Classification of tilting objects on a weighted projective line of type $(2 ...

CLASSIFICATION OF TILTING OBJECTS ON A WEIGHTED PROJECTIVE LINE OF TYPE (2, 2, 2, 2; λ)

arXiv:1303.1323v3 [math.RT] 10 Nov 2013

JIANMIN CHEN, YANAN LIN, PIN LIU, AND SHIQUAN RUAN† Abstract. Using cluster tilting theory, we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of type (2, 2, 2, 2; λ). Furthermore, we classify the endomorphism algebras of tilting objects in the category of coherent sheaves and in its derived category.

1. Introduction Let X be a weighted projective line of genus one over an algebraically closed field k. A famous theorem of Lenzing and Meltzer [17] pointed that X is of weight type (2,3,6), (2,4,4), (3,3,3) or (2, 2, 2, 2; λ), where λ is a parameter in the projective line P1 (k) different from 0, 1, ∞. Kussin, Lenzing and Meltzer [16] proved that the category vect X of vector bundles on X, under the distinguished exact structure, is a Frobenius category with the system L of all line bundles as the system of all indecomposable projective-injectives. A general result of Happel [11] asserts that the attached stable category vectX = vect X/[L] is triangulated, which is equivalent to the bounded derived category Db (coh X) of the category of coherent sheaves in each case of genus one. Much of the work on this triangulated category vectX has focused on the tilting objects (see for instance [16, 8]). Among other things, Kussin, Lenzing and Meltzer [16] showed the existence of a tilting object in vectX for the weighted projective lines of weight triple. Chen, Lin and Ruan [8] constructed a tilting object in vectX for weight type (2, 2, 2, 2; λ). In this way, it seems a little hard to find more tilting objects directly in vectX. The main idea of this article is to use cluster tilting theory to investigate the tilting objects in vectX. As we known, the Fomin-Zelevinsky mutation of quivers plays an important role in the theory of cluster algebras initiated in [9]. Motivated by this theory, a mutation of cluster tilting objects in cluster categories, and more generally Hom-finite triangulated 2-Calabi-Yau categories has been investigated in [6, 12]. This has turned out to give a categorical model for the quiver mutation in certain cases [7, 5]. From the viewpoint of mutation, an advantage of cluster tilting theory over classical tilting theory is that in cluster tilting theory it is always to mutate. Thus the usual procedure of going from a tilting object to another one by exchanging just one indecomposable direct summand gets more regular. 2000 Mathematics Subject Classification. 14F05, 16G70, 16S99, 18E30. Key words and phrases. Tilting object; cluster tilting object; weighted projective line; vector bundle. This work was supported by the National Natural Science Foundation of China (Grants No. 11201381, 11201386, 11201388, 10931006, 11071040) and the Natural Science Foundation of Fujian Province of China (Grant No. 2012J05009). † Corresponding author. 1

2

CHEN, LIN, LIU, AND RUAN

For this purpose, we define the associated cluster category vectX/GZ as the orbit category of vectX under the action of the unique auto-equivalence G satisfying DHom(X, Y [1]) ∼ = Hom(GY, X[1]), where D denotes the usual duality functor Homk (−, k) and [1] denotes the suspension functor of vectX. We describe the relationship between the tilting objects in vectX and the cluster tilting objects in vectX/GZ , and realize, independently by Meltzer [18], a complete classification of the endomorphism algebras of tilting objects in vectX for the weight type (2, 2, 2, 2; λ). Furthermore, we classify all the endomorphism algebras of tilting sheaves in the category coh X for the weight type (2, 2, 2, 2; λ). The paper is organized as follows: In section 2, we recall some basic results on the category of vector bundles on a weighted projective line. In section 3, we study the relationship between the tilting objects in the stable category of vector bundles on a weighted projective line of genus one and the cluster tilting objects in the associated cluster category. We focus on the weighted projective line of type (2, 2, 2, 2; λ) in section 4. We show that each tilting object in the stable category pushes to a cluster tilting object in its cluster category, and describe all the tilting objects corresponding to a given cluster tilting object in the cluster category. Moreover, we realize the classifications of all the endomorphism algebras of the tilting objects in the category of coherent sheaves and in its derived category. In the rest of the paper, we view the isomorphism as equality. Acknowledgments. The authors would like to thank H. Lenzing for useful comments and X.W. Chen for helpful discussions. In particular, Question 4.9 is raised by H. Lenzing and Lemma 4.22 is proved by X.W. Chen. This work started life while the third named author visited Xiamen in May 2012. He would like to thank the people at Xiamen University for hospitality. 2. Stable category of vector bundles In this section, we present some materials concerning the category of vector bundles over a weighted projective line. 2.1. Coherent sheaves on a weighted projective line. Let k be an algebraically closed field. A weighted projective line X over k is specified by giving a collection λ = (λ1 , λ2 , · · · , λt ) of distinct points in the projective line P1 (k), and a weight sequence p = (p1 , p2 , · · · , pt ) of positive integers. Let L be the rank one abelian group with generators ~x1 , ~x2 , · · · , ~xt and the relations p1 ~x1 = p2 ~x2 = · · · = pt ~xt =: ~c. The element ~c is called the canonical element of L, and each element ~x ∈ L can be uniquely written in normal form ~x =

t X

li ~xi + l~c, where 0 ≤ li < pi and l ∈ Z.

i=1

For any ~x ∈ L, define ~x ≥ 0 if l ≥ 0 in the normal form of ~x. Then L becomes a partial order group, and each ~x ∈ L satisfies exactly one of the two possibilities: ~x ≥ 0 or ~x ≤ ~ω + ~c, here ~ ω = (t − 2)~c −

t P

~xi is called the dualizing element of L.

i=1

Denote by S the commutative algebra S = k[X1 , X2 , · · · , Xt ]/I := k[x1 , x2 , · · · , xt ],

CLASSIFICATION OF TILTING OBJECTS OF TYPE (2, 2, 2, 2; λ)

3

where I = (f3 , · · · , ft ) is the ideal generated by fi = Xipi −X2p2 +λi X1p1 , i = 3, · · · , t. Then S is L-graded by setting deg(xi ) = ~xi (i = 1, 2, · · · , t), and S carries a decomposition into k-subspaces: M S= S~x . ~ x∈L

The category of coherent sheaves on X can be defined as the quotient of the category of finitely generated L-graded S-modules over the Serre subcategory of finite length modules as follows coh X := modL (S)/modL0 (S). The free module S gives the structure sheaf O, and each line bundle is given by the grading shift O(~x) for a uniquely determined element ~x ∈ L. Moreover, there is a natural isomorphism Hom(O(~x), O(~y )) = Sy~−~x . Denote by vect X the full subcategory of coh X formed by all vector bundles, i.e. torsion free sheaves, and by coh0 X the full subcategory formed by all sheaves of finite length, i.e. torsion sheaves. Geigle and Lenzing [10] showed that each coherent sheaf is the direct sum of a vector bundle and a finite length sheaf, and there are no non-zero morphisms from coh0 X to vectX. Moreover, coh X is a hereditary abelian category with Serre duality of the form D Ext1 (X, Y ) = Hom(Y, X(~ω)). It implies the existence of almost split sequences for coh X with the AuslanderReiten translation τ given by the grading shift with ~ω . The Grothendieck group K0 (X) of coh X was computed by Geigle and Lenzing [10], and it was proved to be the vector space with basis indexed by elements O(~x) with 0 ≤ ~x ≤ ~c, where we still write X ∈ K0 (X) for the class of an object X ∈ coh X. Let p = l.c.m.(p1 , p2 , · · · , pt ) and δ : L → Z be the homomorphism defined by δ(~xi ) =

p . pi

The determinant map is the group homomorphism det : K0 (X) → L given by det(O(~x)) = ~x. There are two important Z-linear functions, rank rk and degree deg, on K0 (X). The degree function is the composition of δ and det, that is, it is determined by deg(O(~x)) = δ(~x). The rank function rk : K0 (X) → Z is characterized by rk(O(~x)) = 1. For each non-zero object X ∈ coh X, define the slope of X as µX =

deg(X) . rk(X)

Notice that the rank is strictly positive for a non-zero vector bundle and vanishes for a sheaf of finite length. The slope of a vector bundle belongs to Q, while it is infinity for a finite-length sheaf. From now on, we shall always assume that - X is a weighted projective line of genus one, or equivalently, - X is of weight type (2,3,6), (2,4,4), (3,3,3) or (2, 2, 2, 2; λ), where λ is a parameter in the projective line P1 (k) different from 0, 1, ∞.

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Then by [10], for any two indecomposable objects X, Y in coh X, (2.1)

Hom(X, Y ) 6= 0 implies µX ≤ µY.

