This article was downloaded by: [Iowa State University]On: 10 November 2014, At: 18:08Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/lagb20

On Modules over Endomorphism Algebrasof Maximal Rigid Objects in 2-Calabi-YauTriangulated CategoriesPin Liu a & Yunli Xie aa Department of Mathematics , Southwest Jiaotong University ,Chengdu , ChinaPublished online: 14 May 2014.

To cite this article: Pin Liu & Yunli Xie (2014) On Modules over Endomorphism Algebras of MaximalRigid Objects in 2-Calabi-Yau Triangulated Categories, Communications in Algebra, 42:10, 4296-4307,DOI: 10.1080/00927872.2013.809531

To link to this article: http://dx.doi.org/10.1080/00927872.2013.809531

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoever as tothe accuracy, completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims,proceedings, demands, costs, expenses, damages, and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with, in relation to or arisingout of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Communications in Algebra®, 42: 4296–4307, 2014Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2013.809531

MAXIMAL RIGID OBJECTS IN 2-CALABI-YAUTRIANGULATED CATEGORIES

Pin Liu and Yunli Xie

Department of Mathematics, Southwest Jiaotong University, Chengdu, China

This note investigates the modules over the endomorphism algebras of maximal rigidobjects in 2-Calabi-Yau triangulated categories. We study the possible complementsfor almost complete tilting modules. Combining with Happel’s theorem, we show thatthe possible exchange sequences for tilting modules over such algebras are induced bythe exchange triangles for maximal rigid objects in the corresponding 2-Calabi-Yautriangulated categories. For the modules of infinite projective dimension, we generalizea recent result by Beaudet–Brüstle–Todorov for cluster-tilted algebras.

Key Words: 2-Calabi-Yau category; Maximal rigid object; Tilting module.

2010 Mathematics Subject Classification: 18E30; 16D90.

1. INTRODUCTION

The study of the possible complements for an almost complete tilting objecthas been the central point of many investigations during the past decades. Let Abe a finite-dimensional k-algebra over an algebraically closed field k. We denote bymod A the category of finitely generated right A-modules. Let n be the number ofisomorphism classes of simple A-modules. For an A-module T , let add T denotethe full subcategory of mod A with objects all direct summands of direct sums ofcopies of T . Then T is called a tilting module in mod A if the following statementshold:

(1) proj�dimA T ≤ 1;(2) Ext1A�T� T�= 0;(3) There is an exact sequence 0 → A → T 0 → T 1 → 0, with T 0� T 1 in add T .

Received May 9, 2012; Revised April 7, 2013. Communicated by Q. Wu.Address correspondence to Yunli Xie, Department of Mathematics, Southwest Jiaotong

University, 610031 Chengdu P. R. China; Email: xieyunli@home.swjtu.edu.cn

4296

ON MODULES OVER ENDOMORPHISM ALGEBRAS OF

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

ENDOMORPHISM ALGEBRAS IN 2-CY CATEGORIES 4297

This is the original definition of tilting modules from [13], and it was proved in [5]that the third axiom can be replaced by the following one:

(3’) The number of indecomposable direct summands of T (up to isomorphism)is n.

Tilting modules have been found to be very useful in the representation theoryof finite-dimensional algebras (We refer to [11, 24] for an outline of tilting theory.)

An A-module T satisfying the axioms (i) and (ii) of the definition above iscalled a partial tilting module, and if moreover T has n− 1 indecomposable directsummands (up to isomorphism), then we call T an almost complete tilting module.Let M be an A-module such that T ⊕M is a tilting module and addT

⋂addM = 0.

Then M will be called a complement to T . It is known that T always admits acomplement [5, 13]. And it was shown in [14, 23] that T has at most two non-isomorphic indecomposable complements. It has been proved by Happel in [12] (see[14] for the case of hereditary algebras) that an almost complete tilting module Thas exactly two nonisomorphic complements if and only if T is faithful. Moreover,in the case that T has exactly two complements M and M ′, there exists an exchangesequence, a short exact sequence describing the exchange relation for the almostcomplete tilting module T

0 → M → E → M ′ → 0�

whose middle term belongs to add T .On the other hand, tilting theory in cluster categories and more generally,

Hom-finite Calabi–Yau triangulated categories have recently been widelyinvestigated. The study of such categories was originally motivated by their links tocluster algebras, and indeed there has been a considerable amount of activities anda lot of results in this direction. We refer the reader to the surveys [17, 22].

