CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES

IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN

Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfies τ^{2} = id. In particular, we compute 2-CY tilted algebras for simple/minimally elliptic curve singuralities.

Introduction

Motivated by the Fomin-Zelevinsky theory of cluster algebras [FZ1, FZ2, FZ3], a tilting theory in cluster categories was initiated in [BMRRT]. For a finite dimensional hereditary algebra H over a field k, the associated cluster category C_{H} is the orbit category D^{b}(H)/F, where D^{b}(H) is the bounded derived category of finite dimensional H-modules and the functor F: D^{b}(H) → D^{b}(H) is τ^{−1}[1] = S^{−1}[2]. Here τ denotes the translation associated with almost split sequences/triangles and S the Serre functor on D^{b}(H) [BK]. (See [CCS] for an independent definition of the cluster category when H is of Dynkin type A_{n}).

An object T in a cluster category C_{H} was defined to be a (cluster) tilting object if Ext_{C_{H}}^{1} (T, T) = 0, and if Ext_{C_{H}}^{1} (X, X ∐ T) = 0, then X is in add T. The corresponding endomorphism algebras, called cluster tilted algebras, were investigated in [BMR1] and subsequent papers. A useful additional property of a cluster tilting object was that even the weaker condition Ext_{C_{H}}^{1} (X, T) = 0 implies that X is in add T, called Ext-configuration in [BMRRT]. Such a property also appears naturally in the work of the second author on a higher theory of almost split sequences in module categories [I1, I2] and the corresponding modules were called maximal 1-orthogonal. For a category mod Λ of finite dimensional modules over a preprojective algebra of Dynkin type Λ over an algebraically closed field k, the property corresponding to the above definition of cluster tilting object in a cluster category was called maximal rigid [GLSc]. Also in this setting it was shown that being maximal 1-orthogonal was a consequence of being maximal rigid. The same result holds for the stable category \underline{mod}Λ.

The categories C_{H} and \underline{mod}Λ are both triangulated categories [Ke, H], with finite dimensional homomorphism spaces, and they have Calabi-Yau dimension 2 (2-CY for short) (see [BMRRT, Ke];[AR, 3.1,1.2][C][Ke, 8.5]). The last fact means that there is a Serre functor S = Σ^{2}, where Σ is the shift functor in the triangulated category.

For an arbitrary 2-CY triangulated category C with finite dimensional homomorphism spaces over a field k, a cluster tilting object T in C was defined to be an object satisfying the stronger property discussed above, corresponding to the property of being maximal 1-orthogonal/Ext-configuration [KR]. The corresponding class of algebras, containing the cluster tilted ones, have been called 2-CY tilted. With this concept many results have been generalised from cluster categories, and from the stable categories \underline{mod}Λ, to this more general setting in [KR], which moreover contains several results which are new also in the first two cases.

One of the important applications of classical tilting theory has been the construction of derived equivalences: Given a tilting bundle T on a smooth projective variety X, the total right derived functor of Hom(T, ) is an equivalence from the bounded derived category of coherent sheaves on X to the bounded derived category of finite dimensional modules over the endomorphism algebra of T. Analogously, cluster tilting theory allows one to establish equivalences between very large factor categories appearing in the local situation of Cohen-Macaulay modules and categories of modules over finite-dimensional algebras. Namely, if \underline{CM}(R) is the stable category of maximal Cohen-Macaulay modules over an isolated hypersurface singularity, then \underline{CM}(R) is 2-CY. If it contains a cluster tilting object T, then the functor Hom(T, ) induces an equivalence between the quotient of \underline{CM}(R) by the ideal of morphisms factoring through τT and the category of finite-dimensional modules over the endomorphism algebra B = End(T). It is then not hard to see that B is symmetric and the indecomposable nonprojective B-modules are τ-periodic of τ-period at most 2. In this article, we study examples of this setup arising from finite, tame and wild CM-type isolated hypersurface singularities R. The endomorphism algebras of the cluster tilting objects in the tame case occur in lists in [BS, Er, Sk]. We also obtain a large class of symmetric finite dimensional algebras where the stable AR-quiver consists only of tubes of rank one or two. Examples of selfinjective algebras whose stable AR-quiver consists only of tubes of rank one or three were known previously [AR].

The first author was supported by the DFG project Bu 1866/1-1, the last author by a Storforsk grant from the Norwegian Research Council.

In the process we investigate the relationship between cluster tilting and maximal rigid objects. It is of interest to know if the first property implies the second one in general. In this paper we provide interesting examples where this is not the case. The setting we deal with are the simple isolated hypersurface singularities R in dimension one over an algebraically closed field k, with the stable category \underline{CM}(R) of maximal Cohen-Macaulay R-modules being our 2-CY category. These singularities are indexed by the Dynkin diagrams, and in the cases D_{n} for odd n and E_{7} we give examples of maximal rigid objects which are not cluster tilting.

We also investigate the other Dynkin diagrams, and it is interesting to notice that there are also cases with no nonzero rigid objects (A_{n}, n even, E_{6}, E_{8}), and cases where the maximal rigid objects coincide with the cluster tilting objects (A_{n}, n odd and D_{n}, n even). In the last case we see that both loops and 2-cycles can occur for the associated 2-CY tilted algebras, whereas this never happens for the cases C and \underline{mod}Λ [BMRRT, BMR2, GLSc]. The results are also valid for any odd dimensional simple hypersurface singularity, since the stable categories of Cohen-Macaulay modules are all triangle equivalent (see [Kn, So]).

We shall construct a large class of one-dimensional hypersurface singularities R having a cluster tilting object including examples coming from simple singularities and minimally elliptic singularities. We classify all rigid object in CM(R) for these R, in particular, we give a bijection between cluster tilting objects in CM(R) and elements in a symmetric group. Our method is based on a higher theory of almost split sequences [I1, I2], and a crucial role is played by the endomorphism algebras End_{R}(T) (called ‘three dimensional Auslander algebras’) of cluster tilting objects T in CM(R). These algebras have global dimension three, and have 2-CY tilted algebras as stable factors. The functor Hom_{R}(T, ) : CM(R) → mod End_{R}(T) sends cluster tilting objects in CM(R) to tilting modules over End_{R}(T). By comparing cluster mutations in CM(R) and tilting mutation over End_{R}(T), we can apply results on tilting mutation due to Riedtmann-Schofield [RS] to get information on cluster tilting objects in CM(R).

We focus on the interplay between cluster tilting theory and birational geometry. In [V1, V2], Van den Bergh established a relationship between crepant resolutions of singularities and certain algebras called non-commutative crepant resolutions via derived equivalence. It is known that three dimensional Auslander algebras of cluster tilting objects of three dimensional normal Gorenstein singularities are 3-CY in the sense that the bounded derived category of finite length modules is 3-CY, and form a class of non-commutative crepant resolution [I2, IR]. Thus we have a connection between cluster tilting theory and birational geometry. We translate Katz’s criterion [Kat] for three dimensional cA_{n}-singularities on existence of crepant resolutions to a criterion for one-dimensional hypersurface singularities on existence of cluster tilting objects. Consequently the class of hypersurface singularities, which are shown to have cluster tilting objects by using higher almost split sequences, are exactly the class having cluster tilting objects. However we do not know whether the number of cluster tilting objects has a meaning in birational geometry.

In section 2 we investigate maximal rigid objects and cluster tilting objects in \underline{CM}(R) for simple one-dimensional hypersurface singularities. We decide whether extension spaces are zero or not by using covering techniques. In section 3 we point out that we could also use the computer program Singular [GP] to accomplish the same thing. In section 4 we construct a cluster tilting objects for a large class of isolated hypersurface singularities, where the associated 2-CY tilted algebras can be of finite, tame or wild representation type. We also give a formula for the number of cluster tilting and indecomposable rigid objects. In section 5 we establish a connection between existence of cluster tilting objects and existence of small resolutions. In section 6 we give a geometric approach to the results in section 4. Section 7 is devoted to computing some concrete examples of 2-CY tilted algebras. In section 8 we generalize results from section 2 to 2-CY triangulated categories with only a finite number of indecomposable objects.

We refer to [Y] as a general reference for representation theory of Cohen-Macaulay rings, and [AGV] for classification of singularities.

Acknowledgment

The first author would like to thank Duco van Straten for helpful discussions. The second author would like to thank Atsushi Takahashi and Hokuto Uehara for stimulating discussions.

1. Main results

Let (R, 𝔪) be a local complete d-dimensional commutative noetherian Gorenstein isolated singularity and R/𝔪 = k ⊂ R, where k is an algebraically closed field. We denote by CM(R) the category of maximal Cohen-Macaulay modules over R. Then CM(R) is a Frobenius category, and so the stable category \underline{CM}(R) is a triangulated category with shift functor Σ = Ω^{−1} [H].

We collect some fundamental results.

• We have AR-duality

\underline{Hom}(X, Y) ≃ D Ext^{1}(Y, τX)

with τ ≃ Ω^{2−d} [Au]. In particular, \underline{CM}(R) is (d − 1)-CY.

• If R is a hypersurface singularity, then Σ^{2} = id [Ei].

• (Knörrer periodicity)

\underline{CM}(k[[x_{0}, ⋯, x_{d}, y, z]]/(f + yz)) ≃ \underline{CM}(k[[x_{0}, ⋯, x_{d}]]/(f))

for any f ∈ k[[x_{0}, ⋯, x_{d}]] [Kn] ([So] in characteristic two).

Consequently, if d is odd, then τ = Ω and \underline{CM}(R) is 2-CY. If d is even, then τ = id and \underline{CM}(R) is 1-CY, hence any non-free CM R-module M satisfies Ext_{R}^{1}(M, M) ≠ 0.

Definition 1.1. Let C = CM(R) or \underline{CM}(R). We call an object M ∈ C

• rigid if Ext_{R}^{1}(M, M) = 0,

• maximal rigid if it is rigid and any rigid N ∈ C satisfying M ∈ add N satisfies N ∈ add M,

• cluster tilting if add M = {X ∈ C | Ext_{R}^{1}(M, X) = 0} = {X ∈ C | Ext_{R}^{1}(X, M) = 0}.

Cluster tilting objects are maximal rigid, thought the converse does not necessarily hold. If C is 2-CY, then M ∈ C is cluster tilting if and only if add M = {X ∈ C | Ext_{R}^{1}(M, X) = 0}.

Let M ∈ C be a basic cluster tilting object and X an indecomposable summand of M = X⊕N. Then there exist triangles/short exact sequences (called exchange sequences)

X →^{g_{1}} N_{1} →^{f_{1}} Y and Y →^{g_{0}} N_{0} →^{f_{0}} X

such that N_{i} ∈ add N, f_{i} is a minimal right (add N)-approximation, and g_{i} is a minimal left (add N)-approximation. Then Y ⊕ N is a basic cluster tilting object again called (cluster) mutation of M [BMRRT]. It is known that there are no more basic cluster tilting objects containing N [IY].

Let R = k[[x, y, x_{2}, ⋯, x_{d}]]/(f) be a simple hypersurface singularity so that in characteristic zero f is one of the following polynomials,

(A_{n}) x^{2} + y^{n+1} + z_{2}^{2} + z_{3}^{2} + ⋯ + z_{d}^{2} (n ≥ 1)
(D_{n}) x^{2}y + y^{n−1} + z_{2}^{2} + z_{3}^{2} + ⋯ + z_{d}^{2} (n ≥ 4)
(E_{6}) x^{3} + y^{4} + z_{2}^{2} + z_{3}^{2} + ⋯ + z_{d}^{2}
(E_{7}) x^{3} + xy^{3} + z_{2}^{2} + z_{3}^{2} + ⋯ + z_{d}^{2}
(E_{8}) x^{3} + y^{5} + z_{2}^{2} + z_{3}^{2} + ⋯ + z_{d}^{2}

Then R is of finite Cohen-Macaulay representation type [Ar, GK, Kn].

We shall show the following result in section 2 using additive functions on the AR quiver. We shall explain another proof using Singular in section 3.

Theorem 1.2. Let k be an algebraically closed field of characteristic zero and R a simple hypersurface singularity of dimension d ≥ 1.

(1) Assume that d is even. Then \underline{CM}(R) does not have non-zero rigid objects.

(2) Assume that d is odd. Then the number of indecomposable rigid objects, cluster tilting objects, maximal rigid objects, and indecomposable summands of maximal rigid objects in \underline{CM}(R) are as follows:

f

indec. rigid cluster tilting max. rigid

summands of max. rigid

(A_{n}) n : odd  2  2  2  1
(A_{n}) n : even  0  0  1  0
(D_{n}) n : odd  2  0  2  1
(D_{n}) n : even  6  6  6  2
(E_{6})  0  0  1  0
(E_{7})  2  0  2  1
(E_{8})  0  0  1  0

We also consider a minimally elliptic curve singularity T_{p,q}(λ) (p ≤ q). Assume for simplicity that our base field k is algebraically closed of characteristic zero. Then these singularities are given by the equations

x^{p} + y^{q} + λx^{2}y^{2} = 0,

where \frac{1}{p} + \frac{1}{q} ≤ \frac{1}{2} and certain values of λ ∈ k have to be excluded. They are tame Cohen-Macaulay representation type [D, Kah, DG]. We divide into two cases.

(i) Assume \frac{1}{p} + \frac{1}{q} = \frac{1}{2}. This case occurs if and only if (p, q) = (3, 6) or (4, 4), and T_{p,q}(λ) is called simply elliptic. The corresponding coordinate rings can be written in the form

T_{3,6}(λ) = k[[x, y]]/(y(y − x^{2})(y − λx^{2}))

and

T_{4,4}(λ) = k[[x, y]]/(xy(x − y)(x − λy)),

where in both cases λ ∈ k \ {0, 1}.

