arXiv:1608.04179v2 [math.RT] 22 Jan 2018
AND PRECLUSTER TILTING
OSAMU IYAMA AND ØYVIND SOLBERG To the memory of Maurice Auslander.
Abstract. We generalize the notions ofn-cluster tilting subcategories andτ- selfinjective algebras inton-precluster tilting subcategories andτn-selfinjective algebras, where we show that a subcategory naturally associated ton-precluster tilting subcategories has a higher Auslander–Reiten theory. Furthermore, we give a bijection betweenn-precluster tilting subcategories andn-minimal Auslander–Gorenstein algebras, which is a higher dimensional analog of Auslander–
Solberg correspondence (Auslander–Solberg, 1993) as well as a Gorenstein analog ofn-Auslander correspondence (Iyama, 2007). The Auslander–Reiten theory associated to an n-precluster tilting subcategory is used to classify then-minimal Auslander–Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander–Solberg.
Contents
1. Introduction 1
2. Preliminaries on relative homological algebra 5
2.1. Relative homological algebra 5
2.2. The Auslander–Reiten translation revisited 7
2.3. Relative tilting theory 9
3. Elementary properties ofn-precluster tilting subcategories 12
4. Auslander–Solberg correspondence 19
5. n-fold almost split extensions 22
6. Four classes ofn-minimal Auslander–Gorenstein algebras 29
References 31
1. Introduction
Higher Auslander–Reiten theory was introduced in [21] by looking at n-cluster tilting subcategories instead of the whole module category. It is known that any n-cluster tilting subcategory hasn-almost split sequences, and that finiten-cluster
Key words and phrases. Auslander-Reiten theory; cluster tilting; Auslander algebra; relative homological algebra.
Part of the results were presented at the meeting ICRA XIV, Tokyo, Japan, on August 11th, 2010, and at the meeting “Miniconference on selfinjective algebras”, Torun, Poland, on December 15th 2010. Osamu Iyama’s work on this project was partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03923, (C) 23540045 and (S) 15H05738. Øyvind Solberg’s work on this project was partially supported by FRINAT grant number 23130 from the Norwegian Research Council.
1
tilting subcategories correspond to n-Auslander algebras [22]. Moreover Artin al- gebras Λ which haven-cluster tilting modules are calledn-representation-finite [13]
and have been studied. (We do not assume gldim Λ≤nin this paper in contrast with several earlier papers [18, 24, 25].) See for example [1, 19, 20, 26, 28, 29, 30, 31, 34, 36] for further results in higher Auslander–Reiten theory.
In this paper we introduce the weaker notions of n-precluster tilting subcate- gories (Definition 3.2) andτn-selfinjective algebras (Definition 3.4). These notions generalize and unify two seemingly different concepts, namely (finite)n-cluster tilt- ing subcategories [21] and τ-selfinjective algebras (or DTr-selfinjective algebras) [11]. Both of these concepts are generalizations of algebras of finite representation type and they are linked through the Wedderburn correspondence introduced in [2]. The first class corresponds to n-Auslander algebras Γ, that is, Artin algebras Γ satisfying
domdim Γ≥n+ 1≥gldim Γ.
The latter class corresponds to algebras Γ that are a Gorenstein analog of Auslander algebras which satisfy
domdim Γ≥2≥idΓΓ.
Here domdim, gldim and id denote dominant dimension, global dimension and in- jective dimension, respectively. A central notion in higher dimensional Auslander–
Reiten theory is n-cluster tilting subcategories, which are the whole module cat- egories in the classical case n = 1. As in the process of going from classical Auslander–Reiten theory to higher Auslander–Reiten theory, we demonstrate that the number 2 is quite symbolic, as we show thatn-precluster-tilting subcategories correspond to the following class of Artin algebras.
Definition 1.1. We call an Artin algebra Γ ann-minimal Auslander–Gorenstein algebra if it satisfies
domdim Γ≥n+ 1≥idΓΓ.
We will observe in Proposition 4.1(b) that this condition is left-right symmet- ric. Recall that an Artin algebra Γ is called Gorenstein (or Iwanaga–Gorenstein more precisely) if idΓΓ and id ΓΓ are finite [15], and a Gorenstein algebra is called Auslander–Gorenstein if the minimal injective coresolution
0→Γ→I0→I1→ · · ·
of the Γ-module Γ satisfies pdΓIi ≤ i holds for everyi ≥ 0 [16, 5, 6]. Thus our n-minimal Auslander–Gorenstein algebras can be regarded as the most basic class among Auslander–Gorenstein algebras since pdΓIi has the ‘minimal’ value 0 for 0≤i≤n.
We have the following diagrams of generalizations of Auslander algebras and representation-finite algebras:
Auslander algebras❴ ❴ ❴//
✤✤
Gorenstein algebras Γ with domdim Γ≥2≥idΓΓ
✤✤
n-Auslander algebras❴ ❴ ❴ ❴ ❴//n-minimal Auslander–Gorenstein algebras representation-finite algebras❴ ❴❴ ❴//
✤✤
τ-selfinjective algebras
✤✤
n-representation-finite algebras❴ ❴ ❴//τn-selfinjective algebras
We introduce the concept of n-precluster tilting subcategories to describe these algebras, where such a subcategoryCis defined to be a functorially finite generating- cogenerating subcategory stable under the n-th Auslander–Reiten translate and selforthogonal in the interval [1, n−1] (that is, Exti(C,C) = 0 fori= 1,2, . . . , n−1).
More precisely, we show the following results.
Theorem 4.5. Fixn≥1. There is a bijection between Morita-equivalence classes of n-minimal Auslander–Gorenstein algebras and equivalence classes of finite n- precluster tilting subcategories C of Artin algebras, where the correspondences are given in Propositions 4.3and 4.4.