Let � be a Calabi–Yau triangulated category of CY-dimension 2. An objectR of � is rigid if Ext1��R�R�= 0. It is maximal rigid if it is rigid and Ext1��X ⊕R�X ⊕ R�= 0 implies that X ∈ add R. A rigid object R is called cluster tilting ifExt1��X�R�= 0 implies that X ∈ add R. Cluster-tilting objects are always maximalrigid objects, while the converse is not true in general. There exist 2-Calabi-Yautriangulated categories in which maximal rigid objects are not cluster-tilting [3, 6, 8].However, the maximal rigid objects enjoy some very nice properties as cluster tiltingobjects do. Remarkably, one also has exchange triangles to determine mutations ofmaximal rigid objects. More precisely, recall that an object L is an almost completemaximal rigid object if there is an indecomposable object M which is not isomorphicto any object in add L such that L=L⊕M is a maximal rigid object in �. Such M iscalled a complement of an almost complete maximal rigid object. It has been provedin [28] that there are exactly two complements of an almost complete maximal rigidobject L, say M and M∗, and there are non split triangles

M∗ g−→ Ef−→ M

h−→ SM∗ and Mf∗−→ E′ g∗−→ M∗ h∗−→ SM�

such that f is a minimal right add L-approximation and f ∗ a minimal leftadd L-approximation.

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

4298 LIU AND XIE

Meanwhile the endomorphism algebras of cluster-tilting objects in the2-Calabi-Yau triangulated categories have been further developed and revealedthat they have very nice properties (see [7, 18] for instance). And in general, theendomorphism algebras of maximal rigid objects in the 2-Calabi-Yau triangulatedcategories are studied in [26–28]. It is an interesting idea to investigate algebrasderived equivalent to such algebras. Understanding their tilting modules is a stepin this direction. In Section 2, we study the possible complements for an almostcomplete tilting module. We apply Happel’s theorem to study the relationshipbetween exchange sequences and exchange triangles. In Section 3, we characterizepartial tilting modules. Because the endomorphism algebras of maximal rigid objectsare Gorenstein of Gorenstein dimension at most 1, this is equivalent to characterizethe modules of infinite projective dimension. We generalize recent work of Beaudet–Brüstle–Todorov [4] for cluster-tilted algebras to the endomorphism algebras ofmaximal rigid objects in 2-Calabi-Yau triangulated categories.

2. ALMOST COMPLETE TILTING MODULES

Let k be an algebraically closed field and � be a Krull–Schmidt triangulatedk-linear category with split idempotents and suspension functor S. We suppose thatall Hom-spaces of � are finite-dimensional and that � admits a Serre functor ��i.e., for any X� Y in �, we have the following bifunctorial isomorphisms:

Hom��X� Y� � DHom��Y� �X��

where D=Homk�−� k� is the usual duality. We suppose that � is Calabi–Yau ofCY-dimension 2; i�e� there is an isomorphism of triangle functors

S2 ∼→ ��

For X� Y ∈ � and n ∈ �, we put as usual

Extn��X� Y� = Hom��X� SnY��

Thus the Calabi–Yau property can be written as the following bifunctorialisomorphisms:

DExt1�X� Y� � Ext1�Y� X�� for any X� Y�

2.1. Endomorphism Algebras of Maximal Rigid Objects in2-Calabi-Yau Triangulated Categories

Let R be a maximal rigid object in �. The following result about the numberof indecomposable direct summands of maximal rigid objects is very helpful.

Lemma 2.1 ([28]). All maximal rigid objects in a 2-Calabi-Yau triangulated categoryhave the same number of indecomposable direct summands (up to isomorphism).

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

ENDOMORPHISM ALGEBRAS IN 2-CY CATEGORIES 4299

An object M of � is finitely presented by R if there is a triangle in �

R1 → R0

f−→ M → SR1

with R0� R1 in add R. The morphism f is necessarily a right add R-approximationof M , and, conversely, the cone of any add R-approximation of an object M finitelypresented by R belongs to add SR (see [27]). Let pr�R� denote the (full) subcategoryof � of objects finitely presented by R.

Let � be the endomorphism algebra of R. Let mod� denote the categoryof finite-dimensional right modules over � . The following result expresses aclose relationship between 2-Calabi-Yau triangulated categories and the modulecategories of endomorphism algebras of maximal rigid objects in 2-Calabi-Yautriangulated categories (see [16, 27]).

