T-structures and torsion pairs in a 2 Calabi-Yau triangulated category 1

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1 T-structures and torson pars n a 2 Caab-Yau tranguated category 1 Yu Zhou and Bn Zhu Department of Mathematca Scences Department of Mathematca Scences Tsnghua Unversty Tsnghua Unversty Bejng, P. R. Chna Bejng, P. R. Chna E-ma: yu-zhou06@mas.tsnghua.edu.cn E-ma: bzhu@math.tsnghua.edu.cn December 19, 2012 Abstract For a Caab-Yau tranguated category C of Caab-Yau dmenson d wth a d custer ttng subcategory T, the decomposton of C s determned by the decomposton of T satsfyng vanshng condton of negatve extenson groups, namey, C = I C, where C, I are tranguated subcategores, f and ony f T = I T, where T, I are subcategores wth Hom C (T [t], T j ) = 0, 1 t d 2 and j. Ths nduces that for any two custer ttng objects T, T n a 2 Caab-Yau tranguated category C, the Gabre quvers of endomorphsm agebra End C T of T s connected f and ony f so s End C T. As an appcaton, we prove that ndecomposabe 2 Caab-Yau tranguated categores wth custer ttng objects have no non-trva t-structures and no non-trva co-t-structures. Ths aows us to gve a cassfcaton of torson pars n those tranguated categores, and to determne further the hearts of torson pars n the sense of Nakaoka, whch are equvaent to the modue categores over the endomorphsm agebras of the cores of the torson pars. Key words. Caab-Yau tranguated category; d custer ttng subcategory; (co)torson par; t-structure; mutaton of cotorson par, heart. Mathematcs Subject Cassfcaton. 16E99; 16D90; 18E30 1 Introducton Torson pars (equvaenty, cotorson pars) gve a way to construct the whoe categores from certan speca subcategores. They are mportant n the study of tranguated categores and abean categores. We reca the defnton here. Let X, Y be (addtve) subcategores n a tranguated category C wth shft functor [1]. The par (X, Y ) s caed a torson par n C provded the foowng condtons are satsfed: 1. Hom(X, Y) = 0 for any X X, Y Y ; and 2. for any C C, there s a trange X C Y X[1] wth X X, Y Y. Ths noton was ntroduced by Iyama-Yoshno [IY], see aso [KR1], whch s the tranguated verson of the noton wth the same name n abean categores ntroduced by Dckson [D] (see the ntroducton to [ASS] for further detas). The noton of torson pars unfes the noton of t-structures n the sense of [BBD], co-t-structures n the sense of Paukszteo [P] (see [Bon] for another name), and the noton of custer ttng subcategores (objects) n the sense of Keer- Reten [KR1], see aso [BMRRT]. 1 Supported by the NSF of Chna (Grants ) 1

2 Torson pars are mportant n the study of the agebrac structure and geometrc structure of tranguated categores. Iyama and Yoshno [IY] use them to study the mutaton of custer ttng subcategores n tranguated categores, see aso [KR1, BR]. Nakaoka [N] use them to unfy the constructons of abean categores appearng as quotents of tranguated categores by custer ttng subcategores [BMR, KR1, KZ], and the constructon of abean categores as hearts of t- structures [BBD]. There s a reaton between t-structures and stabty condtons n tranguated categores, see [Br] for detas. As one of mportant speca cases, custer ttng objects (or subcategores) appeared naturay n the study on the categorfcaton of custer agebras [BMRRT]. They have many nce agebrac propertes and combnatora propertes whch have been used n the categorfcaton of custer agebras and studed n recent years (see the surveys [K2, Re] and the references theren). In ths categorfcaton, the custer ttng objects n the custer category of an acycc quver (or more genera a quver wth potenta) corresponds to the custers of the correspondng custer agebra. Custer ttng subcategores n tranguated categores are the torson casses of some speca torson pars. A tranguated category (even a 2 Caab-Yau tranguated category) may not admt any custer ttng subcategores [KZ, BIKR]. In contrast, they aways admt torson pars, for exampe, the trva torson par: (the whoe category, the zero category). In a tranguated category C wth shft functor [1], when (X, Y ) s a torson par, we ca the par (X, Y [ 1]) a cotorson par, and ca the subcategory X Y [ 1] the core (denoted by I) of ths cotorson par (and of the correspondng torson par (X, Y )). It foows that (X, Y ) s a cotorson par n C f and ony f (X, Y [1]) s a torson par. Recenty there are severa works on the cassfcaton of torson pars (or equvaenty, cotorson pars) of a 2 Caab-Yau tranguated category. Ng gves a cassfcaton of torson pars n the custer categores of A [Ng] by defnng Ptoemy dagrams of an gon P. Hom-Jørgensen- Rubery [HJR1] gves a cassfcaton of cotorson pars n custer category C An of type A n va Ptoemy dagrams of a reguar (n + 3) gon P n+3. They aso do the same thng for custer tubes [HJR2]. In [ZZ2], we defne the mutaton of torson pars to produce new torson pars by generazng the mutaton of custer ttng subcategores [IY], and show that the mutaton of torson pars has the geometrc meanng when the categores have geometrc modes. In [ZZZ], together wth zhang, we gve the cassfcaton of (co)torson pars n the (generazed) custer categores assocated wth marked Remann surfaces wthout punctures. For cassfcaton of torson pars n an abean category, we refer to the recent work of Baur-Buan-Marsh [BBM]. In ths paper, we show that an ndecomposabe 2 Caab-Yau tranguated category C wth a custer ttng object has ony trva t-structures,.e. (C, 0), or (0, C). For ths, we prove the fact that the decomposton of C s determned by the decomposton of the custer ttng subcategory. Ths decomposton resut hods for arbtrary d Caab-Yau tranguated categores, where d > 1 s an nteger. As an appcaton of the resut on t-structures, we gve a cassfcaton of (co)torson pars n C and determne the hearts of (co)torson pars n the sense of Nakaoka [N], whch are equvaent to the modue categores of ther cores. We aso dscuss the reaton between mutaton of (co)torson pars [ZZ2] wth mutaton of custer ttng objects. Ths paper s organzed as foows: In Secton 2, some basc defntons and resuts on (co)torson pars are recaed. In Secton 3, the defnton of decomposton of tranguated categores s recaed. The decomposton of d custer ttng categores s defned, whch s not ony the decomposton of addtve categores, but aso wth some addtona vansh condton on negatve extenson groups (appeared frst n Secton 4.2, n [KR2]; and for d = 2, ths condton s empty). An exampe s gven to expan n genera the decomposton of tranguated categores s not 2