(ii) Assume \frac{1}{p} + \frac{1}{q} < \frac{1}{2}. Then T_{p,q}(λ) does not depend on the continuous parameter λ, and is called a cusp singularity. In this case the corresponding coordinate rings can be written in the form

T_{p,q} = k[[x, y]]/((x^{p−2} − y^{2})(x^{2} − y^{q−2})).

We shall show the following result in section 6 by applying a result in birational geometry.

Theorem 1.3. Let k be an algebraically closed field of characteristic zero and R a minimally elliptic curve singularity T_{p,q}(λ).

(a) R has a cluster tilting object if and only if p = 3 and q is even or if both p and q are even.

(b) The number of indecomposable rigid objects, cluster tilting objects, and indecomposable summands of cluster tilting objects in \underline{CM}(R) are as follows:

p, q

indec. rigid

cluster tilting summands of cluster tilting

p = 3, q : even  6  6  2
p, q : even  14  24  3

We also prove the following general theorem, which includes both Theorem 1.2 (except the assertion on maximal rigid objects) and Theorem 1.3. The ‘if’ part in (a) and the assertion (b) are proved in section 4 by a purely homological method. The proof of (a), including another proof of the ‘if’ part, is given in section 6 by applying Katz’s criterion in birational geometry.

Theorem 1.4. Let R = k[[x, y]]/(f) (f ∈ (x, y)) be a one-dimensional reduced hypersurface singularity.

(a) R has a cluster tilting object if and only if f is a product f = f_{1} ⋯ f_{n} with f_{i} ∉ (x, y)^{2}.

(b) The number of indecomposable rigid objects, cluster tilting objects, and indecomposable summands of cluster tilting objects in \underline{CM}(R) are as follows:

indec. rigid

cluster tilting summands of cluster tilting

2^{n} − 2  n!  n − 1

The following result gives a bridge between cluster tilting theory and birational geometry. The terminologies are explained in section 5.

Theorem 1.5. Let (R, 𝔪) be a three dimensional isolated cA_{n} singularity defined by the equation g(x, y)+zt and R′ a one dimensional singularity defined by g(x, y). Then the following conditions are equivalent.

(a) Spec R has a small resolution.

(b) Spec R has a crepant resolution.

(c) (R, 𝔪) has a non-commutative crepant resolution.

(d) \underline{CM}(R) has a cluster tilting object.

(e) \underline{CM}(R′) has a cluster tilting object.

(f) The number of irreducible power series in the prime decomposition of g(x, y) is n + 1.

We end this section by giving an application to finite dimensional algebras. A 2-CY tilted algebra is an endomorphism ring End_{C}(M) of a cluster tilting object T in a 2-CY triangulated category C. In section 7, we shall show the following result and compute 2-CY tilted algebras associated with minimally elliptic curve singularities.

Theorem 1.6. Let (R, 𝔪) be an isolated hypersurface singularity and Γ a 2-CY tilted algebra coming from \underline{CM}(R). Then we have the following.

(a) Γ is a symmetric algebra.

(b) All components in the stable AR-quiver of Γ are tubes of rank 1 or 2.

For example, put

R = k[[x, y]]/((x − λ_{1}y) ⋯ (x − λ_{n}y)) and M = ⨁_{i=1}^{n} k[[x, y]]/((x − λ_{1}y) ⋯ (x − λ_{i}y))

for distinct elements λ_{i} ∈ k. Then M is a cluster tilting object in CM(R) by Theorem 4.1, so Γ = \underline{End}_{R}(M) satisfies the conditions in Theorem 1.6. Since CM R has wild Cohen-Macaulay representation type if n > 4 [DG, Th. 3], we should get a family of examples of finite dimensional symmetric k-algebras whose stable AR-quiver consists only of tubes of rank 1 or 2, and are of wild representation type.

2. Simple hypersurface singularities

Let R be a one-dimensional simple hypersurface singularity. In this case the AR-quivers are known for CM(R) [DW], and so also for \underline{CM}(R). We use the notation from [Y].

In order to locate the indecomposable rigid modules M, that is, the modules M with Ext^{1}(M, M) = 0, the following lemmas are useful, where part (a) of the first one is proved in [HKR], and the second one is a direct consequence of [KR] (generalizing [BMR1]).

Lemma 2.1. (a) Let C be an abelian or triangulated k-category with finite dimensional ho-

momorphism spaces. Let A \underrightarrow{( \begin{matrix}f_{1} \\ f_{2} \end{matrix} )} B_{1} ⊕ B_{2} ⟶^{(g_{1},g_{2})} C be a short exact sequence or a triangle, where A is indecomposable, B_{1} and B_{2} nonzero, and (g_{1}, g_{2}) has no nonzero indecomposable summand which is an isomorphism. Then Hom(A, C) ≠ 0 .

(b) Let 0 → A ⟶^{f} B ⟶^{g} C → 0 be an almost split sequence in CM(R), where R is an isolated hypersurface singularity, and B has at least two indecomposable nonprojective summands in a decomposition of B into a direct sum of indecomposable modules. Then Ext^{1}(C, C) ≠ 0.

Proof. (a) See [HKR].

(b) Using (a) together with the above AR-formula and τ^{2} = id, we obtain D Ext^{1}(C, C) ≃ \underline{Hom}(τ^{−1}C, C) = \underline{Hom}(τC, C) ≃ \underline{Hom}(A, C) ≠ 0, where D = Hom_{k}( , k). □

Lemma 2.2. Let T be a cluster tilting object in the Hom-finite connected 2-CY category C, and Γ = End_{C}(T).

(a) The functor G = Hom_{C}(T, ): C → mod Γ induces an equivalence of categories \overline{G}: C/add(τT) → mod Γ.

(b) The AR-quiver for Γ is as a translation quiver obtained from the AR-quiver for C by removing the vertices corresponding to the indecomposable summands of τT.

(c) Assume τ^{2} = id. Then we have the following. (i) Γ is a symmetric algebra. (ii) The indecomposable nonprojective Γ-modules have τ-period one or two. (iii) If C has an infinite number of nonisomorphic indecomposable objects, then all com-

ponents in the stable AR-quiver of Γ are tubes of rank one or two.

(d) If C has only a finite number n of nonisomorphic indecomposable objects, and T has t nonisomorphic indecomposable summands, then there are n − t nonisomorphic indecomposable Γ-modules.

Proof. For (a) and (b) see [BMR1][KR]. Since C is 2-CY, we have τ = Σ, and a functorial isomorphism

D Hom_{C}(T, T) ≃ Hom_{C}(T, Σ^{2}T) = Hom_{C}(T, τ^{2}T) ≃ Hom_{C}(T, T).

This shows that Γ is symmetric. Let C be an indecomposable nonprojective Γ-module. Viewing C as an object in C we have τ_{C}^{2}C ≃ C, and τC is not a projective Γ-module since C is not removed. Hence we have τ_{Γ}^{2}C ≃ C. If C has an infinite number of nonisomorphic indecomposable objects, then Γ is of infinite type. Then each component of the AR-quiver is infinite, and hence is a tube of rank one or two. Finally, (d) is a direct consequence of (a). □

We also use that in our cases we have a covering functor Π: k(ℤQ) → \underline{CM}(R), where Q is the appropriate Dynkin quiver and k(ℤQ) is the mesh category of the translation quiver ℤQ [Rie][Am], (see also [I1, Section 4.4] for another explanation using functorial method).

For the one-dimensional simple hypersurface singularities we have the cases A_{n} (n even or odd), D_{n} (n odd or even), E_{6}, E_{7} and E_{8}. We now investigate them case by case.

Proposition 2.3. In the case A_{n} (with n even) there are no indecomposable rigid objects.

Proof. We have the stable AR-quiver

[figure: I_{1}; I_{2}; ⋯; I_{n/2}]

Here, and later, a dotted line between two indecomposable modules means that they are connected via τ.

Since τI_{j} ≃ I_{j} for each j, Ext^{1}(I_{j}, I_{j}) ≠ 0 for j = 1, ⋯, n/2. Hence no I_{j} is rigid. □

Proposition 2.4. In the case A_{n} (with n odd) the maximal rigid objects coincide with the cluster tilting objects. There are two indecomposable ones, and the corresponding 2-CY tilted algebras are k[x]/(x^{\frac{(n+1)}{2}} )

Proof. We have the stable AR-quiver

[figure: N_{−}; M_{1}; M_{2}; ⋯; M_{(n−1)/2}; N_{+}]

Since τM_{i} ≃ M_{i} for i = 1, ⋯, n − 1/2, we have

Ext^{1}(M_{i}, M_{i}) ≃ \underline{Hom}(M_{i}, τM_{i}) ≃ \underline{Hom}(M_{i}, M_{i}) ≠ 0.

So only the indecomposable objects N_{−} and N_{+} could be rigid. We use covering techniques and additive functions to compute the support of \underline{Hom}(N_{−},). For simplicity, we write l = (n − 1)/2

[figure: M_{1}; M_{1}; M_{1}; M_{2}; M_{2}; M_{2}; ⋯; M_{3}; M_{3}; M_{3}; M_{l−1}; M_{l−1}; M_{l−1}; M_{l}; N_{−}; M_{l}; N_{+}; M_{l}; N_{−}; ⋯; N_{−}; N_{+}; N_{−}] [figure: 1; 0; 1; 1; 0; 1; 1; 1; 0; 1; 1; 1; 1; 1; 1; 0; 1; 1; 1; 0; 0; 1; 0; 1; 0; 1; 0]

We see that \underline{Hom}(N_{−}, N_{+}) = 0, so Ext^{1}(N_{+}, N_{+}) = Ext^{1}(N_{+}, τN_{−}) = 0, and Ext^{1}(N_{−}, N_{−}) = 0. Since Ext^{1}(N_{+}, N_{−}) ≠ 0, we see that N_{+} and N_{−} are exactly the maximal rigid objects. Further \underline{Hom}(N_{−}, M_{i}) ≠ 0 for all i, so Ext^{1}(N_{+}, M_{i}) ≠ 0 and Ext^{1}(N_{−}, M_{i}) ≠ 0 for all i. This shows that N_{+} and N_{−} are also cluster tilting objects.

The description of the cluster tilted algebras follows directly from the above picture. □

Proposition 2.5. In the case D_{n} with n odd we have two maximal rigid objects, which both are indecomposable, and neither one is cluster tilting.

Proof. We have the AR-quiver

[figure: B; Y_{1}; M_{1}; Y_{2}; M_{2}; ⋯; M_{(n−3)/2}; X_{(n−1)/2}; A; X_{1}; N_{1}; X_{2}; N_{2}; ⋯; N_{(n−3)/2}]

Using Lemma 2.1, the only candidates for being indecomposable rigid are A and B. We compute the support of \underline{Hom}(A, )

[figure: A; B; Y_{1}; X_{1}; ⋯; Y_{l}; M_{1}; N_{l}; M_{l}; N_{1}; X_{l+1}; X_{1}; Y_{1}; ⋯; A; B; 1; 0; 1; 0; 1; 0; 1; 0; 1; 0; 1; 0; 1; 0]

where B = τA and l = (n − 3)/2. We see that \underline{Hom}(A, B) = 0, so that Ext^{1}(A, A) = 0. Then A is clearly maximal rigid. Since \underline{Hom}(A, M_{1}) = 0, we have Ext^{1}(A, N_{1}) = 0, so A is not cluster tilting. Alternatively, we could use that we see that \underline{End}(A)^{op} ≃ k[x]/(x^{2}), which has two indecomposable modules, whereas \underline{CM}(R) has 2n − 3 indecomposable objects. If A was cluster tilting, \underline{End}(A)^{op} would have had 2n − 3 − 1 = 2n − 4 indecomposable modules, by Lemma 2.2. □

Proposition 2.6. In the case D_{2n} with n a positive integer we have that the maximal rigid objects coincide with the cluster tilting ones. There are 6 of them, and each is a direct sum of two nonisomorphic indecomposable objects.

The corresponding 2-CY-tilted algebras are given by the quiver with relations \xymatrix{· \ar[r]^{α} & · \ar[l]^{β}} αβα =

0 = βαβ in the case D_{4}, and by [figure: ·; γ; α; ·; β] with γ^{n−1} = βα, γβ = 0 = αγ and \xymatrix{· \ar[r]^{α} & · \ar[l]^{β}} with

(αβ)^{n−1}α = 0 = (βα)^{n−1}β for 2n > 4.