Lemma 2.2. The functor F =Hom��R�−� � � → mod� induces an equivalence

pr�R�/add SR∼→ mod��

where the category on the left has the same objects as pr�R�, with morphisms given bymorphisms in � modulo maps factoring through objects in add SR.

Thus the functor F induces an equivalence from add R to the category ofprojective modules in mod� and each �-module has the form FM for M ∈ pr�R�.Since our focus is on the quotient category pr�R�/add SR, when we mention a �-module FM we shall always assume that add M ∩ add SR= 0. We write ��X� Y� forthe set of morphisms from X to Y in the category � and ��X� Y� for the one in thecategory mod� .

It was proved in [28] that � is Gorenstein of dimension at most 1, that is,inj�dim�� ≤ 1 and inj�dim�� ≤ 1. As a consequence, each �-module is either ofinfinite projective (respectively injective) dimension or of projective (respectivelyinjective) dimension at most 1. We study almost complete tilting modules in thissection and characterize the modules of infinite projective dimension in section 3.

The following result proved partly by the authors in [21] is fundamental for thepresent note to investigate the relationship between possible exchange sequences foralmost complete tilting modules over � and exchange triangles for almost completemaximal rigid objects in �.

Lemma 2.3. Let R be a maximal rigid object in the 2-Calabi-Yau triangulatedcategory �� � be the endomorphism algebra of R, and FL be a tilting module over � .Then FL lifts uniquely to a maximal rigid object in �.

Proof. The lifting part has been proved in [21]. Here we prove that the projectionfunctor pr�R� → pr�R�/addSR does provide a unique lift of a tilting �-module. Thepreimage of FL has the form L⊕M for some M ∈ addSR and L⊕M is a maximalrigid object in �.

On the other hand, because FL is a tilting module over � , the numberof indecomposable direct summands of FL (up to isomorphism) is the same as

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

4300 LIU AND XIE

the number of simple �-modules which is the number of indecomposable directsummands of R. Thus the number of pairwise non-isomorphic indecomposabledirect summands of the rigid object L⊕M is greater than that of the maximalrigid object R if M ∈ add L, which is a contradiction to Lemma 2.1. Hence M ∈add L

⋂add SR, which is 0 by assumption. This finishes the proof. �

2.2. Almost Complete Tilting Modules

Let FL be an almost complete tilting �-module and FM be its complement;i.e., FL⊕ FM is a tilting �-module where M is indecomposable. Thus by Lemma 2.3we know that L⊕M is a maximal rigid object in �. So there are the correspondingexchange triangles in �.

Lemma 2.4. [16, 28] Up to isomorphism, there is a unique indecomposable objectM∗ not isomorphic to M such that L⊕M∗ is maximal rigid in � and there are nonsplittriangles

M∗ g−→ Ef−→ M

h−→ SM∗ and Mf∗−→ E′ g∗−→ M∗ h∗−→ SM�

such that f is a minimal right add L-approximation and f ∗ a minimal left add L-approximation.

The main result in this section is the following theorem which shows thatthe possible complements of an almost complete tilting �-module are given by thecorresponding exchange triangles in �.

Theorem 2.5. Use the above notation. If the image FM∗ of the indecomposable objectM∗ is not zero and of projective dimension at most 1, then FL⊕ FM∗ is a tilting �-module.

Proof. We check the tilting conditions.First by assumptions, the �-module FL⊕ FM∗ is of projective dimension at

most one.Secondly, we prove that

Ext1� �FL⊕ FM∗� FL⊕ FM∗� = 0�

For this, let X� Y be any indecomposable objects in pr�R� such that

Ext1��X� Y� = 0�

Note that � is 2-Calabi-Yau, we also have

Ext1��Y� X� = 0�

Let

RX1

pX1−→ RX0

pX0−→ X → SRX1 (∗)

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

ENDOMORPHISM ALGEBRAS IN 2-CY CATEGORIES 4301

be the “minimal add R-approximation triangle” of X. Applying F =Hom��R�−� on�∗�, we get the following exact sequence in mod� :

FRX1

FpX1−−→ FRX0

FpX0−−→ FX → 0�

Since we consider the approximation triangle and note that FRX0 and FRX

1 areprojectives in mod� , we obtain the following projective resolution for FX:

