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Journal of Algebra 444 (2015) 367–503

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Relative homology, higher cluster-tilting theory and

categorified Auslander–Iyama correspondence

Apostolos BeligiannisDepartment of Mathematics, University of Ioannina, 45110 Ioannina, Greece

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 March 2014Communicated by Luchezar L. Avramov

MSC:18G2518G2018E3018E3518E1016G1016G5013F60

Keywords:Triangulated categoriesAbelian categoriesRelative homologyCluster tilting objectsGorenstein categoriesCalabi–Yau categoriesGorenstein-projective objectsHigher Auslander categories and ringsAuslander correspondence

We study homological properties of contravariantly finite rigid subcategories of an arbitrary triangulated or abelian category, concentrating at the class of higher cluster tilting subcategories. In both the triangulated and abelian context we present several new characterizations of such subcategories and we give conditions ensuring that the associated cluster tilted category is Gorenstein and/or stably Calabi–Yau. Thus we generalize and improve on several recent results of the literature. In the context of abelian categories, we prove a categorified version of higher Auslander correspondence, as developed by Iyama in the case of module categories. In this connection we investigate in detail homological properties of (stable) Auslander categories associated to higher cluster tilting subcategories in an abelian category, and we present applications to (possibly infinitely generated) cluster tilting modules over an arbitrary ring. Finally we give bounds on the global and representation dimension of certain categories of coherent functors associated to Cohen–Macaulay objects.

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368 A. Beligiannis / Journal of Algebra 444 (2015) 367–503

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368Part 1. Relative homology of rigid subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3741. Relative homology: Adams resolutions and cellular towers . . . . . . . . . . . . . . . . . . . . . . . 3752. Ghosts and extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3823. Rigid and corigid subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3934. Resolutions and homological dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Part 2. Higher cluster tilting subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115. Cluster-tilting and maximal rigid subcategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4116. Some cluster tilting subcategories are Gorenstein. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4197. . . .and stably Calabi–Yau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Part 3. Categorified higher Auslander correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4398. Categorified Auslander correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4399. Cluster tilting versus cluster cotilting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

10. Stable Auslander categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46811. Cluster tilting subcategories of Cohen–Macaulay objects . . . . . . . . . . . . . . . . . . . . . . . . 492Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

Introduction

Cluster tilting theory, depending on the context, emerged from two rather different sources. On one hand, in the setting of finitely generated modules over an Artin algebra, Iyama [40,41] introduced cluster tilting modules in order to build a higher Auslander correspondence, generalizing the classical correspondence between algebras of finite rep-resentation type and Auslander algebras as developed by Auslander in the early seventies [3]. On the other hand and approximately at the same time, Buan, Marsh, Reineke, Re-iten and Todorov, see [21–23], [24], [55,56], inspired by classical tilting theory, introduced cluster tilting objects in the context of cluster categories associated to a hereditary alge-bra, in order to categorify certain phenomena occurring in the recently developed theory of cluster algebras by Fomin and Zelevinsky [32,33]. There is a nice interplay between the two approaches, which converge in some important special cases, by providing links be-tween representation theory of hereditary Artin algebras and the combinatorics of cluster algebras, and by helping to understand several hidden features in classical tilting theory, see [59]. Originally cluster tilting objects in the triangulated setting, were defined and investigated for the cluster category associated to a hereditary Artin algebra, and in the abelian setting for the category of finitely generated modules over an Artin algebra and slight generalizations of those. Since then cluster tilting theory was rapidly developed in both settings and at present it plays an increasingly important role in categorifying, and providing new perspectives for, several phenomena of geometric, combinatorial, or homological nature in various different areas of research. It should be noted that in the course of time many new ideas and techniques have emerged and new connections to other areas have been established.

The principal aims of the present paper are the following:

A. Beligiannis / Journal of Algebra 444 (2015) 367–503 369

(A) Our first aim is to develop a categorified version of Iyama’s higher Auslander cor-respondence in the context of arbitrary abelian categories, to study in detail the structure of the associated (stable) Auslander categories, and to give applications to cluster tilting theory in categories of, possibly infinitely generated, modules over associative rings, not necessarily Artin algebras.

(T) Our second aim is to develop a relative homological theory in a triangulated category based on rigid subcategories, concentrating at the class of higher cluster tilting subcategories, and in this context to extend several results of homological nature about them which are known to be true in lower dimensions or under strong finiteness conditions. In particular we are interested in the Gorenstein and the Calabi–Yau property of the cluster tilted category associated to a higher-dimensional cluster tilting subcategory.

In order to explain further our motivation, we present below, as Theorems A and B, two of the main results of the paper in the abelian (A) and triangulated (T) setting, which generalize and extend related results of Keller and Reiten [46,47], Iyama [40], and others, referring to the text for definitions of all unexplained terms:

(A) Auslander in the early seventies proved the following remarkable result which played a fundamental role for much of the later developments in the representation theory of Artin algebras. Firstly, we recall that the dominant dimension dom.dimF of an abelian category F is the largest n ∈ N ∪ {∞} such that any projective object of Fhas an injective resolution whose first n terms are projective; the dominant dimension dom.dim Γ of an Artin algebra Γ is the dominant dimension of the abelian category mod-Γof finitely generated Γ-modules.

Theorem 1 (Auslander). (See [3].) There is a bijective correspondence between Morita equivalence classes of Artin algebras Λ of finite representation type and Artin algebras Γsuch that gl.dimΓ � 2 � dom.dim Γ.

A categorified version of Auslander’s correspondence was proved in [13]. First recall that the free abelian category of an additive category C is an abelian category F(C )together with an additive functor C −→ F(C ) which is universal for additive functors out of C to abelian categories in the sense that any additive functor C −→ A to an abelian category A factorizes uniquely through F(C ) via an exact functor. The free abelian category F(C ) of any additive category C always exists, it is unique up to equivalence, and it admits the following descriptions: mod-(C -mod)op ≈ F(C ) ≈ ((mod-C )-mod)op, where mod-C denotes the category of (contravariant) coherent functors C op −→ Ab from C to the category Ab of abelian groups, and C -mod = mod-C op. On the other hand we call an abelian category F with enough projectives and injectives, an Auslander categoryif gl.dimF � 2 � dom.dimF. A main example is the category of finitely generated modules over the Auslander algebra Γ of a representation finite Artin algebra Λ.


The following result recovers and gives a categorified version of classical Auslander’s correspondence, the latter is achieved by evaluating the bijections below at the category C = projΛ of finitely generated projective modules over a representation finite Artin algebra Λ and the category F = mod-Γ of finitely generated modules over an Auslander algebra Γ. For more details we refer to Theorem 8.2.

Theorem 2. (See [13, Theorems 6.1 and 6.6].) An abelian category F is free if and only if F is an Auslander category. Moreover the maps

C �−→ F(C ) and F �−→ ProjF ∩ InjF

give, up to equivalence, mutually inverse bijections between:

(I) Additive categories C (with split idempotents).(II) Auslander categories F.

Recently Iyama proved, as one of the main results of [40], the following far reach-ing generalization of Auslander’s correspondence which in addition gives interesting connections with unstable cluster tilting theory. Recall that a finitely generated mod-ule M over a finite-dimensional algebra Λ is called an n-cluster tilting module if {X ∈ mod-Λ | ExtkΛ(M, X) = 0, 1 � k � n} = addM = {X ∈ mod-Λ | ExtkΛ(X, M) = 0,1 � k � n}, see [40,41], where the terminology (n +1)-cluster tilting module or maximal n-orthogonal module is used.

Theorem 3 (Iyama). (See [40].) For any n � 0, the map M �−→ EndΛ(M) gives a bijec-tion between equivalence classes of n-cluster tilting Λ-modules M for finite-dimensional algebras Λ and Morita-equivalence classes of finite-dimensional algebras Γ such that gl.dim Γ � n + 2 � dom.dim Γ.

Our first aim in this paper is to refine and generalize Theorems 1, 2 and 3 in the context of abelian categories. In fact we shall prove the following categorification of Theorem 3which shows that Theorem 2 is the zero case of a natural correspondence at a higher level. First we need some definitions. A full subcategory M of an abelian category A is called an n-cluster tilting subcategory, if (α) M is contravariantly finite in A and any right M-approximation is an epimorphism, and (β) M coincides with the full subcategory M⊥

n := {A ∈ A | ExtkA (M, A) = 0, 1 � k � n}. An abelian category B is called an n-Auslander category if B has enough projectives and satisfies gl.dim B � n + 2 �dom.dim B. A ring Λ is called a (right) n-Auslander ring if Mod-Λ is an n-Auslander category. Finally an object X in an additive category A with infinite coproducts is called self-compact if the functor A (X, −) : AddX −→ Ab preserves coproducts, where AddXdenotes the full subcategory of A consisting of all direct factors of arbitrary coproducts of copies of X.

Theorem 3 provides motivation for the following result.


Theorem A (Generalized Auslander correspondence [Theorems 8.23 and 9.9]). For any integer n � 0, the map

M �−→ mod-M

gives, up to equivalence, a bijective correspondence between:

(I) n-cluster tilting subcategories M in abelian categories A with enough injectives.(II) n-Auslander categories B.

Moreover the map

{A ,M

}�−→ Λ = EndA (M)

gives a bijective correspondence between (cluster/Morita) equivalence classes of

(III) Pairs {A , M} where A is a Grothendieck category and M ∈ A is a self-compact n-cluster tilting object.

(IV) Left coherent and right perfect n-Auslander rings Λ.

Note that “Grothendieck” can be replaced by “cocomplete abelian” in Theorem A; the latter when specialized to M = addM for an n-cluster tilting Λ-module M over an Artin algebra Λ, gives Iyama’s Theorem 3.

Note that the abelian category A in the above correspondence is uniquely determined, up to equivalence, by the n-cluster tilting subcategory M it contains: in fact A is equiv-alent to the dual of the category of coherent covariant functors over the full subcategory of projective–injective objects of mod-M. We also define n-cluster cotilting subcategories and n-coAuslander categories and we prove a dual version of the above result. In fact we show that in most cases a subcategory is n-cluster tilting iff it is n-cluster cotilting. Concerning functoriality of the construction, we define the (large) category ClustTilt(n)consisting of pairs (A , M) consisting of an abelian category A with enough injectives and an n-cluster tilting subcategory M of A , and the (large) category AuslCat(n) with objects n-Auslander categories, and we prove that there is an equivalence of categories

ClustTilt(n) ∼=−−→ AuslCat(n), (A ,M) �−→ mod-M

(T) Let T be a triangulated category. Recall that a full subcategory X of T is called (n + 1)-cluster tilting subcategory if: (α) X is contravariantly finite in T, and (β) X

coincides with the full subcategory X�n := {A ∈ T | T(X, A[k]) = 0, 1 � k � n}.

The associated n-cluster tilted category is defined to be the (abelian) category mod-Xof coherent functors over X. In this context we give several characterization of when a subcategory of T is n-cluster tilting. It should be noted that this definition of cluster


tilting subcategory is more general than the definition used in the current literature, e.g. see [44], but we prove that both definitions are equivalent.

Usually one is interested in homological properties of the associated cluster tilted category mod-X which reflect combinatorial or geometric properties of X and T. In this connection, Keller and Reiten [46] proved the following remarkable result, see also [16]for a more general version.

Theorem 4 (Keller–Reiten). (See [46].) Let T be a 2-Calabi–Yau triangulated category and X a 2-cluster tilting subcategory of T. Then the cluster tilted category mod-X is 1-Gorenstein and stably 3-Calabi–Yau.

Recall that an abelian category A with enough projectives and injectives, is called n-Gorenstein if the projective, resp. injective, dimension of any injective, resp. projective, object of A is bounded by n. Such a category is called stably k-Calabi–Yau if the stable triangulated category CM(A ) of Cohen–Macaulay objects of A is k-Calabi–Yau. It is known that the conclusions of Theorem 4 are false for n-cluster tilting subcategories X, if n > 2. This provides motivation for the next result which generalizes, and gives a higher dimensional analog of, Theorem 4. First we say that a full subcategory X of T is k-corigid, k � 1, if T(X, X[−t]) = 0, 1 � t � k.

Theorem B (Theorems 6.4 and 7.6). Let T be a triangulated category and X an (n +1)-cluster tilting subcategory of T which is (n −k)-corigid, for some non-negative integer k with: 0 � k � n+1

2 . Then:

1. The cluster tilted category mod-X is k-Gorenstein.2. If T is (n + 1)-Calabi–Yau and any object of mod-X of the form T(−, A)|X, where

A lies in the extension category X[−n + 1] � · · · � X[−n + k], has finite projective dimension, then the cluster tilted category mod-X is stably (n + 2)-Calabi–Yau, i.e. the triangulated category CM(mod-X) is (n + 2)-Calabi–Yau.

In particular for any (n − 1)-corigid (n +1)-cluster tilting subcategory X of T, i.e. the case k = 1 in Theorem B, the cluster tilted category mod-X is 1-Gorenstein and mod-Xis stably (n + 2)-Calabi–Yau. This special case was proved independently by Iyama and Oppermann [43].

Now we briefly describe the organization and the contents of the paper, which is divided into three parts.

1. In the first part, which consists of the first four sections, we develop a homological theory in a general triangulated category T based on a contravariantly or covariantly finite subcategory X of T, concentrating at n-rigid subcategories, in the sense that T(X, X[k]) = 0, for 1 � k � n. We analyze the structure of T in connection with homological properties of the associated abelian category mod-X of coherent functors over X.


The main tools for this study is the ideal GhX(T) of X-ghost maps in T with respect to X, i.e. maps that are invisible by the homological functor H : T −→ mod-X, H(A) =T(−, A)|X, and the category X � X[1] � · · · � X[n] of extensions, in the sense of [10], of (positively) shifted copies of X. In this connection we characterize when the ideal of X-ghost maps is nilpotent in an appropriate sense or when the category of shifted extensions of X covers T, in terms of finiteness of the global dimension gl.dimX T of Twith respect to X, see Theorem 3.4. Also we construct suitable resolutions of objects of T by objects of (shifted copies of) X, see Theorem 4.2, and we give necessary conditions in order to realize the category of objects of finite projective or injective dimension of mod-X as a full subcategory of T, see Theorems 4.6 and 4.7.

The results of this part are used in the second part for deriving the main properties of a cluster tilting subcategory and its associated cluster tilted category. It should be noted that a similar relative homological theory was developed in [12], see also [29], in case X is stable in the sense that X = X[1]. However the non-stable case behaves differently and, besides some formal analogies, the obtained results are of a different nature.

2. The second part of the paper, consisting of Sections 5, 6 and 7 is devoted to the analysis of the structure of a cluster tilting subcategory X in a triangulated category T, and also to deriving the main homological properties of the cluster tilted category mod-Xassociated to X.

First in Section 5 we present a host of characterizations of when a subcategory X of T is (n +1)-cluster tilting, see Theorem 5.3. These characterizations are new in this generality. In particular we show that half of the conditions in the definition of cluster tilting used in the current literature can be removed. In Section 6 we show in Theorem 6.4 half of Theorem B concerning the Gorenstein property of the cluster tilted category, we prove its analogue for maximal rigid subcategories, and we analyze some direct consequences. More precisely we give applications to the Morita Theorem for cluster categories due to Keller and Reiten [47], to representation dimension of the cluster tilted category, and also we present conditions ensuring that the latter has finite global dimension.

In Section 7, we prove in Theorem 7.6 the second half of Theorem B concerning the stable Calabi–Yau property of the cluster tilted category and we present some of its immediate consequences.

3. The third part of the paper, which consists of Sections 8, 9, 10, and 11, is de-voted to the categorification of the Auslander–Iyama correspondence in the context of abelian categories and to the analysis of the structure of (stable) Auslander categories B associated to a cluster tiling subcategory M of an abelian category A .

More precisely in Section 8 we prove the first part of Theorem A and we discuss its variations: for instance we prove in Theorem 8.32 that suitably defined big categories of n-cluster tilting subcategories in abelian categories with injectives and n-Auslander categories are equivalent, so the categorified Auslander–Iyama correspondence is func-torial. We also discuss universality of the correspondence and we give some applications to K-theory of Auslander categories.


In Section 9, we first show in Theorem 9.6 that cluster tilting subcategories coin-cide with cluster cotilting ones, so half of the conditions of the definition of cluster tilting subcategories in abelian categories can be removed. Then we characterize in The-orem 9.9, left coherent and right perfect n-Auslander rings as the endomorphism rings of self-compact n-cluster tilting objects in Grothendieck categories, or more generally in cocomplete abelian categories with enough injectives, thus completing the proof of Theorem A.

Section 10 is devoted to analyzing the structure of the Auslander category mod-Mand its stable analogue mod-M, associated to an n-cluster tilting subcategory M in an abelian category A . After observing that the n-Auslander category mod-M appears in the middle of a natural recollement situation with end terms the stable n-Auslander category mod-M and the abelian category A , we use duality between mod-M and M-modto describe the projective/injective objects of mod-M and we present the latter as a suitable subquotient category of A , see Theorem 10.7. Next in Theorems 10.13, 10.17and 10.14, we give necessary conditions for a stable Auslander category mod-M to be of finite global dimension or Gorenstein, and when mod-M is itself an Auslander category. Finally in Theorem 10.22 we give applications to n-Auslander rings.

In the final Section 11 of the paper we investigate homological properties of sub-categories M ⊆ A arising from n-cluster tilting subcategories M ⊆ CM(A ) of the triangulated category CM(A ) of Cohen–Macaulay objects in a Gorenstein abelian cat-egory A . In the main result in this section, see Theorem 11.3, we describe homological properties of mod-M and we compute its global dimension as the maximum of n +2 and the Gorenstein dimension of A . In addition we characterize when mod-M is an Auslan-der category and we apply our results to the category of finitely generated modules over an Artin algebra, see Theorem 11.10.

Conventions. A general convention used in the paper is that the composition of morphisms in a given category, but not the composition of functors, is meant in the diagrammatic order. Our additive categories contain finite direct sums and their subcat-egories are assumed to be closed under isomorphisms and direct summands.

Part 1. Relative homology of rigid subcategories

In [12] a relative homological theory in a triangulated category T was developed based on a contra(co)variantly finite subcategory X which is stable, that is, closed under the suspension functor: X = X[1]. In the first part of the paper we develop an analogous, but in many aspects radically different, homological theory on T based on a contra(co)vari-antly finite subcategory X which is not stable but usually is rigid, that is, T(X, X[1]) = 0. In this context the, appropriately defined, ideal of ghost maps and categories of exten-sions with respect to X play an important role in connection with the behavior of certain (co)resolutions and (co)homological dimensions.


1. Relative homology: Adams resolutions and cellular towers

Let C be an additive category and X a full additive subcategory of C . Recall that X is called contravariantly finite in C if for any object A in C there exists a map fA : XA −→ A, where XA lies in X, such that the induced map C (X, fA) : C (X, XA) −→C (X, A) is surjective. In this case the map fA is called a right X-approximation of A. Dually we have the notions of covariant finiteness and left approximations. Then X is called functorially finite if it is both contravariantly and covariantly finite.

For a class of objects V ⊆ C we denote by addV the full subcategory of C consisting of the direct factors of finite direct sums of copies of objects from V. If C contains all small coproducts, resp. products, then AddV, resp. ProdV, denotes the full subcategory of Cconsisting of the direct factors of coproducts, resp. products, of copies of objects from V. If C is an abelian category, then we denote by ProjC , resp. InjC , the full subcategory of projective, resp. injective, objects of C . We denote by Ab the category of abelian groups.

The stable category of C with respect to a full subcategory X is denoted by C /X. Re-call that the objects of C /X are the objects of C , and the morphism space HomC/X(A, B)is the quotient C (A, B)/CX(A, B), where CX(A, B) is the subgroup of C (A, B) consist-ing of all maps A −→ B factorizing through an object from X. We then have a full dense additive functor π : C −→ C /X, π(A) = A and π(f) = f .

An additive functor F : C op −→ Ab is called coherent, see [2], if there exists an exact sequence C (−, A1) −→ C (−, A0) −→ F −→ 0, where the Ai lie in C . We denote by mod-C the category of coherent functors over C . Recall that a weak kernel of a morphism B −→ C in C is a map A −→ B such that the sequence of functors C (−, A) −→C (−, B) −→ C (−, C) is exact. The additive category C is called right coherent if Chas weak kernels, i.e. any morphism in C has a weak kernel. It is easy to see that a contravariantly finite subcategory of a right coherent category is right coherent. Note that the additive category C is right coherent if and only if the category mod-C is abelian; in this case mod-C has enough projectives and if idempotents split in C , then the Yoneda full embedding C −→ mod-C , A �−→ C (−, A), induces an equivalence C ≈Projmod-C . Weak cokernels and left coherent categories are defined dually and they have dual properties. Finally if C is skeletally small, we denote by Mod-C the category of contravariant functors from C to the category Ab of abelian groups.

From now on and for the rest of the paper we denote by T a triangulated category with split idempotents; its suspension functor is denoted by A �−→ A[1]. We also fix a full additive contravariantly finite subcategory X of T which is closed under isomorphisms and direct summands.

For any object A in T we construct inductively triangles, ∀t � 0:

Ωt+1X (A) gt

A−−→ XtA

ftA−−→ Ωt

X(A) htA−−→ Ωt+1

X (A)[1] (T tA)

where Ω0X(A) := A, and the middle map Xt

A −→ ΩtX(A) is a right X-approximation

of ΩtX(A). Applying the homological functor H : T −→ Mod-X, defined by H(A) =


T(−, A)|X, to the above triangles we have exact sequences H(Ωt+1X A) −→ H(Xt

A) −→H(Ωt

XA) −→ 0, ∀t � 0. In particular we have an exact sequence

H(X1A) −→ H(X0

A) −→ H(A) −→ 0 (1.1)

which is a projective presentation of H(A). Since T, as a triangulated category, is right coherent and X is contravariantly finite in T, it follows that X is right coherent and therefore mod-X is abelian with enough projectives. Since T has split idempotents and Xis closed under direct summands, the functor H induces an equivalence X ≈ Proj mod-X. It follows that the functor H(A) is coherent and therefore the restricted Yoneda functor H induces a homological functor

H : T −→ mod-X, H(A) = T(−, A)|X

Clearly we have: Ker H = X� :={A ∈ T | T(X, A) = 0

}.

Remark 1.1. Dually if X is covariantly finite in T, then X is left coherent and therefore the category X-mod := mod-Xop of covariant coherent functors is abelian. For any object A in T we have triangles, ∀t � 0:

Σt+1X (A)[−1] hA

t−−→ ΣtX(A) fA

t−−→ XAt

gAt−−→ Σt+1

X (A) (TAt )

where Σ0X(A) := A, and the middle map Σt

X(A) −→ XAt is a left X-approximation of

ΣtX(A). Further we have a contravariant cohomological functor

Hop : Top −→ X-mod, Hop(A) = T(A,−)|X

which induces a duality Xop ≈ Proj(X-mod) and clearly: Ker Hop = �X := {A ∈ T |T(A, X) = 0}.

We denote by T/X the stable category of T with respect to X; recall that the objects of T/X are the objects of T, and the morphism space HomT/X(A, B) is the quotient T(A, B)/TX(A, B), where TX(A, B) is the subgroup of T(A, B) consisting of all maps A −→ B factorizing through an object from X. We then have an additive functor π : T −→T/X, π(A) = A and π(f) = f .

Remark 1.2. Although the object Ω1X(A) appearing in the triangle (T 0

A) depends on the choice of the right X-approximation of A, it is not difficult to see that it is uniquely determined in the stable category T/X. Moreover the assignment A �−→ Ω1

X(A) induces a well-defined additive functor Ω1

X : T/X −→ T/X. Dually if X is covariantly finite, then the object Σ1

X(A) defined by the triangle (TA0 ) is uniquely determined in T/X and the

assignment A �−→ Σ1X(A) induces a well-defined additive functor Σ1

X : T/X −→ T/X. If X is functorially finite then we have an adjoint pair of endofunctors

(Σ1

X, Ω1X

): T/X −→

T/X; we refer to [16] for more details.


1.1. Ghost and cellular towers

Fix an object A in T and consider as before the triangles, ∀t � 0:

Ωt+1X (A) gt

A−−→ XtA

ftA−−→ Ωt

X(A) htA−−→ Ωt+1

X (A)[1] (T tA)

Then we have as in [12, Section 5], a tower of triangles in T, henceforth denoted by (C•A):

Ω1X(A) β0

A−−−−→ Cell0(A) γ0A−−−−→ A

ω0A−−−−→ Ω1

X(A)[1] (C0A)

h1A

⏐⏐� ⏐⏐�α1A

∥∥∥ ⏐⏐�h1A[1]

⏐⏐�Ω2

X(A)[1] β1A−−−−→ Cell1(A) γ1

A−−−−→ Aω1

A−−−−→ Ω2X(A)[2] (C1

A)

h2A[1]

⏐⏐� ⏐⏐�α2A

∥∥∥ ⏐⏐�h2A[2]

⏐⏐�Ω3

X(A)[2] β2A−−−−→ Cell2(A) γ2

A−−−−→ Aω2

A−−−−→ Ω3X(A)[3] (C2

A)⏐⏐� ⏐⏐� ∥∥∥ ⏐⏐� ⏐⏐�...

......

......⏐⏐� ⏐⏐� ∥∥∥ ⏐⏐� ⏐⏐�

ΩnX(A)[n− 1]

βn−1A−−−−→ Celln−1(A)

γn−1A−−−−→ A

ωn−1A−−−−→ Ωn

X(A)[n] (Cn−1A )

hnA[n−1]

⏐⏐� ⏐⏐�αnA

∥∥∥ ⏐⏐�hnA[n]

⏐⏐�Ωn+1

X (A)[n] βnA−−−−→ Celln(A) γn

A−−−−→ Aωn

A−−−−→ Ωn+1X (A)[n + 1] (Cn

A)⏐⏐� ⏐⏐� ∥∥∥ ⏐⏐� ⏐⏐�...

......

......

(1.2)

where: Cell0(A) := X0A, ω0

A := h0A, γ0

A := f0A, and β0

A := g0A. The tower (C•

A) is con-structed inductively by forming cobase change, in the sense of [12, Proposition 2.1], or taking homotopy push-outs in the sense of [51], of the triangles (Cn−1

A ) along the maps hnA[n − 1] : Ωn

X(A)[n − 1] −→ Ωn+1X (A)[n], ∀n � 1.

Definition 1.3. For any object A ∈ T, the map γnA : Celln(A) −→ A constructed above is

called the nth-cellular approximation of A and the induced tower (Cell•A):

X0A = Cell0(A) α1

A−−→ Cell1(A) α2A−−→ Cell2(A) −→ · · · · · ·

−→ Celln−1(A) αn−1A−−−−→ Celln(A) −→ · · ·


is called the cellular tower of A with respect to X. The tower of objects and X-ghost maps (Gh•A):

A = Ω0X(A) h0

A−−→ Ω1X(A)[1] h1

A[1]−−−−→ Ω2X(A)[2] −→ · · · · · ·

−→ ΩnX(A)[n] hn

A[n]−−−−→ Ωn+1X (A)[n + 1] −→ · · ·

is called the Ghost tower, or an Adams resolution, of A with respect to X. It is convenient to represent an Adams resolution by the following tower of triangles

Ah0A Ω1

X(A)[1]

+1

h1A[1]

Ω2X(A)[2]

+1

h2A[2]

Ω3X(A)[3]

+1

h3A[3]

Ω4X(A)[4]

+1

· · ·

X0A X1

A[1] X2A[2] X3

A[3]

where ΩnX(A)[n] +1−→ Xn−1

A [n − 1] denotes a map of degree +1, i.e. a map ΩnX(A)[n] −→

Xn−1A [n], ∀n � 1.

Recall from [10] that if U and V are subcategories of T, then the extension categoryU � V consists of all direct summands of objects A for which there exists a triangle U −→ A −→ V −→ U [1], where U ∈ U and V ∈ V. Note that if T(U, V) = 0, then U � Vis closed under direct summands, see [44]. The extension category U1 � U2 � · · · � Un is defined inductively for subcategories Ui of T, 1 � i � n.

Remark 1.4.

(i) By the construction of the tower (C•A) we have triangles, ∀t � 1:

XtA[t− 1] −→ Cellt−1(A) αt

A−−→ Cellt(A) −→ XtA[t] (1.3)

and

XtA[t] −→ Ωt

X(A)[t] htA[t]−−−−→ Ωt+1

X (A)[t + 1] −→ XtA[t + 1] (1.4)

Since Cell0(A) = X0A ∈ X, it follows that we have a triangle X0

A −→ Cell1(A) −→X1

A[1] −→ X0A[1] and therefore Cell1(A) ∈ X � X[1]. By induction we infer that:

Cellt(A) ∈ X � X[1] � · · · � X[t], ∀t � 0

(ii) If ΩnX(A) lies in X, then Ωn−t

X (A) lies in X � X[1] � · · · � X[t] and in particular A ∈ X �X[1] � · · · �X[n]. In this case we may choose Xn

A = ΩnX(A) and Ωn+t

X (A) = 0, ∀t � 1.


(iii) Splicing together the triangles (T tA) and setting εnA = fn

A ◦ gn−1A , ∀n � 1, we obtain

a complex

· · · −→ XnA

εnA−−→ Xn−1A −→ · · · · · · −→ X1

Aε1A−−→ X0

Af0A−−→ A −→ 0 (X•

A)

over A by objects of X. Note that although the sequence H(X1A) −→ H(X0

A) −→H(A) −→ 0 is exact, in general the induced complex H(X•

A) in mod-X is not neces-sarily exact.

1.2. Cocellular and Coghost towers

Dually if X is covariantly finite in T, then as in Remark 1.1, for any object A ∈ T we may construct inductively triangles, t � 0:

Σt+1X (A)[−1] hA

t−−→ ΣtX(A) fA

t−−→ XAt

gAt−−→ Σt+1

X (A) (TAt )

where the middle map is a left X-approximation of Σt(A), and Σ0X(A) := A. As in

Subsection 1.1, taking inductively homotopy pull-backs of the triangle (TAn ) along the

maps hAn [−n + 1], ∀n � 0, we obtain the following inverse tower of triangles, henceforth

denoted by (CA• ):

......

......⏐⏐� ∥∥∥ ⏐⏐� ⏐⏐�

ΣnX(A)[−n] ωA

n−−−−→ AβAn−−−−→ Celln−1(A) γA

n−−−−→ ΣnX(A)[−n + 1]

hAn−1[−n+1]

⏐⏐� ∥∥∥ ⏐⏐�αAn−1

⏐⏐�hAn−1[−n+2]

Σn−1X (A)[−n + 1]

ωAn−1−−−−→ A

βAn−1−−−−→ Celln−2(A)

γAn−1−−−−→ Σn−1

X (A)[−n + 2]⏐⏐� ∥∥∥ ⏐⏐� ⏐⏐�...

......

...⏐⏐� ∥∥∥ ⏐⏐� ⏐⏐�Σ3

X(A)[−3] ωA2−−−−→ A

βA2−−−−→ Cell2(A) γA

2−−−−→ Σ3X(A)[−2]

hA2 [−2]

⏐⏐� ∥∥∥ ⏐⏐�αA2

⏐⏐�hA2 [−1]

Σ2X(A)[−2] ωA

1−−−−→ AβA1−−−−→ Cell1(A) γA

1−−−−→ Σ2X(A)[−1]

hA1 [−1]

⏐⏐� ∥∥∥ ⏐⏐�αA1

⏐⏐�hA1

1 ωA0 βA

0 0 γA0 1

(1.5)

ΣX(A)[−1] −−−−→ A −−−−→ Cell (A) −−−−→ ΣX(A)


where we set: Cell0(A) := XA0 , ωA

0 := hA0 , βA

0 := fA0 , and γA

0 := gA0 . The inverse tower of objects (CellA• ):

· · · −→ Celln+1(A)αA

n+1−−−−→ Celln(A) −→ · · · · · ·

−→ Cell2(A) αA2−−→ Cell1(A) αA

1−−→ Cell0(A) = XA0

is called the cocellular tower of A with respect to X, and the inverse tower of objects (CoGhA• ):

· · · −→ Σn+1X (A)[−n− 1] hA

n [−n]−−−−−→ ΣnX(A)[−n] −→ · · ·

−→ Σ2X(A)[−2] hA

1 [−1]−−−−−→ Σ1X(A)[−1] hA

0−−→ Σ0X(A) = A

is called the coghost tower, or an Adams coresolution, of A with respect to X. We leave to the reader to state the dual version of Remark 1.4.

1.3. The functor H : T −→ mod-X

We close this section by analyzing the main properties of the homological functor H : T −→ mod-X, H(A) = T(−, A)|X, which will play a basic role in the rest of the paper.

We call an additive functor F : A −→ B between additive categories A and Balmost full if for any map α : F (A) −→ F (B) in B, there are objects A∗, B∗ in Aand maps α∗ : A∗ −→ B∗, ωA : A∗ −→ A and ωB : B∗ −→ B, such that the maps F (ωA) : F (A∗) −→ F (A) and F (ωB) : F (B∗) −→ F (B) are invertible and the following square commutes:

F (A∗)F (α∗)

F (ωA) ∼=

F (B∗)

F (ωB)∼=

F (A)α

F (B)

Clearly F is full if and only if F is almost full and F is full on isomorphisms, i.e. any isomorphism g : F (X) −→ F (Y ) in B is of the form F (f) for some (not necessarily invertible) map f : X −→ Y in A .

Lemma 1.5. Assume that T(X, X[1]) = 0.

(i) ∀A ∈ T: H(ΩkX(A)[1]) = 0, ∀k � 1.

(ii) For any object A ∈ T, the map H(γ1A) : H(Cell1(A)) −→ H(A) is invertible.

(iii) The functor H : T −→ mod-X is almost full and essentially surjective.


(iv) (See [46].) The functor H : T −→ mod-X induces an equivalence

H : (X � X[1])/X[1] ≈−→ mod-X

Proof. (i) Since H(hkA) = 0, ∀k � 0, applying the homological functor H to the triangle

ΩkX(A) −→ Xk−1

A −→ Ωk−1X A

hk−1A−→ Ωk

X(A)[1] and using that T(X, X[1]) = 0, we see that H(Ωk

X(A)[1]) = 0, ∀k � 1.

(ii) Since H(h0A) = 0, applying H to the triangle (C1

A) : Ω2X(A)[1] −→ Cell1(A) γ1

A−→A

ω1A−→ Ω2

X(A)[2] and using that H(ΩkX(A)[1]) = 0 and H(ω1

A) = H(h0A) ◦ H(h1

A[1]) = 0, we infer that the map H(γ1

A) is invertible.(iii) Let H(X1) −→ H(X0) −→ F −→ 0 be a projective presentation of F ∈ mod-X.

If X1 −→ X0 −→ A −→ X1[1] is a triangle in T, then applying H and using that T(X, X[1]) = 0, we have H(A) ∼= F , so H is essentially surjective. Let α : H(A) −→H(B) be a map in mod-X and let H(X1

A) −→ H(X0A) −→ H(A) −→ 0 and H(X1

B) −→H(X0

B) −→ H(B) −→ 0 be projective presentations of H(A) and H(B) as in (1.1). Since H|X is full and H(X) = Projmod-X, we get a map of projective presentations in mod-Xas in diagram (∗) below

H(X1A)

H(ε1A)

H(γ)

H(X0A)

H(f0A)

H(β)

H(A)

α

0

H(X1B)

H(ε1B)H(X0

B)H(f0

B)H(B) 0

(∗)

X1A

ε1A

γ

X0A

α1A

β

Cell1(A)

α∗

X1A[1]

γ[1]

X1B

ε1BX0

B

α1B Cell1(B) X1

B [1]

(∗∗)

which, since H|X is faithful, lifts to a morphism of triangles as in diagram (∗∗) above. By the construction of the towers (C•

A) and (C•B) we have: α1

A ◦ γ1A = f0

A and α1B ◦ γ1

B = f0B,

and by part (i) the maps H(γ1A) and H(γ1

B) are invertible. Consider the diagram:

H(X0A)

H(α1A)

H(β)

H(Cell1(A))H(γ1

A)

∼=

H(α∗)

H(A)

α

H(X0B)

H(α1B)

H(Cell1(B))H(γ1

B)

∼=H(B)


where the left square is commutative. Then H(α1A) ◦ H(γ1

A) ◦ α = H(f0A) ◦ α = H(β) ◦

H(f0B) = H(β) ◦ H(α1

B) ◦ H(γ1B) = H(α1

A) ◦ H(α∗) ◦ H(γ1B). Since the cone of α1

A lies in X[1], H(α1

A) is an epimorphism. Then H(γ1A) ◦α = H(α∗) ◦H(γ1

B) and the right square is commutative. This shows that H is almost full.

(iv) If A lies in X � X[1] then clearly Ω1X(A) := X1

A ∈ X and A = Cell1(A). Then (ii) shows that the map T(A, B) −→ Hom(H(A), H(B)) is surjective, ∀B ∈ T, i.e. H|X�X[1] is full, and essentially surjective by (i). If α : A −→ B is a map such that H(α) = 0, then the composition X0

A −→ A −→ B is zero and therefore it factorizes through the cone X1

A[1] of X0A −→ A. It follows that Ker H|X�X[1] = X[1] and the assertion follows. �

We refer to [16], see also Corollary 2.6 below, for necessary and sufficient conditions ensuring that H is full.

2. Ghosts and extensions

Let as before X be a contravariantly finite subcategory of T. Our aim here is to analyze the structure of maps in T which are invisible by the functor H : T −→ mod-X, in the sense of the following definition.

Definition 2.1. A map f : A −→ B in T is called X-ghost1 if T(X, f) = 0; equivalently H(f) = 0. Dually the map f is called X-coghost if T(f, X) = 0, or equivalently Hop(f) = 0.

We let GhX(A, B) be the subset of T(A, B) consisting of all X-ghost maps, and let CoGhX(A, B) be the subset of T(A, B) consisting of all X-coghost maps. Note that GhX(A, −) = 0 if and only if A ∈ X if and only if CoGhX(−, A) = 0. Clearly GhX(A, B)is a subgroup of T(A, B) and it is easy to see that in this way we obtain an ideal GhX(T)of T. For n � 1, we denote by Gh[n+1]

X (A, B) the subset of T(A, B) consisting of all maps f : A −→ B which can be written as a composition f = f0 ◦ f1 ◦ · · · ◦ fn, where

each fi is X[i]-ghost, 0 � i � n. Then for n � 0, we have an ideal Gh[n+1]X (T), where

Gh[1]X (T) := GhX(T). In other words, Gh[n+1]

X (T) is the product of the ideals GhX[i](T), 0 � i � n:

Gh[n+1]X (T) = GhX(T) ◦ GhX[1](T) ◦ GhX[2](T) ◦ · · · ◦ GhX[n](T)

Note that if X is stable, that is, X = X[1], then Gh[n]X (T) is the genuine nth power

GhnX(T) of the ideal GhX(T). We call the elements of Gh[n]X (T), X-ghost maps of depth n.

For instance, for any object A ∈ T the tower of triangles (C•A) shows that, for any n � 1,

1 Here we follow the current terminology. A more appropriate terminology would be X-phantom map [12]instead of X-ghost map, reserving the terminology X-ghost map for the case X consists of a single object or more generally X = addT for some T ∈ T.


the map

(−1)n+1h0A ◦ h1

A[1] ◦ · · · ◦ hn−1A [n− 1] := ωn−1

A : A −→ ΩnX(A)[n]

is an X-ghost map of depth n, i.e. lies in Gh[n]X (A, Ωn

X(A)[n]). Sometimes ωn−1A is called

the nth Atiyah class of A with respect to X and is denoted by atnX(A), see [27] for the terminology in the stable case.

2.1. The Ghost Lemma

The following version of the Ghost Lemma (see for instance [12, Proposition 5.1 and Corollary 5.5] in the stable case) shows that

Gh[n]X (A,B) = ωn−1

A ◦ T(Ωn

X(A)[n], B)

In other words the left ideal Gh[n]X (A, −) of X-ghost maps out of A of depth n is prin-

cipal generated by the nth Atiyah class atnX(A) = ωn−1A . As another consequence of the

Ghost Lemma it follows that the complexity of the ideal of X-ghost maps measures the possibility for building T from X using positive shifts and extensions.

Proposition 2.2 (The Ghost Lemma).

(i) For a map f : A −→ B in T, the following are equivalent:(a) f ∈ Gh[n]

X (A, B).(b) f factors through ωn−1

A , that is, there exists a map g : ΩnX(A)[n] −→ B such

that f = ωn−1A ◦ g.

(ii) Let A be an object in T, and consider the following statements:(a) Ωn

X(A) ∈ X.(b) A ∈ X � X[1] � · · · � X[n].(c) Gh[n+1]

X (A, −) = 0.Then (a) ⇒ (b) ⇔ (c). In particular: T = X � X[1] � · · · � X[n] if and only if Gh[n+1]

X (T) = 0.

Proof. (i) If f factors through ωn−1A , then clearly f lies in Gh[n]

X (A, B) since by construc-tion hi

A[i] is X[i]-ghost, 0 � i � n − 1, cf. (1.4). To show the converse, let n = 1 and f : A −→ B be X-ghost. Then the composition X0

A −→ A −→ B is zero and therefore it factorizes through ω0

A = h0A : A −→ Ω1

X(A)[1]. If n = 2 and f : A −→ B lies in Gh[2]

X (A, B), then f admits a factorization f = f0 ◦ f1, where f0 : A −→ B0 is X-ghost and f1 : B0 −→ B is X[1]-ghost. Then there exists a map g0 : Ω1

X(A)[1] −→ B0 such that f0 = h0

A ◦ g0. Since f1 is X[1]-ghost, so is g0 ◦ f1 and therefore its composition with the map X1

A[1] −→ Ω1X(A)[1] is zero. Then g0 ◦ f1 factorizes through the map

−h1A[1] : Ω1

X(A)[1] −→ Ω2X(A)[2], so there exists a map g : Ω2

X(A)[2] −→ B such that


(−h1A[1]) ◦ g = g0 ◦ f1. Then f = f0 ◦ f1 = h0

A ◦ g0 ◦ f1 = h0A ◦ (−h1

A[1]) ◦ g. Hence f = −(h0

A ◦ h1A[1]) ◦ g = ω1

A ◦ g, and this completes the proof for the case n = 2. The assertion for n � 3 follows easily by induction.

(ii) (a) ⇒ (b) If ΩnX(A) lies in X, then since Celln−1(A) lies in X �X[1] �· · ·�X[n −1], from

the triangle (Cn−1A ) in the tower of triangles (C•

A) it follows that A lies in X �X[1] �· · ·�X[n].(b) ⇔ (c) If n = 0, i.e. A ∈ X, then clearly Gh[1]

X (A, −) = GhX(A, −) = 0. Let n = 1, so that A ∈ X � X[1]. Let X0

α−→ A β−→ X1[1] −→ X0[1] be a triangle, where Xi ∈ X,

and f ∈ Gh[2]X (A, B), i.e. f admits a factorization f = f0 ◦ f1, where f0 : A −→ B0 is

X-ghost and f1 : B0 −→ B is X[1]-ghost. Then f0 = β ◦g for some map g : X1[1] −→ B0. Since the map f1 is X[1]-ghost, the composition g ◦ f1 : X1[1] −→ B is zero. Then f = f0 ◦ f1 = β ◦ g ◦ f1 = 0, hence Gh[2]

X (A, B) = 0. For n � 3 the assertion follows by induction. Now assume Gh[n+1]

X (A, −) = 0. Then ωnA = 0 since by definition ωn

A is in

Gh[n+1]X

(A, Ωn+1

X (A)[n + 1]). This implies that A lies in X � X[1] � · · · � X[n] as a direct

summand of Celln(A). �As a consequence of the Ghost Lemma, we have the following result which gives an

alternative description of the “powers” of the ideal of X-ghost maps. Let A be an object of T and consider the triangle

Ωt+1X (A)[t] βt

A−−→ Cellt(A) γtA−−→ A

ωtA−−→ Ωt+1

X (A)[t + 1] (Ct+1A )

Corollary 2.3.

(i) For any t � 0, the map γtA : Cellt(A) −→ A is a right (X �X[1] � · · ·�X[t])-approxima-

tion of A. In particular X � X[1] � · · · � X[t] is contravariantly finite in T.(ii) For any objects A, B ∈ T and any t � 1, we have an equality:

Gh[t+1]X (A,B) = GhX�X[1]�···�X[t](A,B)

Proof. (i) Let g : C −→ A be a map, where C lies in X �X[1] � · · ·�X[t]. Since the map ωtA

lies in Gh[t+1]X (A, Ωt+1

X (A)[t + 1]), it follows that g ◦ωtA lies in Gh[t+1]

X (C, Ωt+1X (A)[t + 1]).

By the Ghost Lemma, we have g ◦ ωtA = 0 and therefore g factorizes through Cellt(A),

i.e. γtA is a right (X �X[1] � · · ·�X[t])-approximation of A. We infer that X �X[1] � · · ·�X[t]

is contravariantly finite in T.(ii) If f : A −→ B lies in GhX�X[1]�···�X[t](A, B), then the composition γt

A ◦ f is zero since Cellt(A) ∈ X �X[1] � · · ·�X[t]. Hence f factorizes through ωt

A : A −→ Ωt+1X (A)[t +1],

say as f = ωtA ◦g. Then f lies in Gh[t+1]

X (A, B) since ωtA lies in Gh[t+1]

X (A, Ωt+1X (A)[t +1]).