For a Gorenstein algebra Γ, we denote by
CM Γ ={X∈mod Γ|ExtiΓ(X,Γ) = 0 fori >0}
the category of maximal Cohen–Macaulay Γ-modules. We denote by CM Γ the stable category of CM Γ, that is, CM Γ modulo the ideal in CM Γ generated by add Γ. It is basic that CM Γ forms a triangulated category. On the other hand, for ann-precluster tilting subcategoryCof mod Λ, let
Z(C) = {X ∈mod Λ|ExtiΛ(C, X) = 0 fori= 1,2, . . . , n−1}, U(C) = Z(C)/[C],
whereZ(C)/[C] denotes the categoryZ(C) modulo the ideal [C] inZ(C) generated by the subcategoryC. Note thatZ(C) = mod Λ for the casen= 1. We show that the categoryZ(C) is a Frobenius category whose stable category isU(C) (Proposition 3.12). HenceU(C) forms a triangulated category. In fact it is an analog of Calabi–
Yau reduction of triangulated categories [27].
Theorem 4.7. Given an Artin algebra Λ with a finiten-precluster tilting subcat- egory C = addM, let Γ = EndΛ(M) be the corresponding n-minimal Auslander–
Gorenstein algebra (see Theorem 4.5). Then Z(C) and CM Γ are dual categories via the functors HomΛ(−, M) :Z(C) → CM Γ and HomΓ(−, M) : CM Γ→ Z(C).
Moreover these functors induce triangle equivalences between U(C)and(CM Γ)op. Furthermore we show that there is a higher Auslander–Reiten theory also for n-precluster tilting subcategories, though with some differences. The first differ- ence is that one cannot definen-fold almost split sequences as forn-cluster tilting subcategories, but one is forced to introduce n-fold almost split extensions (see Definition 5.1). This is because in this more general setting there does not exist a unique exact sequence representing this extension. Namely, we have the follow- ing results, where we denote by P(Λ) (respectively I(Λ)) the category of finitely generated projective (respectively injective) Λ-modules.
Theorem 5.10. Let C be ann-precluster tilting subcategory ofmod Λ,X an inde- composable module inZ(C)\ P(Λ), andY :=τn(X)the corresponding indecompos- able module in Z(C)\ I(Λ).
(a) For each0≤i≤n−1, an n-fold almost split extension inExtnΛ(X, Y)can be represented as
0→Y →Cn−1→ · · · →Ci+1→Zi→Ci−1→ · · · →C0→X →0 withZi inZ(C) andCj inC for each j6=i.
(b) The following sequences are exact.
0→HomΛ(C, Y)→HomΛ(C, Cn−1)→ · · · →HomΛ(C, Ci+1)→HomΛ(C, Zi)
→HomΛ(C, Ci−1)→ · · · →HomΛ(C, C0)→radΛ(C, X)→0, 0→HomΛ(X,C)→HomΛ(C0,C)→ · · · →HomΛ(Ci−1,C)→HomΛ(Zi,C)
→HomΛ(Ci+1,C)→ · · · →HomΛ(Cn−1,C)→radΛ(Y,C)→0.
(c) IfX andY do not belong toC, then then-fold almost split extension in (a)can be given as a Yoneda product of a minimal projective resolution ofX in Z(C)
0→ΩiZ(C)(X)→Ci−1→ · · · →C0→X →0, an almost split sequence inZ(C)
0→Ω−(nZ(C)−i−1)(Y)→Zi→ΩiZ(C)(X)→0, and a minimal injective coresolution ofY inZ(C)
0→Y →Cn−1→ · · · →Ci+1→Ω−Z(n(C−)i−1)(Y)→0.
We use this Auslander–Reiten theory to classify then-minimal Auslander–Goren- stein algebras into four disjoint classes, as was done for the casen= 1 in [11].
We refer to related results. In [12], the authors studied (n, m)-ortho-symmetric modules, where our n-precluster tilting modules are precisely (n−1,0)-ortho- symmetric. In [32], the author proved Theorem 4.5 independently.
The paper is organized as follows. In the second section we recall the relative ho- mological algebra over Artin algebras we need (see [8, 9, 10]) and some unpublished results of Maurice Auslander and the second author. In Section 3 the notions of n-precluster tilting subcategories andτn-selfinjective algebras are introduced and their basic properties are discussed. We show in the next section that there is a one-to-one correspondence between finiten-precluster subcategories andn-minimal Auslander–Gorenstein Artin algebras, where then-Auslander algebras are charac- terized within this class. In the fifth section we show that there is a meaningful higher Auslander–Reiten theory inn-precluster tilting subcategories also. This the- ory is transferred in the next section over to the subcategory of maximal Cohen–
Macaulay modules over the n-minimal Auslander–Gorenstein Artin algebras. In the final section we use higher Auslander–Reiten theory to classify the n-minimal Auslander–Gorenstein Artin algebras into four disjoint classes.
Notations. Throughout the paper, all modules are left modules. The composition of morphismsf: X→Y andg:Y →Z is denoted by gf:X →Z.
LetRbe a commutative Artinian ring and Λ an ArtinR-algebra. We denote by mod Λ the category of finitely generated Λ-modules, byP(Λ) (respectively I(Λ)) the category of finitely generated projective (respectively injective) Λ-modules, and byD: mod Λ↔mod Λopthe duality HomR(−, E), whereE is the injective hull of theR-moduleR/radR. We denote byτ: mod Λ→mod Λ the Auslander–Reiten translation.
Let X be a full subcategory of mod Λ. We call X a generator (respectively cogenerator) if Λ ∈ X (respectively DΛ ∈ X). A right X-approximation of A ∈ mod Λ is a morphism f:X →A withX ∈ X such that any morphism g:Y →A withY ∈ X factors through f. We callX contravariantly finite if anyA∈mod Λ has a right X-approximation. Dually, we define a left X-approximation and a
covariantly finitesubcategory. A contravariantly and covariantly finite subcategory is calledfunctorially finite.
2. Preliminaries on relative homological algebra
A systematic study of relative homological algebra over Artin algebras was car- ried out in [8, 9, 10]. We recall the relevant background and results, and in addition we give some unpublished results of Maurice Auslander and the second author.