Define the Euler form of X, Y by hX, Y i = dimk Hom(X, Y ) − dimk Ext1 (X, Y ). We have the following result. Theorem 2.1 (Riemann-Roch Formula, [17]). For each X, Y ∈ coh X, we have p−1 X

hτ i X, Y i = rk(X) deg(Y ) − deg(X) rk(Y ).

i=0

In particular, if X, Y ∈ vect X, then (2.2)

p−1 X

hτ i X, Y i = rk(X) rk(Y )(µY − µX).

i=0

The following lemma was given in [16]. Lemma 2.2. [16, Lemma 5.3] There is a bijective, monotonous map α : Q → Q with α(q) > q for all q ∈ Q and such that µ(X[1]) = α(µ(X)) for each indecomposable X ∈ vect X. Moreover we have the following tubular factorization property. Lemma 2.3. [16, Theorem A.6] Let X and Y be indecomposable in vect X with slopes µ(X) = q and µ(Y ) = q ′ . If q ′ > α(q) then every morphism X → Y factors through a direct sum of line bundles. 2.2. Stable category of vector bundles. Recall from [16] that a sequence 0 → X ′ → X → X ′′ → 0 in vectX is called distinguished exact if for each line bundle L the induced sequence 0 → Hom(L, X ′ ) → Hom(L, X) → Hom(L, X ′′ ) → 0 is exact. Kussin, Lenzing and Meltzer [16] proved that the distinguished exact sequences define a Frobenius exact structure on the category vect X, such that the system of all line bundles is the system of all indecomposable projective-injectives. By a general result of [11], the related stable category vectX = vect X/[L], is a triangulated category. For simplification of notations, in the rest of the paper we denote the stable category vectX by D and denote the homomorphism space between X and Y in D by D(X, Y ). Lemma 2.4 ([16]). (1) D is Hom-finite: For any two objects X and Y in D, the space of morphisms D(X, Y ) is finite-dimensional. It also ensures that D is a Krull-Schmidt category. (2) D is homologically finite: For any two objects X and Y in D, D(X, Y [n]) = 0 for |n| ≫ 0. (3) D admits a Serre functor: For any two objects X and Y in D, D(X, Y [1]) = DD(Y, X(~ω )), In particular, D has Auslander-Reiten triangles, and the shift by ~ω also serves as the AR-translation for D. The following result is also taken from [16].


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Lemma 2.5 (Interval category, [16]). For any a ∈ Q, the interval category D(a, α(a)], the full subcategory of D obtained as the additive closure of all the indecomposable objects with slopes in the interval (a, α(a)], is an abelian category and equivalent to the category coh X. An object T is called tilting in D if - T is extension-free, i.e. D(T, T [n]) = 0 for each non-zero integer n. - T generates the triangulated category D, i.e., the smallest triangulated subcategory hT i containing T is D. In our case, the indecomposable direct summands of extension-free object T in D can be ordered in such a way that they form an exceptional sequence. Then by [3, Theorem 3.2] and [4, Proposition 1.5], D is generated by hT i together with hT i⊥ . Hence the 2nd axiom above can be replaced by the following statement (see for instance [16]): - for each object X ∈ D, there exists some integer n, such that D(T, X[n]) 6= 0. Lemma 2.6. Let T = ⊕Ti be an object in D with indecomposable direct summand Ti ∈ D(a, α(a)] for some a ∈ Q. Then T is extension-free in D if and only if D(T, T [1]) = 0. Proof. For any Ti , Tj ∈ D(a, α(a)], if n ≤ −1, then by Lemma 2.2, µ(Tj [n]) ≤ a < µTi . Hence from (2.1), Homcoh X (Ti , Tj [n]) = 0, which implies that (2.3)

D(Ti , Tj [n]) = 0.

If n ≥ 2, then µ(Tj [n]) ≥ µ(Tj [2]) > α(µ(Ti )). By Lemma 2.3, each morphism Ti → Tj [n] factors through a direct sum of line bundles, which implies that (2.4)

D(Ti , Tj [n]) = 0.

Combining 2.3 and 2.4, we finish the proof.

3. Relationship to cluster tilting objects Cluster categories were defined in [6] in order to use categorical methods to give a conceptual model for the combinatorics of cluster algebras [9]. For this purpose, a tilting theory was developed in the cluster category and indeed there has been a considerable amount of activity and a lot of results in this direction. We refer to the survey [14, 20]. In this section, we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of genus one via cluster tilting theory. 3.1. Cluster categories. Let H be a hereditary algebra. In [6], the cluster category CH is defined to be the orbit category of the bounded derived category of finitedimensional right H-modules under the action of the auto-equivalence G = τ −1 [1]. Following this, Barot, Kussin and Lenzing [2] did manage to study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. Namely they investigated the orbit category Db (coh X)/GZ . According to [19], the bounded derived category of coherent sheaves over the weighted projective line Db (coh X) is triangular equivalent to the stable category of vector bundles D. Thus parallel to [2], we define the cluster category C to be

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the orbit category of the stable category D under the action of the unique autoequivalence G satisfying DD(X, Y [1]) = D(GY, X[1]). According to section 2.2, the functor G is τ −1 [1]. The cluster category C has the same objects as D, and for any objects X, Y , morphism spaces are given by M C (X, Y ) = D(X, Gn Y ) n∈Z

with the obvious composition. This orbit category is triangulated and Calabi-Yau of CY-dimension 2, and the canonical functor π : D → C is a triangulated functor, cf. [13]. We still denote by [1] its suspension functor. 3.2. Relationship to cluster tilting objects. Recall from [15] that, an object T in C is called rigid if C (T, T [1]) = 0; and a rigid object T in C is called a cluster tilting object if C (T, X[1]) = 0 implies X ∈ add(T ). Throughout this subsection, T = ⊕Ti is an object in D with each indecomposable direct summand Ti ∈ D(a, α(a)] for some a ∈ Q, and we use the same notation for T and its image in C under the canonical functor π : D → C . Lemma 3.1. The object T is extension-free in D if and only if T is rigid in C . Proof. By definition and Lemma 2.3, we have M C (T, T [1]) = D(T, Gi T [1]) = D(T, T (~ω)) ⊕ D(T, T [1]). i∈Z

According to Lemma 2.4, D(T, T [1]) = DD(T, T (~ω)). Hence D(T, T [1]) = 0 if and only if C (T, T [1]) = 0. By Lemma 2.6, T is extension-free in D if and only if T is rigid in C .

The following lemma is easy but necessary for us to describe the relationship between the tilting objects in D and the cluster tilting objects in C . Lemma 3.2. If T is a cluster tilting object in C , then T (~ω) ⊕ T (−~ω) ∈ hT i. Proof. It suffices to show that for each indecomposable direct summand Ti of T , Ti (~ω ) ⊕ Ti (−~ω ) ∈ hT i. The assumption implies that T is rigid in C , hence it is extension-free in D by Lemma 3.1. Therefore, for any indecomposable object X ∈ D, X ∈ hT i if and only if there exists some integer n, such that D(T, X[n]) 6= 0. By Lemma 2.4, D(Ti , Ti (~ ω )[1]) = DD(Ti , Ti ) 6= 0, which implies Ti (~ω ) ∈ hT i. On the other hand, assume that Ti (−~ω) fits into the following Auslander-Reiten triangle in D: Ti → X → Ti (−~ω ) → Ti [1]. Then for each indecomposable direct summand Xj of X, we have D(Ti , Xj ) 6= 0, which implies that Xj ∈ hT i. Thus X ∈ hT i follows. Notice that hT i is closed under the third term of a triangle, hence Ti (−~ω ) ∈ hT i. This finishes the proof. The main result of this section is as follows. Theorem 3.3. The object T is a tilting object in D if and only if T is a cluster tilting object in C .


7

Proof. Firstly, we show the “if ” part. By Lemma 3.1 and Lemma 2.2, it is sufficient to show X ∈ hT i for each indecomposable object X ∈ D(a, α(a)]. Obviously, X ∈ hT i if X ∈ add(T ). So we only need to consider the case that X∈ / add(T ). It follows that 0 6= C (T, X[1]) = D(T, X(~ω)) ⊕ D(T, X[1]). If D(T, X[1]) 6= 0, then X ∈ hT i. If else, we get D(T, X(~ω)) 6= 0. Let θ : TX (−~ ω) → X be a right add(T (−~ω))-approximation of X and (3.1)

θ

TX (−~ ω) − → X → Y → TX (−~ω )[1]

be the induced triangle in D. By Lemma 3.2, TX (−~ω) ∈ hT i. Hence we only need to show that Y ∈ hT i. In fact, by similar arguments in Lemma 3.2, we obtain (3.2)

D(T (−~ω), Y ) = 0.