Hence GhX�X[1]�···�X[t](A, B) ⊆ Gh[t+1]X (A, B). Conversely if f lies in Gh[t+1]

X (A, B), then by the Ghost Lemma we have f = ωt

A ◦ g for some g : Ωt+1X (A)[t +1] −→ B. This implies

that γtA ◦ f = 0. Since, by part (i), γt

A is a right (X � X[1] � · · · � X[t])-approximation of A, we have T(C, f) = 0, ∀C ∈ X �X[1] � · · · �X[t], so f lies in GhX�X[1]�···�X[t](A, B). �


Corollary 2.3 shows that for any contravariantly finite subcategory X of T and any n � 0, the pair

(X � X[1] � · · · � X[n], Gh[n+1]

X (T))

forms a projective class in T in the sense of [29] and [31], see also [12] in the stable case.

Remark 2.4.

(i) For any integer n � 1, we define Gh[−n]X (A, B) to be the subgroup of T(A, B)

consisting of all maps f : A −→ B which can be written as a composition f = f−n+1 ◦ f−n+2 ◦ · · · ◦ f−1 ◦ f0, where f−i : B−i−1 −→ B−i is X[−i]-ghost, A = B−n and B0 = B. Clearly then the map

Gh[−n]X (A,B) −→ Gh[n]

X (A[n− 1], B[n− 1]), f �−→ f [n− 1]

is an isomorphism. By Corollary 2.3 we then have GhX[−n]�···�X[−1]�X(A, B) ∼=GhX�X[1]�···�X[n](A[n], B[n]).

(ii) The above results as well as the results that follow concerning X-ghost maps have dual versions for X-coghost maps. We leave their formulation to the reader.

Clearly if X ⊆ Y, then GhY(A, B) ⊆ GhX(A, B). Hence the increasing filtration by subcategories of T

X ⊆ X � X[1] ⊆ X � X[1] � X[2] ⊆ · · · ⊆ X � X[1] � · · · � X[t] ⊆ · · · ⊆ T (2.1)

induces, by Corollary 2.3, a decreasing filtration of the Hom-functor T(−, −) of T by X-ghost ideals:

· · · ⊆ Gh[t+1]X (T) ⊆ · · · ⊆ Gh[2]

X (T) ⊆ GhX(T) ⊆ T(−,−) (2.2)

By the Ghost Lemma the above filtrations have the same length.For any objects A, B ∈ T, we denote by HA,B the canonical map

HA,B : T(A,B) −→ Hom(H(A),H(B)), f �−→ H(f)

A version of the next useful result was proved in [17] in case X is stable, i.e. X = X[1].

Proposition 2.5. For any objects A, B ∈ T, there exists a natural exact sequence

0 −→ GhX(A,B) −→ T(A,B) −→ Hom(H(A),H(B))

−→ GhX(Ω1X(A), B) −→ Gh[2]

X (A,B[1]) −→ 0


Proof. We construct maps

ϑA,B : Hom(H(A),H(B)) −→ GhX(Ω1X(A), B) &

ζA,B : GhX(Ω1X(A), B) −→ Gh[2]

X (A,B[1])

as follows. Let α : H(A) −→ H(B) be a map and consider the triangle Ω1X(A) −→ X0

A −→A −→ Ω1

X(A)[1]. Then we have an exact sequence H(Ω1X(A)) −→ H(X0

A) −→ H(A) −→ 0in mod-X and clearly the composition H(f0

A) ◦ α : H(X0A) −→ H(A) −→ H(B) is of the

form H(α∗) for a unique map α∗ : X0A −→ B. Define ϑA,B(α) = g0

A ◦ α∗ : Ω1X(A) −→ B.

Clearly g0A ◦α∗ is X-ghost since H(g0

A ◦α∗) = H(g0A) ◦H(α∗) = H(g0

A) ◦H(f0A) ◦α = 0. Now

if α ∈ Hom(H(A), H(B)) lies in Ker ϑA,B , then ϑA,B(α) = g0A ◦α∗ = 0. Hence α∗ = f0

A ◦βfor some map β : A −→ B and then H(f0

A) ◦ α = H(α∗) = H(f0A) ◦ H(β) and therefore

α = H(β). Conversely if α = H(β), for some map β : A −→ B, then α∗ = f0A ◦β and then

ϑA,B(α) = g0A ◦ α∗ = g0

A ◦ f0A ◦ β = 0. Hence Ker ϑA,B = Im HA,B . On the other hand,

if α : Ω1X(A) −→ B is an X-ghost map, then since h0

A[−1] is clearly X[−1]-ghost and αis X-ghost, it follows by Corollary 2.3 that h0

A[−1] ◦ α ∈ GhX[−1]�X(A[−1], B), hence by

Remark 2.4, we have a map ζA,B : GhX(Ω1X(A), B) −→ Gh[2]

X (A, B[1]), α �−→ ζA,B(α) =h0A ◦α[1]. We show that ζA,B is surjective. Let α : A −→ B[1] be a map in Gh[2]

X (A, B[1]), so α = β ◦ γ, where β ∈ GhX(A, C) and γ ∈ GhX[1](C, B[1]). By the Ghost Lemma, β = h0

A ◦ ρ, for some map ρ : Ω1X(A)[1] −→ C. Then the map (ρ ◦ γ)[−1] : Ω1

X(A) −→B is X-ghost and ζA,B((ρ ◦ γ)[−1]) = α, so ζA,B is surjective. Finally we show that Ker ζA,B = ImϑA,B . Let β : Ω1

X(A) −→ B be X-ghost such that ζA,B(β) = h0A ◦β[1] = 0.

Then β = g0A◦γ, for some map γ : X0

A −→ B. Since 0 = H(β) = H(g0A) ◦H(γ), there exists

a unique map α : H(A) −→ H(B) such that H(f0A) ◦α = H(γ). It follows that ϑA,B(α) =

g0A ◦ γ = β, i.e. Ker ζA,B ⊆ ImϑA,B . Finally if δ : Ω1

X(A) −→ B is in the image of ϑA,B, then δ = g0

A ◦ δ∗ for some map δ∗ : X0A −→ B. Plainly ζA,B(δ) = h0

A ◦ g0A[1] ◦ δ∗[1] = 0,

i.e. δ ∈ Ker ζA,B . Hence Ker ζA,B = ImϑA,B and the sequence is exact. �The obstruction group OH(A, B) associated to the objects A, B ∈ T is defined to be

the cokernel of the natural map HA,B : T(A, B) −→ Hom(H(A), H(B)). In this way we obtain the obstruction bifunctor OH : Top × T −→ Ab, and clearly OH(A, −) = 0 iff any map H(A) −→ H(−) is in the image of H, and OH = 0, i.e. OH(A, B) = 0, ∀A, B ∈ T, iff His full. The next result describes the largest subcategory U of T such that the restriction H|U is full. First we need the following notation: if A , B are classes of objects of T, then A ⊕ B denotes the full subcategory of T consisting of the direct factors of coproducts A ⊕B, where A ∈ A and B ∈ B.

Corollary 2.6. If T(X, X[1]) = 0, then:

X�X[1] ={A ∈ T | Gh[2](A,−) = 0

}⊆

{A ∈ T | OH(A,−) = 0

}= (X�X[1]) ⊕ X�


In particular: Gh[2]X (T) = 0 ⇐⇒ T = X � X[1] and OH = 0 ⇐⇒ H is full ⇐⇒ T =

(X � X[1]) ⊕ X�.

Proof. The first equality follows directly by using that Gh[2]X (A, −) ∼= GhX�X[1](A, −), see

Corollary 2.3. If A ∈ X �X[1], then there is a triangle X1 −→ X0 −→ A −→ X1[1], where the Xi lie in X. Since T(X, X[1]) = 0, the map X0 −→ A is a right X-approximation, hence X1 = Ω1

X(A). It follows that GhX(Ω1X(A), −) = 0 and then OH(A, −) = 0 by

Proposition 2.5. For the proof of the second equality we refer to [16]. �2.2. Extensions of coherent functors

The following results extend the exact sequence of Proposition 2.5 and involves ex-tensions of coherent functors.

Proposition 2.7. For any objects A, B ∈ T such that T(X, A[−1]) = 0, there exists a 7-term exact sequence:

0 −→ GhX(A,B) −→ T(A,B) HA,B−−−−→ Hom(H(A),H(B)) ϑA,B−−−−→ GhX(Ω1X(A), B)

ηA,B−−−−→ GhX(A,B[1]) ξA,B−−−−→ Ext1(H(A),H(B)) −→ OH(Ω1X(A), B) −→ 0 (2.3)

Moreover: Im(ϑA,B) = OH(A, B) and Im(ηA,B) ∼= Gh[2]X (A, B[1]).

Proof. Applying H to the triangle Ω1X(A) −→ X0

A −→ A −→ Ω1X(A)[1] and using that

H(A[−1]) = 0, we have an exact sequence 0 −→ H(Ω1X(A)) −→ H(X0

A) −→ H(A) −→ 0and an exact commutative diagram

· · · −−−−→ T(A,B) −−−−→ T(X0A, B)

T(g0A,B)

−−−−−−−→ T(Ω1X(A), B) −−−−→ ImT(h0

A[−1], B) −−−−→ 0

HA,B

⏐⏐� HX0

A,B

⏐⏐�∼= HΩ1X

(A),B⏐⏐� φ

⏐⏐�

0 −−−−→ (H(A),H(B)) −−−−→ (H(X0A),H(B)) −−−−→ (H(Ω1

X(A)),H(B)) −−−−→ Ext1(H(A),H(B)) −−−−→ 0

We know that Ker HA,B = GhX(A, B), Coker HA,B = OH(A, B), and Ker HΩ1X(A),B =

GhX(Ω1X(A), B), and Coker HΩ1

X(A),B = OH(Ω1X(A), B). By chasing the above commuta-

tive diagram, it is not difficult to see that we have an exact sequence

0 −→ OH(A,B) −→ GhX(Ω1X(A), B) −→ ImT(h0

A[−1], B)φ−→ Ext1(H(A),H(B)) −→ Cokerφ −→ 0

and identifications: Cokerφ ∼= Coker HΩ1X(A),B = OH(Ω1

X(A), B), Ker φ =GhX[−1]�X(A[−1], B), and ImT(h0

A[−1], B) = GhX[−1](A[−1], B). Since, by Remark 2.4, we have isomorphisms: GhX[−1](A[−1], B) ∼= GhX(A, B[1]) and GhX[−1]�X(A[−1], B) ∼=Gh[−2]

X (A[−1], B) ∼= Gh[2]X (A, B[1]), the exact sequence (2.3) follows by splicing together

the above exact sequence with the exact sequence of Proposition 2.5. �


Corollary 2.8. Let t � 2 and assume that T(X, X[−i]) = 0, 1 � i � t − 1. Then for any A ∈ T such that T(X, A[−i]) = 0, 1 � i � t, and any object B ∈ T, there exists an exact sequence, for any 1 � i � t − 1:

0 −→ GhX(ΩiX(A), B) −→ T(Ωi

X(A), B) −→ Hom(H(Ωi

X(A)),H(B))

−→ GhX(Ωi+1X (A), B) −→ GhX(Ωi

X(A), B[1]) −→ Exti+1(H(A),H(B))

−→ OH(Ωi+1X (A), B) −→ 0 (2.4)

Proof. Applying the functor H to the triangles ΩiX(A) −→ Xi−1

A −→ Ωi−1X (A) −→

Ωi(A)[1], i � 1, and using the vanishing conditions of the statement, we see easily that T(X, Ωk

X(A)[−t +k]) = 0, 1 � k � t −1. As a consequence we have short exact sequences 0 −→ H(Ωk

X(A)) −→ H(Xk−1A ) −→ H(Ωk−1

X (A)) −→ 0, 1 � k � t, so ΩkH(A) ∼=H(Ωk

X(A)), 1 � k � t. Since T(X, ΩkX(A)[−1]) = 0, 1 � k � t − 1, the existence of (2.4)

follows from the exact sequence (2.3) of Proposition 2.7, by replacing A with ΩiX(A). �

Corollary 2.9. Let A ∈ T be such that T(X, A[−1]) = 0.

(i) If Ω2X(A) ∈ X, then, ∀B ∈ T, there is an isomorphism

Ext1(H(A),H(B)) ∼=−−→ GhX(A,B[1])Gh[2]

X (A,B[1])

(ii) If A ∈ X � X[1] and T(X, X[1]) = 0, then, ∀B ∈ T, there is an isomorphism:

Ext1(H(A),H(B)) ∼=−−→ GhX(A,B[1])

(iii) If A, B ∈ X � X[1] and T(X, X[1]) = 0 = T(X, X[2]), then there is an isomorphism:

Ext1(H(A),H(B)) ∼=−−→ T(A,B[1])

Proof. (i) By Proposition 2.7 we have an exact sequence

GhX(Ω1X(A), B) ηA,B−−−−→ GhX(A,B[1]) ξA,B−−−−→ Ext1(H(A),H(B))

−→ OH(Ω1X(A), B) −→ 0 (2.5)

and an isomorphism Im(ηA,B) ∼= Gh[2]X (A, B[1]). Hence we have an exact sequence

0 −→ GhX(A,B[1])Gh[2]

X (A,B[1])−→ Ext1(H(A),H(B)) −→ OH(Ω1

X(A), B) −→ 0

and it suffices to show that OH(Ω1X(A), B) = 0. Since Ω2

X(A) ∈ X, it follows that there is a triangle X1 −→ X0 −→ Ω1

X(A) −→ X1[1], where the Xi lie in X, such that the sequence


H(X1) −→ H(X0) −→ H(Ω1X(A)) −→ 0 is exact. Applying the functor Hom(−, H(B)) to

this exact sequence we have an exact sequence

0 −→ Hom(H(Ω1X(A)),H(B)) −→ Hom(H(X0),H(B)) −→ Hom(H(X1),H(B))

Since the objects X0, X1 lie in X, the canonical map HXi,B : T(Xi, B) −→ Hom(H(Xi),H(B)), i = 1, 2, is invertible and we have an exact commutative diagram

· · · −−−−→ T(Ω1X(A), B) −−−−→ T(X0, B) −−−−→ T(X1, B)

HΩ1X

(A),B

⏐⏐� HX0,B

⏐⏐�∼= HX1,B

⏐⏐�∼=

0 −−−−→ Hom(H(Ω1X(A)),H(B)) −−−−→ Hom(H(X0),H(B)) −−−−→ Hom(H(X1),H(B))

Chasing the above exact commutative diagram, it follows directly that the map HΩ1X(A),B

is an epimorphism and then by definition we have OH(Ω1X(A), B) = 0 as desired.

(ii) If A ∈ X �X[1] and T(X, X[1]) = 0, then Ω1X(A) ∈ X. Hence GhX(Ω1

X(A), B) = 0 =OH(Ω1

X(A), B) and the map GhX(A, B[1]) −→ Ext1(H(A), H(B)) in (2.5) is invertible.(iii) Under the imposed assumptions, there exists a triangle X1 −→ X0 −→ B −→

X1[1] where Xj ∈ X. Applying H and using that T(X, X[i]) = 0, 1 � i � 2, it follows that H(B[1]) = 0. In particular any map A −→ B[1] is X-ghost. Hence GhX(A, B[1]) ∼=T(A, B[1]) and the assertion follows from (ii). �

To get an analogous result for the higher extension functors we need the following.

Lemma 2.10. For any objects A, B ∈ T and any k � 0, there is an epimorphism:

ϕkA,B : GhX(Ωk

X(A), B[1]) −→ Gh[k+1]X (A,B[k + 1]) −→ 0

which is an isomorphism if T(X, B[i]) = 0, 1 � i � k.

Proof. Define ϕkA,B(α) = ωk−1

A ◦α[k]. Since ωk−1A lies in Gh[k]

X (A, ΩkX(A)[k]) and since α[k]

is X[k]-ghost, it follows that ϕkA,B(α) lies in Gh[k+1]

X (A, B[k+1]). If β : A −→ B[k+1] lies in Gh[k+1]

X (A, B[k+1]), by the Ghost Lemma, there exists a map γ : Ωk+1X (A) −→ B[k+1]

such that β = ωkA ◦ γ = ωk−1

A ◦ hkA[k] ◦ γ. Clearly the map hk

A ◦ γ[−k] : ΩkX(A) −→ B[1]

is X-ghost and ϕkA,B(hk

A ◦ γ[−k]) = ωk−1A ◦ hk

A[k] ◦ γ = ωkA ◦ γ = β, so ϕk

A,B is surjective. Assume now that T(X, B[i]) = 0, 1 � i � k and let α ∈ Kerϕk

A,B . Then ωk−1A ◦ α[k] = 0

and therefore α[k] factorizes through the cone Cellk−1(A)[1] of ωk−1A . It is easy to see that

the condition T(X, B[i]) = 0, 1 � i � k, forces any map from an object in X[1] � · · ·�X[k]to B[k + 1] to be zero. Since Cellk−1(A)[1] lies in X[1] � · · · �X[k], we therefore have that α[k] or equivalently α is zero. We infer that ϕk

A,B is injective. �Corollary 2.11. Let t � 2 and assume that T(X, X[i]) = 0, −t + 1 � i � t, i �= 0. If A ∈ X �X[1] � · · · �X[t] and T(X, A[−i]) = 0, 1 � i � t, then Extt+1(H(A), H(−)) = 0, so pd H(A) � t. Moreover:


Extt(H(A),H(B)

) ∼=−−→ GhX(Ωt−1X (A), B[1]) &

Extt−1(H(A),H(B)) ∼=−−→ GhX(Ωt−2

X (A), B[1])Gh[2]

X (Ωt−2X (A), B[1])

If in addition T(X, B[i]) = 0, for 1 � i � t −1, then: Extt(H(A), H(B)

) ∼=−−→ Gh[t]X (A, B[t]).

Proof. The assumptions on A imply easily that ΩtX(A) ∈ X. Then Ωt−1

X (A) ∈ X � X[1], and therefore we have OH(Ωk

X(A), −) = 0, for k = t, t − 1, and GhX(ΩtX(A), −) = 0. It

follows from (2.4), for i = t, that Extt+1(H(A), H(B)) = 0, ∀B ∈ T. Since any object of mod-X lies in the image of H it follows that pd H(A) � t. Next setting i = t − 1in (2.4), we have the first isomorphism, and setting i = t − 2 in (2.4) we have the second isomorphism. Finally if T(X, B[i]) = 0, 1 � i � t − 1, Lemma 2.10 shows that GhX(Ωt−1

X (A), B[1]) ∼= Gh[t]X (A, B[t]) and therefore the last isomorphism follows. �

Remark 2.12. Assume that T(X, X[1]) = 0. If A ∈ X �X[1] satisfies T(X, A[−1]) = 0, then combining Proposition 2.7 and Corollary 2.9(ii), we have for any B ∈ T a “universal coefficient” exact sequence:

0 −→ Ext1(H(A),H(B[−1])

)−→ T(A,B) −→ Hom

(H(A),H(B)

)−→ 0

For instance if X[2] ⊆ X �X[1], then the above exact sequence gives Ext1(H(X[2]), H(X)) ∼=T(X, X[−1]). Hence if X is 1-rigid and X[2] ⊆ X � X[1], then T(X, X[−1]) = 0 if and only if Ext1(H(X[2]), H(X)) = 0.

2.3. Algebraic ghosts

We close this section by analyzing ghost maps in algebraic triangulated categories. In this context ghost maps of any finite depth admit a concrete realization as elements of relative extension groups.

Let T be an algebraic triangulated category, i.e. T = C is the stable category, modulo the full subcategory P of projective–injective objects, of some exact Frobenius category (C , E ), where E denotes its exact structure. We denote by π : C −→ T = C , π(A) = A, the projection functor, by Ω(A), resp. Ω−1(A), the first syzygy, resp. cosyzygy, of an object A in C , and by [1] = Ω−1 the suspension functor in T.

Let X be a contravariantly finite subcategory of T. Then setting

Mn := π−1(X � X[1] � · · · � X[n]), n � 0

we obtain a chain of full subcategories of C

P ⊆ M := M0 ⊆ M1 ⊆ M2 ⊆ · · · ⊆ Mn ⊆ · · · ⊆ C


Since by Corollary 2.3 each subcategory X �X[1] � · · ·�X[n] is contravariantly finite in T, it is easy to see that Mn is contravariantly finite in C , ∀n � 0. As a consequence the subcategories Mn define new exact structures En(M) on C by declaring a diagram A −→B −→ C in C to be in En(M) if the induced sequence 0 −→ C (Mn, A) −→ C (Mn, B) −→C (Mn, C) −→ 0 is exact in Ab. Then the identity functor IdC of C induces an exact functor (C , En(M)) −→ (C , E ), ∀n � 0. The exact category (C , En(M)) has enough projective objects and the projective objects are the objects of the subcategory Mn. We denote by Ext∗E (A, B) and Ext∗Mn

(A, B) the extension functors of the exact categories (C , E ) and (C , En(M)) respectively, ∀n � 0. Note that since the exact category (C , E )is Frobenius, ∀A, B ∈ T we have isomorphisms ExtnE (A, B) ∼= T(A, B[n]), ∀n � 1. In particular ExtkE (M, M) = 0, for 1 � k � n, if and only if T(X, X[k]) = 0, for 1 � k � n.

The M-resolution dimension of A ∈ C , denoted by res.dimM A, is defined to be the minimum n � 0 (if it exists) such that there exists an exact sequence (∗) : 0 −→ Mn −→Mn−1 −→ · · · −→ M0 −→ A −→ 0 in C , where the Mi lie in M. The M-resolution dimension of C , denoted by res.dimM C , is defined as sup{res.dimM A | A ∈ C }. We also denote by pdM A the projective dimension of A as an object of the exact category (C , E0(M)) and by gl.dimM C = sup{pdM A | A ∈ C } its global dimension. Clearly pdM A is the minimum n � 0 (or infinity) such that there exists an exact sequence (∗)where the Mi lie in M, which remains exact after applying C (M, −).

The next result collects basic properties of ghost maps in the context of algebraic triangulated categories.

Proposition 2.13. With the above notations we have the following.

(i) There are isomorphisms: Gh[n+1]X (A, B) ∼=−−→ Ext1Mn

(A, ΩB), ∀n � 0. In particular:

GhX(A,B) ∼=−−→ Ext1M(A,ΩB)

(ii) Mn ={A ∈ C | res.dimM A � n

}.

(iii) Gh[n+1]X (T) = 0 if and only if res.dimM C � n.

(iv) If T(X, X[k]) = 0, 1 � k � n, then: Mk ={A ∈ C | pdM A � k

}, for 0 � k � n.

(v) If T(X, X[k]) = 0, 1 � k � n, then: Gh[n+1]X (T) = 0 if and only if gl.dimM C � n.

(vi) For any objects A, B in C , there exists an exact sequence:

0 −→ Ext1M(A,ΩB) −→ C (A,B) −→ Hom(C (−, A)|M,C (−, B)|M

)−→ Ext2M(A,ΩB) −→ Ext1M1

(A,B) −→ 0

(vii) Assume that T(X, X[k]) = 0, 1 � k � n, and Gh[n+1]X (T) = 0. If any map between

objects of M admits a kernel in C , then mod-M is an abelian category of finite global dimension, in fact:

gl.dimmod-M � n + 2


Proof. (i) Let h : A −→ B be an X-ghost map in T = C and let ΩB −→ K −→ A −→ B

be a triangle in T. By the construction of the triangulation of T, this triangle is induced by a short exact sequence (Eh) : 0 −→ ΩB −→ K −→ A −→ 0 in C , i.e. an element of E . This sequence is M-exact, i.e. is an element of E0(M); indeed since X = π(M) and his X-ghost, it follows easily that any map M −→ A, where M ∈ M, factorizes through K. Therefore the extension (Eh) is an element of Ext1M(A, ΩB). We leave to the reader to show that the assignment GhX(A, B) −→ Ext1M(A, ΩB), h �−→ (Eh) is an isomorphism. The isomorphisms Gh[n+1]

X (A, B) ∼= Ext1Mn(A, ΩB), ∀n � 1, are proved similarly using

Corollary 2.3.(ii) We show by induction that Mn =

{A ∈ C | res.dimM A � n

}, the case n = 0

being clear. Let n = 1 and let A ∈ M1, so A lies in X �X[1]. Then there exists a triangle M1 −→ A −→ Ω−1M2 −→ Ω−1M1 in T, where the Mi lie in M. The last triangle is induced by a short exact sequence 0 −→ M1 −→ A −→ Ω−1M2 −→ 0 in C and we have a short exact sequence 0 −→ M2 −→ P −→ Ω−1M2 −→ 0, where P is projective–injective in C . It is easy to see then that there is induced an exact sequence 0 −→ M2 −→M1 ⊕P −→ A −→ 0 in C and this shows that res.dimM A � 1, since P ⊆ M. Conversely if res.dimM A � 1 and 0 −→ M2 −→ M1 −→ A −→ 0 is an exact sequence in C , where the Mi lie in M, then the induced triangle M1 −→ A −→ Ω−1M2 −→ Ω−1M1 in Tshows that A ∈ X �X[1] and this means that A lies in M1 = π−1(X �X[1]). We infer that M1 =

{A ∈ C | res.dimM A � 1

}. The general case follows easily by induction.

(iii) By Proposition 2.2 and part (ii) we have Gh[n+1]X (T) = 0 if and only if T =

X � X[1] � · · · � X[n] if and only if T = Mn if and only if res.dimM C � n.(iv) Using (ii), we have clearly an inclusion

{A ∈ C | pdM A � k

}⊆ Mk, ∀k � 0.

Assume that T(X, X[1]) = 0 and let A ∈ M1, so there exists an exact sequence (E) :0 −→ M2 −→ M1 −→ A −→ 0 in C , where the Mi lie in M. Then we have a triangle M2 −→ M1 −→ A −→ Ω−1M2 in T, where the Mi lie in M. Since T(X, X[1]) = 0, any map M −→ A, where M ∈ M, factorizes through the map M1 −→ A. This easily implies that any map M −→ A, where M ∈ M, factorizes through the map M1 −→ A and therefore the extension (E) is M-exact. This means that (E) is a projective resolution of A of length � 1, so pdM A � 1. We infer that

{A ∈ C | pdM A � 1

}= M1. The inclusion

Mk ⊆{A ∈ C | pdM A � k

}, for 0 � k � n, provided that T(X, X[k]) = 0, 1 � k � n,

follows easily by induction.(v), (vi) Part (v) follows directly from (iv) and Proposition 2.2, and part (vi) follows

from Proposition 2.5.(vii) Since any map between objects of M has a kernel in C and M is contravariantly

finite in C , it follows that M has weak kernels: in fact a weak kernel of the map f : M2 −→M1 in M is the composition fA ◦ g : MA −→ M1 of the kernel g : A −→ M1 in C of ffollowed by a right M-approximation fA : MA −→ A of A. We infer that mod-M is abelian. Let A be in C . Then by (iii) and (iv) there exists an exact sequence 0 −→Mn −→ Mn−1 −→ · · · −→ M0 −→ A −→ 0 in C , where the Mi lie in M which remains exact after applying C (M, −). It follows that the sequence 0 −→ C (−, Mn)|M −→C (−, Mn−1)|M −→ · · · −→ C (−, M0)|M −→ C (−, A)|M −→ 0 is a projective resolution


of C (−, A)|M in mod-M and therefore pdC (−, A)|M � n. Finally let F be in mod-Mand let C (−, M1)|M −→ C (−, M0)|M −→ F −→ 0 be a projective presentation of F . If A is the kernel of the map M1 −→ M0 in C , then C (−, A)|M = Ω2F and therefore pdF � n + 2. Hence gl.dimmod-M � n + 2. �

We leave as an exercise to the reader the interpretation of the results of this and subsequent sections in the setting of algebraic triangulated categories.

3. Rigid and corigid subcategories

Throughout this section we fix as before a triangulated category T with split idem-potents and a contravariantly finite subcategory X of T which is closed under direct summands and isomorphisms.

3.1. Rigid subcategories

For n � 1, we consider the following subcategories naturally associated to X:

X�n :=

{A ∈ T | T(X, A[i]) = 0, 1 � i � n

}and

�nX :=

{A ∈ T | T(A,X[i]) = 0, 1 � i � n

}

We also set: X�0 := X� and �0 X := �X. Note that we have filtrations:

T ⊇ X�1 ⊇ X�

2 ⊇ · · · ⊇ X�n ⊇ · · · (3.1)

X ⊆ X � X[1] ⊆ · · · ⊆ X � X[1] � · · · � X[n] ⊆ · · · ⊆ T (3.2)

Definition 3.1. A full subcategory X ⊆ T is called n-rigid, n � 1, if: T(X, X[i]) = 0, 1 � i � n.

It follows that X is n-rigid if and only if X ⊆ X�n or equivalently X ⊆ �

nX. Recall from [44] that a pair (X, Y) of full subcategories of T is called a torsion pair in T, if T(X, Y) = 0 and for any object A in T there is a triangle XA −→ A −→ Y A −→ XA[1], where XA ∈ X and Y A ∈ Y. It follows in particular that T = X � Y for any torsion pair (X, Y) in T. We call a triple (X, Y, Z) of subcategories of T a torsion triple if (X, Y) and (Y, Z) are torsion pairs. If (X, Y) is a torsion pair in T, then clearly the map XA −→ A

is a right X-approximation of A, and the map A −→ Y A is a left Y-approximation of A. Hence X is contravariantly finite and Y is covariantly finite in T. Moreover it is easy to see that X� = Y and �Y = X.

Throughout this subsection: X denotes a contravariantly finite n-rigid subcategory of T, n � 1. We begin with the following result which will be useful in the rest of the paper.


Proposition 3.2. For any object A in T, we have the following:

(i) ΩtX(A) ∈ X�

1 , t � 1, and ΩtX(A) ∈ X�

t , 1 � t � n. Moreover:

H(X[1] � X[2] � · · · � X[k]

)= 0 = Hop(X[−1] � X[−2] � · · · � X[−k]), 1 � k � n

(ii) There is a torsion pair(X � X[1] � X[2] � · · · � X[t− 1], X�

t [t]), 1 � t � n (∗)

The map γt−1A : Cellt−1(A) −→ A is a right (X �X[1] � · · · �X[t − 1])-approximation

of A, the map H(γt−1A ) is invertible, and the map ωt−1

A : A −→ ΩtX(A)[t] is a left

X�t [t]-approximation of A.

(iii) ∀t = 1, 2, · · · , n: ΩtX(A) ∈ X if and only if A ∈ X � X[1] � · · · � X[t].

Proof. (i) Applying the homological functor H : T −→ mod-X to the triangles

ΩtX(A) −→ Xt−1

A −→ Ωt−1X (A) −→ Ωt

X(A)[1] (T tA)

and using that X is n-rigid, it follows that H(ΩtX(A)[1]) = 0, ∀t � 1, and H(Ωt

X(A)[i]) ∼=H(Ωt+1

X (A)[i + 1]), for 1 � i, t � n − 1. Hence for any t = 1, 2, · · · , n, we have: H(Ωt

X(A)[t]) ∼= H(Ωt−1X (A)[t − 1]) ∼= · · · ∼= H(Ω2

X(A)[2]) ∼= H(Ω1X(A)[1]) = 0. It follows

that ΩtX(A) ∈ X�

t , 1 � t � n, and ΩtX(A) ∈ X�

1 , ∀t � 1.We show by induction that H(X[1] � X[2] � · · · � X[k]) = 0, or equivalently T(X, X[1] �

X[2] � · · · � X[k]) = 0, for 1 � k � n. Since X is 1-rigid we have T(X, X[1]) = 0. Assume that T(X, X[1] � X[2] � · · · � X[k − 1]) = 0 and let A ∈ X[1] � X[2] � · · · � X[k]. Then there exists a triangle B −→ A −→ X ′[k] −→ B[1], where B ∈ X[1] � X[2] � · · · � X[k − 1] and X ′ ∈ X. Since X is n-rigid and k � n, we have T(X, X[k]) = 0, hence for any object X ∈ X

and any map f : X −→ A, the composition X −→ A −→ X ′[k] is zero and therefore it factorizes through B, say via a map g : X −→ B. Then by hypothesis g = 0 and therefore f = 0. We infer that any map from an object of X to an object in X[1] �X[2] � · · ·�X[k] is zero, so H(X[1] �X[2] � · · ·�X[k]) = 0. The proof that Hop(X[−1] �X[−2] � · · ·�X[−k]) = 0is similar.

(ii) Since for any object A ∈ T we have a triangle

Cellt−1(A) γt−1A−−−−→ A

ωt−1A−−−−→ Ωt

X(A)[t] −βt−1A [1]−−−−−−→ Cellt−1(A)[1] (Ct

A)

where Cellt−1(A) ∈ X � X[1] � · · · � X[t − 1] and ΩtX(A)[t] ∈ X�

t [t], it suffices to show that T(A, B) = 0, for any object A in X � X[1] � · · · � X[t − 1] and any object B ∈ X�

t [t]. Equivalently we show by induction that any map f : A −→ B is zero provided that Alies in X[1] �X[2] � · · ·�X[t] and B lies in X�

t [t +1], i.e. T(X, B[−i]) = 0, 1 � i � t. If t = 1, then A = X[1] for some X ∈ X and B satisfies T(X, B[−1]) = 0. Hence T(A, B) = 0. Assume now that A ∈ X[1] �X[2] � · · · �X[t] and B ∈ X�

t [t + 1]. Then there is a triangle


A′ g−→ A h−→ A′′ −→ A′[1], where A′ ∈ X[1] � X[2] � · · · � X[t − 1] and A′′ ∈ X[t], i.e. A′′ = X[t] for some X ∈ X. Since clearly B ∈ X�

t−1[t], by the induction hypothesis, the composition g ◦ f = 0 and therefore f = h ◦ φ for some map φ : A′′ −→ B. Since A′′ = X[t] and T(X, B[−t]) = 0, we have T(A′′, B) = 0. Hence φ = 0 and therefore f = 0. We infer that (∗) is a torsion pair in T. Finally using (i) and the construction of the tower of triangles (C•

A) associated to A, we see directly that H(γtA) is invertible,

1 � t � n − 1.(iii) By Remark 1.4(ii), it follows that A ∈ X � X[1] � · · · � X[t]. Conversely assume

that A ∈ X � X[1] � · · · � X[t]. Then the right (X � X[1] � · · · � X[t])-approximation γtA :

Cellt(A) −→ A splits and therefore from the tower of triangles (C•A) we have ωt

A =ωt−1A ◦ht

A[t] = 0. Hence the map htA[t] : Ωt

X(A)[t] −→ Ωt+1X (A)[t +1] factorizes through the

cone Cellt−1(A)[1] of ωt−1A , say as ht

A[t] = βt−1A [1] ◦ f , for some map f : Cellt−1(A)[1] −→

Ωt+1X (A)[t + 1]. We show that f = 0. Observe first that Ωt+1

X (A) ∈ X�t . Indeed this

follows form (i) if t � n − 1 and by applying H to the triangle Ωn+1X (A) −→ Xn

A −→Ωn

X(A) −→ Ωn+1X (A)[1] and using that Ωn

X(A) ∈ X�n , if t = n. Since Cellt−1(A)[1] lies in

X[1] � · · · � X[t] and Ωt+1X (A) ∈ X�

t [t + 1], by (ii) it follows that f is zero and therefore htA[t] = 0. This implies that Ωt

X(A)[t] is a direct summand of the left cone XtA[t] of ht

A[t], see Remark 1.4(i). We infer that Ωt

X(A) lies in X as a direct summand of XtA. �

Combining Remark 1.4, the Ghost Lemma (Proposition 2.2) and Proposition 3.2 we have the following.

Corollary 3.3. For any object A ∈ T and 1 � t � n, the following are equivalent:

(i) ΩtX(A) ∈ X.

(ii) A ∈ X � X[1] � · · · � X[t].(iii) Gh[t+1]

X (A, −) = 0.

Moreover the maps Cellt−1(A) αtA−→ Cellt(A) γt

A−→ A in the cellular tower of A induce isomorphisms:

H(A) ∼=−−→ H(Cell1(A)) ∼=−−→ H(Cell2(A)) ∼=−−→ · · · ∼=−−→ H(Celln(A))

and exact sequences:

⎧⎨⎩

H(X1A) −→ H(Cell0(A)) −→ H(Cell1(A)) −→ 0,

0 −→ H(Celln(A)) −→ H(Celln+1(A)) −→ H(Xn+1A [n + 1]).

The following main result of this section characterizes when the filtrations (2.2), (3.1)and (3.2) stabilize:


Theorem 3.4. If X is a contravariantly finite n-rigid subcategory of T, then the following are equivalent:

(i) X�n = X.

(ii) ΩnX(A) ∈ X, ∀A ∈ T.

(iii) Gh[n+1]X (T) = 0.

(iv) T = X � X[1] � · · · � X[n].(v) X� = X[1] � X[2] � · · · � X[n].(vi) For any 1 � t � n: X�

t [t] = X[t] � X[t + 1] � · · · � X[n].

If (i) holds, then X and X� are functorially finite and X�n+k = 0, ∀k � 1.

Proof. Since by Proposition 3.2, ΩnX(A)[n] lies in X�

n [n], condition (i) implies that Ωn

X(A) ∈ X, ∀A ∈ T. Hence (i) ⇒ (ii), and by Corollary 3.3 we have (ii) ⇔ (iii) ⇔(iv). We show that (iv) ⇒ (i). By Proposition 3.2 we know that, for any object A ∈ T, the map ωn−1

A : A −→ ΩnX(A)[n] is a left X�

n [n]-approximation of A and ΩnX(A)[n] lies

in X[n] since ΩnX(A) ∈ X by Corollary 3.3. Hence if A ∈ X�

n [n], the map ωn−1A is split

monic and therefore A lies in X[n] as a direct summand of ΩnX(A)[n]. We infer that

X�n [n] ⊆ X[n], or equivalently X�

n ⊆ X. Since X is n-rigid, we have X ⊆ X�n , hence

X�n = X, i.e. (iv) ⇒ (i).By Proposition 3.2, for t = 0, we have a torsion pair (X, X�), hence T = X �X�. Clearly

then (v) implies (iv). Conversely if (iv) holds, then by (ii) we have ΩnX(A) ∈ X, ∀A ∈ T.

This implies that Ωn−1X (A) ∈ X � X[1] and inductively Ω1

X(A) ∈ X � X[1] � · · · � X[n − 1]. Hence Ω1

X(A)[1] ∈ X[1] � X[2] � · · · � X[n]. If A lies in X�, then the map X0A −→ A is

zero and therefore A lies in X[1] � · · · � X[n] as a direct summand of Ω1X(A)[1]. Hence

X� ⊆ X[1] � · · · � X[n]. Since by Proposition 3.2(ii) we have X[1] � · · · � X[n] ⊆ X�, we infer that X� = X[1] � · · · �X[n]. Taking t = n in (vi) we get (i). Conversely assume that (i) holds and let 1 � t � n. If A lies in X �X[1] � · · ·�X[n − t], then using n-rigidity of X it follows easily that A ∈ X�

t . This implies that X �X[1] � · · · �X[n − t] ⊆ X�t and therefore

X[t] � X[t + 1] � · · · � X[n] ⊆ X�t [t]. If A lies in X�

t [t], then the left X�t [t]-approximation

ωt−1A : A −→ Ωt

X(A)[t] splits. Since, by (ii), ΩnX(A) lies in X, by Remark 1.4(iii), Ωt

X(A)[t]lies in X[t] � X[t + 1] � · · · � X[n]. Hence A lies in X[t] � X[t + 1] � · · · � X[n] as a direct summand of Ωt

X(A)[t]. We infer that X�t [t] ⊆ X[t] � X[t + 1] � · · · � X[n]. This concludes

the proof of the equivalence of conditions (i)–(vi).If (i) holds, and A ∈ X�

n+1, then clearly A ∈ X�n = X, so A[1] ∈ X[1], and A[1] ∈

X�n = X. It follows that A[1] ∈ X ∩X[1] = 0, hence A = 0. Therefore X�

n+1 = 0 and then X�

n+k = 0, ∀k � 1. From Proposition 3.2(ii), for t = 0, it follows that the subcategory X�

is covariantly finite, and, for t = n, the subcategory X �X[1] �· · ·�X[n −1] is contravariantly finite. This clearly implies that (X �X[1] � · · ·�X[n −1])[1] = X[1] �X[2] � · · ·�X[n] = X� is contravariantly finite. Since, by Proposition 3.2, X�

n [n] is covariantly finite, condition (i) implies that X[n] or equivalently X, is covariantly finite. We infer that X is functorially finite. �


Remark 3.5. By duality, if X is a covariantly finite subcategory of T, n � 1, such that �nX = X, then �X = X[−n] � X[−n + 1] � · · · � X[−1] = Ker Hop is functorially finite. We shall show later that: X is contravariantly finite and X�

n = X if and only if X is covariantly finite and �nX = X, see Proposition 5.1.

We close this subsection with the following vanishing results which will be useful later.

Corollary 3.6. Let X be a contravariantly finite n-rigid subcategory of T, and A ∈ X �X[1] � · · · � X[k].

(i) If k = n, then ωtA = 0, ∀t � n, there is a decomposition: Cellt(A) ∼= A ⊕Ωt+1

X (A)[t], ∀t � n, and:

T(ΩtX(A),X[n− t + 1]) = 0, 1 � t � n

(ii) If 0 � k � n, then for any object B ∈ T such that T(X, B[−i]) = 0, 1 � i � k − 1, we have:

GhX(Ω1

X(A), B)

= 0

and the map HA,B : T(A, B) −→ Hom(H(A), H(B)

), f �−→ H(f), is surjective.

Proof. (i) Since A ∈ X � X[1] � · · · � X[n], we have ΩnX(A) ∈ X by Corollary 3.3, and

then ΩtX(A) lies in X � X[1] � · · · � X[n − t]. On the other hand, since X ⊆ X�

n−t+1, we have X[n − t + 1] ⊆ X�

n−t+1[n − t + 1]. Then the torsion pair (X � X[1] � · · · � X[n −

t], X�n−t+1[n − t +1]

)in T, see Proposition 3.2(ii), shows that T(Ωt

X(A), X[n − t +1]) = 0, 1 � t � n. Finally note that the right (X �X[1] � · · · �X[n])-approximation γn

A of A splits and therefore ωn

A = 0. Then from the tower of triangles (C•A), we have ωt

A = 0, ∀t � n.(ii) Since A ∈ X � X[1] � · · · � X[k], k � n, it follows that Ωk

X(A) ∈ X and then by induction Ω1

X(A) ∈ X � X[1] � · · · � X[k − 1]. Hence there is a triangle X −→ Ω1X(A) −→

C −→ X[1], where X ∈ X and C ∈ X[1] �· · ·�X[k−1]. If α : Ω1X(A) −→ B is X-ghost, then

the composition X −→ Ω1X(A) −→ B is zero, hence Ω1

X(A) −→ B factorizes through Ω1

X(A) −→ C. Since any map from an object from X[1] � · · · � X[k − 1] to an object Bsatisfying T(X, B[−i]) = 0, 1 � i � k−1, is clearly zero, it follows that T(C, B) = 0. This implies that the map Ω1

X(A) −→ B is zero and consequently GhX(Ω1

X(A), B)

= 0. �3.2. Homological dimensions

Theorem 3.4 naturally suggests to introduce the following homological invariants of objects of T with respect to a contravariantly finite subcategory X of T. First recall that by Remark 1.2 we have additive functors Ωn

X : T/X −→ T/X, ∀n � 0, showing in particular that the homological dimensions pdX A and gl.dimX T introduced below are well-defined.


Definition 3.7. The X-projective dimension pdX A of A ∈ T is the smallest n � 0 such that Ωn

X(A) = 0 in T/X or equivalently ΩnX(A) ∈ X. If Ωn

X(A) �= 0, i.e. ΩnX(A) /∈ X,

∀n � 0, then we set pdX A = ∞.The X-global dimension of T is defined by: gl.dimX T = sup{pdX A | A ∈ T}.

Dually if X is covariantly finite, the X-injective dimension idX A is defined by using the functors Σn

X : T/X −→ T/X, ∀n � 0. If X is functorially finite in T, then the adjoint pair (Σn

X, ΩnX) in T/X, ∀n � 0, shows that

sup{

pdX A |A ∈ T}

= gl.dimX T = sup{idX A |A ∈ T

}

Recall from [16] that the triangulated category T is called X-Frobenius, for a functorially finite subcategory X of T, if GhX(−, X[1]) = 0 = CoGhX(X[−1], −). It is shown in [16, Proposition 3.1] that T is X-Frobenius if and only if the stable category T/X carries a natural triangulated structure with suspension functor Σ1

X.

Lemma 3.8. Let A be an object of T.

(i) If pdX A = m < ∞, then A ∈ X � X[1] � · · · � X[m].(ii) If T is X-Frobenius, then for any object A ∈ T: pdX A = 0 or ∞, and idX A = 0

or ∞. In particular if X �= T, then gl.dimX T = ∞.(iii) If X is n-rigid, then for any m � n we have: pdX A � m if and only if A ∈

X � X[1] � · · · � X[m].(iv) If X is n-rigid, and pdX A = m < n, then A ∈ X�

n−m, i.e. T(X, A[i]) = 0, 1 � i �n −m.