Proof. We have the AR-quiver

[figure: C_{+}; B; Y_{1}; M_{1}; Y_{2}; M_{2}; ⋯; Y_{n−1}; D_{+}; C_{−}; A; X_{1}; N_{1}; X_{2}; N_{2}; ⋯; X_{n−1}; D_{−}]

By Lemma 2.1, the only possible indecomposable rigid objects are: A, B, C_{+}, C_{−}, D_{+}, D_{−}. We compute the support of \underline{Hom}(C_{+}, ):

[figure: D_{−}; C_{−}; D_{−}; C_{−}; D_{−}; C_{+}; Y_{l}; D_{+}; X_{l}; C_{+}; Y_{l}; Y_{l}; D_{+}; X_{l}; C_{+}; Y_{l}; ⋯; N_{l−1}; M_{l−1}; N_{l−1}; M_{l−1}; N_{l−1}; M_{l−1}; Y_{2}; X_{2}; Y_{2}; X_{2}; Y_{2}; N_{1}; M_{1}; N_{1}; M_{1}; Y_{1}; X_{1}; Y_{1}; ⋯; A; B]

where l = n − 1

[figure: 1; 0; 1; 0; 0; 1; 1; 0; 1; 1; 1; 1; 0; 1; 1; 0; 1; 1; 1; 1; 1; 0; 1; 1; 1; 1; 0; 1; 1; 1; 0; 1; 1; 0; 1; 0]

We see that \underline{Hom}(C_{+}, D_{+}) = 0, so Ext^{1}(C_{+}, C_{+}) = 0 = Ext^{1}(D_{+}, D_{+}). Further, \underline{Hom}(C_{+}, C_{−}) = 0, so Ext^{1}(C_{+}, D_{−}) = 0. By symmetry Ext^{1}(D_{−}, D_{−}) = 0 = Ext^{1}(C_{−}, C_{−}) and Ext^{1}(D_{+}, C_{−}) = 0. Also Ext^{1}(C_{+}, A) = 0, Ext^{1}(C_{+}, B) ≠ 0, so Ext^{1}(D_{+}, B) = 0, Ext^{1}(D_{+}, A) ≠ 0. Further Ext^{1}(C_{+}, X) ≠ 0 for X ≠ A, D_{−}, C_{+}.

We now compute the support of \underline{Hom}(A, )

[figure: C_{−}; D_{−}; X_{l}; C_{+}; Y_{l}; D_{+}; X_{l}; ⋯; N_{l−1}; M_{l−1}; N_{l−1}; M_{l−1}; N_{1}; M_{1}; M_{1}; N_{1}; M_{1}; X_{1}; Y_{1}; X_{1}; Y_{1}; X_{1}; ⋯; A; B; B; A; B]

where l = n − 1 and we have an odd number of columns and rows.

[figure: 1; 0; 1; 1; 1; 0; 0; 1; 0; 1; 0; 1; 0; 0; 1; 0; 1; 0; 0; 1; 0; 1; 0; 0; 1; 0]

We see that \underline{Hom}(A, B) = 0, so Ext^{1}(A, A) = 0, hence also Ext^{1}(B, B) = 0. Since \underline{Hom}(A, D_{−}) = 0, we have Ext^{1}(A, C_{−}) = 0, hence Ext^{1}(B, D_{−}) = 0. Since \underline{Hom}(A, C_{−}) ≠ 0, we have Ext^{1}(A, D_{−}) ≠ 0, so Ext^{1}(B, D_{+}) ≠ 0.

It follows that C_{+} ∐ D_{−}, C_{−} ∐ D_{+}, C_{+} ∐ A, D_{+} ∐ B, A ∐ C_{−} and B ∐ D_{−} are maximal rigid.

These are also cluster tilting: We have \underline{Hom}(A, X_{i}) ≠ 0, \underline{Hom}(A, N_{i}) ≠ 0, so Ext^{1}(B, X_{i}) ≠ 0, Ext^{1}(B, N_{i}) ≠ 0. Similarly, Ext^{1}(A, Y_{i}) ≠ 0, Ext^{1}(A, M_{i}) ≠ 0. Also \underline{Hom}(C_{+}, Y_{i}) ≠ 0, \underline{Hom}(C_{+}, N_{i}) ≠ 0, so Ext^{1}(D_{+}, Y_{i}) ≠ 0, Ext^{1}(D_{+}, N_{i}) ≠ 0. Hence Ext^{1}(C_{+}, X_{i}) ≠ 0, Ext^{1}(C_{+}, M_{i}) ≠ 0. So Ext^{1}(D_{−}, Y_{i}) ≠ 0, Ext^{1}(D_{−}, N_{i}) ≠ 0, Ext^{1}(C_{−}, X_{i}) ≠ 0, Ext^{1}(C_{−}, M_{i}) ≠ 0. We see that each indecomposable rigid object can be extended to a cluster tilting object in exactly two ways, which we would know from a general result in [IY].

The exchange graph is as follows:

[figure: {C_{+}, D_{−}}; {B, D_{−}}; {A, C_{+}}; {B, D_{+}}; {A, C_{−}}; {C_{−}, D_{+}}]

Considering the above pictures, we get the desired description of the corresponding 2-CY tilted algebras in terms of quivers with relations. □

Proposition 2.7. In the case E_{6} there are no indecomposable rigid objects.

Proof. We have the AR-quiver

[figure: B; M_{1}; M_{2}; X; A; N_{1}]

The only candidates for indecomposable rigid objects according to Lemma 2.1 are M_{1} and N_{1}. We compute the support of \underline{Hom}(M_{1}, ).

\xymatrix{ &  &  &  & N_{1} \ar@{-->}[dr] &  & N_{1} &  &  &  &  &  & 1 \ar[dr] &  & 0 \\  &  &  & A \ar@{-->}[dr] \ar@{-->}[ur] &  & B \ar@{-->}[dr] \ar@{-->}[ur] &  & ⋯ &  &  &  & 1 \ar[dr] \ar@{-->}[ur] &  & 1 \ar[dr] \ar@{-->}[ur] &  & ⋯ \\  &  & X \ar@{-->}[ur] \ar@{-->}[dr] \ar[r] & M_{2} \ar[r] & X \ar@{-->}[ur] \ar@{-->}[dr] \ar[r] & M_{2} \ar[r] & X &  &  &  & 1 \ar@{-->}[ur] \ar[dr] \ar[r] & 1 \ar[r] & 1 \ar@{-->}[ur] \ar[r] \ar[dr] & 0 \ar[r] & 1 \\  & A \ar@{-->}[ur] \ar[dr] &  & B \ar@{-->}[ur] \ar@{-->}[dr] &  & A \ar@{-->}[ur] \ar[dr] &  & ⋯ &  & 1 \ar@{-->}[ur] \ar[dr] &  & 0 \ar@{-->}[ur] \ar[dr] &  & 1 \ar[dr] \ar@{-->}[ur] &  & ⋯ \\ M_{1} \ar@{-->}[ur] &  & N_{1} \ar@{-->}[ur] &  & M_{1} \ar@{-->}[ur] &  & N_{1} &  & 1 \ar@{-->}[ur] &  & 0 \ar@{-->}[ur] &  & 0 \ar@{-->}[ur] &  & 1}

We see that \underline{Hom}(M_{1}, N_{1}) ≠ 0, so that Ext^{1}(M_{1}, M_{1}) ≠ 0 and Ext^{1}(N_{1}, N_{1}) ≠ 0. □

Proposition 2.8. In the case E_{7} there are two maximal rigid objects, which both are indecomposable, and neither of them is cluster tilting.

Proof. We have the AR-quiver

[figure: C; D; A; M_{2}; Y_{2}; Y_{3}; Y_{1}; M_{1}; B; N_{2}; X_{2}; X_{3}; X_{1}; N_{1}]

Using Lemma 2.1, we see that the only candidates for indecomposable rigid objects are A, B, M_{1}, N_{1}, C and D. We first compute the support of \underline{Hom}(A, ).

[figure: M_{1}; Y_{1}; X_{1}; ⋯; Y_{3}; C; X_{3}; D; Y_{2}; X_{2}; Y_{2}; M_{2}; N_{2}; M_{2}; ⋯; A; B; A; B; 1; 0; 0; 1; 0; 1; 1; 0; 1; 1; 0; 1; 1; 1; 0; 1; 1; 1; 0; 1; 1; 1; 0; 0; 1; 0; 1; 1; 1; 0; 1; 0; 1; 0; 0; 1; 1; 0; 0; 1; 0; 1; 0; 0; 0; 1; 0; 0; 0; 1; 0]

We see that Ext^{1}(A, A) = 0, and so also Ext^{1}(B, B) = 0, so A and B are rigid. Next we compute the support of \underline{Hom}(M_{1}, ).

[figure: M_{1}; N_{1}; M_{1}; M_{1}; X_{1}; Y_{1}; X_{1}; ⋯; Y_{3}; C; X_{3}; D; Y_{3}; X_{2}; Y_{2}; M_{2}; N_{2}; ⋯; B; 1; 0; 0; 1; 1; 0; 1; ⋯; 1; 1; 1; 0; 1; 1; 1; 1; 1; ⋯; 1]

We see that Ext^{1}(M_{1}, M_{1}) ≠ 0 and Ext^{1}(N_{1}, N_{1}) ≠ 0, so that M_{1} and N_{1} are not rigid. Then we compute the support of \underline{Hom}(C, ).

[figure: N_{1}; M_{1}; X_{1}; Y_{1}; X_{1}; ⋯; C; X_{3}; D; Y_{3}; C; X_{3}; D; Y_{2}; X_{2}; Y_{2}; N_{2}; M_{2}; ⋯; A; B; 1; 0; 1; 1; 1; ⋯; 1; 1; 0; 1; 1; 2; 1; 1; 1; 2; 1; 1; ⋯; 1; 0]

We see that Ext^{1}(C, C) ≠ 0 and Ext^{1}(D, D) ≠ 0, so that C and D are not rigid. Hence A and B are the rigid indecomposable objects, and they are maximal rigid.

Since Ext^{1}(A, C) = 0, we see that A and hence B is not cluster tilting. □

Proposition 2.9. In the case E_{8} there are no indecomposable rigid objects.

Proof. We have the AR-quiver

[figure: A_{2}; B_{2}; N_{2}; D_{2}; X_{1}; X_{2}; C_{1}; B_{1}; N_{1}; M_{2}; C_{2}; Y_{1}; Y_{2}; D_{1}; A_{1}; M_{1}]

The only candidates for indecomposable rigid objects are M_{1}, N_{1}, M_{2}, N_{2}, A_{2} and B_{2}, by Lemma 2.1. We first compute the support of \underline{Hom}(M_{1}, ):

[figure: M_{2}; N_{2}; D_{2}; C_{2}; ⋯; Y_{1}; B_{2}; X_{1}; A_{2}; Y_{1}; X_{2}; Y_{2}; X_{2}; D_{1}; C_{1}; D_{1}; C_{1}; B_{1}; A_{1}; B_{1}; A_{1}; ⋯; M_{1}; N_{1}; M_{1}; N_{1}; M_{1}; 1; 0; 1; 1; 0; ⋯; 1; 1; 1; 0; 1; 1; 1; 0; 1; 1; 1; 0; 0; 1; 1; 0; 0; 0; 1; ⋯; 1; 0; 0; 0; 0; 1]

We see that Ext^{1}(M_{1}, M_{1}) ≠ 0, and hence Ext^{1}(N_{1}, N_{1}) ≠ 0. Next we compute the support of \underline{Hom}(M_{2}, ):

[figure: M_{2}; N_{2}; M_{2}; N_{2}; C_{2}; D_{2}; C_{2}; ⋯; Y_{1}; B_{2}; X_{1}; A_{2}; Y_{1}; Y_{2}; X_{2}; D_{1}; C_{1}; A_{1}; ⋯; M_{1}; 1; 0; 0; 1; 1; 0; 1; ⋯; 1; 1; 1; 0; 1; 1; 1; 1; 1; 1; ⋯; 1]

We see that Ext^{1}(M_{2}, M_{2}) ≠ 0, and hence Ext^{1}(N_{2}, N_{2}) ≠ 0. Finally we compute the support of \underline{Hom}(A_{2}, ):

[figure: M_{2}; N_{2}; D_{2}; C_{2}; D_{2}; ⋯; A_{2}; Y_{1}; B_{2}; X_{1}; A_{2}; Y_{1}; B_{2}; Y_{2}; X_{2}; Y_{2}; D_{1}; C_{1}; A_{1}; B_{1}; ⋯; M_{1}; 1; 0; 1; 1; 1; ⋯; 1; 1; 0; 1; 1; 2; 1; 1; 1; 2; 1; 1; 1; 1; ⋯; 1]

It follows that Ext^{1}(A_{2}, A_{2}) ≠ 0, and similarly Ext^{1}(B_{2}, B_{2}) ≠ 0. Hence there are no indecomposable rigid objects. □

3. Computation with Singular

An alternative way to carry out computations of Ext^{1}–spaces in the stable category of maximal Cohen-Macaulay modules is to use the computer algebra system Singular, see [GP]. Let

R = k[x_{1}, x_{2}, …, x_{n}]_{⟨x_{1},x_{2},...,x_{n}⟩}/I

be a Cohen-Macaulay local ring which is an isolated singularity, and M and N two maximal Cohen-Macaulay modules. Denote by R̂ the completion of R. Since all the spaces Ext_{R}^{i}(M, N) (i ≥ 1) are finite-dimensional over k and the functor mod R → mod R̂ is exact, maps the maximal Cohen-Macaulay modules to maximal Cohen-Macaulay modules and the finite length modules to finite length modules, we can conclude that

dim_{k}(Ext_{R}^{i}(M, N)) = dim_{k}(Exti ; R̂(M̂, N̂)).

As an illustration we show how to do this for the case E_{7}.

Proposition 3.1. In the case E_{7} there are two maximal rigid objects, which both are indecomposable and neither of them is cluster tilting.

