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⋆ Generalized luster-tilted algebras 9 0 Shunhua Zhang 0 2 Department of Mathemati s, Shandong University, Jinan 250100, P. R. China b e F E-mail: shzhangsdu.edu. n 8 1 ] Abstra t T H R Let be a (cid:28)nite dimensional hereditary algebra over an algebrai ally h. losed (cid:28)eld k and CFm be the generalized luster ategory of H with m ≥ 1. t C a Fm Weinvestigatethepropertiesofgeneralized lustertiltingobje tsin and m [ the stru ture of generalized luster-tilted algebras. Moreover, we generalized 1 Theorem 4.2 in [A. Buan, R. Marsh, I. Reiten. Cluster-tilted algebra. Trans. C v Fm Amer. Math. So ., 359(1)(2007), 323-332.℄ to the situation of , and 7 K C 4 prove that the tilting graph CFm of Fm is onne ted. 0 3 . 2 Keywords. generalized luster ategory, generalized lustertiltingobje t, gen- 0 9 eralized luster-tilted algebra. 0 : Mathemati s Subje t Classi(cid:28) ation. 16G20, 16G70. v i X r a 1 Introdu tion H Let be a (cid:28)nite dimensional hereditary algebra over an algebrai ally losed (cid:28)eld k H . The endomorphism algebra of a tilting module over is alled tilted algebra. H C = Db(H)/ < F > Cluster ategory of type is the orbit ategory of derived Db(H) H F = τ−1[1] τ ategories of byanautomorphismgroupgeneratedby ,where Db(H) [1] Db(H) is the Auslander-Reiten translation in and is the shift fun tor of . ⋆ Supported by the NSF of China (Grant No. 10771112) and of Shandong provin e (Grant No. Y2008A05) 1 C 2 is triangulated ategory and is Calabi-Yau ategories of CY-dimension , see C [BMRRT℄ and [K℄. It was shown that any luster tilting obje t of is indu ed H′ H from a tilting module of a hereditary algebra whi h is derived equivalent to . C The endomorphism algebra of a tilting obje t in is now alled a luster-tilted algebra. Now luster-tilted algebras and luster ategories provide an algebrai under- standing of ombinatori s of luster algebras de(cid:28)ned and studied by Fomin and Zelevinsky in [FZ℄. In this onne tion, the inde omposable ex eptional obje ts in luster ategories orrespond to the luster variables, and luster tilting obje ts(= maximal 1-orthogonal sub ategories [I1, I2℄) to lusters of orresponding luster algebras, see [CK1, CK2℄. Moreover by [KR℄ or [KZ℄, luster-tilted algebras pro- 1 vide a lass of Gorenstein algebras of Gorenstein dimension , whi h is important in representation theory of algebras [Rin2℄. m CFm For any positive integer , a generalized luster ategory , whi h is de(cid:28)ned Db(H) < Fm > by [Zhu℄ as the orbit ategories of derived ategories by the group Fm generated by , is a triangulated ategory by Keller [K℄, whi h is also a Calabi- Yau ategories of Calabi- Yau dimension 2m/m. The generalized luster tilting obje ts in this generalized luster ategory are shown to orrespond one-to-one to those in the lassi al luster ategories. The endomorphism algebras of generalized C Fm luster tilting obje ts in are alled generalized luster-tilted algebras. They all have the same representation type and share an universal overing: the