3 determned by that of custer ttng subcategores. It s proved that for any d Caab-Yau tranguated category, ts decomposton s determned by the decomposton of a d custer ttng subcategory. In Secton 4, the frst man resut s that the ndecomposabe 2 Caab-Yau tranguated categores wth custer ttng objects have no non-trva t-structures (Theorem 4.1). Ths aows us to gve a cassfcaton of (co)torson pars n these categores (Theorem 4.4), whch s the second man resut n ths secton. In Secton 5, we dscuss the reaton between mutaton of cotorson pars and mutaton of custer ttng objects. For any cotorson par (X, Y ) wth core I, any basc custer ttng object T contanng I as a drect summand can be wrtten unquey as T = T X I T Y such that T X I (or T Y I) s custer ttng n X (Y respectvey), whch we sha defne n ths secton, and any trpe (M, I, N) of objects M, I, N n C wth the property above gves a custer ttng object M I N contanng I as a drect summand n C. The mutaton of such T n the ndecomposabe object T 0 can be made nsde T X I or T Y I, dependng on that T 0 s a drect summand of T X or T Y respectvey, f T 0 s not the drect summand of I. If T 0 s the drect summand of I, then the mutaton T of T n T 0 s the custer ttng object whch can be wrtten as T = T X I T Y, where (X, Y ) s the mutaton of (X, Y ) and I s the core of (X, Y ). In the fna secton, for any cotorson par (X, Y ) wth core I n a 2 Caab-Yau tranguated category C, we prove that the heart A of the t structure (X, Y ) n the subquotent tranguated category (I[1])/I s an abean subcategory of the heart H of (X, Y ) defned by Nakaoka, and that the quotent category H/A s equvaent to the modue category over the core I. If the 2 Caab-Yau tranguated category C admts a custer ttng object, then H s equvaent to the modue category over the endomorphsm agebra EndI. 2 Premnares Throughout ths paper, k denotes a fed. When we say that C s a tranguated category, we aways assume that C s a Hom-fnte Kru-Schmdt k near tranguated category over a fed k. Denote by [1] the shft functor n C, and by [-1] the nverse of [1]. For a subcategory D, we mean D s a fu subcategory of C whch s cosed under somorphsms, fnte drect sums and drect summands. In ths sense, D s determned by the set of ndecomposabe objects n t. By X C, we mean that X s an object of C. We denote by addx the addtve cosure generated by object X, whch s a subcategory of C. Sometmes, we dentfy an object I wth the subcategory addi. Moreover, f a subcategory D s cosed under [1], [-1] and extensons, then D s a tranguated subcategory of C (n fact t s a thck subcategory). We ca that a tranguated category C has a Serre functor provded there s an equvaent functor S such that Hom C (X, Y) DHom C (Y, S X), whch are functoray n both varabes, where D = Hom k (, k). If the Serre functor s [d], an nteger, C s caed a d Caab-Yau (d CY, for short) tranguated category. We aways use Hom(X, Y) to denote Hom-space of objects X, Y n C. We denote by Ext n (X, Y) the space Hom(X, Y[n]). For a subcategory X of C, denoted by X C, et X = {Y C Hom(X, Y) = 0 for any X X } and X = {Y C Hom(Y, X) = 0 for any X X }. For two subcategores X, Y, by Hom(X, Y ) = 0, we mean that Hom(X, Y) = 0 for any X X and any Y Y. Smar for the notaton Ext n (X, Y ) = 0. A subcategory X of C s sad to be a 3

4 rgd subcategory f Ext 1 (X, X ) = 0. Let X Y = {Z C a trange X Z Y X[1] n C wth X X, Y Y }. It s easy to see that X Y s cosed under takng somorphsms and fnte drect sums. A subcategory X s sad to be cosed under extensons (or an extenson-cosed subcategory) f X X X. Note that X Y s cosed under takng drect summands f Hom(X, Y ) = 0 (Proposton 2.1(1) n [IY]). Therefore, X Y can be understood as a subcategory of C n ths case. We reca the defnton of cotorson pars n a tranguated category C from [IY, N]. Defnton 2.1. Let X and Y be subcategores of a tranguated category C. 1. A par (X, Y ) of subcategores of C s caed a torson par f Hom(X, Y ) = 0 and C = X Y. The subcategory I = X Y [ 1] s caed the core of the torson par. 2. The par (X, Y ) s a cotorson par f Ext 1 (X, Y ) = 0 and C = X Y [1]. The subcategory I = X Y the core of the cotorson par (X, Y ). 3. A t-structure (X, Y ) n C s a cotorson par such that X s cosed under [1] (equvaenty Y s cosed under [ 1]). In ths case X Y [2] s an abean category, whch s caed the heart of (X, Y ) [BBD, BR]. 4. A co-t-structure (X, Y ) n C s a cotorson par such that X s cosed under [ 1] (equvaenty Y s cosed under [1]) [Bon, P]. 5. The subcategory X s sad to be a custer ttng subcategory f (X, X ) s a cotorson par [KR1, KZ, IY]. We say that an object T s a custer ttng object f addt s a custer ttng subcategory. Remark 2.2. A par (X, Y ) s a cotorson par f and ony f (X, Y [1]) s a torson par. In any case, the core I s a rgd subcategory of C. Remark 2.3. (C, 0) and (0, C) are t-structures n C, whch are caed trva t-structures. They are aso co-t-structures and are caed trva co-t-structures n C. Lemma 2.4. [ZZ1] Let (X, Y ) be a cotorson par n C wth core I. Then 1. (X, Y ) s a t-structure f and ony f I = 0 2. X s a rgd subcategory f and ony f X = I 3. X s a custer ttng subcategory f and ony f X = I = Y. Reca that a subcategory X s sad to be contravaranty fnte n C, f any object M C admts a rght X approxmaton f : X M, whch means that any map from X X to M factors through f. The eft X approxmaton of M and covaranty fnteness of X can be defned duay. X s caed functoray fnte n C f X s both covaranty fnte and contravaranty 4

5 fnte n C. Note that f (X, Y ) s a cotorson par, then X = (Y [1]), Y = (X [ 1]), and t foows that X (or Y ) s a contravaranty (covaranty, respectvey) fnte and extenson-cosed subcategory of C. Let (X, Y ) be a cotorson par wth core I n a tranguated category C. Denote by H the subcategory (X I[1]) (I Y [1]). The mage of H under the natura projecton C C/I, whch denoted by H, s caed the heart of the cotorson par (X, Y ). It s proved by Nakaoka that the heart H s an abean category, see [N] for more detaed constructon. 3 Decompostons of Caab-Yau tranguated categores In ths secton, we dscuss how the decomposton of tranguated categores s determned by that of a custer ttng subcategory. We reca the defnton of d custer ttng subcategores from [KR1, IY] n the foowng: Defnton 3.1. Let C be a tranguated category, d > 1, an nteger. A subcategory T of C s caed d rgd provded Ext (T, T ) = 0 for a 1 d 1. A d rgd subcategory T s caed d custer ttng provded that T s functoray fnte, and satsfes the property: T T f and ony f Ext (T, T) = 0 for a 1 d 1 f and ony f Ext (T, T ) = 0 for a 1 d 1; An object T s caed a d custer ttng (respectvey d rgd) object f addt s d custer ttng (respectvey d rgd). The man exampes of d custer ttng subcategores are d custer ttng subcategores n d custer categores (see [IY, T, Zhu]). Other exampes can be found n [K1, BIKR]. Note that when d = 2, the d custer ttng subcategores (or d custer ttng objects) are caed custer ttng subcategores (custer ttng objects respectvey). Defnton 3.2. Let C be a tranguated category, and C, I be tranguated subcategores of C. We ca that C s a drect sum of tranguated subcategores C, I, provded that 1. Any object M C s a drect sum of fntey many objects M C ; 2. Hom(C, C j ) = 0, j. In ths case, we wrte C = I C. We say C s ndecomposabe f C cannot be wrtten as a drect sum of two nonzero tranguated subcategores. Defnton 3.3. Let T be a d custer ttng subcategory of a tranguated category C, and T, I, be subcategores of T. We ca that T s a drect sum of subcategores T, I, provded that 1. Any object T T s a drect sum of fntey many objects T T ; 2. Hom(T, T j ) = 0, j; 3. Hom(T [k], T j ) = 0, j, 1 k d 2; In ths case, we wrte T = I T. We say T s ndecomposabe f T cannot be wrtten as a drect sum of two nonzero subcategores. 5