By [Y] the AR-quiver of \underline{CM}(R) has the form

[figure: C; D; A; M_{2}; Y_{2}; Y_{3}; Y_{1}; M_{1}; B; N_{2}; X_{2}; X_{3}; X_{1}; N_{1}]

By Lemma 2.1 only the modules A, B, C, D, M_{1}, N_{1} can be rigid. Since N = τ(M), B = τ(A), N_{1} = τ(M_{1}), the pairs of modules (A, B), (C, D) and (M_{1}, N_{1}) are rigid or not rigid simultaneously. By [Y] we have the following presentations:

R ⟶^{x^{2}+y^{3}} R ⟶^{x} R ⟶ A ⟶ 0,

R \underrightarrow{( \begin{matrix}x & y \\ y^{2} & −x \end{matrix} )} R ⟶^{x( \begin{matrix}x & y \\ y^{2} & −x \end{matrix} )} C ⟶ 0,

R \underrightarrow{( \begin{matrix}x^{2} & y \\ xy^{2} & −x \end{matrix} )} R \underrightarrow{( \begin{matrix}x & y \\ xy^{2} & −x^{2} \end{matrix} )} M_{1} ⟶ 0,

so we can use the computer algebra system Singular in order to compute the Ext^{1}–spaces between these modules.

> Singular (call the program ``Singular'')

> LIB ``homolog.lib''; (call the library of homological algebra) > ring S = 0,(x,y),ds; (defines the ring S = ℚ[x, y]_{⟨x,y⟩}) > ideal I = x3 + xy3; (defines the ideal x^{3} + xy^{3} in S) > qring R = std(I); (defines the ring ℚ[x, y]_{⟨x,y⟩}/I)

> module A = [x];
> module C = [x2, xy2], [xy, -x2];
> module M1 = [x2, xy2], [y, -x2]; (define modules A, C, M_{1})
> list l = Ext(1,A,A,1);

// dimension of Ext^{1}: -1 (Output: Ext_{R}^{1}(A, A) = 0)

> list l = Ext(1,C,C,1);
// ** redefining l **

// dimension of Ext^{1}: 0 (the Krull dimension of Ext_{R}^{1}(C, C) is 0) // vdim of Ext^{1}: 2 (dim_{k}(Ext_{R}^{1}(C, C)) = 2)

> list l = Ext(1,M1,M1,1);
// ** redefining l **
// dimension of Ext^{1}:  0
// vdim of Ext^{1}:  10
> list l = Ext(1,A,C,1);
// ** redefining l **
// dimension of Ext^{1}:  -1

This computation shows that the modules A and B are rigid, C, D, M_{1} and N_{1} are not rigid and since Ext_{R}^{1}(A, C) = 0, there are no cluster tilting objects in the stable category \underline{CM}(R).

4. One-dimensional hypersurface singularities

We shall construct a large class of one-dimensional hypersurface singularities having a cluster tilting object, then classify all cluster tilting objects. Our method is based on higher theory of almost split sequences and Auslander algebras studied in [I1, I2]. We also use a relationship between cluster tilting objects in CM(R) and tilting modules over the endomorphism algebra of a cluster tilting object [I2]. Then we shall compare cluster mutation with tilting mutation by using results due to Riedtmann-Schofield [RS].

In this section, we usually consider tilting objects in CM(R) instead of \underline{CM}(R).

Let k be an infinite field, S := k[[x, y]] and 𝔪 := (x, y). We fix f ∈ 𝔪 and write f = f_{1} ⋯ f_{n} for irreducible formal power series f_{i} ∈ 𝔪 (1 ≤ i ≤ n). Put

S_{i} := S/(f_{1} ⋯ f_{i}) and R := S_{n} = S/(f).

We assume that R is reduced, so we have (f_{i}) ≠ (f_{j}) for any i ≠ j.

Our main results in this section are the following.

Theorem 4.1. (a) ⨁_{i=1}^{n} S_{i} is a rigid object in CM(R).

(b) ⨁_{i=1}^{n} S_{i} is a cluster tilting object in CM(R) if the following condition (A) is satisfied. (A) f_{i} ∉ 𝔪^{2} for any 1 ≤ i ≤ n.

Let 𝔖_{n} be the symmetric group of degree n. For w ∈ 𝔖_{n} and I ⊆ {1, ⋯, n}, we put

S_{i}^{w} := S/(f_{w(1)} ⋯ f_{w(i)}), M_{w} := ⨁_{i=1}^{n} S_{i}^{w} and S_{I} := S/(∏_{i∈I} f_{i}).

Theorem 4.2. Assume that (A) is satisfied.

(a) There are exactly n! cluster tilting objects M_{w} (w ∈ 𝔖_{n}) and exactly 2^{n} − 1 indecomposable rigid objects S_{I} (∅ ≠ I ⊆ {1, ⋯, n}) in CM(R).

(b) For any w ∈ 𝔖_{n}, there are exactly n! Cohen-Macaulay tilting End_{R}(M_{w})-modules Hom_{R}(M_{w}, M_{w′}) (w′ ∈ 𝔖_{n}) of projective dimension at most one. Moreover, all algebras End_{R}(M_{w}) (w ∈ 𝔖_{n}) are derived equivalent.

It is interesting to compare with results in [IR], where two-dimensional (2-Calabi-Yau) algebras Γ are treated and a bijection between elements in an affine Weyl group and tilting Γ-modules of projective dimension at most one is given. Here the algebra is one-dimensional, and Weyl groups appear.

Here we consider three examples.

(a) Let R be a curve singularity of type A_{2n−1} or D_{2n+2}, so

R = S/((x − y^{n})(x + y^{n})) or R = S/(y(x − y^{n})(x + y^{n})).

By our theorems, there are exactly 2 or 6 cluster tilting objects and exactly 3 or 7 indecomposable rigid objects in CM(R), which fits with our computations in section 1.

(b) Let R be a curve singularity of type T_{3,2q+2}(λ) or T_{2p+2,2q+2}(λ), so

R = S/((x − y^{2})(x − y^{q})(x + y^{q})) (R = S/(y(y − x^{2})(y − λx^{2})) for q = 2),
R = S/((x^{p} − y)(x^{p} + y)(x − y^{q})(x + y^{q})) (R = S/(xy(x − y)(x − λy)) for p = q = 1).

By our theorems, there are exactly 6 or 24 cluster tilting objects and exactly 7 or 15 indecomposable rigid objects in CM(R).

(c) Let λ_{i} ∈ k (1 ≤ i ≤ n) be mutually distinct elements in k. Put

R := S/((x − λ_{1}y) ⋯ (x − λ_{n}y)).

By our theorems, there are exactly n! cluster tilting objects and exactly 2^{n} − 1 indecomposable rigid objects in CM(R).

First of all, Theorem 4.1(a) follows immediately from the following observation.

Proposition 4.3. For g_{1}, g_{2} ∈ 𝔪 and g_{3} ∈ S, put R := S/(g_{1}g_{2}g_{3}). If g_{1} and g_{2} have no common factor, then Ext_{R}^{1}(S/(g_{1}g_{3}), S/(g_{1})) = 0 = Ext_{R}^{1}(S/(g_{1}), S/(g_{1}g_{3})).

Proof. We have a projective resolution

R →^{g_{2}} R →^{g_{1}g_{3}} R → S/(g_{1}g_{3}) → 0.

Applying Hom_{R}( , S/(g_{1})), we have a complex

S/(g_{1}) ⟶^{g_{1}g_{3}=0} S/(g_{1}) →^{g_{2}} S/(g_{1}).

This is exact since g_{1} and g_{2} have no common factor. Thus we have the former equation, and the other one can be proved similarly. □

Our plan of proof of Theorem 4.1(b) is the following.

(i) First we shall prove Theorem 4.1 under the following stronger assumption: (B) 𝔪 = (f_{1}, f_{2}) = ⋯ = (f_{n−1}, f_{n}).

(ii) Then we shall prove the general statement of Theorem 4.1.

We need the following general result in [I1, I2].

Proposition 4.4. Let R be a complete local Gorenstein ring of dimension at most three and M a rigid Cohen-Macaulay R-module which is a generator. Then the following conditions are equivalent.

(a) M is a cluster tilting object in CM(R).

(b) gl. dim End_{R}(M) ≤ 3.

(c) For any X ∈ CM(R), there exists an exact sequence 0 → M_{1} → M_{0} → X → 0 with M_{i} ∈ add M.

(d) For any indecomposable direct summand X of M, there exists an exact sequence 0 → M_{2} → M_{1} → M_{0} →^{a} X with M_{i} ∈ add M and a is a right almost split map in add M.

Proof. (a)⇔(b) Apply [I2, Th. 5.1(3)] for d = m = 1 and n = 2 there.

(a)⇔(c) See [I1, Prop. 2.2.2].

(a)⇒(d) See [I1, Th. 3.3.1].

(d)⇒(b) For any simple End_{R}(M)-module S, there exists an indecomposable direct summand X of M such that S is the top of the projective Hom_{R}(M, X). Then the sequence in (d) gives a projective resolution 0 → Hom_{R}(M, M_{2}) → Hom_{R}(M, M_{1}) → Hom_{R}(M, M_{0}) → Hom_{R}(M, X) → S → 0. Thus we have pd S ≤ 3 and gl. dim End_{R}(M) ≤ 3. □

The sequence in (d) is called a 2-almost split sequence when X is non-projective. There is a close relationship between 2-almost split sequences and exchange sequences [IY].

We shall construct exact sequences satisfying the above condition (d) in Lemma 4.5 and Lemma 4.6 below.

We use the equality

Hom_{R}(S_{i}, S_{j}) ≃ \{ \begin{matrix}(f_{i+1} ⋯ f_{j})/(f_{1} ⋯ f_{j}) & i < j \\ S/(f_{1} ⋯ f_{j}) & i ≥ j. \end{matrix}

Lemma 4.5. (a) We have exchange sequences

0 → S_{i} ⟶^{(f_{i+1} −1)} S_{i+1} ⊕ S_{i−1} \underrightarrow{\binom{1}{f_{i+1}}} S/(f_{1} ⋯ f_{i−1}f_{i+1}) → 0,

0 → S/(f_{1} ⋯ f_{i−1}f_{i+1}) ⟶^{(f_{i} 1)} S_{i+1} ⊕ S_{i−1} \underrightarrow{\binom{−1}{f_{i}}} S_{i} → 0.

(b) Under the assumption (B), we have a 2-almost split sequence

0 → S_{i} ⟶^{(f_{i+1} −1)} S_{i+1} ⊕ S_{i−1} \underrightarrow{\binom{f_{i} 1}{f_{i}f_{i+1} f_{i+1}}} S_{i+1} ⊕ S_{i−1} \underrightarrow{\binom{−1}{f_{i}}} S_{i} → 0

in add ⨁_{i=1}^{n} S_{i} for any 1 ≤ i < n.

Proof. (a) Consider the map a := \binom{−1}{f_{i}} : S_{i+1} ⊕ S_{i−1} → S_{i}. Any morphism from S_{j} to S_{i} factors through 1 : S_{i+1} → S_{i} (resp. f_{i} : S_{i−1} → S_{i}) if j > i (resp. j < i). Thus a is a minimal right (add ⨁_{j≠i} S_{j})-approximation.

It is easily checked that Ker a = \{s ∈ S_{i+1} | \overline{s} ∈ f_{i}S_{i}\} = (f_{i})/(f_{1} ⋯ f_{i+1}) ≃ S/(f_{1} ⋯ f_{i−1}f_{i+1}), where we denote by \overline{s} the image of s via the natural surjection S_{i+1} → S_{i}.

Consider the surjective map b := \binom{1}{f_{i+1}} : S_{i+1} ⊕ S_{i−1} → S/(f_{1} ⋯ f_{i−1}f_{i+1}). It is easily checked that Ker b = \{s ∈ S_{i+1} | \overline{s} ∈ (f_{i+1})/(f_{1} ⋯ f_{i−1}f_{i+1})\} = (f_{i+1})/(f_{1} ⋯ f_{i+1}) ≃ S_{i}, where we denote by \overline{s} the image of s via the natural surjection S_{i+1} → S/(f_{1} ⋯ f_{i−1}f_{i+1}).

(b) This sequence is exact by (a). Any non-isomorphic endomorphism of S_{i} is multiplication with an element in 𝔪, which is equal to (f_{i}, f_{i+1}) by (B). Since f_{i+1} (resp. f_{i}) : S_{i} → S_{i} factors through 1 : S_{i+1} → S_{i} (resp. f_{i} : S_{i−1} → S_{i}), we have that a is a right almost split map. □

Now we choose f_{n+1} ∈ 𝔪 such that 𝔪 = (f_{n}, f_{n+1}), and f_{n+1} and f_{1} ⋯ f_{n} has no common factor.

Lemma 4.6. Under the assumption (B), we have an exact sequence

0 → S_{n−1} ⟶^{(f_{n} −f_{n+1})} S_{n} ⊕ S_{n−1} \underrightarrow{\binom{f_{n+1}}{fn}} S_{n}

with a right almost split map \binom{f_{n+1}}{f_{n}} in add ⨁_{i=1}^{n} S_{i}.

Proof. Consider the map a := \binom{f_{n+1}}{f_{n}} : S_{n} ⊕ S_{n−1} → S_{n}. Any morphism from S_{j} (j < n) to S_{n} factors through f_{n} : S_{n−1} → S_{n}.

Any non-isomorphic endomorphism of S_{n} is multiplication with an element in 𝔪 = (f_{n+1}, f_{n}). Since f_{n} : S_{n} → S_{n} factors through f_{n} : S_{n−1} → S_{n}, we have that a is a right almost split map.

It is easily checked that Ker a = {s ∈ S_{n−1} | f_{n}s ∈ f_{n+1}S_{n}} = (f_{n+1}, f_{1} ⋯ f_{n−1})/(f_{1} ⋯ f_{n−1}), which is isomorphic to S_{n−1} by the choice of f_{n+1}. □

Thus we finished the proof of Theorem 4.1 under the stronger assumption (B).

To show the general statement of Theorem 4.1, we need some preliminary observations.