endo- Db(H) morphism algebra of orresponding luster tilting sub ategory in , see [Zhu℄ for details. In this paper, we investigate the properties of generalized luster tilting obje ts and the stru ture of generalized luster-tilted algebras. The arti le is organized as follows: In Se tion 2 we prove some basi fa ts about generalized luster tilting obje ts. In se tion 3, we investigate the stru ture of generalized luster-tilted C Fm algebras,andgeneralizeTheorem4.2in[BMR℄tothesituationof ,seeTheorem K C 3.4. Furthermore, we also prove that the tilting graph CFm of Fm is onne ted. k Throughout this paper, we (cid:28)x an algebrai ally losed (cid:28)eld , and denote by 2 H k n a (cid:28)nite dimensional hereditary -algebra with simple modules. We denote C = Db(mod H)/τ−1[1] H C by the luster ategory of . A basi tilting obje t in T n is an obje t with non-isomorphi inde omposable dire t summands su h that Ext1(T,T) = 0 C . We follow the standard terminology and notation used in the representation theory of algebras, see [ARS℄, [ASS℄, [Hap℄ and [Rin1℄. 2 Properties of generalized luster tilting obje ts In this se tion, we (cid:28)rst re allsomede(cid:28)nitionsand olle tsome known results whi h will be used later, then we prove some basi fa ts about generalized luster tilting obje ts. A B A Let be a Krull-Remark-S hmidt ategory and a full sub ategory of . indB B We denote by the set of all inde omposable obje ts in . For any obje t M A addM A in , we denote by the full sub ategory of onsisting of (cid:28)nite dire t M δ(M) sumsofinde omposablesummandsof andby thenumberofon-isomorphi M inde omposable summands of . m m ≥ 1 CFm = Db(H)/(Fm) Let be an integer with and be the generalized F = τ−1[1] CF1 = C luster ategory de(cid:28)ned in [Zhu℄ with . Note that is the lassi al luster ategory. The following de(cid:28)nition is de(cid:28)ned in [I1,I2,Zhu℄. T CFm De(cid:28)nition 2.1. An obje t of is a generalized luster tilting obje t X ∈ addT Ext1 (X,T) = 0 X ∈ addT provided if and only if CFm and if and only if Ext1 (T,X) = 0 CFm . ρm : CFm → C The triangle fun tor is de(cid:28)ned in [Zhu℄ whi h is also a overing fun tor. The following lemma is taken from [Zhu℄ whi h will be used in the sequel. C Fm Lemma 2.2. (1). is a Krull-remark-S hmidt ategory. m−1 indCFm = (indFi(C)) (2). S . i=0 T C ρ−1(T) m (3). is a luster tilting obje t in if and only if is a generalized C Fm luster tilting obje t in . 3 m−1 H T FiT (4). For any tilting -module , L is a generalized luster tilting obje t i=0 C C Fm Fm in , and any generalized luster tilting obje t in arises in this way, i.e., H′ H there is a hereditary algebra , whi h is derived equivalent to , and a tilting H′ T′ T′ -module su h that the generalized luster tilting obje t is indu ed from . δ(T) = mn T Remark. It is easy to see that provided is a generalized luster C Fm tilting obje t in . Let M be an obje t of CFm. M issaid to be 1-orthogonalifExt1CFm(M,M) = 0. 