Cluster-tilting objects in higher cluster categories

Abstract

We consider the existence of cluster-tilting objects in a d-cluster category such that its endomorphism algebra is self-injective, and also the properties for cluster-tilting objects in d-cluster categories. We get the following results: (1) When d>1, any almost complete cluster-tilting object in d-cluster category has only one complement. (2) Cluster-tilting objects in d-cluster categories are induced by tilting modules over some hereditary algebras. We also give a condition for a tilting module to induce a cluster-tilting object in a d-cluster category. (3) A 3-cluster category of finite type admits a cluster-tilting object if and only if its type is A_1,A_3,D_{2n-1}\left(n>2\right). (4) The (2m+1)-cluster category of type D_{2n-1} admits a cluster-tilting object such that its endomorphism algebra is self-injective, and its stable category is equivalent to the (4m+2)-cluster category of type A_{4mn-4m+2n-1}.

Graphical abstract

Keywords

Almost complete cluster-tilting object / Calabi−Yau triangulated category / cluster-tilting object / complement / d-cluster category

1 Introduction

In order to develop a set of combined methods to study and understand the totality of Iusztig in algebraic groups and the conclusions obtained in the model base of quantum groups, Fomin and Zelevinsky introduced a new class of algebras in the spring of 2000—cluster algebras [7, 8, 10]. Since then, it has been found that it is related to many branches of mathematics, such as Poisson geometry [11, 12], integrable system [9], combinatorial mathematics, especially the study of polyhedrons such as Stasheff combination [4, 6], and the algebraic representation theory mainly concerned in this paper [1−3, 14, 16−18, 21]. These connections have further promoted the development of cluster algebra.

The most important structures in cluster algebra are cluster and cluster variables. All cluster questions are linked by cluster transformations, algebraic representation and the connection of cluster algebra follow the idea of categorization: hope to find some suitable categories so that its combined invariants (such as K-theory) can not only characterize the structure of cluster variables, clusters, etc., and the relationship between these combination invariants correspond to the cluster transformation, but also the properties of cluster algebra studied through the rich structure of these categories. It turns out that when we associate the cluster algebra with the orbital category, which is derived category structure of the corresponding type of hereditary algebra, algebraic representation theory plays a great role in the study of cluster algebra.

Buan et al. was the first to get this category [1]. They introduced an orbital category of the derived category of hereditary algebra to categorize cluster algebra, so this category is called the cluster category. In short, the cluster category {\mathcal{C}}_H of the hereditary algebra H is defined as the F-orbital category {\mathcal{D}}^b\left(H\right)/F of the derived category {\mathcal{D}}^b\left(H\right), where F={\tau }^{-1}[1], \tau is Auslander-Reiten transformation (referred to as AR-transformation), [1] is the translation functor of the derived category. Keller [15] proves that the cluster category is a triangulated category. In this category, there is a very important category of objects: the cluster-tilting object, which corresponds to the cluster of the cluster algebra, the non-decomposable direct sum term corresponds to the cluster variable of the cluster algebra, and the mutation transformation of the cluster-tilting object corresponds to the mutation transformation of the cluster, so that the cluster category becomes a very successful category model.

Influenced by the combination theory, Fornin and Reading [6] generalized the concept of cluster complex river defined by Fomin-Zelevinsky, and introduced the d-cluster, where d is a natural number. When d=1, the original cluster is obtained. In terms of category, while proving that the cluster category is triangulated category, Keller [15] defined the category {\mathcal{C}}_d={\mathcal{D}}^b(H)/{\tau }^{-1}[d], and proved that {\mathcal{C}}_d is also a triangulated category. Soon afterwards Thomas [9] and Zhu [22] prove that there is a combination of the correspondence between the category {\mathcal{C}}_d and the d-cluster of Fomin-Reading, and this correspondence is the same as the correspondence between the cluster category and the cluster. Therefore, the category {\mathcal{C}}_d is called the higher cluster category or the d-cluster category [14, 16, 19].

We study the case where the existence of cluster-tilting object in the finite d-cluster category satisfies its endomorphism algebra as an self-injective algebra, and discuss the existence and properties of the cluster-tilting object in the d-cluster category. In fact, the nature of the cluster-tilting object in the d-cluster category is very different from that of in the 2-Callabi-Yau category (cluster category). The almost complete cluster-tilting object in the 2-Calabi-Yau category has two complements, and their question exists as a connected triangle [3, 14]. For the d-cluster category, we prove that when d>1, the almost complete cluster-tilting object of the d-cluster category has only one complement. Buan et al. [3] proved that the cluster-tilting objects in the cluster categories are induced by the tilting modules on the hereditary algebras, and the tilting modules can induce the cluster-tilting objects in the cluster categories. However, there is not necessarily a cluster-tilting object in the d-cluster category. We prove that the cluster-tilting object in the d-cluster category is induced by the tilting module on the hereditary algebra, and give a sufficient condition for the tilting module to induce the cluster-tilting object from the d-cluster category. For the finite d-cluster category, we obtain that there is cluster-tilting object in finite 3-cluster category if and only if the 3-cluster category is of type A_1,A_3,D_{2n-1}(n>2). In addition, the (2m+1)-cluster category of type D_{2n-1} exists in a cluster-tilting object to satisfy its endomorphism algebra as an self-injective algebra, and its stable category is equivalent to the d-cluster category of type A_{4mn-4m+2n-1}.