Lemma 4.7. Let R and R′ be complete local Gorenstein rings with dim R = dim R′ and M a rigid object in CM(R) which is a generator. Assume that there exists a surjection R′ → R, and we regard CM(R) as a full subcategory of CM(R′). If R′ ⊕ M is a cluster tilting object in CM(R′), then M is a cluster tilting object in CM(R).

Proof. We use the equivalence (a)⇔(c) in Proposition 4.4. For any X ∈ CM(R), there exists

an exact sequence 0 → N_{1} → N_{0} →^{f} X → 0 with N_{i} ∈ add(R′ ⊕ M) and a minimal right add(R′ ⊕ M)-approximation f of X. Since f is right minimal, we have N_{1} ∈ add M. Since M is a generator of R, we have that N_{0} ∈ add M. Thus M satisfies condition (c) in Proposition 4.4. □

Next let us consider cluster mutation in CM(R). We use the notation introduced at the beginning of this section.

Lemma 4.8. For w ∈ 𝔖_{n}, we assume that M_{w} is a cluster tilting object in CM(R). Then, for 1 ≤ i < n and s_{i} = (i i + 1), we have exchange sequences

0 → S_{i}^{w} → S_{i+1}^{w} ⊕ S_{i−1}^{w} → S_{i}^{ws_{i}} → 0 and 0 → S_{i}^{ws_{i}} → S_{i+1}^{w} ⊕ S_{i−1}^{w} → S_{i}^{w} → 0.

Proof. Without loss of generality, we can assume w = 1. Then the assertion follows from Lemma 4.5(a). □

Immediately, we have the following.

Proposition 4.9. Assume that M_{w} is a cluster tilting object in CM(R) for some w ∈ 𝔖_{n}. (a) The mutations of M_{w} are M_{ws_{i}} (1 ≤ i < n).

(b) M_{w′} is a cluster tilting object in CM(R) for any w′ ∈ 𝔖_{n}.

Proof. (a) This follows from Lemma 4.8.

(b) This follows from (a) since 𝔖_{n} is generated by s_{i} (1 ≤ i < n). □

Now we shall prove Theorem 4.1. Since k is an infinite field, we can take irreducible formal power series g_{i} ∈ 𝔪 (1 ≤ i ≤ n) such that h_{2i−1} := f_{i} and h_{2i} := g_{i} satisfy the following conditions:

• (h_{i}) ≠ (h_{j}) for any i ≠ j.

• 𝔪 = (h_{1}, h_{2}) = (h_{2}, h_{3}) = ⋯ = (h_{2i−1}, h_{2i}). Put R′ := S/(h_{1} ⋯ h_{2n−1}). This is reduced by the first condition. Since we have already proved Theorem 4.1 under the assumption (B), we have that ⨁_{i=1}^{2n−1} S/(h_{1} ⋯ h_{i})

is a cluster tilting object in CM(R′). By Proposition 4.9, ⨁_{i=1}^{2n−1} S/(h_{w(1)} ⋯ h_{w(i)}) is a cluster tilting object in CM(R′) for any w ∈ 𝔖_{2n−1}. In particular,

(⨁_{i=1}^{n} S/(f_{1} ⋯ f_{i})) ⊕ (⨁_{i=1}^{n−1} S/(f_{1} ⋯ f_{n}g_{1} ⋯ g_{i}))

is a cluster tilting object in CM(R′). Moreover we have surjections

R′ → ⋯ → S/(f_{1} ⋯ f_{n}g_{1}g_{2}) → S/(f_{1} ⋯ f_{n}g_{1}) → R.

Using Lemma 4.7 repeatedly, we have that ⨁_{i=1}^{n}(S/(f_{1} ⋯ f_{i})) is a cluster tilting object in CM(R). Thus we have proved Theorem 4.1. □

In the rest we shall show Theorem 4.2. We recall results on tilting mutation due to Riedtmann-Schofield [RS]. For simplicity, a tilting module means a tilting module of projective dimension at most one.

Let Γ be a module-finite algebra over a complete local ring with n simple modules. Their results remain valid in this setting. Recall that, for basic tilting Γ-modules T and U, we write

T ≤ U

if Ext_{Γ}^{1}(T, U) = 0. Then ≤ gives a partial order. On the other hand, we call a Γ-module T almost complete tilting if pd_{Γ}T ≤ 1, Ext_{Γ}^{1}(T, T) = 0 and T has exactly (n − 1) non-isomorphic indecomposable direct summands.

Proposition 4.10. (a) Any almost complete tilting Γ-module has at most two complements.

(b) T and U are neighbors in the partial order if and only if there exists an almost complete tilting Γ-module which is a common direct summand of T and U.

(c) Assume T ≤ U. Then there exists a sequence T = T_{0} < T_{1} < T_{2} < ⋯ < U satisfying the following conditions.

(i) T_{i} and T_{i+1} are neighbors.

(ii) Either T_{i} = U for some i or the sequence is infinite.

If the conditions in (b) above are satisfied, we call T a (tilting) mutation of U.

We also need the following easy observation on Cohen-Macaulay tilting modules.

Lemma 4.11. Let Γ be a module-finite algebra over a complete local Gorenstein ring R such that Γ ∈ CM(R), and T and U tilting Γ-modules. Assume U ∈ CM(Γ).

(a) If T ≤ U, then T ∈ CM(Γ).

(b) Let P be a projective Γ-module such that Hom_{R}(P, R) is a projective Γ^{op}-module. Then P ∈ add U.

Proof. (a) Recall that T ≤ U holds if and only if there exists an exact sequence 0 → T → U_{0} → U_{1} → 0 with U_{i} ∈ add U. Thus the assertion holds.

(b) There exists an exact sequence 0 → P → U_{0} → U_{1} → 0 with U_{i} ∈ add U, which must split since Ext_{R}^{1}(U, P) = 0. □

Finally, let us recall the following relation between cluster tilting and tilting (see [I2, Th. 5.3.2] for (a), and (b) is clear).

Proposition 4.12. Let R be a complete local Gorenstein ring and M, N and N′ cluster tilting objects in CM(R).

(a) Hom_{R}(M, N) is a tilting End_{R}(M)-module of projective dimension at most one.

(b) If N′ is a mutation of N, then Hom_{R}(M, N′) is a mutation of Hom_{R}(M, N).

Now we shall prove Theorem 4.2. Fix w ∈ 𝔖_{n} and put Γ := End_{R}(M_{w}). The functor Hom_{R}(M_{w}, ) : CM(R) → CM(Γ) is fully faithful since M_{w} is a generator of R. By Theorem 4.1, M_{w} is a cluster tilting object in CM(R). By Proposition 4.12(a), Hom_{R}(M_{w}, M_{w′}) (w′ ∈ 𝔖_{n}) is a Cohen-Macaulay tilting Γ-module.

(b) Take any Cohen-Macaulay tilting Γ-module U. Since P := Hom_{R}(M_{w}, R) is a projective Γ-module such that Hom_{R}(P, R) = M_{w} = Hom_{R}(R, M_{w}) is a projective Γ^{op}-module, we have P ∈ add U by Lemma 4.11(b). In particular, each Cohen-Macaulay tilting Γ-module has at most (n − 1) mutations which are Cohen-Macaulay by Proposition 4.10(a)(b). Conversely, any Cohen-Macaulay tilting Γ-module of the form Hom_{R}(M_{w}, M_{w′}) (w′ ∈ 𝔖_{n}) has precisely (n − 1) mutations Hom_{R}(M_{w}, M_{w′s_{i}}) (1 ≤ i < n) which are Cohen-Macaulay by Proposition 4.9 and Proposition 4.12(b).

Now we shall show that U is isomorphic to Hom_{R}(M_{w}, M_{w′}) for some w′. Since Γ ≤ U, there exists a sequence

Γ = T_{0} < T_{1} < T_{2} < ⋯ < U

satisfying the conditions in Proposition 4.10(c). By Lemma 4.11(a), each T_{i} is Cohen-Macaulay. Since Γ has the form Γ = Hom_{R}(M_{w}, M_{w}), the above argument implies that each T_{i} has the form Hom_{R}(M_{w}, M_{w′}) for some w′ ∈ 𝔖_{n}. Since 𝔖_{n} is a finite group, the above sequence must be finite. Thus U = T_{i} holds for some i, hence the proof is completed.

(a) Let U be a cluster tilting object in CM(R). Again by Proposition 4.12(a), Hom_{R}(M_{w}, U) is a Cohen-Macaulay tilting Γ-module. By part (b) which we already proved, Hom_{R}(M_{w}, U) is isomorphic to Hom_{R}(M_{w}, M_{w′}) for some w′ ∈ 𝔖_{n}. Thus U is isomorphic to M_{w′}, and the former assertion is proved.

For the latter assertion, we only have to show that any rigid object in CM(R) is a direct summand of some cluster tilting object in CM(R). This is valid by the following general result in [BIRS, Th. 1.9]. □

Proposition 4.13. Let C be a 2-CY Frobenius category with a cluster tilting object. Then any rigid object in C is a direct summand of some cluster tilting object in C.

We end this section with the following application to dimension three.

Now let S′′ := k[[x, y, u, v]], f_{i} ∈ 𝔪 = (x, y) (1 ≤ i ≤ n) and R′′ := S′′/(f_{1} ⋯ f_{n} + uv). For w ∈ 𝔖_{n} and I ⊆ {1, ⋯, n}, we put

U_{i}^{w} := (u, f_{w(1)} ⋯ f_{w(i)}) ⊂ R′′, M_{w} := ⨁_{i=1}^{n} U_{i}^{w} and U_{I} := (u, ∏_{i∈I} f_{i}) ⊂ S′′.

We have the following result.

Corollary 4.14. Under the assumption (A), we have the following.

(a) There are exactly n! indecomposable rigid objects M_{w} (w ∈ 𝔖_{n}) and exactly 2^{n} − 1 indecomposable rigid objects U_{I} (∅ ≠ I ⊂ {1, ⋯, n}) in CM(R′′).

(b) There are non-commutative crepant resolutions End_{R′′}(M_{w}) (w ∈ 𝔖_{n}) of R′′, which are derived equivalent.

Proof. (a) This follows from Knörrer periodicity \underline{CM}(R) → \underline{CM}(R′′).

(b) Any cluster tilting object gives a non-commutative crepant resolution. See 5.4 below. □

For example,

k[[x, y, u, v]]/((x − λ_{1}y) ⋯ (x − λ_{n}y) + uv)

has a non-commutative crepant resolution for distinct elements λ_{1}, ⋯, λ_{n} ∈ k.

5. Link with birational geometry

There is another approach to the investigation of cluster tilting objects for maximal Cohen-Macaulay modules, using birational geometry. More specifically there is a close connection between resolutions of three dimensional Gorenstein singularities and cluster-tilting theory, provided by the so-called non-commutative crepant resolutions of Van den Bergh. This provides at the same time alternative proofs for geometric results, using cluster tilting objects. The aim of this section is to establish this link with small resolutions. We give relevant criteria for having small resolutions, and apply them to give an alternative approach to most of the results in the previous sections.

Let (R, 𝔪) be a complete normal Gorenstein algebra of Krull dimension 3 over an algebraically closed field k, and let X = Spec(R). A resolution of singularities Y ⟶^{π} X is called

• crepant, if ω_{Y} ≅ π^{∗}ω_{X} ≅ O_{Y}.

• small, if the relative dimension of the exceptional locus of π is smaller than one. A small resolution is automatically crepant, but the converse is in general not true. However, both types of resolutions coincide for certain important classes of three-dimensional singularities.

A cDV (compound Du Val) singularity is a three dimensional singularity given by the equation

f(x, y, z) + tg(x, y, z, t) = 0,

where f(x, y, z) defines a simple surface singularity and g(x, y, z, t) is arbitrary.

It is called cA_{n} (respectively, cD_{n}, cE_{n}) if the intersection of f + tg with a generic hyperplane in k^{4} is an A_{n} (respectively, D_{n}, E_{n}) surface singularity. By definition, a generic hyperplane means that the coefficients defining this hyperplane belong to a Zariski (dense) open subset of k^{4}. Note that any cDV singularity is terminal.

Theorem 5.1. [Re, Cor. 1.12, Th. 1.14] Let X be a Gorenstein threefold singularity. (a) If X has a small resolution, then it is cDV.

(b) If X is an isolated cDV singularity, then any crepant resolution of X is small.

There is a close connection with the non-commutative crepant resolutions of Van den Bergh defined as follows.

Definition 5.2. [V2, Def. 4.1] Let (R, 𝔪) be a normal Gorenstein domain of Krull dimension three. An R-module M gives rise to a non-commutative crepant resolution if

(i) M is reflexive, (ii) A = End_{R}(M) is Cohen-Macaulay as an R–module, (iii) gl.dim(A) = 3.

The following result establishes a useful connection.

Theorem 5.3. [V1, Cor. 3.2.11][V2, Th. 6.6.3] Let (R, 𝔪) be an isolated cDV singularity of Krull dimension three. Then there exists a crepant resolution of X = Spec R if and only if there exists a non-commutative one in the sense of Definition 5.2.

The existence of a non-commutative crepant resolution turns out to be equivalent to the existence of a cluster tilting object in the triangulated category \underline{CM}(R).

Theorem 5.4. [I2, Th. 5.2.1][IR, Th. 8.9] Let (R, 𝔪) be a normal Gorenstein isolated singularity of Krull dimension three. Then the existence of a non-commutative crepant resolution is equivalent to the existence of a cluster tilting object in the stable category of maximal Cohen-Macaulay modules \underline{CM}(R).

Proof. For convenience of the reader, we give an outline of the proof (see also Proposition 4.4).