Proof. (i), (ii), (iii) Part (i) follows from Proposition 2.2 and part (iii) follows from Proposition 3.2. Since T is X-Frobenius, part (ii) follows directly from the fact that Ω1

X is an equivalence with quasi-inverse Σ1X, see [16, Proposition 3.1]. For part (iv), let

pdX A = m. Then A ∈ X � X[1] � · · · � X[m], hence there exists a triangle X0 −→A −→ B −→ X0[1], where X0 ∈ X and B ∈ X[1] � X[2] � · · · � X[m]. It follows that B[i] ∈ X[i +1] �· · ·�X[i +m] and this implies that for 1 � i � n −m we have T(X, B[i]) = 0since X is n-rigid and m < n. Then applying T(X, −) to the above triangle we see directly that T(X, A[i]) = 0, 1 � i � n −m. �

If X is n-rigid, then by Lemma 3.8 we have: gl.dimX T � m if and only if ΩmX (A) ∈ X,

∀A ∈ T, if and only if Gh[m+1]X (A, −) = 0, ∀A ∈ T, if and only if T = X �X[1] � · · · �X[m].

The following result shows that n is a lower bound for the global dimension of T with respect to contravariantly finite n-rigid subcategories.


Corollary 3.9. Let T be a non-trivial triangulated category and X a contravariantly finite n-rigid subcategory of T, where n � 1. Then gl.dimX T � n. Moreover:

gl.dimX T = n ⇐⇒ T = X � X[1] � · · · � X[n]

Proof. Assume gl.dimX T = m < ∞. If m < n, then T = X �X[1] �· · ·�X[m] by Lemma 3.8. Since X is m-rigid, by Theorem 3.4 we have X�

m+1 = 0. Since pdX A[t] � m, ∀t ∈ Z, ∀A ∈ T, it follows by Lemma 3.8 that A[t] ∈ X�

n−m, in particular T(X, A[t +n −m]) = 0, ∀t ∈ Z. Choosing successively t = 2m −n +1, 2m −n, · · · , m −n +1, we have T(X, A[1]) =· · · = T(X, A[m + 1]) = 0. Then A ∈ X�

m+1 = 0, hence A = 0. Since T �= 0, we infer that gl.dimX T � n. Finally by Lemma 3.8 we have gl.dimX T = n if and only if T = X � X[1] � · · · � X[n]. �3.3. Corigid subcategories

Let T be a triangulated category and X a full subcategory of T. It is easy to see that for any t � 1:

X�t [t + 1] =

{A ∈ T | T(X, A[−k]) = 0, 1 � k � t

}

and we have the following chain of full subcategories of T:

· · · ⊆ X�n [n + 1] ⊆ X�

n−1[n] ⊆ · · · · · · ⊆ X�2 [3] ⊆ X�

1 [2] ⊆ X�0 [1] (∗)

In the sequel we shall need the following notion of corigidity.

Definition 3.10. A full subcategory X of T is called t-corigid, where t � 1, if:

X ⊆ X�t [t + 1], i.e. T(X,X[−1]) = · · · = T(X,X[−t]) = 0

A t-corigid subcategory X is called strictly t-corigid, if X is not (t + 1)-corigid.

Clearly if X is s-corigid, then X is t-corigid for any t � s. The next two results give convenient characterizations of when rigid subcategories are corigid.

Proposition 3.11. Let X be a contravariantly finite n-rigid subcategory of T, where n � 2. Then for 1 � k � n − 1, the following statements are equivalent:

(i) X is (n − k)-corigid.(ii) OH(X[−i], −) = 0, 1 � i � n − k.

In particular if the functor H : T −→ mod-X is full, then X is (n − k)-corigid, for any kwith 1 � k � n − 1.


Proof. If X is (n − k)-corigid, then we have H(X[−i]) = T(X, X[−i]) = 0 and this clearly implies that OH(X[−i], −) = 0, for 1 � i � n − k. Conversely if this holds, then by Corollary 2.6, we have X[−i] ⊆ (X � X[1]) ⊕ X�. Assume that there exists X ∈ X

such that X[−i] lies in X � X[1] for some 1 � i � n − k. Then there exists a triangle X0 −→ X[−i] −→ X1[1] −→ X0[1] and therefore a triangle X0[i] −→ X −→ X1[i +1] −→ X0[i + 1] in T. Since X is n-rigid and 1 � i � n − 1, applying the functor H to the last triangle, we see directly that H(X) = 0, i.e. X = 0. We infer that X[−i] ⊆ X�

for 1 � i � n − k and this means that X is (n − k)-corigid.The last assertion is clear since by Corollary 2.6, the functor H is full if and only if

OH = 0. �Proposition 3.12. Let T be triangulated category and X a contravariantly finite n-rigid subcategory of T. If T = X � X[1] � · · · � X[n], then, for 1 � t � n, the following are equivalent:

(i) X is t-corigid.(ii) X[n + 1] ⊆ X � X[1] � · · · � X[n − t].(iii) X ⊆ X[t + 1] � · · · � X[n] � X[n + 1].(iv) Ghn−t+1

X (X[n + 1], −) = 0.

Proof. (i) ⇒ (ii) Since X[n + 1] ⊆ T = X � X[1] � · · · � X[n], for any object X ∈ X, there exists a triangle A −→ X[n + 1] −→ B −→ A[1], where A ∈ X � X[1] � · · · � X[n − t] and B ∈ X[n − t + 1] � · · · � X[n]. Since X is t-corigid, any map from an object from X to an object from X[−t] � · · ·�X[−1] is zero and this implies that any map from an object from X[n +1] to an object from X[n − t +1] � · · ·�X[n] is zero. Hence the map X[n +1] −→ B

is zero and then X[n + 1] lies in X � X[1] � · · · � X[n − t] as a direct summand of A.(ii) ⇒ (i) The hypothesis implies that X ⊆ X[−n −1] �X[−n] � · · ·�X[−t −1]. Hence for

any object X ∈ X, there exists a triangle A −→ X −→ B −→ A[1], where A ∈ X[−n −1]and B ∈ X[−n] �· · ·�X[−t −1]. Let 1 � k � t and consider any map X −→ X ′[−k], where X ′ ∈ X. The composition A −→ X −→ X ′[−k] is zero since it lies in T(X, X[n + 1 − k])which is zero since X is n-rigid. Hence the map X −→ X ′[−k] factorizes through the map X −→ B. However using that 1 � k � t and X is n-rigid, it follows easily that any map from an object from X[−n] � · · · � X[−t − 1] to an object from X[−k] is zero. Hence the map X −→ X ′[−k] is zero, i.e. T(X, X[−k]) = 0, 1 � k � t. We infer that Xis t-corigid.

Finally, since X is t-corigid iff X ⊆ X�t [t + 1], the equivalence (i) ⇔ (iii) follows from

Theorem 3.4(vi), and the equivalence (ii) ⇔ (iv) is a consequence of Corollary 3.3. �The following gives another characterization of corigid subcategories.

Corollary 3.13. Let X be a contravariantly n-rigid subcategory of T. If T = X �X[1] � · · ·�X[n], then, for 0 � k � n − 1, the following conditions are equivalent.


(i) X is (n − k)-corigid.(ii) X = X�

k

⋂(X[n − k + 1] � X[−k + 1] � · · · � X[−1] � X

).

Proof. (ii) ⇒ (i) By Proposition 3.12 it suffices to show that X[n +1] ⊆ X �X[1] �· · ·�X[k]. By Proposition 3.2(i) we have Ωk

X(X[n + 1]) ⊆ X�k . On the other hand we clearly have

Ω1X(X[n + 1]) ∈ X[n] � X. This implies that Ω2

X(X[n + 1]) ⊆ X[n − 1] � X[−1] � X. Then by induction we have Ωk

X(X[n + 1]) ⊆ X[n − k + 1] � X[−k + 1] � · · · � X[−1] � X. Hence the condition in (ii) implies that Ωk

X(X[n + 1]) ⊆ X. It follows from Corollary 3.3 that X[n + 1] ⊆ X � X[1] � · · · � X[k] and therefore X is (n − k)-corigid.

(i) ⇒ (ii) Clearly X ⊆ (X[n − k + 1] � X[−k + 1] � · · · � X[−1] � X) ∩ X�k . Let A

be in (X[n − k + 1] � X[−k + 1] � · · · � X[−1] � X) ∩ X�k . Then there exists a triangle

M −→ A −→ X −→ M [1], where X ∈ X and M ∈ X[n − k + 1] �X[−k + 1] � · · · �X[−1]. Since A ∈ X�

k , by Theorem 3.4(vi) it follows that A lies in X �X[1] � · · · �X[n − k]. Using that X is (n − k)-corigid, it is easy to see that any map from an object of X[n − k + 1] �X[−k + 1] � · · · � X[−1] to an object of X � X[1] � · · · � X[n − k] is zero. Hence the map M −→ A is zero and then A lies in X as a direct summand of X. �

The following consequence, which will be useful in the next section, characterizes the (n − k)-corigid subcategories satisfying the conditions of Theorem 3.4.

Corollary 3.14. Let X be contravariantly finite in T, and 0 � k � n −1. Then the following are equivalent.

(i) X is n-rigid, (n − k)-corigid, and T = X � X[1] � · · · � X[n].(ii) X = X�

n ⊆ X[n − k + 1] � X[n − k + 2] � · · · � X[n] � X[n + 1].(iii) X = X�

n = X�k

⋂(X[n − k + 1] � X[−k + 1] � · · · � X[−1] � X

).

4. Resolutions and homological dimensions

In this section we construct (co)resolutions of objects of a triangulated category by objects of certain subcategories and we study the associated homological invariants in connection with the homological dimensions defined in Section 3.

4.1. (Co)resolutions

Let T be a triangulated category with split idempotents and X a contravariantly finite subcategory of T. Let H : T −→ mod-X be the associated homological functor.

Definition 4.1. Let U and V be full subcategories of T and let A be an object of T.

(i) A U-coresolution of A (with respect to H) is a complex

0 −→ A −→ U0 −→ U1 −→ · · · · · · −→ Uk −→ · · · (†)


in T, where U i ∈ U, ∀i � 0, such that the induced complex in mod-X

0 −→ H(A) −→ H(U0) −→ H(U1) −→ · · · · · · −→ H(Uk) −→ · · ·

is exact. The minimum k ∈ N ∪ {∞} such that there exists a U-coresolution (†) of A with U i = 0, ∀i > k, is called the U-coresolution dimension of A and is denoted by cores.dimU A.

(ii) A V-resolution of A (with respect to H) is a complex

· · · −→ V k −→ · · · −→ V 1 −→ V 0 −→ A −→ 0 (††)

in T, where V i ∈ V, ∀i � 0, such that the induced complex in mod-X

· · · −→ H(V k) −→ · · · −→ H(V 1) −→ H(V 0) −→ H(A) −→ 0

is exact. The minimum k ∈ N ∪ {∞} such that there exists a V-resolution (††) of A with V i = 0, ∀i > k, is called the V-resolution dimension of A and is denoted by res.dimV A.

(iii) If W ⊆ T, then we set:

res.dimU W = sup{

res.dimU W | W ∈ W}

and

cores.dimU W = sup{

cores.dimV W | W ∈ W}

Observe that res.dimX A = pd H(A). We are mostly interested in the following cases: (α) U = X and W = X�

n [n + 1], and (β) V = X�n [n + 1] and W = X. In this connection

we have the following main result of this section which shows existence of X-resolutions and X�

n [n + 1]-coresolutions for certain objects of T.

Theorem 4.2. Let X be a contravariantly finite n-rigid subcategory of T, n � 1. If n � 2, we assume further that X is (n − k)-corigid, for some k with 0 � k � n+1

2 .

(i) ∀ A ∈(X � X[1] � · · · � X[k]

)⋂X�

k [k + 1] : res.dimX A � k

(ii) If X�n ⊆ X[−n − 1] � X[−n] � · · · � X[−n + k − 1], then:

∀A ∈ X�n−k[n− k + 1]

⋂X�

k : cores.dimX�n [n+1] A � k

Note that if n = 1, then the only values for the non-negative integer k in the condition “X is (n −k)-corigid, for some k with 0 � k � n+1

2 ” appearing in Theorem 4.2, are k = 0and k = 1. Clearly the above condition is vacuous in case n = 1 = k. Note also that if n � 2 and 0 � k � n+1

2 , then 0 � k � n − 1.We split the proof of Theorem 4.2 into several steps. The first one is a direct conse-

quence of Corollary 2.11.


Proposition 4.3. Let X be a contravariantly finite subcategory of T and A ∈ T.

(i) If X is 1-rigid and A ∈(X � X[1]

)∩ X�

1 [2], then: pd H(A) � 1.(ii) Let t � 2 and assume that X is t-rigid and (t − 1)-corigid. Then:

A ∈(X � X[1] � · · · � X[t]

)∩ X�

t [t + 1] =⇒ pd H(A) � t.

Proof. Part (ii) follows from Corollary 2.11. For part (i), let X be 1-rigid and A ∈(X � X[1]

)∩ X�

1 [2]. Then T(X, A[−1]) = 0 and there exists a triangle X0 −→ A −→X1[1] −→ X0[1], where the Xi lie in X. Applying H we have an exact sequence 0 −→H(X1) −→ H(X0) −→ H(A) −→ 0. Hence pd H(A) � 1. �Proposition 4.4. Let X be a contravariantly finite n-rigid subcategory of T, n � 1. If n � 2, we assume further that X is (n − k)-corigid, for some k with 0 � k � n − 1. Let A be an object in X�

n−k[n − k + 1].

(i) There exist triangles:

Ci −→ Ei −→ Ci+1 −→ Ci[1], 0 � i � k − 1, C0 = A, Ck = Ek

where each Ei lies in X�n [n + 1], 0 � i � k, which induce exact sequences in mod-X

0 −→ H(Ci) −→ H(Ei) −→ H(Ci+1)

(ii) If A ∈ X�k , and if X�

n ⊆ X[−n − 1] �X[−n] � · · · �X[−n + k − 1] in case k � 2, then we have an exact sequence:

0 −→ H(A) −→ H(E0) −→ H(E1) −→ · · · · · · −→ H(Ek) −→ 0

provided that k � n+12 .

Proof. We split the proof in two parts, dividing the second part into several steps.

1. Case n = 1: We use Proposition 3.2: since X is 1-rigid, we have a triangle

Ω1X(A[−1])[1] −→ Cell0(A[−1])[1] −→ A −→ Ω1

X(A[−1])[2]

where Cell0(A[−1]) = X0A[−1] ∈ X ⊆ X�

1 . Setting E0 = Ω1X(A[−1])[2] and E1 =

Cell0(A[−1])[2] it follows directly that the Ei lie in X�1 [2], that is H(Ei[−1]) = 0, for

0 � i � 1, so we have the desired triangle A −→ E0 −→ E1 −→ A[1], and an exact sequence

0 −→ H(A) −→ H(E0) −→ H(E1)


Observe that if A ∈ X�1 , i.e. H(A[1]) = 0, then we have an exact sequence

0 −→ H(A) −→ H(E0) −→ H(E1) −→ 0

2. Case n � 2: We assume that X is (n −k)-corigid, for some k with 0 � k � n+12 . For the

convenience of the reader, we first treat the cases 0 � k � 3. Let A ∈ X�n−k[n −k+1].

(a) Let k = 0. Then A lies in X�n [n +1], so we may choose E0 = A, and Ci = 0 = Ei

for i � 1.(b) Let k = 1. Then A ∈ X�

n−1[n], i.e. T(X, A[−i]) = 0, 1 � i � n − 1. Consider the triangle arising from the tower of triangles (C•

A[−1]) associated to the object A[−1]:

ΩnX(A[−1])[n− 1] −→ Celln−1(A[−1]) −→ A[−1] −→ Ωn

X(A[−1])[n]

By Proposition 3.2, ΩnX(A[−1]) ∈ X�

n and Celln−1(A[−1]) ∈ X �X[1] �· · ·�X[n −1]. Then we have a triangle

ΩnX(A[−1])[n] −→ Celln−1(A[−1])[1] −→ A −→ Ωn

X(A[−1])[n + 1] (4.1)

where Celln−1(A[−1])[1] lies in X[1] � · · · � X[n], so H(Celln−1(A[−1])[1]) = 0and Ωn

X(A[−1])[n + 1] lies in X�n [n + 1]. We set E0 := Ωn

X(A[−1])[n + 1] and E1 := Celln−1(A[−1])[2]. Applying the homological functor H to the triangle A −→ E0 −→ E1 −→ A[1] and using that A lies in X�

n−1[n], and E0 lies in X�

n [n + 1], we see directly from the induced long exact sequence that E1 lies in X�

n [n + 1] and we have an exact sequence in mod-X:

0 −→ H(A) −→ H(E0) −→ H(E1)

Observe that if A ∈ X�1 , i.e. we have H(A[1]) = 0, then we get, as required, an

exact sequence:

0 −→ H(A) −→ H(E0) −→ H(E1) −→ 0

where the Ei lie in X�n [n + 1].

(c) Let k = 2. Then n � 3 and X is (n − 2)-corigid, and A ∈ X�n−2[n − 1]. Equiv-

alently H(X[−i]) = 0, 1 � i � n − 2. Consider the triangle (4.1) and set E0 := Ωn

X(A[−1])[n + 1] and C1 := Celln−1(A[−1])[2]. Then as before E0 lies in X�

n [n + 1]. Applying H to the triangle

A −→ E0 −→ C1 −→ A[1] (4.2)

and using that A lies in X�n−2[n − 1] and H(C1[−1]) = 0, we have an exact

sequence

0 −→ H(A) −→ H(E0) −→ H(C1)


and H(C1[−i]) = 0 for 1 � i � n − 1. It follows that C1 lies in X�n−1[n]. Setting

E1 := ΩnX(C1[−1])[n + 1] and E2 := Celln−1(C1[−1])[2], this implies, as in the

case k = 1, the existence of a triangle

C1 −→ E1 −→ E2 −→ C1[1] (4.3)

where the Ei lie in X�n [n +1]. Since E2[−1] = Celln−1(C1[−1])[1] ∈ X[1] �· · ·�X[n],

by Proposition 3.2 we have H(E2[−1]) = 0 and therefore we have an exact sequence

0 −→ H(C1) −→ H(E1) −→ H(E2)

Observe that if A ∈ X�2 , so H(A[i]) = 0 for 1 � i � 2, then applying H to the

triangle (4.2) we get an exact sequence

0 −→ H(A) −→ H(E0) −→ H(C1) −→ 0

and an isomorphism H(E0[1]) ∼= H(C1[1]). We infer that we have an exact se-quence

0 −→ H(A) −→ H(E0) −→ H(E1) −→ H(E2)

If moreover X�n ⊆ X[−n −1] �X[−n] �X[−n +1], then X�

n [n +1] ⊆ X �X[1] �X[2], so X�

n [n + 2] ⊆ X[1] �X[2] �X[3]. Since n � 3, it follows that X[1] �X[2] �X[3] ⊆ X�

and therefore H(X�n [n + 2]) = 0. Since E0[1] lies in X�

n [n + 2], we infer that H(C1[1]) ∼= H(E0[1]) = 0. It follows that the map H(E1) −→ H(E2) is an epimorphism and therefore we have the desired exact sequence

0 −→ H(A) −→ H(E0) −→ H(E1) −→ H(E2) −→ 0

(d) Now we discuss the case k = 3. Then n � 2 · 3 − 1 = 5 and X is (n − 3)-corigid, i.e. H(X[−i]) = 0, 1 � i � n − 3. Let A ∈ X�

n−3[n − 2]. Then as above, we have triangles:

A −→ E0 −→ C1 −→ A[1] & C1 −→ E1 −→ C2 −→ C1[1] (4.4)

where as before: E0 := ΩnX(A[−1])[n + 1], C1 := Celln−1(A[−1])[2], E1 :=

ΩnX(C1[−1])[n + 1], and C2 := Celln−1(C1[−1])[2]. Clearly Ei lie in X�

n [n + 1]for 0 � i � 1. Applying the homological functor H to the triangles (4.4), we see directly that C1 ∈ X�

n−2[n − 1] and C2 ∈ X�n−1[n]. Now setting

E2 := ΩnX(C2[−1])[n + 1] and E3 := Celln−1(C2[−1])[2], we have as before a

triangle

C2 −→ E2 −→ E3 −→ C2[1] (4.5)


where the objects Ei lie in X�n [n + 1], for 2 � i � 3, and exact sequences:

0 −→ H(A) −→ H(E0) −→ H(C1) &

0 −→ H(C1) −→ H(E1) −→ H(C2) (4.6)

0 −→ H(C2) −→ H(E2) −→ H(E3) (4.7)

Now assume that A ∈ X�3 , i.e. H(A[i]) = 0, for 1 � i � 3. Then applying H

to the first triangle in (4.4) we have an exact sequence (∗) : 0 −→ H(A) −→H(E0) −→ H(C1) −→ 0 and isomorphisms H(E0[i]) ∼= H(C1[i]), for 1 � i � 2. Assume moreover that X�

n ⊆ X[−n − 1] � X[−n] � X[−n + 1] � X[−n + 2]. Then X�

n [n + 2] ⊆ X[1] � X[2] � X[3] � X[4] and X�n [n + 3] ⊆ X[2] � X[3] � X[4] � X[5].

Since n � 5, it follows that X[1] � X[2] � X[3] � X[4] ⊆ X� ⊇ X[2] � X[3] � X[4] �X[5] and therefore H(X�

n [n + 2]) = 0 = H(X�n [n + 3]). Since the Ei[1] lie in

X�n [n + 2], we have H(Ei[1]) = 0 and since the Ei[2] lie in X�

n [n + 3], we have H(Ei[2]) = 0. It follows easily then that H(C1[1]) ∼= H(E0[1]) = 0 and H(C1[2]) ∼=H(E0[2]) = 0. Therefore applying H to the second triangle in (4.4) we obtain a short exact sequence (∗∗) : 0 −→ H(C1) −→ H(E1) −→ H(C2) −→ 0. On the other hand applying H to the second triangle in (4.4) we have a long exact sequence · · · −→ H(E1[1]) −→ H(C2[1]) −→ H(C1[2]) −→ H(E1[2]) −→ · · · . Since H(E1[1]) = 0 = H(E1[2]), the map H(C2[1]) −→ H(C1[2]) is invertible and therefore H(C2[1]) = 0. Hence applying H to the triangle (4.5) we obtain that the map H(E2) −→ H(E3) in (4.7) is an epimorphism and we have a short exact sequence (∗∗∗) : 0 −→ H(C2) −→ H(E2) −→ H(E3) −→ 0. Splicing together the short exact sequences (∗), (∗∗), and (∗∗∗), we obtain, as required, the following exact sequence, where the objects Ei lie in X�

n [n + 1], 0 � i � 3:

0 −→ H(A) −→ H(E0) −→ H(E1) −→ H(E2) −→ H(E3) −→ 0

(e) Continuing in this way for n − 1 � k � 4, under the hypotheses of part (ii) and working as above, we construct inductively triangles

Ci −→ Ei −→ Ci+1 −→ Ci[1], 0 � i � k − 1, C0 = A, Ck = Ek

Ei = ΩnX(Ci[−1])[n + 1] and Cj = Celln−1(Cj−1[−1])[2],

for 1 � j � k − 1, 0 � i � k − 1, such that

Ei ∈ X�n [n + 1], 0 � i � k − 1 and Ci ∈ X�

n−k−i[n− k − i + 1]

for 1 � i � k, and the induced sequences

0 −→ H(Ci) −→ H(Ei) −→ H(Ci+1), 0 � i � k − 1


are exact in mod-X. Moreover if A ∈ X�k and X�

n ⊆ X[−n − 1] � X[−n] � · · · �X[−n + k − 1], then as above the maps H(Ei) −→ H(Ci+1) are epimorphisms. Therefore we have an exact sequence

0 −→ H(A) −→ H(E0) −→ H(E1) −→ · · · −→ H(Ek−1) −→ H(Ek) −→ 0

We infer that the induced complex 0 −→ A −→ E0 −→ E1 −→ · · · −→ Ek −→ 0is an X�

n [n + 1]-coresolution of A and therefore cores.dimX�n [n+1] A � k. �

Now we can give the proof of Theorem 4.2.

Proof of Theorem 4.2. Part (ii) follows from Proposition 4.4. For part (i) first note that the case n = 1 follows from Proposition 4.3(i). Assume now that n � 2, and X is (n −k)-corigid, where 0 � k � n+1

2 . Observe that since n � 2 ·k−1, we have n −k � k−1. Since X is (n − k)-corigid, this easily implies that X is (k − 1)-corigid. Since k � n, X is also k-rigid and therefore the assertion follows from Proposition 4.3(ii). �

As a consequence we have the following interesting symmetry between X and X�n [n +1].

Corollary 4.5. Let X be a contravariantly finite n-rigid subcategory of T, n � 1. If n � 2, assume that X is (n − k)-corigid, 0 � k � n+1

2 , and X�n [n + 1] ⊆ X � X[1] � · · · � X[k].

Then:

res.dimX X�n [n + 1] � k and cores.dimX�

n [n+1] X � k.

Proof. Since k � n, it follows that X�n [n + 1] ⊆ X�

k [k + 1]. Now the condition X�n ⊆

X[−n −1] �X[−n] � · · ·�X[−n +k−1] implies that X�n [n +1] ⊆ X �X[1] � · · ·�X[k], hence

X�n [n + 1] ⊆

(X � X[1] � · · · � X[k]

)∩ X�

k [k + 1]. Then the bound res.dimX X�n [n + 1] � k

follows from Theorem 4.2(ii). Since X is (n −k)-corigid and k-rigid, we have X ⊆ X�n−k[n −

k + 1] ∩ X�k and the bound cores.dimX�

n [n+1] X � k follows from Theorem 4.2(i). �Although the condition X�

n ⊆ X[−n − 1] � X[−n] � · · · � X[−n + k − 1] or equiva-lently X�

n [n + 1] ⊆ X � X[1] � · · · � X[k], looks somewhat artificial, we shall see later, cf. Theorem 5.3, that it is directly connected with X being an (n + 1)-cluster tilting subcategory.

4.2. Finite projective dimension

We give a description of the full subcategories Proj�k A , where A = mod-X or X-mod, consisting of all objects with projective dimension � k.

Theorem 4.6. Let X be a contravariantly finite t-rigid subcategory of T, t � 1, and assume that X is (t − 1)-corigid, if t � 2. Then the functor H : T −→ mod-X induces


a full embedding

H :(X � X[1] � · · · � X[t]

) ⋂X�

t [t + 1] Proj�t mod-X

which is an equivalence if X is t-corigid.

Proof. (α) Set Ut :=(X � X[1] � · · · � X[t]

)∩ X�

t [t + 1] and Ht := H|Ut. First note that

by Proposition 4.3, we have pdH(A) � t, for any object A ∈ Ut, so H induces a functor Ht : Ut −→ Proj�k mod-X.

Let α : A −→ B be a map in Ut such that H(α) = 0. Then α factorizes through the map h0

A : A −→ Ω1X(A)[1], say via a map β : Ω1

X(A)[1] −→ B. Since X is t-rigid and A lies in X �X[1] � · · ·�X[t], it follows that Ωt

X(A) lies in X and then it is easy to see that Ω1X(A)

lies in X �X[1] � · · ·�X[t −1], hence ΩtX(A)[1] ∈ X[1] �X[2] � · · ·�X[t]. Since B ∈ X�

t [t +1], we have T(X, B[−i]) = 0, 1 � i � t, and then β = 0 by Proposition 3.2(ii). Therefore α = 0 and Ht is faithful. Next let α : Ω1

X(A) −→ B be an X-ghost map. Then α factorizes through the map h1

A : Ω1X(A) −→ Ω2

X(A)[1], say via a map β : Ω2X(A)[1] −→ B. As above,

Ω2X(A) lies in X � X[1] � · · · � X[t − 2], hence Ω2

X(A)[1] ∈ X[1] � X[2] � · · · � X[t − 1]. Since B ∈ X�

t [t + 1], again by Proposition 3.2(ii) it follows that β = 0, and therefore α = 0. Consequently GhX(Ω1

X(A), B) = 0, and then, by Proposition 2.5, the map T(A, B) −→Hom(H(A), H(B)) is surjective. We infer that Ht is full.

(β) Let X be t-corigid and let F be in mod-X with pdF � t. We have a projective resolution in mod-X

0 −→ H(Xt) −→ H(Xt−1) −→ · · · · · · −→ H(X1) −→ H(X0) −→ F −→ 0

where each Xi lies in X. The last map of the resolution gives us a map Xt −→ Xt−1

which induces a triangle Xt −→ Xt−1 −→ At−1 −→ Xt[1]. Applying H to this triangle and using that X is t-rigid and t-corigid, we see easily that H(At−1[i]) = 0, 1 � i � t − 1, and H(At−1[−i]) = 0, 1 � i � t, so At−1 lies in X�

t [t + 1]. Moreover we have an exact sequence 0 −→ H(Xt) −→ H(Xt−1) −→ H(At−1) −→ 0 and Im

(H(Xt−1) −→

H(Xt−2))

= H(At−1). Since Xt ∈ X, we have At−1 ∈ X �X[1]; hence by Corollary 2.6 the monomorphism H(At−1) −→ H(Xt−2) is induced by a map At−1 −→ Xt−2. Consider a triangle At−1 −→ Xt−2 −→ At−2 −→ At−1[1]. As above, applying H to this triangle and using that X is t-rigid and t-corigid, we see easily that T(X, At−2[i]) = 0, 1 �i � t − 2, and At−2 ∈ X�

t [t + 1]. Moreover we have a short exact sequence 0 −→H(At−1) −→ H(Xt−2) −→ H(At−2) −→ 0 and Im

(H(Xt−2) −→ H(Xt−3)

)= H(At−2).

Since At−1 ∈ X � X[1], we have At−2 ∈ X � X[1] � X[2] and then by Corollary 3.6(ii) the obstruction group OH(At−2, Xt−3) vanishes, since X is (t − 1)-corigid. Hence the natural monomorphism H(At−2) −→ H(Xt−3) is induced by a map At−2 −→ Xt−3. Considering a triangle At−2 −→ Xt−3 −→ At−3 −→ At−2[1] and continuing in this way by using Corollary 3.6(ii) and the fact that X is t-(co)rigid, we may construct inductively triangles Aj −→ Xj−1 −→ Aj−1 −→ Aj [1], with Xj ∈ X, and exact sequences 0 −→


H(Aj) −→ H(Xj−1) −→ H(Aj−1) −→ 0, for 1 � j � t, where At = Xt, and the objects Aj satisfy T(X, Aj [i]) = 0, 1 � i � j, and Aj ∈ (X � X[1] � · · · � X[t − j]) ∩X�

t [t + 1]. In particular for j = 1, we have a triangle A1 −→ X0 −→ A0 −→ A1[1], where A := A0 ∈ X � X[1] � · · · � X[t] ∩ X�

t [t + 1] which induces a short exact sequence 0 −→ H(A1) −→ H(X0) −→ H(A) −→ 0. This implies that H(A) ∼= F , where A ∈ Ut. We infer that H : Ut −→ Proj�t mod-X is essentially surjective, hence an equivalence. �

For latter use we include without proof the following dual version of Theorem 4.6.

Theorem 4.7. Let X be a covariantly finite t-rigid subcategory of T and assume that X is (t − 1)-corigid, if t � 2. Then the functor Hop : Top −→ X-mod induces a full contravariant embedding

Hop :(X[−t] � · · · � X[−1] � X

)⋂⊥t X[−t− 1] Proj�t X-mod

which is a duality if X is t-corigid.

4.3. Serre duality and Calabi–Yau categories

Assume that T is a k-linear triangulated category over a field k. Recall that T is Hom-finite if T(A, B) is a finite-dimensional vector space over k, ∀A, B ∈ T. In this case T admits Serre duality, if there exists a Serre functor S : T −→ T for T, [20]. Thus S is a triangulated auto-equivalence of T, and for any objects A, B in T there are natural isomorphisms

D HomT(A,B) ∼=−−→ HomT(B,SA)

where D denotes the k-dual functor. If T admits a Serre functor S, then T is called d-Calabi–Yau, for some d � 1, provided that S(?) ∼= (?)[d] as triangulated functors.

If X is functorially finite subcategory of T and T has a Serre functor, then it is easy to see that X is a dualizing variety in the sense of [5], compare [9, Theorem 2.3]. In particular the duality D = Homk(−, k) with respect to the base field induces a duality D : (mod-X)op ≈−→ X-mod.

For later use we point-out the following observation.

Lemma 4.8. Let T be a triangulated category with a Serre functor S over a field k, and let X be a functorially finite 1-rigid subcategory of T. Then we have the following:

(i) The category mod-X has enough projectives and enough injectives.(ii) The homological functor H : T −→ mod-X induces equivalences:

S(X) ≈−−→ Inj mod-X and X ≈−−→ Projmod-X


Proof. Since X is a dualizing variety, the duality functor D induces an equivalence be-tween (ProjX-mod)op and Injmod-X and therefore Injmod-X = DHop(X) = DT(X, −)|X =T(−, S(X))|X = H(S(X)). Since X is covariantly finite, for any object A in T there ex-ists a triangle Σ1

X(S−1(A))[−1] −→ S−1(A) −→ XS−1(A)

0 −→ Σ1X(S−1(A)), where the

middle map is a left X-approximation of S−1(A). Then we have a map A −→ SXS−1(A)

0

which induces a map H(A) −→ H(SXS−1(A)

0 ) and H(SXS−1(A)

0 ) is injective in mod-X. This map is a monomorphism since H(SΣ1

X(S−1(A))[−1]) = T(−, SΣ1X(S−1(A))[−1])|X ∼=

DT(Σ1X(S−1(A))[−1], X) and Σ1

X(S−1(A)) ∈ �1 X by the dual of Proposition 3.2(i). Since

any object of mod-X is of the form H(A) for some A ∈ T, this shows that mod-X has enough injectives. Hence we have the functor H : S(X) −→ Inj mod-X which is surjec-tive on objects. If α : S(X1) −→ S(X2) is a map such that H(α) = 0, then α factorizes through X�. Since T(X�, S(X2)) ∼= DT(X2, X�) = 0, it follows that α is zero and there-fore H|S(X) is faithful. We show that H|S(X) is full. By Proposition 2.5 it suffices to show that any X-ghost map α : Ω1

X(S(X1)) −→ S(X2), where the Xi lie in X, is zero. However α factorizes through the universal X-ghost map Ω1

X(S(X1)) −→ Ω2X(S(X1))[1] out of

Ω1X(S(X1)), say via a map β : Ω2

X(S(X1))[1] −→ S(X2). By Serre duality, the last map corresponds to an element in DT(X2, Ω2

X(S(X1))[1]) which is zero by Proposition 3.2(i). It follows that β, hence α, is zero. Hence GhX(Ω1

X(S(X1)), S(X2)) = 0, ∀X1, X2 ∈ X and therefore the functor H : S(X) −→ Inj mod-X is full, hence an equivalence. The case of projectives was treated in Section 1. �

The following consequence of Theorems 4.6 and 4.7 extends Lemma 4.8 to objects of finite projective or injective dimension.

Corollary 4.9. Let T be a triangulated category with Serre duality over a field k, and let X be a functorially finite n-rigid subcategory of T which is (n − 1)-corigid, if n � 2. Let S be the Serre functor of T.

(i) The homological functor H : T −→ mod-X induces a full embedding

H :(X � X[1] � · · · � X[n]

) ⋂X�

n [n + 1] Proj�n mod-X

which is an equivalence if X is n-corigid.(ii) The homological functor D ◦ Hop = H ◦ S : T −→ mod-X induces a full embedding

H ◦ S :(X[−n] � · · · � X[−1] � X

) ⋂⊥nX[−n− 1] Inj�n mod-X

which is an equivalence if X is n-corigid.


Part 2. Higher cluster tilting subcategories

In this part we study higher cluster tilting and maximal rigid subcategories in an ar-bitrary triangulated category. After presenting several characterizations and properties of cluster tilting subcategories, we concentrate at the Gorenstein and the Calabi–Yau property of the associated cluster tilted category of coherent functors. In this context and under some natural assumptions, we give conditions ensuring finiteness of global dimen-sion, and the Gorenstein and the stably Calabi–Yau property, and we give applications to representation dimension.

5. Cluster-tilting and maximal rigid subcategories

Let as before T be a triangulated category with split idempotents and X a full sub-category of T which is closed under direct summands and isomorphisms. The results of Section 4 show that if X is contravariantly finite and satisfies X = X�

n , then X enjoys special properties. In this section we analyze further the structure of such subcategories by giving several characterizations. We also study briefly n-rigid subcategories X which are maximal among those which are contained in X�

n .

5.1. Cluster tilting subcategories

First we observe the following symmetry.

Proposition 5.1. Let X be a full subcategory of T, and n � 1. Then the following are equivalent:

(i) X is contravariantly finite and X = X�n .

(ii) X is contravariantly finite and both X and X�n are n-rigid.

(iii) X is covariantly finite and X = �nX.

(iv) X is covariantly finite and both X and �nX are n-rigid.

Proof. (i) ⇒ (ii) and (iii) ⇒ (iv) The proof is trivial.(ii) ⇒ (i) ⇐ (iv) Assume that (ii) holds. Since X is n-rigid, we have X ⊆ X�

n . By Proposition 3.2, for any object A ∈ T, the map ωn−1

A : A −→ ΩnX(A)[n] in the tower of

triangles (C•A), is a left X�

n [n]-approximation of A. If A lies in X�n , then since the latter

is n-rigid it follows that ωn−1A = 0 and therefore ωn−2

A factorizes through the left cone Xn−1

A [n − 1] of hn−1A [n − 1]. Since both A and Xn−1

A lie in X�n and the latter is n-rigid,

it follows that T(A, Xn−1A [n − 1]) = 0 and this implies that ωn−2

A = 0. Continuing in this way after n − 1 steps we deduce that ω1

A = 0 and therefore the map ω0A factorizes

through the left cone X1A[1] of h1

A[1]. Since both A and X1A lie in X�

n and the latter is n-rigid, we have T(A, X1

A[1]) = 0 and this implies that h0A = ω0

A = 0. Then A lies in X as a direct summand of X0

A. Hence X�n ⊆ X and therefore X = X�

n . The proof that


(iv) ⇒ (i) is dual to the proof of the implication (ii) ⇒ (i) using cocellular towers, cf. Subsection 1.2, and is left to the reader.

(i) ⇒ (iii) By Theorem 3.4, X is covariantly finite, and clearly X ⊆ �nX since X is

n-rigid. Let A be in �nX and consider the map ωn−1A : A −→ Ωn

X(A)[n]. Then ΩnX(A)[n] ∈

X[n] and therefore ωn−1A = 0 since A ∈ �

nX. Hence ωn−2A ◦ hn−1

A [n − 1] = 0 and therefore ωn−1A factors through the left cone Xn−1

A [n − 1] of hn−1A [n − 1], say via a map A −→

Xn−1A [n − 1]. Since A ∈ �

nX, the last map is zero and therefore ωn−2A = 0. Continuing in

this way we deduce after n − 1 steps that ω1A = 0 and therefore ω0

A ◦h1A[1] = 0. Then ω0

A

factorizes through the left cone X1A[1] of h1

A[1]. However T(A, X1A[1]) = 0, since A ∈ �

nX. This implies that h0

A = ω0A = 0, so A lies in X as a direct summand X0

A. Hence �nX ⊆ X

and therefore X = �nX. �

The rest of this section will be devoted to the investigation of the structure and the main properties of the following important class of contravariantly finite rigid subcate-gories of T.

Definition 5.2. (See [39,46,44].) A full subcategory X of T is called a (n +1)-cluster tilting, n � 1, if:

(i) X is contravariantly finite and X = {A ∈ T | T(X, A[i]) = 0, 1 � i � n}, i.e. X = X�

n .(ii) X is covariantly finite and X = {A ∈ T | T(A, X[i]) = 0, 1 � i � n}, i.e. X = �

nX.

In this case the category of coherent functors mod-X is called the cluster tilted categoryof X.

The following main result of this section gives, among other things, several convenient characterizations of (n + 1)-cluster subcategories and generalizes many results of the literature which are known only in case n = 1 or n > 1 and T is (n + 1)-Calabi–Yau, see e.g. [46,48]. For instance if n = 1, then the equivalence between (i) and (ii) and the last assertion concerning injective objects of mod-X is due to Keller and Reiten [46]if T is 2-Calabi–Yau (see [48] if T is not necessarily Calabi–Yau). It should be noted that the characterizations (ii), (iii) below show that in the definition of (n + 1)-cluster subcategories we need only one of the two conditions in Definition 5.2.

Theorem 5.3. Let T be a triangulated category and X a full subcategory of T. For an integer n � 1, the following statements are equivalent.

(i) X is a (n + 1)-cluster tilting subcategory of T.(ii) X is contravariantly finite and X = X�

n .(iii) X is covariantly finite and X = �

nX.(iv) X is contravariantly finite and both X and X�

n are n-rigid.


(v) X covariantly finite and both X and �nX are n-rigid.(vi) X is contravariantly finite n-rigid and: X� = X[1] � X[2] � · · · � X[n].(vii) X is contravariantly finite n-rigid and for 1 � t � n: X�

t [t] = X[t] �X[t + 1] � · · · �X[n].

(viii) X is covariantly finite n-rigid and: �X = X[−n] � X[−n + 1] � · · · � X[−1].(ix) X is contravariantly (or covariantly) finite n-rigid and: gl.dimX T = n.(x) X is contravariantly (or covariantly) finite n-rigid and: T = X � X[1] � · · · � X[n].(xi) X is contravariantly (or covariantly) finite n-rigid and: Gh[n+1](T) = 0.(xii) X is contravariantly (or covariantly) finite n-rigid and, ∀A ∈ T: Ωn

X(A) ∈ X.(xiii) X is covariantly (or contravariantly) finite n-rigid and, ∀A ∈ T: Σn

X(A) ∈ X.(xiv) X is contravariantly finite n-rigid, any object of X�

n [n + 1] is injective in mod-Xand the functor H : X�

n [n + 1] −→ mod-X is full and reflects isomorphisms.

If X is a (n + 1)-cluster subcategory of T, then the abelian category mod-X has enough projectives and enough injectives, the functors X[n + 1] −→ mod-X ←− X are fully faithful and induce equivalences

X[n + 1] ≈−−→ Inj mod-X & X ≈−−→ Projmod-X

By Proposition 5.1 and the results of Sections 3 and 4, the first thirteen conditions are equivalent. So to complete the proof of Theorem 5.3, it remains to show that (xiv) is equivalent to (i). This requires several steps.

Lemma 5.4. Let X be an (n + 1)-cluster tilting subcategory of T. Let B ∈ T be such that T(X, B[−i]) = 0, 1 � i � n − 1, i.e. B ∈ X�

n−1[n]. Then for any object A ∈ T, there exists a short exact sequence

0 −→ Gh[n]X (A,B) −→ T(A,B) −→ Hom[H(A),H(B)] −→ 0

If, in addition, T(X, B[−n]) = 0, i.e. if B ∈ X�n [n + 1], then the map HA,B is invertible.

Proof. By Corollary 3.6(ii), with k = n, the map HA,B is surjective. We show that the inclusion Gh[n]

X (A, B) ⊆ GhX(A, B) = Ker HA,B is an equality. Let α : A −→ B be such that H(α) = 0, i.e. α is X-ghost. Then α factorizes through h0

A : A −→ Ω1X(A)[1], say via

a map β : Ω1X(A)[1] −→ B. Since Ω1

X(A)[1] ∈ X[1] � X[2] � · · · � X[n], there are triangles Xi[i]

li−→ Miξi−→ Mi+1 −→ Xi[i + 1], for 1 � i � n − 1, where M1 = Ω1

X(A)[1], X1 ∈ X and Mi+1 ∈ X[i + 1] � · · · � X[n]; in particular Mn = Xn[n], where Xn ∈ X. Since T(X, B[−1]) = 0, we have l1 ◦ β = 0, and therefore β = ξ1 ◦ β2 for some map β2 : M2 −→ B. Using that T(X, B[−i]) = 0, 1 � i � n − 1, by induction there exists a factorization β = ξ1◦ξ2◦· · ·◦ξn−1◦βn, where βn : Xn[n] −→ B. Since Mi ∈ X[i] �· · ·�X[n]and since clearly any map from an object of X[i] to an object in X[i +1] � · · ·�X[n] is zero, the map ξi : Mi −→ Mi+1 is X[i]-ghost. In particular the map ξn−1 ◦ βn : Mn−1 −→ B


is X[n − 1]-ghost. Since α = h0A ◦ β = h0

A ◦ ξ1 ◦ ξ2 ◦ · · · ◦ ξn−1 ◦ βn, it follows that α lies in Gh[n]

X (A, B), hence GhX(A, B) ⊆ Gh[n]X (A, B). If in addition T(X, B[−n]) = 0, then the

map βn above is zero. So α = 0 and then Gh[n]X (A, B) = 0. �

Since any object B ∈ X[n + 1] satisfies the assumptions of Lemma 5.4 we have the following.

Corollary 5.5. Let X be an (n + 1)-cluster tilting subcategory of T. Then the functor H : X[n + 1] −→ mod-X is fully faithful and for any object A ∈ T and any object X ∈ X, we have a natural isomorphism

HA,X[n+1] : T(A,X[n + 1]) ∼=−−→ Hom(H(A),H(X[n + 1])

)

The following result gives the implication (i) ⇒ (xiv) in Theorem 5.3.