FRX2 → FRX

1

FpX1−−→ FRX0

FpX0−−→ FX → 0�

where FRX2 is projective in mod� . Applying the functor Hom� �−� FY�, we have the

following complex:

0 → ��FX� FY� → ��FRX0 � FY�

��FpX1 �FY�−−−−−→ ��FRX1 � FY� → ��FRX

2 � FY��

By the definition, to prove Ext1� �FX� FY�= 0, it is enough to prove that ��FpX1 � FY�

is a surjective map. Note that R is rigid, it follows that

��FpX1 � FY� = ��pX

1 � Y�

by the definition of quotient category. Thus it suffices for us to prove that ��pX1 � Y�

is surjective. For this, applying the functor Hom��−� Y� to �∗�, one gets thefollowing exact sequence:

��X� Y� → ��RX0 � Y�

��pX1 �Y�−−−−→ ��RX1 � Y� → ��S−X� Y��

But

��S−X� Y� � ��X� SY� = 0�

Hence ��pX1 � Y� is a surjective map, which implies that

Ext1� �FX� FY� = 0�

Similarly, we have that

Ext1� �FY� FX� = 0�

Because all the rigid objects of � lie in pr�R� ([6, 28]), we obtain that FL⊕ FM∗ isa rigid module over � .

Finally, we compare the number of indecomposable direct summands of FL⊕FM∗ (up to isomorphism) with the number of simple �-modules. Since � is theendomorphism algebra of R in �, the number of simple �-modules is same asthe number of indecomposable direct summands of R. But by Lemma 2.1, up toisomorphism, the maximal rigid objects L⊕M∗ and R have the same number ofindecomposable direct summands. This finishes the proof. �

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

4302 LIU AND XIE

Applying Happel’s theorem, we have the following theorem to indicate thatthe exchange sequences for tilting modules over � are induced by the correspondingexchange triangles in �. Please note that the existence of the exchange triangles inthe statement follows from Lemma 2.3 as mentioned above.

Theorem 2.6. Let � be a 2-Calabi-Yau triangulated category with a maximal rigidobject R, and � be the endomorphism algebra of R. Let FL be an almost tilting moduleover � with a complement FM . Let

M∗ g−→ Ef−→ M

h−→ SM∗ and Mf∗−→ E′ g∗−→ M∗ h∗−→ SM

be the corresponding exchange triangles in �. Then the following statements areequivalent:

(1) FL⊕ FM∗ is a tilting �-module;(2) FL is a faithful �-module.

Proof. It has been proved by Happel [12, Corollary 2.6] that the almost completetilting module FL has two non-isomorphic indecomposable complements if and onlyif FL is faithful. Thus for the equivalence of (1) and (2), we only need to provethat (2) implies (1). If FL is faithful, there exists FM ′ � FM such that FL⊕ FM ′

is a tilting �-module. So by Lemma 2.3 we know that L⊕M ′ is a maximal rigidobject in �. Because, up to isomorphism, M ′ =M , it follows that M ′ =M∗, in viewof Lemma 2.4. This finishes the proof. �

2.3. Cluster Tilting Case

As an application, we present here some slight generalization of a result in [25](see also [20]). For the sake of completeness we will provide full proofs.

Cluster-tilting objects in 2-Calabi-Yau triangulated categories and thecorresponding endomorphism algebras were originally defined and studied in [18]([7] for cluster category case).

Cluster-tilting objects are obviously maximal rigid objects. But the converse isnot true in general. There are 2-Calabi-Yau triangulated categories which containno cluster-tilting objects (see more in [8, 9, 19]). However it was shown in [28] thatthe converse is true if the 2-Calabi-Yau triangulated category admits a cluster-tiltingobject.

Lemma 2.7. If the 2-Calabi-Yau triangulated category admits a cluster-tilting object,then every maximal rigid object is cluster tilting.

Thus if the 2-Calabi-Yau triangulated category � admits a cluster-tiltingobject, the endomorphism algebra of a maximal rigid object should read 2-Calabi-Yau tilted algebra.

Let � be a 2-Calabi-Yau triangulated category with a cluster-tilting object T ,and let � be the 2-Calabi-Yau tilted algebra corresponding to T . Denote the functor

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

ENDOMORPHISM ALGEBRAS IN 2-CY CATEGORIES 4303

Hom��T�−� � � → mod� by F . Note that under the condition that T is clustertilting, the subcategory pr�T� coincides with �. In view of Lemma 2.2, the functorF induces an equivalence of categories

�/addST∼→ mod��

This result, which generalises a corresponding result of [7, Theorem 2.2] in the caseof cluster categories, was proved by Keller and Reiten [18, Proposition 2.1]. See also[19, Corollary 4.4].