6 Remark 3.4. The thrd condton n Defnton 3.3 appeared frst n [KR1] for the study of Gorensten property of d custer ttng subcategores (see the subsecton 4.6 there for detas). When d = 2, ths condton s empty. The foowng exampe shows that there are ndecomposabe d CY tranguated categores admttng d custer ttng subcategores, those custer ttng subcategores can be decomposed as sum of subcategores satsfyng the condtons 1, 2, but 3 n Defnton 3.3. Exampe 1. Let Q : be the quver of type A 3 wth near orentaton, and C be the 4 custer category of Q,.e. C = D b (kq)/τ 1 [3] (compare [K1]). Let P 1, P 2, P 3 be the ndecomposabe projectve modues assocated to the vertces of Q, and S 1, S 2, S 3 the correspondng smpe modues. Then T = P 1 P 2 P 3 s a 4 custer ttng object, P 1 P 3 s an amost compete 4 custer ttng object, t has 4 compements (compare [Zhu, T]), one s P 2, the others are S 3, S 3 [1], and S 3 [2]. Denote by T = add(p 1 P 3 S 3 [1]), whch s a 4 custer ttng subcategory of C. Set T 1 = add(p 1 P 3 ), T 2 = adds 3 [1]. Both are subcategores of T. It s easy to see that T, T 1, T 2 satsfy the frst two condtons of Defnton 3.3, but the thrd one, an easy computaton shows Hom(P 3 [1], S 3 [1]) 0. We note that ths 4 custer category C s ndecomposabe. We w dscuss the reaton between the decomposton of tranguated categores and the decomposton of d custer ttng subcategores. Frsty we ook at two exampes: Exampe 2. Let Q be a connected quver wthout orented cyces, C = D b (kq) the bounded derved category of kq. It s an ndecomposabe tranguated category. We know T = add{τ n [ n]kq n Z } s a custer ttng subcategory contanng nfntey many ndecomposabe objects n C. Let T = add{τ [ ]kq} for Z. It s easy to check that T = Z T. Exampe 3. Let Q be a connected quver wthout orented cyces, F = τ 1 [1] an automorphsm of the derved category D b (kq). The repettve custer category of Q s defned for any postve nteger m, namey, the orbt tranguated category C = D b (kq)/(f m ) [K1]. It s an ndecomposabe tranguated category. Let m = 2. Then kq F(kQ) s a custer ttng object n C. Let T = add(kq F(kQ)), T 1 = add(kq), T 2 = add(f(kq)). Then T s a custer ttng subcategory and T = T 1 T 2. The two exampes above show that n genera the ndecomposabe tranguated category may admt a decomposabe d custer ttng subcategory. In the foowng, we w prove that the decomposton of d CY tranguated categores s determned by the decomposton of a d custer ttng subcategory. Reca that a k near tranguated category C s d CY f [d] s the Serre functor. Proposton 3.5. Let C be a d CY tranguated category wth a d custer ttng subcategory T. Suppose that T = I T wth T, I, nonzero subcategores, and et C = T T [1] T [d 1] for any I. Then C s a tranguated subcategory of C and C = I C. Note that by Proposton 2.1 [IY], C, I, are cosed under drect summands, so they are subcategores of C. We dvde our proof nto severa steps: Lemma 3.6. Under the same assumpton as n Proposton 3.5, every object X n C has a decomposton X = I X wth fnte many nonzero X C, I. In partcuar, every ndecomposabe object of C es n some C, I. 6

7 Proof. Snce T = I T s a d custer ttng subcategory, by Coroary 3.3 n [IY], for each ndecomposabe object X n C, there are d tranges: (n) f (n) X J B (n 1) X (n 1) X (n) [1], n = 1,, d, where J s a fnte subset of I, B (n 1) T X (0) = X and X (d) = 0. Then X (d 1) J B (d 1). We want to prove that X J X wth X C. Assume that X (n) J X (n) wth X (n) T T [1] T [d 1 n] for some 1 n d 1. By Defnton 3.3, Hom(T [k], T j ) = 0 for j, 0 k d 1 n d 2, then Hom(X (n), T j ) = 0 f for j. So f (n) s a dagona map, say , where f : X (n) B (n 1). Extend each f 0 0 f J to trange: X (n) f B (n 1) X (n 1) X (n) [1]. Then we have that X (n 1) J X (n 1) and X (n 1) on n (from d 1 to 0), X = X (0) J X (0) wth X (0) C. Lemma 3.7. Under the same assumpton as n Proposton 3.5, C = j I. T T [1] T [d n], J. By nducton 2d 2 k=1 T j [k] hods for any Proof. By Defnton 3.3, Hom(T, T j []) = 0 and Hom(T j, T []) = 0, for (d 2) d 1, j. Then for 0 m d 1, 1 k d 1, we have that Hom(T [m], T j [k]) Hom(T, T j [k m]) = 0 due to (d 2) k m d 1, and Hom(T [m], T j [d + k 1]) DHom(T j [k 1], T [m]) = 0 as (d 2) m k + 1 d 1. So Hom(C, T j [k]) = Hom(T T [1] T [d 1], T j [k]) = 0 for 1 k 2d 2, j. Ths mpes C 2d 2 T j [k]. j k=1 Fx an eement I. Let X be an object satsfyng Hom(X, T j [k]) = 0 for 1 k 2d 2, j. By Lemma 3.6, X has a decomposton X = J X, X C, for some fnte subset J of I. By the defnton of C, there are d tranges: X (n) A (n 1) g (n 1) X (n 1) X (n) [1], n = 1,, d, where A (n 1) T, X (0) = X, X (d) = 0 and g (n 1) s the mnma rght addt approxmaton of X (n 1) (compare Coroary 3.3 n [IY]). If, then g (0) = 0 and B (0) = 0 thanks to Hom(T, X) = DHom(X, T [d]) = 0. So X (1) X (0) [ 1] = X [ 1]. Assume that X (n) X [ n] for some 1 n d 2. Then g (n) = 0 by Hom(T, X[ n]) DHom(X, T [d + n]) = 0 and then X (n+1) X (n) [ 1] X [ (n + 1)]. By nducton on n, we have that g (n 1) = 0 and X (n) = X [ n], for 1 n d 1,. Note that X (d 1) A (d 1) by X (d) = 0. From that Hom(X (d 1), X (d 1) ) Hom(X [ (d 1)], A (d 1) ) Hom(X, B (d 1) [d 1]) = 0, we have X (d 1) = 0. Then X X (d 1) [d 1] = 0 ( ). Hence X X C. Lemma 3.8. Under the same assumpton as n Proposton 3.5, a C, I, are tranguated subcategores of C. 7