Proposition 5.6. Let X be an (n + 1)-cluster tilting subcategory of T. Then mod-X has enough injectives, and the functor H : T −→ mod-X induces an equivalence

H : X[n + 1] ≈−−→ Inj mod-X

Proof. Recall that by Lemma 1.5 the functor H : T −→ mod-X is almost full, i.e. setting A∗ = Cell1(A), for any object A ∈ T, we have a canonical map γ1

A : A∗ −→ A such that H(γ1

A) is invertible and, for any map μ : H(A) −→ H(B) in mod-X, there exists a map μ : A∗ −→ B∗ in T and a commutative diagram

H(A∗)H(μ)

H(γ1A) ∼=

H(B∗)

H(γ1B)∼=

H(A)μ

H(B)

Now let μ : H(A) −→ H(B) be a monomorphism in mod-X and let α : H(A) −→H(X[n + 1]) be a map, where X ∈ X. Clearly the map H(μ) in the above commutative diagram is a monomorphism, so if C −→ A∗ −→ B∗ −→ C[1] is a triangle in T, then the map C −→ A∗ is X-ghost and therefore it factorizes through X� = X[1] � · · · � X[n]. By Corollary 5.5, there is a map α : A −→ X[n + 1] such that H(α) = α. Since any map from an object of X[1] � · · · �X[n] to an object of X[n + 1] is clearly zero, it follows that the composition C −→ A∗ −→ A −→ X[n + 1] is zero and therefore γ1

A ◦ α

factorizes through μ, i.e. γ1A ◦α = μ ◦ ρ for some map ρ : B∗ −→ X[n +1]. Then we have

H(γ1A) ◦H(α) = H(μ) ◦H(ρ) and therefore H(γ1

A) ◦H(α) = H(γ1A) ◦μ◦H(γ1

B)−1◦H(ρ), hence α = H(α) = μ ◦ H(γ1

B)−1 ◦ H(ρ), i.e. α factors through μ. This shows that H(X[n + 1])is injective, ∀X ∈ X.


We show that any object H(A) of mod-X is a subobject of an object from H(X[n +1]). From the tower of triangles (C•

A[−1]) associated to A[−1], there is a map ωn−1A[−1] : A −→

ΩnX(A[−1])[n + 1] and a triangle

ΩnX(A[−1]) −→ Celln−1(A[−1])[1] −→ A −→ Ωn

X(A[−1])[n + 1]

where Celln−1(A[−1])[1] lies in X[1] � · · · � X[n]. Since ΩnX(A[−1]) ∈ X, we have

ΩnX(A[−1])[n + 1] ∈ X[n + 1]. Hence H(Ωn

X(A[−1])[n + 1]) is injective in mod-X, and the map H(ωn−1

A[−1]) : H(A) −→ H(ΩnX(A[−1])[n + 1]) is a monomorphism, since

H(X[1] � · · · � X[n]) = 0. Hence mod-X has enough injectives.Now let H(A) be an injective object of mod-X. By the above there exists a split

monomorphism H(μ) : H(A) −→ H(X[n +1]), where X ∈ X. Hence there exists a map α :H(X[n +1]) −→ H(A) such that H(μ) ◦α = 1H(A). Now the map α◦H(μ) is an idempotent endomorphism of H(X[n + 1]) and therefore since the functor H : X[n + 1] −→ mod-X is fully faithful, there exists an idempotent endomorphism e : X[n + 1] −→ X[n + 1] such that H(e) = α ◦H(μ). Since idempotents split in T, there exist maps κ : X[n + 1] −→ D

and λ : D −→ X[n + 1] such that e = κ ◦ λ and λ ◦ κ = 1X[n+1]. Plainly D is of the form X ′[n + 1], for some object X ′ ∈ X, as a direct summand of X[n + 1]. Clearly the map φ := H(μ) ◦H(κ) : H(A) −→ H(X ′[n +1]) is an isomorphism with inverse the map ψ := H(λ) ◦ α. Hence the functor H : X[n + 1] −→ Inj mod-X is surjective on objects and therefore an equivalence. �

Finally the next result shows the implication (xiv) ⇒ (i) and completes the proof of Theorem 5.3.

Proposition 5.7. Let X be a contravariantly finite n-rigid subcategory of T. If the functor H : X�

n [n +1] −→ mod-X has image in Injmod-X, is full and reflects isomorphisms, then X is (n + 1)-cluster tilting.

Proof. It suffices to show that X�n ⊆ X. Let A ∈ X�

n ; applying H to the trian-gle Ω1

X(A) −→ X0A −→ A −→ Ω1

X(A)[1] it follows that Ω1X(A) ∈ X�

n , and we have a monomorphism H(g0

A[n + 1]) : H(Ω1X(A)[n + 1]) −→ H(X0

A[n + 1]). On the other hand, since X ⊆ X�

n it follows that X0A[n + 1] ∈ X�

n [n + 1]. Since the objects H(Ω1

X(A)[n + 1]) and H(X0A[n + 1]) are injective in mod-X, the above monomorphism

splits. Since H|X�n [n+1] is full and reflects isomorphisms, this implies that the map

g0A[n + 1] : Ω1

X(A)[n + 1] −→ X0A[n + 1], or equivalently the map Ω1

X(A)[1] −→ X0A[1], is

split monic. Then the map A −→ Ω1X(A)[1] is zero and therefore A lies in X as a direct

summand of X0A ∈ X. Hence X�

n = X. �From now on: X denotes an (n + 1)-cluster tilting subcategory of T, where n � 1.


Corollary 5.8.

(i) The abelian category mod-X is Frobenius if and only if X = X[n + 1].(ii) For any integer k, there exists a torsion triple in T:

(X[k − n] � · · · � X[k − 2] � X[k − 1], X[k], X[k + 1] � X[k + 2] � · · · � X[k + n]

)

(iii) If T admits a Serre functor S : T −→ T, then S induces an equivalence: S : X ≈−→X[n + 1].

Proof. Part (i) follows from Theorem 5.3, and part (iii) follows from Lemma 4.8 and Proposition 5.6.

(ii) Since X�n = X, Proposition 3.2 implies that

(X � X[1] � · · · � X[n − 1], X[n]

)is a

torsion pair in T, and then so is (X[k − n] � · · · �X[k − 2] �X[k − 1], X[k]

), ∀k ∈ Z. The

proof that (X[k], X[k + 1] �X[k + 2] � · · · �X[k + n]

)is a torsion pair in T is dual, using

cocellular towers, see Subsection 1.2. �Let K0(X, ⊕) be the Grothendieck group of the exact category X endowed with the split

exact structure and let K0(T) be the Grothendieck group of the triangulated category T. If n = 1, i.e. X is a 2-cluster tilting, and if T is algebraic, then Palu [53] proved that K0(T) is a quotient of K0(X, ⊕) by a certain subgroup. For n � 2 and for arbitrary T we have the following. Let G be the subgroup of K0(X, ⊕) generated by all elements [X], X ∈ X, for which there exist triangles A′ −→ Mi −→ A′′ −→ A′[1] in T, i = 1, 2, and an isomorphism X ⊕M1 ∼= M2.

Corollary 5.9. The inclusion X −→ T induces an isomorphism: K0(X, ⊕)/G

∼=−→K0(T).

Proof. Clearly the inclusion i : X −→ T induces a homomorphism K0(i) : K0(X, ⊕) −→K0(T), by K0(i)[X] = [X]. Let A be in T and consider the triangle Ω1

X(A) −→ X0A −→

A −→ Ω1X(A)[1]. Then in K(T) we have a relation [A] = [X0

A] − [Ω1X(A)]. Similarly the

triangle Ω2X(A) −→ X1

A −→ Ω1X(A) −→ Ω2

X(A)[1] gives the relation [Ω1X(A)] = [X1

A] −[Ω2

X(A)] and therefore we have [A] = [X0A] − [X1

A] + [Ω2X(A)]. Continuing in this way we

have a relation in K0(T): [A] =∑n−1

k=0(−1)k[XkA] +(−1)n[Ωn

X(A)]. By Theorem 5.3 we have Ωn

X(A) := XnA ∈ X, hence [A] =

∑nk=0(−1)k[Xk

A], so [A] =∑n

k=0(−1)kK0(i)([Xk

A])

=K0(i)

(∑nk=0(−1)k[Xk

A]), i.e. for any object A in T, the generator [A] lies in the image

of K0(i). This clearly implies that K0(i) is surjective. It is easy to see that the kernel of the map K0(i) coincides with the subgroup G. �

Let T be a Hom-finite triangulated category over a field k, and T an (n + 1)-cluster tilting object of T, i.e. addT is an (n + 1)-cluster tilting subcategory of T, n � 1. Since K0(addT, ⊕) is free of rank the number of indecomposable direct summands of T , by


Corollary 5.9, K0(T) is finitely generated. Hence any Hom-finite triangulated category over a field whose Grothendieck group is not finitely generated has no cluster tilting object.

5.2. Maximal rigid subcategories

Throughout this subsection we fix an (n + 1)-Calabi–Yau triangulated category T. We also fix a functorially finite n-rigid subcategory X of T, n � 1. Note that as a direct consequence of the (n + 1)-Calabi–Yau property, if Y is an object of T then: Y ∈ X�

n if and only if Y ∈ �

nX.The following result, which generalizes a result of Zhou and Zhu, see [61], shows that

for n = 1 the functor Ω1X preserves 1-rigidity, and for n > 1 the functor Ωk

X|X�n−1

preserves n-rigidity for 0 � k � n.

Proposition 5.10. Let A be an n-rigid object of T, where we assume that A ∈ X�n−1, if

n > 1. Then the object ΩkX(A) is n-rigid, 1 � k � n, and lies in X�

n .

Proof. First assume that n > 1. For 0 � k � n, consider the triangles, whereΩ0

X(A) := A:

Ωk+1X (A) gk

A−−→ XkA

fkA−−→ Ωk

X(A) hkA−−→ Ωk+1

X (A)[1] (Tk(A))

Applying H to the triangle T0(A) and using that A ∈ X�n−1, hence T(X, A[i]) = 0,

1 � i � n − 1, we have T(X, Ω1X(A)[i]) = 0, 1 � i � n, hence Ω1

X(A) ∈ X�n . Using this

and applying H to the triangle T1(A) we see directly that Ω2X(A) ∈ X�

n . It follows by induction that Ωk

X(A) ∈ X�n , ∀k � 1.

Applying the functor T(Ω1X(A), −) to the triangle T0(A), we have a long exact sequence

· · · −→ T(Ω1X(A), X0

A[k]) −→ T(Ω1X(A), A[k])

−→ T(Ω1X(A),Ω1

X(A)[k + 1]) −→ T(Ω1X(A), X0

A[k + 1]) −→ · · ·

Since T(Ω1X(A), X0

A[k + 1]) ∼= DT(X[k − n], Ω1X(A)) = DT(X, Ω1

X(A)[n − k]) = 0, for 0 � k � n − 1, it follows that we have an isomorphism

T(Ω1X(A), A[k]) ∼=−−→ T(Ω1

X(A),Ω1X(A)[k + 1]) (5.1)

and an exact sequence

T(Ω1X(A), X0

A) −→ T(Ω1X(A), A) −→ T(Ω1

X(A),Ω1X(A)[1]) −→ 0 (5.2)

Applying the functor T(−, A) to the triangle T0(A), we have a long exact sequence


· · · −→ T(X0A[−k], A) −→ T(Ω1

X(A)[−k], A)

−→ T(A[−k − 1], A) −→ T(X0A[−k − 1], A) −→ · · ·

Since A is n-rigid and T(X, A[k]) = 0, for 1 � k � n − 1, we have T(Ω1X(A), A[k]) = 0


T(A,A) −→ T(X0A, A) −→ T(Ω1

X(A), A) −→ 0 (5.3)

Hence from (5.1) we have T(Ω1X(A), Ω1

X(A)[k]) = 0, for 2 � k � n. On the other hand from (5.3) it follows that any map α : Ω1

X(A) −→ A is of the form α = g0A ◦ ρ for some

map ρ : X0A −→ A. Then ρ factorizes through the right X-approximation f0

A : X0A −→ A,

i.e. there is an endomorphism σ : X0A −→ X0

A such that ρ = σ ◦ f0A. Then α = g0

A ◦ ρ =g0A ◦ σ ◦ f0

A, i.e. α factorizes through f0A and this means that the leftmost map in (5.2) is

surjective. It follows that T(Ω1X(A), Ω1

X(A)[1]) = 0. We infer that Ω1X(A) is n-rigid and

as noted above Ω1X(A) lies in X�

n . Hence replacing A with Ω1(A), we have that Ω2X(A)

is n-rigid and lies in X�n , and then by induction we conclude that Ωk

X(A) is n-rigid and lies in X�

n for 1 � k � n.If n = 1, then the above proof shows that Ω1

X(A) is 1-rigid and of course Ω1X(A) lies

in X�1 . �

Definition 5.11. A functorially finite subcategory X of T is called maximal n-rigid if X is n-rigid and one of the following equivalent conditions is satisfied:

(i) X contains any n-rigid subcategory Y such that Y ⊆ X�n .

(ii) X contains any n-rigid subcategory Y such that Y ⊆ �nX.

Corollary 5.12. Let X be a maximal n-rigid subcategory of T, and let A be an n-rigid object of T.

(i) Assume that A ∈ X�n−1, if n > 1. Then Ωn

X(A) ∈ X and A ∈ X � X[1] � · · · � X[n].(ii) Assume that A ∈ �

n−1X, if n > 1. Then ΣnX(A) ∈ X and A ∈ X[−n] � · · · �X[−1] �X.

In particular: {A ∈ X�

n | A is n-rigid}

= X ={A ∈ �

nX | A is n-rigid}.

Proof. (i) Let A be in X�n−1. Then, as observed in Proposition 5.10, we have that Ωn

X(A)is n-rigid and Ωn

X(A) ∈ X�n . Since X is maximal n-rigid, it follows that Ωn

X(A) lies in Xand then clearly A lies in X � X[1] � · · · � X[n]. Hence there is a triangle M −→ A −→X[n] −→ M [1], where M ∈ X � X[1] � · · · � X[n − 1] and X ∈ X. If A ∈ X�

n , i.e. we have in addition T(X, A[n]) = 0, then using Calabi–Yau duality we also have T(A, X[n]) = 0. Hence the above triangle splits and then A lies in X � X[1] � · · · � X[n − 1] as a direct summand of M . Using that A ∈ X�

n−1 and induction it follows that A ∈ X. Hence {A ∈ X�

n | A is n-rigid}

= X. Part (ii) is dual. �


The following consequence will be useful in the next section in connection with the Gorenstein property.

Corollary 5.13. Let X be a maximal n-rigid subcategory of T. If X is (n −1)-corigid, then

X[n + 1] ⊆ X � X[1] ⊇ X[−n]

Proof. Clearly X[n + 1] is n-rigid. Since X is (n − 1)-corigid, it follows directly that X[n + 1] ⊆ X⊥

n−1 and therefore, by Corollary 5.12, X[n + 1] ⊆ X � X[1] � · · · � X[n]. Then using that X is (n − 1)-corigid, as in the proof of (i) ⇒ (ii) in Proposition 3.12, we infer that X[n +1] ⊆ X �X[1]. On the other hand, ∀X ∈ X, the object X[−n] is clearly n-rigid and since X is (n − 1)-corigid, we have X[−n] ⊆ X�

n−1. By Corollary 5.12 it follows that X[−n] ∈ X �X[1] � · · · �X[n]. Hence there exists a triangle A −→ X[−n] −→ B −→ A[1], where A ∈ X � X[1] and B ∈ X[2] � · · · � X[n]. Using that X is (n − 1)-corigid and the (n + 1)-Calabi–Yau property it is easy to see that T(X[−n], X[2] � · · · � X[n]) =T(X, X[n +2] � · · ·�X[2n]) = 0. Hence the map X[−n] −→ B is zero and therefore X[−n]lies in X � X[1] as a direct summand of A. We infer that X[−n] ⊆ X � X[1]. �6. Some cluster tilting subcategories are Gorenstein. . .

Keller and Reiten proved in [46] that the cluster tilted category of a 2-cluster tilting subcategory in a 2-Calabi–Yau triangulated category is Gorenstein, see also [48] and [16]. However, by an example of Iyama, see [46], the Gorenstein property fails for higher cluster tilting subcategories. Our aim in this section is to show that a special class of n-cluster tilting subcategories of an arbitrary triangulated category, where n > 2, enjoys the property that the associated cluster tilted category is Gorenstein, and then to derive some direct consequences.

6.1. Gorenstein categories and Gorenstein-projective objects

Let A be an abelian category with enough projectives and enough injectives. We recall from [19] the following invariants attached to A :

silpA = sup{idP |P ∈ ProjA } & spli A = sup{pd I | I ∈ InjA }G-dimA := max

{silpA , spliA

}

We call G-dim A the Gorenstein dimension of A and then A is called Gorenstein if G-dim A < ∞. If G-dim A � n < ∞, then we say that A is n-Gorenstein.

Remark 6.1. It is not difficult to see that if spliA < ∞ and silpA < ∞, then spliA = silpA ; moreover G-dim A � gl.dimA with equality if gl.dimA < ∞, see [19, Proposition VII.1.3].


For future use we also recall the notion of Gorenstein-projective objects and their basic properties.

A complex of projective objects P • : · · · −→ P−1 −→ P 0 −→ P 1 −→ · · · in Ais called totally acyclic if P • and the induced complex A (P •, Q) are acyclic, for any projective object Q of A . Dually a complex of injective objects I• : · · · −→ I−1 −→I0 −→ I1 −→ · · · is called totally acyclic if I• and the induced complex A (J, I•) are acyclic for any injective object J of A .

Definition 6.2.

(i) An object G ∈ A is called Gorenstein-projective if G ∼= Coker(P−1 −→ P 0) for some totally acyclic complex P • of projectives.

(ii) An object G ∈ A is called Gorenstein-injective if G ∼= Ker(I0 −→ I1) for some totally acyclic complex I• of injectives.

The full subcategory of Gorenstein-projective, resp. Gorenstein-injective, objects of A is denoted by GProjA , resp. GInjA . Also we denote by GProjA , resp. GInjA , the stable category of GProjA , resp. GInj A , modulo projectives, resp. injectives.

In the following remark we remind the reader of the basic properties of Gorenstein categories which will be used in the sequel. For proofs and a more detailed discussion, we refer to [11,19].

Remark 6.3. Let A be an abelian category with enough projective and/or injective objects. In this context, we denote by ΩkA, resp. ΣkA, k � 1, the usual k-syzygy, resp. k-cosyzygy, object of A ∈ A , which is uniquely determined modulo projectives, resp. injectives. We also denote by ΩkA , resp. ΣkA , the full subcategory of A consisting of the k-syzygy, resp. k-cosyzygy, objects of A , and by Proj<∞ A , resp. Inj<∞ A , the full subcategory of A consisting of all objects with finite projective, resp. injective, dimension. If C ⊆ A is a full subcategory of A , then the stable category C / ProjA , resp. C / InjA , is denoted by C , resp. C , and the morphism spaces C (C1, C2), resp. C (C1, C2), are denoted by HomA (C1, C2), resp. HomA (C1, C2).

1. For any objects G1 ∈ GProj A , G2 ∈ GInj A , A ∈ A , and any k � 1, there are isomorphisms:

ExtkA (G1, A) ∼=−−→ HomA (ΩkG1, A) and ExtkA (A,G2)∼=−−→ HomA (A,ΣkG2)

2. The categories GProjA and GInjA are exact Frobenius subcategories of A . Hence the stable categories GProjA and GInjA are triangulated.

3. If A is Gorenstein then Proj<∞ A = Inj<∞ A ; if G-dim A = d, then Proj<∞ A =Proj�d A (and dually Inj<∞ A = Inj�d A ). Moreover we have GProjA = ΩdA

and GInjA = ΣdA , see [11, Theorem 4.16]. It follows that if G-dim A � 1, then


GProjA = ΩA consists of the subobjects of the projective objects and GInjA = ΣA

consists of the factors of the injectives.4. The inclusion functor GProjA −→ A admits the functor Ω−dΩd : A −→ GProj A as

a right adjoint, and the inclusion functor GInj A −→ A admits the functor Σ−dΣd :A −→ GInjA as a left adjoint.

5. If A is Gorenstein, then GProjA = {A ∈ A | ExtnA (A, P ) = 0, ∀P ∈ ProjA ,

∀n � 1}. Dually GInjA = {A ∈ A | ExtnA (I, A) = 0, ∀I ∈ Inj A , ∀n � 1}. In this case the Gorenstein-projective objects are usually called Cohen–Macaulay ob-jects and GProjA is denoted by CM(A ).

6.2. Gorensteinness of corigid cluster tilting subcategories

Our main aim in this subsection is to prove, and discuss the consequences of, the following result. Note that part (i) is due to Keller and Reiten [46] if T is 2-Calabi–Yau and to Koenig and Zhu [48] if T is not necessarily Calabi–Yau.

Theorem 6.4. Let X be an (n + 1)-cluster tilting subcategory of T, where n � 1.

(i) If n = 1, then G-dim mod-X � 1.(ii) G-dim mod-X = 0 if and only if X is n-corigid if and only if X = X[n + 1].(iii) Assume that n � 2 and X is (n − k)-corigid, where 0 � k � n − 1. Then:

0 � k � n + 12 =⇒ G-dim mod-X � k

Moreover if X is strictly (n − k)-corigid, then: G-dimmod-X = k.

Proof. Since X is (n + 1)-cluster tilting, we have X�n [n + 1] = X[n + 1]. It follows that

for any object A ∈ T we have: cores.dimX�n [n+1] A = cores.dimX[n+1] A = id H(A), and

clearly res.dimX A = pd H(A). Hence

supX∈X

cores.dimX�n [n+1] X = silpmod-X and sup

A∈X�n [n+1]

res.dimX A = spli mod-X

and the assertions (i)–(iii) follow from Corollary 4.5.It remains to show that if X is strictly (n − k)-corigid, then: G-dim mod-X = k. It

suffices to show that if pdH(X[n + 1]) � k − 1, ∀X ∈ X, then X is (n − k + 1)-corigid. If 0 � k � 1, then the assertion is clear. So assume that k � 2. We consider triangles (Tt) : At −→ Xt−1 −→ At−1 −→ At[1], where each map Xt−1 −→ At−1 is a right X-approximation, t � 1, and A0 = X[n +1], so that At = Ωt

X(X[n +1]). Applying H to the triangle (T1), we obtain an exact sequence 0 −→ H(A1) −→ H(X0) −→ H(X[n + 1]) −→0 and T(X, A1[−i]) = 0, 1 � i � n − k. Using this and applying H to the triangle (T2), we obtain an exact sequence 0 −→ H(A2) −→ H(X1) −→ H(A1) −→ 0 and


T(X, A2[−i]) = 0, 1 � i � n − k − 1. Continuing in this way, we finally obtain an exact sequence 0 −→ H(Ak−1) −→ H(Xk−2) −→ H(Ak−2) −→ 0 and T(X, Ak−1[−i]) = 0, 1 � i � n − 2k + 2. Since pd H(X[n + 1]) � k − 1, the object H(Ak−1) is projective. Hence there is a map α : X∗ −→ Ak−1, where X∗ ∈ X, inducing an isomorphism H(α) : H(X∗)

∼=−→ H(Ak−1). Let X∗ −→ Ak−1 −→ B −→ X∗[1] be a triangle. Applying H to this triangle and using that X is (n − k)-corigid, the fact that H(α) is invertible, and the vanishing condition T(X, Ak−1[−i]) = 0, 1 � i � n − 2k + 2, we infer that T(X, B[−k + 1]) = T(X, B[−k + 2]) = · · · = T(X, B[−k + n]) = 0, i.e. B[−k] ∈ X�

n = X. Hence B ∈ X[k] and this implies that the map B −→ X∗[1] is zero since it lies in T(X, X[−k + 1]) and this space is zero since 1 � k − 1 � n − k, X is (n − k)-corigid and n � 2k− 1. We infer that Ak−1 admits a direct sum decomposition Ak−1 ∼= X∗ ⊕X ′[k]. On the other hand, since Ak−1 = Ωk−1

X (X[n +1]), we know by Remark 1.4 that Ak−1 lies in X �X[1] � · · · �X[k− 1]. Since clearly any map from an object of X �X[1] � · · · �X[k− 1]to an object from X[k] is zero, it follows that the projection Ak−1 −→ X ′[k] is zero and this implies that Ak−1 ∼= X∗ ∈ X. Then Ak−2 lies in X � X[1] and using that Ai = Ωi

X(X[n +1]), ∀i � 1, it follows inductively that A1 ∈ X �X[1] � · · ·�X[k−2]. Then X[n + 1] lies in X � X[1] � · · · � X[k − 1], hence X[n + 1] ⊆ X � X[1] � · · · � X[k − 1]. Then Proposition 3.12 shows that X is (n − k + 1)-corigid as required. �

A special case of part (a) of the next consequence was observed independently by Iyama and Oppermann [43].

Corollary 6.5. Let X be an (n + 1)-cluster tilting subcategory of T.

(a) If X is (n − 1)-corigid, then: G-dimmod-X � 1.(b) If n � 3 is odd and X is strictly (n−1

2 )-corigid, then: G-dim mod-X = n+12 .

(c) If n � 2 is even and X is strictly (n2 )-corigid, then: G-dim mod-X = n2 .

Using Remark 6.1 we have the following consequence.

Corollary 6.6. Let X be an (n + 1)-cluster tilting subcategory of T, where n � 1. Let 0 �k � n −1 and assume that X is (n −k)-corigid. If n � 2k−1, then either gl.dimmod-X =∞ or else gl.dim mod-X � k. Moreover if gl.dim mod-X < ∞ and if X is strictly (n −k)-corigid, then gl.dimmod-X = k.

We have seen that if X is a 2-cluster tilting subcategory of T, then the functor H : T −→mod-X is full. Generalizing [16, Theorem 5.2] we provide a partial converse and we show that, for an (n + 1)-cluster tilting subcategory X of T, fullness of H implies that mod-Xis 1-Gorenstein. Note that H is not always full; we refer to [16], [46] for an example of a 3-cluster tilting subcategory X in a 3-Calabi–Yau triangulated category T for which the functor H : T −→ mod-X is not full and the cluster tilted category mod-X is not Gorenstein.


Corollary 6.7. Let X be a (n + 1)-cluster tilting subcategory of T, where n � 1. If the functor H : T −→ mod-X is full, then the category mod-X is 1-Gorenstein.

Proof. If n = 1, then mod-X is 1-Gorenstein by Theorem 6.4(i). If n � 2, then by Proposition 3.11, X is (n − 1)-corigid, and therefore mod-X is 1-Gorenstein by Theo-rem 6.4(iii). �

It is shown in [61] that the category of coherent functors over any maximal 1-rigid subcategory of a 2-Calabi–Yau triangulated category is 1-Gorenstein. We close this sub-section by treating the case of maximal n-rigid subcategories.

Theorem 6.8. Let T be an (n +1)-Calabi–Yau triangulated category over a field k and X a maximal n-rigid subcategory of T. If X is an (n −1)-corigid, then mod-X is 1-Gorenstein.

Proof. By Lemma 4.8 the abelian category mod-X has enough projectives and enough injectives; moreover we have proj(X) = H(X) and inj(X) = H(X[n +1]). By Corollary 5.13for any object X ∈ X, there are triangles X1 −→ X0 −→ X[n + 1] −→ X1[1] and X −→ X1[n + 1] −→ X0[n + 1] −→ X[1], where the Xi and the Xi lie in X. Then clearly we have short exact sequences 0 −→ H(X1) −→ H(X0) −→ H(X[n + 1]) −→ 0and 0 −→ H(X) −→ H(X1[n + 1]) −→ H(X0[n + 1]) −→ 0 in mod-X. This means that pd H(X[n + 1]) � 1 and id H(X) � 1. We infer that silpmod-X � 1 and spli mod-X � 1. Hence mod-X is 1-Gorenstein. �6.3. Keller–Reiten’s Morita Theorem for cluster categories

Prominent examples of d-Calabi–Yau triangulated categories over a field k are the d-cluster categories C

(d)H , d � 1, associated to a finite-dimensional hereditary k-algebra H

over a field k, see [45]. In this context Keller and Reiten [47] proved that the (n +1)-cluster categories C (n+1)

H are, up to a triangle equivalence, the algebraic (n +1)-Calabi–Yau tri-angulated categories admitting an (n − 1)-corigid (n + 1)-cluster tilting object whose endomorphism algebra is hereditary. Corollary 6.5 allows us to give a slight improve-ment of [47, Theorem 4.2] by replacing hereditary with finite global dimension, see the comments after Theorem 4.2 in [47].

Theorem 6.9. (See [47].) Let T be a k-linear triangulated category with finite-dimensional Hom-spaces over an algebraically closed field k. Then for an integer n � 1, the following statements are equivalent.

(i) T is triangle equivalent to the (n +1)-cluster category C (n+1)H of a finite-dimensional

hereditary k-algebra H.(ii) T is algebraic (n +1)-Calabi–Yau and admits a (n −1)-corigid (n +1)-cluster tilting

object T such that the endomorphism algebra EndT(T ) has finite global dimension.


6.4. Abelian subcategories

We show that if the cluster tilted category mod-X of an (n −k)-corigid (n +1)-cluster tilting subcategory X has finite global dimension, for some 0 � k � n

2 , then mod-X can be realized as a full subcategory of T, in some cases via a ∂-functor T. Note that if n = k = 1 in the next result, then the corigidity condition is vacuous and T gives the full embedding Injmod-X ≈ X[2] −→ T.

Theorem 6.10. Let X be an (n − k)-corigid (n + 1)-cluster tilting subcategory of T, where 0 � k � n − 1.

(i) If k � n2 , then the following are equivalent:

(a) The cluster tilted category mod-X has finite global dimension.(b) The functor H : T −→ mod-X induces an equivalence:

H :(X � X[1] � · · · � X[k]

) ⋂X�

k [k + 1] ≈−−→ mod-X

If T admits a Serre functor S, then the above conditions are equivalent to:(c) The functor H ◦ S : T −→ mod-X induces an equivalence

H ◦ S :(X[−k] � · · · � X[−1] � X

) ⋂⊥k X[−k − 1] ≈−−→ mod-X

In any of the above cases we have: gl.dimmod-X � k with equality if X is strictly (n − k)-corigid.

(ii) If k = 1, then the induced full embedding T : mod-X −→ T is a ∂-functor, which extends uniquely to an additive functor Db(mod-X) −→ T commuting with the shifts.

Proof. (i) (a) ⇒ (b) Since X is (n −k)-corigid and n � 2k, it follows that X ⊆ X�n−k[n −

k + 1] ⊆ X�k [k + 1], i.e. X is k-corigid. Then by Theorem 4.6, the functor H induces an

equivalence Hk :(X � X[1] � · · · � X[k]

)∩ X�

k [k + 1] ≈−→ Proj�k mod-X. By Theorem 6.4, mod-X is k-Gorenstein, so finiteness of gl.dimmod-X implies that gl.dim mod-X � k, and therefore mod-X = Proj�k mod-X. The implication (b) ⇒ (a) and the equivalence (a) ⇔(c) follow from Corollary 4.9 and Theorem 6.4.

(ii) Assume that k = 1, so n � 2, X is (n − 1)-corigid, and mod-X is hereditary. By Corollary 2.9 we have isomorphisms T(A, B[1]) ∼= Ext1(H(A), H(B)), ∀A, B ∈

(X �

X[1])∩ X�

1 [2]. This implies easily that the induced fully faithful functor T : mod-X ≈(X � X[1]

)∩ X�

1 [2] −→ T is a ∂-functor; details are left to the reader. By a result of Amiot [1], T extends uniquely to an additive functor Db(mod-X) −→ T commuting with the shifts. �


6.5. Representation dimension

Recall that an additive category is called Krull–Schmidt if any of its objects is a finite coproduct of indecomposable objects and any indecomposable object has local endomorphism ring. A Krull–Schmidt category has finite representation type if there are only finitely many indecomposable objects up to isomorphism. Now let A be an abelian category with enough projectives. We say that A is of finite Cohen–Macaulay type, finite CM-type for short, if GProjA is of finite representation type. Finally recall that the dominant dimension, dom.dim A , of A is the largest n ∈ N ∪ {∞} such that any projective object has an injective resolution whose first n terms are projective. The codominant dimension, codom.dim A , is defined dually. For an Artin algebra Λ we set: dom.dim Λ = dom.dim mod-Λ.

Our aim is to show that if X is an (n − 1)-corigid (n + 1)-cluster tilting subcategory of a triangulated category T and the associated cluster tilted category mod-X is of finite CM-type, then mod-X is equivalent to the category of finitely presented modules over a coherent ring of representation dimension � 3 in the sense of Auslander [3].

We begin with the following result which generalizes, and is inspired by, a result of Ringel proved in [58] in the context of finitely generated modules over an Artin algebra.

Proposition 6.11. Let A be an abelian category with enough projectives and let U be a contravariantly finite subcategory of A which is closed under subobjects and contains the projectives. Also let V be a contravariantly finite subcategory of A which is closed under factors and set:

E := U⊕ V = add{X ⊕ Y ∈ A | X ∈ U & Y ∈ V

}

Then E is contravariantly finite in A and the abelian category mod-E has global dimen-sion at most 3. If in addition A has enough injectives and V contains them, then mod-Ehas dominant dimension at least 2.

Proof. Let A be in A and let U0A −→ A be a right U-approximation of A. Since U

contains the projectives and is closed under subobjects, we have an exact sequence, where U0

A and U1A lie in U:

0 −→ U1A −→ U0

AfA−−→ A −→ 0 (6.1)

Let gA : V 0A −→ A be a right V-approximation of A and set V 0

A = Im(V 0A −→ A). Note

that V 0A lies in V since V is closed under factors. Clearly then the inclusion gA : V 0

A −→ A

is a right V-approximation of A. Taking the pull-back of (6.1) along the monomorphism


gA we have the following exact commutative diagram

0 −−−−→ U1A −−−−→ U1

A −−−−→ V 0A −−−−→ 0∥∥∥ ⏐⏐� ⏐⏐�gA

0 −−−−→ U1A −−−−→ U0

A

fA−−−−→ A −−−−→ 0

(6.2)

where the middle vertical map is a monomorphism, so U1A ∈ U ⊆ E , and (6.2) induces a

short exact sequence

0 −→ U1A −→ U0

A ⊕ V 0A

ϕ−→ A −→ 0

where ϕ = t(fA, gA) is easily seen to be a right E -approximation of A. Hence E is contravariantly finite in A and therefore the category mod-E of coherent functors over E is abelian. Moreover the exact sequence

0 −→ A (−, U1A)|E −→ A (−, U0

A ⊕ V 0A)|E ϕ∗

−−→ A (−, A)|E −→ 0 (6.3)

is a projective resolution in mod-E of the object A (−, A)|E and therefore pd A (−,

A)|E � 1. On the other hand, let F be an arbitrary object of mod-E and choose a pro-jective presentation A (−, W1)|E −→ A (−, W0)|E −→ F −→ 0 in mod-E , where the Wi

lie in E . If A = Ker(W1 → W0), then setting W3 = U1A ∈ U ⊆ E and W2 = U0

A ⊕V 0A ∈ E

and using (6.3) we have a projective resolution of F in mod-E of the form:

0 −→ A (−,W3)|E −→ A (−,W2)|E −→ A (−,W1)|E −→ A (−,W0)|E −→ F −→ 0

Hence pdF � 3 and therefore gl.dimmod-E � 3. Since ProjA ⊆ E , it is easy to see that the functor A −→ mod-E , A �−→ A (−, A)|E admits an exact left adjoint, see Lemma 8.5, so the above functor preserves injectives. If A has enough injectives and Inj A ⊆ V, then for any object E ∈ E , let 0 −→ E −→ I0 −→ I1 be exact, where the Ii are injective. Then the objects A (−, Ii) are projective–injective in mod-E and the exact sequence 0 −→ A (−, E)|E −→ A (−, I0)|E −→ A (−, I1)|E shows that the projective object A (−, E)|E admits an injective copresentation whose first two terms are projective. We infer that dom.dimmod-E � 2. �Remark 6.12. Assume in Proposition 6.11 that A has enough injectives and is not semi-simple. Then it is not difficult to see that in fact dom.dimmod-E = 2. Indeed if dom.dim mod-E � 3, then as we shall see in Section 8, E = U ⊕ V is a 1-cluster tilting subcategory of A , in particular Ext1A (E , E ) = 0. This easily implies that A is semi-simple and U = ProjA = Inj A = V. On the other hand if gl.dimmod-E � 2, then it is easy to see that A = E . Hence if A is not semi-simple and E �= A , then gl.dim mod-E = 3 and dom.dim mod-E = 2.


A full subcategory C of A is called generator, resp. cogenerator, if any object of A is a factor, resp. subobject, of an object from C . Then an object T of A is called generator, resp. cogenerator, if the full subcategory addT is a generator, resp. cogenerator. Recall that an additive category is called right, resp. left, coherent if it has weak kernels, resp. weak cokernels. Then an object T of A is called right, resp. left, coherent if the full subcategory addT is right, resp. left, coherent. For instance T ∈ A is right, resp. left, coherent, if addT is contravariantly, resp. covariantly, finite in A . Of course T is called coherent if T is left and right coherent object. The representation dimension rep.dim A

of A in the sense Auslander [3] is defined as follows.

rep.dim A = inf{

gl.dim mod-EndA (T ) | T is a coherent generator–cogenerator of A}

Note that if the object T is coherent, then both the categories mod- addT ≈mod-EndA (T ) and addT -mod ≈ (mod-EndA (T )op)op are abelian and gl.dim mod-EndA (T ) = w.gl.dimEndA (T ) = gl.dimmod-EndA (T )op.

Corollary 6.13. Let A be a 1-Gorenstein abelian category and assume that ProjA is covariantly finite and Inj A is contravariantly finite. Then E := GProjA ⊕ GInj A is functorially finite in A , and gl.dimmod-E � 3 and dom.dim mod-E � 2. If in addition A is of finite Cohen–Macaulay type, then rep.dimA � 3; more precisely there exists a coherent ring Λ with rep.dimΛ � 3 and an equivalence mod-Λ ≈ A .

Proof. Since A is Gorenstein, by [19, Theorem VII.2.2], GProjA and GInjA are func-torially finite and GProjA = ΩA and GInjA = ΣA . Hence GProjA is closed under subobjects and GInjA is closed under factors. Now the first assertion follows from Propo-sition 6.11. Assume that A is of finite Cohen–Macaulay type, so GProj A = addX, for some Gorenstein-projective object X. Then X is coherent since addX is functorially finite. By [19] there is a triangle equivalence GProjA ≈ GInj A . This implies that GInjAis also of finite representation type, so GInjA = addY for some, necessarily coherent, Gorenstein-injective object Y . We infer that E := GProj A ⊕ GInj A = add(X ⊕ Y )and the object X ⊕ Y is coherent. Since clearly this object is a generator–cogeneratorof A and since mod-E = mod-EndA (X ⊕ Y ), we infer that rep.dim A � 3. Finally since Proj A ⊆ GProjA , clearly Proj A is of finite representation type and we have ProjA = addT for some, necessarily coherent, object T ∈ Proj A . Then mod-A ≈ mod-Λwhere Λ = End(T ). �

For simplicity of notation from now on and throughout the rest of the paper we set:

Projmod-X := projX, Inj mod-X := injX,

GProjmod-X := GprojX = CM(X), GInj mod-X := GinjX

and GProjmod-X := GprojX = CM(X) and GInj mod-X := GinjX.


Lemma 6.14. Let X be a full subcategory of T, and assume that X is (n − 1)-corigid if n > 1. Then the subcategories projX, GprojX, injX, GinjX are functorially finite in mod-X in the following cases:

(i) X is (n + 1)-cluster tilting.(ii) T is (n + 1)-Calabi–Yau and X is maximal n-rigid.

Proof. (i) Since mod-X has enough projectives and injectives, projX is contravariantly finite and injX is covariantly finite. Let F ∼= H(A) be in mod-X, where we may assume that A lies in X �X[1]. Since X is contravariantly finite, so is X[n +1]; let f : X[n +1] −→ A

be a right X[n +1]-approximation of A which induces a map H(f) : H(X[n +1]) −→ H(A)in mod-X. Let α : H(X ′[n + 1]) −→ H(A) be a map, where X ′ ∈ X. If n > 1, so X is (n − 1)-corigid, then by Proposition 3.12 we have X[n + 1] ⊆ X �X[1]. By Corollary 2.6, α = H(α) for some map α : X ′[n + 1] −→ A which then factorizes through f . It follows that H(α) factorizes through H(f). Hence H(f) is a right inj(X)-approximation of H(A)and inj(X) is contravariantly finite in mod-X. If n = 1 the above argument works using that T = X � X[1] in this case. Next let g : A −→ X be a left X-approximation of Awhich induces a map H(g) : H(A) −→ H(X). If β : H(A) −→ H(X ′) is a map in mod-X, where X ′ ∈ X, then, since A ∈ X �X[1], β is induced by a map β : A −→ X ′ which then factorizes through g. It follows that H(β) factorizes through H(g). Hence H(g) is a left proj(X)-approximation of H(A) and proj(X) is covariantly finite in mod-X. Case (ii) is similar using Corollary 5.13.

Finally by Corollary 6.5, the category mod-X is 1-Gorenstein. Then by [19, Theo-rem VII.2.2] functorial finiteness of projX and injX implies that GprojX and GinjX are functorially finite. �

Combining Theorem 6.4, Corollary 6.13, and Lemma 6.14 we have the following con-sequence.

Theorem 6.15. Let T be a triangulated category and X a contravariantly finite subcate-gory of T. Assume X is (n − 1)-corigid, if n > 1. If mod-X is of finite CM-type, then rep.dim mod-X � 3 in the following cases:

(i) X is an (n + 1)-cluster tilting subcategory of T.(ii) T is (n + 1)-Calabi–Yau and X is maximal n-rigid.

More precisely mod-X ≈ mod-EndT(T ) for some T ∈ X and rep.dim EndT(T ) � 3 and dom.dim EndT(T ) � 2.

7. . . .and stably Calabi–Yau

We have seen, in Theorem 6.4, that the cluster tilted category mod-X associated to an (n − k)-corigid (n + 1)-cluster tilting subcategory X of a triangulated category


T is k-Gorenstein, provided that either n = 1, or n > 1 and 0 � k � 1, or k �n+1

2 , if 2 � k � n − 1. In this section we show that if the triangulated category T is (n +1)-Calabi–Yau over a field, then the triangulated stable category modulo projectives of the Gorenstein-projective objects of mod-X is (n + 2)-Calabi–Yau in case 0 � k � 1, and under an additional assumption if 2 � k � n+1

2 . This generalizes a basic result of Keller and Reiten [46] who treated the case n = 1.

Throughout this section: we fix a k-linear triangulated category T with split idempo-tents and finite-dimensional Hom-spaces over a field k, and let X be an (n + 1)-cluster tilting subcategory of T, n � 1.

7.1. Serre functors

Assume that T admits a Serre functor S. So S : T −→ T is a triangulated equivalence and for any objects A and B in T there are natural isomorphisms

DT(A,B) ≈−−→ T(B,S(A)) (∗)

where D denotes duality with respect to the base field k. For any object A ∈ T we consider the triangles

X1A −→ X0

A −→ Cell1(A) −→ X1A[1] &

XA1 [−1] −→ Cell1(A) −→ XA

0 −→ XA1 (7.1)

where the map X1A −→ X0

A is the composition of the right X-approximation X1A −→

Ω1X(A) and the map Ω1

X(A) −→ X0A, and the map XA

0 −→ XA1 is the composition of the

map XA0 −→ Σ1

X(A) and the left X-approximation Σ1X(A) −→ XA

1 . Then

Cell1(A) ∈ X � X[1] and H(Cell1(A)) ∼= H(A),

Cell1(A) ∈ X[−1] � X and Hop(A) ∼= H(Cell1(A))

Recall that the transpose Tr(F ), in the sense of Auslander and Bridger [4], of an object F in mod-X is defined as follows. Let H(X1) −→ H(X0) −→ F −→ 0 be a projective presentation of F . Consider the duality functor dr := Hom(−, H(?)|X) : mod-X −→X-mod, defined by dr(F ) = Hom(F, H(?)|X) : X −→ Ab, where Hom(F, H(?)|X)(X) =Hom(F, H(X)). Similarly the duality functor dl : X-mod −→ mod-X is defined, and it is well-known that (dr, dl) : mod-X � X-mod is an adjoint on the right pair of contravariant functors inducing a duality between projX and projXop, a duality between GprojX and GprojXop, and finally a duality between GprojX and GprojXop. Now the transpose Tr(F )of F is defined by Tr(F ) = Coker

(drH(X0) −→ drH(X1)

). Since drH(X) ∼= Hop(X), we

have an exact sequence

0 −→ dr(F ) −→ Hop(X0) −→ Hop(X1) −→ Tr(F ) −→ 0


In this way we obtain a functor Tr : mod-X −→ mod-Xop which is well-known to be a duality. Note that the duality D with respect to the base field k acts on mod-X by D(F )(X) = D(F (X)).

Let F ∈ mod-X with presentation H(X1) −→ H(X0) −→ F −→ 0 and G ∈ X-modwith presentation Hop(X1) −→ Hop(X0) −→ G −→ 0. Then F ∼= H(A∗

F ) and G ∼=Hop(AG

∗ ), where the objects A∗F and AG

∗ are defined by the following triangles in T:

X1 −→ X0 −→ A∗F −→ X1[1] and

X1[−1] −→ AG∗ −→ X0 −→ X1 (†)

Note that if F = H(A), resp. G = Hop(A), then we may choose: A∗F

∼= Cell1(A), resp. AG

∗ = Cell1(A).