1 CFm δ(M) = A basi -orthogonal obje t of is said to be an almost tilting obje t if nm−1 X CFm M ⊕X and there exists an inde omposable obje t of su h that is C Fm a generalized luster tilting obje t of . M CFm F M De(cid:28)nition 2.3. An obje t of is said to be -stable if an be written M = X ⊕FX ⊕···⊕Fm−1 X C as for some obje t in . In this ase, we say that F M X the -stable obje t is determined by . M F CFm M = X ⊕ FX ⊕ Lemma 2.4. Let be a -stable obje t of and ··· ⊕ Fm−1X X C Ext1 (M,M) = 0 for some obje t in . Then CFm if and only if Ext1(X,X) = 0 C . Proof. It follows from that Ext1 (M,M) = Hom (M,M[1]) CFm CFm = Hom (M,(Fm)iM[1]) M Db(H) i∈Z m−1 m−1 = Hom ( FjX,Fmi( FlX)[1]) M Db(H) M M i∈Z j=0 l=0 = Hom (X,FmiX[1]) MM Db(H) m i∈Z = Hom (X,X[1]) C M m = Ext1(X,X). M C m X m X 2 Where L denotes the dire t sum od opies of . m 4 X Y C M = X ⊕FX ⊕···⊕Fm−1 Lemma 2.5. Let and be obje ts in , and be the obje t in CFm determined by X. Then HomCFm(M,FjY) ≃ HomCFm(M,Y) 0 ≤ j ≤ m−1 for . FjM ≃ M CFm 2 Proof. It follows from that in . M CFm m ≥ 2 Proposition 2.6. Let be a basi almost tilting obje t in with . M X CFm Then has only one inde omposable omplement in . X M Y Proof. Let be an inde omposable omplement to . Assume that is M Y ≃ X another inde omposable omplement to , we want to show that . Sin e M ⊕X CFm M ⊕X is a basi tilting obje t in , we may assume that is determined T C M⊕X = T⊕FT⊕···⊕Fm−1T by a basi tiltingobje t in , that is, . Note that m ≥ 2 δ(M) = nm−1 T ⊕FT ⊕···⊕Fm−1T = M ⊕Y and , we have that . Then Y ≃ X CFm 2 follows from Lemma 2.2 sin e is a Krull-Remark-S hmidt ategory. M F CFm X C Let be the -stable obje t in determined by an obje t in . We O(M) F M denote by the number of -orbit in determined by the inde omposable X O(M) = δ(X) summands of . It is easy to see that . Lemma 2.7. Let M be a F-stable obje t of CFm and Ext1CFm(M,M) = 0. M CFm O(M) = n Then is a generalized luster tilting obje t of if and only if . M X C M = X ⊕FX ⊕ Proof. Assume that is determined by in , that is, ··· ⊕ Fm−1 Ext1(X,X) = 0 C . A ording to Lemma 2.4, we have that . It is well X C δ(X) = n known that is a tilting obje t in if and only if . The onsequen e 2 follows from Lemma 2.2. M CFm M F De(cid:28)nition 2.8. An obje t of is said to be rigid if is -stable and Ext1CFm(M,M) = 0. A rigid obje t M of CFm is said to be an almost near tilting O(M) = n−1 obje t if . M CFm M Proposition 2.9. Let be an almost near tilting obje t of . Then F M 1 has exa tly two kinds of omplements. That is, there exist two -stable obje ts M X X C 2 1 2 and , determined by non-isomorphi inde omposable obje ts and of M ⊕M M ⊕M 1 2 respe tively, su h that and are generalized luster tilting obje ts 5 C Fm in . M X C M = X ⊕ Proof. Assume that is determined by an obje t in and FX ⊕ ··· ⊕ Fm−1X δ(X) = n − 1 . Then , and by using Lemma 2.4 we have Ext1(X,X) = 0 X C C that . Therefore, is an almost tilting obje t of . A ording X X X C 1 2 to [BMRRT℄, has exa tly two non-isomorphi omplements and in . M = X ⊕FX ⊕···⊕Fm−1X i = 1,2 i i i i Assume that for , then it is easy to see that M ⊕M1 M ⊕M2 CFm 2 and are generalized luster tilting obje ts in . 