8 Proof. Let X Z Y X[1] be a trange wth X, Y C. By Lemma 3.7, we have that Hom(X, T j [k]) = 0 and Hom(Y, T j [k]) = 0 and then Hom(Z, T j [k]) = 0 for 1 k 2d 2, j. By Lemma 3.7 agan, we have that Z C. Therefore, C s cosed under extensons. For any, T, T [1],, T [d 1] are ncuded n C. We cam that T [d] s a subcategory of C. Otherwse, there s an ndecomposabe object of T, say X, such that X[d] s not an object of C. Then by Lemma 3.6, X[d] s n C j for some j. Note that Hom(X, X[d]) DHom(X, X) 0 whch contradcts wth Hom(T, C j ) = 0 by Lemma 3.7 and d CY property. Then we prove that T [d] s ncuded n C. Hence C [1] = T [1] T [d] C, that s, C s cosed under [1]. Duay, one can prove that C s cosed under [-1]. Therefore, C s a tranguated subcategory of C. Proof of Proposton 3.5. It s suffcent to verfy that Hom(C, C j ) = 0, for j. By Lemma 3.8, C [ 1] = C, then Hom(C, T j ) = Hom(C [ 1], T j ) = Hom(C, T j [1]) = 0 for j, where the ast equaty s due to Lemma 3.7. Then Hom(C, C j ) = Hom(C, T j T j [1] T j [d 1]) = 0. The foowng emma s a generazaton of Remark 2.3 n [ZZ1]. Lemma 3.9. Let C be a tranguated category and T be a d rgd subcategory of C satsfyng C = T T [1] T [d 1]. Then T s a d custer ttng subcategory of C. Proof. Note that (T, T [1] T [d 1]) and (T T [d 2], T [d 1]) form two torson pars. So T s contravaranty fnte n C and T [d] s covaranty fnte n C. Therefore T s functoray fnte n C. Take an object X n C wth Hom(X, T [t]) = 0 for 1 t d 1. Then Hom(X, T [1] T [d 1]) = 0. Hence X T. Smar proof for X T f Hom(T, X[t]) = 0 for 1 t d 1. Hence T s d custer ttng n C. Now we prove our man resut n ths secton. Theorem Let C be a d CY tranguated category wth a d custer ttng subcategory T. Then C s a drect sum of ndecomposabe tranguated subcategores C, I f and ony f the custer ttng subcategory T s a drect sum of ndecomposabe subcategores T, I. Moreover C = T T [1] T [d 1] and T s a d custer ttng subcategory n C, I. Proof. We frst show the ony f part. By the defnton of drect sums of tranguated subcategores, any object T n T has a decomposton T = J T wth J a fnte subset of I, T C and Hom(T [k], T j ) = 0 for 0 k d 2, j. Then T = I T where T = T C. By Defnton 3.2, for any object X C, Hom(X, T j [k]) = 0 for j and any k. Then by Lemma 3.7, X T T [1] T [d 1]. Hence C = T T [1] T [d 1]. By Lemma 3.9, T s a d custer ttng subcategory of C. It foows from that of C and Proposton 3.5 that T s ndecomposabe. To prove the f part. It foows from Proposton 3.5 that there s a decomposton C = I C, where C = T T [1] T [d 1] s a tranguated subcategory of C. By Lemma 3.9, T s a d custer ttng subcategory n C. If C s not ndecomposabe, say C = C C wth nonzero tranguated subcategores C, C, then by the proof of the ony f part, we have T = T T, and C = T T [1] T [d 1], C = T T [1] T [d 1]. It foows that T, T are nonzero subcategores, a contradcton to the ndecomposabeness of T. The other asserton foows from Lemma 3.9. We gve a smpe exampe for d = 2. 8

9 Exampe 4. Let Q : , C = C Q, the custer category of Q whose Ausander-Reten quver s the foowng: P 4 [1] P 4 P 1 [1] P 3 [1] P 3 I 2 P 2 [1] P 2 [1] P 2 E I 3 P 3 [1] P 1 [1] P 1 S 2 S 3 S 4 P 4 [1] We take X = add(e), (X[1]) = add({e, P 3, P 4 [1], P 4, I 2, P 1 [1], S 2, S 3 } By [IY], the subquotent category (X[1])/X = add({p 3, P 4 [1], P 4, I 2, P 1 [1], S 2, S 3 }) s tranguated, and 2 CY. Ths subquotent category admts custer ttng objects, for exampe, the object T = P 4 [1] P 3 E S 3. We have that n ths subquotent category, addt = add(s 3 ) add(p 3 P 4 [1]). Then by Theorem 3.10, ths subquotent category (X[1])/X = add({s 2, S 3 }) add({p 3, P 4 [1], P 4, I 2, P 1 [1]}), n whch, the frst drect summand s equvaent to the custer category of type A 1, the second one s equvaent to the custer category of type A 2. Coroary Let C be a d CY tranguated category admttng a d custer ttng subcategory T. Then C s ndecomposabe f and ony f T s ndecomposabe. Coroary Let C be a d CY tranguated category, T and T be two d custer ttng subcategores. Then T s ndecomposabe f and ony f T s ndecomposabe. Coroary Let C be a d CY tranguated category wth a d custer ttng object T. Then C s a drect sum of fntey many ndecomposabe tranguated subcategores C, = 1, m. Moreover the custer ttng subcategory T = addt s a drect sum of ndecomposabe subcategores T, = 1,, m, and C = T T [1] T [d 1] and T s a d custer ttng subcategory n C, = 1, m. Proof. Any tranguated category can be decomposed as a drect sum of tranguated subcategores. For the d CY tranguated category C wth a d custer ttng object T, the number of drect summands of the decomposton of C s fnte snce that the number of ndecomposabe drect summands of T s fnte. Then we have the decomposton of C = m =1 C. The other asserton foows drecty from Theorem For the speca case of d = 2,.e., C s 2 CY tranguated category wth a custer ttng object T, the decomposton of C corresponds to the partton of connected components of the Gabre quver of End(T). Defnton A basc rgd object T n C s caed connected provded T cannot wrtten as T = T 1 T 2 wth property that T 0, and Hom(T, T j ) = 0, for j {1, 2}. Any custer ttng object n C can be decomposed as a drect sum of connected summands: T = m =1 T wth T beng connected. We ca such decomposton a compete decomposton of T. 9

10 Every C can be decomposed unquey to a drect sum of nonzero ndecomposabe tranguated subcategores. We ca ths decomposton s the compete decomposton of C and denote by ns(c) the number of ndecomposabe drect summands of such decomposton of C. For a custer ttng object T n C, the Gabre quver of End(T) s denoted by Γ T and the number of connected components of Γ T s denoted by nc(γ T ). Note that the compete decomposton of T corresponds to the connected components of Gabre quver of the 2 CY tted agebra End(T). So by appyng the theorem above, we have the foowng resut mmedatey. Coroary Let C be a 2 CY tranguated category admttng a custer ttng object T. Then the number nc(γ T ) of connected components of the quver Γ T s equa to ns(c). In partcuar, C s ndecomposabe f and ony f Γ T s connected. Coroary Let C be a 2 CY tranguated category and et T, T be custer ttng objects n C. Then Γ T s connected f and ony f Γ T s connected. Proof. Γ T s connected C s ndecomposabe Γ T s connected. Remark Let (S, M) be a marked surface and nc(s ) denote the number of connected components of S. Then nc(s ) = ns(c(s, M)) (compare [ZZ2]). 4 Cassfcaton of Cotorson pars n 2-Caab-Yau categores From now on, except Proposton 4.6, we aways suppose that the tranguated category C s 2 Caab-Yau (2 CY for short),.e. [2] s the Serre functor of C. The man exampes of 2 CY tranguated categores are the foowngs: 1. Custer categores of heredtary abean k categores n the sense of [BMRRT] (aso [CCS] for type A); and generazed custer categores of agebras wth goba dmenson at most 2 (ncudng the case of quvers wth potentas) n the sense of Amot [Am]. A these 2 CY tranguated categores have custer ttng objects. 2. The stabe categores of preprojectve agebras of Dynkn quvers. They aso have custer ttng objects [GLS, BIRS]. 3. The custer category of type A. It has custer ttng subcategores, whch contans nfntey many ndecomposabe objects [KR1, HJ, Ng]. 4. The bounded derved categores D b (mod f.. Λ) of modues wth fnte ength over preprojectve agebras Λ of non-dynkn quvers. They have no custer ttng subcategores. There are many stabe subcategores of mod f.. Λ assocated to eements n the Coxeter groups of the quvers. Ther stabe categores are 2 CY, and have custer ttng objects. See [GLS, BIRS] for detas. 5. Stabe categores of Cohen-Macauay modues over three-dmensona compete oca commutatve noetheran Gorensten soated snguarty contanng the resdue fed [BIKR]. We sha frst decde a speca knd of cotorson pars: t-structures. Reca that (X, Y ) s a t-structure n C, f Ext 1 (X, Y ) = 0, C = X Y [1] and X [1] X, Y [ 1] Y. The frst man resut n ths secton s the foowng resut. 10

