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On generalized cluster categories

Claire Amiot

Abstract.Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten andTodorov in order to categorify Fomin-Zelevinsky cluster algebras. This survey motivatesand outlines the construction of a generalization of cluster categories, and explains dif-ferent applications of these new categories in representation theory.

2010 Mathematics Subject Classification.16Gx; 18Ex; 13F60; 16W50; 16E35; 16E45.

Keywords. cluster categories, 2-Calabi-Yau triangulated categories, cluster-tilting the-ory, quiver mutation, quivers with potentials, Jacobian algebras, preprojective algebras.

Introduction

In 2003, Marsh, Reineke and Zelevinsky attempted in[79] to understand Fomin-Zelevinsky cluster algebras (defined in[40]) in terms of representations of quivers.This article was immediately followed by the fundamental paper [24] of Buan,Marsh, Reineke, Reiten and Todorov which stated the definition of cluster cate-gories. In this paper, the authors associate to each finite dimensional hereditaryalgebra a triangulated category endowed with a special set of objects called cluster-tilting. The combinatorics of these ob jects is closely related to the combinatoricsof acyclic cluster algebras, and especially with the mutation of quivers.

The same kind of phenomena appear naturally in the stable categories of mod-ules over preprojective algebras of Dynkin type, and have been studied by Geiss,Leclerc and Schroer in [45], [47]. Actually most of the results on cluster-tiltingtheory of [24],[26],[27], [45],[47] have been proved in the more general setting ofHom-finite, triangulated, 2-Calabi-Yau categories with cluster-tilting objects (see[70], [63],[22]).

Since then, other categories with the above properties have been constructedand investigated. One finds stable subcategories of modules over a preprojectivealgebra of type Q associated with any element in the Coxeter group of Q (see[22], [23], [50], [51]); stable Cohen-Macaulay modules over isolated singularities

(see [14], [60], [29]); and generalized cluster categories associated with finite di-mensional algebras of global dimension at most two, or with Jacobi-finite quiverswith potential (introduced in[2], [3]).

The context of this survey is this last family of triangulated 2-Calabi-Yau cat-egories: the generalized cluster categories. The aim, here, is first to give somemotivation for enlarging the family of cluster categories (Sections 1 and 2). Thenwe will explain the general construction of these new cluster categories (Section
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3), and link these new categories with the categories cited above (Section 4). Fi-

nally, we give some applications in representation theory of the cluster categoriesassociated with algebras of global dimension at most 2 (Section 5).

Contents

1 Motivation

1.1 Mutation of quivers1.2 Cluster category of typeA31.3 Stable module category of a preprojective algebra of typeA3

2 2-Calabi-Yau categories and cluster-tilting theory

2.1 Iyama-Yoshino mutation2.2 2-Calabi-Yau-tilted algebras2.3 Two fundamental families of examples2.4 Quivers with potential

3 From 3-Calabi-Yau DG-algebras to 2-Calabi-Yau categories

3.1 Graded algebras and DG algebras3.2 General construction3.3 Ginzburg DG algebras3.4 Application to surfaces with marked points3.5 Derived preprojective algebras3.6 Link between the two constructions3.7 Higher generalized cluster categories

4 Stable categories as generalized cluster categories

4.1 Preprojective algebras of Dynkin type4.2 Stable categories associated to words4.3 Cohen Macaulay modules over isolated singularities

5 On the Z-grading on the 3-preprojective algebra

5.1 Grading of 3()5.2 Mutation of graded QPs5.3 Application to cluster equivalence5.4 Algebras of acyclic cluster type5.5 Gradable 3-modules and piecewise hereditary algebras

Acknowledgments. I deeply thank Andrzej Skowronski for encouraging me towrite this survey. Some of the results presented here are part of my Ph.D. thesisunder the supervision of Bernhard Keller. I thank him for his constant supportand helpful comments and suggestions on a previous version of this survey. I amalso grateful to Steffen Oppermann for helpful comments. I thank him and myother coauthors Osamu Iyama, Gordana Todorov and Idun Reiten for pleasantcollaborations.

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On generalized cluster categories 3

Notation. k is an algebraically closed field. All categories considered in this

paper arek-linear additive Krull-Schmidt categories. By Hom-finite categories, wemean, categories such that Hom(X, Y) is finite dimensional for any objectsX andY. We denote byD = Homk(, k) the k-dual. The tensor products are over thefield k when not specified. For an object T in a categoryC, we denote by add (T)the additive closure ofT, that is the smallest full subcategory ofCcontaining Tand stable under taking direct summands.

All modules considered here are right modules. For a k-algebra A, we denoteby ModA the category of right modules and by modA the category of finitelypresented rightA-modules.

A quiverQ = (Q0, Q1, s , t) is given by a set of vertices Q0, a set of arrows Q1,a source maps : Q1 Q0and a target mapt : Q1 Q0. We denote by ei, i Q0the set of primitive idempotents of the path algebra kQ. By a Dynkin quiver, wemean a quiver whose underlying graph is of simply laced Dynkin type.

1. Motivation

1.1. Mutation of quivers. In the fundamental paper[40], Fomin and Zelevinskyintroduced the notion of mutation of quivers as follows.

Definition 1.1(Fomin-Zelevinsky[40]). LetQbe a finite quiver without loops andoriented cycles of length 2 (2-cycles for short). Letibe a vertex ofQ. Themutationof the quiverQ at the vertexi is a quiver denoted by i(Q) and constructed fromQ using the following rule:

(M1) for any couple of arrowsj i k, add an arrow j k;

(M2) reverse the arrows incident withi;

(M3) remove a maximal collection of 2-cycles.

This definition is one of the key steps in the definition ofCluster Algebras. Evenif the initial motivation of Fomin and Zelevinsky to define cluster algebras was togive a combinatorial and algebraic setup to understand canonical basis and totalpositivity in algebraic groups (see[77], [78],[50]), these algebras have led to manyapplications in very different domains of mathematics:

representations of groups of surfaces (higher Teichmuller theory [37], [38],[39]);

discrete dynamical systems, (Y-systems, integrable systems [43],[69]);

non commutative algebraic geometry (Donaldson-Thomas invariants [74],Calabi-Yau algebras [53]);

Poisson geometry[52];

quiver and finite dimensional algebras representations.

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4 C. Amiot

This paper is focused on this last connection and on the new insights that cluster

algebras have brought in representation theory. However, in order to make thissurvey not too long, we will not discuss cluster algebras and their precise linkswith representation theory, but we will concentrate on the link between certaincategories and quiver mutation. We refer to [67], [25], [82] for nice overviews ofcategorifications of cluster algebras.

Example 1.2.

2

2

2 2

1

3

2

1 3

1

1

3

3

1

3

.

One easily checks that the mutation at a given vertex is an involution. If iis a source (i.e. there are no arrows with target i) or a sink (i.e. there are noarrows with source i), the mutation i consists only in reversing arrows incidentwithi (step (M2)). Hence it coincides with the reflection introduced by Bernstein,Gelfand, and Ponomarov [17]. Therefore mutation of quivers can be seen as ageneralization of reflections (which are just defined ifi is a source or a sink).

The BGP reflections have really nice applications in representation theory, andare closely related to tilting theory of hereditary algebras. For instance, combiningthe functorial interpretation of reflections functors by Brenner and Butler [20], theinterpretation of tilting modules for derived categories by Happel[56, Thm 1.6] andits description of the derived categories of hereditary algebras[56,Cor. 4.8], we ob-tain that the reflections characterize combinatorially derived equivalence between

hereditary algebras.

Theorem 1.3 (Happel). LetQ andQ be two acyclic quivers. Then the algebraskQ andkQ are derived equivalent if and only if one can pass fromQ to Q usinga finite sequence of reflections.

One hopes therefore that the notion of quiver mutation has also nice applica-tions and meaning in representation theory.

In the rest of the section, we give two fundamental examples of categories wherethe combinatorics of the quiver mutation of 1 2 3 appear naturally. Thesecategories come from representation theory, they are very close to categories offinite dimensional modules over finite dimensional algebras.

1.2. Cluster category of type A3. Let Q be the following quiver 1 2 3. We consider the category modkQ of finite dimensional (right) modules overthe path algebra kQ. We refer to the books [12], [15], [44], [84] for a wealth ofinformation on the representation theory of quivers and finite dimensional algebras.

Since A3 is a Dynkin quiver, this category has finitely many indecomposable

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modules. The Auslander-Reiten quiver is the following:

1 2 3

21

32

321

Here the simples associated with the vertices are symbolized by 1 , 2 , 3 . The

module 3

21

is an indecomposable module M such that there exists a filtration

M1 M2 M3 = M with M1 1 , M2/M1 2 and M3/M2 3 . In this

situation, it determines the module M up to isomorphism. There are three pro-jective indecomposable objects corresponding to the vertices ofQ and which are

P1 = e1kQ = 1 , P2 = e2kQ = 21 and P3 = e3kQ = 3

21

. The dotted lines describe

the AR-translation that we denote by; it induces a bijection between indecompos-able non projective kA3-modules and indecomposable non injective kA3-modules.Dotted lines also correspond to minimal relations in the AR-quiver ofmodkQ.

The bounded derived categoryDb(A), where A is a finite dimensional algebraof finite global dimension, is a k-category whose ob jects are bounded complexes offinite dimensional (right)A-modules. Its morphisms are obtained from morphismsof complexes by inverting formally quasi-isomorphisms. We refer to [57](see also

[66]) for more precise description. For an ob jectM Db(A) we denote by M[1]the shift complex defined by M[1]n :=Mn+1 and dM[1] = dM. This category isa triangulated category, with suspension functor MM[1] corresponding to theshift. The functor modA Db(A) which sends a A-module M on the complex. . . 0 M0 . . . concentrated in degree 0 is fully faithful. Moreover for anyM and N in mod (A) and i Z we have:

HomDb(A)(M, N[i]) ExtiA(M, N).

The categoryDb(A) has a Serre functor S = L

A DA(which sends the projectiveA-module eiA on the injective A-module Ii =eiDA) and an AR-translation =

L

A DA[1] which extends the AR-translation ofmodAand that we still denoteby.