Lemma 7.1. With the above notations, for any F ∈ mod-X and G ∈ X-mod, there are isomorphisms:

DTrF ∼= H(SA∗F [−1]) & TrDF ∼= H(Cell1(S−1A∗

F )[1]) (7.2)

DTrG ∼= Hop(S−1AG∗ )[1]) & TrDG ∼= Hop(Cell1(SAG

∗ )[−1]) (7.3)

In particular: DTrH(A) ∼= H(SCell1(A)[−1]) and TrDH(A) ∼= H(Cell1(S−1Cell1(A))[1]).

Proof. Applying the X-dual functor dr = Hom(−, H(?)|X) : mod-X −→ X-mod to the projective presentation of F , we have an exact sequence 0 −→ drF −→ Hop(X0) −→Hop(X1) −→ TrF −→ 0. Then applying the cohomological functor Hop to the first triangle in (†) and using that X is rigid, we see directly that we have an exact se-quence Hop(X0) −→ Hop(X1) −→ Hop(A∗

F [−1]) −→ 0. Hence TrF ∼= Hop(A∗F [−1]).

Using Serre duality it follows that: DTr(F ) ∼= DHop(A∗F [−1]) ∼= DT(A∗

F [−1], −)|X ∼=T(−, SA∗

F [−1])|X = H(SA∗F [−1]).

On the other hand by using Serre duality we have: DF ∼= DH(A∗F ) = DT(−, A∗

F )|X ∼=T(A∗

F , S−)|X = T(S−1A∗F , −)|X = Hop(S−1A∗

F ). Set B := S−1A∗F , so we have Hop(B) =

Hop(S−1A∗F ) ∼= Hop(Cell1(B)). From the triangle (+) : XB

1 [−1] −→ Cell1(B) −→ XB0 −→

XB1 we have a projective presentation Hop(XB

1 ) −→ Hop(XB0 ) −→ Hop(Cell1(B)) −→ 0

inducing an exact sequence H(XB0 ) −→ H(XB

1 ) −→ TrHop(Cell1(B)) −→ 0. Applying Hto the triangle (+) and using that X is rigid, we have an isomorphism TrHop(Cell1(B)) ∼=H(Cell1(B)[1]). Putting things together we have isomorphisms TrDF ∼= TrHop(S−1A∗

F ) ∼=H(Cell1(S−1A∗

F )[1]).The isomorphisms (7.3) are proved similarly. �

Lemma 7.2. The cluster tilted category mod-X has Auslander–Reiten sequences. In par-ticular AR-formula


D Hom(H(A),H(B)) ∼=−−→ Ext1(H(B),DTr(H(A))∼=−−→ Ext1(H(B),H(SCell1(A)[−1])) (7.4)

holds for any objects H(A) and H(B) in mod-X. Further if A ∈ X � X[1], then:

D Hom(H(A),H(B)

) ∼=−−→ Ext1(H(B),H(SA[−1])

)

Proof. Since X is (n + 1)-cluster tilting, it follows that X is functorially finite in T, so Xhas weak kernels and weak cokernels. On the other hand by using Serre duality (∗), we have isomorphisms, ∀X ∈ X:

DH(X) ∼=−−→ DT(X, X) ∼=−−→ T(X, SX) ∼=−−→ T(S−1X,X) ∼=−−→ Hop(S−1X)

DHop(X) ∼=−−→ DT(X,X) ∼=−−→ T(X, SX) ∼=−−→ H(SX)

Hence k-duals of contravariant or covariant representable functors over X are coherent. This clearly implies that X, and therefore mod-X, is a dualizing k-variety, see [5]. In particular mod-X has Auslander–Reiten sequences and AR-formula (7.4) holds. The last isomorphism holds since Cell1(A) = A, if A ∈ X � X[1]. �7.2. (n + 1)-Calabi–Yau categories

From now on we assume that: T is (n + 1)-Calabi–Yau, i.e. there is an isomorphism

of triangulated functors: S(?) ∼=−→ (?)[n + 1]. Then we have natural isomorphisms

D HomT(A,B) ∼=−−→ HomT(B,A[n + 1]) (7.5)

Lemma 7.3. Let A be an object in X � X[1]. Then DTrH(A) ∼= H(A[n]), and for any Gorenstein-projective object H(B) of mod-X, there is a natural isomorphism:

D Hom(H(A),H(B)

) ∼=−−→ Hom(Ω1H(B),H(A[n])

)

Proof. Since A ∈ X � X[1], we may take A = Cell1(A) and then by Lemma 7.1, we have an isomorphism DTr(H(A)) ∼= H(A[n]). By Lemma 7.2 we have an isomorphism DHom

(H(A), H(B)

) ∼= Ext1(H(B), H(A[n])

). Since H(B) ∈ GprojX, by Remark 6.3 we

have an isomorphism Ext1(H(B), H(A[n])

) ∼= Hom(Ω1H(B), H(A[n])

)and the last asser-

tion follows. �To proceed further we need the following two preliminary results.

Lemma 7.4. Let n � 2, and A be an object in X � X[1] such that H(A) is Gorenstein-projective. Let

C −→ A −→ B −→ C[1]


be a triangle in T. If C lies in X[−n] � X[−n + 1] � · · · � X[−1], then there exists a short exact sequence

0 −→ H(A) −→ H(B) −→ H(C[1]) −→ 0 (7.6)

which remains exact after the application of the functor Hom(−, H(X)), ∀X ∈ X. More-over if H(C[1]) is Gorenstein-projective, then so is H(B). The converse holds if mod-Xis Gorenstein.

Proof. We have A[1] ∈ X[1] � X[2], and this implies that H(A[1]) = 0 since n � 2. Applying the homological functor H to the triangle C −→ A −→ B −→ C[1], we have therefore an exact sequence H(C) −→ H(A) −→ H(B) −→ H(C[1]) −→ 0. We show that the map H(C) −→ H(A) is zero. Since H(A) is Gorenstein-projective, there is a monomorphism H(A) −→ H(X), where X ∈ X. Since A lies in X � X[1], this map is induced by a map A −→ X. Using that X is n-rigid, it follows that the composition C −→ A −→ X is zero, since C lies in X[−n] � X[−n + 1] � · · · � X[−1] and X ∈ X. As a consequence the composition H(C) −→ H(A) −→ H(X) is zero and then the map H(C) −→ H(A) is zero since H(A) −→ H(X) is a monomorphism. We infer that the map H(A) −→ H(B) is a monomorphism and therefore the sequence (7.6) is exact.

As the above argument shows, any map H(A) −→ H(X), X ∈ X, factorizes through H(B). It follows that the exact sequence (7.6) remains exact after the application of the functor Hom(−, H(X)), ∀X ∈ X. If H(C[1]) is Gorenstein-projective, then so is H(B), since GprojX is well-known to be closed under extensions. Conversely let mod-X be Gorenstein, and let H(B) be Gorenstein-projective. Applying Hom(−, H(X)), ∀X ∈ X, to the exact sequence (7.6) we have trivially Extk(H(C[1]), H(X)) = 0, ∀X ∈ X, ∀k � 1. It follows that H(C[1]) is Gorenstein-projective by Remark 6.3. �Lemma 7.5. Assume that the cluster tilted category mod-X is k-Gorenstein, k � 0, and for any object A ∈ X � X[1] such that H(A) is Gorenstein-projective, there is a natural isomorphism in GprojX:

Ω−(n+1)H(A) ∼=−−→ Ω−kΩkH(A[n]), or

H(A) ∼=−−→ Ωn+1H(A[n]) if k � n + 1 (††)

Then GprojX is (n + 2)-Calabi–Yau.

Proof. Since mod-X is k-Gorenstein, it follows from Remark 6.3 that ΩkH(C) ∈ GprojX, ∀C ∈ T, and moreover the functor Ω−kΩk : mod-X −→ GprojX is a right adjoint of the inclusion mod-X −→ GprojX. Then by Lemma 7.3 we have natural isomorphisms

D Hom(H(A),H(B)

) ∼=−−→ Hom(Ω1H(B),H(A[n])

)∼=−−→ Hom

(Ω1H(B),Ω−kΩkH(A[n])

)


∼=−−→ Hom(Ω1H(B),Ω−(n+1)H(A)

)∼=−−→ Hom

(H(B),Ω−(n+2)H(A)

)

for any Gorenstein-projective objects H(B) and H(A), with A ∈ X � X[1]. Since any object F = H(C) of mod-X is isomorphic to an object of the form H(A), where A ∈ X � X[1], namely A = Cell1(C), the above natural isomorphisms hold for all Gorenstein-projective objects. Hence the functor Ω−(n+2) : GprojX −→ GprojX serves as a Serre functor in GprojX, i.e. GprojX is (n + 2)-Calabi–Yau. Finally if n + 1 � k, then since Ωkmod-X = GprojX and since Ω−kΩk|Gproj X ∼= IdGproj X, we have an isomorphism of functors Ωn+1Ω−kΩk ∼= Ωn+1. This implies that the existence of an isomorphism Ω−(n+1)H(A)

∼=−→ Ω−kΩkH(A[n]) is equivalent to the existence of an isomorphism

H(A) ∼=−→ Ωn+1H(A[n]). �

Now we can prove the main result of this section. Note that the case n = 1 in Theorem 7.6 is due to Keller and Reiten [46, Theorem 3.3]. We give below a short proof of their result for the convenience of the reader.

Theorem 7.6. Let T be an (n +1)-Calabi–Yau triangulated category over a field k. Let Xbe an (n +1)-cluster tilting subcategory of T and assume that X is (n − k)-corigid, where n � 2k − 1, if n � 2.

If any object H(C), where C ∈ X[−n + 1] � · · · � X[−n + k], has finite projective or injective dimension, then the stable triangulated category GprojX is (n + 2)-Calabi–Yau.

Proof. First note that, since by Theorem 6.4 the cluster tilted category mod-X is k-Gorenstein, from Remark 6.3 it follows that ΩkH(A) lies in GprojX, ∀A ∈ T. We use throughout that if an object of mod-X has finite projective dimension, then its pro-jective dimension is at most k. By Lemma 7.5 it suffices to show the existence of a natural isomorphism (††) for any object A ∈ X � X[1] such that H(A) is Gorenstein-projective.

1. Case n = 1. Then T = X �X[1], and X is a 2-cluster tilting subcategory of the 2-Calabi–Yau category T. By Lemma 6.14, there exists a triangle X1

A[1][−1] −→ X0A[1][−1] −→

A −→ X1A[1], where the last map is a left X-approximation of A and the Xi

A[1] lie

in X, and moreover the map H(A) −→ H(X1A[1]) is a left projX-approximation of

H(A). Since H(A) is Gorenstein-projective, it follows that H(A) is a subobject of a projective and this implies that the map H(A) −→ H(X1

A[1]) is a monomorphism. Hence applying H to the above triangle and using that T(X, X[1]) = 0, we have an exact sequence

0 −→ H(A) −→ H(X1A[1]) −→ H(X0

A[1]) −→ H(A[1]) −→ 0


which shows that we have an isomorphism: H(A) ∼= Ω2H(A[1]) in mod-X. Since Ω1H(A[1]) is Gorenstein-projective, we have an isomorphism Ω−2H(A) ∼=Ω−1Ω1H(A[1]) as required.

2. Case n � 2. We divide the proof into several steps.Step 1: Any object H(C), where C ∈ X[−n + 1] � · · · � X[−1], has finite projective

dimension.

Proof. Indeed we have a triangle A −→ C −→ B −→ A[1], where A ∈ X[−n +1] � · · · � X[−n + k] and B ∈ X[−n + k + 1] � · · · � X[−1]. Hence B[−1] ∈X[−n + k] � · · · �X[−2] and by hypothesis, H(A) has finite projective dimension. Since X is (n − k)-corigid, it follows that H(B) = 0 = H(B[−1]). It follows from the above triangle that we have an isomorphism H(A) ∼= H(C) and therefore H(C) has finite projective dimension.

Step 2: There are isomorphisms: ΩkH(A[n]) ∼=−→ H(Ωk

X(A[n])).

Proof. We use the triangles, 1 � t � n:

ΩtX(A[n]) −→ Xt−1

A[n] −→ Ωt−1X (A[n]) −→ Ωt

X(A[n])[1] (T tA[n])

where ΩtX(A[n]) ∈ X and Ω0

X(A[n]) = A[n]. Using that T(X, A[i]) = 0, 1 � i �n − 1, and T(X, X[−i]) = 0, 1 � i � n − k, from the triangle (T 1

A[n]) we deduce a short exact sequence:

0 −→ H(Ω1X(A[n])) −→ H(X0

A[n]) −→ H(A[n]) −→ 0

and isomorphisms:

H(Ω1X(A[n])[−t]) = 0, 1 � t � n− k

Using this and applying H to the triangle (T 2A[n]), we have an exact sequence

0 −→ H(Ω2X(A[n])) −→ H(X1

A[n]) −→ H(Ω1X(A[n])) −→ 0

and isomorphisms:

H(Ω2X(A[n])[−k]) = 0, 1 � k � n− k − 1

Continuing in this way, and finally applying H to the triangle (T k−1A[n]), we have

an exact sequence

0 −→ H(Ωk−1X (A[n])) −→ H(Xk−2

A[n]) −→ H(Ωk−2X (A[n])) −→ 0

and an isomorphism:

H(Ωk−1X (A[n])[−1]) = 0


Finally applying H to the triangle (T kA[n]), we have an exact sequence

0 −→ H(ΩkX(A[n])) −→ H(Xk−1

A[n]) −→ H(Ωk−1X (A[n])) −→ 0

From the above exact sequences we deduce that:

ΩkH(A[n]) ∼=−−→ H(ΩkX(A[n])) (7.7)

Step 3: There are isomorphisms: ΩkH(A) ∼=−→ Ωk+2H(Ωn−1

X (A[n])) ∼=−→

Ωk+3H(Ωn−2X (A[n])).

Proof. Consider the following triangles, for 1 � t � n:

ΩtX(A[n])[t− 1] −→ Cellt−1(A[n]) −→ A[n] −→ Ωt

X(A[n])[t] (Ct−1A[n])

arising from the tower of triangles (C•A[n]) associated to A[n]. Applying H and

using that H(A[i]) = 0, 1 � i � n −1, and, by Proposition 3.2, H(ΩtX(A[n])[i]) =

0, 1 � i � t, we deduce an exact sequence

H(Cellt−1(A[n])[−n]

)−→ H(A) −→ H

(Ωt

X(A[n])[−n + t])

−→ H(Cellt−1(A[n])[−n + 1]

)−→ 0

Since Cellt−1(A[n])[−n] lies in (X � X[1] � · · · � X[t − 1])[−n] = X[−n] � X[−n +1] � · · ·�X[−n + t −1] which is contained in X[−n] �X[−n +1] � · · ·�X[−1], since t � n, by Lemma 7.4 we deduce short exact sequences

0 −→ H(A) −→ H(Ωt

X(A[n])[−n + t])

−→ H(Cellt−1(A[n])[−n + 1]

)−→ 0 (7.8)

for 2 � t � n. Moreover we infer isomorphisms:

H(ΩtX(A[n])[−n + t + 1]) ∼=−−→ H(Cellt−1(A[n])[−n + 2])

H(ΩtX(A[n])[−n + t + 2]) ∼=−−→ H(Cellt−1(A[n])[−n + 3])

...

H(ΩtX(A[n])[t− 2]) ∼=−−→ H(Cellt−1(A[n])[−1]) (7.9)

Consider the short exact sequence (7.8) for t = n −1. Since Celln−2(A[n])[−n +1]lies in X[−n +1] � · · · �X[−1], by Step 1 it follows that pdH(Celln−2(A[n])[−n +1]) � k. Then (7.8) gives us an isomorphism in mod-X:

ΩkH(A) ∼=−−→ ΩkH(Ωn−1X (A[n])[−1]) (7.10)


From the triangle (TnA[n]), we have an exact sequence

0 −→ H(Ωn−1X (A[n])[−1]) −→ H(Ωn

X(A[n])) −→ H(Xn−1A[n])

−→ H(Ωn−1X (A[n])) −→ 0

Since ΩnX(A[n]) := Xn

A[n] lies in X, it follows that

H(Ωn−1X (A[n])[−1]) ∼=−−→ Ω2H(Ωn−1

X (A[n])) (7.11)

Consider the short exact sequence

0 −→ H(Ωn−2X (A[n])[−1]) −→ H(Ωn−1

X (A[n]))

−→ H(Xn−2A[n]) −→ H(Ωn−2

X (A[n])) −→ 0

induced from the triangle (Tn−2A[n] ). Then we have a short exact sequence

0 −→ H(Ωn−2X (A[n])[−1]) −→ H(Ωn−1

X (A[n]))

−→ ΩH(Ωn−2X (A[n])) −→ 0 (7.12)

Setting t = n − 2 in (7.9), we have an isomorphism H(Ωn−2X (A[n])[−1]) ∼=

H(Celln−3(A[n])[−n +2]). Since Celln−3(A[n])[−n +2] lies in X[−n +2] �· · ·�X[−1], as above we have pd H(Ωn−2

X (A[n])[−1]) � k. Hence from (7.12) we get an

isomorphism Ωk+1H(Ωn−1X (A[n]))

∼=−→ Ωk+2H(Ωn−2X (A[n])). Since Ωkmod-X =

GprojX, combining (7.10) and (7.11) gives us the desired isomorphisms:

ΩkH(A) ∼=−−→ ΩkH(Ωn−1X (A[n])[−1]) ∼=−−→ Ωk+2H(Ωn−1

X (A[n]))∼=−−→ Ωk+3H(Ωn−2

X (A[n])) (7.13)

Step 4: There are isomorphisms: ΩkH(A) ∼=−→ Ωn+1H(Ωk

X(A[n])).

Proof: Consider the exact sequence

0 −→ H(Ωn−3X (A[n])[−1]) −→ H(Ωn−2

X (A[n])) −→ H(Xn−3A[n])

−→ H(Ωn−3X (A[n])) −→ 0

induced from the triangle (Tn−3A[n] ). Then we have a short exact sequence

0 −→ H(Ωn−3X (A[n])[−1]) −→ H(Ωn−2

X (A[n])) −→ ΩH(Ωn−3X (A[n])) −→ 0

Setting t = n − 3 in (7.9), we have an isomorphism H(Ωn−3X (A[n])[−1]) ∼=

H(Celln−4(A[n])[−n +3]). Since Celln−4(A[n])[−n +3] lies in X[−n +3] �· · ·�X[−1],


by Step 1 we have pdH(Ωn−2X (A[n])[−1]) � k. Hence from the above exact se-

quence we obtain as above an isomorphism:

ΩkH(Ωn−2X (A[n])) ∼=−−→ Ωk+1H(Ωn−3

X (A[n])) (7.14)

Combining (7.13) and (7.14) we arrive at an isomorphism:

ΩkH(A) ∼=−−→ Ωk+3H(Ωn−2X (A[n])) ∼=−−→ Ωk+4H(Ωn−3

X (A[n]))

Continuing in this way we obtain inductively isomorphisms:

ΩkH(A) ∼=−−→ Ωk+3H(Ωn−2X (A[n])) ∼=−−→ Ωk+4H(Ωn−3

X (A[n]))∼=−−→ · · · ∼=−−→ Ωn+1H(Ωk

X(A[n]))

Step 5: There is an isomorphism: H(A) ∼=−→ Ωn+1H(A[n]).

Proof: By Step 4 we have an isomorphism ΩkH(A) ∼=−→ Ωn+1H(Ωk

X(A[n]))and by Step 2 we have an isomorphism H(Ωk

X(A[n])) ∼=−→ ΩkH(A[n]). Com-

bining the two, we have an isomorphism ΩkH(A) ∼=−→ Ωn+1ΩkH(A[n])

∼=−→ΩkΩn+1H(A[n]). However since Ωkmod-X = Gproj(X) and n + 1 � k, the object Ωn+1H(A[n]) is Gorenstein-projective. Since H(A) is also Gorenstein-projective, we infer an isomorphism in mod-X:

H(A) ∼=−−→ Ωn+1H(A[n])

By Step 5 and Lemma 7.5 the assertion follows. �If k = 1 in Theorem 7.6, then clearly H(C) = 0, for any C ∈ X[−n +1] �· · ·�X[−1]. As a

consequence we have the following result, case k = 1 of which was obtained independently by Iyama and Oppermann, see [43, Theorem 5.11].

Corollary 7.7. Let X be an (n − k)-corigid (n + 1)-cluster tilting subcategory of T, n � 2, 0 � k � 1. If T is (n + 1)-Calabi–Yau, then the stable triangulated category GprojX is (n + 2)-Calabi–Yau.

Corollary 7.8. Let T be an (n +1)-Calabi–Yau triangulated category over a field k. Let Xbe an (n +1)-cluster tilting subcategory of T and assume that X is (n − k)-corigid, where n � 2k− 1, if n � 2. If X[−n + 1] � · · · �X[−n + k] ⊆

(X �X[1] � · · · �X[k]

)∩X�

k [k + 1]then GprojX is (n + 2)-Calabi–Yau.

Proof. Since n � 2k−1, it follows X is (k−1)-corigid and k-rigid. Then by Corollary 4.9, the functor H(C) has finite projective dimension (in fact pdH(C) � k) for any object C


in (X � X[1] � · · · � X[k]

)∩ X�

k [k + 1]. Hence the assertion follows from the assumption and Theorem 7.6. �

Let T be an (n +1)-Calabi–Yau triangulated category T over a field k, for instance T =C

(n+1)H the (n + 1)-cluster category of a finite-dimensional hereditary k-algebra H. We

also fix a full subcategory X of T satisfying the conditions of Theorem 7.6. Recall that the triangulated category of singularities, Db

sing(Λ), in the sense of Orlov, see [52], associated to a finite-dimensional k-algebra Λ, is the Verdier quotient Db(mod-Λ)/Kb(projΛ) of the bounded derived category of finite-dimensional Λ-modules by the thick subcategory of perfect complexes.

Corollary 7.9.

(i) The triangulated category GprojX has AR-triangles with AR-translation

τ = Ω−(n+1) : GprojX ≈−−→ GprojX,

τH(A) = Ω−(n+1)H(A) = Ω−kΩkH(Cell1(A)[n])

(ii) The stable category mod-GprojX of coherent functors over the stable category GprojX of Gorenstein-projective coherent functors over X is a triangulated cate-gory which is (3n + 5)-Calabi–Yau.

(iii) If X = addT for some object T ∈ T, then the triangulated category of singularities Db

sing(EndT(T )

)associated to the cluster-tilted k-algebra EndT(T ) is (n +2)-Calabi–

Yau.

Proof. Part (i) follows from Theorem 7.6 and the fact that the AR-translation in a trian-gulated category with Serre functor S is given by τ = S[−1], see [57]. For (ii) note that the category mod- GprojX of coherent functors over GprojX is Frobenius and therefore its sta-ble category mod- GprojX is triangulated. Then the assertion follows from Theorem 7.6and a result of Keller which says that if a triangulated category C is d-Calabi–Yau, then the stable category mod-C of coherent functors over C is (3d − 1)-Calabi–Yau, see [45]. Part (iii) follows from the fact that, since by Theorem 6.4 the endomorphism alge-bra EndT(T ) is Gorenstein, the triangulated category of singularities Db

sing(EndT(T )) of EndT(T ) is triangle equivalent to the stable category Gproj EndT(T ) of finitely generated Gorenstein-projective modules over EndT(T ), so Theorem 7.6 applies. �Remark 7.10. Using Serre duality, it is easy to see that a d-Calabi–Yau triangulated category T, 1 � d � 2n, may contain a non-trivial (n + 1)-cluster tilting subcategory, only if d � n + 1, and may contain a non-trivial (n − k)-corigid (n + 1)-cluster tilting subcategory, only if d = n + 1.


Remark 7.11. Let X be an (n +1)-cluster tilting subcategory of a triangulated category T. In the special case where X satisfies X = X[n +1], more pleasant results can be obtained. First observe that by Corollary 5.8, the category mod-X of coherent functors over X is Frobenius so the stable category mod-X is triangulated. It follows then from the work of Geiss, Keller and Oppermann in [36], that if T is d-Calabi–Yau, where d = k(n + 1), k ∈ Z, which is easily seen to be the only possible value of d for T to be d-Calabi–Yau, then the stable category mod-X is (d + 2k − 1)-Calabi–Yau. The proof uses that in this case X carries a natural (n +3)-angulated structure in the sense of [36]. For k = 0, 1 this also follows from Corollary 7.7.

Part 3. Categorified higher Auslander correspondence

In this part we develop, in the context of abelian categories, a categorified version of higher Auslander correspondence between n-Auslander algebras and n-cluster tilting modules, which is due to Auslander for n = 0 and to Iyama for n > 0. We also study cluster tilting subcategories in this setting and we investigate homological properties of the associated stable Auslander categories. We also give applications to (infinitely generated) cluster tilting modules over any ring and to Cohen–Macaulay modules over an Artin algebra.

8. Categorified Auslander correspondence

Auslander in the early seventies proved the following remarkable result which played a fundamental role for much of the later developments in the representation theory of Artin algebras:

Theorem 8.1. (See [3].) There is bijective correspondence between Morita equivalence classes of Artin algebras Λ of finite representation type and Artin algebras Γ such that gl.dim Γ � 2 � dom.dim Γ.

A categorified version of Auslander’s correspondence was proved in [13]. First recall that the free abelian category of an additive category C is an abelian category F(C )together with an additive functor F : C −→ F(C ) satisfying the following universal prop-erty: for any additive functor G : C −→ A to an abelian category A , there exists a unique exact functor G∗ : F(C ) −→ A such that G∗ ◦ F = G. The free abelian category F(C ) of any additive category C always exists, it is unique up to equivalence, and it admits the following descriptions: F(C ) ≈ mod- (C -mod)op ≈ ((mod-C )-mod)op; we refer to [35], [13] for details and more information. We call an abelian category F with enough projectives and injectives an Auslander category if gl.dimF � 2 � dom.dimF.

The following result recovers and gives a categorified version of Auslander’s corre-spondence.


Theorem 8.2. (See [13, Theorems 6.1 and 6.6; Corollary 6.9].) 1. An abelian category F is free if and only if F is an Auslander category. Moreover the maps

C �−→ F(C ) and F �−→ ProjF ∩ InjF

give, up to equivalence, mutually inverse bijections between:

(I) Additive categories C (with split idempotents).(II) Auslander categories F.

2. The maps

A �−→ mod-A and F �−→ ProjF

give, up to equivalence, mutually inverse bijections between

(a) (Representation finite) Abelian categories A with enough injectives.(b) Auslander categories F such that ProjF is abelian (and representation finite).(c) Free abelian categories F such that ProjF ∩ InjF is left coherent (and ProjF is rep-

resentation finite).

3. The maps

B �−→ (B-mod)op and F �−→ InjF

give, up to equivalence, mutually inverse bijections between

(d) (Representation finite) Abelian categories B with enough projectives.(e) Auslander categories F such that InjF is abelian (and representation finite).(f) Free abelian categories F such that ProjF ∩ InjF is right coherent (and InjF is rep-

resentation finite).

Recently Iyama proved, as one of the main result of [40], the following far reaching generalization of Auslander’s correspondence which in addition supports interesting con-nections with cluster tilting theory in module categories. Recall that a finitely generated module M over a finite-dimensional algebra Λ is called an n-cluster tilting module if {X ∈ mod-Λ | ExtkΛ(M, X) = 0, 1 � k � n} = addM = {X ∈ mod-Λ | ExtkΛ(X, M) = 0,1 � k � n}, see [40,41], where the terminology (n + 1)-cluster tilting or maximal n-orthogonal module is used.

Theorem 8.3. (See [40].) For any n � 0, the map M �−→ EndΛ(M) gives a bijec-tion between equivalence classes of n-cluster tilting Λ-modules M for finite-dimensional


algebras Λ and Morita-equivalence classes of finite-dimensional algebras Γ such that gl.dim Γ � n + 2 � dom.dim Γ.

Our aim in this section is to refine and generalize Theorems 8.2, 8.3 in the context of abelian categories. In fact we shall prove the following categorification of Theorem 8.3, referring to [14] for a relative version, which shows that (part 1. of) Theorem 8.2 is the zero case of a natural correspondence at a higher level. For the notions of n-cluster tilting objects or subcategories and n-Auslander categories, we refer to Subsection 8.3 below.

Theorem 8.4 (Generalized Auslander correspondence). For any integer n � 0, the map M �−→ mod-M gives, up to equivalence, a bijective correspondence between:

(I) n-cluster tilting subcategories M in abelian categories A with enough injectives.(II) n-Auslander categories B.

Note that the abelian category A in the above correspondence is uniquely determined, up to equivalence, by the n-cluster tilting subcategory M it contains: in fact A is equiv-alent to the dual of the category of coherent covariant functors over the full subcategory of projective–injective objects of mod-M, see Proposition 8.12.

Specializing Theorem 8.4 to M = addM for an n-cluster tilting Λ-module M over a finite-dimensional algebra Λ, we obtain Iyama’s Theorem 8.3. The proof of (a more precise form of) Theorem 8.4 will be given in Theorem 8.23 after developing the necessary tools and giving definitions of all unexplained terms.

8.1. From cluster tilting subcategories to Auslander categories

Throughout this subsection we fix an abelian category A with enough injectives. We also fix a contravariantly finite subcategory M of A which is closed under direct summands and isomorphisms. Then the category mod-M of coherent functors over M is abelian and we have a left exact functor

H : A −→ mod-M, H(A) = A (−, A)|M

From now on: we assume that any right M-approximation in A is an epimorphism. Note that if A has enough projectives, then this happens if and only if ProjA ⊆ M.

Lemma 8.5. There is an adjoint pair of functors

(R,H) : mod-M −→←− A

where the functor R is exact and the functor H is fully faithful and induces an equivalence M ≈−→ Projmod-M.


Proof. Define a functor R : mod-M −→ A as follows. If F ∈ mod-M admits a presen-tation (−, M1) −→ (−, M0) −→ F −→ 0, where f : M1 −→ M0 is a map in M, then R(F ) = Coker f , i.e. R is defined by the exact sequence M1 −→ M0 −→ R(F ) −→ 0. We leave to the reader the easy proof that in this way we obtain a left adjoint of H. Since right M-approximations are epics, it follows directly that the natural map RH −→ IdA is invertible, so H is fully faithful. By Yoneda, H induces an equivalence M ≈ Proj mod-Msince M is closed under direct summands. Let AF = Ker(M1 −→ M0) in the presentation of F above and let MAF

−→ AF be a right M-approximation. Then we have an exact sequence H(MAF

) −→ H(M1) −→ H(M0) −→ F −→ 0 which is part of a projective resolution of F . Since the map MAF

−→ AF is an epimorphism, applying R we have an exact sequence MAF

−→ M1 −→ M0. This means that L1R(F ) = 0. We infer that R is exact. �

By Lemma 8.5, KerR := mod-M is a localizing subcategory of mod-M, so

0 −→ mod-M −→ mod-M −→ A −→ 0 (8.1)

is a short exact sequence of abelian categories and mod-M consists of all coherent functors F : Mop −→ Ab with a projective presentation H(M1) −→ H(M0) −→ F −→ 0 where the map M1 −→ M0 is epic in A .

Let F ∈ mod-M, and fix throughout a projective presentation H(M1) −→ H(M0) −→F −→ 0 of F . Then we have an exact sequence

0 −→ H(AF ) −→ H(M1) −→ H(M0) −→ F −→ 0 (8.2)

where AF = Ker(M1 −→ M0) and then Ω(F ) = Im(H(M1) −→ H(M0)) and H(AF ) =Ω2(F ).

Let A ∈ A . Since right M-approximations are epics, there exists an exact sequence

· · · −→ MnA −→ Mn−1

A −→ · · · −→ M1A −→ M0

A −→ A −→ 0 (8.3)

called an M-resolution of A, such that its image under H is a projective resolution of H(A):

· · · −→ H(MnA) −→ H(Mn−1

A ) −→ · · · −→ H(M1A)

−→ H(M0A) −→ H(A) −→ 0 (8.4)

This is defined as the Yoneda composition of short exact sequences 0 −→ Ki+1A −→

M iA −→ Ki

A −→ 0 constructed inductively, where each map M iA −→ Ki

A is a right M-approximation of Ki, ∀i � 0, and K0

A = A. It follows from (8.2) and (8.4) that any coherent functor F ∈ mod-M admits a projective resolution of the form


· · · −→ H(MnAF

) −→ H(Mn−1AF

) −→ · · · −→ H(M0AF

)

−→ H(M1) −→ H(M0) −→ F −→ 0 (8.5)

Now in analogy with the triangulated case we define

M⊥n =

{A ∈ A | Extk(M, A) = 0, 1 � k � n

}and

⊥nM =

{A ∈ A | Extk(A,M) = 0, 1 � k � n

}

Then M is called n-rigid if M ⊆ M⊥n or equivalently M ⊆ ⊥

nM.

Lemma 8.6. For any object F ∈ mod-M: F ∈ mod-M if and only if Hom(F, H(B)) = 0, ∀B ∈ A . If F ∈ mod-M, then Ext1(F, H(B)) = 0, ∀B ∈ A . If in addition M ⊆ M⊥

n , then there are isomorphisms:

F ∼= Ext1(−, AF )|M ∼= Ext2(−,K1AF

)|M ∼= · · ·∼= Extn−1(−,Kn−2

AF)|M ∼= Extn(−,Kn−1

AF)|M (∗)

Proof. The first part follows from the adjunction isomorphism Hom(F, H(B)) ∼=Hom(R(F ), B) and the fact that KerR = mod-M. Let F ∈ mod-M; applying the exact functor R to (8.2) we have an isomorphism R(ΩF ) ∼= RH(M0) = M0. Then the exact sequence (H(M0), H(B)) −→ (ΩF, H(B)) −→ Ext1(F, H(B)) −→ 0 is iso-morphic to (M0, B) −→ (R(ΩF ), B) −→ Ext1(F, H(B)) −→ 0, where the first map is invertible. Hence Ext1(F, H(B)) = 0, ∀B ∈ A . If M ⊆ M⊥

n , then from the extension 0 −→ AF −→ M1 −→ M0 −→ 0 we have F ∼= Ext1(−, AF )|M, since Ext1(−, M1)|M = 0. Considering the short exact sequences 0 −→ Ki+1

AF−→ M i

AF−→ Ki

AF−→ 0, i � 0,

where K0AF

= AF , and using that M ⊆ M⊥n , as above the isomorphisms (∗) follow easily

by induction. �Lemma 8.7. For any object B ∈ A , and any n � 1, the following are equivalent:

(i) B ∈ M⊥n .

(ii) The functor H induces isomorphisms:

Extk(−, B)∼=−→ Extk(H(−),H(B)), 1 � k � n

Proof. If (ii) holds, then using that H(M) is projective in mod-M, for any M ∈ M, we infer that Extk(M, B) = 0, 1 � k � n. Hence B lies in M⊥

n . For the converse, considering an M-resolution of A and using that M is n-rigid and that the left exact functor H is fully faithful and induces an equivalence between M and Projmod-M, it follows directly by induction that the hypothesis B ∈ M⊥

n implies that the naturally induced maps Extk(A, B) −→ Extk(H(A), H(B)), for 1 � k � n, are invertible, ∀A ∈ A . �


For any object M ∈ M, consider the functor (−, M) : Mop −→ Ab, M ′ �−→Hom(M ′, M). If A has enough projectives and 0 −→ ΩM −→ PM −→ M −→ 0 is an exact sequence in A where PM is projective, then it is easy to see that we have an exact sequence of functors

0 −→ H(ΩM) −→ H(PM ) −→ H(M) −→ (−,M) −→ 0 (†)

Hence, for any M ∈ M, the functor (−, M) lies in mod-M.Observe that since for F = (−, M) we have AF = ΩM , it follows that we have

isomorphisms, ∀M ′ ∈ M, ∀k � 1:

Extk+2((−,M), (−,M ′)) ∼=−−→ Extk(H(ΩM),H(M ′)) (‡)

We use the family of functors (−, M), M ∈ M, to give the following characterization of rigidity:

Lemma 8.8. If n � 1 then for the following statements:

(i) M is n-rigid;(ii) For any F ∈ mod-M ⊆ mod-M, we have: Extk(F, H(M)) = 0, 0 � k � n + 1,

we have (i) ⇒ (ii). If A has enough projectives, then the statements (i) and (ii) are equivalent.

Proof. (i) ⇒ (ii) Since M ⊆ M⊥n , from (8.2) and Lemma 8.7 we have Extk+2(F, H(M)) ∼=

Extk(H(AF ), H(M)) ∼= Extk(AF , M), 1 � k � n. Since F ∈ mod-M, we have a short exact sequence 0 −→ AF −→ M1 −→ M0 −→ 0. Applying A (−, X), with X ∈ M, the induced long exact sequence gives Extk(AF , X) = 0, 1 � k � n − 1. Hence Extk+2(F, H(X)) ∼=Extk(AF , X) = 0, 1 � k � n − 1, and then by Lemma 8.6: Extk(F, H(X)) = 0, 0 �k � 1 and 3 � k � n + 1. Applying (−, H(X)) to the short exact sequence 0 −→H(AF ) −→ H(M1) −→ Ω(F ) −→ 0 we have an exact sequence (H(M1), H(X)) −→(H(AF ), H(X)) −→ Ext2(F, H(X)) −→ 0. Since (H(M1), H(X)) −→ (H(AF ), H(X)) is isomorphic to the map (M1, X) −→ (AF , X) which is epic since Ext1(M0, X) = 0, we infer that Ext2(F, H(X)) = 0. Hence Extk(F, H(X)) = 0, 0 � k � n + 1, as required.

(ii) ⇒ (i) Assume that A has enough projectives. For M ∈ M we consider the functor (−, M) ∈ mod-M. Let G = Im(H(PM ) −→ H(M)). Then 0 = Ext2((−, M), H(M)) ∼=Ext1(G, H(M)) ∼= Coker[A (PM , M) −→ A (ΩM, M)] ∼= Ext1(M, M), hence M ⊆ M⊥

1 . Using this, the isomorphism (‡) and Lemma 8.7, we have: 0 = Ext3((−, M), H(M)) ∼=Ext1(H(ΩM), H(M)) ∼= Ext1(ΩM, M) ∼= Ext2(M, M). Hence M ⊆ M⊥

2 . Continuing in this way and using Lemma 8.7 and the isomorphism (‡) we have M ⊆ M⊥

n as required. �To proceed further we need the following observation.


Lemma 8.9. Let C be an abelian category with enough projectives, and C ∈ C . Then: C = 0 if and only if C has finite projective dimension and Extk(C, P ) = 0, ∀P ∈ ProjC , 0 � k � pdC.

Proof. Let pdC = n and let 0 −→ Pn −→ Pn−1 −→ · · · −→ P 0 −→ C −→ 0 be a projective resolution of C. Then 0 = Extn(C, Pn) ∼= Ext1(Ωn−1C, Pn), so the extension 0 −→ Pn −→ Pn−1 −→ Ωn−1C −→ 0 splits and therefore Ωn−1C is projective. Then Ext1(Ωn−2C, Ωn−1C) ∼= Extn−1(C, Ωn−1C) = 0, so the extension 0 −→ Ωn−1C −→Pn−2 −→ Ωn−2C −→ 0 splits and therefore Ωn−2C is projective. Continuing in this way we see that C is projective. Then C = 0 since C (C, C) = 0. �Proposition 8.10. If M �= ProjA and M is n-rigid, i.e. M ⊆ M⊥

n , then gl.dimmod-M �n + 2. Moreover:

M⊥n = M �= ProjA =⇒ ⊥

nM = M �= InjA & gl.dim mod-M = n + 2

Proof. Since M �= ProjA , we have mod-M �= 0. Indeed if mod-M = 0 then the functor R induces an equivalence mod-M ≈ A and this implies that A has enough projec-tives and M = Proj A . This contradiction shows that there exists 0 �= F ∈ mod-M. If gl.dim mod-M � n +1, then by Lemma 8.7 we have Extk(F, H(X)) = 0, 0 � k � n +1, and then F = 0 by Lemma 8.9. This contradiction shows that gl.dimmod-M � n +2. Now as-sume that M⊥

n = M. Then clearly InjA ⊆ M⊥n = M. To show that gl.dimmod-M = n +2,

it suffices to show that gl.dimmod-M � n + 2. By using the exact sequences (8.2), (8.4)this holds if pdH(A) � n, ∀A ∈ A . Since the exact sequence (8.3) becomes a projective resolution of H(A) after applying H, it suffices to show that H(Kn

A) is projective, or equivalently that Kn

A ∈ M. Since M⊥n = M, it suffices to show that Kn

A ∈ M⊥n . Since M

is n-rigid, we have Ext1(M, KiA) = 0, 1 � i � n. In particular Ext1(M, Kn

A) = 0 and we have an isomorphism Ext2(M, Kn

A) ∼= Ext1(M, Kn−1A ) = 0, hence Kn

A ∈ M⊥2 . Continuing

in this way we have finally isomorphisms Extn(M, KnA) ∼= Extn−1(M, Kn−1

A ) ∼= · · · ∼=Ext1(M, K1

A) = 0. Hence KnA ∈ M⊥

n = M.Now let A ∈ ⊥

nM. Applying A (−, M) to the extension 0 −→ K1A −→ M0

A −→ A −→ 0, we have Exti(K1

A, M) = 0, 1 � i � n − 1. Using this and applying A (−, M) to the extension 0 −→ K2

A −→ M1A −→ K1

A −→ 0, we have Exti(K2A, M) = 0, 1 � i � n − 2.

Continuing in this way we finally have Ext1(Kn−1A , M) = 0. Since pdH(A) � n, we have

KnA ∈ M and therefore the extension 0 −→ Kn

A −→ Mn−1A −→ Kn−1

A −→ 0 splits, hence Kn−1

A ∈ M. Since Ext1(Kn−2A , M) = 0, the extension 0 −→ Kn−1

A −→ Mn−2A −→

Kn−2A −→ 0 splits and therefore Kn−2

A ∈ M. Continuing in this way we see that the objects Ki

A lie in M and therefore since Ext1(A, M) = 0, we infer that the extension 0 −→ K1

A −→ M0A −→ A −→ 0 splits. Then A ∈ M as a direct summand of M0. We

conclude that A ∈ M, i.e. ⊥nM = M. If InjA = M, then clearly M = A and therefore A is semisimple. It follows that A = M = ProjA and this is not the case. Hence Inj A � M. �


Remark 8.11. The following observations will be useful later.

1. Assume that M⊥n = M. Then M⊥

n+1 = Inj A . Indeed if A ∈ M⊥n+1, then clearly A ∈ M

and ΣA ∈ M⊥n = M. Hence any extension 0 −→ A −→ I −→ ΣA −→ 0, where

I ∈ Inj A , splits and therefore A is injective. Hence M⊥n+1 ⊆ Inj A and therefore

M⊥n+1 = Inj A since always InjA ⊆ M⊥

n+1. If A , in addition, has enough projectives, then since ⊥nM = M by Proposition 8.10, we have similarly that ⊥n+1M = ProjA . It follows directly from this that M = Proj A = Inj A , in particular A is Frobenius.

2. It follows from Lemmas 8.7 and 8.8, and Proposition 8.10 that the following are equivalent:(a) M is n-rigid.(b) grademod-M F � n + 2, ∀F ∈ mod-M, F �= 0,with equality in (b) if M⊥

n = M. This generalizes a result of Buchweitz [25, Theo-rem 2.13].We recall that if C is an abelian category, then the grade gradeC X of X ∈ C is defined by gradeC X = inf{i � 0 | Exti(X, P ) �= 0, ∀P ∈ ProjC }.

3. If M⊥n = M, then for any F ∈ mod-M we have: Extkmod-M(F, (−, M)) = 0,

∀k � 0, except possibly for k = n + 2. In this case Extn+2mod-M(F, (−, M)) ∼=

Extnmod-M((−, AF )|M, (−, M)) ∼= ExtnA (AF , M). In particular for F = (−, M), M ∈ M, we have: Extn+2

mod-M((−, M), (−, M)) ∼= Extn+1A (M, M).

4. If M⊥n = M, then any object A ∈ A admits an M-resolution

0 −→ MnA −→ Mn−1

A −→ · · · −→ M1A −→ M0

A −→ A −→ 0

Indeed as in the proof of Proposition 8.10 we have pdH(A) � n. Using the projective resolution of H(A) which arises by applying H to the M-resolution (8.3), we infer that the image of the map H(Mn

A) −→ H(Mn−1A ) is projective. Then Im(Mn

A −→ Mn−1A ) in

(8.3) lies in M and our claim follows.

Proposition 8.12. Let A be an abelian category with enough injectives and M a con-travariantly finite subcategory of A such that any right M-approximation is epic. If Proj A �= M = M⊥

n for some n � 1, then:

1. We have equalities:

gl.dimmod-M = n + 2 = dom.dim mod-M (8.6)

2. Inj A �= M = ⊥nM, and A , Inj A , and M can be recovered from B := mod-M as

follows:

A ≈−−→ (U-mod)op and M ≈−−→ ProjB and Inj A ≈−−→ U (8.7)


where U denotes the full subcategory of projective–injective objects of mod-M and is covariantly finite.