2.1. Relative homological algebra. We start with the setup for relative homo- logical algebra, where we assume throughout that Λ is an Artin algebra. Relative homological algebra for us begins with defining a set of exact sequences, and this is done through giving an additive sub-bifunctor of Ext1Λ(−,−) (see [8] for further details). Let
F ⊆Ext1Λ(−,−) : (mod Λ)op×mod Λ→Ab
be an additive sub-bifunctor. Such an additive sub-bifunctor is nothing else than, for each pair of Λ-modulesCandA, a chosen set of short exact sequences,F(C, A), starting inAand ending inC, which is closed under pullbacks, pushouts and Baer sums (or direct sums of short exact sequences).
Definition 2.1. An exact sequenceη: 0→A→B →C→0 is said to beF-exact ifη is inF(C, A).
In the rest, we fix a subcategoryX of mod Λ. We consider the following collection FX(C, A) of short exact sequences given for a pair of modulesAandC in mod Λ:
FX(C, A) ={0→A→B →C→0∈Ext1Λ(C, A)|
HomΛ(X, B)→HomΛ(X, C)→0 exact for allX in X }.
Dually one defines FX. By [8, Proposition 1.7] these collections induce additive sub-bifunctorsFX andFX of Ext1Λ(−,−), and
FX =FτX and Fτ−X =FX (2.1)
by [8, Proposition 1.8]. If X = addX holds for someX in mod Λ, we denote FX
andFX byFX andFX respectively.
We can endow mod Λ with a new exact structure by the following result, where we call a full subcategory C of mod Λ F-extension closed if, for every F-exact sequence 0→A→B→C→0 such thatAandC are inC,B is in C.
Proposition 2.2([14]). LetX be a full subcategory ofmod ΛandF=FX (respec- tively F =FX). Thenmod Λhas a structure of an exact category whose short ex- act sequences are precisely theF-exact sequences. More generally, anyF-extension closed subcategoryCofmod Λhas a structure of an exact category whose short exact sequences are precisely theF-exact sequences contained inC.
Proof. The first assertion follows from [14, Propositions 1.4 and 1.7]. The second assertion is a general property of extension closed subcategories of an exact category.
We denote by (mod Λ, F) and (C, F) the exact categories given in Proposition 2.2.
Recall the following definitions from [8], which coincide with the corresponding notions in the exact category (mod Λ, F).
Definition 2.3. (i) A Λ-module P is said to be F-projective if all F-exact se- quences 0→A→B→P →0 split. The full subcategory of mod Λ consisting of allF-projective modules is denoted byP(F).
(ii) A Λ-module I is said to be F-injective if all F-exact sequences 0 → I → B →C→0 split. The full subcategory of mod Λ consisting of allF-injective modules is denoted byI(F).
(iii) F hasenough F-projectives (respectivelyF-injectives) if for each C (respec- tivelyA) in mod Λ there exists anF-exact sequence
0→C′→P →C→0
withP inP(F) (respectively 0→A→I→A′→0 with I inI(F)).
TheFX-projectives andFX-injectives are given as
P(FX) = add{X,P(Λ)} and I(FX) = add{τX,I(Λ)} (2.2) by [8, Proposition 1.10]. For example, we have P(FΛ) = P(Ext1Λ(−,−)) =P(Λ) and I(FDΛ) =I(Ext1Λ(−,−)) =I(Λ). Furthermore an additive sub-bifunctor F of Ext1Λ(−,−) has enough F-projectives and F-injectives if and only if P(F) is functorially finite in mod Λ andF =FP(F)(see [8, Corollary 1.13]). In this case we denote by Ω1F(X) the kernel of theF-projective cover ofX. The Λ-module Ω−F1(Y) is defined dually.
Assume from now on that F is an additive sub-bifunctor of Ext1Λ(−,−) with enoughF-projectives and F-injectives. Recall that an exact sequence
· · · →Ci+1 fi+1
−−−→Ci fi
−→Ci−1→ · · ·
is called F-exact if all the short exact sequences 0→Imfi+1 → Ci → Imfi → 0 areF-exact. Given two modulesAandC in mod Λ, there existF-exact sequences
P: · · · →P2→P1→P0→C→0 withPi inP(F) and
I: 0→A→I0→I1→I2→ · · ·
withIjinI(F). We call these exact sequences anF-projective resolutionand anF- injective coresolution ofCandA, respectively. Fori≥1, thei-th homologies of the complexes HomΛ(P, A) and HomΛ(C,I) are isomorphic, and denoted by ExtiF(C, A) and called thei-thF-relative extension group (see [8, Section 2]). Then Ext1F(−,−) is naturally identified with F. Using ExtiF(−,−), we define F-relative projective dimension,F-relative injective dimension andF-relative global dimension as in the absolute setting and the basic properties are the same in theF-relative setting. On the other hand, we denote by
HomF(A, C) (respectively HomF(A, C))
HomΛ(A, C) modulo all the homomorphisms factoring through an F-projective (respectivelyF-injective) module. TheF-stable category
modFΛ (respectively modFΛ)
has the same objects as mod Λ, and the morphism sets are given by HomF(A, C) (re- spectively HomF(A, C)). ForF =FX, the Auslander–Reiten translationτ: mod Λ≃ mod Λ induces the F-relative Auslander–Reiten translation
τ: modFΛ≃modFΛ. (2.3)
Another central result that has an analog in theF-relative setting is the Auslander–
Reiten formula, which we recall next.
Proposition 2.4 ([8, Proposition 2.3]). Let F be an additive sub-bifunctor of Ext1Λ(−,−) with enough F-projectives (and F-injectives). Then for all modules A andC inmod Λwe have an isomorphism
Ext1F(C, τ A)≃DHomF(A, C).