In the case A= kQ, the indecomposable objects are isomorphic to stalk com-plexes, that is, are of the form X[i], where i Z and X is an indecomposablekQ-module. The AR-quiver ofDb(kQ) is the following infinite quiver (cf. [56]):

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6 C. Amiot

321

[1] S2( 1 [1]) 1 2 3321

[1]

32 [1] S2(

21 [1])

21

32

21 [1]

2 [1] S2( 3 ) 3 [1] S2(321

[1]) 3

21

1 [1] 2 [1] S12 ( 1 )

The functor S2:= S[2] =[1] acts bijectively on the indecomposable objectsofDb(kQ). It is an auto-equivalence ofDb(kQ).

The cluster category CQ is defined to be the orbit category ofDb(kQ) by the

functorS2: the indecomposable objects are the indecomposable objects ofDb(kQ).The space of morphisms between two objects in Db(kQ) is given by

HomC(X, Y) =iZ

HomD(X, Si2Y).

Hence the objects X and Si2X become isomorphic in CQ for any i Z. We denoteby (X) the S2-orbit of an indecomposableX. One can see that there are finitelymany indecomposable objects in CQ. They are of the form (X), where X is anindecomposablekQ-module, orXP[1] whereP is an indecomposable projectivekQ-module. Hence the AR-quiver of the category CQ is the following:

( 1 ) ( 2 ) ( 3 ) (321

[1])

( 21 ) (32 ) (

21 [1])

(321

[1]) (321

) ( 1 [1]) ( 1 )

where the two vertical lines are identified. Therefore one can view the categoryCQ

as the module categorymod kQ with extra objects (( 1 [1]), ( 21 [1]),( 321[1])) and

extra morphisms: ifM andN arekQ-modules, then

HomC(M, N) HomkQ(M, N) Ext1kQ(M,

N).

Note that in the categoryCQ, since the functor S2 = S[2] =[1] is isomorphicto the identity, then the functorsand [1] are isomorphic.

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We are interested in the objects ofCQwhich are rigid (i.e. satisfyingHomC(X, X[1]) =

0), basic (i.e. with pairwise non-isomorphic direct summands), and maximal forthese properties. An easy computation yields 14 maximal basic rigid objects whichare :

i(( 1 ) ( 21 ) (321

)), i= 0, . . . , 5; i(( 2 ) ( 21 ) (321

)), i= 0, . . . , 2;

i(( 2 ) ( 32 ) (321

)), i= 0, . . . , 2; i(( 1 ) ( 3 ) (321

)), i= 0, 1.

All these maximal rigid objects have the same number of indecomposable sum-mands. The Gabriel quivers of their endomorphism algebras are the following:

for i(( 1 ) ( 21 ) (321

)), i= 0, . . . , 5;

for i(( 2 ) ( 21 ) (321

)), i= 0, . . . , 2;

for i(( 2 ) ( 32 ) (321

)), i= 0, . . . , 2;

fori(( 1 ) ( 3 ) (321

)), i= 0, 1.

These objects satisfy a remarkable property: given a maximal basic rigid object,one can replace an indecomposable summand by another one in a unique way tofind another maximal basic rigid ob ject. This process can be understood as amutation, and under this process the quiver of the endomorphism algebra changes

according to the mutation rule. For instance, if we write the maximal rigid objecttogether with its quiver we obtain:

21

3

3 3

1

321

( 21 )

1 ( 3 )

321

( 1 )

32

( 32 )

321

(321

)

32

( 1 [1])

1 [1]

.

1.3. Stable module category of a preprojective algebra of typeA3. LetQ be the quiver 1 2 3. Then the preprojective algebra 2(kQ) (see[85]) ispresented by the quiver

1a

2b

a 3

b with relationsaa= 0, aa bb= 0, bb = 0.

It is a finite dimensional algebra. The projective indecomposable 2(kQ)-modules

are (up to isomorphism) P1 = I3 = 1

23

, P2 = I2 = 23 1

2and P3 = I1 =

321

. They

are also injective, since the algebra 2(kQ) is self-injective.

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8 C. Amiot

The category mod2(kQ) has finitely many indecomposable objects (up to

isomorphism) and its AR-quiver is the following:

1 2312 3

123

23 1

22

3 1 2 3 1

2

23 1

2

3 2132 1

321

where the vertical dotted lines are identified. There are 14 different basic maximalrigid objects (that is objectsM satisfying Ext12(kQ)(M, M) = 0). They all have 6direct summands, among which the three projective-injectives.

Therefore, if we work in the stable category mod2(kQ) (see [57]), where allmorphisms factorizing through a projective-injective vanish, we obtain 14 maxi-mal rigid objects with 3 direct summands (indeed the projective-injective objectsbecome isomorphic to zero in mod2(kQ)). Moreover, the categorymod2(kQ)

satisfies also the nice property that, given a maximal rigid object, one can replaceany of its indecomposable summands by another one in a unique way to obtainanother maximal rigid object.

The categorymod2(kQ) has the same AR-quiver as the cluster category CQ,moreover if we look at the combinatorics of the quivers of the maximal rigid objectsin mod2(kQ), we obtain:

23 1

12

12

12

1

21

( 23 1

)

1 ( 1

2)

21

( 1 )

2( 2 )

21

( 21

)

2( 3

2)

32

.

In conclusion, the categories CQ and mod2(kQ) seem to be equivalent, andthe combinatorics of their maximal rigid objects is closely related to the Fomin-Zelevinsky mutation of quivers.

Our aim in this survey is to show how general these phenomena are, and toconstruct a large class of categories in which similar phenomena occur.

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2. 2-Calabi-Yau categories and cluster-tilting theory

The previous examples are not isolated. These two categories have the same prop-erties, namely they are triangulated 2-Calabi-Yau categories, and have cluster-tilting ob jects (which in these cases are the same as maximal rigid). We will seein this section that most of the phenomena that appear in the two cases describedin Section1 still hold in the general setup of 2-Calabi-Yau categories with cluster-tilting objects.

2.1. Iyama-Yoshino mutation. A triangulated category which isHom-finite iscalledd-Calabi-Yau (d-CY for short) if there is a bifunctorial isomorphism

HomC(X, Y) DHomC(Y, X[d])

whereD = Homk(, k) is the usual duality over k .

Definition 2.1. LetC be a Hom-finite triangulated category. An objectT C iscalledcluster-tilting (or 2-cluster-tilting) ifT is basic and if we have

add (T) ={X C |HomC(X, T[1]) = 0}= {X C |HomC(T, X[1]) = 0}.

Note that a cluster-tilting object is maximal rigid (the converse is not always truecf. [29]), and that the second equality in the definition always holds when C is2-Calabi-Yau.

If there exists a cluster-tilting object in a 2-CY category C, then it is possibleto construct others by a recursive process resumed in the following:

Theorem 2.2(Iyama-Yoshino, [63]). LetCbe aHom-finite 2-CY triangulated cat-

egory with a cluster-tilting objectT. LetTi be an indecomposable direct summandof T Ti T0. Then there exists a unique indecomposable Ti non isomorphic

to Ti such that T0Ti is cluster-tilting. Morever Ti and T

i are linked by the

existence of triangles

Tiu B

v Tiw Ti[1] and T

i

u B v Ti

wTi[1]

whereu andu are minimal leftadd(T0)-approximations andv andv are minimal

rightadd (T0)-approximations.

These exchange triangles permit to mutate cluster-tilting objects and have beenfirst described by Buan, Marsh, Reineke, Reiten and Todorov (see [ 24, Proposi-tion 6.9]) for cluster categories. The corresponding exchange short exact sequences

in module categories over a preprojective algebra of Dynkin type appeared also inthe work of Geiss, Leclerc and Schroer (see[45, Lemma 5.1]). The general state-ment is due to Iyama and Yoshino [63, Theorem 5.3]. This is why we decided torefer to this mutation as theIY-mutationof cluster-tilting objects in this article.

This recursive process of mutation of cluster-tilting objects is closely relatedto the notion of mutation of quivers defined by Fomin and Zelevinsky [40]in thefollowing sense.

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10 C. Amiot

Theorem 2.3 (Buan-Iyama-Reiten-Scott [22]). Let C be aHom-finite 2-CY tri-

angulated category with cluster-tilting objectT. LetTi be an indecomposable directsummand of T, and denote by T the cluster-tilting object Ti(T). Denote byQT (resp. QT) the Gabriel quiver of the endomorphism algebra EndC(T) (resp.QT).Assume that there are no loops and no 2-cycles at the vertex i ofQT (resp.QT) corresponding to the indecomposableTi (resp. T

i). Then we have

QT =i(QT),

wherei is the Fomin-Zelevinsky quiver mutation.

We illustrate this result by the following diagram

Tclustertilting

IYmutation

Tclustertilting

QT

FZmutation QT.

The corresponding results have been first shown in the setting of cluster cate-gories in[26]and in the setting of preprojective algebras of Dynkin type in [45].

2.2. 2-Calabi-Yau-tilted algebras. The endomorphism algebrasEndC(T) whereTis a cluster-tilting object are of importance, they are called 2-CY tilted algebras(orcluster-tilted algebras ifCis a cluster category). The following result says thata 2-CY category Cwith cluster-tilting objects is very close to a module categoryover a 2-CY-tilted algebra. Such a 2-CY category can be seen as a recollementof module categories over 2-CY tilted algebras.

Proposition 2.4(Keller-Reiten [70]). LetCbe a 2-CY triangulated category witha cluster-tilting objectT. Then the functor

FT =HomC(T, ) :C modEndC(T)

induces an equivalenceC/add(T[1]) modEndC(T).If the objects T and T are linked by an IY-mutation, then the categories

modEndC(T) andmodEndC(T) are nearly Morita equivalent, that is there existsa simpleEndC(T)-moduleSand a simpleEndC(T)-moduleS, and an equivalenceof categoriesmodEndC(T)/add (S) modEndC(T)/add (S).

These results have been first studied in [26]for cluster categories. The 2-CY-tilted algebras are Gorenstein (i.e. EndC(T) has finite injective dimension as a

module over itself) and are either of infinite global dimension, or hereditary (see[70]). Other nice properties of the functor HomC(T, ) : C modEndC(T) arestudied in [73].