3. M is covariantly finite in A if and only if ProjB is covariantly finite in B = mod-M.

Proof. 1., 2. In view of Proposition 8.10, it remains to show that dom.dim mod-M =n + 2 and the existence of the equivalences (8.7). Let M ∈ M and consider an injective resolution of M :

0 −→ M −→ I0 −→ I1 −→ · · · −→ In −→ In+1 −→ · · ·

Applying the left exact functor H and using that M is n-rigid, we have an exact sequence:

0 −→ H(M) −→ H(I0) −→ H(I1) −→ · · ·

−→ H(In−1) −→ H(In) −→ H(In+1) (†)

Since any projective object of mod-M is of the form H(M) for some M ∈ M and since the functor H preserves injective objects (since its left adjoint R is exact), it follows that (†) is part of an injective resolution of the projective object H(M). Since InjA ⊆ M, the objects H(Ik) above are projective–injective. Hence dom.dim mod-M � n + 2. If dom.dim mod-M � n +3, then for any M ∈ M there is an exact sequence 0 −→ H(M) −→H(I0) −→ · · · −→ H(In+1) −→ H(In+2) −→ F −→ 0 where the Ik are injective in A . Clearly then pdF = n + 3 and this is impossible since gl.dim mod-M = n + 2 by Proposition 8.10. We have a fully faithful functor H : Inj A −→ U. If F lies in U, then F = H(M) for some M ∈ M since F is projective. Let M −→ J be monic, where J is injective; clearly the induced monic H(M) −→ H(J) splits and then M is injective as a direct summand of J . We infer that the functor H : Inj A −→ U is essentially surjective, hence an equivalence. Let F be in mod-M and let R(F ) −→ I be a monomorphism, where I is injective in A . Then H(I) lies in U and it is easy to see that the composition F −→ HR(F ) −→ H(I) is a left U-approximation of F . Hence U is covariantly finite in mod-M. Finally we have A ≈ (Inj A -mod)op ≈ (U-mod)op, the first equivalence being true for any abelian category with enough injectives.

3. If M is covariantly finite in A , let F ∈ B and let R(F ) −→ MR(F ) be a left M-approximation of R(F ). We leave to the reader as an easy exercise to check that the composite map F −→ HR(F ) −→ H(MR(F )) is a left Proj B-approximation of F , where the first map is the unit of the adjoint pair (R, H). Hence ProjB is covariantly finite. If this holds, let B ∈ B and H(B) −→ H(M) be a left ProjB-approximation of H(B). This gives us a map B −→ M which clearly is a left M-approximation of B. Hence M is covariantly finite. �

In Section 9 we shall prove that any full subcategory M of A satisfying the conditions of Proposition 8.12 is covariantly finite in A .


8.2. From Auslander categories to cluster tilting subcategories

Throughout this subsection, we fix an abelian category B with enough projectives.We assume that: the full subcategory

U := ProjB ∩ Inj B

consisting of all projective–injective objects of B is covariantly finite.

Remark 8.13. ProjC ∩ Inj C is covariantly finite in any abelian category C satisfying: (α) Proj C is covariantly finite, and (β) dom.dim C � 1. Indeed consider the composition A −→ P −→ J , where A −→ P is a left ProjC -approximation of A, and P −→ J is monic, where J is projective–injective, which exists since dom.dim C � 1. Then it is easy to see that the map A −→ J is a left (Proj C ∩ Inj C )-approximation of A.

It follows that the category of coherent functors U-mod is abelian with enough pro-jectives. Then setting

A := (U-mod)op

we obtain an abelian category with enough injectives.

Lemma 8.14. There is an adjoint pair of functors

(Rop,Hop) : B −→←− A

where the functor Hop is fully faithful and the functor Rop is exact and induces an equiv-alence U ≈−→ Inj A .

Proof. First let B ∈ B. Since U is covariantly finite, there exists a left U-approximation gB0 : B −→ UB

0 . Let LB1 = Coker gB0 and let gB1 : UB

0 −→ UB1 be the composition of

UB0 −→ LB

1 with a left U-approximation LB1 −→ UB

1 . Then we obtain a complex

BgB0−−→ UB

0gB1−−→ UB

1 (8.8)

in B with the property that the induced complex B(UB1 , −)|U −→ B(UB

0 , −)|U −→B(B, −)|B −→ 0 is exact in U-mod. We set Rop(B) = B(B, −)|U, ∀B ∈ B, so that in A = (U-mod)op we have an exact sequence

0 −→ Rop(B) −→ Rop(UB0 ) −→ Rop(UB

1 ) (8.9)

It is easy to see that the assignment B −→ Rop(B) extends to a functor Rop : B −→ A

which clearly is exact since U consists of injective objects of B. We define a functor


Hop : A −→ B as follows: if F lies in A , then in U-mod, F admits a presentation (U1, −)|U −→ (U0, −)|U −→ F −→ 0. Then we have a map fF : U0 −→ U1 in B where the Ui lie in U. It is easy to see that the assignment F �−→ Hop(F ) = Ker fF :

0 −→ Hop(F ) −→ U0 −→ U1 (8.10)

defines an additive functor Hop : A −→ B. We show that (Rop, Hop) is an adjoint pair. First observe that in (8.10) the maps Hop(F ) −→ U0 and Im(U0 −→ U1) −→ U1 are left U-approximations since both maps are monics and U consists of injective objects. As a consequence applying to (8.10) the exact functor Rop we have an isomorphism

RopHop(F ) ∼=−→ F . Now let B ∈ B and consider a diagram as in (8.8). By construction,

the naturally induced map B −→ Ker gB1 is the map B −→ HopRop(B) which is the evaluation at B of a natural map IdB −→ HopRop. It is easy to see that the isomorphism RopHop ∼=−→ IdA and the natural map IdB −→ HopRop are the counit and the unit respectively of an adjoint pair (Rop, Hop), where the functor Hop is fully faithful, since the counit is invertible. Clearly then Rop induces an equivalence U ≈ Inj(U-mod)op ≈Inj A . �Lemma 8.15. If dom.dim B � 2, then the functor Rop induces an equivalence

ProjB ≈−−→ ImRop

Proof. We have to show that the functor Rop is fully faithful when restricted to ProjB. So let α : P −→ Q be a map in ProjB. Choosing copresentations of P and Q as in (8.8)and using that the maps gP0 and gQ0 are monomorphisms since dom.dimB � 2, we have the following exact commutative diagram

0 PgP0

α

UP0

gP1

β

UP1

γ

0 QgQ0

UQ0

gQ1

UQ1

(∗)

where the maps β and γ exist since UP0 and UP

1 are left U-approximations of P and Coker gP0 respectively, and UQ

0 , UQ1 lie in U. Applying the exact functor Rop, where

Rop(B) = B(B, −)|U, to the diagram (∗) and setting μ = Rop(gP0 ) and ν = Rop(gQ0 ), we have the following exact commutative diagram:

0 Rop(P )μ

Rop(α)

(UP0 ,−)|U

(gP1 ,−)|U

(β,−)|U

(UP1 ,−)|U

(γ,−)|U

0 Rop(Q) ν (UQ0 ,−)|U

(gQ1 ,−)|U

(UQ1 ,−)|U

(∗∗)


If Rop(α) = 0, then using the injectivity of the objects (U, −)|U in A , it follows from diagram (∗∗) that there is a map δ : UP

1 −→ UQ0 such that gP1 ◦δ = β. Then gP0 ◦β = 0 and

this implies that α = 0 since gQ0 is a monic. Hence Rop|Proj B is faithful. If α : Rop(P ) −→Rop(Q) is map in A , then injectivity of the objects (U, −)|U implies that there are maps (β, −)|U : (UP

0 , −)|U −→ (UQ0 , −)|U and (γ, −)|U : (UP

1 , −)|U −→ (UQ1 , −)|U such that

the induced diagram (∗∗), with Rop(α) deleted, is commutative. Clearly then β and γmake the right square of diagram (∗) commutative and this implies the existence of a map α : P −→ Q making the whole diagram (∗) commutative. It follows then that Rop(α) = α and this means that Rop|Proj B is full. �

From now on we set M := Rop(ProjB) ⊆ A . Then by Lemma 8.15, the functor Rop

induces an equivalence

Rop : ProjB ≈−−→ M, if dom.dim B � 2

Proposition 8.16. Let B be an abelian category with enough projectives. Assume that gl.dim B = n + 2 = dom.dim B, where n � 1, and let U = Proj B ∩ Inj B. If Proj B is covariantly finite in B, then U = Proj B ∩ Inj B is covariantly finite in B, and M is functorially finite in A .

If U is covariantly finite, then setting A = (U-mod)op we have the following:

(a) M is contravariantly finite in A and any right M-approximation is an epimorphism.(b) M⊥

n = M �= ProjA .

Moreover B, ProjB, and U can be recovered from A and M as follows:

B ≈−−→ mod-M and U ≈−−→ Inj A and ProjB ≈−−→ M (8.11)

Proof. The first assertion follows from Lemma 8.15 and Remark 8.13. Assume that U is covariantly finite.

(α) Let F ∈ A . Since B has enough projectives, there is an epimorphism P −→Hop(F ), where P is projective in B. Applying the exact functor Rop and using that RopHop ∼= IdA , we then have an epimorphism MF −→ F , where MF := Rop(P ). Clearly this map is a right M-approximation of F .

(β) We first show that M is n-rigid, i.e. ExtkA (M, M) = 0, 1 � k � n. Let Mi be in M, i = 1, 2. Then there are projective objects P and Q in B such that M1 = Rop(P ) and M2 = Rop(Q). Since dom.dim B = n + 2, there exists an exact sequence in B:

0 −→ Q −→ U0 −→ U1 −→ · · · −→ Un −→ Un+1

where the Uk lie in U. Applying the exact functor Rop we then have an exact sequence:


0 −→ M2 −→ (U0,−)|U −→ (U1,−)|U −→ · · · −→ (Un,−)|U −→ (Un+1,−)|U

which is part of an injective resolution of M2. Hence for 0 � k � n, the kth-cohomology of the complex:

0 −→ (M1,M2) −→ (M1, (U0,−)|U) −→ (M1, (U1,−)|U) −→ · · ·

−→ (M1, (Un,−)|U) −→ (M1, (Un+1,−)|U)

gives Extk(M1, M2). Using the adjoint pair (Rop, Hop) the above complex is isomorphic to the complex

0 −→ (M1,M2) −→ (P,U0) −→ (P,U1) −→ · · · −→ (P,Un) −→ (P,Un+1)

which is exact since P is projective. We infer that ExtkA (M1, M2) = 0, 1 � k � n. Hence M is n-rigid.

Now let F ∈ A be in M⊥n , i.e. ExtkA (M, F ) = 0, 1 � k � n. Since from (8.10)

we have an exact sequence 0 −→ Hop(F ) −→ U0 −→ U1 in B, where the U i are projective–injective, and since gl.dimB = n + 2, it follows that pdHop(F ) � n. Hence there exists a projective resolution of Hop(F ) in B of the form:

0 −→ Pn −→ Pn−1 −→ · · · −→ P 1 −→ P 0 −→ Hop(F ) −→ 0

Applying the exact functor Rop we have an exact resolution of F ∼= RopHop(F ) by objects of M:

0 −→ Mn −→ Mn−1 −→ · · · · · · −→ M1 −→ M0 −→ F −→ 0

Applying (M, −) to the above exact sequence, we have Ki ∈ M⊥i , for 1 � i � n −1, where

Ki = Im(M i −→ M i−1). Applying (M, −) to the extension 0 −→ K1 −→ M0 −→ F −→ 0and using that F ∈ M⊥

n we see directly that K1 ∈ M⊥n . Using this and applying (M, −)

to the extension 0 −→ K2 −→ M1 −→ K1 −→ 0 we see that K2 ∈ M⊥n . Continuing

in this way we see finally that Ki ∈ M⊥n , 1 � i � n − 1. It follows that the extension

0 −→ Mn −→ Mn−1 −→ Kn−1 −→ 0 splits and therefore Kn−1 ∈ M. Using this and the fact that Ki ∈ M⊥

n , 1 � i � n − 1, we see finally that K1 ∈ M. Then the extension 0 −→ K1 −→ M0 −→ F −→ 0 splits and therefore F ∈ M. We infer that M⊥

n = M. Since gl.dimB = n + 2 it follows that M �= ProjA .

Finally, by Lemma 8.14, the functor Rop induces an equivalence between U and InjA . Since, by Lemma 8.15, the functor Rop induces an equivalence between ProjB and M =Rop(ProjB) and since B has enough projectives, it follows that there is induced an equivalence between B = mod-ProjB and mod-M. �


8.3. Generalized Auslander correspondence

To state and prove the main result of this section we need the following definitions which generalize, and are inspired by, Iyama’s [39].

Definition 8.17. Let A be an abelian category. A full subcategory M of A is called an n-cluster tilting, resp. n-cluster cotilting, subcategory, n � 1, if:

(i) M is contravariantly, resp. covariantly, finite in A .(ii) Any right, resp. left, M-approximation is an epimorphism, resp. monomorphism.(iii) M⊥

n = M, resp. ⊥nM = M.

We consider M = A as, the unique, 0-cluster tilting, resp. 0-cluster cotilting, subcategory of A .

Specializing, we have the following notions. An object M ∈ A is called an n-cluster tilting, resp. n-cluster cotilting, object, if addM is an n-cluster tilting, resp. n-cluster cotilting, subcategory of A . If A has all small coproducts, resp. products, then M is called an n-cluster tilting, resp. n-cluster cotilting, object, if AddM , resp. ProdM , is an n-cluster tilting, resp. n-cluster cotilting, subcategory.

Remark 8.18. Let A be an abelian category with small coproducts, resp. products, and let M ∈ A .

(i) AddM is contravariantly finite in A , resp. ProdM is covariantly finite in A .(ii) If A has enough projectives and M is a generator, then M is an n-cluster tilting

object if and only if M⊥n = AddM .

(iii) If A has enough injectives and M is a cogenerator, then M is an n-cluster cotilting object if and only if ⊥nM = ProdM .

Definition 8.19. An abelian category B is called n-Auslander, resp. n-coAuslander, cat-egory, n � 0, if:

(i) B has enough projectives, resp. injectives.(ii) The full subcategory ProjB∩ Inj B is covariantly, resp. contravariantly, finite in B.(iii) gl.dim B � n + 2 � dom.dim B, resp. gl.dimB � n + 2 � codom.dim B.

A ring Λ is called a (right) n-Auslander ring, if Mod-Λ is an n-Auslander category.

Remark 8.20. It should be noted that if A is an abelian category with enough projectives and enough injectives, then dom.dimA � n if and only if codom.dimA � n, see [38, Lemma 2.1] for the case of rings, the general case being similar. Also observe that:


(α) M is n-cluster tilting in A if and only if Mop is n-cluster cotilting in A op, and (β) B

is an n-Auslander category if and only if Bop is an n-coAuslander category. Clearly an n-Auslander category B is either semisimple or else satisfies gl.dim B = n + 2 =dom.dim B.

Example 8.21. Let Λ be an Artin algebra. Then Λ is an Auslander algebra, see [8], if and only if Λ is a 0-Auslander ring. Also it is easy to see that Λ is an n-Auslander ring if and only if mod-Λ is an n-Auslander category. On the other hand let P be a dualizing R-variety over a commutative Artin ring, see [5], for instance P = projΛ for an Artin algebra Λ. Then the category mod-P is n-Auslander if and only if mod-P is n-coAuslander.

From now on we consider pairs {A , M} consisting of an abelian category A with enough injectives and an n-cluster tilting subcategory M of A . We call two such pairs {A , M} and {A ′, M′} cluster equivalent, if there is an equivalence M ≈ M′.

By the following lemma, we infer that the pair {A , M} is completely determined, up to equivalence, by M.

Lemma 8.22. If {A , M} and {A ′, M′} are cluster equivalent pairs, then the categories A and A ′ are equivalent.

Proof. Any equivalence M ≈ M′ extends to an equivalence mod-M ≈ mod-M′ and there-fore to an equivalence U ≈ U′, where U, resp. U′, is the full subcategory of projective–injective objects of mod-M, resp. mod-M′. Since, by Proposition 8.12, A ≈ (U-mod)op

and A ′ ≈ (U′-mod)op, it follows that if the pairs {A , M} and {A ′, M′} are cluster equivalent, then the categories A and A ′ are equivalent. �

Now we can prove the following main result of this section which gives a categorified version of higher Auslander–Iyama correspondence.

Theorem 8.23 (Generalized Auslander correspondence). For any integer n � 0, the map M �−→ mod-M induces, up to (cluster) equivalence, a bijective correspondence between:

(I) Pairs {A , M} consisting of an abelian category A with enough injectives and an n-cluster tilting subcategory M of A .

(II) n-Auslander categories B.

The mutually inverse bijections are given as follows:{A ,M

}�−→ mod-M and B �−→

{(U−mod)op,N

}

where U = ProjB ∩ Inj B and N = B(ProjB, −)|U. Under the above correspondence:M is covariantly finite in A if and only if Proj B is covariantly finite in B.


Proof. For n � 1 the assertion follows from Propositions 8.12 and 8.16. It remains to treat the case n = 0. Let M = A be 0-cluster tilting, so A is abelian with enough injectives. Then B := mod-A ≈ mod- (Inj A -mod)op is the free abelian category of the additive category InjA , so by Theorem 8.2, B has enough projectives and gl.dimB �2 � dom.dim B. Next we show that ProjB ∩ Inj B ≈ InjA = U is covariantly finite in

B. Indeed let F be in B = mod-A and let A (−, A) f∗−→ A (−, B) ε−→ F −→ 0 be a

presentation of F , where f∗ = A (−, f). If A f−→ B

c−→ C −→ 0 is exact in A , then clearly there exists a unique map τ : F −→ A (−, C) such that ε ◦τ = A (−, c). It is easy to see that the map τ is a left ProjB-approximation of F , so ProjB ≈ A is covariantly finite in B. Using Remark 8.13, it follows that ProjB ∩ Inj B ≈ InjA = U is covariantly finite in B. We infer that B is a 0-Auslander category and A = (Inj A -mod)op =(U-mod)op. Conversely let B be a 0-Auslander category. By Theorem 8.2, there is an equivalence B ≈ mod- (U-mod)op and the subcategory U = Proj B ∩ Inj B is covariantly finite in B. Then U is left coherent, so A := (U-mod)op is abelian with enough injectives, i.e. it is a 0-cluster tilting subcategory of itself. Clearly the maps A �−→ mod-A and B �−→ (U-mod)op, where U = ProjB∩ Inj B, are mutually inverse bijections. For the last assertion see part 3. of Proposition 8.12. �

Note that, as we shall see in Proposition 9.2 of the next section, any n-cluster tilting subcategory of A is covariantly finite. Hence in the last assertion of Theorem 8.23 the full subcategory of projectives of the associated n-Auslander category is always covariantly finite in B.

For the purpose of the next direct consequence of our previous results, we call a triangulated category T strongly algebraic if T is triangle equivalent to the stable category A modulo projectives of a Frobenius abelian category A .

Corollary 8.24. Under the correspondence of Theorem 8.23: A is Frobenius if and only if U-mod is Frobenius, for U = ProjB∩ Inj B. In this case M is an (n +1)-cluster tilting subcategory of A , and conversely any (n + 1)-cluster tilting subcategory of a strongly algebraic triangulated category arises in this way.

Our next result gives the dual version of Theorem 8.23. The proof is dual and is left to the reader.

Theorem 8.25. For any integer n � 0, the map M �−→ (M-mod)op gives, up to equiva-lence, a bijective correspondence between:

(I) n-cluster cotilting subcategories M in abelian categories A with enough projectives.(II) n-coAuslander categories B.


The following result gives some information on the (higher) K-theory of n-Auslander categories. Recall that an abelian category is a length category if any of its objects has finite (composition) length.

Proposition 8.26. Let B be an n-Auslander category.

(i) There is a 0-Auslander category F, an abelian category C with gl.dimC � 3n − 1, and a short exact sequence of abelian categories

0 −→ C −→ F −→ B −→ 0

which induces isomorphisms in K-theory: Ki(F) ∼= Ki(B) ⊕ Ki(C ), ∀i � 0.(ii) If the abelian category A = ProjF is a length category with enough projectives

and injectives and ProjB is covariantly finite in B, then there exists a short exact sequence of abelian categories

0 −→ D −→ B −→ A −→ 0

which induces isomorphisms in K-theory: Ki(B) ∼= Ki(A ) ⊕ Ki(D), ∀i � 0.

Proof. (i) By Theorem 8.23, there is an abelian category A with enough injectives and an n-cluster tilting subcategory M of A such that B = mod-M. Since A = (Inj A -mod)op, it follows by Theorem 8.2 that F := mod-A = mod- (Inj A -mod)op is a 0-Auslander category and A = ProjF. Now the inclusion M ⊆ A induces a short exact sequence of abelian categories 0 −→ C −→ F −→ B −→ 0, where the functor F −→ B is the restriction mod-A −→ mod-M, and as easily seen C := mod- (A /M) is a Serre subcat-egory of F identified with the full subcategory of F consisting of all coherent functors A op −→ Ab vanishing on M. Since M is contravariantly finite in A , it follows that A /M is a left triangulated category with loop functor Ω1

M : A /M −→ A /M given by Ω1

M(A) = the kernel of a right M-approximation of A; we refer to [18] for details and more information on left triangulated categories. Now let F : (A /M)op −→ Ab

be a coherent functor with presentation (−, B) −→ (−, C) −→ F −→ 0 and let Ω1

M(C) −→ A −→ B −→ C be a left triangle in A /M. Then the exact sequence of functors · · · −→ (−, Ω2

M(C)) −→ (−, Ω1M(A)) −→ (−, Ω1

M(B)) −→ (−, Ω1M(C)) −→

(−, A) −→ (−, B) −→ (−, C) −→ F −→ 0 is a projective resolution of F in mod-A /M. If A ∈ A , then by part 4. of Remark 8.11, there exists an M-resolution of A of the form 0 −→ Mn

A −→ Mn−1A −→ · · · −→ M0

A −→ A −→ 0 and ΩkM(A) = Im(Mk

A −→ Mk−1A ),

∀k � 1. Since MnA = Ωn

M(A) lies in M, we have ΩnM(A) = 0 in A /M. It follows

that ΩnM = 0 and from the projective resolution of F ∈ mod- (A /M) it follows that

pdF � 3n − 1. Hence gl.dimC � 3n − 1. Finally since gl.dimB � n + 2 < ∞, the last assertion follows from [6, Proposition 2.4].

(ii) Under the assumption of part (ii), the n-cluster tilting subcategory M of A is n-cluster cotilting, see Theorem 9.6 below. In particular Proj A ⊆ M and then the


restriction functor B = mod-M −→ A = mod-ProjA induces a short exact sequence 0 −→ D −→ B −→ A −→ 0 of abelian categories, where D = mod-M. Since any object of A has finite length, the assertion follows from [6, Proposition 2.4]. �

In particular any 1-Auslander category B, is a Gabriel quotient of a 0-Auslander category F by an abelian category C with gl.dim C � 2.

The bijective correspondence below is due to Iyama [40]. We consider pairs {Λ, M}, where Λ is an Artin algebra and M is a finitely generated n-cluster (co)tilting module over Λ. Two such pairs {Λ, M} and {Λ′, M ′}, are cluster equivalent, if there is an equiv-alence addM ≈ addM ′. Then as before we have an equivalence mod-Λ ≈ mod-Λ′, so the algebras Λ and Λ′ are Morita equivalent.

Corollary 8.27. The map {Λ, M} �−→ EndΛ(M) gives a bijection between cluster/Morita equivalence classes of:

(i) Pairs {Λ, M}, where Λ is an Artin algebra and M is a finitely generated n-cluster (co)tilting Λ-module.

(ii) n-Auslander algebras Γ.

Moreover: (α) M is an n-cluster (co)tilting Λ-module if and only if M is an n-rigid (co)generator and n +2 = gl.dim EndΛ(M), and (β) rep.dim Λ � n +2 and K∗(addM, ⊕) ∼=K∗(mod-Λ) ⊕ K∗(mod-EndΛ(M)).

8.4. Universality and functoriality

As observed in Theorem 8.2, the 0-Auslander categories are precisely the free abelian categories, so they are characterized by a universal property. More precisely let Abelbe the (large) category of abelian categories and exact functors, and Add the (large) category of additive categories and additive functors. Then Theorem 8.2 implies that the forgetful functor U : Abel −→ Add admits a left adjoint F : Add −→ Abel, where F(C ) is the 0-Auslander category of the additive category C . In other words for any additive category C and any abelian category A , the canonical functor C −→ F(C )induces a natural isomorphism

HomAbel(F(C ),A

) ≈−−→ HomAdd(C ,U(A )

)

However it is not clear if the class of n-Auslander categories are characterized by an analogous universal property. We close this section by analyzing briefly this problem.

If M ⊆ A is a full subcategory of an abelian category A , then an additive functor F : A −→ C is called right M-exact if for any map f : A −→ B in A which is M-epic in A , i.e. A (M, f) is surjective, ∀M ∈ M, the map F (f) is epic in C . If C is an abelian category, then let LexM-ex(A , C ) be the category of left exact and right M-exact functors A −→ C , and let E x(mod-M, C ) be the category of exact functors mod-M −→ C .


First we show that the restricted Yoneda functor H : A −→ mod-M satisfies a univer-sal property:

Proposition 8.28. Let M ⊆ A be a contravariantly finite full subcategory of an abelian category A and let H : A −→ mod-M, H(A) = A (−, A)|M be the restricted Yoneda functor. Then H is left exact and right M-exact, and if S : A −→ C is a left exact and right M-exact functor to an abelian category C , then there exists a unique exact functor S∗ : mod-M −→ C such that S∗ ◦H = S. In other words, H induces an equivalence:

− ◦H : Ex(mod-M,C ) ≈−−→ LexM-ex(A ,C ), F �−→ F ◦H

Proof. First observe that mod-M is abelian since M is contravariantly finite in A . Clearly the functor H is left exact and right M-exact. As in the proof of Proposition 8.26, the inclusion M ⊆ A induces by restriction an exact functor mod-A −→ mod-M and we have a short exact sequence of abelian categories

0 −→ mod-(A /M) −→ mod-A −→ mod-M −→ 0

where the category mod- (A /M) of coherent functors over the stable category A /M

is identified with the full subcategory of mod-A consisting of all coherent functors A op −→ Ab admitting a presentation A (−, B) −→ A (−, C) −→ F −→ 0, where the map B −→ C is M-epic. Now the functor S : A −→ C admits a unique right ex-act extension S† : mod-A −→ C by setting S†(F ) = CokerS(f), where A (−, B) −→A (−, C) −→ F −→ 0 is a presentation of F in mod-A . Since S is left exact, it is easy to see that S† is exact. On the other hand, since S is right M-exact, it is easy to see that S† kills the objects of the Serre subcategory mod- (A /M) ⊆ mod-A and therefore it factorizes through the exact quotient functor mod-A −→ mod-M yielding an exact functor S∗ : mod-M −→ C . Clearly we have S∗ ◦ H = S and the exact functor S∗ is unique with this property. �

As a direct consequence we have the following.

Corollary 8.29. Let M be an n-cluster tilting subcategory of an abelian category A and let B = mod-M be the associated n-Auslander category. For any abelian category C , the functor H : A −→ B induces an equivalence

− ◦H : Ex(B,C ) −→ LexM-ex(A ,C ), F �−→ F ◦H

Our aim now is to show that for any n � 0, appropriately defined, (large) categories of n-cluster tilting categories and n-Auslander categories are equivalent:

• Let AuslCat(n) be the (large) category with objects n-Auslander categories. If Band B′ are n-Auslander categories, then a morphism between them in AuslCat(n) is an exact functor G : B −→ B′ preserving projectives and projective–injective objects.


• Let ClustTilt(n) be the (large) category with object pairs {A , M} consisting of an abelian category A with enough injectives and an n-cluster tilting subcategory M of A . A morphism {A , M} −→ {A ′, M′} in ClustTilt(n) is a left exact functor F : A −→ A ′

such that: (α) F (M) ⊆ M′ and F (Inj A ) ⊆ Inj A ′, and (β) F sends M-epics in A to M′-epics in A ′.

Lemma 8.30. If F : {A , M} −→ {A ′, M′} is a morphism in ClustTilt(n), then there exists a unique exact functor F ∗ : mod-M −→ mod-M′ which preserves projectives and projective-injectives, such that the following diagram commutes:

A

H

FA ′

H′

mod-M F∗

mod-M′

Proof. Since F is left exact and sends M-epics to M′-epics and since H′ is left ex-act and sends M′-epics to epics, the composite functor H′ ◦ F : A −→ mod-M′

is left exact and sends M-epics to epics. Then by Proposition 8.28, there exists a unique exact functor F ∗ : mod-M −→ mod-M′ such that F ∗ ◦ H = H′ ◦ F . In fact if H(M1) −→ H(M0) −→ X −→ 0 is a finite presentation of X in mod-M, then F ∗(X)is defined by the exact sequence H′(F (M1)) −→ H′(F (M0)) −→ F ∗(X) −→ 0. Since F (M) ⊆ M′ and F ∗H(M) = H′F (M)), it follows that F ∗ preserves projectives. On the other hand, since F (InjA ) ⊆ Inj A ′ and F ∗H(InjA ) = H′F (Inj A )), and since the functors H and H′ induce equivalences between InjA , resp. Inj A ′, and the full subcate-gory of projective–injective objects of mod-M, resp. mod-M′, it follows that F ∗ preserves projective-injectives. �

Lemma 8.30 allows us to define a functor

A : ClustTilt(n) −→ AuslCat(n), A({A ,M}

)= mod-M & A(F ) = F ∗

Lemma 8.31. Let G : B −→ B′ be a morphism in AuslCat(n). Let A = (U-mod)op

and A = (U′-mod)op, where U and U′ are the projective–injective objects of B and B′

respectively, and let M = B(ProjB, −)|U and M′ = B′(ProjB′, −)|U′ . Then G induces a left exact functor G† : A −→ A ′ such that G†(U) ⊆ U′ and G†(M) ⊆ M′, and G†

sends M-epics to M′-epics.

Proof. By Theorem 8.23 we may assume that B = mod-M and B′ = mod-M′, where Mand M′ are n-cluster tilting subcategories in the abelian categories A = U-mod and A ′ =U′-mod. Then G : mod-M −→ mod-M′ is an exact functor such that G(Projmod-M) ⊆Proj mod-M′ and G(U) ⊆ U′. The induced functor G : U −→ U′ extends uniquely to a left exact functor G† : A −→ A ′ and clearly G† preserves injectives and sends M to M′.


It remains to show that G† sends M-epics to M′-epics. Consider, as in Lemma 8.30, the left exact functors H : A −→ mod-M and H′ : A ′ −→ mod-M′. By our assumptions for G it follows easily that G ◦ H = H′ ◦ G†. Let f : B −→ C be an M-epic in A . Then H′(G†(f)) = G(H(f)) is epic since H sends M-epics to epics and G is exact. This means that the map G†(f) is M′-epic. �

Lemma 8.31 allows us to define a functor

C : AuslCat(n) −→ ClustTilt(n), C(B) =({(U-mod)op,M}

)& C(G) = G†

where U = ProjB ∩ Inj B and M = B(ProjB, −)|U.

Theorem 8.32. The functors A and C define mutually inverse equivalences

A : ClustTilt(n)≈−→←−≈

AuslCat(n) : C

Proof. Using Propositions 8.12 and 8.16 or Theorem 8.23, for any pair {A , M} in ClustTilt(n), we have:

CA({A ,M}) = C(mod-M) =({(U-mod)op,N}

)≈ {A ,M}

since N := B(ProjB, −)|U ≈ M and U := Proj mod-M ∩ Inj mod-M, so that A ≈(U−mod)op.

Similarly for any n-Auslander category B we have:

AC(B) = A({A ,M}) = mod-M ≈ B

where: M := B(ProjB, −)|U and A := (U-mod)op, where U = ProjB ∩ Inj B.If F, G : {A , M} −→ {A ′, M′} are morphisms in ClustTilt(n) such that A(F ) ∼= A(G),

so F ∗ ∼= G∗, then using Lemma 8.30, we have H′F = F ∗H ∼= G∗H = H′G. Since R′H′ ∼=IdA ′ , we have obviously F ∼= G. On the other hand if F : A({A , M}) = mod-M −→A({A ′, M′}) = mod-M′ is a morphism in AuslCat(n), then it is easy to see that F ∼=A(F †) = A C(F ) = F † ∗. Indeed, since B has enough projectives, the functors F and F † ∗

are isomorphic since both are (right) exact and clearly agree on the projective objects of B. Hence A is an equivalence of large categories with quasi-inverse C. �

We leave to the reader the interpretation of the above results in the context of module over rings and in particular in the context of finitely generated modules over an Artin algebra.

9. Cluster tilting versus cluster cotilting

Our definition of n-cluster tilting subcategories in abelian or triangulated categories in Sections 8 and 5 respectively are slightly different than those of Iyama in [39–41]


and [44]. Recall from [41] that a full subcategory M of an abelian category A , say with enough projectives and/or injectives, is called n-cluster tilting (also (n −1)-cluster tilting or maximal n-orthogonal in the terminology used in [39]) if: (A1) M is contravariantly finite and M⊥

n = M, and (A2) M is covariantly finite and M = ⊥nM. Also recall from [44]

that a full subcategory X of a triangulated category T is called (n + 1)-cluster tilting if: (T1) X is contravariantly finite and X�

n = X, and (T2) X is covariantly finite and X = �

nX.In Section 5 we proved that the conditions (T1) and (T2) are equivalent, so only

half of the conditions in [44] are needed in the definition of cluster tilting subcategories in triangulated categories. Our aim in this section is to show that the conditions (A1)and (A2) are also equivalent in the abelian case. As a consequence n-cluster tilting and n-cluster cotilting are equivalent notions. This improves on related results of Iyama in [39–41] and has some interesting consequences on (infinitely generated) n-cluster tilting modules.

We begin our analysis by fixing some notation. For a full subcategory M of A and any integer k � 0, we denote by M�k, resp. M�k the full subcategory of A consisting of all objects A admitting an exact sequence 0 −→ Mk −→ · · · −→ M1 −→ M0 −→ A −→ 0, where the Mi ∈ M, 0 � i � k, resp. and the sequence remains exact after applying the functor A (M, −). For instance if M is an n-cluster tilting subcategory of A , then by part 4. of Remark 8.11 we have A = M�n. We also denote by M�k, resp. M�k, the full subcategory of A consisting of all objects A such that there exists an exact sequence 0 −→ A −→ M0 −→ M1 −→ · · · −→ Mk −→ 0, where the M i ∈ M, 0 � i � k, resp. and the sequence remains exact after applying the functor A (−, M).

Clearly for any t � 0 we have M�t ⊆ M�t and M�t ⊆ M�t.

Lemma 9.1. Let A be an abelian category with enough injectives and M a contravariantly finite subcategory of A such that right M-approximations are epic and M⊥

n = M. Then for 0 � t � n − 1 we have:

(α) M�t = M�t ⊆ ⊥n−tM & (β) M�t = M�t ⊆ M⊥

n−t

Proof. We only prove (α) since (β) is dual. If t = 0, then clearly M�0 = M = M�0. By Proposition 8.10 we have M = ⊥

nM, so (α) is true for t = 0. Assume that t = 1and let A ∈ M�1. Then there exits an exact sequence 0 −→ A −→ M0 −→ M1 −→ 0. Applying the functor A (−, M) to this exact sequence and using that M is n-rigid, we see directly that Extk(M, A) = 0 for 1 � k � n − 1. Hence A ∈ ⊥

n−1M ∩ M�1 and therefore M�1 ⊆ ⊥

n−1M ∩ M�1. Hence we have M�1 = M�1 ⊆ ⊥n−1M and (α) holds for t = 1.

Assume that M�t−1 = M�t−1 ⊆ ⊥n−t+1M, for 0 � t � n − 1, and let A ∈ M�t. Then

there exists a short exact sequence 0 −→ A −→ M0 −→ B −→ 0, where B ∈ M�t−1. By hypothesis then we have B ∈ ⊥

n−t+1M. Using this and n-rigidity of M, application of the functor A (−, M) to this exact sequence gives that the sequence remains exact and


Extk(M, A) = 0, for 1 � k � n − t. This implies that A ∈ M�t ∩ ⊥n−tM. We infer that

M�t = M�t ⊆ ⊥n−tM. �

Setting for convenience ⊥0 M = A , we have the following:

Proposition 9.2. Let A be an abelian category with enough injectives and M a contravari-antly finite subcategory of A . If any right M-approximation is epic and M⊥

n = M, then M is covariantly finite and

M�t = M�t = ⊥n−tM, 0 � t � n

Proof. Let A be an object of A , consider an injective resolution

0 −→ A −→ I0 −→ I1 −→ · · · · · · −→ In −→ In+1 −→ · · ·

of A, and let ΣiA = Im(Ii−1 −→ Ii), ∀i � 1, be the cosyzygy objects of A. We shall construct a left M-approximation of A in two steps. The first one is the proof of the following:

1. Claim: For any 1 � t � n, there exist objects Xt ∈ A , K1Xt ∈ M�t−1 ∩M⊥

1 and

Zt ∈ M�n−t, and an exact commutative diagram:

0 0

K1Xt K1

Xt

0 Σt−1A Xt−1 Zt 0

0 Σt−1A It−1 ΣtA 0

0 0

(9.1)

Note that in this case, since K1Xt ∈ M⊥

1 and since Inj A ⊆ M, we haveExt1(It−1, K1

Xt) = 0. Therefore the second vertical short exact sequence splits and we have a direct sum decomposition Xt−1 = K1

Xt ⊕ It−1 and then clearly Xt−1 ∈M�t−1 ∩M⊥

1 .Proof of the Claim: By Remark 8.11, any object X ∈ A admits an M-resolution

0 −→ MnX −→ · · · −→ M0

X −→ X −→ 0, such that KiX := Im(M i

X −→ M i−1X ) ∈ M�n−i


and clearly KiX ∈ M⊥

1 , ∀i � 1. Let Zn := M0ΣnA −→ ΣnA be a right M-approximation

of ΣnA with kernel K1Xn := K1

ΣnA, and consider the pull-back diagram

0 0

K1ΣnA K1

ΣnA

0 Σn−1A Xn−1 M0ΣnA 0

0 Σn−1A In−1 ΣnA 0

0 0

which defines the object Xn−1. Then K1ΣnA ∈ M�n−1 ∩M⊥

1 . Since M contains the injec-tives, the middle vertical short exact sequence splits and therefore Xn−1 = K1

ΣnA⊕In−1

lies in M�n−1 ∩M⊥1 . In particular there exists a short exact sequence 0 −→ K1

Xn−1 −→M0

Xn−1 −→ Xn−1 −→ 0 where K1Xn−1 ∈ M�n−2 ∩M⊥

1 .Consider the following exact commutative diagram induced by the composition of

epimorphisms M0Xn−1 � Xn−1 � M0

ΣnA, which defines the object Zn−1:

0 0

K1Xn−1 K1

Xn−1

0 Zn−1 M0Xn−1 M0

ΣnA 0

0 Σn−1A Xn−1 M0ΣnA 0

0 0


We observe that Zn−1 ∈ M�1. Next we form the pull-back of the short exact sequence 0 −→ Σn−2A −→ In−2 −→ Σn−1A −→ 0 along the epimorphism Zn−1 −→ Σn−1A:

0 0

K1Xn−1 K1

Xn−1

0 Σn−2A Xn−2 Zn−1 0

0 Σn−2A In−2 Σn−1A 0

0 0

which defines the object Xn−2. Since K1Xn−1 lies in M⊥

1 and In−2 ∈ M, it follows that the middle vertical sequence splits and therefore Xn−2 = K1

Xn−1 ⊕ In−2. In particular we have Xn−2 ∈ M�n−2 ∩M⊥

1 .Continuing in this way we construct, for 1 � t � n, the exact commutative diagram

(9.1) as desired.2. In particular for t = 1, we have the following pull-back diagram

0 0

M1X1 M1

X1

0 A MA0 Z1 0

0 A I0 ΣA 0

0 0

which defines the object MA0 = X0, and where M1

X1 ∈ M�0 ∩ M⊥1 , hence M1

X1 ∈ M, and Z1 ∈ M�n−1. Since M1

X1 ∈ M and I0 ∈ Inj A ⊆ M, as before the middle vertical sequence splits and therefore MA

0 = M1X1 ⊕ I0 ∈ M.


We claim that the map A −→ MA0 is a left M-approximation of A, i.e. any map

A −→ M , where M ∈ M, factorizes through MA0 . To this end, clearly it suffices to show

that Ext1A (Z1, M) = 0, i.e. Z1 ∈ ⊥1 M. Since Z1 lies in M�n−1, it follows from Lemma 9.1

that indeed Z1 ∈ ⊥1 M. We infer that A −→ MA

0 is a left M-approximation of A, and therefore M is covariantly finite in A .

It remains to show that the inclusion M�t ⊆ ⊥n−tM, 0 � t � n, of Lemma 9.1 is

an equality. First let t = n, so we have to show that M�n = A := ⊥0 M. Let A be an

object of A ; since M is covariantly finite in A and contains the injectives, there exists an M-coresolution 0 −→ A −→ M0 −→ M1 −→ · · · −→ Mk −→ Mk+1 −→ · · · of A. Setting Ai = Im(M i−1 −→ M i), ∀i � 0, and using that M is n-rigid, it is easy to see that Ak ∈ ⊥

k M, ∀k � 1, and in particular An ∈ ⊥nM = M. It follows then that A ∈ M�n

and therefore M�n = A . On the other hand, let A ∈ ⊥n−tM, where 0 � t � n − 1. Then

applying the functor A (−, M) successively to the short exact sequences 0 −→ Ai−1 −→M i−1 −→ Ai −→ 0, we see easily by induction that Ai ∈ ⊥

n−t+iM. In particular we have At ∈ ⊥

nM = M. Then clearly A ∈ M�t. Hence M�t = M�t = ⊥n−tM, for 0 � t � n. �

Setting A := M⊥0 , by duality we have the following.

Proposition 9.3. Let A be an abelian category with enough projectives and M a covari-antly finite subcategory of A . If any left M-approximation is monic and ⊥nM = M, then M is contravariantly finite and

M�t = M�t = M⊥n−t, 0 � t � n

Combining Propositions 9.2, 9.3 and Proposition 8.10 and its dual, we have the follow-ing consequence which shows that n-cluster tilting and n-cluster cotilting are equivalent notions.

Corollary 9.4. Let A be an abelian category with enough projective and injective objects. For a full subcategory M of A , the following are equivalent:

(i) M is contravariantly finite, ProjA ⊆ M, and M⊥n = M.

(ii) M is covariantly finite, InjA ⊆ M, and ⊥nM = M.

In other words: M is an n-cluster tilting subcategory if and only if M is an n-cluster cotilting subcategory. Moreover we have for 0 � t � n − 1:

M�t = M�t = M⊥n−t & M�t = M�t = ⊥

n−tM & M�n = A = M�n

Remark 9.5. Iyama and Oppermann in [43] call an object M in an abelian category Aa right, resp. left, (n + 1)-cluster tilting object if addM = ⊥

nM , resp. addM = M⊥n .

It follows from Corollary 9.4 that if A has enough projectives and addM contains the


projectives, then: M is a right (n +1)-cluster tilting object =⇒ M is a left (n +1)-cluster tilting object.

The following summarizes some of the main results of this and the previous section.

Theorem 9.6. Let A be a non-semisimple abelian category with enough projectives and enough injectives. For a full subcategory M of A and any integer n � 1, the following are equivalent.

(i) M is an n-cluster tilting subcategory of A .(ii) M is an n-cluster cotilting subcategory of A .(iii) mod-M is an n-Auslander category.(iv) M-mod is an n-Auslander category.(v) M is contravariantly finite n-rigid, ProjA ⊆ M, and gl.dim mod-M = n + 2.(vi) M is covariantly finite n-rigid, Inj A ⊆ M, and gl.dimmod-M = n + 2.

If (i) holds, then: M�n = A = M�n. Moreover: dom.dimmod-M = codom.dimM-mod =gl.dimM-mod = n +2. If in addition M is of finite representation type, then: rep.dimA �n + 2.

Proof. The implications (i) ⇔ (iii) ⇒ (v) and (ii) ⇔ (iv) ⇒ (vi) follow from Proposi-tion 8.10, its dual, and Theorems 8.23 and 8.25. The equivalence (i) ⇔ (ii) follows from Corollary 9.4. It remains to prove the implication (vi) ⇒ (ii), since the implication (v) ⇒(i) is dual.

(v) ⇒ (ii) Let A ∈ ⊥nM. Since M is covariantly finite and contains the injectives, there

exists an exact sequence 0 −→ A −→ MA1 −→ MA

0 , where the MAi lie in M. Then we have

an exact sequence 0 −→ H(A) −→ H(MA1 ) −→ H(MA

0 ) −→ F −→ 0 in mod-M. Since gl.dim mod-M = n +2, it follows that pdF � n +2 and therefore pdH(A) � n. Hence we have an M-resolution 0 −→ Mn

A −→ Mn−1A −→ · · · −→ M1

A −→ M0A −→ A −→ 0 of A.