2 Preparatory knowledge

All the categories in this article are k-linear, in which the k is a field, and \mathcal{D} is the dual functor {\mathrm{H}\mathrm{o}\mathrm{m}}_k(-,k). The translation functors of all triangular categories is recorded as [1]. For any additive category \mathcal{A}, the set constructed nondecomposable objects (in an isomorphic sense) with \mathcal{A} are denoted as Ind\mathcal{A}. In particular, for any A\in \mathcal{A}, record Ind A as the set of all nondecomposable direct sum of A. Any A\in \mathcal{A}, record add A as the full subcategory of A, formed by the direct sum of the nondecomposable direct sum of the term of A.

First of all, let's review the definition of the cluster-tilting subcategory in the general triangulated category.

Definition 2.1 [14, 18]　Let \mathcal{C} be a triangulated category. If the subcategory \mathcal{T} of \mathcal{C} satisfies the following conditions:

(1) \mathcal{T} is a finite function,

(2) X\in \mathcal{T} if and only if for any T\in \mathcal{T},{\mathrm{E}\mathrm{x}\mathrm{t}}^1(X,T)=0,

(3) X\in \mathcal{T} if and only if for any T\in \mathcal{T},{\mathrm{E}\mathrm{x}\mathrm{t}}^1(T,X)=0.

Then \mathcal{T} is called the cluster-tilting subcategom (sometimes also called the tilting subcategom). If the cluster-tilting subcategom \mathcal{T} in \mathcal{C} is satisfied with the existence of T\in \mathcal{C} such that \mathcal{T}=\mathrm{a}\mathrm{d}\mathrm{d}T, then T is said to be the cluster-tilting object.

In the following, we agree that the nondecomposable direct sum of the cluster-tilting objects are isomorphic from each other.

Koenig and Zhu [18] proved that if \mathcal{T} is cluster-tilting subcategory in \mathcal{C}, then \mathcal{C}=\mathcal{T}[-1]\ast \mathcal{T}. That is, for any C\in \mathcal{C}, there is T_1,T_2\in \mathcal{T} that meets the existence of triangle T_1[-1]\to C\to T_2\to T_1.

Lemma 2.1 [18]　 Let \mathcal{C} be a triangulated category. If the subcategory \mathcal{T} of \mathcal{C} meets the following conditions: \mathcal{T} is limited inversion, and X\in \mathcal{T} if and only if for any T\in \mathcal{T},{\mathrm{E}\mathrm{x}\mathrm{t}}^1(T,X)=0, then \mathcal{T} is the cluster-tilting subcategory.

Let \mathcal{C} be the triangulated category, and \mathcal{T} is the cluster-tilting subcategory. The object of quotient category \mathcal{A}:=\mathcal{C}/\mathcal{T} is the same as \mathcal{C}, and the morphism from X to Y in the quotient category is a subgroup of morphism modes in \mathcal{C} decomposing along an object in \mathcal{T}.

Theorem 2.1 [16, 18]　(1) Let \mathcal{C} be the triangulated category, and \mathcal{T} is the cluster-tilting subcategory of \mathcal{C}. Then \mathcal{A}=\mathcal{C}/\mathcal{T} is the abelian category. In particular, if T is a cluster-tilting object of \mathcal{C}, then \text{mod}{\mathrm{E}\mathrm{n}\mathrm{d}}_{\mathcal{C}}(T)\simeq \mathcal{C}/\mathcal{T}.

(2) Let \mathcal{C} be a triangulated category with AR-triangle, and the AR-transformation of \mathcal{C} is denoted as \tau. If \mathcal{T} is the cluster-tilting subcategory of \mathcal{C}, then {\tau }^{-1}\mathcal{T}[1]=\mathcal{T}, and the abelian category \mathcal{A}=\mathcal{C}/\mathcal{T} also has the AR-sequence. In particular, \mathcal{A} is Frobenius category only and only if \mathcal{T}=\mathcal{T}[2].

Let H be the hereditary algebras, {\mathcal{D}}^b(H) for its bounded derived category. Note

F_d={\tau }^{-1}[d],\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\tau \mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\text{AR-transformation}\mathrm{o}\mathrm{f}{\mathcal{D}}^b(H).

d-cluster category {\mathcal{C}}_d defined as F_{d^{-}}orbital category of {\mathcal{D}}^b(H), i.e.,

{\mathcal{C}}_d={\mathcal{D}}^b(H)/F_d.

Theorem 2.2 [15]　 {\mathcal{C}}_d is the triangulated category, and the natural projection functor {\pi }_d:{\mathcal{D}}^b(H)\to {\mathcal{C}}_d is the triangulated functor.

So we also note that [1] is a translation functor for {\mathcal{C}}_d, where {\mathcal{C}}_1 is also called the cluster category.

Theorem 2.3 [1, 16, 22]　(1) {\mathcal{C}}_d is the Krull-Schmidt category.

(2) {\mathcal{C}}_d has AR-triangle, and is induced by the AR-triangle of {\mathcal{D}}^b(H). Remember that the AR-transform of {\mathcal{C}}_d is \tau.

(3) The nondecomposable object of {\mathcal{C}}_d (in the isomorphic sense) is

\text{Ind}{\mathcal{C}}_d=\bigcup _{i=0}^{d-1}(\mathrm{I}\mathrm{n}\mathrm{d}(\text{mod}H))[i]\cup \mathrm{I}\mathrm{n}\mathrm{d}H[d]\text{.}

(4) {\mathcal{C}}_d is a triangulated category (also called the (d+1)-Calabi-Yau category) with Calabi-Yau dimension d+1, i.e., for any M,N\in {\mathcal{C}}_d, there is a natural isomorphism:

D{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_d}(M,N)\simeq {\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_d}(N,M[d+1]).