Example 2.5. LetQ be the quiver 1 2 3. The category CQ/add (T[1]) where

T =( 1 ) ( 21 ) (321

) has the same AR-quiver as modkQ. (Note thatkQis an

hereditary algebra.)

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If we take T = ( 1 ) ( 3 ) (321

), then the associated 2-CY-titled algebra

is given by the quiver 2a

1 c 3

bwith relations ab = bc = ca = 0. It is an

algebra of infinite global dimension. The module category of this algebra hasfinitely many indecomposables and its AR-quiver is of the form

13

32

1 2 3

21

13

It is the same quiver as the AR-quiver ofCQ/add (T[1]).

( 1 ) ( 2 ) =( 3 [1]) ( 3 ) (321

[1])

( 21 ) (32 ) (

21 [1])

(321

[1]) (321

) ( 1 [1]) ( 1 )

2.3. Two fundamental families of examples. The two examples studied insection 1 are part of two fundamental families of 2-CY categories with cluster-tilting objects. We give here the general construction of these families.

The acyclic cluster categoryThe first one is given by the cluster category CQ associated with an acyclic

quiverQ. This category was first defined in[24], following the decorated represen-tations of acyclic quivers introduced in [79].

The acyclic cluster category is defined in [24] as the orbit category of thebounded derived category Db(kQ) of finite dimensional modules over kQ by the

autoequivalence S2 = S[2] [1], where S is the Serre functor L

kQDkQ,

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andis the AR-translation ofDb(kQ). The objects of this category are the same

as those ofDb

(kQ) and the spaces of morphisms are given by

HomCQ(X, Y) :=iZ

HomDb(kQ)(X, Si2Y).

The canonical functor : Db(kQ) CQ satisfies S2 . Moreover the k-category CQ satisfies the property that, for any k-category T and functor F :Db(kQ) T with F S2 F, the functor F factors through .

Db(kQ)

S2

F

CQ T

Db(kQ)

F

For any X and Y in Db(kQ), the inifinite sum

iZHomD(X, Si2Y) has finitely

many non zero summands, hence the category CQ is Hom-finite. Moreover theshift functor, and the Serre functor of Db(kQ) yield a shift functor and a Serrefunctor for CQ. The acyclic cluster category CQ satisfies the 2-CY property byconstruction since the functor S2 = S[2] becomes isomorphic to the identity inthe orbit category Db(kQ)/S2.

Furthermore, the category CQhas even more structure as shown in the followingfundamental result.

Theorem 2.6 (Keller [65]). Let Q be an acyclic quiver. The acyclic cluster

category has a natural structure of triangulated category making the functor :Db(kQ) CQ a triangle functor.

The triangles in Db(kQ) yield natural candidates for triangles inCQ, but check-ing that they satisfy the axioms of triangulated categories (see e.g. [57]) is a difficulttask. A direct proof (by axioms checking) thatCQ is triangulated is especially dif-ficult for axiom (T R1): Given a morphism in CQ, how to define its cone if it is notliftable into a morphism in Db(kQ)? Keller obtains Theorem2.6 using differenttechniques. He first embeds CQ in a triangulated category (called triangulatedhull), and then shows that this embedding is dense.

Link with tilting theoryIfTis a tiltingkQ-module, then(T) is a cluster-tilting object as shown in[24].

Moreover the mutation of the cluster-tilting objects is closely related to mutation

of tiltingkQ-modules in the following sense (see [24]): IfTTiT0is a tiltingkQ-module with Ti indecomposable and if there exists an indecomposable module Tisuch thatT0 T

i is a tilting module, then(T

i T0) is the IY-mutation of(T) at

(Ti). The exchange triangles are the images of exchange sequences in the modulecategory. Furthermore, the 2-CY-tilted algebra EndC((T)) (called cluster-tiltedin this case) is isomorphic to the trivial extension of the algebra B = EndkQ(T) bytheB-B-bimodule Ext2B(D(B), B) as shown in[11].

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An advantage of cluster-tilting theory over tilting theory is that in cluster-

tilting theory it is always possible to mutate. In a hereditary module category,an almost complete tilting module does not always have 2 complements (cf. [58],[83]).

Using strong results of tilting theory due to Happel and Unger [59], one hasthe following property.

Theorem 2.7 (Prop. 3.5 [24]). Let Q be an acyclic quiver. Any cluster-tiltingobjectsT andT inCQ are IY-mutation-equivalent, that is one can pass fromT toT using a finite sequence of IY-mutations.

This fact is unfortunately not known to be true or false in general 2-CY cate-gories (see aslo Remark3.15).

The categoryCQ provides a first and very important example which permits to

categorify the acyclic cluster algebras. We refer to [24], [30], [26], [28], [33], [34]for results on the categorification of acyclic cluster algebras. Detailed overviews ofthis subject can be found in [25],[82] and in the first sections of[67].

Remark 2.8. The construction ofCQ can be generalized to any hereditary cate-gory. Therefore, it is possible to consider CH = Db(H)/S2, where H= cohX, andXis a weighted projective line. This orbit category is still triangulated by [ 65](thegeneralization from acyclic quivers to hereditary categories was first suggested byAsashiba). These cluster categories are studied in details by Barot, Kussin andLenzing in[16].

Recognition theorem for acyclic cluster categories.The following result due to Keller and Reiten ensures that the acyclic cluster

categories are the only categories satisfying the properties needed to categorifyacyclic cluster algebras.

Theorem 2.9 (Keller-Reiten [71]). LetC be an algebraic triangulated 2-CY cat-egory with a cluster-tilting object T. If the quiver of the endomorphism algebraEndC(T) is acyclic, then there exists a triangle equivalenceC CQ.

Note that this result implies in particular that the cluster-tilting objects ofthe categoryCare all mutation equivalent. The equivalence of the two categoriesdescribed in subsections1.2 and 1.3 is a consequence of Theorem2.9.

Geometric description of the acyclic cluster category.Another description of the cluster category has been given independently by

Caldero, Chapoton and Schiffler in [31], [32] in the case where the quiver Q isan orientation of the graph An. The description is geometric: in this situation,indecomposable objects inCQ correspond to homotopy classes of arcs joining twonon consecutive vertices of an (n+ 3)-gon, and cluster-tilting objects correspondto ideal triangulations of the (n + 3)-gon. One can associate a quiver to any idealtriangulation : the vertices are in bijection with the inner arcs of , and twovertices are linked by an arrow if and only if the two corresponding arcs are part

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14 C. Amiot

of the same triangle. Mutation of cluster-tilting objects corresponds to the flip of

an arc in an ideal triangulation.A similar description ofCQ exists in the case where Q is some (acyclic) ori-

entation of the graphs Dn, An and Dn. In these cases, the surface with markedpoints is respectively ann-gon with a puncture, an annulus without puncture, andan n-gon with two punctures (see [86], [88], [87]).

QT =Q FZmutation Q =QT

ideal triang.

flip

1:1

ideal triang.

1:1

Tclustertilting

IYmutation

Tclustertilting

Example 2.10. Here is the correspondence between cluster-tilting objects of theexample in subsection1.2 and ideal triangulations of the hexagon.

1

21

321

1

3

321

32

3

321

32

3

1 [1]

2 1 3

2

1 3

2

1 3

21

3

21

3

Preprojective algebras of Dynkin typeA very different approach is given by module categories over preprojective

algebras. Let be a finite graph, and Q be an acyclic orientation of . Thedouble quiver Q is obtained from Q by adding to each arrow a : i j of Q anarrow a : j i. The preprojective algebra 2(kQ) is then defined to be thequotient of the path algebra kQ by the ideal of relations

aQ1

aa aa. Thisalgebra is finite dimensional if and only if Q is of Dynkin type, and its modulecategory is closely related to the theory of the Lie algebra of type Q. When Q isDynkin, the algebra 2(kQ) is selfinjective and the stable category mod2(kQ)

is triangulated (see [57]) and 2-Calabi-Yau. Moreover it does not depend on thechoice of the orientation ofQ, so we denote it by 2().

In their work [45],[46] about cluster algebras arising in Lie theory, Geiss, Leclercand Schroer have constructed special cluster-tilting objects in the module categorymod2(), which helped them to categorify certain cluster algebras. Furtherdevelopments of this link between cluster algebras and preprojective algebras canbe found in[47], [48],[50],[51](see also [49]for an overview).

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Theorem 2.11(Geiss-Leclerc-Schroer[46]). Let be a simply laced Dynkin dia-

gram. Then for any orientationQ of, there exists a cluster-tilting objectTQ inthe categorymod2() so that the Gabriel quiver of the cluster-tilting object TQis obtained from the AR quiver ofmodkQ by adding arrows corresponding to theAR-translation.

Example 2.12. In the example of the first section with =A3, with the threeorientations 1 2 3, 1 2 3 and 1 2 3 of , we obtain respectivelythe following three cluster-tilting objects

1 12

123

21

23 1

2

321

12

123

22

3 1

2

32

321

1123

21 3

21 3

2

3321

CorrespondenceThe link between cluster categories and stable module categories of preprojec-

tive algebras of Dynkin type is given in the following.

Theorem 2.13. Let be a simply laced Dynkin diagram and Q be an acyclicquiver. There is a triangle equivalencemod2() CQ if and only if we are inone of the cases =A2andQ is of typeA1, =A3 andQ is of typeA3, =A4andQ is of typeD6.

Remark 2.14. In the case =A5, the category mod 2(A5) is equivalent to the

cluster categoryCH whereH is of tubular type E(1,1)8 .

Even though the categories mod2() and CQare constructed in a very distinctway, this theorem shows that these categories are not so different in a certain sense.This observation allows us to ask the following.

Question 2.15. Is it possible to generalize the constructions above so that the2-Calabi-Yau categoriesmod 2()andCQare part of the same family of categories?

There are two different approaches to this problem. One consists in generalizingthe constructionmod 2() and the other consists in generalizing the construction

of the cluster category. A construction of the first type is given in [47]and in[22](see subsection4.2) and a construction of the second type is given in [2] and[3](see section3).

2.4. Quivers with potential. Theorem2.3 links the quivers of the 2-CY-tiltedalgebras appearing in the same IY-mutation class of cluster-tilting object. Thisleads to the natural question:

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16 C. Amiot

Is there a combinatorial way to deduce the relations of the2-CY-tilted algebra after

mutation?This question (among others) brought Derksen Weyman and Zelevinsky to

introduce in [36] the notions of quiver with potential (QP for short), Jacobianalgebra and mutation of QP.

