1. Introduction Recently, 2-Calabi-Yau triangulated categories with cluster titling objects have been studied. They play an important role in representation theory of algebras, for instance, categorification of cluster algebras. There exist two classes of 2-Calabi-Yau triangulated categories with cluster titling objects. One was introduced by Amiot [A] and the other was introduced by Buan-Iyama-Reiten-Scott [BIRS]. It was shown that the latter is included by the former [ART]. In this paper, we study the relationship of these categories in the context of derived categories. 2. Preliminary In this section, we define some notation. Throughout this section, let Q = (Q0 , Q1 , s, t) be a finite acyclic quiver. We denote by K an algebraically closed field and D = HomK (−, K). 2.1. Preprojective algebras and Coxeter groups. The double quiver Q = (Q0 , Q1 , s, t) of the quiver Q is defined by Q0 = Q0 , Q1 = Q1 t { α∗ : t(α) → s(α) | α ∈ Q1 }, where for an arrow α, we denote by s(α) the source of α, and by t(α) the target of α. The preprojective algebra Π of Q is definite by the following X Π := KQ/h αα∗ − α∗ αi. α∈Q1

The Coxeter group W = WQ of Q is the group generated by the set { su | u ∈ Q0 } with relations s2u = 1, su sv = su sv if there exist no arrows between u and v, and su sv su = sv su sv if there exists exactly one arrow between u and v. Definition 2.1. Let w ∈ WQ and w = su1 su2 · · · sul be an expression of w. (1) A subword of w is an expression w0 = sui1 sui2 · · · suim such that 1 ≤ i1 < i2 < · · · < im ≤ l. (2) An expression w of w is reduced if l is smallest possible. (3) Let w be a reduced expression of w, put Supp(w) := {u1 , u2 , . . . , ul } ⊂ Q0 . Note that, Supp(w) is independent of the choice of a reduced expression of w. Let u be a vertex of Q. We define a two-sided ideal Iu of Π by Iu := Π(1 − eu )Π. Let w be an element of W and w = su1 su2 · · · sul be a reduced expression of w. We define a two-sided ideal I(w) of Π by I(w) := Iu1 Iu2 · · · Iul . 1

2

Note that I(w) is independent of the choice of a reduced expression w of w by [BIRS, Theorem III. 1.9]. We define an algebra Π(w) by Π(w) := Π/I(w). For an algebra A, we denote by Sub A the full subcategory of mod A of submodules of finitely generated free A-modules. The following is the main result of [BIRS]. Proposition 2.2. [BIRS] For any w ∈ W , we have (a) The algebra Π(w) is finite dimensional and Iwanaga-Gorenstein of dimension at most one, that is, inj. dim Π(w) Π(w) ≤ 1 and inj. dim Π(w)Π(w) ≤ 1. (b) Sub Π(w) is a Frobenius category, and the stable category Sub Π(w) is a 2-CalabiYau triangulated category, that is, for any objects X, Y ∈ Sub Π(w) there exists a functorial isomorphism HomΠ(w) (X, Y ) ' D HomΠ(w) (Y, X[2]). L (c) For any reduced expression w = su1 su2 · · · sul of w, the object T = li=1 Π(su1 su2 · · · sui ) is a cluster tilting object of Sub Π(w), that is, add T = { X ∈ mod Π(w) | Ext1Π(w) (X, T ) = 0 }. 2.2. Grading on preprojective algebras. In this subsection, we introduce a Z-grading on preprojective algebras. We define a map deg : Q1 → {0, 1} as follows: for each β ∈ Q1 , let ( 1 β = α∗ , α ∈ Q1 deg(β) = 0 β = α, α ∈ Q1 . We the path algebra KQ as a graded algebra by this map deg. Since the element P regard (αα∗ − α∗ α) in KQ is homogeneous of degree 1, the grading of KQ naturally gives a α∈Q1 L Πi . grading on the preprojective algebra Π = i≥0

Remark 2.3. (a) We have Π0 = KQ, since Π0 is spanned by all paths of degree 0. (b) For any w ∈ W , the ideal I(w) of Π is a homogeneous ideal of Π since so is each I(u). (c) In particular, the factor algebra Π(w) is a graded algebra. L Let X = i∈Z Xi be a graded module over a graded algebra. For any integer j, we define the graded module X(j) by (X(j))i = Xi+j . For a graded algebra A, we denote by modZ A the category of the finitely generated graded A-modules with degree zero morphisms. Let SubZ A be the full subcategory of modZ A of submodules of graded free A-modules, that is, m M Z Z Sub A = X ∈ mod A | X is a submodule of A(ji ), m, ji ∈ Z, m ≥ 0 . i=1

Since Π(w) is Iwanaga-Gorenstein of dimension at most one, SubZ Π(w) is a Frobenius category. Then we have a triangulated category SubZ Π(w). 3. Main results In this section, we describe the main results of this paper. We first recall the definitions of silting and tilting objects of triangulated categories. Let T be a triangulated category. For an object X of T , we denote by thick X the smallest triangulated full subcategory of T containing X and closed under direct summands.