Applying to the extension 0 −→ K1A −→ M0

A −→ A −→ 0 the functor A (−, M), we see that K1

A ∈ ⊥n−1M and then by induction Kn−i

A ∈ ⊥i M, 1 � i � n − 1. In particular since

Kn−1A ∈ ⊥

1 M, the extension 0 −→ MnA −→ Mn−1

A −→ Kn−1A −→ 0 splits and therefore

Kn−1A ∈ M. Since Kn−2

A ∈ ⊥2 M, it follows that the extension 0 −→ Kn−1

A −→ Mn−2A −→

Kn−2A −→ 0 splits and therefore Kn−2

A ∈ M. Continuing in this way we see that the extension 0 −→ K1

A −→ M0A −→ A −→ 0 splits, hence A ∈ M. We infer that ⊥nM = M.

Assume now that (i) holds. By part 4. of Remark 8.11 we have A = M�n. By Proposition 9.2, with t = n, we have A = M�n. On the other hand since M is functorially finite in A it follows that M is coherent and then by [13] we have gl.dimM-mod =gl.dim mod-M = n + 2. Finally if M = addM , for some M ∈ A , then clearly M is a coherent generator–cogenerator of A . Since mod-EndA (M) ≈ mod-M, it follows that gl.dim mod- EndA (M) = n + 2 and therefore rep.dimA � n + 2. �


Example 9.7. (i) Let A be a cocomplete abelian category with enough projectives and injectives, and let M be a self-compact generator such that M⊥

n = AddM . Recall that Mis self-compact if the functor A (M, −) : AddM −→ Ab preserves coproducts. Our inter-est for self-compact objects, stems from the fact that M is self-compact if and only if the canonical functor mod-AddM −→ Mod-EndA (M), F �−→ F (M) is an equivalence, see [12, Proposition 8.4]. It follows by Remark 8.18 that M is an n-cluster tilting object, i.e. AddM is an n-cluster tilting subcategory, and the category mod- AddT ≈ Mod-EndA (M)is an n-Auslander category, i.e. EndA (M) is an n-Auslander ring. Also by Proposition 9.2, the subcategory AddM is n-cluster cotilting, so AddM is covariantly finite and in par-ticular closed under all small products, i.e. M is by definition product-complete.

(ii) Let A be an abelian R-linear category with enough injectives over a commutative ring R and let M be an object of A such that the R-module A (M, X) is finitely generated for any X ∈ A . If M⊥

n = addM and addM contains a projective generator of A , then M is an n-cluster tilting object and mod-EndA (M) is an n-Auslander category. This follows from the fact that addM is contravariantly finite in A .

The following observation shows that any finitely generated n-cluster tilting module M is product-complete and in particular Σ-pure-injective, i.e. any coproduct of copies of M is pure-injective.

Corollary 9.8. Let Λ be a ring and M a finitely generated n-cluster tilting Λ-module, n � 1. Then EndΛ(M) is a left coherent and right perfect n-Auslander ring and M is a product-complete, in particular Σ-pure-injective, Λ-module which is finitely presented as an EndΛ(M)-module.

Proof. Since finitely generated modules are self-compact, as in Example 9.7, M is product-complete and EndΛ(M) is a left coherent and right perfect n-Auslander ring. By a result of Krause and Saorin [49], product-completeness of M implies that M is Σ-pure-injective and finitely presented over EndΛ(M). �

We say that pairs {A , M} and {A ′, M ′}, consisting of Grothendieck categories Aand A ′ and n-cluster tilting objects M ∈ A and M ′ ∈ A ′, are cluster equivalent if the pairs {A , AddM} and {A ′, AddM ′} are cluster equivalent in the sense of Section 8. Note that in this case the Grothendieck categories A and A ′ are equivalent.

Theorem 9.9. For any n � 0, the map{A ,M

}�−→ Λ = EndA (M)

gives a bijective correspondence between (cluster/Morita) equivalence classes of

(I) Pairs {A , M} where A is a Grothendieck category and M ∈ A is a self-compact n-cluster tilting object.

(II) Left coherent and right perfect n-Auslander rings Λ.


Under this bijection we have an equivalence

A≈−→ (U-mod)op

where U = Proj Λ ⋂

Inj Λ and Λ = EndA (M).

Proof. (I) �→ (II) If A is a Grothendieck category and M is a self-compact n-cluster tilting object of A , then by Example 9.7, EndA (M) is an n-Auslander ring. Since M is self-compact, the product-preserving functor F : A −→ Mod-EndA (M), F (A) =HomA (M, A), induces an equivalence between AddM and Proj EndA (M). Let {Pi}i∈I

be a set of projective EndA (M)-modules. Then Pi = F (Xi) for some Xi ∈ AddM , ∀i ∈ I, and then

∏i∈I Pi =

∏i∈I F (Xi) ∼= F (

∏i∈I Xi). By Proposition 9.2, AddM is

covariantly finite, in particular closed under products, in A . Then ∏

i∈I Xi ∈ AddMand

∏i∈I Pi ∈ F (AddM) = Proj EndA (M). Therefore Proj EndA (M) is closed under

products, hence EndA (M) is left coherent and right perfect by Chase’s Theorem [28].(II) �→ (I) Let Λ be a left coherent and right perfect n-Auslander ring, so Mod-Λ is

an n-Auslander category. By Theorem 8.23 there is an abelian category A with enough injectives and an n-cluster tilting subcategory M such that mod-M ≈ Mod-Λ and the short exact sequence of abelian categories (8.1) takes the form

0 −→ mod-M −→ Mod-Λ R−→ A −→ 0

Then A is a quotient of Mod-Λ by the localizing subcategory mod-M and this implies that A is a Grothendieck category. Since ProjΛ = AddΛ, the equivalence ProjΛ ≈Projmod-M ≈ M induced by R shows that M = AddM , where M = R(Λ), and then Λ ∼= EndA (M). Hence M is an n-cluster tilting object of A and, since the functor mod-M −→ Mod-EndA (M) = Mod-Λ, F �−→ F (M) is an equivalence, we infer that Mis self-compact.

By Proposition 8.12 and Theorem 8.23 the map {A , M

}�−→ Λ = EndA (M) on

cluster/Morita equivalence classes (I) and (II) is a bijection, and the last part follows from Proposition 8.12. �Remark 9.10. It is shown in [14] that any cocomplete abelian category with a set of self-compact generators is a Grothendieck category. Since any n-cluster tilting object is clearly a generator, it follows that the condition “A is Grothendieck” in (I) above can be replaced by “A is cocomplete abelian”.

Remark 9.11. Let us call an object M in an additive category A with infinite coproducts an additive generator if A = AddM . Then the case n = 0 of Theorem 9.9 gives a bijective correspondence between Grothendieck categories with a self-compact additive generator and left coherent and right perfect 0-Auslander rings. As in Remark 9.10, in this correspondence Grothendieck categories can be replaced by cocomplete abelian categories.


For instance if Λ is a ring, then A = Proj Λ contains Λ as a self-compact additive generator. Hence ProjΛ is a Grothendieck category if and only if Λ is a left coherent and right perfect 0-Auslander ring. Since left coherent and right perfect rings are semipri-mary, see [50], we deduce easily the well-known result of Tachikawa [60] characterizing rings whose projective modules form a Grothendieck category: ProjΛ is a Grothendieck category if and only if Λ is a semiprimary QF-3 ring such that gl.dimΛ � 2 � dom.dim Λ.

10. Stable Auslander categories

Let M be an n-cluster tilting subcategory in an abelian category A and let mod-Mand M-mod be the corresponding n-Auslander categories of M. Our aim in this section is to study homological properties of the stable (co-)Auslander categories mod-M and M-mod associated to M. This includes the case n = 0, where A is abelian with injectives considered as a 0-cluster tilting subcategory of itself.

Throughout this section: we fix an abelian category A with enough projective and injective objects. We also fix a functorially finite subcategory M of A and assume that M contains the projectives and the injectives.

10.1. Recollements and duality

Under the above assumptions both the categories mod-M and M-mod are abelian and we have the fully faithful left exact functors

H : A −→ mod-M, H(A) = A (−, A)|M &

Hop : A −→ (M-mod)op, Hop(A) = A (A,−)|M

Consider the stable category M modulo projectives and the stable category M modulo injectives. We identify the category of coherent functors mod-M with the full subcate-gory of mod-M consisting of functors F admitting a presentation H(M1) −→ H(M0) −→F −→ 0, where the map M1 −→ M0 is epic. Similarly we identify the category M-modwith the full subcategory of M-mod consisting of all functors F admitting a presentation Hop(M0) −→ Hop(M1) −→ F −→ 0, where the map M1 −→ M0 is monic. Both cate-gories mod-M and M-mod are abelian because the category M has weak kernels and the category M has weak cokernels. In fact we have the following short exact sequences of abelian categories

0 −→ mod-M −→ mod-M −→ A −→ 0 &

0 −→ (M-mod)op −→ (M-mod)op −→ A −→ 0

induced by the exact adjoints R : mod-M −→ A ←− (M-mod)op : Rop of the Yoneda restriction functors H and Hop respectively. By Lemma 8.6 we have F ∈ mod-M if


and only if Hom(F, H(A)) = 0, for any object A ∈ A . Since M contains the injec-tives and M is covariantly finite, for any object A ∈ A there exists a monic left M-approximation A −→ MA, and then clearly Hom(F, H(A)) = 0, ∀A ∈ A , if and only if Hom(F, H(M)) = 0, ∀M ∈ M. We infer that if M contains the injectives, then F ∈ mod-M if and only if Hom(F, H(A )) = 0 if and only if Hom(F, H(M)) = 0 if and only if Hom(F, H(Inj A )) = 0. Dually since M contains the projectives, then F ∈ M-modif and only if Hom(F, Hop(A )) = 0 if and only if Hom(F, Hop(M)) = 0 if and only if Hom(F, Hop(ProjA )) = 0.

If U is a full subcategory of an abelian category C , then FacU, resp. SubU, denotes the full subcategory of C consisting of all factors, resp. subobjects, of an object from U. Recall that a triple (U, V, W) of full subcategories of C is called a TTF-triple and V a TTF-class, if (U, V) and (V, W) are torsion pairs in C . In this case the TTF-class V is an abelian category, the inclusion V −→ C is exact and admits a left and right adjoint, see [19, Chapter I] for details on torsion pairs and TTF-triples in abelian categories. Recall that the diagram (†) below consisting of abelian categories and additive functors is a recollement diagram, if:

Ai

Be

q

p

C

l

r

(†)

(i) (l, e, r) is an adjoint triple.(ii) (q, i, p) is an adjoint triple.(iii) The functors i, l, and r are fully faithful.(iv) Im i = Ker e.

To proceed further we need the following result which is of independent interest.

Proposition 10.1. Let U : B −→ A be an exact functor between abelian categories.

1. Assume that A and B have enough projectives. If U admits a fully faithful right adjoint R and R(Proj A ) ⊆ Proj B ⊆ Im R, then U admits a fully faithful left adjoint L and there exists a recollement situation

Ci∗

BU

i∗

i!A

L

R

where C = Ker U. Moreover we have L = L0R and there exists a TTF-triple in B:(FacR(ProjA ), C , SubR(A )

)


2. Assume that A and B have enough injectives. If U admits a fully faithful left adjoint L and L(Inj A ) ⊆ Inj B ⊆ Im L, then U admits a fully faithful right adjoint R and there exists a recollement situation

Cj∗

BU

j∗

j!A

L

R

where C = Ker U. Moreover we have R = R0L and there exists a TTF-triple in B:(Fac L(A ), C , Sub L(Inj A )

)

Proof. We prove only part 1. since the proof of 2. is completely dual.We define L = L0R : A −→ B to be the 0th left derived functor of R, so L(A) is defined

by the exact sequence R(P 1) −→ R(P 0) −→ L(A) −→ 0, where P 1 −→ P 0 −→ A −→ 0is a projective presentation of A in A . We show that L is a left adjoint of U. Let Abe in A as above and B be in B. Since B has enough projectives and ProjB ⊆ Im R, there exists a projective presentation R(Y ) −→ R(X) −→ B −→ 0, where X, Y ∈ A . Applying B(R(P i), −) and using that R preserves projectives, we have an exact se-quence B(R(P i), R(Y )) −→ B(R(P i), R(X)) −→ B(R(P i), B) −→ 0 which, since Ris fully faithful, is isomorphic to the exact sequence A (P i, Y ) −→ A (P i, X) −→B(R(P i), B) −→ 0. Since the functor U is exact and the functor R is fully faithful we have UR = IdA and an exact sequence Y −→ X −→ U(B) −→ 0 in A . Since the P i are projective we have an exact sequence A (P i, Y ) −→ A (P i, X) −→ A (P i, U(B)) −→ 0. This shows that we have an isomorphism B(R(P i), B) ∼= A (P i, U(B)). Using this iso-morphism, the exact sequence 0 −→ B(L(A), B) −→ B(R(P 0), B) −→ B(R(P 1), B) is isomorphic to the exact sequence 0 −→ B(L(A), B) −→ A (P 0, U(B)) −→ A (P 1, U(B)). Since the kernel of the last map is isomorphic to A (A, U(B)) we infer that we have an isomorphism B(L(A), B) ∼= A (A, U(B)). Clearly this isomorphism is natural in A ∈ A

and B ∈ B, so we have an adjoint pair (L, U) and therefore an adjoint triple (L, U, R).Since R is fully faithful, standard arguments show that so is L. Denote by i∗ : C −→ B

the inclusion of Ker U := C in B. We define functors i∗ and i! by the exact sequences of functors:

LU −→ IdB −→ i∗ −→ 0 & 0 −→ i! −→ IdB −→ RU

induced from the counit LU −→ IdB of the adjoint pair (L, U) and the unit IdB −→ RUof the adjoint pair (U, R). It is easy to see that in this way we obtain additive functors i∗ : B −→ C and i! : B −→ C and an adjoint triple (i∗, i∗, i!). Hence the diagram in the statement represents a recollement situation. By general facts on recollements there is a TTF-triple (Ker i∗, C = Ker U, Ker i!) in B. We show that

FacR(Proj A ) = Ker i∗ & SubR(A ) = Ker i!


Clearly if B lies in Ker i!, then the unit B −→ R(U(B)) is monic and therefore B lies in Sub R(A ). Conversely if B −→ R(A) is a monomorphism, then since the unit B −→R(U(B)) is the reflection of B in Im R, it follows that B −→ R(A) factors through the unit and therefore the latter is monic. Then its kernel i!(B) is zero and therefore B ∈ Ker i!. We infer that SubR(A ) = Ker i!.

Now let B ∈ Ker i∗. Then we have an epimorphism L(U(B)) −→ B −→ 0. Since by construction we have an exact sequence R(P 1) −→ R(P 0) −→ L(U(B)) −→ 0, where P 1 −→ P 0 −→ U(B) −→ 0 is a projective presentation of U(B) in A , it follows that we have an epimorphism R(P 0) −→ B −→ 0, so B lies in FacR(Proj A ). Conversely if R(P ) −→ B −→ 0 is exact, where P is projective in A , then applying the right exact functor LU we have an epimorphism LUR(P ) −→ LU(B) −→ 0 and therefore an epimorphism L(P ) −→ LU(B) −→ 0 since UR = IdA . Since by naturality the composition L(P ) −→ LU(B) −→ B is equal to the composition L(P ) −→ R(P ) −→ B and since clearly the map L(P ) −→ R(P ) is invertible because P is projective, we infer that L(P ) −→ LU(B) −→ B is epic and therefore the counit LU(B) −→ B is epic. This means by construction that i∗(B) = 0, so B ∈ Ker i∗. We infer that FacR(ProjA ) = Ker i∗. �

Returning to our default situation, let A be an abelian category with enough pro-jectives and injectives and M a functorially finite subcategory of A containing the projectives and the injectives. Then considering the abelian categories mod-M and (M-mod)op, it is clear that the assumptions of Proposition 10.1 hold for the exact left adjoint R : mod-M −→ A of the fully faithful functor H and for the exact right adjoint Rop : (M-mod)op −→ A of the fully faithful functor Hop. Hence by Proposition 10.1 we have the following consequence.

Proposition 10.2. Let A be an abelian category with enough projective and injective objects and M a functorially finite subcategory of A containing the projectives and the injectives. Then there are recollements:

mod-MΠ∗ mod-M R

Π∗

Π!A

T

H

&

(M-mod)op Π∗ (M-mod)op Rop

Π!

Π∗A

Hop

Top

and TTF-triples in mod-M and in (M-mod)op respectively:

(FacH(ProjA ), mod-M, SubH(A )

)&

(FacHop(A ), (M-mod)op, SubHop(Inj A )

)


Remark 10.3. We give more explicit descriptions of the functors Π∗, Π! : mod-M −→mod-M. Fix a projective presentation H(M1) −→ H(M0) −→ F −→ 0 of F ∈ mod-M.

• Let PM0 −→ M0 be an epimorphism, where PM0 is projective. Then the induced map M1 ⊕ PM0 −→ M0 is an epimorphism and it is easy to see that there is an exact sequence

H(M1 ⊕ PM0) −→ H(M0) −→ Π∗(F ) −→ 0

If K is defined by the exact sequence 0 −→ K −→ M1 ⊕ PM0 −→ M0 −→ 0 and if Mis 1-rigid, then clearly

Π∗(F ) = R1H(K) = Ext1A (−,K)|M

In particular if M ∈ M, then Π∗H(M) = M(−, M) and if M is 1-rigid, then Π∗H(M) =R1H(ΩM) ∼= M(−, M).

• Consider the exact sequence 0 −→ AF −→ M1 −→ M0 −→ R(F ) −→ 0 and let L = Im(M1 → M0). By diagram chasing it is not difficult to see that there exists an exact sequence

0 −→ Coker[H(M1) → H(L)] −→ F −→ HR(F )

−→ Ker[R1H(L) → R1H(M0)] −→ 0

Hence Π!(F ) = Coker[H(M1) → H(L)]. If M is 1-rigid, then clearly Coker[H(M1) →H(L)] = R1H(AF ) and Ker[R1H(L) → R1H(M0] = R1H(L). Hence in this case we have an isomorphism and an exact sequence

Π!(F ) = R1H(AF ) &

0 −→ Π!(F ) −→ F −→ HR(F ) −→ R1H(L) −→ 0

which describes the kernel and the cokernel of the unit of the adjoint pair (R, H). Note that if M is 2-rigid, then the exact sequence 0 −→ AF −→ M1 −→ L −→ 0 shows that R1H(L) ∼= R2H(AF ) = Ext2A (−, AF )|M.

Remark 10.4. It is not difficult to see that for the left adjoint T = L0H : A −→ mod-Mof the exact functor R : mod-M −→ A , we have the following:

LmT : A −→ mod-M, LmT(A) = Hom(−,Ωm+1(A))|M, ∀m � 1

In particular Im LmT ⊆ mod-M, ∀m � 1. If M is 1-rigid, then Ext1A (−, Ωm+2(?))M =R1HΩm+2(?) ∼= LmT(?), ∀m � 1, and there exists an exact sequence:

0 −→ R1H(Ω2A) −→ T(A) −→ H(A) −→ R1H(ΩA) −→ 0


As a consequence T is exact if and only if L1T = 0 if and only if L1T(A) =Hom(−, Ω2A)M = 0, ∀A ∈ A , if and only if for any object A ∈ A , any map M −→ Ω2A, where M ∈ M, factorizes through a projective object. In particular T is exact if gl.dim A � 2. We leave to the reader to check that the natural map T −→ H is a monomorphism when evaluated at objects A with pdA � 1, and is an isomorphism if in addition Ext1(M, P ) = 0, ∀M ∈ M and P ∈ ProjA , for instance if M is 1-rigid.

Hence if M is functorially finite and ProjA ⊆ M ⊇ InjA , then there are identifications of TTF classes:

mod-M = Tors(mod-M) & M-mod = Tors(M-mod) (10.1)

where

Tors(mod-M) ={F ∈ mod-M | Hom(F,H(M)) = 0

}&

Tors(M-mod) ={F ∈ M-mod | Hom(F,Hop(M)) = 0

}

are the full subcategories of mod-M and M-mod respectively consisting of the torsion functors, i.e. those functors annihilated by the M-dual functors Hom(−, H(M)) and/or Hom(−, Hop(M)), ∀M ∈ M.

Definition 10.5. Let A be an abelian category with enough projective and injective objects and M an n-cluster tilting subcategory of A . The TTF-class mod-M in the Auslander category mod-M is called the stable n-Auslander category of A or M, and the TTF-class M-mod in the coAuslander category M-mod is called the stable n-coAuslander category of A or M.

For instance if M is an n-cluster tilting module over an Artin algebra Λ, then the stable Auslander category mod-M, where M = addM , is the category mod-EndΛ(M)of finitely generated modules over the stable n-Auslander algebra EndΛ(M) and dually (M-mod)op ≈ mod-EndΛ(M)op.

To proceed further we shall need the following useful result which shows that there is a duality between the stable n-Auslander category and the stable n-coAuslander category. This result is due to Iyama and proved in [39, Theorem 3.6.1] in a more restrictive context, see also Buchweitz’s treatment in [26] for a special case and [34, Section 3]. For the convenience of the reader we give the proof in our context.

Proposition 10.6. Let A be an abelian category with enough projective and injective objects and let M be an n-cluster tilting subcategory of A . Then the adjoint on the right pair (dr, dl) of M-dual functors

dr = Hom(−,H(?)) : mod-M −→ M-mod &

dl = Hom(−,Hop(?)) : M-mod −→ mod-M


induce quasi-inverse dualities between the stable n-(co)Auslander categories:

Rn+2dr : mod-M≈−→←−≈

(M-mod)op : Rn+2dl

Proof. Let H(M1) −→ H(M0) −→ F −→ 0 be a presentation of F ∈ mod-M, where the sequence 0 −→ AF −→ M1 −→ M0 −→ 0 is exact in A . Since M is n-cluster tilting, there exists an M-resolution 0 −→ Mn

AF−→ Mn−1

AF−→ · · · −→ M0

AF−→ AF −→ 0

of AF . Applying H we have a deleted projective resolution

0 −→ H(MnAF

) −→ H(Mn−1AF

) −→ · · ·

−→ H(M0AF

) −→ H(M1) −→ H(M0) −→ 0

of F . Applying dr and using that drH = Hop, we have a complex

0 −→ Hop(M0) −→ Hop(M1) −→ Hop(M0AF

) −→ · · ·−→ Hop(Mn−1

AF) −→ Hop(Mn

AF) −→ 0

whose cohomology gives the derived functors Rkdr(F ). Since M is n-rigid and dr(F ) = 0, it follows that the above complex is acyclic everywhere except in the last position on the right which corresponds to (n +2)-degree. Hence the above complex is a deleted projective resolution of Rn+2dr(F ). This functor is coherent and lies in M-mod since it has a presen-tation Hop(Mn−1

AF) −→ Hop(Mn

AF) −→ Rn+2dr(F ) −→ 0 where the map Mn

AF−→ Mn−1

AF

is a monomorphism. Hence we have a contravariant functor Rn+2dr : mod-M −→ M-mod. Applying dl to the deleted projective resolution of Rn+2dr(F ) constructed above, we obtain the original deleted projective resolution of F and this means that we have a nat-ural isomorphism F ∼= Rn+2dlRn+2dr(F ). By symmetry we have a natural isomorphism G ∼= Rn+2drRn+2dl(G), ∀G ∈ M-mod. Hence Rn+2dr is a duality with quasi-inverse Rn+2dl. �

In the sequel we shall use the fact that if M is n-rigid and ProjA ⊆ M ⊇ Inj A , then we have in particular ExtkA (M, P ) = 0 = Extk(I, M), 1 � k � n for M ∈ M, P ∈ ProjAand I ∈ Inj A . As a consequence of [4, Proposition 2.46] and its dual, we have canonical isomorphisms, ∀M ∈ M and ∀X ∈ A :

ExtkA (M,X) ∼=−−→ A (ΩkM,X) &

A (M,X) ∼=−−→ A (ΩkM,ΩkX), 1 � k � n (10.2)

ExtkA (X,M) ∼=−−→ A (X,ΣkM) &

A (X,M) ∼=−−→ A (ΣkX,ΣkM), 1 � k � n (10.3)

On the other hand if M is an n-cluster tilting subcategory of A , then any object A ∈ A has an M-resolution and an M-coresolution of length � n:


0 −→ MnA −→ Mn−1

A −→ · · · −→ M1A −→ M0

A −→ A −→ 0 (10.4)

0 −→ A −→ MA0 −→ MA

1 −→ · · · −→ MAn−1 −→ MA

n −→ 0 (10.5)

10.2. The subcategories M�1 and M�1

Let M be a functorially finite 1-rigid subcategory of A . We analyze the structure of the full subcategories M�1 and M�1 of A appearing in Propositions 9.2 and 9.3 in connection with the categories mod-M and M-mod.

Recall that a functor F in mod-M, resp. in M-mod, is called torsion, if dr(F ) = 0, resp. dl(F ) = 0. We denote by Tors(mod-M), resp. Tors(M−mod), the full subcategory of mod-M, resp. M-mod, consisting of the torsion functors. By an observation of Auslander and Reiten, see [7, Lemma 4.1], and using that any map between torsion functors which factorizes through a projective functor is clearly zero, it follows that the Auslander–Bridger transpose duality functors Tr : M-mod −→ mod-M and Tr : mod-M −→ M-mod, see Subsection 7.1, induce dualities

Proj�1M-mod ≈ Tors(mod-M)op & Proj�1mod-M ≈ Tors(M-mod)op (10.6)

Combining (10.1) and (10.6), we have dualities

(Proj�1M-mod)op ≈ mod-M & (Proj�1mod-M)op ≈ M-mod (10.7)

Next observe that the functors Hop and H clearly induce equivalences M�1 ≈(Proj�1 M-mod)op and M�1 ≈ Proj�1 mod-M respectively, which in turn induce equiva-lences

M�1

M≈−−→ (Proj�1M-mod)op & M�1

M≈−−→ Proj�1mod-M (10.8)

Summarizing the above observations, we have the following result, part (i) of which gives an alternative description of the stable n-Auslander categories. Note also that part (i) gives, in our context, a different and shorter proof of a result due to Demonet and Liu, see [30, Theorem 3.2] in the setting of exact categories, and part (ii) follows from (i) and Proposition 10.6.

Theorem 10.7. Let A be an abelian category with enough projective and injective objects and M a functorially finite n-rigid subcategory of A containing the projectives and the injectives.

(i) The functors H : A −→ mod-M and Hop : A −→ (M-mod)op and the duality Tr : mod-M −→ M-mod induce equivalences:


Tr ◦Hop : M�1

M=

⊥n−1M

M≈−−→ mod-M &

Tr ◦H :M⊥

n−1M

= M�1

M≈−−→ (M-mod)op

In particular the category M�1/M is abelian with enough projectives and the category M�1/M is abelian with enough injectives.

(ii) If M is an n-cluster tilting subcategory of A , then there are equivalences

R ◦ Tr ◦ Rn+2dr ◦ Tr ◦Hop : M�1

M=

⊥n−1M

M

≈−→←−≈

M⊥n−1M

= M�1

M: Rop ◦ Tr ◦ Rn+2dl ◦ Tr ◦H

and the involved abelian categories have enough projectives and enough injectives.

It should be noted that in case M is an n-cluster tilting subcategory of A , then the equivalences of Theorem 10.7 can be suitably generalized to include the subcategories M�t and M�t, for 2 � t � n, by using, in the Frobenius case, Proposition 2.13 and [44, Theorem 6.3]. Details are left to the interested reader.

10.3. Projective objects in mod-M and M-mod

Let M be a functorially finite 1-rigid subcategory of A . We are interested in the description of the projective objects of mod-M, resp. M-mod, inside the category mod-M, resp. M-mod.

Let F ∈ mod-M with presentation H(M1) −→ H(M0) −→ F −→ 0, where the map M1 −→ M0 is an epimorphism in M. Applying the left exact functor H to the short exact sequence 0 −→ AF −→ M1 −→ M0 −→ 0 and using that M is 1-rigid, so that R1H(M) = Ext1(−, M)|M = 0, we have the exact sequence

0 −→ H(AF ) −→ H(M1) −→ H(M0) −→ R1H(AF ) −→ 0 (10.9)

and therefore F = R1H(AF ). Hence any object F ∈ mod-M is of the form R1H(AF ) for some object AF which is a kernel of an epimorphism in M. Now let M ∈ M and let 0 −→ ΩM −→ PM −→ M −→ 0 be exact, where PM is projective. Then we have the following exact sequence

0 −→ H(ΩM) −→ H(PM ) −→ H(M) −→ R1H(ΩM) −→ 0 (10.10)

showing that the functor R1H(ΩM) lies in mod-M and, using (10.2), we have isomor-phisms


R1H(ΩM) = Ext1(−,ΩM)|M∼=−−→ A (Ω(−),ΩM)|M

∼=−−→ A (−,M)|M = M(−,M)

showing that the functor R1H(ΩM) is projective in mod-M and any projective in mod-Mis of this form. Note that by Remark 10.3 we have Π∗H(M) = R1H(ΩM), ∀M ∈ M.

Let F = R1H(AF ) be an object in mod-M. Let 0 −→ ΩM0 −→ PM0 −→ M0 −→ 0be an exact sequence, where PM0 is projective, and consider the pull-back short exact sequence 0 −→ ΩM1 −→ ΩM0 −→ AF −→ 0 of the projective presentation of M0 along the monomorphism AF −→ M0. Then applying the left exact functor H to this short exact sequence we have a long exact sequence:

0 −→ H(ΩM1) −→ H(ΩM0) −→ H(AF ) −→ R1H(ΩM1)

−→ R1H(ΩM0) −→ R1H(AF ) −→ R2H(ΩM1) −→ · · · (10.11)

Clearly the map R1H(AF ) −→ R2H(ΩM1) factorizes through R1H(M1) = Ext1(−, M1)|Mwhich is zero since M is 1-rigid. Therefore (10.11) becomes an exact sequence

0 −→ H(ΩM1) −→ H(ΩM0) −→ H(AF )

−→ R1H(ΩM1) −→ R1H(ΩM0) −→ R1H(AF ) −→ 0 (10.12)

and as noted above the objects R1H(ΩM i) are projective in mod-M. This proves the following.

Lemma 10.8. The abelian category mod-M has enough projectives and:

Projmod-M ={Ext1A (−,ΩM)|M | M ∈ M

}=

{R1H(ΩM) | M ∈ M

}=

{M(−,M)| M ∈ M

}= Π∗H(M)

If F ∈ mod-M and H(M1) −→ H(M0) −→ F −→ 0 is a finite projective presentation of F in mod-M, where the sequence 0 −→ AF −→ M1 −→ M0 −→ 0 is exact in A , then the exact sequence

R1H(ΩM1) −→ R1H(ΩM0) −→ F −→ 0

is a projective presentation of F in mod-M.

Working with covariant coherent functors, by duality we have the following

Lemma 10.9. The abelian category M-mod has enough projectives and:

ProjM-mod ={Ext1A (ΣM,−)|M | M ∈ M

}=

{R1Hop(ΣM) | M ∈ M

}=

{M(M,−)| M ∈ M

}


If F ∈ M-mod and Hop(M0) −→ Hop(M1) −→ F −→ 0 is a projective presentation of Fin M-mod, where the sequence 0 −→ M1 −→ M0 −→ AF −→ 0 is exact in A , then the exact sequence

R1Hop(ΣM0) −→ R1Hop(ΣM1) −→ F −→ 0

is a projective presentation of F = R1Hop(AF ) in M-mod.

10.4. Injective objects in mod-M and in M-mod

From now on and until the rest of this section, we assume that M is an n-cluster tilting subcategory of A . We are interested in the description of the injective objects of mod-M, resp. M-mod, inside the category mod-M, resp. M-mod.

Let M be in M and let 0 −→ M −→ IM −→ ΣM −→ 0 be exact, where IM is injective. Consider, as in (10.5), an M-coresolution 0 −→ ΣM −→ MΣM

0 −→ MΣM1 −→ · · · −→

MΣMn −→ 0 of ΣM , i.e. each MΣM

k is a left M-approximation of Im(MΣMk−1 −→ MΣM

k ). Hence we have an exact sequence

0 −→ M −→ IM −→ MΣM0 −→ MΣM

1 −→ · · · −→ MΣMn −→ 0 (10.13)

Applying the left exact functor H and using that M is n-rigid, it follows that we have an exact sequence

0 −→ H(M) −→ H(IM ) −→ H(MΣM0 ) −→ · · ·

−→ H(MΣMn ) −→ Rn+1H(M) −→ 0 (10.14)

which is a projective resolution of Rn+1H(M) in mod-M. Since the map MΣMn−1 −→ MΣM

n

is an epimorphism, we infer that Rn+1H(M) lies in mod-M. We show that this functor is injective in mod-M, by invoking Proposition 10.6. Indeed by Proposition 10.6 it follows that:

• The functor Rn+2dr : Inj mod-M −→ ProjM-mod is an equivalence with quasi-inverse Rn+2dl.

• The functor Rn+2dl : InjM−mod −→ Proj mod-M is an equivalence with quasi-inverse Rn+2dr.

Therefore, using Lemma 10.9, any injective object of mod-M is of the formRn+2dlR1Hop(ΣM), M ∈ M. In order to compute Rn+2dlR1Hop(ΣM), we apply Hop

to the exact sequence (10.13) to obtain the complex

0 −→ Hop(MΣMn ) −→ Hop(MΣM

n−1) −→ · · ·−→ Hop(MΣM

0 ) −→ Hop(IM ) −→ Hop(M) −→ 0


which is a deleted projective resolution of R1Hop(ΣM). Applying dl we obtain the exact complex (10.14), so

Rn+2dl(R1Hop(ΣM)) ∼= Rn+1H(M)

This proves the following result.

Lemma 10.10. The abelian category mod-M has enough injectives and

Inj mod-M ={Extn+1

A (−,M)|M | M ∈ M}

={Rn+1H(M) | M ∈ M

}=

{M(Ωn(−),M)|M | M ∈ M

}

If F ∈ mod-M and H(M1) −→ H(M0) −→ F −→ 0 is a projective presentation of Fin mod-M, where the sequence 0 −→ AF −→ M1 −→ M0 −→ 0 is exact in A , then the exact sequence

0 −→ F −→ Rn+1H(MnAF

) −→ Rn+1H(Mn−1AF

)

is an injective copresentation of F = R1H(AF ) in mod-M.

By duality we have the following.

Lemma 10.11. The abelian category M-mod has enough injectives and

InjM-mod ={Extn+1

A (M,−)|M | M ∈ M}

={Rn+1Hop(M) | M ∈ M

}=

{M(M,Σn(−))|M | M ∈ M

}

If F ∈ M-mod and Hop(M0) −→ Hop(M1) −→ F −→ 0 is a projective presentation of Fin M-mod, where the sequence 0 −→ M1 −→ M0 −→ AF −→ 0 is exact in A , then the exact sequence

0 −→ F −→ Rn+1Hop(MAF

n ) −→ Rn+1Hop(MAF

n−1)

is an injective copresentation of F = R1Hop(AF ) in M-mod.

10.5. mod-M as an Auslander category

Let M ∈ M and 0 −→ M −→ I0 −→ · · · −→ In+1 −→ · · · be an injective resolution of M in A . Since M is n-rigid, the sequence 0 −→ H(M) −→ H(I0) −→ · · · −→ H(In) −→H(In+1) is exact. We set EM := Coker(H(In) → H(In+1)), so EM = Σn+2H(M), and we have an exact sequence

0 −→ H(M) −→ H(I0) −→ · · · −→ H(In) −→ H(In+1) −→ EM −→ 0 (10.15)


Considering the full subcategory EM ={EM ∈ mod-M | M ∈ M

}of mod-M consisting

of all objects EM arising in this way, we have inclusions:

EM ⊆ EM ⊕H(InjA ) ⊆ Inj B, where B = mod-M

Indeed let EM be in EM, so there is an M ∈ M and an exact sequence (10.15). Since each H(Ik) is injective and gl.dimB � n +2, it follows that EM is injective in B. Hence add{H(I) ⊕ EM | I ∈ Inj B & EM ∈ EM} := EM ⊕ H(InjA ) ⊆ Inj B since H(I) is projective–injective in B, ∀I ∈ Inj A .

Remark 10.12. 1. If B = mod-M is an n-Auslander category and B has enough injectives, then using that dom.dim B � m if and only if codom.dim B � m, see [38, Lemma 2.1], it follows that Inj B = EM ⊕H(Inj A ).

Indeed let E be an injective object of mod-M and assume that E is non-projective, so E /∈ ImH. Since dom.dim B � n +2, by Remark 8.20 it follows that codom.dim B � n +2, and this means that there exists an exact sequence Pn+1 −→ Pn −→ · · · −→ P 1 −→P 0 −→ E −→ 0 in B, where each P k is projective–injective, hence of the form H(Ik)for some Ik ∈ Inj A . Since gl.dimB � n + 2, the kernel of Pn+1 −→ Pn is projective, hence of the form H(M), for some M ∈ M. Summarizing we have an exact sequence

0 −→ H(M) −→ H(In+1) −→ · · · −→ H(I1) −→ H(I0) −→ E −→ 0

Since H is fully faithful and M is n-rigid, we have that the sequence

0 −→ M −→ In+1 −→ · · · −→ I1 −→ I0

is exact. This means that E = EM for M ∈ M and therefore E ∈ EM. On the other hand if E is projective–injective, then E = H(I) for some I ∈ Inj A . We infer that Inj B = EM ⊕H(InjA ).

2. Assume that B = mod-M has injective envelopes. This is the case, for instance, if A is cocomplete and M consists of self-compact objects since then mod-M ≈ Mod-M. Since mod-M is a localizing subcategory of mod-M by [54] it follows that any injective object of mod-M is of the form E = E1 ⊕E2 where E1 is the injective envelope E(F ) of the largest subobject F of E which lies in mod-M and E2 lies in SubH(A ). However the largest subobject of E lying in mod-M is the functor Π!(E) which is injective in mod-Msince Π! preserves injectives as a right adjoint of the exact functor Π∗. Then Lemma 10.10gives Π!(E) = Rn+1H(M) for some M ∈ M and then E1 = E(Rn+1H(M)). Since clearly E2 is of the form H(I) for some injective object I of A we infer that

Inj mod-M = E(Rn+1H(M)) ⊕H(InjA )

={E(Rn+1H(M)) ⊕H(I) ∈ mod-M | M ∈ M & I ∈ Inj A

}In particular if the hereditary torsion pair (mod-M, Sub(H(A )) is stable, that is mod-Mis closed under injective envelopes in mod-M, then Inj mod-M = Rn+1H(M) ⊕H(InjA ).


The following generalizes a basic result of Iyama, see [42, Proposition 4.2].

Theorem 10.13. The map {A , M} �−→ mod-M gives a bijection between (cluster) equiv-alence classes of:

(i) Pairs {A , M} where A is an abelian category with gl.dimA � n + 1 and M is an n-cluster tilting subcategory of A .

(ii) Auslander categories B such that

ExtkB(E,P ) = 0, 1 � k � n + 1, ∀E ∈ EM & ∀P ∈ Proj B

Under this bijection the n-Auslander category B = mod-M has enough injectives, EM =Rn+1H(M), and

Inj B = Rn+1H(M) ⊕H(Inj A )

Proof. By Theorem 8.23, Lemmas 10.8 and 10.10, in order to establish the bijective correspondence between (i) and (ii), it suffices to show that if M is an n-cluster tilting subcategory of A , then, ∀M, M ′ ∈ M:

gl.dimA � n + 1 ⇐⇒ Extk(EM ,H(M ′)

)= 0, 1 � k � n + 1

“=⇒” Assume that gl.dim A � n +1. Let M ∈ M and consider an injective resolution of M :

0 −→ M −→ I0 −→ I1 −→ · · · −→ In −→ In+1 −→ 0

Applying the functor H and using that M is n-rigid, we have the following exact complex

0 −→ H(M) −→ H(I0) −→ H(I1) −→ · · ·−→ H(In) −→ H(In+1) −→ EM −→ 0 (10.16)

which is a projective resolution of EM in mod-M. Note that the object EM is torsion since the map In −→ In+1 is an epimorphism, in fact EM

∼= Rn+1H(M). Applying to this resolution the functor Hom(−, H(M ′)), M ′ ∈ M, we have the following complex

0 −→ (H(In+1),H(M ′)) −→ · · · −→ (H(I1),H(M ′))

−→ (H(I0),H(M ′)) −→ (H(M),H(M ′)) −→ 0

which, using that H is fully faithful, is isomorphic to the complex

0 −→ A (In+1,M ′) −→ A (In,M ′) −→ · · · −→ A (I1,M ′)

−→ A (I0,M ′) −→ A (M,M ′) −→ 0 (10.17)


whose cohomologies give Ext∗(EM , H(M ′)). Since EM is torsion, Inj A ⊆ M and M is n-rigid, this complex is acyclic everywhere except in the last position on the right which corresponds to (n + 2)-degree where the cohomology is Ext1(ΣM, M ′). We infer that Extk(EM , H(M ′)) = 0, 1 � k � n + 1, M, M ′ ∈ M.

“⇐=” Assume the vanishing condition: Extk(EM , H(M ′)) = 0, 1 � k � n + 1, ∀M, M ′ ∈ M.

Consider an injective resolution of M ′:

0 −→ M ′ −→ J0 −→ J1 −→ · · · −→ Jn −→ Jn+1 −→ · · ·

Then, as before, n-rigidity of M implies that the following complex is exact

0 −→ H(M ′) −→ H(J0) −→ H(J1) −→ · · ·

−→ H(Jn) −→ H(Jn+1) −→ EM ′ −→ 0 (10.18)

and therefore it is an injective resolution of H(M ′) in mod-M, since EM ′ is injective and the objects H(Jk) are projective–injective. Applying the functor Hom(EM , −) to (10.18)we have the following complex

0 −→ Hom(EM ,H(J0)) −→ Hom(EM ,H(J1)) −→ · · ·

−→ Hom(EM ,H(Jn+1)) −→ Hom(EM , EM ′) −→ 0 (10.19)

whose kth cohomology gives Extk(EM , H(M ′)), 0 � k � n +1. By hypothesis the complex (10.19) is acyclic, except possibly in degrees 0 and n + 2, where the cohomologies are Hom(EM , H(M ′)) and Extn+2(EM , H(M ′)) respectively. Using the adjoint pair (R, H), it follows that the complex (10.19) is isomorphic to the complex

0 −→ Hom(R(EM ), J0) −→ Hom(R(EM ), J1) −→ · · ·

−→ Hom(R(EM ), Jn+1)) −→ Hom(EM , EM ′) −→ 0 (10.20)

Clearly the kth cohomology, for 0 � k � n, of the complex (10.20) computes Extk(R(EM ), M ′). Applying the exact functor R to (10.15) and using that RH = IdA , we have an exact sequence In −→ In+1 −→ R(EM ) −→ 0 which shows that R(EM ) =Σn+2M . We infer that Extk(Σn+2M, M ′) = 0, 1 � k � n. Hence Σn+2M ∈ ⊥

nM = M. On the other hand Extn+1(EM , H(M ′)) = Ext1(EM , ΣnH(M ′)) = 0, so the complex (10.19) is exact in degree n + 1, and this implies by diagram chasing that the sequence 0 −→ Hom(EM , H(Σn(M ′)) −→ Hom(EM , H(Jn)) −→ Hom(EM , H(Σn+1(M ′)) −→ 0 is exact. Using the adjoint pair (R, H) and the fact that R(EM ) = Σn+2M , we have the following exact commutative diagram


0 Hom(EM ,H(Σn(M ′))

∼=

Hom(EM ,H(Jn))

∼=

0 A (Σn+2M,Σn(M ′)) A (Σn+2M,Jn)

Hom(EM ,H(Σn+1(M ′))

∼=

0

A (Σn+2M,Σn+1(M ′)) Ext1A (Σn+2M,ΣnM ′) 0

which shows that Ext1(Σn+2M, ΣnM ′) ∼= Extn+1(Σn+2M, M ′) = 0. Since this happens for any M ′ ∈ M and since Σn+2M already lies in ⊥nM, we infer that Σn+2M ∈ ⊥

n+1M. Since by Remark 8.11, ⊥n+1M = ProjA , we infer that Σn+2M is projective in A . Then the exact sequence 0 −→ Σn+1M −→ In+1 −→ Σn+2M −→ 0 splits and therefore Σn+1M is injective as a direct summand of In+1. Hence idM � n + 1, for any object Mof M. Finally let A ∈ A and consider an M-resolution 0 −→ Mn

A −→ Mn−1A −→ · · · −→

M1A −→ M0

A −→ A −→ 0 of A. Since idM � n + 1, we have directly by dimension shifting that idA � n + 1 and therefore gl.dimA � n + 1.

We saw in the proof of the direction “=⇒” that if gl.dimA � n +1, then the injective object EM is isomorphic to the object Rn+1H(M) which lies, and is injective, in mod-Mby Lemma 10.8. Now let F ∈ mod-M. Since, by Proposition 10.2, (mod-M, SubH(A )) is a torsion pair in mod-M, there exists an exact sequence 0 −→ XF −→ F −→ Y F −→ 0, where XF lies in mod-M and there is a monomorphism Y F −→ H(A) for some A ∈ A . If A −→ I is a monomorphism, where I is injective in A , then Y F −→ H(I) is a monomorphism from Y F to a (projective–)injective in mod-M. Since mod-M has enough injectives and any injective object of mod-M is of the form Rn+1H(M), for M ∈ M, there exists a monomorphism XF −→ Rn+1H(M) and as noted above Rn+1H(M) is injective in mod-M. Then by standard arguments we may construct a monomorphism F −→ Rn+1H(M) ⊕H(I) and Rn+1H(M) ⊕H(I) is injective in mod-M. Hence mod-M has enough injectives and any injective of mod-M is a direct summand of Rn+1H(M) ⊕H(I), where M ∈ M and I ∈ Inj A . �Theorem 10.14. Let A be an abelian category with enough projective and injective objects and let M be an n-cluster tilting subcategory of A . Then we have the following.