Let us first assume that M is a cluster tilting object in \underline{CM}(R). Then M is automatically reflexive. From the exact sequence

0 ⟶ Ω(M) ⟶ F ⟶ M ⟶ 0

we obtain

(1) 0 ⟶ End_{R}(M) ⟶ Hom_{R}(F, M) ⟶ Hom_{R}(Ω(M), M) ⟶ Ext_{R}^{1}(M, M) ⟶ 0.

Since M is rigid, Ext_{R}^{1}(M, M) = 0. Moreover, depth(Hom_{R}(F, M)) = depth(M) = 3 and depth(Hom_{R}(Ω(M), M) ≥ 2, and hence depth(End_{R}(M)) = 3 and A = End(M) is maximal Cohen-Macaulay over R.

For the difficult part of this implication, claiming that gl.dim(A) = 3, we refer to [I1, Th. 3.6.2].

For the other direction, let M be a module giving rise to a non-commutative crepant resolution. Then by [IR, Th. 8.9] there exists another module M′ giving rise to a non-commutative crepant resolution, which is maximal Cohen-Macaulay and contains R as a direct summand.

By the assumption, depth(End_{R}(M′)) = 3 and we can apply [IR, Lem. 8.5] to the exact sequence(1) to deduce that Ext_{R}^{1}(M′, M′) = 0, so that M′ is rigid. The difficult part saying that M′ is cluster tilting is proven in [I2, Th. 5.2.1]. □

We now summarize the results of this section.

Theorem 5.5. Let (R, 𝔪) be a three dimensional isolated cDV singularity. Then the following are equivalent.

(a) Spec R has a small resolution.

(b) Spec R has a crepant resolution.

(c) (R, 𝔪) has a non-commutative crepant resolution.

(d) \underline{CM}(R) has a cluster tilting object.

We have an efficient criterion for existence of a small resolution of a cA_{n} singularity.

Theorem 5.6. [Kat, Th. 1.1] Let X = Spec(R) be an isolated cA_{n} singularity.

(a) Let Y ⟶ X be a small resolution. Then the exceptional curve in Y is a chain of n projective lines and X has the form g(x, y) + uv, where the curve singularity g(x, y) has n + 1 distinct branches at the origin.

(b) If X has the form g(x, y) + uv, where the curve singularity g(x, y) has n + 1 distinct branches at the origin, then X has a small resolution.

Using the criterion of Katz together with Knörrer periodicity, we get additional equivalent conditions in a special case.

Theorem 5.7. Let (R, 𝔪) be a three dimensional isolated cA_{n} singularity defined by the equation g(x, y) + zt. Then the following conditions are equivalent in addition to (a)-(d) in Theorem 5.5.

(e) Let R′ be a one dimensional singularity defined by g(x, y). Then \underline{CM}(R′) has a cluster tilting object.

(f) The number of irreducible power series in the prime decomposition of g(x, y) is n + 1.

Proof. (a)⇔(f) This follows from Theorem 5.6.

(d)⇔(e) By the Knörrer correspondence there is an equivalence of triangulated categories between the stable categories \underline{CM}(R) ≅ \underline{CM}(R′). For, the equivalence of these stable categories given in [Kn], [So] is induced by an exact functor taking projectives to projectives. □

Theorem 5.8. Assume that the equivalent conditions in Theorem 5.7 are satisfied. Then the following numbers are equal.

(a) One plus the number of irreducible components of the exceptional curve of a small resolution of Spec R.

(b) The number of irreducible power series in the prime decomposition of g(z, t).

(c) The number of simple modules of non-commutative crepant resolutions of (R, 𝔪).

(d) One plus the number of non-isomorphic indecomposable summands of cluster tilting objects in \underline{CM}(R).

Proof. (a) and (b) are equal by Theorem 5.6.

(a) and (c) are equal by [V1, Th. 3.5.6].

(c) and (d) are equal by [IR, Cor. 8.8]. □

6. Application to curve singularities

In this section we apply results in the previous section to some curve singularities to investigate whether they have cluster tilting object or not. In addition to simple singularities, we study some other nice singularities. In what follows we refer to [AGV] as a general reference for classification of singularities.

To apply results in previous sections to minimally elliptic singularities, we also consider a three-dimensional hypersurface singularity

T_{p,q,2,2}(λ) = k[[x, y, u, v]]/(x^{p} + y^{q} + λx^{2}y^{2} + uv).

To apply Theorem 5.7 to a curve singularity, we have to know that the corresponding three dimensional singularity is cA_{n}. It is given by the following result, where we denote by ord(g) the degree of the lowest term of a power series g.

Proposition 6.1. We have the following properties of three-dimensional hypersurface singularities:

(a) A_{n} (n ≥ 1) is a cA_{1}–singularity,

(b) D_{n} (n ≥ 4) and E_{n} (n = 6, 7, 8) are cA_{2}–singularities,

(c) T_{3,q,2,2}(λ) (q ≥ 6) is a cA_{2}–singularity,

(d) T_{p,q,2,2}(λ) (p ≥ q ≥ 4) is a cA_{3}–singularity,

(e) k[[x, y, z, t]]/(x^{2} + y^{2} + g(z, t)) (g ∈ k[[z, t]]) is a cA_{m}–singularity if m = ord(g) − 1 ≥ 1.

We shall give a detailed proof at the end of this section. In view of Theorem 5.7 and Proposition 6.1, we have the following main result in this section.

Theorem 6.2. (a) A simple three dimensional singularity satisfies the equivalent conditions

in Theorem 5.8 if and only if it is of type A_{n} (n is odd) or D_{n} (n is even).

(b) A T_{p,q,2,2}(λ)–singularity satisfies the equivalent conditions in Theorem 5.8 if and only if p = 3 and q is even or if both p and q are even.

(c) A singularity k[[x, y, u, v]]/(uv + f_{1} ⋯ f_{n}) with irreducible and mutually prime f_{i} ∈ (x, y) ⊂ k[[x, y]] (1 ≤ i ≤ n) satisfies the equivalent conditions in Theorem 5.8 if and only if f_{i} ∉ (x, y)^{2} for any i.

Proof. Each singularity is cA_{m} by Proposition 6.1, and defined by an equation of the form g(x, y) + uv. By Theorem 5.8, we only have to check whether the number of irreducible power series factors of g(x, y) is m + 1 or not.

(a) For an A_{n}–singularity, we have m = 1 and g(x, y) = x^{2} + y^{n+1}. So g has two factors if and only if n is odd.

For a D_{n}–singularity, we have m = 2 and g(x, y) = (x^{2} + y^{n−2})y. So g has three factors if and only if n is even.

For an E_{n}–singularity, we have m = 2 and g(x, y) = x^{3} + y^{4}, x(x^{2} + y^{3}) or x^{3} + y^{5}. In each case, g does not have three factors.

(b) First we consider the simply elliptic case. We have m = 2 and g(x, y) = y(y −x^{2})(y −λx^{2}) for (p, q) = (3, 6), and m = 3 and g(x, y) = xy(x − y)(x − λy) for (p, q) = (4, 4). In both cases, g has m + 1 factors.

Now we consider the cusp case. We have m = 2 for p = 3 and m = 3 for p > 3, and g(x, y) = (x^{p−2} − y^{2})(x^{2} − y^{q−2}). So g has m + 1 factors if and only if p = 3 and q is even or if both p and q are even.

(c) We have m = ∑_{i=1}^{n} ord(f_{i}) − 1 and g = f_{1} ⋯ f_{n}. So g has m + 1 factors if and only if ord(f_{i}) = 1 for any i. □

Immediately we have the following conclusion.

Corollary 6.3. (a) A simple curve singularity has a cluster tilting object if and only if it is of type A_{n} (n is odd) or D_{n} (n is even). The number of non-isomorphic indecomposable summands of cluster tilting objects is 1 for type A_{n} (n is odd) and 2 for type D_{n} (n is even).

(b) A T_{p,q}(λ)-singularity has a cluster tilting object if and only if p = 3 and q is even or if both p and q are even. The number of non-isomorphic indecomposable summands of cluster tilting objects is 2 if p = 3 and q is even, and 3 if both p and p are even.

(c) A singularity R = k[[x, y]]/(f_{1} ⋯ f_{n}) with irreducible and mutually prime f_{i} ∈ (x, y) ⊂ k[[x, y]] (1 ≤ i ≤ n) has a cluster tilting object if and only if f_{i} ∉ (x, y)^{2} for any i. In this case, the number of non-isomorphic indecomposable summands of cluster tilting objects in \underline{CM}R is n.

Summarizing with Theorem 4.2, we have completed the proof of Theorem 1.4.

In the rest of this section, we shall prove Proposition 6.1.

Let k be an algebraically closed field of characteristic zero, R = k[[x_{1}, x_{2}, …, x_{n}]] the local ring of formal power series and 𝔪 its maximal ideal. We shall need the following standard notions.

Definition 6.4. For f ∈ 𝔪^{2} we denote by J(f) = ⟨\frac{∂f}{∂x_{1}} , …, \frac{∂f}{∂x_{n}} ⟩ its Jacobi ideal. The Milnor number μ(f) is defined as

μ(f) := dim_{k}(R/J(f)).

The following lemma is standard (see for example [AGV, GLSh]):

Lemma 6.5. A hypersurface singularity f = 0 is isolated if and only if μ(f) < ∞.

Definition 6.6 ([AGV]). Two hypersurface singularities f = 0 and g = 0 are called right equivalent (f ∼^{r} g) if there exists an algebra automorphism φ ∈ Aut(R) such that g = φ(f).

Note, that f ∼^{r} g implies an isomorphism of k–algebras

R/(f) ≅ R/(g).

The following lemma is straightforward, see for example [GLSh, Lem. 2.10].

Lemma 6.7. Assume f ∼^{r} g, then μ(f) = μ(g).

In what follows, we shall need the next standard result on classification of singularities, see for example [GLSh, Cor. 2.24].

Theorem 6.8. Let f ∈ 𝔪^{2} be an isolated singularity with Milnor number μ. Then

f ∼^{r} f + g

for any g ∈ 𝔪^{μ+2}. One also says that f is (μ + 1) – determined meaning that the equivalence class of f is determined by its (μ + 1) – jet.

We shall need the following easy lemma.

Lemma 6.9. Let f = x^{2} + y^{2} + p(x, y, z), where

p(x, y, z) = z^{n} + p_{1}(x, y)z^{n−1} + ⋯ + p_{n}(x, y)

is a homogeneous form of degree n ≥ 3. Then

f ∼^{r} x^{2} + y^{2} + z^{n}.

Proof. Write p(x, y, z) = z^{n} + xu + yv for some homogeneous forms u and v of degree n − 1. Then

x^{2} + y^{2} + z^{n} + xu + yv = (x + u/2)^{2} + (y + v/2)^{2} + z^{n} − (u^{2} + v^{2})/4.

After a change of variables x ↦ x + u/2, y ↦ y + v/2 and z ↦ z we reduce f to the form

f = x^{2} + y^{2} + z^{n} + h,

where h ∈ 𝔪^{2(n−1)} ⊂ 𝔪^{n+1}. Note that μ(x^{2} + y^{2} + z^{n}) = n − 1, hence by Theorem 6.8 we have

f ∼^{r} x^{2} + y^{2} + z^{n}.

□

Now we are ready to give a proof of Proposition 6.1. We only have to show the assertion (e) since other cases are special cases of this. We denote by H the hyperplane in a four-dimensional space defined by the equation t = αx + βy + γz, α, β, γ ∈ k. We put

g(z, t) = a_{0}z^{m+1} + a_{1}z^{m}t + ⋯ + a_{m+1}t^{m+1} + (higher terms).

Then the intersection of H with the singularity defined by the equation x^{2} + y^{2} + g(z, t) is given by the equation f = h + (higher terms), where

h = x^{2} + y^{2} + a_{0}z^{m+1} + a_{1}z^{m}(αx + βy + γz) + ⋯ + a_{m+1}(αx + βy + γz)^{m+1}.

Now we consider the case m = 1. We have h ∼^{r} x^{2} + y^{2} + z^{2} since any quadratic form can be diagonalized using linear transformations. By Lemma 6.7, we have μ(h) = μ(x^{2} + y^{2} + z^{2}) = 1. Hence f ∼^{r} h ∼^{r} x^{2} + y^{2} + z^{2} by Theorem 6.8.

Next we consider the case m ≥ 2. Assume α ∈ k satisfies a_{0} + a_{1}α + ⋯ + a_{m+1}α^{m+1} ≠ 0. By Lemma 6.9, we have h ∼^{r} x^{2} +y^{2} +z^{m+1}. By Lemma 6.7, we have μ(h) = μ(x^{2} +y^{2} +z^{m+1}) = m. Hence f ∼^{r} h ∼^{r} x^{2} + y^{2} + z^{m+1} by Theorem 6.8.

Consequently, x^{2} + y^{2} + g(z, t) is cA_{m}. □

7. Examples of 2-CY tilted algebras

Since the 2-CY tilted algebras coming from maximal Cohen-Macaulay modules over hypersurfaces have some nice properties, it is of interest to have more explicit information about such algebras. This section is devoted to some such computations for algebras coming from minimally elliptic singularities. We obtain algebras appearing in classification lists for some classes of tame self-injective algebras [Er, BS].

We start with giving some general properties which are direct consequences of Lemma 2.2.

Theorem 7.1. Let (R, 𝔪) be an isolated hypersurface singularity and Γ a 2-CY tilted algebra coming from \underline{CM}(R). Then we have the following.

(a) Γ is a symmetric algebra.