Definition 2.16. A potential W on a quiver Q is an element inkQ/ [kQ,kQ]wherekQ is the completion of the path algebra kQ for the J-adic topology (Jbeing the ideal ofkQ generated by the arrows) where [kQ,kQ] is the subspace ofkQ generated by the commutators of the algebra kQ.In other words, a potential is a (possibly infinite) linear combination of cycles ofQ, up to cyclic equivalence (a1a2 . . . an a2a3 . . . ana1).

Definition 2.17. LetQbe a quiver. The partial derivative:kQ/ [kQ,kQ] kQis defined to be the unique continuous linear map which sends the class of a pathp to the sum

p=uavvu taken over all decompositions of the path p.

Let (Q, W) be a quiver with potential. The Jacobian algebra of (Q, W) is

defined to be Jac(Q, W) :=kQ/aW, a Q1, where aW, a Q1 is the idealofkQ generated by aW for all a Q1.

A QP (Q, W) is Jacobi-finiteif its Jacobian algebra is finite dimensional.

In [36]the authors introduced the notion ofreductionof a QP. The reductionof (Q, W) consists in finding a QP (Qr, Wr) whose key properties are that theJacobian algebras Jac(Q, W) and Jac(Qr, Wr) are isomorphic and Wr has no 2-cycles as summands. In the case of a Jacobi-finite QP, the quiver Qr is the Gabrielquiver of the Jacobian algebra and is then uniquely determined. The potentialWr

is not uniquely determined. The notion of reduction is defined up to an equivalencerelation of QPs called right equivalence (see[36]).

Derksen, Weyman and Zelevinsky also refined the notion of the FZ-mutation ofa quiver into the notion of mutation of a QP (that we will call DWZ-mutation) ata vertex ofQ without loop. Steps (M1) and (M2) are the same as in FZ-mutation.The new potential is defined to be a sum [W] + W, constructed from W usingthe new arrows. Step (M3) consists in reducing the QP we obtained. For genericQPs, step (M3) of the DWZ-mutation coincides with step (M3) of the FZ-mutationwhen restricted to the quiver. Let us illustrate this process in an example.

Example 2.18. Let us define (Q, W) as follows:

2

b

Q= 1 c

a

3

d

with W =edc.

4e

The Jacobian ideal is defined by the relations ed = ce = dc = 0. One easily checksthat (Q, W) is Jacobi-finite.

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Let us mutate (Q, W) at the vertex 3. After steps (M1) and (M2) we obtain

the QP (Q

, W

) defined as follows:

2

[db]

Q = 1

[dc]

a

3c

b

with W =e[dc] + d[dc]c + d[db]b.

4

e

d

After Step (M3) we obtain the QP:

2

(Q)r = 1

a

3

with (W)r =.

4

The map kQ k(Q)r sending a, b, c, d e, [dc], [db] on respectively , , ,,, 0, , sendsW on (W)r and induces an isomorphism between Jac(Q, W)and Jac((Q)r, (W)r).

The mutation of (Q, W) at 3 is defined to be ((Q

)r

, (W

)r

) (up to right equiv-alence).

If there do not occur 2-cycles at any iterate mutation of (Q, W), then thepotential is non-degenerate [36] and the DWZ-mutation (when restricted to thequiver) coincides with the FZ-mutation of quivers. This happens in particularwhen the potential isrigid, that is when all cycles ofQ are cyclically equivalent toan element in the Jacobian ideal aW, a Q1. This notion of rigidity is stableunder mutation[36].

Moreover the following theorem gives an answer to the previous question in thecase where the 2-CY-tilted algebra is Jacobian.

Theorem 2.19(Buan-Iyama-Reiten-Smith[23]). LetT be a cluster-tilting object

in an Hom-finite 2-CY category C and let Ti be an indecomposable direct sum-mand of T. Assume that there is an algebra isomorphismEndC(T) Jac(Q, W),for some QP (Q, W) and assume that there are no loops nor2-cycles at vertex i(corresponding to Ti) in the quiver ofEndC(T) and a technical assumption calledglueing condition. Then there is an isomorphism of algebras

Jac(i(Q, W)) EndC(Ti(T)).

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18 C. Amiot

Tclustertilting

IYmutation

Tclustertilting

EndC(T) Jac(Q, W) EndC(T

) Jac(Q, W)

(Q, W) DWZmutation

(Q, W)

This result applies in particular for acyclic cluster categories. IfQ is an acyclicquiver, the endomorphism algebra of the canonical cluster-tilting object (kQ)CQis isomorphic to the Jacobian algebraJac(Q, 0) and 0 is a rigid potential for thequiver Q. Moreover the categories CQsatisfy the glueing condition [23]. Therefore,combining Theorems2.19and2.7, all cluster-tilted algebras are Jacobian algebrasassociated with a rigid QP and can be deduced one from each other by mutation.

The endomorphism algebras of the canonical cluster-tilting object of the cate-gorymod2(), where is a Dynkin graph, are also Jacobian algebras with rigidQP and satisfy the glueing condition [23,Thm 6.5].

The two natural questions are now the following:

Question 2.20. 1. Are all2-CY-tilted algebras Jacobian algebras?

2. Are all Jacobian algebras2-CY-tilted algebras?

The first question is still open, but the answer is expected to be negative by

recent results of Davison [35] and Van den Bergh [92], which pointed out theexistence of algebras which are bimodule 3-CY but which do not come from apotential. The second one has a positive answer in the Jacobi-finite case (seesubsection3.3).

Derksen Weyman and Zelevinsky also describe the mutation of (decorated) rep-resentation of Jacobian algebras in [36]. Their aim was to generalize the construc-tion of mutations of decorated representations[79]. Given aJac(Q, W)-module M,they associate a Jac(i(Q, W))-module i(M). This mutation preserves indecom-posability.

3. From 3-Calabi-Yau DG-algebras to 2-Calabi-Yau categories

3.1. Graded algebras and DG algebras. In this section, we recall some def-initions concerning differential graded algebras, differential graded modules, andderived categories. We refer to [64] for definitions and properties on differentialgraded algebras and associated triangulated categories.

Graded algebras

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We denote by Gr k the category of graded k-vector spaces. Recall that given

twoZ

-graded vector spaces M = pZ Mp and N = pZ Np, a morphism f HomGr k(M, N) is of the form f = pZ(fp) where fp Homk(Mp, Np) for anyp Z.

Let A =

iZ Ai be a Z-graded algebra. A Z-graded A-module M is a Z-graded k-vector space M=

iZ Mi with a morphism ofZ-graded algebrasA

pZHomGr k(M, M(p)), whereM(p) is the Z-gradedk-module such thatM(p)i=Mp+i for any i Z. In other words, for any n, p Z there is a k-linear mapMn Ap Mn+p sending (m, a) to m.a such that m.1 = m for all m M andsuch that the following diagram commutes

Mn Ap Aq

Mn Ap+q

Mn+p Aq Mn+p+q,

where the maps are induced by the multiplication in A for the first row and bythe action ofA onMrespectively for the others. Then the degree shift M(1) of agraded module M is still a graded A-module. A morphism of graded A-modulesis a morphism f : M Nhomogeneous of degree 0, that is f =

nZ fn where

fn Homk(Mn, Nn) and which commutes with the action of A, that is for anya Ap there is a commutative diagram

Mnfn

.a

Nn

.a

Mn+pfn+pNn+p.

The category of graded A-modules GrA is an abelian category. Therefore wecan define the derived category of graded A-modules D(GrA) as usual (see [66]).For a complex of graded A-modules, we denote by M[1] the complex such thatM[1]n =Mn+1 and dM[1] = dM.

Definition 3.1. Letd 2. We say thatA is bimoduled-Calabi-Yau of Gorensteinparameter 1 if there exists an isomorphism

RHomAe (A, Ae)[d](1) A in D(GrAe)

whereAe is the graded algebra Aop A.

DG algebras

Definition 3.2. A differential graded algebra(=DG algebra for short) A is a Z-graded k-algebra A =

nZ A

n with a differential dA, that is a k-endomorphismofA homogeneous of degree 1 satisfying the Leibniz rule: for any a Ap and anyb A we have dA(ab) =dA(a)b+ (1)padA(b).

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20 C. Amiot

A DG A-module M is a graded A-module M =

nZ Mn, endowed with a

differentialdM, that is dMHomGrA(M, M(1)) such thatdM dM= 0. Moreoverthe differential dM of the complex M is compatible with the differential of A inthe following sense

m Mn, a Ap, dM(m.a) =m.dA(a) + (1)ndM(m).a

Mn Ap (dM,1)

(1,dA)

Mn+1 Ap

Mn+pdM

Mn Ap+1 Mn+p+1

.

Note that this equality for p = 0 implies that Mhas a structure of complex of

Z

0

(A)-modules. A morphism of DGA-modules is a morphism of graded A-moduleswhich is a morphism of complexes. Note that if we endow the gradedA-moduleM(1) with the differential dM(1) = dM, it is a DG A-module and there is acanonical isomorphism of DGA-modules M(1) M[1].

Thederived categoryD(A) is defined as follows. The objects are DGA-modules,and morphisms are equivalence classes of diagrams

s1f : M

f

N

s

N

wheref is a morphism of DG A-modules, ands is a morphism of DG A-modulessuch that for any n Z, the morphism Hn(s) : Hn(N) Hn(N) is an iso-morphism of H0(A)-module (s is a quasi-isomorphism). This is a triangulatedcategory. We denote by per A the thick subcategory (= the smallest triangulatedcategory stable under direct summands) generated byA. We denote byDb(A) thesubcategory ofD(A) whose objects are the DGA-modules with finite dimensionaltotal cohomology.

Remark 3.3. IfA is a k-algebra, we can view it as a DG algebra concentratedin degree 0, and with differential 0. Then we recover the usual derived categoriesD(A), perA andDb(A).

Definition 3.4. A DG algebra is bimodule d-Calabi-Yau, if there exists an iso-morphism

RHomAe (A, Ae) A[d] in D(Ae)

whereAe is the DG algebra Aop A.