In general the higherF-relative extension groups ExtiF(C, A) are not necessarily related to the higher absolute extension groups ExtiΛ(C, A). However, in some situations one can compute the absolute extensions byF-relative ones, as described in the next result.
Proposition 2.5([33, Proposition 1.3]). LetX be a functorially finite subcategory of mod Λ.
(a) A module C in mod Λ satisfies ExtiΛ(C,X) = 0 for 0 < i < n if and only if ExtiFX(C, A) = ExtiΛ(C, A) holds for0< i < nand for all Ainmod Λ.
(b) A module A in mod Λsatisfies ExtiΛ(X, A) = 0 for 0 < i < n if and only if ExtiFX(C, A) = ExtiΛ(C, A)for 0< i < n and for allC inmod Λ.
Proof. We only show (a). This is proved in [33, Proposition 1.3] under the assump- tion that X is a cogenerator. We can drop it by considering X′ = add{X,I(Λ)}
and usingFX =FX′.
2.2. The Auslander–Reiten translation revisited. Next we recall some un- published results of Maurice Auslander and the second author that we need later.
We show that the Auslander–Reiten translationτgives a bijection between Ext1F(C, A) and Ext1G(τ(C), τ(A)) for certainF andG, and moreoverτcommutes with relative syzygies.
The transpose is a duality Tr : mod Λ→mod Λop [3]. However, from an exact sequence
0→A→B→C→0
in mod Λ, there is not necessarily a naturally associated exact sequence 0→TrC→TrB⊕X→TrA→0
in mod Λop. We show that when restricting to appropriate classes of exact sequences we have such a natural correspondence.
Proposition 2.6. Let X be a functorially finite generator-cogenerator in mod Λ, and letF =FX,Gop=FD(X)andG=FX.
(a) If 0 → A → B → C → 0 is F-exact in mod Λ, then there are a Gop-exact sequence
0→TrC→TrB⊕P→TrA→0
inmod Λop for some projectiveΛop-moduleP and aG-exact sequence 0→τ(A)→τ(B)⊕I→τ(C)→0
inmod Λfor some injectiveΛ-module I.
(b) For allAandC in mod Λ, we have functorial isomorphisms F(C, A)≃Gop(TrA,TrC)≃G(τ(C), τ(A)).
Proof. (a) Letη: 0→A→B−→f C→0 be anF-exact sequence. By the Horseshoe Lemma we have the following commutative diagram
0 //P1 //
P1⊕Q1 //
Q1 //
0 0 //P0 //
P0⊕Q0 //
Q0 //
0
0 //A //
B //
C //
0
0 0 0
withPi and Qi projective Λ-modules fori= 0,1. Let (−)∗= HomΛ(−,Λ). Then this induces the following commutative diagram
0
0
0
0 //C∗ //
B∗ //
A∗ //
0 0 //Q∗0 //
P0∗⊕Q∗0 //
P0∗ //
0 0 //Q∗1 //
P1∗⊕Q∗1 //
P1∗ //
0 η′: 0 //TrC //
TrB⊕P g //
TrA //
0
0 0 0
whereP is a projective Λop-module. The columns are exact by the definition of Tr, and the top row is exact becauseηisF-exact andX contains Λ. Since the second and the third rows are split exact, the Snake Lemma shows that the bottom row η′ is also exact. It remains to show that η′ is Gop-exact. Then the dual of this sequence isG-exact.
Sinceη isF-exact, the mapf: HomΛ(B,X)→HomΛ(A,X) is surjective. Since Tr is a duality, the map g = Trf: HomΛop(TrX,TrB) → HomΛop(TrX,TrA) is surjective. By a standard argument, the map g: HomΛop(TrX,TrB ⊕P) → HomΛop(TrX,TrA) is also surjective. SinceτTrX =D(X) and henceFTrX =Gop holds,η′ isGop-exact.
(b) By (a), we have a morphismF(C, A)→Gop(TrA,TrC) of bifunctors. Since the same argument gives the inverse morphism Gop(TrA,TrC) →F(C, A) of bi- functors, we have the first desired isomorphism. The second isomorphism follows
immediately by applying the dualityD.
We end this subsection by showing how this induces isomorphisms on relative extension groups and relative stable homomorphism sets.
Theorem 2.7. i)LetX ⊆mod Λbe a functorially finite generator-cogenerator, and letF=FXandG=FX be the corresponding additive sub-bifunctors ofExt1Λ(−,−).
Then the following is true.
(a) τ gives an equivalence modFΛ≃modGΛ.
i)Theorem 2.7 was obtained by Maurice Auslander and the second author, but they never got to be published in printed form before now. However, the results were presented at a seminar at the University of Bielefeld, Germany.
(b) The following diagrams commute up to isomorphisms of functors.
modFΛ τ //
ΩF
modGΛ
ΩG
modFΛ τ //modGΛ
modFΛoo τ
−
OO
Ω−F
modGΛ
OO
Ω−G
modFΛoo τ
−
modGΛ (c) τ induces a functorial isomorphism in both variables
ϕn =ϕC,A,n: ExtnF(C, A)≃ExtnG(τ(C), τ(A)) for all pairs ofA andC inmod Λandn≥1.
Proof. (a) We have an equivalence τ: modFΛ ≃ modFΛ in (2.3). Since X is a generator-cogenerator,I(F) = P(G) holds by (2.2). Thus modFΛ = modGΛ holds, and the assertion follows.
(b) We only prove commutativity of the left diagram. Let 0→ΩF(A)→P →A→0
beF-exact withP in P(F). By Proposition 2.6, we have aG-exact sequence 0→τΩF(A)→τ(P)⊕I→τ(A)→0
with I in I(Λ). Since τ(P)⊕I is in P(G) by (2.2), we have an isomorphism τΩF ≃ΩGτ of functors.
(c) We have the following functorial isomorphisms ExtnF(C, A)≃Ext1F(ΩnF−1(C), A)
≃Ext1G(τΩnF−1(C), τ(A))
≃Ext1G(ΩnG−1τ(C), τ(A))
≃ExtnG(τ(C), τ(A))
by dimension shift, Proposition 2.6(b) and (b), where all the involved isomorphisms
are functorial. The claim follows.