It was proved in [18, Lemma 3.5] that the suspension functor in � induces theAuslander-Reiten translation of mod� .

The following lemma, which will be useful in the future, was shown in [19].

Lemma 2.8. Let f � X → Y be a morphism in � which is a part of a triangle S−1Zh−→

Xf−→ Y

g−→ Z. Then Ff is a monomorphism in mod� if and only if Fh= 0; Ff is anepimorphism in mod� if and only if Fg = 0. Furthermore, if Fh= 0=FSh, then 0 →FX

Ff−→ FYFg−→ FZ → 0 is an exact sequence in mod� .

As before, let FL be an almost complete tilting �-module and FM be itscomplement; i.e., FL⊕ FM is a tilting �-module where M is indecomposable.

It follows from Lemma 2.3 and Lemma 2.7 that a tilting module over � liftsto a cluster-tilting object in �. This result, which generalises a corresponding resultof [25] for cluster categories, was proved by Fu and Liu [10, Theorem 3.3]. See also[15]. Hence we have that L = L⊕M is a cluster-tilting object in �. By Lemma [28],up to isomorphism, there is a unique indecomposable object M∗ not isomorphic toM such that L⊕M∗ is cluster tilting in � and there are nonsplit triangles

M∗ g−→ Ef−→ M

h−→ SM∗ and Mf∗−→ E′ g∗−→ M∗ h∗−→ SM�

such that f is a minimal right add L-approximation and f ∗ a minimal left add L-approximation.

Proposition 2.9. Let � be a 2-Calabi-Yau triangulated category with a cluster-tiltingobject T� � be the 2-Calabi-Yau tilted algebra corresponding to T , and FL be an almostcomplete tilting �-module with a complement FM . Let

M∗ g−→ Ef−→ M

h−→ SM∗ and Mf∗−→ E′ g∗−→ M∗ h∗−→ SM

be the corresponding exchange triangles in �. Then the following statements areequivalent:

(1) In mod� , FL⊕ FM∗ is a tilting �-module;(2) The image FM∗ of the indecomposable object M∗ is not zero and of projective

dimension at most 1;(3) In mod� , either Ff is an epimorphism or Ff ∗ is a monomorphism;(4) The almost complete tilting �-module FL is faithful.

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

4304 LIU AND XIE

Proof. We only need to prove the equivalence of (2) and (3).We assume that (2) holds. Thus by Theorem 2.5 the almost complete tilting

�-module FL admits two non-isomorphic complements FM and FM∗. And byHappel’s theorem [12, Theorem 2.5], there exists a short sequence of the form

0 → FM∗ → FXj−→ FM → 0 (∗)

or

0 → FMj∗−→ FY → FM∗ → 0� (∗∗)

where X� Y ∈ add L. Assume that the exact sequence �∗� exists, and let j � X → M bea morphism in � such that Fj= j in mod� . Now, because f � E → M is a minimalright add L-approximation, there is f ′ � X → E

such that j= ff ′. Then j=Ff · Ff ′. Since j is an epimorphism, Ff is anepimorphism. Similarly, if the short exact sequence �∗∗� exists, one can prove thatFf ∗ is a monomorphism. This implies (3).

Conversely, without loss of generality, we assume Ff is an epimorphism. ByLemma 2.8, we know Fh= 0. Thus h factors through add ST , and hence, S−1hfactors through add T in �; i.e., S−1h is the composition

S−1h � S−1M → T ′ → M∗ for some T ′ ∈ addT�

On the other hand, proj�dim�FM ≤ 1 means that there exists the exact sequence

0 → FTM1 → FTM

0 → FM → 0�

We know that the projective �-modules FTM1 � FTM

0 are of injective dimension atmost 1. So the injective dimension of �-module FM is finite, which implies that FMis of injective dimension at most 1. Thus, by [1, Lemma IV.2.7 (b)] (or the dual of[2, Proposition IV.1.16]), we have