Since µ(X) ∈ (a, α(a)] and µ(TX (−~ω )[1]) ∈ (α(a), α2 (a)], we know that for each indecomposable direct summand Yi of Y , µYi ∈ (a, α2 (a)]. Case 1: µYi ∈ (a, α(a)]. Then from (3.2), C (T, Yi [1]) = D(T, Yi [1]). Applying D(T [−1], −) to the triangle (3.1), we get D(T [−1], Y ) = 0, which implies C (T, Yi [1]) = 0. Hence Yi ∈ add(T ). Case 2: µYi ∈ (α(a), α2 (a)]. Then from (3.2), C (T, Yi (~ω )) = D(T, Yi (2~ω )[−1]). Applying D(T (−2~ ω)[1], −) to the triangle (3.1), we get D(T (−2~ω)[1], Y ) = 0, which implies C (T, Yi (~ ω )) = 0. Thus Yi (~ω )[−1] ∈ add(T ), i.e., there exists some T ′ ∈ add(T ) such that Yi = T ′ (−~ω )[1]. Hence by Lemma 3.2, Yi ∈ hT i, as claimed. Conversely, let T be a tilting object in D. By Lemma 3.1, we only need to prove that C (T, X[1]) 6= 0 with an indecomposable X not occurring as a direct summand of T . Since there exists some n ∈ Z such that µGi X ∈ (a, α(a)], without loss of generality, we assume µX ∈ (a, α(a)]. Then by Lemma 2.3 C (T, X[1]) = D(T, X(~ω)) ⊕ D(T, X[1]). It is sufficient to show that D(T, X[1]) = 0 implies D(T, X(~ω)) 6= 0. Note that hT i = D and D(T, X[n]) = 0 for n 6= 0, 1, the assumption D(T, X[1]) = 0 implies that D(T, X) 6= 0. Let θ : TX → X be a right add(T )-approximation of X and (3.3)

β

θ

→X − → Y → TX [1] TX −

be the induced triangle in D. As in Lemma 3.2, it is easy to know that (3.4)

D(T, Y ) = 0.

Moreover, by applying D(T [n], −) to the triangle (3.3), we get D(T [n], Y ) = 0 for any n 6= 0, 1. Then combining with (3.4), we have D(T [1], Yi ) 6= 0 for each indecomposable direct summand Yi of Y . Hence µYi ∈ (α(a), α2 (a)]. Let δ : TY [1] → Y be the minimal right add(T [1])-approximation of Y and (3.5)

δ

TY [1] − → Y → Z → TY [2]

be the induced triangle in D. Applying D(T [1], −) to the triangle (3.5), we have D(T [1], Z) = 0. Moreover, it is easy to check that D(T [n], Z) = 0 for any n 6= 1, 2.

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Again since T is tilting, for each indecomposable direct summand Zj of Z, we have D(T [2], Zj ) 6= 0. Thus µ(Zj ) > α2 (a). By Lemma 2.3, we have D(X, Z) = 0. Hence there exists a morphism γ : X → TY [1], such that β = δγ 6= 0. γ θ

TX

/X

④

④

④

β

TY [1] ④= δ

/Y

/ TX [1]

Z TY [2] Then by Lemma 2.4, D(TY , X(~ω)) = DD(X, TY [1]) 6= 0. This finishes the proof.

4. Tilting objects for weight type (2, 2, 2, 2; λ) This section dedicates to investigate the tilting objects in the stable category of vector bundles on a weighted projective line of weight type (2, 2, 2, 2; λ). Throughout D is the stable category vectX of vector bundles and C is the attached cluster category. 4.1. Canonical tilting object. In [8], Chen, Lin and Ruan constructed a tilting object in vectX for type (2, 2, 2, 2; λ), whose endomorphism algebra is a canonical algebra of the same type. We recall some notations and results there. Let E be the Auslander bundle determined by the almost-split sequence 0 → O(~ω ) → E → O → 0. and Ei be the central term of the non-split exact sequence for each i = 1, · · · , 4 0 → O(~ω ) → Ei → O(~xi ) → 0. Set F = E(w ~ + ~c)[−1]. We have the following description for F : Lemma 4.1. [8] For each i = 1, · · · , 4, 0 → O(~ω ) → F → E(~ω + ~xi ) → 0 is an exact sequence in coh X. The following is the main result in [8]. Lemma 4.2. [8] The object 4 M Ei ) ⊕ F Tcan = E ⊕ ( i=1

is a tilting object in D and the endomorphism algebra End(Tcan ) is a canonical algebra of type (2,2,2,2).


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Consequently, we have Lemma 4.3. The object Tcan is a cluster tilting object in C . Proof. Since Tcan is a direct sum of indecomposable vector bundles with slopes in the interval (α−1 (1), 1], by Theorem 3.3, we get what we want. 4.2. Quiver mutation and cluster tilting mutation. In the fundamental paper [9], Fomin and Zelevinsky introduced the notion of mutation of quivers as follows. Let Q be a finite quiver without loops and oriented cycles of length 2 (2-cycles for short). Let i be a vertex of Q. The mutation of the quiver Q at the vertex i is a quiver denoted by Mi (Q) and constructed from Q using the following rule: (M1) for any couple of arrows j → i → k, add an arrow j → k; (M2) reverse the arrows incident with i; (M3) remove a maximal collection of 2-cycles. Example 4.4. Let Q be the quiver B2✿ ☎☎ ✿✿✿ ☎ ✿ ☎ ☎☎ r9 3 ▲▲ ✿✿✿ ☎ ▲ r ☎ r ▲▲ ✿✿ ▲▲ ☎r☎rrr % s k 1 ✿▲▲ r☎9 B 6 . r ✿✿▲▲▲ r ✿✿ ▲▲% rrrr ☎☎☎ ✿✿ 4 ☎ ✿✿ ☎☎ ✿ ☎☎☎ 5 It is easy to compute that the mutation class (the set of all quivers obtained from Q by iterated mutations) consists of the following 4 quivers up to isomorphism.

(4.1)

2 ✝B ✽✽✽ ✝ ✽ ✝ ✝✝ss9 3 ❑❑❑✽✽✽ ✝ ❑❑% ✝sss ks 1 ✽❑ 96 ❑ ✽✽❑❑❑ sss✝s✝B % s ✽✽ ✝ ✽✽ 4 ✝✝✝ ✝ 5

2 ✝ \✽✽✽ ✝ ✽ ✝ ✝✝ss9 3 ❑❑❑✽✽✽ ✝ ❑❑% ✝sss 1 ✽o❑❑ 96 ✽✽❑❑❑ sss✝s✝B s % ✽✽ ✝ ✽✽ 4 ✝✝✝ ✝ 5

2 ✝ \✽✽✽ ✝ ✽ ✝ ✝✝ss 3 e❑❑❑✽✽✽ ✝ ❑❑ ✝ss ys 1 ✽❑❑ 96 ✽✽❑❑❑ sss✝s✝B % s ✽✽ ✝ ✽✽ 4 ✝✝✝ ✝ 5

Q1 = Q

Q2 = M2 (Q1 )

Q3 = M3 (Q2 )

z /2 /4 1 ❂l ❂❂ ✁✁@ ❂❂❂ ✁✁@ ✁❂✁❂ ✁❂✁❂ ~✁✁ ❂ ✁✁ ❂ /3 /5 6d Q4 = M6 (Q3 )

An advantage of cluster tilting theory over classical tilting theory is that from a cluster tilting object in a 2-Calabi-Yau triangulated category C, it is possible to construct others by a recursive process resumed in the following. Theorem 4.5 ([6, 12]). Let C be a Hom-finite 2-CY triangulated category with a cluster tilting object T . Let Ti be indecomposable and T = T0 ⊕ Ti . Then there exists a unique indecomposable Ti∗ non-isomorphic to Ti such that T0 ⊕ Ti∗ is cluster tilting. Moreover Ti and Ti∗ are linked by the existence of exchange triangles u

v

w

→ Ti [1] →B− → Ti∗ − Ti −

and

u′

v′

w′

Ti∗ −→ B ′ −→ Ti −→ Ti∗ [1]

where u and u′ are minimal left add T0 -approximations and v and v ′ are minimal right add T0 -approximations. This recursive process of mutation of cluster tilting objects is closely related to the notion of mutation of quivers in the following sense.

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Theorem 4.6 ([5]). Let C be a Hom-finite 2-CY triangulated category with a cluster tilting object T . Let Ti be an indecomposable direct summand of T , and denote by T ′ the cluster tilting object MTi (T ). Denote by QT (resp. QT ′ ) the quiver of the endomorphism algebra EndC (T ) (resp. EndC (T ′ )). Assume that there are no loops and no 2-cycles at the vertex i of QT (resp. QT ′ ) corresponding to the indecomposable Ti (resp. Ti∗ ). Then we have QT ′ = Mi (QT ), where Mi is the Fomin-Zelevinsky quiver mutation. We shall make frequent use of the following results. Lemma 4.7. For each i = 1, · · · , 4, there is a triangle in C a

i F → E(~ω + ~xi ) → Ei [1]. Ei −→

Proof. For any i, there is an almost split sequence in coh X i

(4.2)

0 → O(~xi ) − → E(~ω + ~xi ) → O(~ω + ~xi ) → 0.

Applying the functor Hom(−, O(~ω )) to (4.2), we get Ext1 (E(~ ω + ~xi ), O(~ω )) = Ext1 (O(~xi ), O(~ω )). By Lemma 4.1, there exists a commutative diagram induced by pullback of i and φ: / O(~ / Ei / O(~xi ) /0 0 ω)

ai

0

/F

/ O(~ ω)

φ

i

/ E(~ω + ~xi )

/0.