2.3. Relative tilting theory. Tilting theory is an important topic in representa- tion theory of Artin algebras and elsewhere. It also has a relative version, which we recall from [9, 10]. Here we always assume that our additive sub-bifunctor F has enoughF-projectives and enoughF-injectives. For a subcategoryC in mod Λ, let
C⊥F = {M ∈mod Λ|ExtiF(C, M) = 0 for alli >0},
⊥F
C = {M ∈mod Λ|ExtiF(M,C) = 0 for alli >0}.
WhenF = Ext1Λ(−,−), we simply denoteC⊥F and⊥FCbyC⊥and⊥Crespectively.
Definition 2.8. We call T in mod Λ anF-cotilting module if (i) idFT <∞,
(ii) ExtiF(T, T) = 0 fori >0,
(iii) for allI inI(F) there exists anF-exact sequence 0→Tn→Tn−1→ · · · →T1→T0→I→0 withTi in addT.
It is shown in [9, Corollary 3.14] that if there exists an F-cotilting module T, thenP(F) and henceI(F) are of finite type. In fact, they contain the same number of non-isomorphic indecomposable objects as addT.
Next we collect the basic results on relative cotilting modules that we need later.
Theorem 2.9 ([9]). Let X be a generator in mod Λ and F =FX. Let T be an F-cotiltingΛ-module andΓ = EndΛ(T). Then we have the following.
(a) Λ≃EndΓ(T).
(b) AnyC in ⊥FT has an F-exact sequence 0→C→T0
f0
−→T1 f1
−→T2 f2
−→T3→ · · · withTi inaddT andImfi in ⊥FT for alli≥0.
(c) ExtiF(C, A)≃ExtiΓ(HomΛ(A, T),HomΛ(C, T))for all modulesAandCin⊥FT andi≥0.
(d) The module U = HomΛ(X, T) is a cotilting Γ-module with idFT ≤ idΓU ≤ idFT + 2ii).
(e) HomΛ(−, T) : mod Λ→mod ΓandHomΓ(−, T) : mod Γ→mod Λinduce quasi- inverse dualities HomΛ(−, T) :⊥FT →⊥U andHomΓ(−, T) :⊥U →⊥FT.
Proof. (a) is [9, Corollary 3.4], (b) is [9, Theorem 3.2(a)], (c) is [9, Proposition 3.7], (d) is [9, Theorem 3.13(d)], and (e) is [9, Corollary 3.6(a), Proposition 3.8(b)].
Now let F be an additive sub-bifunctor of Ext1Λ(−,−), and X a generator of mod Λ. ThenX-resdimFM is defined to be the infimum ofnsuch that there exists anF-exact sequence
0→Xn→Xn−1→ · · · →X1→X0→M →0,
where Xi is in X for all i ≥ 0. We denote by Xb the full subcategory of mod Λ consisting of allM in mod Λ withX-resdimFM <∞. LetX-resdimF(mod Λ) :=
sup{X-resdimFM |M ∈mod Λ}.
The following result is taken from [10] (with the exception of (c)), which con- nects special direct summands of absolute cotilting modules with relative cotilting modules. Let Γ be an Artin algebra andX in mod Γ. Recall that a direct summand Y ofX is said to be adualizing summand of X if there exists an exact sequence
0→X −→f Y0→Y1
withf a left (addY)-approximation andYiin addY. This is shown to be equivalent to the natural homomorphism
X →HomΛ(HomΓ(X, Y), Y)
being an isomorphism, where Λ = EndΓ(Y) [10, Proposition 2.1].
Theorem 2.10. Let Γ be an Artin algebra and U a cotilting Γ-module. For a dualizing summand T of U, let Λ = EndΓ(T),X = HomΓ(U, T) andF =FX an additive sub-bifunctor of Ext1Λ(−,−). Then the following assertions hold.
(a) Γ≃EndΛ(T).
(b) T is anF-cotilting Λ-module with idFT ≤max{idΓU ,2}.
(c) IfT is injective as a Γ-module, thenidFT ≤max{idΓU−2,0}.
ii)This inequality is a corrected version of [9, Theorem 3.13(d)] due to an error in the statement of [9, Proposition 3.11], where the bound should be idFT + 2, instead of max{idFT ,2}.
Proof. (a) is [10, Proposition 2.4(b)], and (b) is [10, Proposition 2.7(c)].
(c) Letr= idΓUandt= max{r−2,0}. We prove that (⊥FT)-resdimF(mod Λ)≤ t, as this implies that idFT ≤t. This is done by showing that ΩtF(C) is in⊥FT for allC in mod Λ.
Assume that ΓT is injective. Then by [11, Lemma 2.4] the module ΛT is a cogenerator in mod Λ, and therefore
C≃HomΓ(HomΛ(C, T), T)
for all modulesC in mod Λ. In particular,C≃HomΓ(B, T) for some Γ-moduleB.
Let
P: B f
0
−→U0 f
1
−→U1 f
2
−→U2 f
3
−→ · · ·,
be a sequence of minimal left (addU)-approximations of B, and let B0 =B and Bj = Cokerfj−1 forj ≥1. Then anF-projective resolution ofC is given by
HomΓ(P, T) : · · · →HomΓ(U2, T)→HomΓ(U1, T)→HomΓ(U0, T)→C→0, and therefore we have, for everyj≥0,
ΩjF(C) = HomΓ(Bj, T).
Let Σ = EndΓ(U). Then U is a cotilting Σ-module with idΣU = idΓU. The above complexPgives rise to a projective resolution of the Σ-module HomΓ(B, U),
· · · →HomΓ(U2, U)→HomΓ(U1, U)→HomΓ(U0, U)→HomΓ(B, U)→0.