Hom� �−1FM� �� = 0�

Since the Auslander–Reiten translation on mod� is induced by the suspensionfunctor on � ([18, Lemma 3.5]), we know that all the morphisms from S−1M toadd T in � do factor through addST . It follows that S−1M → T ′ factors throughaddST . Hence FS−1h= 0. Thus by Lemma 2.8, we have the exact sequence in mod�

0 → FM∗ → FEFf−→ FM → 0�

Note that E ∈ add L and FL⊕ FM is a tilting module, so proj�dim�FM ≤ 1 andproj�dim�FE ≤ 1, which implies that the �-module FM∗ is of finite projective

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

ENDOMORPHISM ALGEBRAS IN 2-CY CATEGORIES 4305

dimension and so proj�dim�FM∗ ≤ 1. Furthermore, Ff is not an isomorphism,

which implies that FM∗ = 0. Dually, if Ff ∗ is a monomorphism, one can use thetriangle

Mf∗−→ E′ g∗−→ M∗ h∗−→ SM

to prove that (2) holds. This finishes the proof. �

3. MODULES OF INFINITE PROJECTIVE DIMENSION

In this section we characterize the modules of infinite projective dimensionover the endomorphism algebras of maximal rigid objects in 2-Calabi-Yautriangulated categories. Let � be a 2-Calabi-Yau triangulated category with amaximal rigid object R. As in [4], for any indecomposable object M in � which doesnot belong to addSR, we denote by IM the factorization ideal of End��SR� given byall endomorphisms that factor through M .

The main result in this section is the following theorem which is ageneralization of Theorem 1 in [4].

Theorem 3.1. Let � be a 2-Calabi-Yau triangulated category with a maximal rigidobject R, and � be the endomorphism algebra of R. Then a �-module FM is of infiniteprojective dimension if and only if the factorization ideal IM is nonzero.

Proof. Let

RM1

pM1−→ RM0

pM0−→ M → SRM1 (∗)

be the “minimal addR-approximation triangle” of M .First, we assume �-module FM has infinite projective dimension. Applying the

functor F on �∗�, we get the following exact sequence in mod� :

FS−MFS−−−→ FRM

1

FpM1−−→ FRM0

FpM0−−→ FM → 0�

It follows that the morphism FS− = 0, since otherwise the projective dimension ofFM would be at most one. Choose a morphism S−� in ��R� S−M� whose imageunder FS− is nonzero, that is, the composition

RS−�−→ S−M

S−−−→ RM1

is nonzero. It follows that there is a nonzero element � in the factorization ideal IM .Conversely, if the factorization ideal IM is nonzero, we prove that the

�-module FM is not of projective dimension at most 1. Otherwise, applying F on�∗�, we have the following exact sequence:

0 → FRM1

FpM1−−→ FRM0

FpM0−−→ FM → 0�

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

4306 LIU AND XIE

Let � � SR�−→ M

−→ SR be any element in IM . Note that R is rigid, pM0 ∈

��RM0 � SR� = 0. Hence we have the following commutative diagram:

Therefore, there are morphisms x � R → RM1 and y � RM

1 → R which make thefollowing diagram commutative:

Thus pM1 x= 0, and hence x= 0 since FpM

1 is a monomorphism. This implies Sx= 0,and, therefore, �= SySx= 0. It follows that IM = 0, which is a contradiction to ourassumption. This finishes the proof. �

ACKNOWLEDGMENTS

The authors are grateful to the anonymous referee for the careful reading ofthe manuscript and very helpful advice.

FUNDING

Supported by the NSF of China (Grant 11201381).

REFERENCES

[1] Assem, I., Simson, D., Skowronski, A. (2006). Elements of the representation theoryof associative algebras, volumn 1: techniques of representation theory. London Math.Soc. Student Texts 65, Cambridge: Cambridge University Press.

[2] Auslander, M., Reiten, I., Smalo, S. O. (1995). Representation Theory of Artin Algebras.Cambridge: Cambridge University Press.

[3] Barot, M., Kussin, D., Lenzing, H. (2008). The Grothendieck group of a clustercategory. J. Pure Appl. Algebra 212(1):33–46.

[4] Beaudet, L., Brüstle, T., Todorov, G. (2011) Projective dimension of modules overcluster-tilted algebras. Preprint, arXiv:1111.2013v1.

[5] Bongartz, K. (1981). Tilted algebras. Representations of algebras (Puebla, 1980),Lecture Notes in Math. 903, Berlin-New York: Springer, pp. 26–38.