Moreover, we know that the right square is also a pushout. Hence ai : Ei → F is injective and Coker(ai ) = Coker(i) = O(~ω + ~xi ). Then we obtain the following exact sequence: a

i F → O(~ω + ~xi ) → 0. 0 → Ei −→

(4.3)

Denote by I(Ei ) the injective hull of Ei . By [8], M I(Ei ) = O(~xi ) ⊕ ( O(~ω + ~xj )). j6=i

There is the following pushout 0

/ Ei

0

/ I(Ei )

/F

/ O(~ω + ~xi )

/0

/C

/ O(~ω + ~xi )

/ 0.

Notice that for any j 6= i, Ext1 (O(~ω + ~xi ), O(~ω + ~xj )) = 0. Combining with (4.2), we get

M C = E(~ω + x~i ) ⊕ ( O(~ω + ~xj )). j6=i

Hence, there exists a triangle in D Ei → F → E(~ω + ~xi ) → Ei [1]. Since π : D → C is a triangulated functor, we get what we want.


11

Lemma 4.8. The following is a triangle in C F [−1] → E(~x1 − ~x2 ) → E3 ⊕ E4 → F. Proof. From (4.3), we obtain the following commutative diagram induced by the pullback of a3 and a4 0

/A b3

0

b4

a4

/ E3

/B

/ E4

a3

/0

b

/ O(~ω + ~x3 )

/F

/ 0,

where the induced maps b3 , b4 and b are all injective. Notice that 1 and µ(O(~ω + ~x3 )) = 1. 2 We get B = 0 or B = O(~ ω + ~x3 ). The fact Hom(E4 , E3 ) = 0 ensures that µE4 =

B = O(~ω + ~x3 ). It follows that the kernel A satisfying det A = ~x3 − ~x4 and rk A = 1. Hence A = O(~x3 − ~x4 ), and the left square is also a pushout. Thus there is an exact sequence in coh X 0 → O(~x3 − ~x4 ) → E3 ⊕ E4 → F → 0. Denote by P (F ) the projective cover of F . By [8], M P (F ) = O(~ω )2 ⊕ ( O(~xi − ~xj )). j6=i

Now consider the following pullback diagram 0

/ O(~x3 − ~x4 )

/C

/ O(~x3 − ~x4 )

/ E3 ⊕ E4

/ P (F )

/0

/F

/ 0.

0

It is easy to see that for any ~x ∈ L with δ(~x) = 0, Ext1 (O(~x), O(~x3 − ~x4 )) 6= 0 if and only if ~x = ~x1 − ~x2 . Thus in D, C = E(~x1 − ~x2 ). Similarly to the proof of Lemma 4.1, it is easy to know that there exists the following triangle in C F [−1] → E(~x1 − ~x2 ) → E3 ⊕ E4 → F. In [16], particular attention was given to the indecomposable vector bundles of rank two, which led to all major results there. The following question was raised by H. Lenzing. Question 4.9. Whether one can construct a tilting object in vectX of weight type (2, 2, 2, 2; λ), consisting of rank two bundles? The following is the main result in this subsection, which gives a positive answer to this question.

12


Theorem 4.10. The object 2 4 M M E(~ω + ~xj )) Ei ) ⊕ ( Trk = E ⊕ E(~x1 − ~x2 ) ⊕ ( i=3

j=1

is a tilting object in D, with the quiver of the endomorphism algebra EndD (Trk ) as the following ΓD :

/ / E(~ω + ~x1 ) E■ : E3 ■■ ■■ ✉: ■■ ■■ ✉✉✉✉ ■■ ✉✉✉✉ ■■✉✉ ■■ ✉✉ ✉✉■■ ✉✉ ■■■■ ✉✉ ■■■■ ✉ ✉ ✉ $ ✉ ✉ $ / E(~ω + ~x2 ). / E4 E(~x1 − ~x2 )

Proof. By Lemma 4.3, 4 M Ei ) ⊕ F = E1 ⊕ T Tcan = E ⊕ ( i=1

is a cluster tilting object in C . And the quiver of endomorphism algebra EndC (Tcan ) has the form

Q1 :

E1 ✄A ❀❀❀ ✄ ❀❀ ✄ ❀ a1 ✄✄ E ✄ 2 ▲▲ ❀❀❀ ✄ rr8 ✄ ▲▲▲ ❀ ✄ r ❀ ✄r✄rrr a2 ▲▲& ks E ❀▼ 8 AF . ❀❀▼▼▼▼ a3 rrr✄ ✄ r r ❀❀ ▼▼ rr ✄✄✄ ❀❀ & E 3 ❀❀ ✄✄a4 ❀❀ ✄✄✄ ✄ E4

By Lemma 4.7, there exists a triangle in C a

1 F → E(~ω + ~x1 ) → E1 [1], E1 −→

where a1 : E1 → F is the minimal left add T -approximation easily known from the above quiver. Then by Theorem 4.5, 4 M Ei ) ⊕ F ⊕ E(~ω + ~x1 ) T⋆ = E ⊕ ( i=2

is a cluster tilting object in C . By Theorem 4.6, we obtain the quiver of EndC (T⋆ ) as follows

Q2 :

E(~ω + ~x1 ) a❈❈ ④ ❈❈ ④④ ❈❈ ④ ④ ④④ ♠6 E2 ◗◗◗ ❈❈❈ ④ ◗◗◗ ❈❈ ④④ ♠♠♠ ◗ ❈❈ ④♠④♠♠♠♠♠ a2 ◗◗◗◗ }④ ♠ ( o E ❉◗◗◗ ♠6 = F . ❉❉ ◗◗◗ a3 ♠♠♠♠③③ ❉❉ ◗◗◗◗ ♠♠ ③ ◗ ❉❉ ♠♠♠ ③③ ❉❉ ( E3 ♠ ③③③ ❉❉ ③ a4 ❉❉ ③③ ③ ! ③ E4


13

Write T⋆ as T⋆ = E2 ⊕ T⋆ . Similarly, we obtain that a2 : E2 → F is the minimal left add T⋆ -approximation in C and T⋆⋆

2 4 M M E(~ω + ~xj )) Ei ) ⊕ F ⊕ ( =E⊕( i=3

j=1

is a cluster tilting object, whose endomorphism algebra EndC (T⋆⋆ ) has the quiver

Q3 :

E(~ω + ~x1 ) a❈❈ ④ ❈❈ ④④ ❈❈ ④ ④ ④ ④ E(~ω + ~x2 ) ❈❈❈ ④ h◗◗◗ ❈❈ ④ ◗◗◗ ❈❈ ④④ ♠♠♠♠ ◗◗◗ ④④♠♠♠♠ }v♠ E ❉◗◗◗ ♠6 = F . ❉❉ ◗◗◗ a3 ♠♠♠♠③③ ◗ ♠ ❉❉ ◗◗◗ ♠ ③ ◗ ❉❉ ♠♠♠ ③③ ❉❉ ( E3 ♠ ③③③ ❉❉ ③ a4 ❉❉ ③③ ! ③③ E4

Write T⋆⋆ as F ⊕ T⋆⋆ . By Lemma 4.8, there exists a triangle in C (a3 ,a4 )

F [−1] → E(~x1 − ~x2 ) → E3 ⊕ E4 −−−−→ F, where (a3 , a4 ) : E3 ⊕ E4 → F is the minimal right add T⋆⋆ -approximation easily known from the above quiver. Hence Trk is a cluster tilting object in C whose endomorphism algebra EndC (Trk ) is given by Q4 :

/ E(~ω + ~x1 ) / E3 E lt■■ ■■ : ✉: ■■ ■ ✉ ■■ ✉✉✉✉ ■■ ✉✉✉ ■■ ■✉■✉ ✉ ✉✉■ ✉ ■ ✉✉ ■■■■ ✉✉ ■■■■ ✉ ✉ ✉w $ $ ✉✉ / E(~ω + ~x2 ). / E4 E(~x1 − ~x2 ) l

Notice that Trk ∈ D(α−1 (1), 1], applying Theorem 3.3 again completes the proof. Remark 4.11. In the proof, we obtain four cluster tilting objects Tcan , T⋆ , T⋆⋆ , Trk in C . As Example 4.4 indicated, the set of quivers of their endomorphism algebras is the mutation class of Q. Notice that the tilting graph of C is connected (see for instance [2]), hence for any cluster tilting object T in C , the quiver of the endomorphism algebra EndC (T ) belongs to list (4.1). 4.3. Further relationship between tilting and cluster tilting objects. In this subsection, we investigate further relationship between the tilting objects in D and the cluster tilting objects in C for X of weight type (2,2,2,2;λ). Namely, under the canonical functor π : D → C , tilting becomes cluster tilting. And on the other hand, we describe all the tilting objects corresponding to a given cluster tilting object. For any a ∈ Q, let Φa : D(a, α(a)] → coh X be the equivalence introduced in Lemma 2.5. The following lemma shows that Φa preserves the order of slopes. Lemma 4.12. For any indecomposable objects F1 , F2 ∈ D(a, α(a)], µF1 < µF2 if and only if µ(Φa (F1 )) < µ(Φa (F2 )). Consequently, µF1 = µF2 if and only if µ(Φa (F1 )) = µ(Φa (F2 )).