The next proposition says that the bimodule d-Calabi-Yau property for A im-plies thed-CY property for the bounded derived categoryDb(A).

Proposition 3.5([53,67]). IfA is a DG algebra which is bimoduled-Calabi-Yau,then there exists a functorial isomorphism

HomDA(X, Y) DHomDA(Y, X[d]), for anyX DA andY Db(A).

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Remark 3.6. If A is a graded algebra, we can view it as a DG algebra with

differential 0. Then if X is a complex of graded A-modules, for any n Z

,the differential dX : Xn = iZ Xni Xn+1 = iZ Xn+1i is homogeneous of

degree 0, it goes from Xni toXn+1i . If we setM

n :=

iZ Xnii , then the complex

(M, dX) has naturally a structure of DG A-module. This yields a canonical functorD(GrA) DA, which induces a fully faithful functor per GrA/(1) perA. Thecategory per (A) is generated, as a triangulated category, by the image of theorbit category per (GrA)/(1) through this functor. More precisely, perA is thetriangulated hull of the orbit category per (GrA)/(1).

If the graded algebra A is bimodule d-Calabi-Yau of Gorenstein parameter 1,then the DG algebra A endowed with the zero differential is bimodule d-Calabi-Yau.

3.2. General construction. The next result is the main step in the constructionof new 2-CY categories with cluster-tilting object which generalize the acycliccluster categories.

Theorem 3.7(Thm 2.1[3]). Let be a DG-algebra with the following properties:

(a) is homologically smooth (i.e. per (e)),

(b) Hp() = 0 for allp 1,

(c) H0() is finite dimensional ask-vector space,

(d) is bimodule3-CY.

Then the triangulated categoryC() =per/Db

() isHom-finite, 2-CY and theobject is a cluster-tilting object withEndC() H0().

Let us make a few comments on the hypotheses and on the proof of this theorem.Hypothesis (a) implies that Db() per ([67]), so that the category C() =per /Db() exists. Hypothesis (b) implies the existence of a natural t-structure(coming from the usual truncation) on per , which is an essential ingredient forthe proof. Moreover, hypotheses (c) and (d) imply thatHp() is finite dimensionaloverk for all p Z. Thus for any Xper, there is a triangle

nX X >nX nX[1],

where>nX is in Db(). Moreover we have

HomC(X, Y) limn

Homper(nX, nY),

which implies the Hom-finiteness of the category C.Now we define a full subcategoryF() ofper by

F() = (per )0 ((per )2),

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22 C. Amiot

where (per)p (resp. (per )p) is the full subcategory ofper consisting of

objects having their homology concentrated in degrees p (resp. p). Animportant step of the proof consists in showing that the composition

F() per per /Db= C()

is an equivalence. Notice that the subcategory F() is not stable under the shiftfunctor.

This equivalence implies in particular the following.

Proposition 3.8. Let be as in Theorem 3.7. Then the following diagram iscommutative:

per F()

H0

C()

HomC(,)=F

modH0()

3.3. Ginzburg DG algebras. The general theorem above applies to GinzburgDG algebras associated with a Jacobi-finite QPs.

Definition 3.9(Ginzburg[53]). Let (Q, W) be a QP. LetQG be the graded quiverwith the same set of vertices as Q and whose arrows are:

the arrows ofQ (of degree 0);

an arrowa :j i of degree1 for each arrowa : i j ofQ;

a loopti: i i of degree2 for each vertex i Q0.

The completed Ginzburg DG algebra(Q, W) is the DG algebra whose under-lying graded algebra is the completion (for the J-adic topology) of the graded

path algebra kQG. The differential of(Q, W) is the unique continuous lin-ear endomorphism homogeneous of degree 1 which satisfies the Leibniz rule (i.e.d(uv) = (du)v+ (1)pudv for all homogeneous u of degreep and allv), and takesthe following values on the arrows ofQG:

d(a) = 0 and d(a) =aW a Q1; d(ti) =ei(aQ1

[a, a])ei i Q0.

Theorem 3.10 (Keller [68]). The completed Ginzburg DG algebra(Q, W) ishomologically smooth and bimodule3-Calabi-Yau.

It is immediate to see that(Q, W) is non zero only in negative degrees, andthatH0((Q, W)) Jac(Q, W). Therefore by Theorem3.7 we get the following.Corollary 3.11. Let(Q, W) be a Jacobi-finite QP. Then the category

C(Q,W) :=per(Q, W)/Db((Q, W))isHom-finite, 2-Calabi-Yau, and has a canonical cluster-tilting object whose endo-morphism algebra is isomorphic to Jac(Q, W).

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This categoryC(Q,W) is called the cluster category associated with a QP. This

corollary gives in particular an answer to Question 2.20 (2) in the Jacobi-finitecase.

Remark 3.12. If (Q, W) is not Jacobi-finite, a generalization of the categoryC(Q,W), which is not Hom-finite, is constructed in[81].

Note that in a recent paper [91], Van den Bergh has shown that complete DGalgebras in negative degrees which are bimodule 3-Calabi-Yau are quasi-isomorphicto deformations of Ginzburg DG algebras in the sense of [68]. Hence a completeDG algebra as in Theorem 3.7 is a (deformation) of a Ginzburg DG algebraassociated with a Jacobi-finite QP.

The following two results give a link between Ginzburg DG algebras associatedwith QPs linked by mutation.

Theorem 3.13. Let(Q, W)be a QP without loops andi Q0 not on a2-cycle in

Q. Denote by :=(Q, W) and :=(i(Q, W)) the completed Ginzburg DGalgebras.

(a) [72], [68] There are triangle equivalences

per per

Db

Db.

Hence we have a triangle equivalenceC(Q, W) C(i(Q, W)).

(b) [81] We have a diagram

per

H0

per

H0

modJac(Q, W)

DWZmutation

for representationsmodJac(i(Q, W)).

Combining (b) together with Proposition3.8,we obtain that in the Jacobi-finitecase, for any cluster-tilting object T C(Q,W) which is IY-mutation equivalent tothe canonical one, we have:

T

IYmutation

C(Q,W)

FT

FT

T

modEndC(T) DWZmutation

for representationsmodEndC(T).

(3.3.1)

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24 C. Amiot

This is not clear from the definition wether the cluster categories C(Q,W)satisfy

the glueing condition of Theorem 2.19. However Theorem3.13(a) ensures that if(Q, W) is a non degenerate Jacobi-finite QP, then IY-mutation is compatible withDWZ-mutation for cluster-tilting objects mutation equivalent to the canonical one(Q, W).

3.4. Application to surfaces with marked points. Let (, M) be a pairconsisting of a compact Riemann surface with non-empty boundary and a set Mof marked points of , with at least one marked point on each boundary component.By an arc on , we mean the homotopy class of a non crossing simple curve on with endpoints inM(which may coincide), which does not cut out an unpuncturedmonogon, or an unpunctured digon.

To each ideal triangulationof the surface (, M), Labardini associates in[75]a QP (Q, W) which is rigid and Jacobi-finite. Moreover he shows that the flip ofthe triangulation coincides with the DWZ-mutation of the quiver with potential.Since any two triangulations are linked by a finite sequence of flips, the generalizedcluster categories obtained from the QPs associated with the triangulations areall equivalent. More precisely, combining results in [75] with Corollary3.11andTheorem3.13(a) we get the following.

Corollary 3.14. Let (, M) be a surface with marked points with non emptyboundary. Then there exists aHom-finite triangulated 2-CY categoryC(,M) witha cluster-tilting objectTcorresponding to each ideal triangulationsuch that wehave the following commutative diagram

triangulation

flip

triangulation

T

clustertilting IYmutation T

clustertilting .

All the cluster-tilting objectsT foran ideal triangulation are IY-mutation equiv-alent.

In [76]Labardini moreover associates to each arc j on the surface a module overthe Jacobian algebra Jac(Q, W) for each triangulation , in a way compatiblewith DWZ-mutation for representations. More precisely, ifX(j) (resp. X(j))) isthe Jac(Q, W)-module (resp. Jac(Q, W)-module), where is a flip of, thenone can pass from X(j) to X(j) using the DWZ-mutation for representations

on the corresponding vertex:

triangulation

flip

triangulation

X(j)

DWZmutation

for representationsX(j).

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Denote by IY(, M) C(,M) the set of all cluster-tilting objects ofC(,M)

which are in the IY-mutation class of a cluster-tilting object T, where is anideal triangulation (it does not depend on the choice ofby Corollary3.14). Ifa triangulation contains a self-folded triangle, it is not possible to flip at theinside radius of the self-folded triangle. To avoid this problem, Fomin, Shapiro andThurston introduced in [39] the notions of tagged arcs and tagged triangulationswhich generalize the notions of arcs and triangulation. Then it is possible to mutateany tagged arc of any tagged triangulations. Hence we obtain a bijection:

IY(, M) 1:1 {tagged triangulations of (, M)}.

Let ind IY(, M) be the set of all indecomposable summands of IY(, M).

Then since any (tagged) arc on (, M) can be completed into a (tagged) triangu-lation, there is a bijection:

ind IY(, M) 1:1 {tagged arcs on (, M)}.

Now fix an ideal triangulation of (, M) and denote by T the associatedcluster-tilting ofC(,M). Using results of [76] together with the diagram3.3.1weobtain the following commutative diagram:

ind IY(, M) \ ind (T)

{arcs on (, M) not in }

X

C(,M)

FT[1]

modJac(Q, W)

,

whereFT[1]= HomC(T[1], ) HomC(T, [1]).

Remark 3.15. In the case of an unpunctured surface (, M), Brustle and Zhangstudy the category C(,M) very precisely in [21](see also[10]), and show an ana-logue of Theorem2.7, that is any cluster-tilting object ofC(,M) is in I Y(, M).

Example 3.16. Let be a surface of genus 0 with boundary having two con-nected components, and let Mbe a set of marked points on consisting of threemarked points on the boundary, and one puncture. Let be the following ideal

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26 C. Amiot

triangulation

1

5

3

2

4

6

The quiver with potential (Q, W) is the following

Q =

1 2

3

45

6

a

bc

d

e

f

g

h

i

W =cba +f ed + gdb.