3 Generalized luster-tilted algebras M CFm Let be a generalized luster tilting obje t in . Then the endomorphism End (M) algebra CFm is alled a generalized luster-tilted algebra. T H M F Proposition 3.1. Let be a basi tilting module of and be the -stable CFm T M obje t in determined by . Then is a generalized luster tilting obje t in CFm and the endomorphism algebra EndCFm(M) is isomorphi to C  0  E C  1 1 .    ... ...     Em−1 Cm−1 C = C = End T E = Ext2(DC,C) 0 ≤ i ≤ m − 1 i H i C Where and for , all the remaining oe(cid:30) ients are zero and multipli ation is indu ed from the anoni al ∼ ∼ C ⊗ E = E = E ⊗ C E ⊗ E −→ 0 C C C C isomorphisms and the zero morphism . M = T ⊕FT ⊕···⊕Fm−1T Proof. By the assumption, . As a ve tor spa e, we have m−1 m−1 End (M) = Hom ( FjT,Fmi( FlT)). CFm M Db(H) M M i∈Z j=0 l=0 T H Hom (FiT,FjT) = 0 i = j i = Db(H) Sin e is a -module, we have that unless or j−1 Hom (FiT,FiT) = Hom (T,T) = C Hom (FiT,Fi+1T) = Db(H) H Db(H) . Moreover, and Hom (T,FT) = Ext2(DC,C) 2 Db(H) C . 6 Remark. A ording to Lemma 2.2.(4), every generalized luster-tilted algebra an be des ribed as Proposition 3.1. M CFm m ≥ 2 Lemma 3.2. Let be an inde omposable obje t in with . Assume M = FjX X C 0 ≤ j ≤ m−1 that with , an inde omposable obje t in , and that . M = FjX ⊕ ··· ⊕ Fm−1X ⊕ Fj−1 ⊕ ··· ⊕ X CFm If is a rigid obje t in , then c End (M) CFm is a (cid:28)eld. Ext1 (M,M) = 0 Proof. By assumption, we have that CFm . A ording to Ext1(X,X) = 0 Ecndc (X,X) ≃ k C Db(H) Lemma2.4, we have that . Hen e . Sin e m ≥ 2 , we have that End (M) = End (X) CFm CFm = Hom (X,FmiX) M Db(H) i∈Z = Hom (X,X) = k. Db(H) 2 M′ CFm Let be a basi almost near tilting obje t of . We may assume that M′ = X ⊕FX ⊕···⊕Fm−1X X C with be a basi almost tilting obje t in . Let N N F M′ M = M′ ⊕N 1 2 1 1 and be non-isomorphi -stable omplements of . Then M2 = M′ ⊕ N2 CFm and are generalized luster tilting obje ts in . We denote by Λ = End (M ) i = 1,2 N N i CFm i for . If 1 and 2 are determined by non-isomorphi X X C N = X ⊕FX ⊕···⊕ 1 2 1 1 1 inde omposable obje ts and of respe tively, then Fm−1X N = X ⊕FX ⊕···⊕Fm−1X 1 2 2 2 2 and . S = Hom (M ,N [1]) Lemma 3.3. Take the notation as above. Then 12 CFm 1 2 is Λ S = Hom (M ,N [1]) Λ a semisimple 1-module and 21 CFm 2 1 is a semisimple 2-module. S Λ 12 1 Proof. By duality, we only need to prove that is a semisimple -module. j 0 ≤ j ≤ m − 1 For an integer with , by Lemma 2.5 we have the following k Hom (M ,FjX [1]) ≃ Hom (M ,X [1]) isomorphism of -spa es: CFm 1 2 CFm 1 2 . Hom (M ,X [1]) Λ We laim that CFm 1 2 is a simple 1-module. m = 1 In fa t, if , our laim is proved in [BMR℄. m ≥ 2 X → B → X → X [1] 2 1 2 If , we onsider the following triangle , where B → X addX Hom (M ,−) 1 is the minimal right -approximation. Applying CFm 1 7 Hom (M ,B) → Hom (M ,X ) → we obtain the following exa t sequen e CFm 1 CFm 1 1 Hom (M ,X [1]) → 0 CFm 1 2 Hom (M ,X [1]) = Hom (M′ ⊕N ,X [1]) CFm 1 2 CFm 1 2 = Hom (N ,X [1]) CFm 1 2 = Hom (X ⊕FX ⊕···⊕Fm−1X ,X [1]) CFm 1 1 1 2 = Hom (X ⊕FX ⊕···⊕Fm−1X ,FjmX [1]) M Db(H) 1 1 1 