14


Proof. Notice that the rank of each tube in the Auslander-Reiten quiver of D or coh X is less than or equal to two. We know that F1 and F2 are in the same tube if and only if D(Fi , τ Fj ⊕ Fj ) 6= 0 for i 6= j, equivalently, Hom(Φa (Fi ), Φa (Fj ) ⊕ τ (Φa (Fj ))) 6= 0 for i 6= j, that is, Φa (F1 ) and Φa (F2 ) are in the same tube. For the necessity, firstly we claim that µF1 < µF2 implies µ(Φa (F1 )) ≤ µ(Φa (F2 )). Otherwise, by Riemann-Roch Formula (2.2), dimk Hom(Φa (F2 ), Φa (F1 ) ⊕ τ (Φa (F1 ))) = hΦa (F2 ), Φa (F1 ) ⊕ τ (Φa (F1 ))i = rk(Φa (F1 )) rk(Φa (F2 ))(µ(Φa (F1 )) − µ(Φa (F2 ))) > 0, Hence by Lemma 2.5, D(F2 , F1 ⊕ τ F1 ) 6= 0. It follows that µF1 ≥ µF2 , which is a contradiction. Secondly, we show that µ(Φa (F1 )) = µ(Φa (F2 )) implies µF1 = µF2 . There are two cases to consider: Case 1 : Φa (F1 ) and Φa (F2 ) are in the same tube. We have done. Case 2 : Φa (F1 ) and Φa (F2 ) are in different tubes. Without loss of generality, we suppose that µF1 < µF2 for contradiction. By the structure of the Auslander-Reiten quiver of D, there exists F3 ∈ D such that µF1 < µF3 < µF2 and D(F1 , F3 ) 6= 0. It follows that µ(Φa (F1 )) ≤ µ(Φa (F3 )) ≤ µ(Φa (F2 )), which implies µ(Φa (F1 )) = µ(Φa (F3 )). Moreover, since Hom(Φa (F1 ), Φa (F3 )) = D(F1 , F3 ) 6= 0, we obtain that Φa (F1 ) and Φa (F3 ) are in the same tube. It follows that F1 and F3 are in the same tube, which is a contradiction. We now prove the other hand. It suffices to prove that µF1 = µF2 implies µ(Φa (F1 )) = µ(Φa (F2 )), since we have shown above that µF1 > µF2 implies µ(Φa (F1 )) > µ(Φa (F2 )). For contradiction, we assume µ(Φa (F1 )) < µ(Φa (F2 )) without loss of generality. Then D(F1 , F2 ⊕ τ F2 ) = Hom(Φa (F1 ), Φa (F2 ) ⊕ τ (Φa (F2 ))), which is not zero by Riemann-Roch Formula (2.2). It follows that F1 and F2 , furthermore Φa (F1 ) and Φa (F2 ), are in the same tube, a contradiction to the assumption. The following result is crucial. Lemma 4.13. Let F1 , F2 be indecomposable objects in D(a, α(a)]. If µF1 < µF2 , then D(F1 , F2 ⊕ τ F2 ) 6= 0.


15

Proof. By Lemma 2.5, Lemma 4.12 and Riemann-Roch Formula (2.2), dimk D(F1 , F2 ⊕ τ F2 ) = dimk Hom(Φa (F1 ), Φa (F2 ) ⊕ τ (Φa (F2 ))) = hΦa (F1 ), Φa (F2 ) ⊕ τ (Φa (F2 ))i = rk(Φa (F1 )) rk(Φa (F2 ))(µ(Φa (F2 )) − µ(Φa (F1 ))) > 0. This finishes the proof.

4.3.1. Projecting a tilting object. Recall from [8] that each tilting object in D con6 L Ti be a tilting object sists of just six indecomposable direct summands. Let T = i=1

in D. Without loss of generality, we assume µTi ≤ µTi+1 , i = 1, · · · , 5. Lemma 4.14. For all the indecomposable direct summands Ti , the slope µTi is in the range a ≤ µTi ≤ α(a) for some a ∈ Q. Proof. In fact, we show that for any i, µTi ≤ µ(T1 [1]). Otherwise, there exists some i and n ≥ 2 such that µTi ≤ µ(T1 [n]) < µ(Ti [1]). If µTi 6= µ(T1 [n]), by Lemma 4.13, D(Ti , T1 [n] ⊕ τ (T1 [n])) 6= 0. But the fact that T is a tilting object in D implies that D(Ti , T1 [n]) = 0, and D(Ti , τ (T1 [n])) = DD(Ti [n − 1], Ti ) = 0. These give a contradiction. Thus (4.4)

µTi = µ(T1 [n]).

For those j < i with µTj < µ(T1 [1]), we claim that (4.5)

µTj = µT1 .

In fact, since µTi = µ(T1 [n]) > µ(Tj [n − 1]) ≥ µ(Tj [1]), by the similar arguments above, there exists some kj ≥ 2 such that µTi = µ(Tj [kj ]). Comparing with (4.4), we get µ(Tj ) = µ(T1 [n − kj ]). Then (4.5) follows from the assumption µT1 ≤ µTj < µ(T1 [1]). By means of (4.4) and (4.5), we have shown that the slope of each indecomposable direct summand of T has the form µ(T1 [m]) for some integer m. By [8] the exceptional objects in D lie in the bottom of tubes of rank two. The indecomposable direct summands of T are orthogonal to each other if they have the same slope. Combining with Lemma 2.3, we obtain D(Tj , Tk ) 6= 0 if and only if Tk = τ Tj [1]. It is not hard to see that the endomorphism algebra End(T ) is not connected, which is impossible. This finishes the proof. The next lemma shows that we can obtain a series of tilting objects from T .

16


Lemma 4.15. For 1 ≤ i ≤ 5, the object 6 i M M Tj ) GTj ) ⊕ ( ( j=1

j=i+1

is tilting in D. Proof. We only prove the object 6 M Ti ) T ′ = GT1 ⊕ ( i=2

is tiling in D, the others are similar. The following two equalities, D(GT1 , Ti [n]) = DD(Ti [n − 2], T1 ) = 0 for any n ∈ Z and D(Ti , GT1 [n]) = DD(T1 [n], Ti ) = 0 for any n 6= 0, imply that T is extension-free. Moreover, we know that T1 ∈ hT ′ i since ′

D(GT1 , T1 [2]) = DD(T1 , T1 ) 6= 0. ′

Hence D = hT i ⊆ hT i ⊆ D, which implies hT ′ i = D.

The following shows that each tilting object in D pushes to a cluster tilting object in C . Theorem 4.16. The image of T in C is a cluster tilting object. Proof. Lemma 4.14 implies that µT6 ≤ µ(T1 [1]). There are two cases to consider. Case 1 : µT6 < µ(T1 [1]). Then all the indecomposable direct summands are of slopes in the interval (µ(T6 [−1]), µ(T6 )]. Hence π(T ) is a cluster tilting object in C by Theorem 3.3. Case 2 : µT6 = µ(T1 [1]). Let i be the largest index satisfying µT1 = µTi . Lemma 4.15 implies that 6 i M M Tj ) GTj ) ⊕ ( T ′′ = ( j=1

j=i+1

is tilting in D. Clearly, the slope of each indecomposable direct summand is in the interval (µT1 , µ(T1 [1])]. Then by Theorem 3.3, π(T ′′ ) is a cluster tilting object in C . Note that T and T ′′ have the same image in C . We get what we want. 4.3.2. Lifting a cluster tilting. By Theorem 3.3, there is a one to one correspondence from the set of tilting objects in D(a, α(a)] to the set of cluster tilting objects in 6 L Ti is tilting C . Each cluster tilting object in C has the form π(T ), where T = i=1

in D. As before, we always assume µTi ≤ µTi+1 for 1 ≤ i ≤ 5. A lifting of π(T ) to D is an object X in D with π(X) = π(T ). Obviously, T is a lifting of π(T ), and any other lifting has the form 6 M i=1

Gki Ti , where ki ∈ Z.