Letj be an arc of (, M) which is not in . In the most simple cases, the com-position series of the moduleX(j) correspond to the arcs ofcrossed transversallybyj . For instance, ifj is the following arc

1

5

3

2

4

6

the module X(j) has the following composition series X(j) = 6

2 54

.

Letj be an arc ofandj be the flip of this arc with respect to . Then the arcj crosses the triangulationuniquely through the arc j, thus the correspondingJac(Q, W)-module X(j) is the simple module associated with the vertex ofQcorresponding to the arc j of.

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Remark 3.17. Labardini also associates in [75] QPs to ideal triangulations of

surfaces without boundary. For instance if (, M) is a once punctured torus, thenall triangulations give the same QP:

2

aa

Q= 1c c3

b

b

W =abc + abc + abcabc.

This QP is not rigid, but is non-degenerate and Jacobi-finite (cf. [76, Section 8]).Using Corollary3.11,it is possible to associate a generalized cluster category withcluster-tilting objects to this surface. However the non-degeneracy and Jacobi-finiteness are not known to be true or false for general surfaces without boundary.

3.5. Derived preprojective algebras. Derived preprojective algebras give an-other application of Theorem3.7.

Definition 3.18 (Keller [68]). Let be a finite dimensional algebra of globaldimension at most 2. Denote by 2 a cofibrant resolution ofRHome (, e)[2]D(e). Then thederived3-preprojective algebrais defined as the tensor DG algebra

3() :=T2= 2 (2 2) . . .

and the algebra 3() :=H0(3()) is called the 3-preprojective algebra.

Since the algebra is a finite dimensional algebra of finite global dimension,there is an isomorphism of --bimodules RHome (, e) RHom(D, ).Moreover the functor

L

RHom(D, ) :Db() Db()

is a quasi inverse for the Serre functor ofDb() (see[64] or [2,Chapter 1]). Hencewe have an isomorphism 3()

p0HomDb(, S

p2 ). Since is of global

dimension at most 2, then HomDb(, Sp2) vanishes for p 1. Therefore we have

isomorphisms of algebras:

3()pZ

HomDb(, Sp2 ) TExt

2(D, ) (cf. [3,Prop. 4.7]). (3.5.1)

Definition 3.19 (Iyama). An algebra of global dimension at most 2 is called2-finite if 3() is finite dimensional. This is equivalent to the fact that the

endofunctor2:= H0(S2) ofmod is nilpotent.

Note that if is an hereditary algebra, then 3() since Ext2(D, )

vanishes. Hence for any acyclic quiverQ, the algebra kQ is 2-finite.

Theorem 3.20(Keller,[68]). Let be an algebra of global dimension at most 2.Then the derived preprojective algebra3()is homologically smooth and bimodule3-CY.

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28 C. Amiot

Applying then Theorem3.7 we get the following construction.

Corollary 3.21. Let be as in Theorem 3.20. If is moreover 2-finite, thenthe category

C2() :=per3()/Db(3())

isHom-finite and2-Calabi-Yau.Moreover the object3() is cluster-tilting inC2() with endomorphism alge-

bra3().

This category is called the cluster category associated with an algebra of globaldimension at most 2. The next result gives the link between the category C2()and the orbit categoryDb()/S2.

Theorem 3.22. Let be a finite dimensional algebra of global dimension at

most 2.

(1) For anyXinDb(), there is an isomorphismS2(X)L

3() XL

3()inC2() functorial inX. Thus there is a commutative diagram

Db()L

3()

per 3()

Db()/S2 C2() .

Moreover, the factorizationDb()/S2 C2() is fully faithful.

(2) The smallest triangulated subcategory ofC2() containing the orbit categoryDb()/S2 isC2().

LetJbe the idealJ= 2 (2 2) . . .of3(). Then using the isomor-

phism3.5.1,we obtain an isomorphism S2(X)L

JXL

3(). The morphismin (1) of the above theorem is induced by the inclusion J3() inper (3()e).The cone ofJ3() is inDb(3()e) since is finite dimensional. Thereforethe morphism

S2(X)L

3() XL

JXL

3() in per 3()

has its cone in Db(3()), so is an isomorphism in C2().

Point (2) of this theorem says that C2() can be understood as the triangu-lated hull of the orbit category Db()/S2. Keller introduced this notion with DGcategories. A DG category is a category where morphisms have a structure ofk-complexes. We refer to [64], [65] for precise definitions and constructions (seealso[62,Appendix], and[7, Section 6]).

The philosophy is the following: any algebraic triangulated category is triangleequivalent to H0(X), whereXis a DG category. But for a given DG category X,

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the categoryH0(X) is not always triangulated. However the categoryH0(X) can

be viewed (via the Yoneda functor) as a full subcategory ofH0

(DGModX) which isa triangulated category. Here DGModXis the DG category of the DG X-modules.The triangulated hull ofH0(X) is defined to be the smallest triangulated subcat-egory ofH0(DGModX) containing H0(X). In our situation, the categoryDb()has a canonical enhancement in a DG category X (that is Db() is equivalentto some category H0(X) for a canonical X ). The functorS2 can be canonicallylifted to a DG functor SofX. One can define the orbit category X/S; it is a DGcategory. The orbit category Db()/S2is equivalent to the categoryH0(X/S). Itstriangulated hull is defined to be the triangulated hull ofH0(X/S) following theprevious construction. Since the enhancementXand the lift S are canonical, onecan speak of the triangulated hull. But in general, triangulated hulls depend onthe choice of the enhancement.

From Theorem3.22one deduces the following fact.

Corollary 3.23. Let = kQ for an acyclic quiver Q. Then kQ is2-finite andthere is a triangle equivalenceC2(kQ) CQ.

This corollary associated with Theorem 3.22 shows that C2() is the mostnatural generalization of the cluster category when replacing the algebra kQ ofglobal dimension 1 by the algebra of global dimension 2.

Remark 3.24. If T is a tilting kQ-module, where Q is an acyclic quiver, thenthe algebra B = EndkQ(T) has global dimension at most 2 and is 2-finite. The

triangle functor L

B T : Db(B) Db(kQ) is an equivalence (see [56]) sendingB Db(B) on T Db(kQ). This equivalence induces a triangle equivalencebetween the cluster categories C2(B) andCQ sending(B) on(T). Moreover the

tensor productExt2

(DB,B) BExt2

(DB,B) vanishes, hence the algebra 3(B) isisomorphic to the trivial extension B Ext2(DB,B). The isomorphism 3(B)EndCQ(T) recovers then a result of[11].

Note that in general, a 2-finite algebra B of global dimension 2 can satisfyExt2(DB,B) BExt

2(DB,B)= 0, even ifC2(B) is an acyclic cluster category, asshown in the following example. Let B be the Auslander algebra ofmod kQ, whereQ is the linear orientation ofA3

1

2

3

4

5

6 .

Then an easy computation givesExt2B(e4DB,e1B)= 0 and Ext2B(e6DB,e4B)= 0,hence we have

e1Ext2B(DB,B) BExt

2B(DB,B)e6= Ext

2B(DB,e1B) BExt

2B(e6DB,B)= 0.

However, the category C2(B) is equivalent to the acyclic cluster category CD6 ,since the algebra B is an iterated tilted algebra of type D6 and of global dimen-sion 2.

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30 C. Amiot

From Kellers results on triangulated hulls[65], one deduces the following prop-

erty for the category C2() (see Appendix of [62] for details).Theorem 3.25. Let be a finite dimensional algebra of global dimension at most2. LetE be a Frobenius category. Let M be an object in Db(op E). Assume

that there is a morphismf : RHome (, e)L

M[2] M inDb(op E) suchthat the cone off, viewed as an object inDb(E) is perfect. Then there is trianglefunctorF :C2() E making the following diagram commutative:

Db()

L

M

Db(E)

C2()

F

Db

(E)/per (E) E.

Remark 3.26. A very powerful application of cluster categories associated withalgebras of global dimension at most 2 is made in [69], and allows to prove theperiodicity conjecture for pairs of Dynkin diagrams (see also last sections of [67]for an overview of this result).

3.6. Link between the two constructions. In the last two subsections we havegeneralized the notion of cluster category, first using the Ginzburg DG algebra, andthen derived 3-preprojective algebra. So a natural question is the following:

What is the link between the categoriesC(Q,W), where(Q, W)is a Jacobi-finiteQP, and the categoriesC2(), whereis a2-finite algebra of global dimension at

most 2?Let = kQ/I (where I is an admissible ideal ofkQ) be a finite dimensional

algebra of global dimension at most 2. LetR be a minimal set of relations, i.e.the lift to Iof a basis ofI/(IJ+J I) (where J is the ideal ofkQ generated bythe arrows) compatible with the decomposition I =

i,jejIei. Let

Q be thequiver obtained from Q by adding an arrow ar : t(r) s(r) for each relationr: s(r) t(r). Let W be the potential W =

rR rar.

Example 3.27. Let be the algebra presented by the quiver

2a

1 c 3

band the relation ab = 0.

Then the associated quiver with potential (Q, W) is

2a

1c

rab3

b

and W =rabab.

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The following result is due to Keller [68,Theorem 6.12].

Theorem 3.28(Keller). Let be a finite dimensional algebra of global dimensionat most 2. There exists a isomorphism of DG algebras f : (Q, W) 3()such thatHn(f)is an isomorphism for eachn Z.

This quasi-equivalence induces triangle equivalencesper(Q, W) per3()sending the object(Q, W) to the object3() andDb(Q, W) Db3().

This result implies in particular an isomorphism of algebras

3() TExt2(D, ) Jac(Q, W). (3.6.1)

It also gives an answer to the above question.

Corollary 3.29. Let and (Q, W) be as in Theorem3.28. Assume moreover

that is2-finite. Then there exists a triangle equivalenceC2() C(Q,W) send-ing the cluster-tilting object 3() C2() on the cluster-tilting object 3() C(Q,W).

Jacobi-finite QPs

algebras of global dimension at most 2

acyclic quivers

Remark 3.30. IfQis an acyclic quiver, and =kQ. Then we have Q= Q andW= 0, hence we recover 3(kQ) kQ Jac(Q, 0).

IfTis a tiltingkQ-module (whereQis an acyclic quiver), and if = EndkQ(T),then the description ofQ was already given in[11].