Given a projective presentation F1 →F0→B →0 of the Γ-module B, it induces an exact sequence
0→HomΓ(B, U)→HomΓ(F0, U)→HomΓ(F1, U)→B′ →0
of Σ-modules with HomΓ(F0, U) and HomΓ(F1, U) in addΣU. Then Ωj+2Σ (B′) = HomΓ(Bj, U) holds for everyj≥0.
Now we show HomΓ(Bt, U) is in⊥ΣU. Ifr≤2, then HomΓ(Bt, U) = HomΓ(B, U) = Ω2Σ(B′) is in⊥ΣU since idΓU =r≤2. Ifr >2, then HomΓ(Bt, U) = HomΓ(Br−2, U) = ΩrΣ(B′) is in⊥ΣU since idΓU =r.
By (a), (b) and Theorem 2.9(e), we have dualities
⊥
ΣU ←−−−−−−−HomΓ(−,U) ⊥ΓU −−−−−−−→HomΓ(−,T) ⊥FT .
We takeB′′ in⊥ΓU such that HomΓ(B′′, U)≃HomΓ(Bt, U) as Σ-modules. Then ΩtF(C) = HomΓ(Bt, T)≃HomΓ(B′′, T)∈⊥FT
as Λ-modules. Thus the claim holds.
Let F be an additive sub-bifunctor of Ext1Λ(−,−). A full subcategory X of mod Λ isF-resolving (respectivelyF-coresolving) if
(i) X isF-extension closed,
(ii) P(F) (respectivelyI(F)) is contained inX,
(iii) if 0→A →B → C →0 is F-exact and B and C are in X (respectively A andB are in X), thenA(respectivelyC) is inX.
We need the following preparation from Auslander–Buchweitz theory.
Proposition 2.11 ([9, Theorems 2.4, 2.5, Proposition 2.2]). Let F be an additive sub-bifunctor ofExt1Λ(−,−), andX an F-resolving subcategory ofmod Λ. Assume that the exact category(X, F)given in Proposition 2.2has enoughF-injectives and Xb= mod Λ. Then the following assertions hold.
(a) X is a contravariantly finite subcategory of mod ΛandY :=X⊥F is a covari- antly finite subcategory ofmod Λ.
(b) X-resdimF(mod Λ) = idF(X⊥F)holds.
3. Elementary properties of n-precluster tilting subcategories Recall that forn≥1 a subcategoryC of mod Λ is called n-cluster tilting ifC is functorially finite and
C=⊥n−1C=C⊥n−1,
where⊥n−1C is the full subcategory of mod Λ given by the modules
⊥n−1C={X ∈mod Λ|ExtiΛ(X,C) = 0 for 0< i < n}.
The full subcategoryC⊥n−1 is defined dually. In particular, it follows immediately from the definition thatCis a generator-cogenerator for mod Λ and ExtiΛ(C,C) = 0 for 0< i < n. In [21] the functors
τn=τΩnΛ−1: mod Λ→mod Λ and τn−=τ−Ω−Λ(n−1): mod Λ→mod Λ are defined as then-Auslander–Reiten translations. By [22, Theorem 2.3.1], the pair (τn−, τn) forms an adjunction. By [21, Theorem 1.4.1], they induce equivalences
τn:⊥n−1Λ→DΛ⊥n−1 and τn−:DΛ⊥n−1→⊥n−1Λ. (3.1) In particular τn and τn− give bijections between indecomposable non-projective modules in ⊥n−1Λ and indecomposable non-injective modules in DΛ⊥n−1, while they do not preserve indecomposability for arbitrary modules. Moreover, for any n-cluster tilting subcategoryC of mod Λ, they restrict to equivalences
τn:C → C and τn−:C → C.
The next result gives a higher analog of Auslander–Reiten duality.
Lemma 3.1 ([21, Theorem 1.5]). We have the following.
(a) HomΛ(C, A) ≃ DExtnΛ(A, τn(C)) and ExtiΛ(C, A) ≃ DExtnΛ−i(A, τn(C)) for 0< i < n, for all modulesC in ⊥n−1Λ and all modulesA inmod Λ.
(b) HomΛ(C, A) ≃DExtnΛ(τn−(A), C) and ExtiΛ(C, A) ≃DExtnΛ−i(τn−(A), C) for 0< i < n, for all modulesC in mod Λand all modules Ain DΛ⊥n−1.
Proof. We give a proof of the second isomorphisms in our language of relative homological algebra.
(a) LetF =FΛbe an additive sub-bifunctor of Ext1Λ(−,−). LetCbe in⊥n−1Λ, and let
η: 0→Ωn−1Λ (C)→Pn−2→ · · · →P0→C→0
be a minimal projective resolution of C. Since C is in ⊥n−1Λ, we have that the exact sequence η is F-exact. Using this we have for 0 < i < n and an arbitrary
Λ-moduleAthat
DExtiF(C, A)≃DExt1F(ΩiΛ−1(C), A)
≃DHomF(ΩiΛ(C), A)
≃Ext1F(τ−(A),ΩiΛ(C))
≃ExtnF−i(τ−(A),ΩnΛ−1(C))
≃ExtnΛ−i(A, τn(C)),
where the second isomorphism uses thatPi is inI(F), the third is the Auslander–
Reiten formula, the first and the fourth use dimension shift and the last one is given by Theorem 2.7 andG=FΛ= Ext1Λ(−,−).
SinceC is in ⊥n−1Λ, we have that ExtiΛ(C, A)≃ExtiF(C, A) for 0< i < n and for all modulesA in mod Λ by Proposition 2.5(a). Thus the assertion follows.
(b) Similar proof as in (a).
We introduce the notion of an n-precluster tilting subcategory by relaxing the conditions for ann-cluster tilting subcategory.
Definition 3.2. A subcategory Cof mod Λ is called ann-precluster tilting subcat- egory if it satisfies the following conditions.