If the d-cluster category {\mathcal{C}}_d only has a finite number of non-decomposable objects, it is said to be finite d-cluster category. From the above theorem, it can be seen that at this time H is also a finite hereditary algebras, so H corresponds to the arrow diagram Q for the Dynkin diagram. In this article, we always give the Dynkin plot the fixed arrow direction as follows:

3 Master conclusion and its proof

First, the properties of almost complete cluster-tilting objects in the d-cluster category are discussed.

Definition 3.1 [1]　Assuming that \mathcal{C} is the Krull-Schmidt triangulated category, if \overline{T}\in \mathcal{C} satisfies the existence of a nondecomposable object T_1\notin Ind \overline{T} such that \overline{T}⨿T_1 is a cluster-tilting object of \mathcal{C}, then \overline{T} is said to be an almost complete cluster-tilting object. At this time, T_1 is called the almost complete complement of the cluster-tilting object \overline{T}.

Similar to the property of an almost complete cluster-tilting object in the 2-Calabi-Yau triangulated category [1, 14], we have the following proposition.

Proposition 3.1　 Suppose \mathcal{C} is the n-Calabi-Yau triangulated category (n>2),\overline{L} is almost complete cluster-tilting object of \mathcal{C}. If L=\overline{L}⨿M is cluster-tilting object in \mathcal{C}, and M≆M[n-2], then in the isomorphic sense, \overline{L} exists only one complement M. In particular, if all nondecomposable objects X in \mathcal{C} satisfy {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(X,X[2])=0, then there is only one complement for all almost complete cluster-tilting objects in \mathcal{C}.

Proof　Because L=\overline{L}⨿M is \mathcal{C} in cluster-tilting objects, it is known by Theorem 2.1 {\tau }^{-1}L[1]\cong L. Since \mathcal{C} is the triangulated category of n-Calabi-Yau, \tau \simeq [n-1], thus L[1]\cong L[n-1], i.e., L\cong L[n-2].

If \overline{L} also exists complement N, note L^{\mathrm{\prime }}=\overline{L}⨿N, then L^{\mathrm{\prime }} is a cluster-tilting object, L^{\mathrm{\prime }}\cong L^{\mathrm{\prime }}[n-2]. Because M≆M[n-2]\in \mathrm{I}\mathrm{n}\mathrm{d}L[n-2], and \mathrm{I}\mathrm{n}\mathrm{d}\overline{L}[n-2]\cup M[n-2]=\mathrm{I}\mathrm{n}\mathrm{d}L[n-2]=\mathrm{I}\mathrm{n}\mathrm{d}L=\mathrm{I}\mathrm{n}\mathrm{d}\overline{L}\cup M, so M[n-2]\in \mathrm{I}\mathrm{n}\mathrm{d}\overline{L}, i.e., exists L_1\in \mathrm{I}\mathrm{n}\mathrm{d}\overline{L} makes M[n-2]\cong L_1. And because L^{\mathrm{\prime }} is also a cluster-tilting object, so L_1\in \mathrm{I}\mathrm{n}\mathrm{d}\overline{L}\subset \mathrm{I}\mathrm{n}\mathrm{d}{L}^{\mathrm{\prime }}=\mathrm{I}\mathrm{n}\mathrm{d}{L}^{\mathrm{\prime }}[n-2]=\mathrm{I}\mathrm{n}\mathrm{d}\overline{L}[n-2]\cup N[n-2]. If there is L_2\in \mathrm{I}\mathrm{n}\mathrm{d}\overline{L} makes L_1\cong L_2[n-2], then M[n-2]\cong L_1\cong L_2[n-2], thus M\cong L_2\in \mathrm{I}\mathrm{n}\mathrm{d}\overline{L}, contradict the fact that M is the complement of \overline{L}, so L_1\notin \mathrm{I}\mathrm{n}\mathrm{d}\overline{L}[n-2]. Then L_1\cong N[n-2], therefore M[n-2]\cong L_1\cong N[n-2], i.e., M\cong N, so \overline{L} has only one complement M.

If all nondecomposable objects M in \mathcal{C} satisfy {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[2])=0, then {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[n-2])\cong D{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M[-2],M)=0, thus M≆M[n-2]. As shown from the above, there is now only one complement for all almost complete cluster-tilting objects in \mathcal{C}.

Now prove the first main conclusion of this article.

Theorem 3.1　 Let \mathcal{C} be d-cluster category (d>1),\overline{L} for almost complete cluster-tilting objects in \mathcal{C}. Then in the isomorphic sense, \overline{L} exists only one complement M.

Proof　Let \mathcal{C}={\mathcal{D}}^b(H)/{\tau }^{-1}[d], where {\mathcal{D}}^b(H) is the bounded derived category of the hereditary algebra H. Then for any nondecomposable object M\in \mathcal{C}, there is {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[2])=\coprod _i{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(M,{\tau }^{-i}M[id+2]\right).

Assuming that M\in \text{mod}H, since H is a hereditary algebra, it can be seen that if there exists i\in \mathbb{Z} satisfies {\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(M,{\tau }^{-i}M[id+2]\right)\ne 0, then there exists N\in \text{mod}H such that {\tau }^{-i}M[id+2]\cong N or {\tau }^{-i}M[id+2]\cong N[1]. Easy to know i\ne 0.

Situation (1) i>0. Easy to know there exists n⩾0 makes {\tau }^{-i}M\in \text{mod}H[n]. Suppose {\tau }^{-i}M[id+2]\cong N\in \text{mod}H, then n+id+2=0, that is, , contradicts i>0,d>1. Suppose {\tau }^{-i}M[id+2]\cong N[1]\in \text{mod}H[1], then id+2+n=1, that is, id+1⩽0, also contradicts i>0,d>1.