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14

ENDOMORPHISM ALGEBRAS IN 2-CY CATEGORIES 4307

[6] Buan, A. B., Iyama, O., Reiten, I., Scott, J. (2009). Cluster structure for 2-Calabi-Yaucategories and unipotent groups. Compos. Math. 145(4): 1035–1079.

[7] Buan, A. B., Marsh, R. J., Reiten, I. (2007). Cluster-tilted algebras. Trans. Amer. Math.Soc. 359(1):323–332.

[8] Buan, A. B., Marsh, R. J., Vatne, D. F. (2010). Cluster structure from 2-Calabi-Yaucategories with loops. Math. Z. 265(4):951–970.

[9] Burban, I., Iyama, O., Keller, B., Reiten, I. (2008). Cluster tilting for one-dimensionalhypersurface singularities. Adv. Math. 217(6):2443–2484.

[10] Fu, C., Liu, P. (2009). Lifting to cluster-tilting objects in 2-Calabi-Yau triangulatedcategories. Comm. Algebra 37:1–9.

[11] Happel, D. (1988). Triangulated categories in the representation theory of finitedimensional algebras. London Math. Soc. Lecture Note Ser. 119, London-New York:Cambridge University Press.

[12] Happel, D. (1992). Partial tilting modules and recollement, in “Proceedings, MalcevConference”, Contemporary Mathematics 131, Providence: pp. 345–361.

[13] Happel, D., Ringel, C. M. (1982). Tilted algebras. Trans. Amer. Math. Soc.274(2):399–443.

[14] Happel, D., Unger, L. (1989). Almost complete tilting modules. Proc. Amer. Math.Soc. 107:603–610.

[15] Holm, T., Jorgensen, P. (2010). On the relation between cluster and classical tilting.J. Pure Appl. Algebra 214(9):1523–1533.

[16] Iyama, O., Yoshino, Y. (2008). Mutation in triangulated categories and rigid Cohen-Macaulay modules. Inv. Math. 172(1):117–168.

[17] Keller, B. (2010). Cluster algebras, quiver representations and triangulated categories,in Triangulated Categories (edited by Holm, T., Jogensen, P., Rouquier, R.). LondonMath. Soc. Lecture Note Ser. 375, Cambridge: Cambridge University Press.

[18] Keller, B., Reiten, I. (2007). Cluster-tilted algebras are Gorenstein and stably Calabi–Yau. Adv. Math. 211(1):123–151.

[19] Koenig, S., Zhu, B. (2008). From triangulated categories to abelian categories-clustertilting in a general framework. Math. Z. 258:143–160.

[20] Liu, P. (2010). Exchange relation for 2-Calabi-Yau tilted algebras (in Chinese). Sci.Sin. Math. 40(11):1039–1044.

[21] Liu, P., Xie, Y. (2013). Lifting to maximal rigid objects in 2-Calabi-Yau triangulatedcategories. Proc. Amer. Math. Soc. 141:3361–3367.

[22] Reiten, I. (2011). Cluster categories. Proceedings of the International Congress ofMathematicians 2010 (ICM 2010), vol. 1, New Delhi: Hindustan Book Agency, pp.558–594.

[23] Riedtmann, C., Schofield, A. (1991). On a simplicial complex associated with tiltingmodules. Comment. Math. Helv. 66:70–78.

[24] Ringel, C. M. (2006). Some remarks concerning tilting modules and tilted algebras.Origin. Relevance. Future. An appendix to the Handbook of Tilting Theory (edited byAngeleri-Huegel, L., Happel, D., Krause, H.). London Mathematical Society LectureNotes Series 332, Cambridge: Cambridge University Press.

[25] Smith, D. (2008). On tilting modules over cluster-tilted algebras. Illinois J. Math.52(4):1223–1247.

[26] Vatne, D. F. (2011). Endomorphism rings of maximal rigid objects in cluster tubes.Colloq. Math. 123:63–93.

[27] Yang, D. (2012). Endomorphism algebras of maximal rigid objects in cluster tubes.Comm. Algebra 40:4347–4371.

[28] Zhou, Y., Zhu, B. (2011). Maximal rigid subcategories in 2-Calabi-Yau triangulatedcategories. J. Algebra 348:49–60.

Dow

nloa

ded

by [

Iow

a St

ate

Uni

vers

ity]

at 1

8:08

10

Nov

embe

r 20

14