ)). The next result contains another characterization of Gorenstein algebras, which is based on the notion of duality of derived categories. It follows that the Gorenstein property is invariant under derived equivalence.

Theorem 6.23. (1) For an Artin algebra Λ, the following are equivalent: (α) Λ is Gorenstein. (β) The functor d = Hom_{Λ}(−, Λ) induces an exact duality:

\underline{\underline{R}}^{b}Hom(−, Λ) = \underline{\underline{R}}^{b}d : D^{b}(mod(Λ)) ⟶ D^{b}(mod(Λ^{op})).

(γ) The functor N^{+} = − ⊗_{Λ} D(Λ) induces a triangle equivalence:

− ⊗_{Λ}^{L} D(Λ) = \underline{\underline{L}}^{b}N^{+} : D^{b}(mod(Λ)) ⟶ D^{b}(mod(Λ)).

In this case we have the following commuting diagram of localization sequences, where all vertical arrows are exact dualities:

0 ⟶ K^{b}(P_{Λ}) ⟶ D^{b}(mod(Λ)) ⟶ G_{P}(\underline{mod}(Λ)) ⟶ 0
K^{b}(d)↓  ^{\underline{\underline{R}}^{b}d}↓  \underline{d}↓
0 ⟶ K^{b}(_{Λ}P) ⟶ D^{b}(mod(Λ^{op})) ⟶ G_{P}(\underline{mod}(Λ^{op})) ⟶ 0

and a commuting diagram of localization sequences, extending the ”exact sequences”

0 ⟶ P_{Λ} ↪ mod(Λ) ⟶ \underline{mod}(Λ) ⟶ 0, 0 ⟶ I_{Λ} ↪ mod(Λ) ⟶ \overline{mod}(Λ) ⟶ 0

where the vertical arrows are triangle equivalences:

0 ⟶ K^{b}(P_{Λ}) ⟶ D^{b}(mod(Λ)) ⟶ G_{P}(\underline{mod}(Λ)) ⟶ 0
K^{b}(N^{+})↓  ^{\underline{\underline{L}}^{b}N^{+}}↓  \underline{N}^{+}↓
0 ⟶ K^{b}(I_{Λ}) ⟶ D^{b}(mod(Λ)) ⟶ G_{I}(\overline{mod}(Λ)) ⟶ 0

(2) If Λ and Γ are derived equivalent, then: Λ is Gorenstein ⇔ Γ is Gorenstein.

Proof. (1) (α) ⇒ (β) If Λ is Gorenstein, then since the functor d induces a duality between the left and right Gorenstein projective modules, condition (β) follows from parts (15), (16) of Theorem 6.9 and Corollary 6.14. (β) ⇒ (γ) Trivial. (γ) ⇒ (α) It is well-known that under condition (γ), the module D(Λ) is a (generalized) tilting module, and this is equivalent to i.d_{Λ}Λ < ∞ and i.dΛ_{Λ} < ∞, i.e. Λ is a Gorenstein algebra. Clearly the above diagrams commute by construction.

(2) If F : D^{b}(mod(Λ)) ⟶ D^{b}(mod(Γ)) is a triangle equivalence, then by [43] we have that F commutes with the total derived functors \underline{\underline{L}}^{b}N_{Λ}^{+} and \underline{\underline{L}}^{b}N_{Γ}^{+}. Hence the assertion follows from part (1).

We note that if Λ is Gorenstein, then a Gorenstein-projective module is called a Cohen-Macaulay module in [5]. Since D induces a duality ⊥ ΛP ≈ I_{Λ}^{⊥}, if Λ is Gorenstein, the Gorenstein-injective modules G_{I}(mod(Λ)) = I_{Λ}^{⊥} coincide with the Co-Cohen-Macaulay modules in the sense of Auslander-Reiten [5], i.e. with the full subcategory D(⊥ ΛP). Note that many of the above results for Artin algebras can be extended easily to (Gorenstein) rings with a Matlis duality in the sense of [21].

We close this section discussing briefly the complete projective or injective extension functors for an Artin algebra Λ and the relative homology induced in mod(Λ) using the contravariant finiteness of I_{Λ} and the covariant finiteness of P_{Λ}. Of course the relative homology using the covariant finiteness of I_{Λ} and the contravariant finiteness of P_{Λ} is the usual (absolute) homology in mod(Λ). Define P_{Λ}−gl.codimΛ = sup{P_{Λ}−codimA; A ∈ mod(Λ)}, I_{Λ}−gl.dimΛ = sup{I_{Λ}−dimA; A ∈ mod(Λ)} as in section 2, using the covariant finiteness of P_{Λ} and the contravariantly finiteness of I_{Λ}. Since P_{Λ}, I_{Λ} are functorially finite the complete functors Ĥ_{P}^{∗}(−, B), Ĥ_{∗}^{I}(A, −) are defined and it is easy to see that:

Ĥ_{P}^{n+1}(A, B) = DĤ_{P}^{−n}(TrDB, A) and Ĥ_{P}^{−n}(A, B) = DĤ_{P}^{n+1}(B, DTrA), ∀n ≥ 0.

Similarly for the complete functors Ĥ_{∗}^{I}(A, −). Having describing the left projective approximation of a right Λ−module, it is not difficult to see that a right injective approximation of A is the composition N^{+}(P) ⟶ N^{+}N^{−}(A) ⟶ A, where P ⟶ N^{−}(A) is an epimorphism with P projective and N^{+}N^{−}(A) ⟶ A is the counit of the adjoint pair (N^{+}, N^{−}). In particular the right adjoint of the usual suspension functor Σ^{n} in \overline{mod}(Λ) is given by DTrΩ^{n}TrD, ∀n ≥ 0, and there exists an exact sequence 0 ⟶ DTrΩTrD(A) ⟶ N^{+}(P) ⟶ A ⟶ DExt_{Λ}^{1}(TrD(A), Λ) ⟶ 0. The proof of our final result is left to the reader, noting that most of the assertions can be generalized to right Noetherian or left coherent and right perfect rings.

Corollary 6.24. (1) P_{Λ}−gl.codimΛ = 0 iff gl.dimΛ ≤ 2 iff I_{Λ}−gl.dimΛ = 0 iff P_{Λ} is a reflective subcategory of mod(Λ) iff I_{Λ} is a coreflective subcategory of mod(Λ). In this case the reflection of A in P_{Λ} is given by the natural morphism A ⟶ A^{∗∗} and the coreflection of A in I_{Λ} is given by the natural morphism N^{+}N^{−}(A) ⟶ A.

(2) If gl.dimΛ ≥ 2, then: P_{Λ}−gl.codimΛ = gl.dimΛ − 2 = I_{Λ}−gl.dimΛ.

Acknowledgement

Part of this work was done between 1993-1995, when I was a Ph.D student in the University of Ioannina, Greece. This work has been completed during a one year stay at the Instituto De Matemáticas, U.N.A.M., Mexico, thanks to a postdoctoral fellowship. The author thanks all the members of the Institute, and especially Professor J. A. de la Peña, for their warm hospitality.

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