2 j∈Z = Hom (X ,X [1])⊕Hom (X ,FX [1]) Db(H) 1 2 Db(H) 1 2 = Hom (X ,X [1]). C 1 2 k Hom (X ,X [1]) C 1 2 Sin e is an algebrai ally losed (cid:28)eld, by [BMMRT℄, is one- k Hom (M ,X [1]) Λ dimensional -spa e. Thus CFm 1 2 is a simple 1-module, our laim is proved. S = Hom (M ,N [1]) By Lemma 2.5 again, we have that 12 CFm 1 2 is a semisimple 2 Λ 1 -module. This ompletes the proof. modΛ1 ≃ CFm/addM1[1] A ording to Corollary 4.4 in [KZ℄, we know that modΛ2 ≃ CFm/addM2[1] M = M′ ⊕ N1 ⊕ N2 S12 and . Let . We denote by the f Λ Hom (M ,N [1]) S Λ semisimple 1-module CFm 1 2 and by 21 the semisimple 2-module HomCFm(M2,N1[1]), then we get equivalen es modΛ1/addS12 ≃ CFm/addM[1] and modΛ2/addS21 ≃ CFm/addM[1] f . f Summarizing the above dis ussions, we get the following theorem whi h is a generalization of Theorem 4.2 in [BMR℄. M′ CFm Theorem 3.4. Let be a basi almost near tilting obje t of determined X C X 1 by a basi almost tilting obje t in with non-isomorphi omplements and X C N = X ⊕ FX ⊕ ··· ⊕ Fm−1X N = X ⊕ FX ⊕ 2 1 1 1 1 2 2 2 be of in . Then and ···⊕Fm−1X F M′ M = 2 1 are non-isomorphi -stable omplements of . Moreover, M′ ⊕ N1 M2 = M′ ⊕ N2 CFm and are generalized luster tilting obje ts in . Let Λ = End (M ) Λ = End (M ) modΛ /addS ≃ Λ /addS 1 CFm 1 and 2 CFm 2 . Then 1 12 2 21, S = Hom (M ,N [1]) S = Hom (M ,N [1]) where 12 CFm 1 2 and 21 CFm 2 1 are semisimple modules. 8 T C Let CFm be the set of all basi generalized luster tilting obje ts in Fm up to K C isomorphism. A ording to [HU℄, the tilting graph CFm of Fm is de(cid:28)ned as the K T following. The verti esof CFm aretheelementsof CFm. There isanedge between M′ M B M′ = B ⊕N 1 and if there exists an almost near tilting obje t su h that M = B⊕N N N X Y 2 1 2 and with and are determined by inde omposables and in C B T C T . That is, is determined by an almost tilting obje t in and has exa tly X Y N N 1 2 two non-isomorphi inde omposable omplements and , su h that and X Y are determined by and respe tively. K C Theorem 3.5. The tilting graph CFm of Fm is onne ted. M M T M M Proof. Let 1 and 2 be two elements in CFm. We suppose that 1 and 2 T T C 1 2 are determined by basi tilting obje ts and of respe tively. A ording to K C C [BMRRT℄ the tilting graph of is onne ted, hen e there exist basi tilting X ,··· ,X C T − X − ··· − X − T 1 t 1 1 t 2 obje ts of su h that there is a path in K C N T X tilting graph C of . We denote by i the element of CFm determined by i, N = X ⊕ FX ⊕ ···⊕ F X 1 ≤ i ≤ t i i i m−1 i i.e., for , a ording to Proposition 2.9, M − N −···− N − M K we obtain a path 1 1 t 2 in tilting graph CFm. The proof is 2 ompleted. Referen es [ABS1℄ I.Assem, T.Brstle, R.S hi(cid:31)er. Cluster-tiltedalgebrasastrivialextensions. Bull.London Math. 40,(2006), 151-162. [ABS2℄ I.Assem, T.Brstle, R.S hi(cid:31)er. On the Galois overings of luster-tilted algebras. Preprint, arXiv:math.RT/0709.0805, 2007. φ [ARS℄ M.Auslander, I.Reiten, S.O.Smal . Representation Theory of Artin Alge- bras. Cambridge Univ. 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