11 Theorem 4.1. Let C be an ndecomposabe 2-CY tranguated category wth a custer ttng object T. Then C has no non-trva t-structures,.e. the t-structures n C are (C, 0) and (0, C). Proof. Let (X, Y ) be a t-structure n C. Put T = addt. Then for each ndecomposabe object T T, I, there s a trange X f T g Y [1] h X [1] wth X X, Y Y. Let R be the subcategory of C generated addtvey by X, Y, I. Then T R R[1]. We sha prove that R s a custer ttng subcategory. For any map α Hom(Y [1], Y j [2]), consder the foowng dagram: g X T Y [1] X [1] α β X j [1] g T j [1] j [1] h Y j [2] j [1] X j [2] The composton h j [1] α Hom(Y [1], X j [2]) DHom(X j, Y [1]) = 0, then α factors through g j [1]. So α g = 0 due to Hom(T, T j [1]) = 0. Therefore α factors through h,.e. there s a morphsm β Hom(X [1], Y j [2]) such that α = β h. But Hom(X [1], Y j [2]) = 0, so α = 0. Then Ext 1 (Y, Y j ) = 0. Duay, we have that Ext 1 (X, X j ) = 0. By the defnton of t-structure and 2-CY property, we aso have Ext 1 (X, Y j ) = 0 and Ext 1 (Y, X j ). Hence R s a rgd subcategory. Gven an object M wth Ext 1 (M, X ) = 0, Ext 1 (M, Y ) = 0 for I. Snce T s custer ttng, there s a trange M tranges w A u B X A X B h v M[1] wth A, B T. Snce T R R[1], there are f A A g A Y A [1] h A X A [1], f B B g B Y B [1] h B X B [1], where f A (resp. f B ) s the mnma rght X approxmaton of A (resp. B) and g A (resp. g B ) s the mnma eft Y [1] approxmaton of A (resp. B). Then the composton u f A factors through f B. So there exsts s such that f B s = u f A. s X A r X B f A f B M u A v B M[1] Due to Hom(X B, M[1]) = 0, we have v f B = 0, then f B factors through u. Snce any morphsm from X B to A factors through f A, then there s a morphsm r Hom(X B, X A ) such that f B = u f A r. Repace u f A by f B s, we have f B = f B s r. Then s r s an somorphsm by the rght s 0 mnmaty of f B. Thus s s a retracton and we have the trange X A X B X C [1] X A [1], where X C s a drect summand of X A. From f B s and u f A respectvey, by the octahedra axom, 11

12 we have the foowng two commutatve dagrams of tranges: and X A = X A s f B s Y B X B f B B g B Y B [1] 0 ( ) Y B X C [1] N Y B [1] X A [1] = X A [1] M = M w f A A g A Y A [1] h A X A [1] u ( ) X A X A u f A B N X A [1] v M[1] = M[1]. Snce the morphsm from Y B to X C [1] n the thrd row of the dagram ( ) s zero, then N X C [1] Y B [1] R[1]. On the other hand, the morphsm from M to Y A [1] n the thrd coumn of the dagram ( ) s zero due to Hom(M, Y A [1]) = 0, then M[1] s somorphc to a drect summand of N, and then t s n R[1]. Hence M s an object n R. The functoray fnteness of R foows from that the number of ndecomposabe objects (up to somorphsm) s fnte and C s Hom-fnte. Therefore R s custer ttng n C. R s ndecomposabe by Coroary Now we repace T by R, repeat the proof above. Namey, we consder the foowng spt tranges: X X 0 0 X [1], 0 Y Y [ 1][1] 0 0. In these tranges, X X, Y [ 1] Y. We have that the subcategory R generated by X, Y [ 1], I s a custer ttng subcategory. It s an ndecomposabe by Coroary For any, j I, Y j [ 2] Y, then Hom(X, Y j [ 1]) = Ext 1 (X, Y j [ 2]) = 0. Note that Hom(Y j [ 1], X ) = Ext 1 (Y j, X ) DExt 1 (X, Y j ) = 0. Therefore X 0 for a or Y 0 for a as R s ndecomposabe. Then R Y or R X. Hence C = Y or C = X. Remark 4.2. The resut s not true for 2 CY tranguated categores wthout custer ttng objects. For exampe: the derved category of coherent sheaves on an agebrac K3 surface s 2-CY and admts no custer ttng objects. It admts a non trva t-structure (the canonca t- structure whose heart s the category of coherent sheaves). There are aso exampes that there are nontrva t-structures n a 2-CY tranguated category admttng custer ttng subcategores whch contans nfntey many ndecomposabes (up to somorphsm). The custer category C A of type A ntroduced by Hom-Jörgensen [HJ, KR1] has non-trva t-structures (see Theorem 4.1 n [Ng]), whch s such an exampe. Ths custer category has custer-ttng subcategores contanng nfntey many ndecomposabe objects (see [Ng] for more detas). 12