Situation (2) . Easy to know there exists {\tau }^{-i}M\in \text{mod}H[n],n⩽0. If {\tau }^{-i}M[id+2]\cong N, then id+2⩾0. However, d⩾2, so there is only one solution to the inequality d=2,i=-1. If {\tau }^{-i}M[id+2]\cong N[1], then id+2⩾1. However, d⩾2, so the inequality has no solution.

In summary, for any M\in \mathrm{I}\mathrm{n}\mathrm{d}(\text{mod}H), if

{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(M,{\tau }^{-i}M[id+2]\right)\ne 0,

then d=2,i=-1.

We know that for the nondecomposable object M in {\mathcal{D}}^b(H), there is always an integer n, satisfying M[n]\in \text{mod}H, so similar to the above proof, if

{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(M,{\tau }^{-i}M[id+2]\right)\ne 0,

then d=2,i=-1.

Thus when d\ne 2, for any nondecomposable object M, there is

{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[2])=\coprod _i{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(M,{\tau }^{-i}M[id+2]\right)=0.

From Proposition 3.1 it follows that \overline{L} has only one complement.

When d=2, as can be seen from the discussion above,

{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[2])={\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}(M,\tau M)\cong D{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}(M[-1],M).

For any almost complete cluster-tilting object \overline{L} in \mathcal{C}, if the nondecomposable object M is a complement to \overline{L}, then \overline{L}\coprod M is the cluster-tilting object. It can be seen that {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[1])=0, thus {\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}(M,M[1])=0. At this time,

{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[2])={\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}(M,\tau M)\cong D{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}(M[-1],M)=0.

From Theorem 2.3, it is known that \mathcal{C} is the (d+1)-Calabi-Yau category,

{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[d+1-2])\cong D{\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(M,M[2])=0.

Thus M\ne M[d+1-2]. From Proposition 3.1, it follows that \overline{L} has the only complement M.

The connection between the cluster-tilting object in the d-category and the tilting module of the hereditary algebra will be considered below.

Definition 3.2 [1]　Let \mathcal{D} be the triangulated category. \mathcal{T} is a set of nondecomposable objects in \mathcal{D} that are not isomorphic from each other, if \mathcal{T} satisfies:

(1) for any X,Y\in \mathcal{T}, there {\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{D}}^1(X,Y)=0,

(2) for any Z\notin \mathcal{T}, all exist T\in \mathcal{T} making {\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{D}}^1(T,Z)\ne 0,

then we call \mathcal{T} an Ext-configuration.□

Now prove the second main conclusion of this article.

Theorem 3.2　 Let H be the hereditary algebra, {\mathcal{C}}_d(H) for its d-cluster category, {\pi }_d:{\mathcal{D}}^b(H)\to {\mathcal{C}}_d(H) for the natural projection functor. If T is a cluster-tilting object in the d-cluster category {\mathcal{C}}_d(H), then T must be induced by the tilt H^{\mathrm{\prime }}-modulo, where H^{\mathrm{\prime }} is hereditary algebra equivalent to H derivation (perhaps H itself), i.e., there is a tilt H^{\mathrm{\prime }}-modulo M, such that

\mathrm{I}\mathrm{n}\mathrm{d}T={\pi }_d\left(\bigcup _{i\in \mathbb{Z}}{\tau }^{-i}(\mathrm{I}\mathrm{n}\mathrm{d}M)[i]\right)\text{.}

If any tilting module M on the hereditary algebra H satisfies M[d-1]\in \mathrm{a}\mathrm{d}\mathrm{d}\left({⨿}_i{\tau }^{-i}M[i]\right), then M induces a cluster-tilting subcategory \mathcal{T} of {\mathcal{C}}_d(H).

Proof　Note \pi ={\pi }_1:{\mathcal{D}}^b(H)\to {\mathcal{C}}_1(H). If T is a cluster-tilting object in the d-cluster category {\mathcal{C}}_d(H), then \mathrm{I}\mathrm{n}\mathrm{d}T constitutes an Ext-combination of {\mathcal{C}}_d(H). The set \tilde{T} of preimages \mathrm{I}\mathrm{n}\mathrm{d}T in {\mathcal{D}}^b(H) is the Ext-combination in {\mathcal{D}}^b(H) from [1, Proposition 2.2], and \pi (\tilde{T}) is the Ext-combination in {\mathcal{C}}_1(H). From [1, Proposition 2.3] it follows that T_1=\coprod _{N\in \pi (\tilde{T})}N is the cluster-tilting object in the cluster category {\mathcal{C}}_1(H). In particular, by [1, Proposition 2.2], it follows that the preimage of \mathrm{I}\mathrm{n}\mathrm{d}{T}_1 under \pi is also the Ext-combination of {\mathcal{D}}^b(H), and contains \tilde{T}. As can be seen by the maximality of the Ext-combination, the preimage of \mathrm{I}\mathrm{n}\mathrm{d}{T}_1 under \pi is \tilde{T}. From [1, Theorem 3.3], it is known that there is a tilt H^{\mathrm{\prime }}-modulo M such that \pi (M)=T_1, where H^{\mathrm{\prime }} is some hereditary algebra equivalent to H derived (probably H itself). Thus, the preimage \left\{{\tau }^{-i}X[i]\mid X\in \mathrm{a}\mathrm{d}\mathrm{d}M,i\in \mathbb{Z}\right\} in {\mathcal{D}}^b(H) of \mathrm{I}\mathrm{n}\mathrm{d}\pi (M) is equivalent to \tilde{T}. Therefore,

\mathrm{I}\mathrm{n}\mathrm{d}T={\pi }_d(\tilde{T})={\pi }_d\left(\bigcup _{i\in \mathbb{Z}}{\tau }^{-i}(\mathrm{I}\mathrm{n}\mathrm{d}M)[i]\right).