(i) gl.dim A � n + 1 =⇒ gl.dimmod-M � n + 2.(ii) gl.dim A � n +1 � dom.dim A =⇒ gl.dim mod-M � n +2 � dom.dim mod-M.

In particular let A be an (n − 1)-Auslander category and M an n-cluster tilting sub-category M. If ProjA is covariantly finite in A , then the stable n-Auslander category mod-M is an n-Auslander category.


Proof. (i) Since Inj mod-M = Rn+1H(M) and since by (the proof of) Theorem 10.13 any object of the form Rn+1H(M), M ∈ M, is isomorphic to EM ∈ Inj mod-M, we infer that Inj mod-M ⊆ Inj mod-M. This means that the inclusion Π∗ : mod-M −→ mod-Mpreserves injectives. Since mod-M has enough injectives, this implies that its left adjoint Π∗ : mod-M −→ mod-M is exact. Let F ∈ mod-M and consider a projective resolution

0 −→ H(Mn+2) −→ H(Mn+1) −→ · · · −→ H(M1) −→ H(M0) −→ Π∗(F ) −→ 0

of Π∗(F ) in mod-M. Applying Π∗ and using that Π∗Π∗ = Idmod-M and Π∗ preserves projectives since its right adjoint Π∗ is exact, we obtain a projective resolution

0 −→ Π∗H(Mn+2) −→ Π∗H(Mn+1) −→ · · ·−→ Π∗H(M1) −→ Π∗H(M0) −→ F −→ 0

of F in mod-M and therefore pdF � n + 2. We infer that gl.dimmod-M � n + 2.(ii) Now assume in addition that dom.dim A � n + 1. Let F be a projective ob-

ject of mod-M, so F = R1H(ΩM) = M(−, M) = Π∗H(M), for some M ∈ M. Since dom.dim mod-M � n +2, there exists an exact sequence in mod-M, where EM ∈ mod-M:

0 −→ H(M) −→ H(I0) −→ · · · −→ H(In) −→ H(In+1) −→ EM −→ 0

where each Ik is injective in A . Applying the exact functor Π∗ we have an exact sequence

0 −→ F −→ Π∗H(I0) −→ · · ·−→ Π∗H(In) −→ Π∗H(In+1) −→ EM −→ 0 (∗)

We show that the functor Π∗H(I) = Ext1A (−, ΩI)|M = M(−, I) is (projective–)injective in mod-M, for any injective object I ∈ A . Since dom.dim A � n + 1, it follows by [38, Lemma 2.1] that codom.dimA � n + 1 and this means that for the injective object Ithere exists an exact sequence

0 −→ Q −→ Pn −→ Pn−1 −→ · · · −→ P 1 −→ P 0 −→ I −→ 0

where the P k are projective–injective in A , 0 � k � n. This means that ΩI = ΣnQ

(up to injective summands). Since gl.dimA � n + 1, it follows that Q is projective, in particular Q ∈ M, and then

Π∗H(I) = Ext1A (−,ΩI)|M ∼= Ext1A (−,ΣnQ)|M ∼= Extn+1A (−, Q)|M = Rn+1H(Q)

Hence Π∗H(I) is projective–injective in mod-M. Then the exact sequence (∗) shows that dom.dim mod-M � n + 2. In order to prove that mod-M is an n-Auslander category, it remains to show that U := Proj mod-M ∩ Inj mod-M or equivalently by Remark 8.13,


Projmod-M, is covariantly finite in mod-M. Let F be in mod-M with projective presen-tation M(−, M1) −→ M(−, M0) −→ F −→ 0. Consider the map M1 −→ M0 in A and let M1 −→ P be a left projective approximation of M1 in A . Then consider the exact sequence M1 −→ M0 ⊕ P −→ X −→ 0 and let X −→ MX be a left M-approximation of X. Then it is easy to see that the map M0 −→ MX is a weak cokernel of M1 −→ M0

in M. Since the composition M(−, M1) −→ M(−, M0) −→ M(−, MX) is clearly zero in mod-M, it follows that the map M(−, M0) −→ M(−, MX) factorizes uniquely through F , say via a map F −→ M(−, MX). It is easy to see that this map is in fact a left Projmod-M-approximation of F , so Projmod-M is covariantly finite in mod-M. We infer that mod-M is an n-Auslander category. �Remark 10.15. Keeping the assumptions of Theorem 10.14, it follows from its proof that any object of the form M(−, I), I ∈ Inj A , is projective–injective in mod-M. Now let Gbe a projective–injective object of mod-M. By projectivity G = M(−, M) = R1H(ΩM)for some M ∈ M. Let M −→ I be a monomorphism in A where I is injective. Then we have a monomorphism H(M) −→ H(I) in mod-M and therefore a monomorphism Π∗H(M) −→ Π∗H(I) in mod-M since the functor Π∗ is exact. Since this monomorphism is isomorphic to M(−, M) −→ M(−, I) and the object M(−, M) is injective, it follows that G is a direct summand of M(−, I). This implies that M is a direct summand of I in A and then clearly M ∈ Inj A . We infer that U := Projmod-M ∩ Inj mod-M ={M(−, I) ∈ mod-M | I ∈ A } = Π∗H(InjA ) and we have an equivalence

InjA ≈−−→ U = Projmod-M ∩ Inj mod-M, I �−→ M(−, I)

Let A ! be the abelian category (U-mod)op ≈ (InjA -mod)op and set N :=mod-M(Projmod-M, −)|U ≈ (M, −)|InjA . By Theorem 8.23, N is an n-cluster tilting subcategory of A ! and its n-Auslander category is equivalent to mod-N ≈ mod-M, in particular N ≈ M. Note that the projection InjA −→ Inj A induces by restriction a full exact embedding of abelian categories (InjA -mod)op = A ! −→ A = (Inj A -mod)op.

10.6. The Gorenstein property

We close this section by studying the Gorenstein property of stable n-Auslander cat-egories mod-M associated to an n-cluster tilting subcategory M of A .

We begin with the following.

Proposition 10.16. Let A be an abelian category with enough projective and injective objects, and let M be an n-cluster tilting subcategory of A . If G-dim A � n, then A is Frobenius and:


(i) If n = 1, then G-dim mod-M � 1.(ii) If n � 2 and Ext1A (M, ΩiM) = 0, where 2 � i � n − k + 1 and 0 � k � n+1

2 , then

G-dimmod-M � k

Proof. Since G-dimA � n, it follows that A is Gorenstein and, by Remark 6.3, GProj Aconsists of all objects A such that ExtkA (A, P ) = 0, for any projective object P and for any k � 1, or equivalently for 1 � k � n since the injective dimension of any projective object is at most n. Since M is n-rigid and M contains the projectives of A , it follows that M ⊆ GProjA . Then clearly M is an (n + 1)-cluster tilting subcategory of the triangulated category GProjA . Since any injective object has projective dimension at most n and since any injective is Gorenstein-projective because InjA ⊆ M ⊆ GProjA , it follows that Inj A ⊆ Proj<∞ A ∩ GProj A = ProjA , hence any injective is projective. If P is projective, let 0 −→ P −→ I −→ ΣP −→ 0 be exact where I is injective. Since Phas finite injective dimension (bounded by n), we have that also ΣP has finite injective dimension. Since any injective is projective, it follows that ExtkA (E, ΣP ) = 0, ∀k � 1and for any injective object E. It follows that ΣP is Gorenstein-injective, and then it is injective since it has finite injective dimension. Then the above sequence splits and P is injective as a direct summand of I. Hence any projective is injective and A is Frobenius.

If n = 1, then M is a 2-cluster tilting subcategory of GProjA and therefore G-dim mod-M � 1, by Theorem 6.4(i). If n � 2, then the vanishing condition Ext1(M, ΩiM) = 0, where 2 � i � n −k+1, implies that the (n +1)-cluster tilting subcate-gory M of A is (n −k)-corigid, since Ext1A (M, ΩiM) ∼= A (ΩM, ΩiM) ∼= A (M, Ωi−1M) =A (M, Σ−i+1M). Then by Theorem 6.4(iii) we have G-dim mod-M � k. �Theorem 10.17. Let A be an abelian category with enough projective and injective ob-jects, and let M be an n-cluster tilting subcategory of A . Assume the following vanishing conditions:

(i) Extn+1A (M, ΩM) = 0 = Extn+1

A (ΣM, M).(ii) ExtkA (M, M) = 0, for n + 2 � k � 2n.

Then the stable Auslander category mod-M is Gorenstein and G-dimmod-M � n.

Proof. We show that splimod-M � n, i.e. the projective dimension of any injective object is at most n, and silpmod-M � n, i.e. the injective dimension of any projective object is at most n. By Lemmas 10.8 and 10.10, we have to show that:

id R1H(ΩM) � n & pd Rn+1H(M) � n

Using the duality Rn+2dl : M-mod −→ mod-M with quasi-inverse Rn+2dr established in Proposition 10.6, in order to show that any injective object F in mod-M has projective


dimension at most n, it suffices to show that any projective object G in M-mod has injective dimension at most n. Since by Lemma 10.11, any projective object G of M-modis of the form G = R1Hop(ΣM), for M ∈ M, it suffices to show that:

id R1H(ΩM) � n & id R1Hop(ΣM) � n

1. Let F = R1H(ΩM) ∈ mod-M be a projective object, where M ∈ M. Then we have exact sequences

0 −→ MnΩM −→ Mn−1

ΩM −→ · · · −→ M1ΩM −→ M0

ΩM −→ ΩM −→ 0 &

0 −→ ΩM −→ P −→ M −→ 0

As usual we set KiΩM = Im(M i

ΩM → M i−1ΩM ), 1 � i � n −1. We shall construct inductively

an injective resolution of F of length at most n of the following form:

0 −→ F −→ Rn+1H(MnΩM ) −→ Rn+1H(Mn−1

ΩM ) −→ · · ·

−→ Rn+1H(M1ΩM ) −→ Rn+1H(M0

ΩM ) −→ 0

Setting KnΩM := Mn

ΩM , we apply the left exact functor H successively to the short exact sequences:

0 −→ KtΩM −→ M t−1

ΩM −→ Kt−1ΩM −→ 0, 1 � t � n &

0 −→ ΩM −→ P −→ M −→ 0 (10.21)

and use n-rigidity of M and the vanishing conditions Extn+1(M, ΩM) = 0 and Extk(M, M) = 0, n + 2 � k � 2n:

(An) For t = n in (10.21), we have isomorphisms:

RkH(Kn−1ΩM ) = 0, 1 � k � 2n− 1, k �= n, n + 1


0 −→ RnH(Kn−1ΩM ) −→ Rn+1H(Mn

ΩM ) −→ Rn+1H(Mn−1ΩM )

−→ Rn+1H(Kn−1ΩM ) −→ 0

(An−1) For t = n − 1 in (10.21), we have isomorphisms:

RkH(Kn−2ΩM ) ∼= Rk+1H(Kn−1

ΩM ), 1 � k � 2n− 1, k �= n, n + 1



0 −→ RnH(Kn−2ΩM ) −→ Rn+1H(Kn−1


−→ Rn+1H(Kn−2ΩM ) −→ Rn+2H(Kn−1

ΩM ) −→ 0

(An−2) For t = n − 2 in (10.21), we have isomorphisms:

RkH(Kn−3ΩM ) ∼= Rk+1H(Kn−2

ΩM ), 1 � k � 2n− 1, k �= n, n + 1


0 −→ RnH(Kn−3ΩM ) −→ Rn+1H(Kn−2


−→ Rn+1H(Kn−3ΩM ) −→ Rn+2H(Kn−2

ΩM ) −→ 0...

(A2) For t = 2 in (10.21), we have isomorphisms:

RkH(K1ΩM ) ∼= Rk+1H(K2

ΩM ), 1 � k � 2n− 1, k �= n, n + 1


0 −→ RnH(K1ΩM ) −→ Rn+1H(K2

ΩM ) −→ Rn+1H(M1ΩM )

−→ Rn+1H(K1ΩM ) −→ Rn+2H(K2

ΩM ) −→ 0

(A1) For t = 1 in (10.21), we have Rn+2H(K1ΩM ) = 0 and isomorphisms:

RkH(ΩM) ∼= Rk+1H(K1ΩM ), 1 � k � 2n− 1, k �= n, n + 1


0 −→ RnH(ΩM) −→ Rn+1H(K1ΩM ) −→ Rn+1H(M0

ΩM ) −→ 0

(A0) Finally from the second short exact sequence in (10.21), we have

RkH(ΩM) = 0, 2 � k � 2n, k �= n + 2

and an exact sequence:

0 −→ Rn+1H(P ) −→ Rn+1H(M) −→ Rn+2H(ΩM) −→ 0

Combining the isomorphisms in (A1)–(An), we have isomorphisms:

F = R1H(ΩM) ∼= R2H(K1ΩM ) ∼= R3H(K2

ΩM ) ∼= · · · ∼= Rn−1H(Kn−2ΩM ) ∼= RnH(Kn−1

ΩM )


On the other hand using the isomorphisms and the exact sequences in (A0)–(An), we obtain:

RnH(Kn−2ΩM ) = RnH(Kn−3

ΩM ) = · · · = RnH(K1ΩM ) = RnH(ΩM) = 0

and

Rn+2H(Kn−1ΩM ) = Rn+2H(Kn−2

ΩM ) = · · · = Rn+2H(K2ΩM ) = 0

As a consequence the exact sequences in (A0)–(An) become respectively an exact sequence

0 −→ F −→ Rn+1H(MnΩM ) −→ Rn+1H(Mn−1

ΩM ) −→ Rn+1H(Kn−1ΩM ) −→ 0 (En−1)

short exact sequences

0 −→ Rn+1H(Kn−1ΩM ) −→ Rn+1H(Mn−2

ΩM ) −→ Rn+1H(Kn−2ΩM ) −→ 0 (En−2)

0 −→ Rn+1H(Kn−2ΩM ) −→ Rn+1H(Mn−3

ΩM ) −→ Rn+1H(Kn−3ΩM ) −→ 0 (En−3)

...

0 −→ Rn+1H(K2ΩM ) −→ Rn+1H(M1

ΩM ) −→ Rn+1H(K1ΩM ) −→ 0 (E1)

and an isomorphism

Rn+1H(K1ΩM ) ∼= Rn+1H(M0

ΩM ) (E0)

Forming the Yoneda composition of the exact sequences (Ei), 1 � i � n − 1 and the isomorphism (E0) we arrive at an exact sequence

0 −→ F −→ Rn+1H(MnΩM ) −→ Rn+1H(Mn−1

ΩM ) −→ · · ·−→ Rn+1H(M1

ΩM ) −→ Rn+1H(M0ΩM ) −→ 0

which is an injective resolution of the projective object F . Hence idF � n and therefore silpmod-M � n.

2. Working in the category of covariant coherent functors M-mod and using the second vanishing condition Extn+1

A (ΣM, M) = 0 of (i) and the vanishing condition ExtkA (M, M) = 0, n +2 � k � 2n, in (ii), we have in a dual way that id R1Hop(ΣM) � n. As explained before this is equivalent to say that pdG � n for any injective object G of mod-M. We infer that spli mod-M � n. �

If n = 1 in Theorem 10.17, then the second condition (ii) is vacuous. Hence we have the following.


Corollary 10.18. Let A be an abelian category with enough projective and injective objects, and let M be a 1-cluster tilting subcategory of A . If Ext2A (M, ΩM) = 0 = Ext2A (ΣM, M), then the stable Auslander category mod-M is Gorenstein and G-dim mod-M � 1.

Remark 10.19. If A is Frobenius, then Ext2A (M, ΩM) ∼= A (Ω2M, ΩM) ∼= A (ΩM, M) ∼=Ext1A (M, M) = 0 and similarly Ext2A (ΣM, M) = 0. Hence the condition Ext2A (M, ΩM) =0 = Ext2A (ΣM, M) of Corollary 10.18 holds always in the Frobenius case and M is a 2-cluster tilting subcategory of A . It follows that the bound G-dimmod-M � 1 of Corollary 10.18 generalizes Theorem 6.4(i) to the non-stable case.

It is easy to see that in case n = 1 the conditions R2H(P ) = 0 = R2Hop(I), P ∈ Proj A , and I ∈ Inj A , imply the conditions of Corollary 10.18. Hence we have the following consequence.

Corollary 10.20. Let A be an abelian category with enough projective and injective objects, and let M be a 1-cluster tilting subcategory of A . If Ext2A (M, P ) = 0 = Ext2A (I, M), ∀M ∈ M, P ∈ Proj A , and I ∈ Inj A , then the stable Auslander category mod-M is Gorenstein and G-dimmod-M � 1.

More generally for n > 1 it is easy to see that the conditions Extn+1A (M, P ) = 0 =

Extn+1A (I, M), ∀M ∈ M, P ∈ Proj A , and I ∈ InjA , which clearly hold in case A is a

Frobenius category or more generally in case G-dim A � n, imply the conditions (i) in Theorem 10.17, and then we have the following.

Corollary 10.21. Let A be an abelian category with enough projective and injective objects, and let M be an n-cluster tilting subcategory of A . Assume the following vanishing conditions:

Extn+1A (M,ProjA ) = 0 = Extn+1

A (Inj A ,M) &

ExtkA (M,M) = 0, n + 2 � k � 2n (†)

(1) The stable Auslander category mod-M is Gorenstein and G-dim mod-M � n.(2) If G-dimA � 2n, then A is Frobenius and M is an (n +1)-cluster tilting subcategory

of A .(3) If G-dimA � 2n and the triangulated category A is (n + 1)-Calabi–Yau, then the

(n + 1)-cluster tilting subcategory M of A is (n − 1)-corigid and G-dim mod-M � 1.

Proof. (1) This follows from Theorem 10.17.(2) The vanishing conditions (†) imply that ExtkA (M, P ) = 0, for 1 � k � 2n, for any

object M ∈ M and any projective object P of A . It follows that if G-dim A � 2n, then M ⊆ GProjA and then as in Proposition 10.16 we infer that A is Frobenius. Clearly then M is an (n + 1)-cluster tilting subcategory of A .


(3) If in addition A is (n + 1)-Calabi–Yau, then for n + 2 � k � 2n:

0 = ExtkA (M,M) ∼= A (M,ΣkM) ∼= A (M,Σn+1Σk−n−1M)

∼= DA (Σk−n−1M,M) ∼= DA (M,Σn−k+1M)

Hence the (n + 1)-cluster tilting subcategory M of A is (n − 1)-corigid. Then by Corol-lary 6.5(a) we have G-dimmod-M � 1. �10.7. The module-theoretic interpretation

We close this section by summarizing our results in this section in the context of categories of modules over an associative ring Λ. Below G-dim Λ means G-dim Mod-Λ, i.e. the Gorenstein dimension of the category of right Λ-modules.

Theorem 10.22. Let M be a self-compact n-cluster tilting Λ-module over a ring Λ and let Γ = EndΛ(M) be, as in Theorem 9.9, the associated left coherent and right perfect n-Auslander ring.

(i) The Γ-dual functor HomΓ(−, Γ) induces a duality Extn+2Γ (−, Γ): mod-EndΛ(M) ≈−→

(EndΛ(M)-mod)op.(ii) r.gl.dimΛ � n +1 ⇐⇒ ExtkΓ(E, Γ) = 0, 1 � k � n +1, for any injective Γ-module E.(iii) If Λ is an (n − 1)-Auslander ring, then:

(a) The stable n-Auslander ring EndΛ(M) is an n-Auslander ring.(b) M is an n-cluster tilting object in (InjΛ-mod)op.(c) If Λ is an Artin algebra, then M is an n-cluster tilting object in mod-EndΛ(DΛ)op

with endomorphism algebra the stable n-Auslander algebra EndΛ(M).(iv) G-dimΛ � n ⇐⇒ Λ is QF. In this case M is finitely generated and:

(a) If n = 1, then G-dimEndΛ(M) � 1.(b) If n � 2 and Ext1Λ(M, ΩiM) = 0, where 2 � i � n − k + 1 and 0 � k � n+1

2 , then G-dim EndΛ(M) � k.

(v) If n = 1 and Ext2Λ(M, P ) = 0 = Ext2Λ(J, M), ∀P ∈ ProjΛ, ∀J ∈ Inj Λ, then: G-dimEndΛ(M) � 1.

(vi) If n � 2, and assuming that: Extn+1Λ (M, P ) = 0 = Extn+1(J, M), ∀P ∈ Proj Λ,

∀J ∈ Inj Λ, and ExtkΛ(M, M) = 0, for n + 2 � k � 2n, then:(a) G-dim EndΛ(M) � n.(b) G-dim Λ � 2n ⇐⇒ Λ is QF; in this case M is a compact (n + 1)-cluster tilting

object in Mod-Λ.(c) If G-dimΛ � 2n and the triangulated category mod-Λ is (n + 1)-Calabi–Yau,

then the (n + 1)-cluster tilting object M of mod-Λ is (n − 1)-corigid and G-dim EndΛ(M) � 1.


11. Cluster tilting subcategories of Cohen–Macaulay objects

In this section we are interested in the homological behavior of the category of co-herent functors over rigid subcategories of an abelian category induced by cluster-tilting subcategories of the triangulated category of Gorenstein-projective (or Cohen–Macaulay) objects.

Throughout this section: we fix an abelian category A and assume that A has enough projectives. We also fix a contravariantly finite subcategory M of A and assume that Mcontains the projectives.

Then, as in the previous sections, right M-approximations of objects of A are epics and we have an adjoint pair (R, H) : mod-M −→←− A , where the functor H : A −→ mod-M, H(A) = A (−, A)|M is fully faithful and the functor R is exact. Moreover KerR = mod-Mis equivalent to the category mod-M of coherent functors over the stable category M of M modulo projectives, and we have an exact sequence:

0 −→ mod-M −→ mod-M −→ A −→ 0 (11.1)

We consider the stable category GProjA of Gorenstein-projective objects of A modulo projectives as a triangulated category with suspension functor Ω−1. For a full subcategory X of GProjA , we denote by M = π−1X the pre-image of X under the projection functor π : GProj A −→ GProjA , so that M = X. Note that then ProjA ⊆ M ⊆ GProj A . In what follows the operations �k X ←−� X �−→ X�

k are taken inside the triangulated category GProjA , while the operations ⊥k M ←−� M �−→ M⊥

k are taken inside the abelian category A .

The following observation is easy and is left to the reader noting that if A is Goren-stein, then GProjA is contravariantly finite, and also covariantly finite if in addition Proj A is covariantly finite, see [11].

Lemma 11.1. X is contravariantly, resp. covariantly, finite in GProjA if and only if M is contravariantly, resp. covariantly, finite in GProjA . If A is Gorenstein, then Xis contravariantly finite in GProjA if and only if M is contravariantly finite in A ; if Proj A is covariantly finite in A , then X is covariantly finite in GProjA if and only if M is covariantly finite in A . Finally X�

n = M⊥n ∩ GProj A and �nX = ⊥

nM ∩ GProjA .

We call the abelian category A (projectively) Gorenstein if A has enough projec-tives and there exists n � 0, such that any object of A admits an exact resolution of length � n by Gorenstein-projective objects. The minimum such n is called the Gorenstein projective dimension G-dimP A of A . Dually one defines the Gorenstein injective dimension G-dimI A of A and when A is (injectively) Gorenstein. Note that if A has enough projectives and enough injectives, then these dimensions coincide with the Gorenstein dimension G-dim A = max{silp A , spliA } as defined in Section 6, so: G-dimP A = G-dimA = G-dimI A , see [19].


Lemma 11.2. Let A be an abelian category with enough projectives. Then A is Gorenstein if and only if A contains a contravariantly finite subcategory M such that ProjA ⊆ M ⊆GProj A and: gl.dim mod-M < ∞.

Proof. If A is Gorenstein, then M := GProjA is contravariantly finite by [19]. Hence mod-M is abelian. Since M is resolving, by the arguments of [15, Corollaries 6.8 and 6.13]it follows that gl.dimmod-M < ∞. Conversely if gl.dimmod-M = t < ∞, then pdH(A) �t, ∀A ∈ A , and since M contains the projectives, this implies that any object A of Aadmits a finite M-resolution of length � t. Since M consists of Gorenstein-projectives, it follows that G-dim A � t, hence A is Gorenstein by [19]. �

Let U, V be full subcategories of A . Then the Gabriel product U � V is defined to be the full subcategory

U � V = add{A ∈ A | ∃ an exact sequence : 0 −→ U −→ A −→ V −→ 0,

where U ∈ U & V ∈ V}

Inductively we define U1 �U2 � · · · �Un, ∀n � 1, for full subcategories Ui of A . It is easy to see that the operation � is associative and clearly U1 � U2 � · · · � Un coincides with the full subcategory Filt(U1, · · · , Un) of A consisting of direct summands of objects Awhich admit a finite filtration

0 = A0 ⊆ A1 ⊆ A2 ⊆ · · · ⊆ An−1 ⊆ An = A

such that Ak/Ak−1 ∈ Uk, 1 � k � n.If M is contained in GProjA , then we denote by Ω−1M the full subcategory of GProjA

consisting of all direct summands of objects A for which there exists an exact sequence 0 −→ M −→ P −→ A −→ 0, where M ∈ M and P is projective. Then Ω−kM is defined inductively for k � 2 by Ω−kM = Ω−1Ω−k+1M. Now we are ready to prove the main result of this section which generalizes, and is inspired by, some results of Iyama [40,39].

Theorem 11.3. Let A be an abelian category with enough projectives.

1. (a) The map X �−→ M = π−1X gives a bijective correspondence between:(i) (n + 1)-cluster tilting subcategories X of GProjA .(ii) Contravariantly finite subcategories M of GProjA such that:

M⊥n ∩ GProj A = M

(iii) Covariantly finite subcategories M of GProjA such that:

⊥nM ∩ GProj A = M


(b) Under the above correspondences:(i) A is Gorenstein if and only if M is contravariantly finite in A and

gl.dimmod-M < ∞.(ii) If A is Gorenstein and ProjA is covariantly finite, then M is functorially

finite in A .2. Let A be Gorenstein, 0 �= X be an (n +1)-cluster tilting subcategory of GProjA , and

M = π−1X.(a) pdH(G) � n, ∀G ∈ GProjA , and setting d := G-dimA there are equalities:

A = M � Ω−1M � · · · � Ω−nM � Proj�dA

= Filt(M, Ω−1M, · · · Ω−nM, Proj�dA

)(11.2)

(b) pdmod-M F = n + 2, ∀F ∈ mod-X, F �= 0, and

gl.dimmod-M = max{n + 2,G-dim A

}(11.3)

In particular, if G-dim A � n + 2, then: gl.dimmod-M = n + 2.(c) If A has enough injectives, then any projective object Q of mod-M admits an

injective resolution

0 −→ Q −→ J0 −→ J1 −→ · · · · · · −→ Jn −→ Jn+1 −→ · · ·

where: pdJk � G-dimA , 0 � k � n + 1.

Proof. 1. Parts (a) and (b)(ii) follow from Lemmas 11.1, 11.2 and Theorem 5.3. For (b)(i) note that if M is contravariantly finite in A and gl.dimmod-M < ∞, then A is Gorenstein by Lemma 11.2. If A is Gorenstein, then M is contravariantly finite in A by Lemma 11.1. Finiteness of gl.dim mod-M will be shown in 2. below.

2.(a) By Theorem 5.3 we have: GProjA = X � Ω−1X � · · · � Ω−nX, and by parts (ii) and (iii) of Proposition 2.13 for any Gorenstein-projective object G there exists an M-resolution of G of the form

0 −→ Mn −→ Mn−1 −→ · · · −→ M1 −→ M0 −→ G −→ 0 (11.4)

Applying H to (11.4) and using that M is n-rigid, we have an exact sequence

0 −→ H(Mn) −→ H(Mn−1) −→ · · · · · · −→ H(M1) −→ H(M0) −→ H(G) −→ 0

which is a projective resolution of H(G). Hence pdH(G) � n, ∀G ∈ GProjA .Since A is Gorenstein, by [19], for any object A ∈ A , there is an exact sequence 0 −→

YA −→ GA −→ A −→ 0, where GA is Gorenstein-projective and YA has finite projective dimension; then pdYA � d since Proj<∞ A = Proj�d A , see Remark 6.3. Since GA

is Gorenstein-projective, there exists an extension 0 −→ GA −→ P −→ Ω−1GA −→ 0,


where P is projective and Ω−1GA is Gorenstein-projective. Then the composition YA −→GA −→ P induces a short exact sequence 0 −→ YA −→ P −→ Y A −→ 0 and clearly Y A has finite projective dimension. By diagram chasing then it is easy to see that there exists a short exact sequence 0 −→ GA −→ A ⊕ P −→ Y A −→ 0. Hence A ∈ GProj A �Proj<∞ A . Using that GProjA = X � Ω−1X � · · · � Ω−nX, it follows that GProjA =M � Ω−1M � · · · � Ω−nM. We infer that A = GProjA � Proj<∞ A = M � Ω−1M � · · · �Ω−nM � Proj<∞ A .

2.(b) Since X �= 0 we have M �= ProjA . Then gl.dimmod-M � n + 2 by Proposi-tion 8.10. If 0 �= F ∈ mod-X, then F admits a presentation 0 −→ H(AF ) −→ H(M1) −→H(M0) −→ F −→ 0, where the map M1 −→ M0 is epic. Since the M i are Gorenstein-projective and GProjA is closed under kernels of epimorphisms, it follows that AF is Gorenstein-projective. Then by 2.(a) we have pdH(AF ) � n and therefore pdF � n +2. Since by Lemma 8.8, Extk(F, H(M)) = 0, 0 � k � n + 1, and F �= 0, we infer by Lemma 8.9 that pdF = n + 2.

To show the equality (11.3), set d := G-dim A . Let F ∈ mod-M and consider the exact sequence 0 −→ H(AF ) −→ H(M1) −→ H(M0) −→ F −→ 0. Consider, as in (8.3), an M-resolution of AF :

· · · −→ MnAF

−→ Mn−1AF

−→ · · · −→ M1AF

−→ M0AF

−→ AF −→ 0 (11.5)

Then clearly KiAF

∈ M⊥i , ∀i � 1. On the other hand composing the exact sequence

0 −→ AF −→ M1 −→ M0 −→ R(F ) −→ 0 (11.6)

with the exact sequence (11.5) we have an exact sequence

· · · −→ MnAF

−→ Mn−1AF

−→ · · ·

−→ M0AF

−→ M1 −→ M0 −→ R(F ) −→ 0 (11.7)

Assume first that d � 3. Since GProjA is resolving and G-dimR(F ) � d, we have clearly that Kd−2

AF∈ GProjA , see [4, Lemma 3.12]. Hence Kd−2

AF∈ M⊥

d−2∩GProj A . If d −2 > n, then Kd−2

AF∈ M⊥

n ∩ GProjA = M. This implies that pdH(AF ) � d − 2 and therefore pdF � d. If d − 2 � n, then using that GProjA is resolving and Kd−2

AF∈ GProjA it

follows that KnAF

is Gorenstein-projective and therefore lies in M⊥n ∩GProjA = M. This

implies that H(AF ) � n and therefore pdF � n + 2. On the other hand if d � 2, then clearly d < n +2 and G-dimR(F ) � 2 and therefore (11.6) shows that AF lies in GProjA . It follows that Kn

AF∈ M⊥

n ∩ GProjA = M and as above we have pdF � n + 2. Hence pdF � max{n +2, d}, for any F ∈ mod-M, and therefore gl.dimmod-M � max{n +2, d}. Summarizing we have shown that n + 2 � gl.dim mod-M � max{n + 2, d}.

• If d � n + 2, then we have n + 2 � gl.dimmod-M � n + 2, hence: gl.dimmod-M =n + 2 = max{n + 2, d}.


• Assume that n + 2 � d, so gl.dim mod-M � d. If gl.dim mod-M �= d, then for any object A ∈ A we have pdH(A) � d − 1 and therefore there exists an M-resolution 0 −→ Md−1 −→ Md−2 −→ · · · −→ M0 −→ A −→ 0 of A of length � d − 1. Since Mconsists of Gorenstein-projective objects it follows that G-dimA � d − 1 and therefore d = G-dim A � d − 1. This contradiction shows that gl.dimmod-M = d = max{n + 2, d}.

2.(c) For any M ∈ M consider an injective resolution 0 −→ M −→ I0 −→ I1 −→ · · · . Applying the functor H, and using that M is n-rigid, we have an exact sequence in mod-M

0 −→ Q −→ J0 −→ J1 −→ · · · −→ Jn −→ Jn+1

where Q = H(M) and Jk = H(Ik), 0 � k � n + 1. Since the left adjoint R of H is exact it follows that the Jk are injective in mod-M. On the other hand, since A is Gorenstein, we have pdE � spliA = G-dimA := d for any injective object E ∈ A . It follows that, for any k � 0, there exists a projective resolution

0 −→ P kd −→ P k

d−1 −→ · · · −→ P k1 −→ P k

0 −→ Ik −→ 0

Since M consists of Gorenstein-projectives, we have Extm(M, P ) = 0, ∀m � 1, and for any projective object P ∈ A . Hence applying H to the above resolution we obtain an exact sequence

0 −→ H(P kd ) −→ H(P k

d−1) −→ · · · −→ H(P k1 ) −→ H(P k

0 ) −→ Jk −→ 0

which is a projective resolution in mod-M of the injective object Jk. Hence pdJk � d. �Corollary 11.4. Let A be a Gorenstein abelian category, X an (n + 1)-cluster tilting subcategory of GProjA , and set M = π−1(X) ⊆ A . If G-dim A � n + 2, then the following are equivalent.

(i) dom.dim mod-M � n + 2.(ii) A is Frobenius.(iii) mod-M is an n-Auslander category.(iv) M is an n-cluster (co)tilting subcategory of A .

Moreover if A is a length category, then the short exact sequence of abelian categories

0 −→ mod-X −→ mod-M −→ A −→ 0

induces isomorphisms in K-theory: K∗(M, ⊕) ∼= K∗(A ) ⊕ K∗(mod-X).

Proof. (i) ⇐⇒ (iii) ⇐⇒ (iv) Since G-dim A � n + 2, by Theorem 11.3 we have gl.dim mod-M � n + 2. Then the assertions follow from Theorem 8.23.


(i) ⇐⇒ (ii) Since G-dim A � n + 2, by Theorem 11.3 it follows that gl.dim mod-M �n + 2 � dom.dim mod-M, hence mod-M is an n-Auslander category and therefore, by Theorem 8.23, M is an n-cluster tilting subcategory of A . Then as in Proposi-tion 10.16 we deduce that A is Frobenius. Conversely if this holds, then G-dimA = 0and dom.dim mod-M � n + 2 by Theorem 11.3(2)(c). �

Keller and Reiten proved that if X is a 2-cluster tilting subcategory of A , where A is a Frobenius abelian category and if A is 2-Calabi–Yau, then gl.dimmod-M = 3, where M = π−1(X), see [46]. The following consequence of Theorems 8.23 and 9.6, and Corollary 9.4, generalizes, among others, the result of Keller and Reiten for any (n + 1)-cluster tilting subcategory, n � 2, without assuming any Calabi–Yau condition.

Corollary 11.5. Let A be an abelian category with enough projectives and enough injec-tives. For a full subcategory ProjA �= M ⊆ GProjA , the following are equivalent.

(i) A is Frobenius and M is an (n + 1)-cluster tilting subcategory of A .(ii) M is an n-cluster tilting subcategory of A .(iii) M is an n-cluster cotilting subcategory of A .(iv) mod-M is an n-Auslander category.(v) M is n-rigid, contravariantly finite in A and contains the projectives, and

gl.dimmod-M = n + 2.(vi) M is n-rigid, covariantly finite in A and contains the injectives, and

gl.dimmod-M = n + 2

If (i) holds, and if M is of finite representation type, then:

rep.dim A � n + 2

We continue with some consequences of our previous results in this context. The first one follows from Corollary 7.7 or Corollary 6.5.

Corollary 11.6. Let A be Gorenstein and assume that GProjA is (n + 1)-Calabi–Yau. Let X be an (n + 1)-cluster tilting subcategory of GProjA and let M = π−1X. If Hom(M, ΩiM) = 0, 1 � i � n − 1, then mod-M is 1-Gorenstein and the stable tri-angulated category GProjmod-M is (n + 2)-Calabi–Yau.

The following result is due to Keller and Reiten, see [46, Theorem 5.4], called relative (n +2)-Calabi–Yau duality, proved in [46] in the context of an algebraic (n +1)-Calabi–Yau triangulated category over a field. In our setting we give a different proof.

Proposition 11.7. Let A be a Gorenstein abelian Hom-finite k-category over a field k, and assume that GProjA is (n + 1)-Calabi–Yau. Let 0 �= X be a (n + 1)-cluster tilting


subcategory of GProjA , where n � 1. If M = π−1X, then for any object F ∈ mod-X ⊆mod-M, there is a natural isomorphism:

D HomDb(mod-M)(F,−

) ∼=−−→ HomDb(mod-M)(−, F [n + 2]

)

In particular, for any two objects F, G ∈ mod-M with F ∈ mod-X:

DExtimod-M(F,G

) ∼=−−→ Extn−i+2mod-M

(G,F

), i ∈ Z

Proof. By Theorem 11.3, we have gl.dimmod-M < ∞. Then by [37], [57], the category Db(mod-M) admits a Serre functor which is given by S = − ⊗L

M D HomA (−, ?)|M, where HomA (−, ?)|M(M) = HomA (−, M) = H(M). Hence we have a natural isomorphism

D HomDb(mod-M)(F,−

) ∼=−−→ HomDb(mod-M)(−, F ⊗L

M D HomA (−, ?)|M)

So it suffices to show that we have an isomorphism F [n +2] ∼=−→ F⊗L

MD HomA (−, ?)|M in Db(mod-M). Applying the triangulated functor − ⊗L

M D HomA (−, ?)|M to the projective resolution of F as in (8.5)

· · · −→ H(MnAF

) −→ H(Mn−1AF

) −→ · · ·

−→ H(M0AF

) −→ H(M1) −→ H(M0) −→ F −→ 0 (11.8)

with F deleted, and noting that we may choose Mn+1AF

= 0 since pdF = n +2, we obtain a complex

0 −→ D HomA (MnAF

,−)|M −→ D HomA (Mn−1AF

,−)|M −→ · · ·

−→ D HomA (M0AF

,−)|M −→ · · ·−→ D HomA (M1,−)|M −→ D HomA (M0,−)|M −→ 0

which is acyclic everywhere except in first position on the left, which corresponds to −n − 2 degree, where the homology is given by DExt1(Kn−1

F , −)|M. Since GProjA is (n + 1)-Calabi–Yau, we have natural isomorphisms:

DExt1(Kn−1F ,−)|M ∼= D HomA (ΩKn−1

F ,−)|M ∼= HomA (−,Ω−n−1ΩKn−1F )|M =

= HomA (−,Ω−nKn−1F )|M ∼= HomA (Ωn(−),Kn−1

F )|M ∼= ExtnA (−,Kn−1F )|M ∼= F

where the last isomorphism follows from Lemma 8.6. We infer that in Db(mod-M) we have isomorphisms

F [n + 2] ∼=−−→ ExtnA (−,Kn−1F )|M[n + 2] ∼=−−→ F ⊗L

M D HomA (−, ?)|M �Under the assumptions of Proposition 11.7, Lemma 8.6 and Remark 8.11 admit the

following consequence:


Corollary 11.8. Let F ∈ mod-M, G ∈ mod-M, and M, M ′ ∈ M. Then there are isomor-phisms:

DExtn+2mod-M(F,G) ∼=−−→ Hom(G,F ) and DExtn+2

mod-M(F, (−,M ′)) ∼=−−→ F (M ′)

DExtn+2mod-M((−,M), (−,M ′)) ∼=−−→ DExtn+1

A (M,M ′) ∼=−−→ Hom(M ′,M)

We close this section with some applications of our previous results to Artin algebras. The first one is a consequence of Theorem 11.3 and Corollary 11.5.

Corollary 11.9. For an Artin algebra Λ the following are equivalent.

(i) Λ is Gorenstein and GprojΛ contains a (n + 1)-cluster tilting object.(ii) mod-Λ contains a generator M such that addM = M⊥

n ∩ Gproj Λ andgl.dim EndΛ(M) < ∞.

If (ii) holds, then rep.dimΛ � n + 2. Further M is a cogenerator if and only if Λ is self-injective.

Our final result illustrates some of the main results of the paper in the context of Gorenstein-projective modules over an Artin algebra Λ. Note that if Λ is Gorenstein, then Gproj Λ admits a Serre functor S which is given by S(A) = Ω−1XDTrA, where XDTrA is the minimal right GprojΛ-approximation of DTrA. It follows that GprojΛ is (n + 1)-Calabi–Yau if and only if there is a natural isomorphism XDTrA

∼= Ω−nA, see [15].

Theorem 11.10. Let Λ be an Artin algebra and let T be an (n + 1)-cluster tilting object of Gproj Λ, n � 1.

1. Λ is Gorenstein if and only if gl.dim EndΛ(T ) < ∞. If Λ is Gorenstein, then:(a) gl.dim EndΛ(T ) = max

{n + 2, idΛ

}.

(b) If idΛ � n + 2, then gl.dimEndΛ(T ) = n + 2 and EndΛ(T ) admits an injective resolution

0 −→ EndΛ(T ) −→ J0 −→ J1 −→ · · · −→ Jn+1 −→ Jn+2 −→ 0

in mod-EndΛ(T ), where pd J t � id Λ, 0 � t � n + 1. Moreover the following are equivalent:

(i) dom.dim EndΛ(T ) � n + 2.(ii) Λ is self-injective.(iii) EndΛ(T ) is an n-Auslander algebra.(iv) T is an n-cluster (co)tilting Λ-module.In this case rep.dimΛ � n + 2.


2. Assume that ExttΛ(T, DTrT ) = 0, 2 � t � n − k + 1, where 0 � k � n − 1.(a) If k � n+1

2 , then the cluster tilted algebra EndΛ(T ) is k-Gorenstein.(b) If 0 � k � 1, and Gproj Λ is (n + 1)-Calabi–Yau, then Gproj EndΛ(T ) is (n +

2)-Calabi–Yau.(c) If k � n

2 , denote by Mk(T ) the full subcategory of GprojΛ:

Mk(T ) :={

addT � addΩ−1T � · · · � addΩ−kT}

⋂ {A ∈ GprojΛ | ExttΛ(A,DTrT ) = 0, 2 � t � k + 1

}

Then gl.dimEndΛ(T ) < ∞ if and only if the functor

HomΛ(T,−) : Mk(T ) ∼=−−→ mod-EndΛ(T )

is an equivalence. In this case gl.dimEndΛ(T ) � k with equality if Extn−k+2(T,DTrT ) �= 0.

3. Assume that ExttΛ(T, DTrT ) = 0, 2 � t � n.(a) If EndΛ(T ) is of finite CM-type, then rep.dimEndΛ(T ) � 3.(b) If Gproj Λ is (n + 1)-Calabi–Yau and EndΛ(T ) has finite global dimension, then

EndΛ(T ) is hereditary and there is a triangle equivalence:

GprojΛ ≈−−→ C n+1EndΛ(T )

sending T to π(EndΛ(T )), where C n+1EndΛ(T ) is the (n +1)-cluster category associated

to EndΛ(T ).4. If GprojΛ is (n + 1)-Calabi–Yau, then for any F ∈ mod-EndΛ(T ) and any G ∈

mod-EndΛ(T ):

DExtiEndΛ(T )(F,G

) ∼=−−→ Extn−i+2EndΛ(T )

(G,F

), ∀i ∈ Z

Extn+2EndΛ(T )

(EndΛ(T ),EndΛ(T )

) ∼=−−→ DEndΛ(T )

Proof. Part 1. follows from Theorem 11.3 and Corollary 11.4. Part 2. follows from The-orems 6.4, 6.10, and 7.6, by observing that T is (n − k)-corigid, i.e. HomΛ(T, ΩtT ) = 0, 1 � t � n − k, if and only if Extt(T, DTrT ) = 0, 2 � t � n − k + 1, where 0 � k � n − 1, if and only if T ∈ Mk(T ), as follows from Auslander–Reiten duality:

Extt+1Λ (T,DTrT ) ∼= Ext1Λ(ΩtT,DTrT ) ∼= D HomΛ(T,ΩtT )

Since the triangulated category GprojΛ is algebraic, part 3. follows from 2.(a) and Theo-rems 6.9 and 6.15. Finally part 4. follows from Proposition 11.7 and Corollary 11.8. �


Acknowledgments

The main results of this paper go back to 2008–2010, and have been presented in a series of lectures in IPM, Tehran, Iran in July 2010, and in Oberwolfach, Germany, in February 2011. I take this opportunity to thank Javad Asadollahi and Shokrollah Salarian for the invitation and the kind hospitality in IPM.

The author is grateful to the referee for the extremely careful reading of the paper and the exemplary report. His/her very useful and helpful remarks resulted in a significant improvement of the paper.

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