Theorem 4.17. Let T ′ =

6 L

17

Gki Ti be a lifting of π(T ). Then T ′ is tilting in D if

i=1

and only if ki ≥ kj ≥ ki − 1 whence µTi < µTj . Proof. Assume T ′ is tilting in D and µTi < µTj . Then Lemma 4.13 implies that D(Ti , τ Tj ⊕ Tj ) 6= 0. Notice that D(Gki Ti , Gkj Tj [ki − kj ]) = D(Ti , τ ki −kj Tj ) and D(Gkj Tj , Gki Ti [kj − ki + 1]) = D(Tj , τ kj −ki Ti [1]) = DD(Ti , τ ki −kj +1 Tj ). Hence T ′ is extension-free implies that either ki − kj = 0 or kj − ki + 1 = 0, that is, ki ≥ kj ≥ ki − 1. Conversely, assume µTi < µTj implies ki ≥ kj ≥ ki − 1. Arrange the indecomposables Ti with same slope by ki to ensure k1 ≥ k2 ≥ · · · ≥ k6 ≥ k1 − 1. If k1 = k2 = · · · = k6 , then T ′ = Gk6 T is a tilting object in D. If else, there exists some 1 ≤ l ≤ 5, such that k1 = · · · = kl > kl+1 = · · · = k6 = k1 − 1. So T ′ = Gk6 (G(T1 ⊕ · · · ⊕ Tl ) ⊕ (Tl+1 ⊕ · · · ⊕ T6 )). By Lemma 4.15, T ′ is a tilting object in D. Corollary 4.18. Let T ′ =

6 L

Gki Ti be a lifting of π(T ). If T ′ is a tilting object

i=1

in D, then µ(Gki Ti ) ∈ (a, α(a)] for any i and some a ∈ Q if and only if ki = kj whence µTi = µTj . Proof. Assume that all the slopes µ(Gki Ti ) belong to (a, α(a)] and µTi = µTj . For contradiction, we assume ki > kj without loss of generality. Then µ(Gki Ti ) = µ(Ti [ki ]) ≥ µ(Tj [kj + 1]) = α(µ(Tj [kj ])) = α(µ(Gkj Tj )), which gives a contradiction. On the contrary, by Theorem 4.17, the tilting object T ′ has the form T ′ = Gk6 (G(T1 ⊕ · · · ⊕ Tl ) ⊕ (Tl+1 ⊕ · · · ⊕ T6 )) for some l. Since µTi = µTj implies ki = kj , we have µTl < µTl+1 . Hence for any 1 ≤ i ≤ 6, we have µ(Gki Ti ) ∈ [µ(Gk6 Tl+1 ), µ(Gk6 +1 Tl )] ⊆ (µ(Gk6 Tl ), µ(Gk6 +1 Tl )]. Set a = µ(Gk6 Tl ). We get what we want.

4.4. Endomorphism algebras of tilting complexes and Meltzer’s list. Recall that the stable category D is triangulated equivalent to the bounded derived category Db (coh X). In this subsection, we will provide complete classifications of endomorphism algebras of tilting complexes in Db (coh X) and tilting sheaves in 6 L Ti be a tilting object in D where Ti ∈ D(a, α(a)] for coh X. As before, let T = i=1

some a ∈ Q. Let ΓC be the quiver of the endomorphism algebra EndC (T ) and ΓD be that of EndD (T ) (in fact EndD(a,α(a)] (T )). Lemma 4.19. For any i 6= j, (1) D(Ti , Tj ) 6= 0 if and only if µTi < µTj ;

18


(2) C (Ti , Tj ) = 0 if and only if µTi = µTj . Proof. (1) By [8], the indecomposable direct summands of T lie in the bottom of tubes of rank two, and they are orthogonal to each other if they have the same slope. Hence D(Ti , Tj ) 6= 0 implies µTi < µTj . Conversely, by Lemma 4.13, µTi < µTj implies that D(Ti , Tj ⊕ τ Tj ) 6= 0. But D(Ti , τ Tj ) = DD(Tj , Ti [1]) = 0. Thus D(Ti , Tj ) 6= 0. (2) Let C (Ti , Tj ) = 0. For contradiction, we assume µTi < µTj without loss of generality. Then D(Ti , Tj ) 6= 0 by (1). It follows that C (Ti , Tj ) 6= 0, a contradiction. On the contrary, µTi = µTj implies that (Ti , Tj ) is an orthogonal pair [8]. Hence C (Ti , Tj ) = D(Ti , Tj ) ⊕ D(Ti , GTj ) = D(Ti , Tj ) ⊕ DD(Tj , Ti ) = 0. Let si : Ti → Tl,i be the minimal left add(T \Ti )-approximation of Ti in C . Lemma 4.20. If ΓC has no 2-cycles and there exist i 6= j satisfying (i) the approximations si and sj have the same target T ′ ; (ii) for each indecomposable direct summand Tm of T ′ , dim C (Ti , Tm ) = dim C (Tj , Tm ) = 1; then µTi = µTj . Proof. For contradiction, we assume µTi < µTj without loss of generality. Then Lemma 4.19(1) implies D(Ti , Tj ) 6= 0. So there exists a path ρ from Ti to Tj in ΓD and then in ΓC . By condition (i), the length of ρ is greater than one. Hence there exists at least one indecomposable direct summand Tm of T ′ , such that µTi < µTm < µTj . Furthermore, we claim that for any indecomposable summand Tn of T ′ , (4.6)

µTi < µTn < µTj .

In fact, if µTn ≥ µTj for some n, according to condition (ii), we get dim D(Ti , Tn ) = dim D(Tj , Tn ) = 1. Then by (i), the composition Ti → Tj → Tn vanishes, which induces an arrow from Tn to Ti in ΓC since C (T, T ) can be explained as a trivial-extension and then a relation-extension algebra of D(T, T ), cf. [21, 1]. Hence a 2-cycle between Ti and Tn appears in ΓC , a contradiction. If µTn ≤ µTi for some n, then D(Tn , Tm ) 6= 0 by Lemma 4.19(1). Moreover, according to condition (ii), dim D(G−1 Tj , Tn ) = dim D(G−1 Tj , Tm ) = 1. Similar arguments show that a 2-cycle between Tj and Tm appears in ΓC , which is a contradiction. Thus the claim (4.6) holds. It follows that C (Tj , T ′ ) = D(Tj , GT ′ ). Hence the approximation sj : Tj → T ′ in C lifts to a triangle in D sj

ε : Tj −→ GT ′ → Tj∗ → Tj [1]. Applying D(Ti , −) to ε, we obtain (4.7)

D(Ti , Tj∗ [−1]) 6= 0.


19

But ε induces a triangle in C : sj

ε : Tj −→ Tl → Tj∗ → Tj [1]. By Theorem 4.5, Tj∗ is a complement of the almost complete cluster tilting object T \Tj . Note that [2] = G2 in D. Therefore C (Ti , Tj∗ [−1]) = C (Ti , Tj∗ [1]) = 0, which gives a contradiction to (4.7). This finishes the proof.

Now we can give a classification of all the endomorphism algebras of tilting complexes in Db (coh X). For a totally different approach, see [18, Theorem 10.4.1]. Theorem 4.21. Let Σ be a finite dimensional k-algebra. Then Σ is an endomorphism algebra of a tilting complex in Db (coh X) if and only if Σ belongs to the following list.

List 1 Endomorphism algebras of tilting complexes of type (2, 2, 2, 2; λ)

algebra

B11

B12

A13

quiver

relations

1 ☎B ✿✿✿ ☎ ✿✿ b ☎ a1 ☎☎ 1 ☎ r9 2 ▲▲ ✿✿✿ ☎ ▲▲ ✿ r ☎ r ▲▲ ✿ ☎☎rrra2 ▲% b2 r 0 ✿▲▲ 95 r b3 rr ☎B ✿✿▲▲a▲3 ☎ r ✿✿ ▲▲% rrr ☎☎ ✿ ☎☎ a4 ✿✿ 3 ✿✿ ☎☎☎ b4 ☎ 4 1✿ ✿✿ ✿✿ b 2 ▲▲ ✿✿✿1 ▲▲ ✿ ▲▲ ✿ ▲% b2 c1 9 5 c2 +3 0′ b3 sss☎B ss ☎ sss ☎☎☎ 3 ☎☎ b ☎ 4 ☎☎ 4 2 ▲▲ ▲▲b2 ▲▲ b3 ▲% /9 5 3 b4 sss s s sss 4

b 3 a3 = b 2 a2 − b 1 a1 b4 a4 = b2 a2 − λb1 a1

c1 b 1 = 0 c2 b 2 = 0 (c2 − c1 )b3 = 0 (c2 − λc1 )b4 = 0

a1 c1 = 0 c1 c2

+3 0′

a1

/ 1′

c2 b 2 = 0 (c2 − c1 )b3 = 0 (c2 − λc1 )b4 = 0

20


A14

3 ❑❑ ❑❑❑b3 ❑❑❑ % 95 s s b4 s s s sss 4

′

81 rrr r r r +3 0′ r ▲▲▲ a ▲▲2▲ ▲& 2′ a1

c1 c2

′

81 rrr r rrra2 / ′ +3 0′ ▲ ▲▲▲a3 2 ▲▲▲ & ′ 3 a1

A15

4

B13

b4

5

/5

c1 c2

c1 c2

1′ ☎A ☎ ☎ a1 ☎☎ ′ ☎ ☎ rr8 2 ☎ r ☎☎rrra2 +3 0′ r ✿▲✿▲▲ a3 ✿✿ ▲▲▲ ✿✿ ▲& ′ a4 ✿✿ 3 ✿✿ 4′