(i) C is a generator-cogenerator for mod Λ, (ii) τn(C)⊆ C andτn−(C)⊆ C,
(iii) ExtiΛ(C,C) = 0 for 0< i < n,
(iv) C is a functorially finite subcategory of mod Λ.
If moreoverC admits an additive generatorM, we say that M is ann-precluster tilting module.
Usingτn andτn−, we define the subcategories
Pn= add{τn−i(Λ)}∞i=0 and In = add{τni(D(ΛΛ))}∞i=0. For anyn-precluster tilting subcategoryC of mod Λ, we have
Pn∨ In⊆ C and C ⊆DΛ⊥n−1∩⊥n−1Λ by (i), (ii) and (i), (iii) respectively.
Recall from [11] that an Artin algebra Λ is calledτ-selfinjectiveifP1is of finite type, which is shown to be equivalent to thatP1is equal toI1. We show next that this is equivalent to the existence of a 1-precluster tilting Λ-module.
Example 3.3. An Artin algebra Λ is τ-selfinjective if and only if Λ has a 1- precluster tilting module.
Proof. If Λ is τ-selfinjective, then clearly the additive generator of P1 is a 1- precluster tilting module since DΛ ∈ I1 = P1. If Λ has a finite 1-precluster subcategoryC, then it is clear from the definition thatP1 is contained inC. Hence
P1is of finite type and Λ isτ-selfinjective.
This observation leads us to the following definition.
Definition 3.4. An Artin algebra Λ is called τn-selfinjective if Λ admits an n- precluster tilting module.
This is a common generalization of selfinjective algebras andn-representation- finite algebras. We continue by asking and giving one answer to the natural ques- tion: When is an Artin algebraτn-selfinjective?
Proposition 3.5. Let Λ be an Artin algebra and n ≥ 1. Then the following conditions are equivalent.
(i) Λisτn-selfinjective.
(ii) Pn∨ In is of finite type and ExtiΛ(Pn∨ In,Pn∨ In) = 0for 0< i < n.
(iii) In is of finite type,In⊂⊥n−1Λ andExtiΛ(In,In) = 0 for 0< i < n.
(iv) Λ∈ In andExtiΛ(In,In) = 0 for 0< i < n.
(v) Pn is of finite type,Pn⊂DΛ⊥n−1 andExtiΛ(Pn,Pn) = 0 for 0< i < n.
(vi) DΛ∈ Pn andExtiΛ(Pn,Pn) = 0for 0< i < n.
Moreover, if these conditions are satisfied, then every additive generator ofPn∨ In
is an n-precluster tiltingΛ-module.
Proof. (i) is equivalent to (ii): Assume that there exists an n-precluster tilting moduleM in mod Λ. Since Pn∨ In ⊂addM, it is immediate that Pn∨ In is of finite type and satisfies ExtiΛ(Pn∨ In,Pn∨ In) = 0 for 0< i < n. Conversely, if Pn∨ Insatisfies (ii), then an additive generator ofPn∨ Inis ann-precluster tilting Λ-module.
(ii) implies (iii): This is immediate.
(iii) implies (iv): For each indecomposable non-projective moduleX in ⊥n−1Λ, we know thatτn(X) is indecomposable again by the equivalence (3.1). SinceIn is of finite type, then τnl(I)6= 0 is projective for some l ≥0 for all indecomposable injective modules I. Since the number of indecomposable projectives and of inde- composable injectives coincide, all indecomposable projective modules must occur in this way, hence Λ is inIn and (iv) is satisfied.
(iv) implies (ii): For each indecomposable projective modules P there exists an indecomposable injective module I such that P ≃τnl(I). Since τn−i(P)≃τnl−i(I) for 0≤i≤landτn−(l+1)(P) = 0 hold, we havePn∨ In=In. Thus (ii) is satisfied.
The equivalences of (ii), (v) and (vi) are shown dually.
Using the above we have the following consequence forn= 2 thanks to a general property ofI2.
Proposition 3.6(cf. [23, Proposition 1.7]). LetΛ be an Artin algebra. Then Λis τ2-selfinjective if and only if Λ∈ I2.
Proof. By [23, Proposition 2.5] the subcategoryI2 satisfies Ext1Λ(I2,I2) = 0. The claim then follows immediately from Proposition 3.5(iv)⇒(i).
The next results give analog of properties of higher Auslander–Reiten translation forn-cluster tilting subcategories.
Proposition 3.7. LetΛ be an Artin algebra andCan n-precluster tilting subcate- gory with n≥1. Then we have the following.
(a) We have mutually quasi-inverse equivalences τn:C → C andτn−:C → C.
(b) τn andτn− give bijections between indecomposable non-projective modules in C and indecomposable non-injective modules inC.
(c) We haveadd{τn−(C),Λ}=C= add{τn(C), DΛ}.
(d) There exists a full subcategory D of mod Λ such that C = add{Pn ∨ In,D}, (Pn∨ In)∩ D={0} andτn(D) =D=τn−(D).
Proof. (a) We haveC ⊆DΛ⊥n−1 ∩⊥n−1Λ. Since τn(C)⊆ C andτn−(C)⊆ C hold, the claim follows from the equivalences (3.1).
(b)(c)(d) These follow immediately from (a).
An n-cluster tilting subcategory C satisfies by definition the equalities C =
⊥n−1C = C⊥n−1. The same is not true for an n-precluster tilting subcategory C of mod Λ (e.g. Λ is non-semisimple selfinjective and C= add Λ). But the equality
⊥n−1C=C⊥n−1 is shown still to be true.
Proposition 3.8. Let C be a subcategory of mod Λ satisfying the conditions (i), (iii)and (iv)in Definition 3.2.
(a) We haveΩ−FC(⊥n−1C)⊆⊥n−1C andΩFC(C⊥n−1)⊆ C⊥n−1.
(b) Assumen >1. ThenC is an n-precluster tilting subcategory of mod Λif and only ifC⊥n−1 =⊥n−1C.
Proof. (a) We only prove the first inclusion since the other one is dual.