Remark 3.31. One can show that the category C(Q,W), where

Q= 2b

1

a 3

c

and W =cbacba,

is not triangle equivalent to a generalized cluster category C2() associated withan algebra of global dimension at most 2. Therefore the inclusion given in Corol-lary3.29is strict.

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32 C. Amiot

Example 3.32.Let (, M) be the surface with marked points as in Example3.16.

Let be the algebra presented by the quiver

Q=

1 2

3

45

6

bc

dfh

i

with relationscb = df =db = 0.

This is a 2-finite algebra of global dimension 2. By Corollary 3.29, there is anequivalence of triangulated categories C2() C(,M) sending the cluster-tiltingobject() on the cluster-tilting object T associated with the triangulation .

We write () = (e2) T0 C2(). One easily checks that there is an

exchange triangle in C2()

(e2) (e3) ( 35 ) (e2)[1],

thus the object( 35 ) T0 is a cluster-tilting object in C2().Denote by 2 the flip of the arc 2 with respect to the triangulation . Then one

checks that the image of the -module 35 through the functors

Db()

C2()

C(,M)

FT[1]

modJac(Q, W)

is isomorphic to the module X(2) constructed by Labardini in [76]. By the

same process, for any arc j of the surface, by making iterated IY-mutations andcorresponding flips, one can construct an object M in Db() such that (M)corresponds toj .

For instance, the summand corresponding to the vertex 6 of the object L6 L5

L4

L2 (()) ofD

b() (see Section 5 for a definition of the graded mutationL) corresponds to the arc j of Example3.16.

3.7. Higher generalized cluster categories. In general, it is of interest tostudy n-Calabi-Yau categoriesCwith n-cluster-tilting objects, i.e. objects T suchthat

add (T) = {X C |ExtiC(X, T) = 0, 1 i n 1}= {X C |ExtiC(T, X) = 0, 1 i n 1}.

Theorem3.7generalizes when replacing the hypothesis (d) by being bimodulen-Calabi-Yau. The formal proof is given in [55]. Associated to a finite dimensional

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algebra of global dimension at most n, Keller also introduced in [68]the derived

(n+ 1)-preprojective algebra n+1() as the DG algebra Tn, where n is acofibrant resolution ofRHome (, e)[n]. Then Corollary3.21 generalizes when

replacing 2 by n.In recent papers [53], [19], [68], [91], generalizations of Ginzburg DG algebras

are introduced and studied.

Special case n= 1.The casen = 1 is notable. A finite dimensional algebra of global dimension at

most one is Morita equivalent to some path algebra of an acyclic quiverQ. The cat-egoryC1(kQ) isHom-finite if and only if the algebra kQ is 1-finite, that is whenQis Dynkin, since1 is isomorphic to the usual Auslander-Reiten translation. In thiscase, the orbit categoryDb(kQ)/S1 is equivalent to the category proj2(kQ) andis naturally triangulated ([65] or [1]). They have finitely many indecomposables.

A 1-cluster-tilting object corresponds to a basic generator-cogenerator, therefore2(kQ) is the only 1-cluster-tilting object in C1(kQ). From results in[18] and[1],we deduce the following.

Theorem 3.33. LetQ be a Dynkin quiver.

(a) [1] The deformed preprojective algebras defined in [18]are1-CY-tilted alge-bras.

(b) [18] Ifchar(k) = 2, then the deformed 2-preprojective algebras are not iso-morphic to 2-preprojective algebras.

Hence we may expect that there also exist some higher deformed preprojectivealgebras that are 2-CY-tilted algebras, which would yield a negative answer to

Question2.20(1).

4. Stable categories as generalized cluster categories

In this section, we see how generalized cluster categories generalize other construc-tions of 2-Calabi-Yau categories via Frobenius categories.

4.1. Preprojective algebras of Dynkin type. Once the notion of generalizedcluster category is established, it is natural to see if it answers Question2.15.

Theorem 4.1 ([3]). Let Q be a Dynkin quiver. Let be the stable Auslanderalgebra of the categorymodkQ. Then there is a triangle equivalence

mod2(kQ) C2(),

sending the canonical cluster-tilting object to the standard cluster-tilting objectMassociated with the orientation ofQ.

The proof consists in constructing a triangle functor Db() Db(2(kQ))and in using Theorem3.25to deduce a triangle functor C2() mod 2(kQ). Weexplain the construction of the functor in the example below.

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34 C. Amiot

Example 4.2. Let Q be the linear orientation ofA3. Let X= 1 21 321

2

21 32 3 be the kQ-generator-cogenerator. Define the algebras = EndkQ(X), := EndkQ(X), and = End(M), whereM is the standard cluster-tilting objectof the module category mod 2(Q). Then one easily checks that is a subalgebraof and that we have natural algebra morphisms:

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6 .

Then we consider the following triangle functor

F :Db() Res Db()

L

MDb(2(kQ))

induced by these algebra morphisms and check that it satisfies the hypotheses ofTheorem3.25.

Remark 4.3. This theorem has been very nicely generalized in [62] for highercluster categories as follows: For n 2, let A be a (n 1)-representation-finitealgebra (in particular it is of global dimension n 1 and n1-finite). Then thereis a triangle equivalence modn(A) Cn(), where is the stable Auslanderalgebra =EndA(n(A)).

4.2. Stable categories associated to words. In this subsection, we describethe alternative answer to Question 2.15 developed in [22] and the link with thegeneralized cluster categories associated to algebras of global dimension at most 2.

Let be a graph and = 2() be the associated preprojective algebra (upto isomorphism, it does not depend on the orientation of ). Define the CoxetergroupCassociated to by the generators si, fori 0and the relationss2i = 1,sisj = sjsi if there are no arrows linking i with j, and sisjsi =sjsisj if there isexactly one edge linking i and j .

To any element w of C, Buan, Iyama, Reiten and Scott have associated analgebra w which is defined to be the quotient /Iw, where

Iw := (1 ei1 )(1 ei2 ) . . . (1 eil)

andsi1 si2. . . sil is a reduced expression of the elementw. The idealIw, hence thealgebra w, does not depend on the choice of this reduced expression.

Theorem 4.4 (Buan-Iyama-Reiten-Scott [22]). For any element w in C thecategory Subw formed by all -submodules of modules in addw is FrobeniusandHom-finite. The categorySub w is2-Calabi-Yau and contains a cluster-tiltingobjectMw for each reduced expressionw ofw.

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All cluster-tilting objects of the form Mw are in the same IY-mutation class.

More precisely, if w and w

are two reduced expressions of the element w linkedby a braid relation, thenMw andMw are linked by a IY-mutation:

wreduced expression

braid flip

wreduced expression

Mw

clustertilting IYmutation Mw

clustertilting.

Note that there are cluster-tilting objects in the IY-mutation class ofMw butwhich are not isomorphic to someMw.

These categories give an answer to Question 2.15. Indeed, if is a Dynkingraph and ifw is the element of maximal length in C, then we have Subw mod. Moreover, there is a natural bijection between orientations of and re-duced expressions of w (with commutativity relation). Using this bijection, thestandard cluster-tilting objects of[45]correspond to the cluster-tilting objectsMw.For any acyclic quiverQ (exceptAnwith the linear orientation), there is a triangleequivalenceSubw CQ when w = c2, where c is the Coxeter element associatedwith the orientation ofQ. See[23], [5]for further results on these categories. Thisconstruction and the constructions in Section 3 lead to the natural question.

What is the link between the categoriesSubw andC2()?

Let be a graph. Then for any acyclic orientation Q of , one can de-fine a Z-grading on , by assigning degree 0 to the arrows of Q and degree 1to the additional arrows. One easily checks that the preprojective relations arehomogeneous of degree 1. For any reduced expressionsi1. . . sil in C, the ideal(1ei1 ) . . . (1eil) is graded. The objectMwis defined as a sum ofei/Iw,therefore it is a graded module.

Theorem 4.5([6]). Let be a graph andw be a reduced expression of an elementw in C. Then there exists an orientation of such that there is a triangleequivalence Subw C2() sending the canonical cluster-tilting object () onMw, where is the stable endomorphism algebraEndgr (Mw) (in the category ofgraded-modulesgr).

As in Theorem 4.1,the proof consists in constructing a triangle functor Db()Db(Subw) satisfying the hypotheses of Theorem3.25. We may visualize the sit-uation as follows

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36 C. Amiot

cluster categoriesC2()

categories Sub w associated with elements w in Coxeter groups

acyclic quivers

mod2() with Dynkin

Example 4.6. Let be the graph A2, and w = s2s1s2s3s2s1s3s2 C bea reduced word. The associated cluster-tilting object constructed in [22] is thefollowing:

Mw = 2 12 21

31 2

2 1

23 1

1 22

1

2 33 1 2

1 2 12

31 2

2 3 13 1 2

1 22

23 1

1 2 32 3 1 2

3 1 2 11 2

2

The last three summands (corresponding to the last s1, s2 and s3 in the reduced

word w) are the projective-injectives of the category Sub w. The endomorphismalgebra End(Mw) is the frozen Jacobian algebra (see [23]) with quiver Q of theform

1 3 5 8 ,

2 6

4 7

p1

p3p2

q

r

a a a a

c c

c c

bb b

with potentialW =p1p2aa qaa +p3aa qbb

+ rbb +p2cc rcc +p3cc, and

frozen arrowsa, band c. That is we have an isomorphism

End(Mw) kQ/xW, x Q1, x=a,b, c.

The orientation of is given by the quiver Q restricted to vertices correspond-ing to projective-injectives, that is in this example 6, 7 and 8. Hence we have

= 3c

1b

a 2

and = 3

0

1

1 0

0

2 .1

1

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With this grading on , the degree of the arrows of the graded algebra End(Mw)

are d(p1) =d(p2) =d(p3) =d(q) =d(r) = 0,

d(a) =d(a) =d(b) =d(b) =d(c) =d(c) = 0,

and d(a) =d(a) =d(b) =d(c) = 1.

Now we have natural algebra morphisms

1 3 5 8

2 6

4 7

0 00

0

0

1 0 1 0

1 0 1 0

0 1 01 3 5 8

2 6

4 7

1 3 5

2

4

End(Mw

)

where := Endgr(Mw) and = Endgr(Mw). Then we consider the followingtriangle functor

F :Db() Res Db()

L

MwDb(Subw)

induced by these algebra morphisms and check that it satisfies the hypotheses ofTheorem3.25.