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Definition 3.1. Let T be a triangulated category. (1) An object X of T is called a silting object if HomT (X, X[i]) = 0 for any 0 < i and thick X = T . (2) An object X of T is called a tilting object if X is a silting object of T and HomT (X, X[i]) = 0 for any i < 0. The first result is the following theorem. Let w = su1 su2 · · · sul be a reduced expression of w ∈ W , and put M (w)i = M i = (Π/I(su1 su2 · · · sui ))eui ,

M (w) = M =

l M

M (w)i .

i=1

Theorem 3.2. Let w ∈ WQ . For any reduced expression w of w, the object M = M (w) is a silting object in SubZ Π(w). Next we give a sufficient condition on w such that the object M (w) is a tilting object of SubZ Πw . Let w = su1 su2 · · · sul be a reduced expression of w. For simplicity, assume that Supp(w) = Q0 . For any u ∈ Q0 , put pu = max{1 ≤ j ≤ l | uj = u}. We call an expression w c-ending if for any u, v ∈ Q0 , pu < pv holds whenever there exists an arrow from u to v in Q. Theorem 3.3. Let w ∈ WQ and w be a reduced expression of w. If the expression w is c-ending, then we have the following. (a) M (w) is a tilting object of SubZ Π(w). (b) The global dimension of A = EndZΠ(w) (M ) is at most two. (c) We have a triangle equivalence SubZ Π(w) ' Db (mod A). Note that the assertion (c) of Theorem 3.3 comes form the result of Keller [Ke94]. Let A be a finite dimensional algebra of global dimension at most two. A cluster category C(A) of A is the triangulated hull of the orbit category Db (mod A)/ − ⊗L A D A[−2] in the sense of Keller [Ke05]. We have the composition of triangle functors πA : Db (mod A) → Db (mod A)/S2 → C(A). Then we recall the result of Amiot-Reiten-Todorov [ART]. Proposition 3.4. [ART] Let w ∈ WQ and w be a reduced expression of w. Assume op b that w is c-ending. Put A = EndZΠ(w) (M ). Let N = M ⊗L A A ∈ D (mod(Π(w) ⊗ A )). Then there exists a triangle equivalence G : C(A) → Sub Π(w) which makes the following diagram commutative up to isomorphism of functors Db (mod A)

N ⊗L A−

/ Db (mod Π(w))

πA

C(A)

G

/ Sub Π(w).

Finally, we have the following theorem which is a graded version of Proposition 3.4.

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Theorem 3.5. Let w ∈ WQ and w be a reduced expression of w. Assume that w is cending. Then we have the following commutative diagram up to isomorphism of functors Db (mod A)

'

/ SubZ Π(w)

πA

C(A)

G

Forget

/ Sub Π(w).

We finish this paper by giving an example. 1 Example 3.6. (a) Let Q be a quiver w ' . Let w be an element of WQ which has / 2 3 a reduced expression w = s3 s2 s1 s3 s2 s3 . The expression w is c-ending. Then we have a graded algebra Π(w) = Π(w)e1 ⊕ Π(w)e2 ⊕ Π(w)e3 , where 2 3 1 3 1 1 2 2 3 Π(w)e3 = , Π(w)e1 = , Π(w)e2 = 1 2 3, 2 3 1 3 2 3 3 3 3 where numbers connected by solid lines are in the same degree, the tops of the Π(w)ei are concentrated in degree 0, and the degree zero parts are denoted by bold numbers. We have a tilting object M = M (w) in SubZ Π(w) as follows: 3 2 1 2. M =3⊕ ⊕ 3 2 3 3 The endomorphism algebra EndZΠw (M ) is given by the following quiver with relations b / a / • • , ab = 0. ∆= • References [A] C. Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 6, 2525-2590. [ART] C. Amiot, I. Reiten, G. Todorov, The ubiquity of the generalized cluster categories, Adv. Math. 226 (2011), no. 4, 3813-3849. [BIRS] A. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math. 145 (2009), no. 4, 1035-1079. ` [Ke94] B. Keller, Deriving DG categories, Ann. Sci. Ecole Norm. Sup. (4) 27 (1994), no. 1, 63-102. [Ke05] B. Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551-581. Graduate School of Mathematics, Nagoya University, Frocho, Chikusaku, Nagoya, 4648602, Japan E-mail address: [email protected]