(b) All components in the stable AR-quiver of Γ are tubes of rank 1 or 2.

We now start with our computations of 2-CY tilted algebras coming from minimally elliptic singularities. We first introduce and investigate two classes of algebras, and then show that they are isomorphic to 2-CY tilted algebras coming from minimally elliptic singularities.

For a quiver Q with finitely many vertices and arrows we define the radical completion k̂Q of the path algebra kQ by the formula

k̂Q = lim_{⟵} kQ/ rad^{n}(kQ).

The reason we deal with completion is the following: Let Q be a finite quiver, J the ideal of k̂Q generated by the arrows and I ⊆ J^{2} a complete ideal such that Λ = k̂Q/I is finite-dimensional.

Lemma 7.2. The ideal I is generated in k̂Q by a minimal system of relations, that is, a set of elements ρ_{1}, ⋯, ρ_{n} of I whose images form a k-basis of I/IJ + JI.

The lemma is shown by a standard argument (cf [BMR3, Section 3]). Its analogue for the non complete path algebra is not always true. For example, for the algebra Λ = B_{2,2}(λ) defined below, the elements ρ_{1}, ⋯, ρ_{n} listed as generators for I form a minimal system of relations. So they generate I in k̂Q. They also yield a k-basis of I′/I′J + JI′ ⟶^{∼} I/IJ + JI, where I′ = I ∩kQ and J′ = J ∩kQ. But they do not generate the ideal I′ of kQ since, as one can show, the quotient kQ/⟨ρ_{1}, ⋯, ρ_{n}⟩ is infinite-dimensional.

On the other hand, the ideal I′ is generated by the preimage ρ_{1}, ⋯, ρ_{n} of a basis of I′/I′J′ + J′I′ if the quotient kQ/⟨ρ_{1}, ⋯, ρ_{n}⟩ is finite-dimensional, since then the ideal ⟨ρ_{1}, ⋯, ρ_{n}⟩ contains a power of J′. This happens for example for the algebra A_{2}(λ) as defined below, cf. also [Sk, 5.9] and [BS, Th. 1].

We know that for all vertices i, j of Q, we have

dim_{k} e_{i}(I/IJ + JI)e_{j} = dim_{k} Ext_{Λ}^{2}(S_{i}, S_{j})

where S_{i} and S_{j} denote the simple Λ-modules corresponding to the vertices i and j. When Λ is 2-CY tilted, then

dim Ext_{Λ}^{1}(S_{j}, S_{i}) ≥ dim Ext_{Λ}^{2}(S_{i}, S_{j})

(see [BMR3, KR]). Thus the number of arrows in Q is an upper bound on the number of elements in a minimal system of relations.

Definition 7.3. (1) For q ≥ 2 and λ ∈ k^{∗} we write A_{q}(λ) = k̂Q/I, where

Q = [figure: ·; φ; α; ·; β; ψ]

and

I = ⟨ψα − αφ, βψ − φβ, φ^{2} − βα, ψ^{q} − λαβ⟩.

If q = 2, then we additionally assume λ ≠ 1. (It can be shown that for q ≥ 3 we have A_{q}(λ) ≅ A_{q}(1), so we drop the parameter λ in this case.)

(2)For p, q ≥ 1 and λ ∈ k^{∗} we write B_{p,q}(λ) = k̂Q/I, where

Q = [figure: ·; φ; α; ·; β; γ; ·; δ; ψ]

and

I = ⟨βα − φ^{p}, γδ − λψ^{q}, αφ − δγα, φβ − βδγ, δψ − αβδ, ψγ − γαβ⟩.

For p = q = 1 we additionally assume λ ≠ 1.

When p = q = 1, the generators φ and ψ can be excluded and B_{1,1}(λ) is given by the completion of the path algebra of the quiver

Q = \xymatrix{· \ar[r]^{α} & · \ar[l]^{β} \ar[r]^{γ} & · \ar[l]^{δ}}

modulo the relations

I = ⟨αβα − δγα, αβδ − λδγδ, γαβ − λγδγ, βδγ − βαβ⟩.

For (p, q) ≠ (1, 1) we have B_{p,q}(λ) ≅ B_{p,q}(1). In particular, for p = 1 and q ≥ 2 the algebra is isomorphic to k̂Q/I, where

Q = [figure: ·; α; ·; β; γ; ·; δ; ψ]

and

I = ⟨γδ − ψ^{q}, αβα − δγα, βαβ − βδγ, δψ − αβδ, ψγ − γαβ⟩.

It turns out that the algebras A_{q}(λ) and B_{p,q}(λ) are finite dimensional. In order to show this it suffices to check that all oriented cycles in k̂Q/I are nilpotent.

Lemma 7.4. In the algebra A_{q}(λ) the following zero relations hold:

αβα = 0, βαβ = 0, αφ^{2} = ψ^{2}α = 0, φ^{2}β = βψ^{2} = 0, φ^{4} = 0, ψ^{q+2} = 0.

Proof. We have to consider separately the cases q = 2 and q ≥ 3.

Let q = 2, then we assumed λ ≠ 1. We have

αβα = αφ^{2} = ψ^{2}α = λ^{−1}αβα,

hence αβα = 0. In a similar way we obtain βαβ = 0. Then αφ^{2} = αβα = 0, φ^{2} = βαβα = 0 and the remaining zero relations follow analogously.

Let q ≥ 3. Then

ψ^{q}α = αβα = αφ^{2} = ψ^{2}α,

so (1 − ψ^{q−2})ψ^{2}α = 0 and hence

ψ^{2}α = αβα = 0

in k̂Q/I. The remaining zero relations follow similarly. □

Lemma 7.5. We have the following relations in B_{p,q}(λ):

φ^{p+2} = 0, ψ^{q+2} = 0, γαφ = ψγα = 0, φβδ = βδψ = 0.

Moreover, αβ · δγ = δγ · αβ. For q ≥ p ≥ 2 we have

(αβ)^{2} = (δγ)^{2} = 0,

for q > p = 1 we have

(αβ)^{3} = 0, (δγ)^{2} = 0, (αβ)^{2} · (δγ) = 0

and for p = q = 1

(αβ)^{3} = (γδ)^{3} = 0, (αβ)^{2} = αβ · δγ = λ(δγ)^{2}.

The proof is completely parallel to the proof of the previous lemma and is therefore skipped. □

The main result of this section is the following

Theorem 7.6. (a) Let R be a T_{3,2q+2}(λ)–singularity, where q ≥ 2 and λ ∈ k^{∗}. Then in the triangulated category \underline{CM}(R) there exists a cluster tilting object with the corresponding 2-CY-tilted algebra isomorphic to A_{q}(λ).

(b)For R = T_{2p+2,2q+2}(λ) the category \underline{CM}(R) has a cluster tilting object with endomorphism algebra isomorphic to B_{p,q}(λ).

Proof. (a) We consider first the case of T_{3,2q+2}(λ).

The coordinate ring of T_{3,6}(λ) is isomorphic to

R = k[[x, y]]/(y(y − x^{2})(y − λx^{2})),

where λ ≠ 0, 1. Consider Cohen-Macaulay modules M and N given by the two-periodic free resolutions

\{ \begin{matrix}M = (R ⟶^{y−x^{2}} R ⟶^{y(y−λx^{2})} R), \\ N = (R ⟶^{y(y−x^{2})} R ⟶^{y−λx^{2}} R). \end{matrix}

It is cluster tilting by Theorem 4.1 or Corollary 6.3. In order to compute the endomorphism algebra \underline{End}(M ∐ N), note that

\underline{End}(M) ≅ k[φ]/⟨φ^{4}⟩,

where φ = (x, x) is an endomorphism of M viewed as a two-periodic map of a free resolution. In \underline{End}(M) we have (y, y) = (x, x)^{2} = φ^{2}. Similarly,

\underline{End}(N) ≅ k[ψ]/⟨ψ^{4}⟩, ψ = (x, x), (y, y) = λ(x, x)^{2} = λψ^{2},

and

\underline{Hom}(M, N) = k^{2} = ⟨(1, y), (x, xy)⟩, \underline{Hom}(N, M) = k^{2} = ⟨(y, 1), (xy, x)⟩.

The isomorphism A_{2}(λ) ⟶ \underline{End}(M ∐ N) is given by

φ ↦ (x, x), ψ ↦ (x, x), α ↦ (1, y), β ↦ (y, 1).

Assume now q ≥ 3 and R = T_{3,2q+2}. By [AGV] we may write

R = k[[x, y]]/((x − y^{2})(x^{2} − y^{2q})).

Consider the Cohen-Macaulay module M ∐ N, where

\{ \begin{matrix}M = (R ⟶^{x−y^{2}} R ⟶^{x^{2}−y^{2q}} R), \\ N = (R ⟶^{(x−y^{2})(x+y^{q})} R ⟶^{x−y^{q}} R). \end{matrix}

Again, by a straightforward calculation

\underline{End}(M) ≅ k[φ]/⟨φ^{4}⟩, φ = (y, y), \underline{End}(N) ≅ k[ψ]/⟨ψ^{q+2}⟩, ψ = (y, y)

and

\underline{Hom}(M, N) = k^{2} = ⟨(1, x + y^{q}), (y, y(x + y^{q}))⟩,
\underline{Hom}(M, N) = k^{2} = ⟨(x + y^{q}, 1), (y(x + y^{q}), y)⟩.

If q ≥ 4 then \underline{End}(M ∐ N) is isomorphic to k̂Q/I, where

Q = [figure: ·; φ; α; ·; β; ψ]

and the relations are

βα = φ^{2}, αβ = 2ψ^{q}, αφ = ψα, φβ = βψ

for

φ = (y, y), ψ = (y, y), α = (1, x + y^{q}), β = (x + y^{q}, 1).

By rescaling all generators α ↦ 2^{a}α, β ↦ 2^{b}β, φ ↦ 2^{f}φ, ψ ↦ 2^{g}ψ for properly chosen a, b, f, g ∈ ℚ one can easily show \underline{End}(M ∐ N) ≅ A_{q}.

The case q = 3 has to be considered separately, since this time the relations are

βα = φ^{2} + φ^{3}, αβ = 2ψ^{q}, αφ = ψα, φβ = βψ.

We claim that there exist invertible power series u(t), v(t), w(t), z(t) ∈ k[[t]] such that the new generators

φ′ = u(φ)φ, ψ′ = v(ψ)ψ, α′ = αw(φ) = w(ψ)α, β′ = βz(ψ) = z(φ)β

satisfy precisely the relations of the algebra A_{3}. This is fulfilled provided we have the following equations in k[[t]]:

\{ \begin{matrix}zw = u^{2}(1 + tu) \\ zw = 2v^{3} \\ uw = vw \\ uz = vz. \end{matrix}

This system is equivalent to

u(t) = (2 − t)^{−1} = \frac{1}{2}(1 + \frac{t}{2} + (\frac{t}{2})^{2} + …)

and hence the statement is proven.

The case of T_{2p+2,2q+2}(λ) is essentially similar. For p = q = 1 we have

R = k[[x, y]]/(xy(x − y)(x − λy)).

Take

\{ \begin{matrix}M = (R ⟶^{x−y} R ⟶^{xy(x−λy)} R), \\ N = (R ⟶^{x(x−y)} R ⟶^{y(x−λy)} R), \\ K = (R ⟶^{xy(x−y)} R ⟶^{x−λy} R). \end{matrix}

By Theorem 4.1 or Corollary 6.3, M ∐ N ∐ K is cluster tilting. Moreover, B_{1,1}(λ) ≃ \underline{End}(M ∐ N ∐ K).

Let now

R = k[[x, y]]/((x^{p} − y)(x^{p} + y)(y^{q} − x)(y^{q} + x)),

where (p, q) ≠ (1, 1) and

\{ \begin{matrix}M = (R ⟶^{x^{p}−y} R ⟶^{(y^{q}+x)(y^{q}−x)(x^{p}+y)} R), \\ N = (R ⟶^{(x^{p}−y)(x^{p}+y)} R ⟶^{(y^{q}−x)(y^{q}+x)} R), \\ K = (R ⟶^{(x^{p}−y)(x^{p}+y)(y^{q}+x)} R ⟶^{y^{q}−x} R). \end{matrix}

By Theorem 4.1 or Corollary 6.3, M ∐ N ∐ K is cluster tilting, and by a similar case-by-case analysis it can be verified that \underline{End}(M ∐ N ∐ K) ≅ B_{p,q}. □

We have seen that the algebras A_{q}(λ) and B_{p,q}(λ) are symmetric, and the indecomposable nonprojective modules have τ-period at most 2, hence Ω-period dividing 4 since τ = Ω^{2} in this case. A direct computation shows that the Cartan matrix is nonsingular. Note that these algebras appear in Erdmann’s list of algebras of quaternion type [Er], see also [Sk], that is, in addition to the above properties, the algebras are tame. Note that for the corresponding algebras, more relations are given in Erdmann’s list. This has to do with the fact that we are working with the completion, as discussed earlier. In our case all relations correspond to different arrows in the quiver. The simply elliptic ones also appear in Białkowski-Skowroński’s list of weakly symmetric tubular algebras with a nonsingular Cartan matrix.

This provides a link between some stable categories of maximal Cohen-Macaulay modules over isolated hypersurface singularities, and some classes of finite dimensional algebras, obtained via cluster tilting theory.

Previously a link between maximal Cohen-Macaulay modules and finite dimensional algebras was given with the canonical algebras of Ringel, via the categories coh𝕏 of coherent sheaves on weighted projective lines in the sense of Geigle-Lenzing [GL]. Here the category of vector bundles is equivalent to the category of graded maximal Cohen-Macaulay modules with degree zero maps, over some isolated singularity. And the canonical algebras are obtained as endomorphism algebras of certain tilting objects in coh𝕏 which are vector bundles.