LetX be in⊥n−1C, andη: 0→X −→f C0→Y →0 be an exact sequence with a minimal leftC-approximationf ofX. Applying HomΛ(−,C), one easily shows that Y is in⊥n−1C.
(b) It suffices to show thatτn(C)⊆ Candτn−(C)⊆ Chold if and only ifC⊥n−1=
⊥n−1
Cholds.
Assumeτn(C)⊆ Candτn−(C)⊆ C. SinceC ⊂⊥n−1Λ, we have ExtiΛ(C,⊥n−1C)≃DExtnΛ−i(⊥n−1C, τn(C)) = 0
for all 0 < i < n by Lemma 3.1. Thus⊥n−1C ⊆ C⊥n−1 holds. Similarly C⊥n−1 ⊆
⊥n−1
Cholds. ConsequentlyC⊥n−1=⊥n−1C.
Assume thatC⊥n−1=⊥n−1C. By Lemma 3.1, we have
ExtiΛ(⊥n−1C, τn(C))≃DExtnΛ−i(C,⊥n−1C) =DExtnΛ−i(C,C⊥n−1) = 0 for all 0 < i < n. Thus τn(C)⊆(⊥n−1C)⊥n−1. We show (⊥n−1C)⊥n−1 ⊆ C. Since C ⊆⊥n−1C, we have⊥n−1C =C⊥n−1 ⊇(⊥n−1C)⊥n−1. For any X in (⊥n−1C)⊥n−1, there exists an exact sequence η: 0 → X → C0 → Y → 0 with C0 ∈ C and Y ∈⊥n−1Cby (a). This splits by the assumptionn >1, andX is a direct summand ofC0. HenceX is inC.
Similarly we prove thatτn−(C)⊆ C.
Now we introduce the following categoryU(C), which is an analog of Calabi–Yau reduction of triangulated categories [27].
Definition 3.9. For ann-precluster tilting subcategoryC in mod Λ, let Z(C) =C⊥n−1=⊥n−1C and U(C) =Z(C)/[C].
Note that whenCis a 1-precluster tilting subcategory, thenZ(C) = mod Λ, since the orthogonality condition is void.
The next result gives basic properties ofZ(C) which generalize those ofn-cluster tilting subcategories [21, Theorems 2.3, 2.3.1, 2.2.3]. In particular it gives higher Auslander–Reiten translation forZ(C) extending Proposition 3.7.
Theorem 3.10. LetΛbe an Artin algebra andCann-precluster tilting subcategory of mod Λwithn≥1. Then we have the following.
(a) We have equivalencesτn:Z(C)→ Z(C) andτn−:Z(C)→ Z(C).
(b) τn and τn− give bijections between indecomposable non-projective modules in Z(C)and indecomposable non-injective modules inZ(C).
(c) We haveadd{τn(Z(C)), DΛ}=Z(C) = add{τn−(Z(C)),Λ}.
(d) For everyX andY in Z(C)and0< i < n, we have functorial isomorphisms HomΛ(X, Y)≃DExtnΛ(Y, τn(X)), ExtiΛ(X, Y)≃DExtnΛ−i(Y, τn(X)) HomΛ(X, Y)≃DExtnΛ(τn−(Y), X), ExtiΛ(X, Y)≃DExtnΛ−i(τn−(Y), X).
(e) For everyX inmod Λ, there exists anFC-exact sequence 0→Zn−1→Cn−2→ · · · →C0→X →0 withCi inC for every iandZn−1 in Z(C).
Proof. (a) Thanks to the equivalences (3.1), it suffices to show τn(Z(C))⊆ Z(C) andτn−(Z(C))⊆ Z(C). Using Lemma 3.1, we have
0 =DExtiΛ(Z(C),C)≃ExtnΛ−i(C, τn(Z(C))) 0 =DExtiΛ(C,Z(C))≃ExtnΛ−i(τn−(Z(C)),C) for 0< i < n. Thus the assertion follows.
(b)(c) Immediate from (a).
(d) This follows from Lemma 3.1.
(e) Let X be in mod Λ, and let 0 → ΩFC(X) → C0 → X → 0 be an FC- exact sequence given by anFC-projective cover. Then Ext1Λ(C,ΩFC(X)) = 0 holds.
Taking anFC-exact sequence 0→Ω2FC(X)→C1 →ΩFC(X)→0 given by an FC- projective cover, it follows that ExtiΛ(C,Ω2FC(X)) = 0 fori= 1,2. Continuing this process we obtain that ExtiΛ(C,ΩnF−C1(X)) = 0 for 0< i < n, and hence ΩnF−C1(X)
is inZ(C).
The following easy property below is useful.
Lemma 3.11. LetC be ann-precluster tilting subcategory of mod Λwithn≥1.
(a) We haveFC|Z(C)×Z(C)=FC|Z(C)×Z(C).
(b) For every0< i < n, we haveExtiFC(−,−)|Z(C)×Z(C)= ExtiΛ(−,−)|Z(C)×Z(C)= ExtiFC(−,−)|Z(C)×Z(C).
(c) The exact categories(Z(C), FC)and(Z(C), FC)are the same. It coincides with (Z(C),Ext1Λ(−,−))ifn≥2.
Proof. (a) (c) Forn= 1, we haveFC =FτC =FCby (2.1) sinceCis 1-precluster tilt- ing, and forn≥2, we haveFC|Z(C)×Z(C)= Ext1Λ(−,−)|Z(C)×Z(C)=FC|Z(C)×Z(C).
(b) This is immediate from Proposition 2.5.
The categoryZ(C) enjoys the following remarkable properties.
Proposition 3.12. Let C be ann-precluster tilting subcategory ofmod Λfor some n≥1.
(a) Z(C)is extension closed.
(b) Z(C) has a structure of a Frobenius category whose short exact sequences are preciselyFC-exact sequences, and projective-injective objects are precisely C.
(c) U(C) has a structure of a triangulated category with the suspension functor [1] = Ω−FC1.