13 Coroary 4.3. Let C be a 2-CY tranguated category wth a custer ttng object T and et C = j J C j be the compete decomposton of C. Then the t-structures n C are of the form ( j L C j, j J L C) where L s a subset of J. In partcuar, each t-structure has a trva heart. The foowng theorem s the second man resut n ths secton, whch gves a cassfcaton of cotorson pars (equvaenty torson pars) n 2 CY tranguated categores C wth custer ttng objects. We note that n any 2 CY tranguated category C wth a custer ttng object, any rgd subcategory I contans ony fntey many ndecomposabes (up to somorphsm) [DK]. So we dentfy I wth the object I obtaned as the drect sum of representatves of socasses of ndecomposabes n t. We aso note that for any rgd subcategory I n C, the subquotent category (I[1])/I s agan a 2 CY tranguated category [IY]. Theorem 4.4. Let C be a 2-CY tranguated category admttng custer ttng objects and I a rgd subcategory of C. Let (I[1])/I = j J I j be the compete decomposton of (I[1])/I. Then a cotorson pars wth core I are obtaned as premages under π : (I[1]) (I[1])/I of the pars ( j L I j, j J L I j ) where L s a subset of J. There are 2 ns( (I[1])/I) cotorson pars wth core I. Proof. Thanks to Theorem 4.7 and Theorem 4.9 n [IY], (I[1])/I s a 2 CY tranguated category wth custer ttng objects. By Theorem 3.5 and Coroary 3.6 n [ZZ2], a par (X 1, X 2 ) of subcategores of C s a cotorson par wth core I f and ony f I X (I[1]), = 1, 2, and (π(x 1 ), π(x 2 )) s a t-structure n (I[1])/I. Then by Coroary 4.3, the t-structures n (I[1])/I are of the form ( j L I j, j J L I j ). Therefore the cotorson pars wth core I are the premages under π : (I[1]) (I[1])/I of the t structure ( j L I j, j J L I j ) n (I[1])/I. Indeed, ths correspondence s the same as that n Theorem II.2.5 n [BIRS] under the foowng resut: every cotorson par s symmetrc,.e. Coroary 4.5. Let C be a 2-CY tranguated category admttng a custer ttng object and et (X, Y ) be a cotorson par wth core I. Then (Y, X ) s aso a cotorson par wth the same core. Proof. By Theorem 4.4, (X, Y ) = (π 1 ( j L I j ), π 1 ( j J L I j )) for some subset J, then (Y, X ) = (π 1 ( j J L I j ), π 1 ( j L I j )) s aso a cotorson par wth core I. Reca that (X, Y ) s a co-t-structure n C, f Ext 1 (X, Y ) = 0, C = X Y [1] and X [ 1] X, Y [1] Y. Usng Theorem 4.4 and Coroary 4.3, one can prove that there are no non-trva co-t-structures n an ndecomposabe 2-CY tranguated category wth a custer ttng object n the smar way as [ZZZ]. Indeed, f (X, Y ) s a co-t-structure n C, then (X, Y ) s a cotorson par by the defnton of co-t-structure. By Coroary 4.5, (Y, X ) s aso a cotorson par. Snce Y s cosed under [1] and X s cosed under [-1], (Y, X ) s a t-structure. Then by Theorem 4.1, X = 0 or Y = 0. In fact, we have the foowng more genera resut on t-structures or co-t-structures n a d CY tranguated category, generazng a recent resut n [HJY]. Proposton 4.6. Let C be an ndecomposabe d CY tranguated category. If d 1, then the co-t-structures n C are (C, 0) and (0, C). Duay, f d 1, Then the t-structures n C are (C, 0) and (0, C). 13

14 Proof. We ony prove the case of d 1. Let (X, Y ) be a co-t-structure n C. For any object M X Y, we have Hom(M, M) DExt 1 (M, M[d 1]) = 0 by M X and M[d 1] Y. Ths mpes the core of (X, Y ) s zero. Then by Lemma 2.3, (X, Y ) s a t-structure. Thus X, Y are tranguated subcategores of C. For any X X, Y Y, we have that Hom(X, Y) = Ext 1 (X, Y[ 1]) = 0 and Hom(Y, X) DExt 1 (X, Y[d 1]) = 0 by Y[ 1], Y[d 1] Y. Due to C = X Y [1], we have C = X Y. Therefore X = 0 or Y = 0. 5 Mutatons In ths secton, a custer ttng objects we consdered are basc. We sha dscuss the reaton between mutaton of cotorson pars and that of custer ttng objects contaned n those cotorson pars n a 2 CY tranguated category wth custer ttng object. Frst we ntroduce a noton of custer ttng subcategores n a subcategory. Defnton 5.1. Let X be a contravaranty fnte (or covaranty fnte) extenson-cosed subcategory of a tranguated category C and et D be a subcategory of X. We ca that D s a X custer ttng subcategory provded that D s functoray fnte n X, and satsfes that for any object D X, M D f and ony f Ext 1 (D, M) = 0 f and ony f Ext 1 (M, D) = 0. An object D n X s caed a X custer ttng object f addd s a X custer ttng subcategory. When X = C, then C custer ttng subcategores are exacty custer ttng n C. When X s a contravaranty fnte (or covaranty fnte) rgd subcategory, then X s the ony X custer ttng subcategory. From now on to the rest of the secton, C denotes a 2 CY tranguated category wth a custer ttng object, (X, Y ) denotes a cotorson par wth core I n C. We sha show that any custer ttng object contanng I as a drect summand n C gves a X custer ttng object and a Y custer ttng object respectvey. Frst we prove a emma. Lemma 5.2. Let C and (X, Y ) be above. Let T be a custer ttng object n C. Suppose T can be wrtten as T = T X I T Y wth T X X and T Y Y. Then T X I s X custer ttng and T Y I s Y custer ttng. Proof. We prove the asserton for X custer ttng, the proof for Y custer ttng s smar. Suppose that Ext 1 (T X I, X) = 0 for X X, then Ext 1 (T, X) = 0 snce Ext 1 (T Y, X) DExt 1 (X, T Y ) = 0, the frst somorphsm dues to 2 CY property and the second one dues to that (X, Y ) s a co-torson par. Hence X addt. It foows that X add(t X I). Then T X I s a X custer ttng object. The foowng resut gves the precse reaton between the custer ttng objects contanng I as a drect summand and the X custer ttng objects, Y custer ttng objects. Proposton 5.3. Let C be a 2 CY tranguated category wth a custer ttng object, and (X, Y ) be a cotorson par n C wth core I. Then 1. Any custer ttng object T contanng I as a drect summand can be wrtten unquey as: T = T X I T Y, such that T X I s X custer ttng, and T Y I s Y custer ttng. 2. Any X custer ttng object M (or Y custer ttng object) contans I as a drect summand, and can be wrtten as M = M X I ( M = M Y I resp.). Furthermore M X I M Y s a custer ttng object n C. 14

15 3. There s a bjecton between the set of custer ttng objects contanng I as a drect summand n C and the product of the set of X custer ttng objects wth the set of Y custer ttng objects. The bjecton s gven by T T X I T Y. Proof. 1. Let T be any custer ttng object contanng I as a drect summand, we wrte T as T = I T 0. Then T 0 (I[1]), and by passng from (I[1]) to the quotent tranguated category (I[1])/I, T 0 s a custer ttng object n ths quotent category by Theorem 4.9 n [IY]. From the proof of Theorem 4.4, (I[1])/I = X /I Y /I as tranguated categores, then T 0 = T X T Y, where T X X, T Y Y such that T X, T Y are custer ttng objects n X /I, Y /I respectvey. Therefore T = T X I T Y. By Lemma 5.2, T X I, T Y I are X custer ttng, Y custer ttng respectvey. 2. Let T 1 be a X custer ttng object. Then by Ext 1 (T 1, I) = 0, we have that I addt 1,.e. I s a drect summand of T 1. Then T 1 = T X I. Smary, any Y custer ttng object T 2 can be wrtten as T 2 = T Y I. Now T X, T Y are custer ttngs n X /I, Y /I respectvey, and then T X T Y s a custer ttng object n (I[1])/I snce (I[1])/I = X /I Y /I. It foows that T X I T Y s a custer ttng object n C. 3. It foows from 1 and 2. We know that one can mutate custer ttng objects to get new ones. In the foowng we sha see that the mutaton of custer ttng objects contanng I as a drect summand s reated to the mutaton of cotorson pars ntroduced n [ZZ2]. We reca the noton of mutaton of cotorson pars n 2 CY tranguated categores. Ths noton s defned n a genera tranguated category n [ZZ2]. Let C be a 2 CY tranguated category wth a custer ttng object T. We denote by δ(m) the number of ndecomposabe drect summands (up to somorphsm) of an object M. We assume that δ(t) = n. Suppose that (X, Y ) be a cotorson par wth core I. Then 0 δ(i) n [DK]. It foows from Lemma 2.4 that δ(i) = 0 f and ony f (X, Y ) s a t-structure n C, whe δ(i) = n f and ony f X = Y = add(i) s a custer ttng n C. In the ater case, I s a custer ttng object n C. Defnton 5.4. Let C be an ndecomposabe 2 CY tranguated category wth a custer ttng object T, and δ(t) = n. Assume that 0 d n s an nteger. A cotorson par (X, Y ) wth core I s caed a d cotorson par f δ(i) = d. Denote by CT N d (C) the set of a d cotorson pars. From the defnton above and Theorem 4.1, CT N 0 (C) = {(C, 0), (0, C)}. CT N n (C) conssts of custer ttng objects n C. Let D be a drect summand of I (maybe zero summand). Denote by D = addd. Put: The foowng proposton s proved n [ZZ2]. µ 1 (X ; D) := (D X [1]) (D[1]); µ 1 (Y ; D) := (D Y [1]) (D[1]); µ 1 (I; D) := (D I[1]) (D[1]). 15