A23

′′

A24

4

b4

/ 1′′

a1 c2 = 0 a2 (c2 − c1 ) = 0 a3 (c2 − λc1 ) = 0

b 4 a4 = b 3 a3 − b 2 a2 d1 (b3 a3 − λb2 a2 ) = 0

cb2 = 0 e1 d1 b3 = 0 (e1 d1 −

λ−1 λ c)b4

=0

cb3 = 0 a2

/ 2′

e1 d1 b4 = 0 a2 (e1 d1 −

′

1 ▲▲e1 2 s9 ▲▲▲ as2sss9 sss c s & s / 0′ ❑ /5 ❑❑a❑3 ❑% 3′ d1

c1 b 4 = 0

a4 (c2 − λc1 ) = 0

B22

3 ■■ b3 d1 9 1′′ ▲▲e1 ■■■ ▲▲▲ ss $ sss c /& 0 ′ 5 b4 tt9 ttt 4

(c2 − λc1 )b4 = 0

a3 (c2 − c1 ) = 0

′′ 2 ❑❑ 8 1 ▼▼▼ e ❑❑❑b2 d1 rrr ▼▼1▼ ❑❑ ▼& rrr b3 ❑% r c /5 / 0′ 3 9 s b4 ss s s sss 4

d1

(c2 − c1 )b3 = 0

a2 c2 = 0

B21

▲▲ ▲▲b2 ▲▲ b3 ▲% /9 5 b4 sss ss sss

a2 c2 = 0

a1 c1 = 0

s9 2 sss s s ss a3 / 0 ▲▲ 3 ▲▲a4 ▲▲ ▲% 4 a2

a1 c1 = 0

λ−1 λ c)

=0

a2 c = 0 a3 e1 d1 = 0 (e1 d1 −

λ−1 λ c)b4

=0


′′ ′ 8 1 ▼▼▼ e1 r8 2 a r 2 r ▼▼▼ r ▼& rrra3 c / 0′ ▲ / ′ ▲▲ a4 3 ▲▲▲ ▲& 4′

d1 rrr

B23

5

rr rr

′

r8 2 rrr r r a3 ′ / 0′ r ▲▲ a / 3 ▲▲▲4 ▲▲& 4′ a2

B24

1′′

e1

▲▲▲ b ▲▲2▲ b3 ▲& ′ 8/ 5 b4 rrr r rrr

B31

′′ 3 81 ss9 ▲▲▲▲b3 d1 qq s q s ▲▲ s ▲% qqqq sss 0 ▲▲ 9 5 ▼▼▼ b4 sss ▲▲a4 ▼d▼2▼ s ▲▲ s ▼& ▲% sss 4 2′′

B32

′′ 3 ❑❑ 81 ❑❑❑b3 d1 rrr ❑❑❑ rrr % r 9 5 ▲▲▲ b4 sss ▲d▲2▲ ss ▲& sss 4 2′′

21

a2 c = 0 a3 e1 d1 = 0 a4 (e1 d1 −

λ−1 λ c)

=0

b 4 a4 = b 3 a3 − b 2 a2 (b3 a3 − λb2 a2 )e1 = 0

a3

▼▼▼ e ▼▼1▼ ▼& ′ 80 q e2 qq q qqq

d1 (b4 a4 − b3 a3 ) = 0 d2 (b4 a4 − λb3 a3 ) = 0

(e2 d2 − e1 d1 )b3 = 0 (e2 d2 − λe1 d1 )b4 = 0

′′

1 s9 sss s /5❑ ❑❑❑d2 ❑% 2′′ d1

A33

B41

4

b4

a3

/ 3′

a3 (e2 d2 − e1 d1 ) = 0 (e2 d2 − λe1 d1 )b4 = 0

f1 a3 /3 / 1′′ 0❄ ❄❄ ⑧⑧? ❄❄❄ ⑧⑧? a4 ❄❄ ⑧⑧ g2 ❄❄ ⑧⑧ ⑧❄ ⑧❄ h3 ⑧⑧ ❄❄ g1 ⑧⑧ ❄❄ ❄ ⑧ ⑧ ❄ ⑧ h4 ⑧⑧ f2 ❄ /4 / 2′ 5′′

g 1 a4 = 0

f1 / 1′′ ▼ 9 3✿ s ▼▼▼k1 s ✿ ☎A s ☎ ✿ s ▼▼▼ s ☎ ✿ s ✿✿☎☎ g1 & ′ ss ☎ ✿ 0 ❑❑ 85 ☎ ✿✿g2 q ❑❑❑a4 ☎ q k 2 q q ✿ ❑❑❑ ☎☎☎ q qq f2 % / 2′ q 4

g 1 a4 = 0

a3

A42

▲▲▲e1 ▲▲& ′ e2 rr8 0 r r r

f 2 a4 = 0 f 1 h3 − g 1 h4 = 0 f2 h4 − λg2 h3 = 0

f 2 a4 = 0 k1 f1 − k2 g2 = 0 k1 g1 − λk2 f2 = 0

22


Proof. Assume Σ = EndDb (coh X) (Tc ) for some tilting complex Tc in Db (coh X). Since Db (coh X) = D by [16], we can view Tc as a tilting object in D and Σ = EndD (Tc ). By Theorem 4.16, π(Tc ) is a cluster tilting object in C . Hence the quiver Γ of the endomorphism algebra EndC (π(Tc )) belongs to list (4.1) according 6 L Ti , to Remark 4.11. By Theorem 3.3, we suppose π(Tc ) = π(T ) for some T = i=1

where Ti ∈ D(a, α(a)] for each i and some a ∈ Q. As before, we assume µTi ≤ µTi+1 for 1 ≤ i ≤ 5. If Γ = Q1 , then by Lemma 4.20 and 4.19, µT1 < µT2 = µT3 = µT4 = µT5 < µT6 . By Theorem 4.17, Tc has the form(under the equivalence Gk6 ) 6 M

5 M Gki Ti ) ⊕ T6 , Ti or GT1 ⊕ (

i=1

i=2

where ki = 0 or 1 for 2 ≤ i ≤ 5. For some choice of the representatives for the arrows, Σ is isomorphic to B11 , B12 , A13 , A14 , A15 or B13 in list 1. Similarly, one can prove that if Γ = Q2 , then Σ is isomorphic to B21 , B22 , A23 , A24 , B23 or B24 ; if Γ = Q3 , then Σ is isomorphic to B31 , B32 , or A33 ; if Γ = Q4 , then Σ is isomorphic to B41 or A42 in list 1. Conversely, using the method applied to prove Theorem 6.2 in [8], combining Theorem 4.17 and Theorem 4.10, in each case in list 2 one proves that the listed objects do form a tilting object in D having the corresponding algebras as endomorphism algebras. List 2 Realization of the endomorphism algebras of tilting complexes of type (2, 2, 2, 2; λ)

algebra

tilting complexes

B11

E1 ✄A ❀❀❀ ✄ ✄ ❀ ✄✄ E2 ❀❀❀ ✄ ▼ 8 ▼▼▼ ❀❀ ✄✄ qqq ▼▼▼❀ ✄q✄qqq & E ❀▼▼ 8 AF q q ▼ ❀❀ ▼▼ q ✄✄ ❀❀ ▼▼& qqq ✄ ❀❀ E q ✄✄✄ ❀❀ 3 ✄✄ ❀❀ ✄ ✄✄ E4

B12

E1 ❀ ❀❀ ❀❀ E2 ▼▼ ❀❀❀ ▼▼▼ ❀❀ ▼▼& 8F qq ✄A q ✄ q qqq ✄✄ E3 ✄✄✄ ✄✄ ✄✄ E4

+3 GE


A13

E2 ▼ ▼▼▼ ▼▼▼ /& 8 F E3 q qqq qqq E4

+3 GE

A14

E3 ▼▼ ▼▼▼ ▼▼& 8F qqq q q qq E4

GE1 ♥♥7 ♥ ♥ ♥♥♥ +3 GE PPP PPP P' GE2

/F

7 GE1 ♦♦♦ ♦ ♦ ♦♦ +3 GE / GE2 ❖❖❖ ❖❖❖ ❖' GE3

A15

B13

B21

B22

E4

F

/ GE1

GE1 ⑦> ⑦ ⑦⑦ ⑦⑦ 7 GE2 ⑦ ⑦⑦ ♥♥♥ ⑦♥⑦♥♥♥ +3 GE ❅❅PPP ❅❅ PPP ❅❅ P' ❅❅ GE3 ❅❅ ❅❅ GE4

8 E2 ▲▲ rr ▲▲ rr ▲▲ r ▲& rr / / E3 E ▲▲ r8 F ▲▲ rr r ▲▲ r ▲& rr E4

/ E(~ω + ~x1 )

E2 ▲ E(~ω + ~x1 ) ▲▲ ❘❘❘ ♠♠6 ▲▲ ❘❘❘ ♠ ♠ ♠ ▲▲ ♠ ❘❘( ♠ &/ ♠♠ / GE F E3 8 qq q q qqq E4

23

24


A23

E(~ω + ~x1 ) E3 ❑ ❑❑❑ ❙❙❙ 6 ❙❙) ❑% ❧❧❧❧❧❧ / GE F 9 r r r rr E4 E(~ω + ~x1 ) 6 ❙❙❙ ❧❧❧ ❙❙) ❧❧❧ /