Note that it is known from work of Dieterich [D], Kahn [Kah], Drozd and Greuel [DG] that minimally elliptic curve singularities have tame Cohen-Macaulay representation type. Vice versa, any Cohen-Macaulay tame reduced hypersurface curve singularity is isomorphic to one of the T_{p,q}(λ), see [DG]. Moreover, simply elliptic singularities are tame of polynomial growth and cusp singularities are tame of exponential growth. Furthermore, the Auslander-Reiten quiver of the corresponding stable categories of maximal Cohen-Macaulay modules consists of tubes of rank one or two, see [Kah, Th. 3.1] and [DGK, Cor. 7.2].

It should follows from the tameness of CM(T_{3,p}(λ)) and CM(T_{p,q}(λ)) that the associated 2-CY tilted algebras are tame.

We point out that in the wild case we can obtain symmetric 2-CY tilted algebras where the stable AR-quiver consists of tubes of rank one and two, and most of them should be wild. It was previously known that there are examples of wild selfinjective algebras whose AR-quivers consist of tubes of rank one or three [AR].

8. Appendix: 2-CY triangulated categories of finite type

In this section, we consider more general situation than in section 2. Let k be an algebraically closed field and C a k-linear connected 2-Calabi-Yau triangulated category with only finitely many indecomposable objects. We show that it follows from the shape of the AR quiver of C whether cluster tilting objects (resp. non-zero rigid objects) exist in C or not. Let us start with giving the possible shapes of the AR quiver of C.

Proposition 8.1. The AR quiver of C is ZΔ/H for a Dynkin diagram Δ and a weakly admissible subgroup H of Aut(ZΔ) which contains F ∈ Aut(ZΔ) defined by the list below. Moreover, H is generated by a single element h ∈ Aut(ZΔ) in the list below.

Δ  Aut(ZΔ)  F  h
(A_{n}) n : odd  Z × Z/2Z (\frac{n+3}{2}, 1)  (k, 1) (k|\frac{n+3}{2}, \frac{n+3}{2k} is odd)
(A_{n}) n : even  Z  n + 3  k (k|n + 3)
(D_{n}) n : odd  Z × Z/2Z (n, 1)  (k, 1) (k|n)
(D_{4})  Z × S_{3}  (4, 0)  (k, σ) (k|4, σ^{\frac{4}{k}} = 1)
(D_{n}) n : even, n > 4 Z × Z/2Z (n, 0) (k, 0) (k|n) or (k, 1) (k|n, \frac{n}{k} is even)
(E_{6})  Z × Z/2Z (7, 1)  (1, 1) or (7, 1)
(E_{7})  Z  10  1, 2, 5 or 10
(E_{8})  Z  16  1, 2, 4, 8 or 16

Proof By [XZ] (see also [Am, 4.0.4]), the AR quiver of C is ZΔ/H for a Dynkin diagram Δ and a weakly admissible subgroup H of Aut(ZΔ). Since C is 2-Calabi-Yau, H contains F. By [Am, 2.2.1], H is generated by a single element h. By the condition F ∈ ⟨h⟩, we have the above list. □

Theorem 8.2. (1) C has a cluster tilting object if and only if the AR quiver of C is ZΔ/⟨h⟩ for a Dynkin diagram Δ and h ∈ Aut(ZΔ) in the list below.

Δ  Aut(ZΔ)  h
(A_{n}) n : odd  Z × Z/2Z  (\frac{n+3}{6}, 1) (3|n) or (\frac{n+3}{2}, 1)
(A_{n}) n : even  Z  \frac{n+3}{3} (3|n) or n + 3
(D_{n}) n : odd  Z × Z/2Z  (k, 1) (k|n)
(D_{4})  Z × S_{3} (k, σ) (k|4, σ^{\frac{4}{k}} = 1, (k, σ) ≠ (1, 1))
(D_{n}) n : even, n > 4 Z × Z/2Z  (k, \overline{k}) (k|n)
(E_{6})  Z × Z/2Z  (7, 1)
(E_{7})  Z  10
(E_{8})  Z  8 or 16

(2) C does not have a non-zero rigid object if and only if the AR quiver of C is ZΔ/⟨h⟩ for a Dynkin diagram Δ and h ∈ Aut(ZΔ) in the list below.

Δ  Aut(ZΔ)  h
(A_{n}) n : odd  Z × Z/2Z −
(A_{n}) n : even Z 1
(D_{n}) n : odd  Z × Z/2Z −
(D_{4})  Z × S_{3} (1, 1)
(D_{n}) n : even, n > 4 Z × Z/2Z (1, 0)
(E_{6})  Z × Z/2Z (1, 1)
(E_{7})  Z  1
(E_{8})  Z  1 or 2

Proof Our method is based on the computation of additive functions in section 2.

(1) Assume that h is on the list. Then one can check that C has a cluster tilting object. For example, consider the (D_{n}) case here. Fix a vertex x ∈ ZΔ corresponding to an end point of Δ which is adjacent to the branch vertex of Δ. Then the subset {(1, 1)^{l}x | l ∈ Z} of ZΔ is stable under the action of h, and gives a cluster tilting object of C.

Conversely, assume that C has a cluster tilting object. Then one can check that h is on the list. For example, consider the (A_{n}) case with even n here. By [I1], cluster tilting objects correspond to dissections of a regular (n + 3)-polygon into triangles by non-crossing diagonals. The action of h shows that it is invariant under the rotation of \frac{2kπ}{n+3}-radian. Since the center of the regular (n + 3)-polygon is contained in some triangle or its edge, we have \frac{2kπ}{n+3} = 2π, \frac{4π}{3} , π or \frac{2π}{3} . Since k|n + 3 and n is even, we have k = n + 3 or \frac{n+3}{3}.

(2) If h is on the list above, then one can easily check that C does not have a non-zero rigid objects. Conversely, if h is not on the list, then one can easily check that at least one indecomposable object which corresponds to an end point of Δ is rigid. □

References

[Am] C. Amiot, On the structure of triangulated categories with finitely many idecomposables, arXiv: math.CT/0612141.

[AGV] V. Arnold, S. Guseĭn-Zade, S. Varchenko, Singularities of differentiable maps, Vol. I. The classification of critical points, caustics and wave fronts, Monographs in Mathematics, 82, Birkhäuser Boston, Inc., Boston, MA, (1985).

[Ar] M. Artin, Coverings of the rational double points in characteristic p, Complex analysis and algebraic geometry, pp. 11–22. Iwanami Shoten, Tokyo, 1977.

[Au] M. Auslander, Functors and morphisms determined by objects, Representation of algebras (Proc. Conf., Temple Univ. Philadelphia, Pa(1976) Dekker, New York 1978, pp 1–244, , Lecture Notes in Pure and Appl. Math., Vol. 37.

[AR] M. Auslander, I. Reiten, DTr-periodic modules and functors, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc. 18, Amer. Math. Soc., 39–50, Providence, RI 1996.

[BK] A. Bondal, M. Kapranov, Representable functors, Serre functors and reconstructions, Izv. Akad. Nauk SSSR ser. Mat. 53(1989) no. 6, 1183–1205.

[BMR1] A. Buan, R. Marsh, I. Reiten, Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(2007), no. 1, 323–332, electronic.

[BMR2] A. Buan, R. Marsh, I. Reiten, Cluster mutation via quiver representations, Comm. Math. Helv. (to appear).

[BMR3] A. Buan, R. Marsh, I. Reiten, Cluster-tilted algebras of finite representation type. J. Algebra 306 (2006), no. 2, 412–431.

[BMRRT] A. Buan, R. Marsh, I. Reiten, M. Reineke, G. Todorov, Tilting theory and cluster combinatorics, Adv. in Math. 204 (2006), no. 2, 572–618.

[BIRS] A. Buan, O. Iyama, I. Reiten, J. Scott, Cluster structure for 2-Calabi-Yau categories and unipotent groups, arXiv: math.RT/0701557.

[BS] J. Białkowski, A. Skowroński, On tame weakly symmetric algebras having only periodic modules, Arch. Math. 81 (2003), no. 2, 142–154.

[CCS] P. Caldero, F. Chapoton, R. Schiffler, Quivers with relations arising from clusters (A_{n} case), Trans. Amer. Math. Soc. 358 (2006) no. 2, 1347–1364.

[C] W. Crawley-Boevey, On the exceptional fibre of Kleinian singularities, Amer. J. Math. 122 (2000) 1027–1037.

[D] E. Dieterich, Lattices over curve singularities with large conductor, Invent. Math. 114 (1993), no. 2, 399–433.

[DG] Yu. Drozd, G.-M. Greuel, Cohen-Macaulay module type, Compositio Math. 89 (1993), no. 3, 315–338.

[DGK] Yu. Drozd, G.-M. Greuel, I. Kashuba, On Cohen-Macaulay modules on surface singularities, Mosc. Math. J. 3 (2003), no. 2, 397–418, 742.

[DW] E. Dieterich, A. Wiedemann, Auslander-Reiten quiver of a simple curve singularity, Trans. Amer. Math. Soc. 294 (1986) 455–475.

[Ei] D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980) 35–64.

[Er] K. Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, 1428. Springer-Verlag, Berlin, 1990.

[FZ1] Fomin S., Zelevinsky A. Cluster Algebras I: Foundations, J. Amer. Math. Soc. 15, no. 2 (2002), 497–529 [FZ2] Fomin S., Zelevinsky A. Cluster algebras II: Finite type classification, Invent. Math. 154, no. 1 (2003),

63–121.

[FZ3] Fomin S., Zelevinsky A. Cluster algebras: Notes for the CDM-03 conference, current developments in mathematics 2003, Int. Press, Somerville, MA, 2003, 1–34.

[GK] G. M. Greuel, H. Knörrer, Einfache Kurvensingularitäten und torsionsfreie Moduln, Math. Ann. 270 (1985), no. 3, 417–425.

[GL] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of finitedimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265– 297, Lecture Notes in Math., 1273, Springer, Berlin, 1987.

[GLSc] C. Geiss, B. Leclerc, J. Schröer, Rigid modules over preprojective algebras, Invent. Math. 165 (2006), no. 3, 589–632.

[GP] G.-M. Greuel, G. Pfister, A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann, Springer-Verlag, Berlin, 2002.

[GLSh] G.-M. Greuel, C. Lossen, E. Shustin, Introduction to Singularities and Deformations, Springer Verlag, (2006), 476 pages.

[H] D. Happel, Triangulated categories in the represantation theory of finite dimensional algebras, London Math. Soc. Lecture Note Series 119, Cambridge Univ. Press, Cambridge 1988.

[HKR] D. Happel, B. Keller, I. Reiten, Bounded derived categories and repetitive algebras, arxiv: math.RT/0702302.

[I1] O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories, arxiv: math.RT/0407052, to appear in Advances in Mathematics.

[I2] O. Iyama, Auslander correspondence, arxiv: math.RT/0411631, to appear in Advances in Mathematics.

[IR] O. Iyama, I. Reiten, Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras, arxiv: math.RT/0605136.

[IY] O. Iyama, Y. Yoshino, Mutation in triangulated categories and rigid Cohen-Macaulay modules, arxiv: math RT/0607736.

[Kah] C. Kahn, Reflexive modules on minimally elliptic singularities, Math. Ann. 285 (1989), no. 1, 141–160.

[Kat] S. Katz, Small resolutions of Gorenstein threefold singularities, Algebraic geometry: Sundance 1988, 61–70, Contemp. Math., 116, Amer. Math. Soc., Providence, RI, 1991.

[Ke] B. Keller, On triangulated orbit categories, Doc. Math. 10 (2005), 551–581.

[KR] B. Keller, I. Reiten, Cluster-tilted algebras are Gorenstein and stably Calabi-Yau, Adv. in Math., to appear [Kn] H. Knörrer, Cohen-Macaulay modules on hypersurface singularities I, Invent. Math. 88 (1987), no. 1, 153–

164.

[Rin] C.-M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer-Verlag, Berlin, (1984).

[Rie] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224.

[Re] M. Reid, Minimal models of canonical 3-folds, Algebraic varieties and analytic varieties (Tokyo, 1981), 131– 180, Adv. Stud. Pure Math., 1, 1983.

[RS] C. Riedtmann, A. Schofield, On a simplicial complex associated with tilting modules, Comment. Math. Helv. 66 (1991), no. 1, 70–78.

[Sk] A. Skowroński, Selfinjective algebras: finite and tame type, Trends in representation theory of algebras and related topics, 169–238, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, 2006.

[So] Ø. Solberg, Hypersurface singularities of finite Cohen-Macaulay type, Proc. London Math. Soc. 58 (1989) 258–280.

[V1] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455.

[V2] M. Van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, 749–770, Springer, Berlin, 2004.

[XZ] J. Xiao, B. Zhu, Locally finite triangulated categories, J. Algebra 290 (2005), no. 2, 473–490.

[Y] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, LMS 146, 1990.

Johannes-Gutenberg Universität Mainz, Fachbereich Physik, Mathematik und Informatik, Institut für Mathematik, 55099 Mainz, Germany

   E-mail address: burban@mathematik.uni-mainz.de

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

   E-mail address: iyama@math.nagoya-u.ac.jp

UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France

   E-mail address: keller@math.jussieu.fr

Institutt for matematiske fag, Norges Teknisk-naturvitenskapelige universitet, N-7491, Trondheim, Norway

   E-mail address: idunr@math.ntnu.no