16 Proposton 5.5. Wth the assumpton above, we have that (µ 1 (X ; D), µ 1 (Y ; D)) s aso a cotorson par wth the core µ 1 (I; D) n C. Moreover (µ 1 (X ; D), µ 1 (Y ; D)) CT N d (C) f and ony f (X, Y ) CT N d (C). Defnton 5.6. We ca the cotorson par (µ 1 (X ; D), µ 1 (Y ; D)) s a D mutaton of cotorson par (X, Y ). Sometmes denote ths cotorson par by (X, Y ), denote ts core by I. Coroary 5.7. Let (X, Y ) be a cotorson par wth core I, and (X, Y ) wth core I be the D mutaton of (X, Y ). Then (X, Y ) = (X, Y ) f and ony f I = I. Proof. The ony f part s obvousy. We prove the f part. Suppose I = I. Then by Theorem 3.11(2) n [ZZ1], D = I. It foows that passng to the quotent category (I[1])/I, (X, Y ) s 0 mutaton of the t-structure (X, Y ) n the quotent tranguated category (I[1])/I. Then ((X, Y ) = (X, Y ) n ths quotent category, snce X, Y are tranguated subcategores of (I[1])/I by the proof of Theorem 4.4. Hence (X, Y ) = (X, Y ) n C. Ths coroary was proved for fnte tranguated categores n [ZZ2]. Note that there are many choces for D. Two extreme cases are: when D = {0}, then the D mutaton of (X, Y ) s (X [1], Y [1]); when D = addi, then the D mutaton of (X, Y ) s (X, Y ) tsef. When D s the drect summand of I wth δ(d) = δ(i) 1, the D mutaton s the usuay one, whch was defned and studed for custer ttng objects (subcategores) n [BMRRT, KR1, IY], for rgd objects(subcategores) n [MP], for maxma rgd objects(subcategores) n [ZZ1]. We ca the D mutaton wth δ(d) = δ(i) 1 just mutaton, for smpcty. Denote ths mutaton by µ I0, where I 0 s the mssng ndecomposabe object of D n I. Remark 5.8. For a custer ttng object T, the mutaton µ s an nvouton. But the mutaton of cotorson pars s not an nvouton n genera (compare [MP]), see the foowng exampe. Exampe 5. Let Q : , and C = D b (kq)/τ 1 [1], the custer category of Q, see the AR-quver beow. Set X = add(p 1 P 2 P 3 S 2 ), Y = add(p 2 P 3 P 4 P 4 [1]), I = P 2 P 3. Then (X, Y ) s a cotorson par wth core I. We mutate the cotorson par (X, Y ) at P 2 to get a new cotorson par (X 1, Y 1 ) wth core I 1, where X 1 = add(e S 3 P 3 P 1 ), Y 1 = add(s 3 P 3 P 4 [1] P 4 ), I 1 = S 3 P 3. Now we contnues to mutate (X 1, Y 1 ) at S 3. We get another new cotorson par (X 2, Y 2 ) wth core I 2, where X 2 = add(p 2 S 2 P 3 E), Y 2 = add(s 2 P 3 P 4 P 4 [1]), I 2 = S 2 P 3. We concude that (X 2, Y 2 ) (X, Y ). P 4 [1] P 4 P 1 [1] P 3 [1] P 3 I 2 P 2 [1] P 2 [1] P 2 E I 3 P 3 [1] P 1 [1] P 1 S 2 S 3 P 4 [1] I 4 16

17 We defne mutaton quver of cotorson pars n C. It s a quver whose vertces are cotorson pars, there s an arrow from the vertex to another vertex f the target cotorson par s a mutaton of the nta one. Ths quver s denoted by M (C). It s not connected from Proposton 5.5. Denoted by M d (C) the subquver of M (C) consstng of vertces beong to CT N d (C). M (C) = n d=0 M d (C). Note that f we repace the each doube ant-arrows by an edge, then M n (C) s the exchange graph of custer ttng objects n C. Ths graph s conjectured to be connected for every ndecomposabe 2 CY tranguated category [Re]. Now we gve the reaton of mutaton of custer ttng objects contanng I as a drect summand wth the mutaton of cotorson pars. Proposton 5.9. Let (X, Y ) be a cotorson par wth core I n C, T = T X I T Y a custer ttng object. Suppose (X, Y ) s a D mutaton of (X, Y ), I s the core of (X, Y ). Then the D mutaton T of T s T X I T Y. Proof. For D = addd, where D s a drect summand of I, we consder the subquotent category (D[1])/D. It s a tranguated category by [IY] wth shft functor < 1 >. In ths subquotent category, (X, Y ) s a cotorson par wth core I n [ZZ2] and T = T X I T Y s a custer ttng object by [IY]. The mages of ther D mutatons are (X, Y ) = (X < 1 >, Y < 1 >), T = T < 1 >= T X < 1 > I < 1 > T Y < 1 > respectvey. It foows that T X < 1 > X < 1 >, T Y < 1 > Y < 1 >. Therefore T = T X I T Y, where T X I, T Y I are X custer ttng object n X, Y custer ttng object n Y respectvey. Now we state and prove the man resut n ths secton. Theorem Let (X, Y ) be a cotorson par wth core I n a 2 CY tranguated category C wth a custer ttng object. Let T = T X I T Y be a custer ttng object contanng I as a drect summand. Suppose that T 0 s an ndecomposabe drect summand of T. We consder the mutaton µ T0 (T) of T n T If T 0 s a drect summand of I, then µ T0 (T) = T I T, where (X, Y ) = µ X Y T0 (X, Y ) s the mutaton of (X, Y ), I s the core of cotorson par (X, Y ). 2. If T 0 s not the drect summand of I, then µ T0 (T) = µ T0 (T X I) T Y when T 0 s a drect summand of T X, and µ T0 (T) = T X µ T0 (I T Y ) when T 0 s a drect summand of T Y. Proof. 1. The asserton foows from Proposton We w prove the case of that T 0 s a drect summand of T X, the proof for the other case s smar. We frst note that any morphsm f : X Y wth X X, Y Y factors through the core I. Ths dues to the fact the mage of f under the projecton π : (I[1]) (I[1])/I s zero snce (I[1])/I = X /I Y /I as tranguated categores. It foows that for the mnma eft T/T 0 approxmaton of T 0, say g : T 0 B, we have B add(t X I). Then g : T 0 B s a g mnma eft (T X I) approxmaton. Extend g to a trange T 0 B T 0 T 0 [1]. It nduces a trange n the subfactor tranguated category g (I[1])/I : T 0 B T 0 T 0 < 1 > [IY]. It foows that T 0 X /I and T 0 X. Then µ T 0 (T) = (T/T 0 ) T 0 = T 0 (T X /T 0 ) I T Y = µ T0 (T X I) T Y. Remark For any cotorson par (X, Y ) wth core I n C. From the theorem above, X (or Y ) has weak custer structure n the sense of [BIRS],.e. the X custer ttngs T X I are the 17