The second conclusion is proven below. For any one of the tilting modulos M on the hereditary algebra H, let \mathcal{D} be the subcategory of {\mathcal{D}}^b(H) obtained by \tilde{T}= \left\{{\tau }^{-i}M[i]\mid i\in \mathbb{Z}\right\} with respect to direct sum terms and finite direct sum closure. M can naturally be seen as a cluster-tilting object in {\mathcal{C}}_1(H). From [1, Proposition 2.2], it follows that \tilde{T} is the Ext-combination in {\mathcal{D}}^b(H), so that \mathcal{D} satisfies that any X,Y\in \mathcal{D}. There is

{\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathcal{D}}^b(H)}^1(X,Y)=0,

which satisfies the third condition for the definition of the cluster-tilting subcategory. Note to arbitrary non-decomposable objects X\in {\mathcal{D}}^b(H), there are only a finite number of i\in \mathbb{Z} satisfying {\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left({\tau }^{-i}M[i],X\right)\ne 0, so \mathcal{D} is a subcategory of the finite inverse. From Lemma 2.1, it follows that \mathcal{D} is the cluster-tilting subcategory of {\mathcal{D}}^b(H).

Since M[d-1]\in \mathcal{D}, then \mathcal{D} is closed with respect to [d-1], i.e., for any X\in \mathcal{D}, there is X[d-1]\in \mathcal{D}. Let \mathcal{T}={\pi }_d(\mathcal{D}). For any X,Y\in \mathcal{T}, there is {\tau }^{-i}Y[i]\in \mathcal{D},i\in \mathbb{Z}. Thereby,

{\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathcal{C}}_d(H)}^1(X,Y)=\coprod _i{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(X,{\tau }^{-i}Y[id]\right)=\coprod _i{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}(X,({\tau }^{-i}Y[i])[i(d-1)]).

Since \mathcal{D} is closed about [d-1], {\tau }^{-i}Y[id]=\left({\tau }^{-i}Y[i]\right)[i(d-1)]\in \mathcal{D}. Since \mathcal{D} is the cluster-tilting subcategory of {\mathcal{D}}^b(H), it is known that {\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathcal{C}}_d(H)}^1(X,Y)=\coprod _i{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{D}}^b(H)}\left(X,{\tau }^{-i}Y[id]\right)=0. For any Z\notin \mathcal{T}, obviously Z\notin \mathcal{D}, thus exist T\in \mathcal{D} making {\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathcal{D}}^b(H)}^1(T,Z)\ne 0. From this, we can see that {\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathcal{C}}_d(H)}^1(T,Z)\ne 0, therefore \mathcal{T} satisfies X\in \mathcal{T} if and only if it is valid for any T\in \mathcal{T}, {\mathrm{E}\mathrm{x}\mathrm{t}}^1(T,X)=0. Since \mathcal{D} is the inverse finite, it can be seen that \mathcal{T} is also the inverse finite, and thus it follows from Lemma 2.1 that \mathcal{T} is the cluster-tilting subcategory of {\mathcal{C}}_d(H).

The finite d-cluster category has cluster-tilting objects in the following is discussed. According to Theorem 2.3, there are three types of finite d-cluster category: A_n(n⩾1),D_n(n⩾4) and E_n(n=6,7,8).

First, we give the necessary conditions for the existence of cluster-tilting objects in the finite d-cluster category.

Proposition 3.2　(a) Let {\mathcal{C}}_d\left(A_n\right) be the d-cluster category of type A_n, n⩾1. If (n+3,d-1)=1, then there is no cluster-tilting object in {\mathcal{C}}_d\left(A_n\right).

(b) Let {\mathcal{C}}_d\left(D_n\right) be the d-cluster category of type D_n, n⩾4. If (2n,d-1)=1, then there is no cluster-tilting object in {\mathcal{C}}_d\left(D_n\right).

(c) Let {\mathcal{C}}_d\left(E_n\right) be the d-cluster category of type E_n, n=6,7,8. If (14,d-1)=1, then there is no cluster-tilting object in {\mathcal{C}}_d\left(E_6\right); if (20,d-1)=1, then there is no cluster-tilting object in {\mathcal{C}}_d\left(E_7\right); if (32,d-1)=1, then there is no cluster-tilting object in {\mathcal{C}}_d\left(E_8\right).

Proof　(a) Let H be a hereditary algebra of type A_n. From (4) in Theorem 2.3, we know {\mathcal{C}}_d\left(A_n\right)={\mathcal{D}}^b(H)/\left({\tau }^{-1}[d]\right) is the (d+1)-Calabi-Yau triangulated category. If {\mathcal{C}}_d\left(A_n\right) has a cluster-tilting object T, then T[1]=\tau T=T[d]. Because the triangle in {\mathcal{C}}_d\left(A_n\right) is induced by the triangle in {\mathcal{D}}^b\left(A_n\right), in {\mathcal{C}}_d\left(A_n\right), it is known that [2]{\tau }^{n+1}=[nd+d+2]=id from [20, Proposition 3.3.2], thus T[nd+d+2]=T. Therefore T[(nd+d+2,d-1)]=T. And (nd+d+2,d-1)=(n+3,d-1)=1, T=T[1], which contradicts {\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_d\left(A_n\right)}(T,T[1])=0.

(b) Similar to provable, just note that for {\mathcal{C}}_d\left(D_n\right), there is [2]={\tau }^{-2n+2}.