4.3. Cohen Macaulay modules over isolated singularities. In this sectionwe assume that the characteristic ofk is zero.

LetSbe the polynomial ring k [x0

, x1

, . . . , xd] in d + 1 variables. LetG be thefinite cyclic subgroup of order n ofSLd+1(k) generated by g = diag(

a0 , . . . , ad ),whereis a primitiven-root of unity. We assume moreover thata0+a1+. . . ad= n.

The group G acts naturally on S. We denote by SG the invariant algebraand by S G the skew group algebra. The algebraSG is a Gorenstein isolatedsingularity of Krull dimension d + 1.

Theorem 4.7. In the setup above, the following assertions hold.

(a) [13] The stable categoryCM (SG) of maximal Cohen-Macaulay SG-modulesis ad-CY triangulated category.

(b) [60] TheSG-moduleS is ad-cluster tilting object inCM (SG).

(c) [14],[93] We haveEndSG(S) S G.

If we want to realize C M(SG) as a cluster category, we have to find a naturalZ-grading onSG. If we set deg(xi) =

ain

, we obtain a structure of Zn

-graded algebraon S =

iZ Sin . Then the invariant subringS

G corresponds to the subalgebraofSof the direct sum of homogeneous components of integral degrees:

SG =iZ

Si.

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38 C. Amiot

Hence we obtain a structure ofZ-graded algebra forSG.

Now we define, for j = 0, . . . , n 1, the graded SG

-modules Tj = iZ Si+ jn .Then we have T0= S

G and T =n1

j=0 Tj Sas SG-modules.

The following result is also proved in[90] using different techniques.

Theorem 4.8 ([4]). LetS andG be as above. LetA be the stable endomorphismalgebraA:= EndGr SG(T). Then the following assertions hold.

(a) The algebra A is a finite dimensional, d-finite algebra of global dimensionat mostd.

(b) There exists a triangle equivalenceCd(A) C M(SG) sending thed-cluster-

tilting objectA to T.

Example 4.9. Let d = 3 and G be the subgroup generated by 15

(1, 2, 2). Then

the skew-group algebraS G EndSG(S) is presented by the McKay quiver

0

1

2

34

yz

z y

yz

yz

zy

x x

x

x

x

with the commutativity relations xy =y x, yz =z y, zx = xz . This is a Jacobianalgebra associated with a non degenerate QP by [89]. The vertex j = 0, . . . , 5corresponds to the summand Tj of T. With the grading defined as above, anarrow i j has degree 0 if i < j and degree 1 if i > j. Hence the algebraEndGr SG(T) is presented by

0

1

2

34

yz

z y

yz

x x

x

x

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with the commutativity relations. Therefore the stable endomorphism algebra

A= EndGrSG(T) EndGrSG(T1 T4) is presented by the quiver1 2

34

zy yz

x

x

x

with the commutativity relations. By the previous theorem the categoryCM (SG)is triangle equivalent to the generalized cluster category C2(A) .

The proof of this theorem uses again Theorem 3.25. One fundamental step in

the proof consists in using the fact that the skew-group algebra with the gradingdefined as above is bimodule d-Calabi-Yau of Gorenstein parameter 1 by [19], andthen in showing the following:

Theorem 4.10([4]). LetB be aZ-graded algebra bimoduled-Calabi-Yau of Goren-stein parameter 1 such that dimkBi is finite for all i, and such that B per Be.Then the DG algebrad(A)has its homology concentrated in degree0. Moreover,there is an isomorphism ofZ-graded algebrasd(A) B.

We end this section by asking the following intriguing questions:

Question 4.11. 1. What can we say about the stable categoryCM (SG) whenG is not cyclic ?

2. Is there an analogue of the categoriesSubw in higher CY-dimensions?

5. On theZ-grading on the 3-preprojective algebra

Throughout the section is a finite dimensional algebra of global dimension atmost two.

As we already saw in section 3, the 3-preprojective algebra 3() is isomorphicto the tensor algebraTExt

2(D, ) (see3.5.1) and thus is naturally positively Z-

graded as a tensor algebra. The algebra can easily be recovered from the gradedalgebra 3() as its degree zero subalgebra. We already used this remark to proveequivalences between certain stable 2-CY categories and cluster categories C2()in subsections4.2and 4.3.

In this section, we study this grading, and outline applications in representationtheory.

5.1. Grading of 3(). Let us describe more explicitly this grading. The alge-bra 3() is isomorphic to the Jacobian algebra Jac(Q, W), where Q and Ware obtained from the quiver Q and the minimal relations of (cf. isomorphism3.6.1). This natural grading comes from a grading on Q defined as follows:

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40 C. Amiot

the arrows ofQ are of degree 0, (indeed, they correspond to elements of

which is the degree 0 subalgebra of 3());

the new arrows, corresponding to minimal relations, are of degree 1, (indeed,they correspond to elements of the bimodule Ext2(D(), )).

This grading on Q makes the potential W homogeneous of degree 1. Hence therelationsa W fora Q1 are homogeneous.

Example 5.1. Let be as in Example 3.27. The 3-preprojective algebra 3()is isomorphic, as a graded algebra, to the graded Jacobian algebra Jac(Q,W,d)with

Q= 2a

1c

rab3

b

d(a) =d(b) =d(c) = 0, d(rab) = 1 and W =rabab.

Since we have an isomorphism of graded algebras 3()

pZHomDb()(, Sp2 )

(cf. 3.5.1), the subcategory

U := 1() ={Sp2 , p Z} D

b()

provides a Z-covering of the graded algebra 3().

Hence the study of the category U =1() is one of the fundamental tools

for the study of the grading of 3().

Theorem 5.2(Thm 1.22[61], Prop 5.4.2[2]). If is2-finite, then the subcategoryU is a cluster-tilting subcategory ofD

b().

The next result shows that, moreover, the category Db() is determined by thecategoryU.

Theorem 5.3 (Recognition Theorem, Thm 3.5 [7]). Let T be an algebraic tri-angulated category, with Serre functor S and with a cluster-tilting subcategoryV.Assume that there exists a 2-finite algebra and an equivalence f : V U =add{Sp2, p Z} such that f S2 S f[2]. Then T is triangle equivalent toDb().

This result is the key step for the proof of all results presented in the next two

sections.

5.2. Mutation of graded QPs. In this section, we assume that is a 2-finitealgebra of global dimension at most 2.

The observation that 3() is a graded Jacobian algebra leads us to study thenotion of graded QP (Q,W,d) with W homogeneous of degree 1. We can definethe notion of reduction similarly to [36](see [7,Section 6]).

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Definition 5.4([7], see also[89]). Let (Q,W,d) be a graded QP such thatQ does

not have any loops. Let i be a vertex ofQ. Then the left mutation L

i(Q,W,d)(resp. right mutation Ri (Q,W,d)) of (Q,W,d) at vertex i is defined to be thereduction of the graded QP (Q, W, d) constructed as follows:

(M1gr) for each pair of arrows j a i

b k , add an arrow j [ba]k and put

d([ba]) =d(a) + d(b),

(M2gr) replace each arrow j a i by an arrow j i

a and putd(a) = 1

d(a) (resp. d(a) =d(a)),

replace each arrow i b k by an arrow i k

b and putd(b) =d(b)

(resp. d(b) = 1 d(b)).

All other arrows remain with the same degree and the potential W = [W] +W

(as defined in[36]) and is again homogeneous of degree 1.

In the derived category Db(), if T is an object such that (T) C2() iscluster-tilting, then one can lift the exchange triangles of Theorem2.2 to trianglesin Db(). Then we obtain for any indecomposable direct summand Ti of T =T0 Ti the following triangles in Db()

Ti B TLiTi[1] and TRi

B Ti TRi [1].

The objects TLi and TRi satisfy that (TLi ) = (TRi ) = (Ti) is the uniquecomplement non isomorphic to (Ti) of the almost complete cluster-tilting object(T0). We call the object T0T

Li (resp. T

Ri ) the left mutation (resp. right

mutation) ofT atTi.

Then we can formulate the graded analogue of Theorem 2.19, which is a firstmotivation for the introduction of mutation of graded QP.

Theorem 5.5 ([7]). Let and T Db() as above. Assume that there exist agraded QP(Q,W,d) with potential homogeneous of degree 1 such that we have anisomorphism of graded algebras

pZHomD(T, S

p2 T)

Z

Jac(Q,W,d).

Let Ti be an indecomposable summand of T Ti T0 and assume that thereare neither loops nor2-cycles incident to i (corresponding to Ti) in the quiver ofEndC(T). Then there is an isomorphism ofZ-graded algebras

pZHomD(T0 T

Li , S

p2 (T0 T

Li ))

Z

Jac(Li(Q,W,d)).

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42 C. Amiot

This result can be illustrated by the following picture

T Db()(T) cluster-tilting

left mutation

T Db()(T) cluster-tilting

right mutation

iZHomD(T,S

i2 T)

Z

Jac(Q,W,d)

iZHomD(T, Si2 T

)Z

Jac(Q, W, d)

(Q,W,d)left graded mutation

(Q, W, d).

right graded mutation

Our main application of the graded mutation is given by the following result,

which gives a combinatorial criterion to see when two algebras of global dimensionat most two are derived equivalent.

Theorem 5.6([7]). Let and be two algebras of global dimension 2 , which are2-finite. Assume that one can pass from the graded QP(Q, W , d) associated with3() to the graded QP (Q

, W, d) (up to graded right equivalence) associatedwith3(

) using a finite sequence of left and right mutations. Then the algebras and are derived equivalent.

This result can be seen as a generalization (in one direction) of Happels The-orem (Thm 1.3). So we could ask what is the meaning of the hypothesis of 2-finiteness for the algebras and and wether this hypothesis is necessary or not.Indeed, cluster categories and cluster-tilting theory are hidden in the statement.However, the proof uses strongly Theorems5.2,5.3 and 5.5, where the hypothesis

of2-finiteness is fundamental.

Example 5.7. Let us illustrate this result with an example. The following gradedquivers with potential are linked by sequences of left and right graded mutations

1

2

3

0 0

0 1

2

3

L2 L2

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