18 canddates of extended custers, where I s the set of coeffcents; one can mutate the X custer ttngs at T 0 to get a new X custer ttng by the above theorem; and one aso have exchange tranges. There s a substructure of C nduced by a X custer ttng and a Y custer ttng: Let T X I be a X custer ttng, T Y I a Y custer ttng. Then T X I T Y s the custer ttng n C by Proposton 5.3. We ca that T X I and T Y I gve a substructure of C (compare [BIRS]) f for any X custer ttng T X I, Y custer ttng T I, both of whch are obtaned Y from T X I and T Y I respectvey va a fnte number of mutatons, the custer ttng object T X I T Y can be obtaned from T X I T Y va a fnte number of mutatons n C. 6 Hearts of cotorson pars As an appcaton of the cassfcaton theorem of cotorson pars, we determne the hearts of cotorson pars n 2 CY tranguated categores wth custer ttng objects n ths secton. Hearts of cotorson pars n any tranguated category were ntroduced by Nakaoka [N], whch unfy the constructon of hearts of t-structures [BBD] and constructon of the abean quotent categores by custer ttng subcategores [BMRRT, KR1, KZ]. We reca the constructon of hearts of cotorson pars from [N]: Let C be a tranguated category and (X, Y ) a cotorson par wth core I n C. Denote by H the subcategory (X [ 1] I) (I Y [1]). The heart of the cotorson par (X, Y ) s defned as the quotent category H/I, denoted by H. It was proved that H s an abean category [N]. There s a cohomoogy functor H = hπ from C to H, where π s the quotent functor from C to C = C/I and h s a functor from C to H. Those constructons were gven n Proposton 3.4 and Proposton 4.2 n [AN] combned wth Constructon 4.2, Proposton 4.3 and Remark 4.5 n [N]. For the convenence of reader, we reca the defntons of the functor h from [AN] as foows. For any M C, there s a trange Y M X M M Y M [1] wth X M X, Y M Y, snce (X, Y ) s a cotorson par. Then there s a trange X M [ 1] X M Y M X M wth X M X, Y M Y, snce (X [ 1], Y [ 1]) s a cotorson par. Composng the morphsm from X M [ 1] to X M and the morphsm from X M to M, we have the foowng commutatve dagram of tranges n C by the octahedra axom, n whch we get M and s M : M M: X M [ 1] = X M [ 1] Y M X M M Y M [1] s M Y M Y M M Y M [1] X M = X M ( ). Usng the defnton of cotorson par (X [ 1], Y [ 1]) agan, we have a trange X M [ 1] M Y M X M and then we have another trange Y M X M Y M Y M [1] wth X M, X M X and Y M, Y M Y. Compose the morphsm from X M to Y M and the morphsm from Y M to X M, by the octahedra axom, we have the foowng commutatve dagram of tranges n C, n whch we 18

19 have M and t M : M M: Y M = Y M X M [ 1] M X M X M t M ( ). X M [ 1] M Y M X M Y M [1] = Y M [1] The mage of M under h s defned as M. Abe and Nakaoka proved that M H. It s easy to see that up to somorphsms n H, M does not depend on the choce of X M, X M, X M, X M and Y M, Y M, Y M, Y M (See Secton 4 n [AN] for detas). For any morphsm f : M N n C, there s a unque morphsm f n C such that the eft square of the foowng dagram commutate (Proposton 4.3 n [N]) and then there s a unque morphsm f n C such that the rght square n the foowng dagram commutate (Remark 4.5 n [N]): t M M s M M M f f f N s N Ñ t N N ( ). The mage of f under h s defned as f. We state two smpe facts foowed from the constructons above. Lemma 6.1. H(X ) = 0 and H(Y ) = 0 hod. Proof. We gve a proof for H(X ) = 0, H(Y ) = 0 can be proved duay. Let M be an object n X. One can choose Y M = 0. Then M Y M. So one can choose X M = 0. Then h(m) = M X M. Note that X M Y Y Y and I = X Y. We have that h(m) I and hence h(m) 0 n H. Lemma 6.2. h H = d H. Proof. By the defnton of h, one ony need to check that h(m) M for any M H. In ths case, we have that X M I by Coroary 3.3 n [N]. One can choose Y M = X M and then M M. By the dua, one can have that M M. Thus ths emma hods. Let (X 1, Y 1 ) and (X 2, Y 2 ) be two cotorson pars wth the same core I n a tranguated category C. Denote by H the heart of (X, Y ), = 1, 2. Let H = h π be the cohomoogy functor from C to H gven n [AN], and ι be the ncuson functor from H to C, = 1, 2. The composton functors h 1 ι 2 and h 2 ι 1 are denoted by E and F respectvey. 19

20 H 1 π C C ι 1 ι2 H 2 H 1 h 1 E F h 2 H 2 Lemma 6.3. If H 1 ( (I[1])) = 0 and H 1 ((I[ 1]) ) = 0, then EF d H1. Proof. For any M H 1, we have the above commutatve dagrams ( ) and ( ) wth X M, X M, X M, X M X 2 and Y M, Y M, Y M, Y M Y 2. Then h 2 (M) = M by the defnton. The frst and the ast morphsms n the thrd coumn of the dagram ( ) and n the second coumn of the dagram ( ) factor through (I[1]) or (I[ 1]) respectvey, by X 2 (I[1]) and Y 2 (I[ 1]). Then the mage of these morphsms under H 1 are zero. Appyng the cohomoogy functor H 1 to these two tranges (n the thrd coumn of the dagram ( ) and n the second coumn of the dagram ( )), one has two somorphsms n H 1 : and H 1 M H 1(s M ) E M EFM H 1(t M ) E M. Snce M H 1, so H 1 M = M by Lemma 6.2. For any morphsm f : M N n H 1, appyng the functor h 1 to the dagram ( ), we have the foowng commutatve dagram n H 1 : h 1 M h 1 f h 1 N H 1 (t M ) 1 H 1 (s M ) H 1 (t N ) 1 H 1 (s N ) EFM EF f EFN. Snce M, N H 1, then h 1 M = M, h 1 N = N and h 1 f = f by Lemma 6.2. Therefore, d H1 EF. From now on to the end of ths secton, we assume that C s a 2-CY tranguated category. We contnue to use the same notatons as above. Fxed a cotorson par (X, Y ) wth core I, whch s assumed functoray fnte n C (e.g. I contans ony fntey many ndecomposabe objects). Let (X 1, Y 1 ) be the cotorson par (I, (I[1])) and (X 2, Y 2 ) = (X, Y ). Then the condton of Lemma 6.3 hods automatcay by Lemma 6.1 and the heart H 1 s equvaent to the modue category over I [IY], denoted by mod I. By Coroary 3.6 n [ZZ2], we have that (X /I, Y /I) s a t-structure n the 2-CY tranguated category (I[1])/I. Reca that the shft functor 1 n the tranguated category (I[1])/I defned n [IY] s obtaned by the foowng trange: M I M M 1 M[1], where M (I[1]), I M I. We denote the heart (X /I) 1 (Y /I) 1 of ths t-structure by A whch s an abean category [BBD]. Lemma 6.4. The category A s an abean subcategory of the heart H of (X, Y ). 20