(c) Similar to provable, just note that for {\mathcal{C}}_d\left(E_6\right), there is [2]={\tau }^{-12}; for {\mathcal{C}}_d\left(E_7\right), there is [2]={\tau }^{-18}; for {\mathcal{C}}_d\left(E_8\right), there is [2]={\tau }^{-30}.□

For the 3-Calabi-Yau triangulated category \mathcal{C}, we have \tau =[2]. If there is a cluster-tilting object T in \mathcal{C}, then {\tau }^{-1}T[1]=T, thus T[-1]=T. But since T is a cluster-tilting object, we know that {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{C}}(T[-1],T)=0, contradicts T[-1]=T. Therefore \mathcal{C} cannot exist for cluster-tilting objects. In particular, there can be no cluster-tilting object in the 2-cluster category.

Below we will describe the existence of cluster-tilting objects in the finite 3-cluster category, first of all we give an example.

Example 3.1　Let Q be an arrow diagram of type A_3, \mathcal{H}=D^b(kQ) and {\mathcal{C}}_3\left(A_3\right)=\mathcal{H}/{\tau }^{-1}[3]. Then {\mathcal{C}}_3\left(A_3\right) is a triangulated category, and its AR-arrow diagram is as follows:

Let \mathcal{T} be a subcategory \mathrm{a}\mathrm{d}\mathrm{d}(M), where M=\underset{i=1}{\overset{7}{⨁}}{T}_i,T_i is the non-decomposable object corresponding to the i position in Fig.1. Easy to verify that \mathcal{T} is a cluster-tilting subcategory. We can see that the quotient category of {\mathcal{C}}_3\left(A_3\right) of this cluster-tilting subcategory is equivalent to the modular category of the endomorphism algebra B=\mathrm{E}\mathrm{n}\mathrm{d}(M) of M from [16]. It is easy to know B\cong kQ/I, where Q is a directed circle with 7 vertices, and I is the ideal generated by a path of length 2. In this case, B is an self-injective algebra, and \text{mod}B is 7-Calabi-Yau.

In general, for any positive and even number d, we can always find the category d-Calabi-Yau \mathcal{C}, which has a cluster-tilting subcategory \mathcal{T}, satisfying \mathcal{A}=\mathcal{C}/\mathcal{T} is a Frobenius category. Actually, similar to the above construction, it can be verified that such a cluster-tilting object exists in the type (d-1)-cluster category of type A_3.

Lemma 3.1　 Suppose {\mathcal{C}}_3\left(A_n\right) is a 3-cluster category of type A_n. Then {\mathcal{C}}_3\left(A_n\right) has a cluster-tilting object if and only if n=1,3.

Proof　When n=1 is trivial.

When n=3, it is obtained from Example 3.1.

It follows from Proposition 3.2 that n must be odd. So we always assume that n is an odd number.

When n>3, if there is a cluster-tilting subcategory \mathcal{T} in {\mathcal{C}}_3\left(A_n\right), then {\tau }^{-1}\mathcal{T}[1]=\mathcal{T}. While {\mathcal{C}}_3\left(A_n\right)= {\mathcal{D}}^b(H)/{\tau }^{-1}[3], where H is a hereditary algebra of type A_n, so {\tau }^{-1}\mathcal{T}[3]=\mathcal{T}, thus \mathcal{T}=\mathcal{T}[2]. This can be known {\tau }^{-2}\mathcal{T}=\mathcal{T}, and {\tau }^{-1}\mathcal{T}=\mathcal{T}[-1]=\mathcal{T}[1]. For any non-decomposable object T\in \mathcal{T}, there is

{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_3\left(A_n\right)}\left(T,{\tau }^{-1}T\right)={\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_3\left(A_n\right)}(T,T[1])=0.

So T must fall on the upper or lower border of the AR- arrow diagram of {\mathcal{C}}_3\left(A_n\right).

In the AR-arrow diagram of {\mathcal{C}}_3\left(A_n\right) described in Fig.2, it is advisable to assume that the non-decomposable modulus corresponding to position a belongs to the cluster-tilting subcategory \mathcal{T}, then the nondecomposable modul corresponding to the ◼ position in the figure belongs to \mathcal{T}. Because n>3, the nondecomposable module M corresponding to the ⧫ position in the figure does not belong to \mathcal{T}[1]. However,

{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_3\left(A_n\right)}(\mathcal{T},M)=0.

This contradicts \mathcal{T} which is a cluster-tilting subcategory.□

Actually, we have the following corollary.

Corollary 3.1　 Let {\mathcal{C}}_{2d+1}\left(A_3\right)(d⩾0) is (2d+1)-cluster category of type A_3. Then there is a cluster-tilting object M in {\mathcal{C}}_{2d+1}\left(A_3\right) satisfying M\cong M[2].

Proof　The AR-arrow diagram for {\mathcal{C}}_{2d+1}\left(A_3\right) is as follows:

Easy to know that the two boundaries of the AR-arrow diagram have 4d+3 vertices each. Let M=\underset{i=1}{\overset{4d+3}{⨁}}{T}_i,T_i is the non-decomposable object corresponding to the i position in Fig.3. Easy to verify that M is a cluster-tilting object. In this case, T_i[2]\cong T_{i+4}, where we equate 4d+4 to 1, thus M\cong M[2].

Corollary 3.2　 Let A=kQ/I, where Q is a directed circle of 4d+3 vertices, and I is the ideal generated by a path of length 2. Then A is a self-injective algebra, and the stable category \underset{\_}{\mathrm{m}\mathrm{o}\mathrm{d}}A is (4d+3)-Calabi-Yau.