/ GE2

GE2 ♦♦7 ♦♦♦ GE ❖❖ ❖❖❖ ' GE3

A24

E4

B23

E(~ω + ~x1 ) GE2 ❘❘❘ ♦♦7 ♠♠6 ♦ ❘ ♠ ♦ ❘ ♠ ❘❘❘ ♠♠ ♦♦♦ ♠♠♠ /( GE / GE3 F ❖❖❖ ❖❖❖ ❖' GE4

B24

B31

B32

A33

/F

GE2 ❖ ❖❖❖ ♦♦7 ♦ ❖❖❖ ♦ ♦ ♦ ♦ ' / GE / / GE3 7 GF ❖❖❖ ♦ ♦ ♦ ❖❖❖ ♦ ❖' ♦♦♦ GE4

E(~ ω + ~x1 )

E3 qq8 q q qqq E ▼▼ ▼▼▼ ▼▼ & E4

E3 ▼ ▼▼▼ ▼▼▼ & 8F q qqq qqq E4

E4

▼▼▼ ▼▼▼ ▼& F qq8 q q qqq

E(~ω + ~x1 ) ♠♠6 ♠ ♠ ♠ ♠♠♠ ◗◗◗ ◗◗◗ ◗◗( E(~ω + ~x2 )

E(~ω + ~x1 ) 6 ❘❘❘ ❘❘❘ ♠♠♠ ♠ ♠ ❘❘) ♠ ♠ ♠ ◗◗◗ ❧5 GE ◗◗◗ ❧❧❧ ❧ ◗◗( ❧ ❧❧ E(~ω + ~x2 )

E(~ω + ~x1 ) ❙❙❙ ❧6 ❧ ❧ ❙❙) ❧ ❧ /F ❧ GE ❘❘❘❘ ❦❦5 ❘❘( ❦❦❦ E(~ω + ~x2 )

/ GE3


B41

/ E3 / E(~ω + ~x1 ) E■ : ❍❍ ■■ ✈: ❍❍ ■■ ✉✉✉✉ ❍❍✈✈✈✈ ■■✉✉ ✈❍ ✉✉■■ ✈✈ ❍❍❍❍ ✉✉ ■■■■ ✈ ✉ ✈ $ ✉ ✈ $ / E4 / E(~ω + ~x2 ) E(~x1 − ~x2 )

A42

/ E(~ω + ~x1 ) 6 E3 ❈ ◗◗◗ ♠♠♠ ❈ ♠ ④= ♠ ❈ ◗◗◗ ♠ ❈❈ ④④④ ◗◗( ♠♠♠ ❈❈④④ ❈ ④❈ E(~x1 − ~x2 ) ④ ♠6 GE ◗◗◗ ④④ ❈❈❈ ♠♠♠ ◗◗◗ ♠ ④ ♠ ❈ ◗◗◗ ♠♠ ! ④④ ( / E(~ω + ~x2 ) E3

25

4.5. Endomorphism algebras of tilting sheaves in coh X. This subsection presents classification result for the endomorphism algebras of tilting sheaves in coh X. Let a be a rational number and Φa be the equivalence from the integral category D(a, α(a)] to coh X. Lemma 4.22. Let T be an object in D(a, α(a)]. Then T is tilting in D if and only if Φa (T ) is a tilting sheaf in coh X. Proof. If Φa (T ) is a tilting sheaf in coh X, then T is clearly tilting in D since D = Db (coh X). On the contrary, if T is a tilting object in D, then Ext1coh X (Φa (T ), Φa (T )) = Ext1D(a,α(a)] (T, T ) = D(T, T [1]) = 0. That is, Φa (T ) is extension-free in coh X. Moreover, for each object X ∈ coh X satisfying Ext1coh X (Φa (T ), X) = 0 = Homcoh X (Φa (T ), X), we have −1 D(T, Φ−1 a (X) ⊕ Φa (X)[1]) = 0,

then it follows D(T,

M

Φ−1 a (X)[n]) = 0.

n∈Z

So the assumption that T is tilting in D implies Φ−1 a (X) = 0, hence X = 0. This finishes the proof. Now we present detailed classification result. Theorem 4.23. Let Σ′ be a finite dimensional k-algebra. Then Σ′ is an endomorphism algebra of a tilting sheaf in coh X if and only if it is isomorphic to some Bij in list 1. Proof. Since coh X is equivalent to the interval category D(a, α(a)], Σ′ can be viewed as the endomorphism algebra of a tilting object in D, with indecomposable direct summands are of slopes in the interval (a, α(a)], which pushes to a cluster tilting object in C by Theorem 3.3. We use the quivers of the endomorphism algebras of cluster tilting objects in C to finish the classification. For a cluster tilting

26


object H in C , we assume H = π(

6 L

Ti ) with µTi ≤ µTi+1 for 1 ≤ i ≤ 5 as before.

i=1

If the quiver Γ of EndC (H) is Q1 , by Lemma 4.20 and 4.19, µT1 < µT2 = µT3 = µT4 = µT5 < µT6 . Then according to Corollary 4.18, a lifting of H in D has one of the following forms(under the equivalence Gk6 ): 6 M i=1

Ti ,

6 M Ti ), GT1 ⊕ ( i=2

5 M GTi ) ⊕ T6 . ( i=1

For some choice of the representatives for the arrows, we obtain that the endomorphism algebras Σ′ is isomorphic to B11 , B12 or B13 in list 1. Similarly, one can prove that if Γ = Q2 , then Σ′ is isomorphic to B21 , B22 , B23 or B24 ; if Γ = Q3 ,then Σ′ is isomorphic to B31 or B32 ; if Γ = Q4 , then Σ′ is isomorphic to B41 . On the other hand, each object corresponding to Bij in list 2 gives a tilting sheaf that we need. References 1. I. Assem, T. Br¨ ustle and R. Schiffler, Cluster-tilted algebras as trivial extensions. Bull. London Math. Soc., 40(1) (2008), 151-162. 2. M. Barot, D. Kussin and H. Lenzing, The cluster category of a canonical algebra. Trans. Amer. Math. Soc., 362 (2010), 4313–4330. 3. A. I. Bondal, Representations of associative algebras and coherent sheaves. Izv.Akad. Nauk SSSR Ser. Mat., 53(1)(1989), 25-44. 4. A. I. Bondal and M. M. Kapranov, Representations functors, Serre functors, and reconstructions. Izv.Akad. Nauk SSSR Ser. Mat., 53(6) (1989), 1183–1205. 5. A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups. Compos. Math. 145(4) (2009), 1035–1079. 6. A. B. Buan, R. J. Marsh, M. Reineke, I. Reiten and G. Todorov, Tilting theory and cluster combinatorics. Adv. Math., 204 (2006), 572–618. 7. A. B. Buan, R. J. Marsh, I. Reiten, Cluster mutation via quiver representations. Comm. Math. Helv., 83 (2008), 143–177. 8. J. Chen, Y. Lin, S. Ruan, Tilting objects in the stable category of vector bundles on a weighted projective line of type (2, 2, 2, 2; λ). J.Algebra., 397 (2014), 570–588. 9. S. Fomin and A. Zelevinsky, Cluster algebras I. Foundations. J. Amer. Math. Soc., 15(2) (2002), 497–529. 10. W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras. Singularities, representations of algebras, and Vector bundles, Springer Lect. Notes Math., 1273 (1987), 265–297. 11. D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 119 (1988). 12. O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules. Invent. Math. 172 (2008), 117–168. 13. B. Keller, On triangulated orbit categories. Doc. Math. 10 (2005), 551–581. 14. B. Keller, Cluster algebras, quiver representations and triangulated categories, in Tri- angulated Categories (edited by Holm, T; Jøgensen, P.; Rouquier, R.), London Math. Soc. Lecture Note Ser., 375(2010), Cambridge University Press, Cambridge. 15. B. Keller, I. Reiten, Acyclic Calabi-Yau categories. Compos. Math. 144 (5) (2008), 1332– 1348. 16. D. Kussin, H. Lenzing and H. Meltzer, Triangle singularities, ADE-chains and weighted projective lines. Advances in Mathematics, 237 (2013), 194-251. 17. H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra. In Representations of algebras, Sixth International Conference, Ottawa 1992. CMS Conf.Proc. 14 (1993), 313–337. 18. H. Meltzer, Exceptional Vector Bundles, Tilting Sheaves and Tilting Complexes for Weighted Projective Lines. Memoirs of the Amer. Math. Soc., 171(808) (2004).


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19. D. Orlov, Derived categories of coherent sheaves and triangulated categories of singularities. arXiv:math.AG/0503632v2. 20. I. Reiten, Cluster categories. To appear in Proceedings of ICM (2010). 21. B. Zhu, Equivalences between cluster categories. J.Algebra, 304 (2006), 832–850. School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China E-mail address: [email protected] School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China E-mail address: [email protected] Department of Mathematics, Southwest Jiaotong University, Chengdu 610031, P.R.China E-mail address: [email protected] School of Mathematical Sciences, Xiamen University, Xiamen 361005, P.R.China E-mail address: [email protected]