Proof　Let M be the cluster-tilting object of {\mathcal{C}}_{2d+1}\left(A_3\right) in Corollary 3.1. Then it is easy to know that the endomorphism algebra of M is isomorphic to A. From Theorem 2.1, it is known that A is an self-injective algebra. By [5, Theorem 2.1], it can be known \underset{\_}{\text{mod}}A\simeq \underset{\_}{\text{mod}}\left({\mathrm{E}\mathrm{n}\mathrm{d}}_{{\mathcal{C}}_{2d+1}\left(A_3\right)}(M)\right) is (4d+3)-Calabi-Yau.

Before discussing the case where there are cluster-tilting objects in the 3-cluster category of type D_n, let us prove a more general conclusion.

Proposition 3.3　Let {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) be (2m+1)-cluster category of type D_{2n-1}. Then there is a cluster-tilting object T in {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) satisfying T=T[2].

Proof　First, we give the AR-arrow diagram (Fig.4) of {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right).

Let's take T=\coprod _i{T}_i, where T_i is the non-decomposable object corresponding to the ◼ position in the figure above. Easy to verify T is a rigid object in {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right), i.e., \mathrm{H}\mathrm{o}\mathrm{m}(T,T[1])=0. Actually, T[1]=\coprod _i{R}_i, where R_i is the non-decomposable object corresponding to the ★ position in the figure above, where \tau T={\tau }^{-1}T=\coprod _i{R}_i=T[1], specifically, T[2]=T. For any non-decomposable object M, we have {\mathrm{E}\mathrm{x}\mathrm{t}}^1(M,T)\cong D\mathrm{H}\mathrm{o}\mathrm{m}\left({\tau }^{-1}T,M\right)=D\mathrm{H}\mathrm{o}\mathrm{m}\left(\coprod _i{R}_i,M\right). If the non-decomposable object M\notin \mathrm{a}\mathrm{d}\mathrm{d}T, in the first case, if M is located at position A in the figure, then \mathrm{H}\mathrm{o}\mathrm{m}\left(R_A,M\right)\ne 0, where R_A\in \mathrm{I}\mathrm{n}\mathrm{d}T[1]=\mathrm{I}\mathrm{n}\mathrm{d}T[-1] is located in the figure; in the second case, if M is located at position B in the figure, then \mathrm{H}\mathrm{o}\mathrm{m}\left(R_B,M\right)\ne 0, where R_B\in \mathrm{I}\mathrm{n}\mathrm{d}T[1]=\mathrm{I}\mathrm{n}\mathrm{d}T[-1] is located in the figure. Thus, any M\notin \mathrm{a}\mathrm{d}\mathrm{d}T has {\mathrm{E}\mathrm{x}\mathrm{t}}^1(T,M)\ne 0. Similarly, we can show that for any non-decomposable object N, if N\notin \mathrm{a}\mathrm{d}\mathrm{d}T has {\mathrm{E}\mathrm{x}\mathrm{t}}^1(N,T)\ne 0. In summary, T is the cluster-tilting object of {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right), and T=T[2].□

Note 3.1　The number of nondecomposable direct sum terms of cluster-tilting objects in {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) constructed by Proposition 3.3 is 4mn-4m+2n-1.

Proof　Under the fixed orientation of D_{2n-1}, the lower boundary of the AR-arrow diagram of \text{mod}\left(k{D}_{2n-1}\right)[2i] has 2n-1 vertices, and the lower boundary of \text{mod}\left(k{D}_{2n-1}\right)[2i+1] has 2n-3 vertices. Since

\text{Ind}{\mathcal{C}}_{2m+1}\left(D_{2n-1}\right)=\underset{i=0}{\overset{2m}{⨆}}\mathrm{I}\mathrm{n}\mathrm{d}\left(\text{mod}\left(k{D}_{2n-1}\right)\right)[i]\bigsqcup \mathrm{I}\mathrm{n}\mathrm{d}\left(k{D}_{2n-1}\right)[2m+1],

the lower bound of {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) has (2n-1+2n-3)m+2n-1+1=(4n-4)m+2n vertices. It is easy to know that the number of non-decomposable direct sum terms of T at the lower boundary is (2n-2)m+n. Similarly, it can be seen that the number of nondecomposable direct sum terms of T in the upper layer is also (2n-2)m+n, but the nondecomposable object corresponding to position a appears in both layers. In that case, the number of nondecomposable direct sum terms of T is 4mn-4m+2n-1.

Corollary 3.3　 Let {\mathcal{C}}_3\left(D_n\right) is 3-cluster category of type D_n. Then there is a cluster-tilting object in {\mathcal{C}}_3\left(D_n\right) if and only if n is odd. If T is a cluster-tilting object of {\mathcal{C}}_3\left(D_n\right), then {\mathcal{C}}_3\left(D_n\right) has only two cluster-tilting objects T,T[1].

Proof　From Proposition 3.2 it is known that n⩾5 is odd. The proof of any cluster-tilting object R in {\mathcal{C}}_3\left(D_n\right), a proof similar to Lemma 3.1 shows that R=R[2], and \mathrm{H}\mathrm{o}\mathrm{m}(R,\tau R)=D\mathrm{H}\mathrm{o}\mathrm{m}(R,R[1])=0, so the non-decomposable direct sum term of R can only fall on two boundaries. As long as R has a non-decomposable direct sum term R_1 falling on the lower boundary, then {\tau }^{2}R=R can be known from {\tau }^{-1}R[1]=R. While repeating {\tau }^2 on R_1, it is easy to see that if R_1\in \mathrm{a}\mathrm{d}\mathrm{d}T, then R is the cluster-tilting object T constructed in Proposition 3.3. If R_1\in \mathrm{a}\mathrm{d}\mathrm{d}T[1], then T[1]. If all non-decomposable direct sum terms of R fall on the upper boundary, it is known that R for some nondecomposable object R_2 obtained by repeated action of {\tau }^2 from {\tau }^{2}R=R. A proof similar to Note 3.1 shows that the upper boundary has 4mn-4m+2n-2 vertices with an even number, so that all non-decomposable objects of the upper boundary are direct sum terms of R. Here, R=\tau R=R[1], which contradicts with R as a rigid body object. So {\mathcal{C}}_3\left(D_n\right) has only two cluster-tilting objects T,T[1].

Below we prove that there is no cluster-tilting object in {\mathcal{C}}_3\left(E_n\right).

Lemma 3.2　 Let {\mathcal{C}}_3\left(E_n\right) is 3-cluster category of type E_n(n=6,7,8). Then there is no cluster-tilting object in {\mathcal{C}}_3\left(E_n\right).

Proof　If T is a cluster-tilting object in the 3-cluster category of type E_n, a proof similar to Corollary 3.3 shows that {\tau }^{2}T=T, and the non-decomposable direct sum term of T can only fall on the upper and lower boundaries of the AR-arrow diagram of {\mathcal{C}}_3\left(E_n\right). Similar to the discussion of type A_n in Lemma 3.1, it is known that there is no cluster-tilting object in {\mathcal{C}}_3\left(E_n\right), and the detailed proof is omitted here.

In summary, we have proved the third main conclusion of this article.

Theorem 3.3　 A finite 3-cluster category exists for cluster-tilting objects if and only if the 3-cluster category is of type A_1,A_3,D_{2n-1}(n>2).

Proof　It can be immediately available from Lemma 3.1, Corollary 3.3 and Lemma 3.2.

Below we will prove that from {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right), we can get {\mathcal{C}}_{4m+2}\left(A_{2n-3}\right), where n⩾2,m⩾0, when n=2, D_3=A_3.

First, review one of the results of Holm-Jørgensen.

For integers n⩾1,N⩾1, define the Nakayama algebra B_{N,n+1} for the path algebra generated by the directional ring {\tilde{A}}_N with N vertices, and the quotient algebra derived from the ideal generated by the path generation of length n+1. In particular, this is a self-injective algebra.

Theorem 3.4 [13]　 Let u⩾2 be even, n⩾1, and

N=\frac{u}{2}(n+1)+1.

Then A_n type u-cluster category is equivalent to the stable category \text{mod}{B}_{N,n+1} of B_{N,n+1}.

In this way, we can get the fourth main conclusion of this article.

Theorem 3.5　 Let {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) be (2m+1)-cluster category of type D_{2n-1}, where n⩾2,m⩾0. Then there is a cluster-tilting object T in {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) and satisfying an endomorphism algebra T, is a self-injective, and its stable category is equivalent to the type A_{2n-3} (4m+2)-cluster category.

Proof　We will prove that the cluster-tilting objects in Corollary 3.1 and Proposition 3.3 of the proof {\mathcal{C}}_{2m+1}\left(D_{2n-1}\right) satisfy the narrative in the theorem. Let D_{2n-1} be the arrow diagram:

Note 3.1 shows that the number of non-decomposable direct sum terms of T is 4mn-4m+2n-1. An endomorphism algebra of T is a path-algebraic modulus with an orientation circle composed of 4mn- 4m+2n-1 vertices to some ideal I, which is a Nakayama algebra. From T=T[2], it follows that an endomorphism algebra of T is a self-injective algebra, and thus I is the ideal generated for paths of equal length.

Note S_i,P_i,I_i for single-mode, projective mode, and incident mode corresponding to vertex i, respectively. In Fig.4, let position a correspond to the non-decomposable object as S_1, and denoted as T_1. Let T_{2i+1}:={\tau }^{-2i}{S}_1,1⩽i⩽2mn-2m+n-1. At this time, P_2 is the direct sum of T, corresponding to the ◼ on the far left of the second layer. Note T_2:=P_2, and let T_{2i+2}={\tau }^{-2i}{P}_2,1⩽i⩽2mn-2m+n-2. Form the AR-arrow diagram of k{D}_{2n-1}, it can be seen that {\tau }^{-2n+1}{S}_1\cong P_2[1], thus {\tau }^{-2n+2}{S}_1\cong I_2=S_2, that is, T_{2n-1}=S_2. So

{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_{2m+1}\left(D_{2n-1}\right)}\left(T_1,T_{2n-1}\right)={\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_{2m+1}\left(D_{2n-1}\right)}\left(S_1,S_2\right)=0.

Secondly, {\tau }^{-2n+3}{P}_2\cong S_1[1], thus {\tau }^{-2n+4}{P}_2\cong I_1, that is, T_{2n-2}=I_1. So

{\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_{2m+1}\left(D_{2n-1}\right)}\left(T_1,T_{2n-2}\right)={\mathrm{H}\mathrm{o}\mathrm{m}}_{{\mathcal{C}}_{2m+1}\left(D_{2n-1}\right)}\left(S_1,I_1\right)\ne 0\text{.}

Therefore, we find a road with a length of 2n-2 that belongs to I, and a true path that does not belong to I. I is generated by all paths of length 2n-2, thus \mathrm{E}\mathrm{n}\mathrm{d}(T)\cong B_{4mn-4m+2n-1,2n-2}. From Theorem 3.4, it is known that the stable category of an endomorphism algebra of T is equivalent to the A_{2n-3} type (4m+2)-cluster category.□

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