Cluster tilting for one-dimensional hypersurface singularities

TL;DR

This paper explores the structure of Cohen-Macaulay modules over one-dimensional hypersurface singularities, linking them to cluster tilting theory and classifying associated 2-Calabi-Yau tilted algebras.

Contribution

It provides criteria for the existence of cluster tilting objects and describes them using homological methods, including applications to specific curve singularities.

Findings

Classified 2-CY tilted algebras for simple singularities

Identified symmetric 2-CY tilted algebras with $ au^2=\id$

Connected Cohen-Macaulay modules with cluster tilting theory

Abstract

In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the 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 methods, 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 satisfy $\tau^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.

Equations

$${\underline{\mathrm{Hom}}}_R(X,Y)\simeq D{\underline{\mathrm{Hom}}}_R(Y,{\Sigma }^{n}X)$$

$${\underline{\mathrm{Hom}}}_R(X,Y)\simeq D{\mathrm{Ext}}_R^1(Y,\tau X)$$

$$\underline{\mathrm{CM}}(k[[x_0,\cdots ,x_d,y,z]]/(f+yz))\simeq \underline{\mathrm{CM}}(k[[x_0,\cdots ,x_d]]/(f))$$

$$X\stackrel{g_1}{\to }{N}_1\stackrel{f_1}{\to }Y\text{mbox}andY\stackrel{g_0}{\to }{N}_0\stackrel{f_0}{\to }X$$

$$\begin{array}[]{ccc}(A_{n})&x^{2}+y^{n+1}+z_{2}^{2}+z_{3}^{2}+\cdots+z_{d}^{2}&(n\geq 1)\\ (D_{n})&x^{2}y+y^{n-1}+z_{2}^{2}+z_{3}^{2}+\cdots+z_{d}^{2}&(n\geq 4)\\ (E_{6})&x^{3}+y^{4}+z_{2}^{2}+z_{3}^{2}+\cdots+z_{d}^{2}&\\ (E_{7})&x^{3}+xy^{3}+z_{2}^{2}+z_{3}^{2}+\cdots+z_{d}^{2}&\\ (E_{8})&x^{3}+y^{5}+z_{2}^{2}+z_{3}^{2}+\cdots+z_{d}^{2}&\end{array}$$

$$\begin{array}[]{|c|c|c|c|c|}\hline\cr f&\mbox{indec. rigid}&\mbox{cluster tilting}&\mbox{max. rigid}&\mbox{summands of max. rigid}\\ \hline\cr\hline\cr(A_{n})\ n:\mbox{odd}&2&2&2&1\\ \hline\cr(A_{n})\ n:\mbox{even}&0&0&1&0\\ \hline\cr(D_{n})\ n:\mbox{odd}&2&0&2&1\\ \hline\cr(D_{n})\ n:\mbox{even}&6&6&6&2\\ \hline\cr(E_{6})&0&0&1&0\\ \hline\cr(E_{7})&2&0&2&1\\ \hline\cr(E_{8})&0&0&1&0\\ \hline\cr\end{array}$$

$$x^p+y^q+\lambda x^2{y}^2=0,$$

$$T_{3,6}(\lambda )=k[[x,y]]/(y(y-x^2)(y-\lambda x^2))$$

$$T_{4,4}(\lambda )=k[[x,y]]/(xy(x-y)(x-\lambda y)),$$

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

$$\begin{array}[]{|c|c|c|c|}\hline\cr p,q&\mbox{indec. rigid}&\mbox{cluster tilting}&\mbox{summands of cluster tilting}\\ \hline\cr\hline\cr p=3,\ q:\mbox{even}&6&6&2\\ \hline\cr p,q:\mbox{even}&14&24&3\\ \hline\cr\end{array}$$

$$\begin{array}[]{|c|c|c|}\hline\cr\mbox{indec. rigid}&\mbox{cluster tilting}&\mbox{summands of cluster tilting}\\ \hline\cr\hline\cr 2^{n}-2&n!&n-1\\ \hline\cr\end{array}$$

$$R=k[[x,y]]/((x-{\lambda }_{1}y)\cdots (x-{\lambda }_{n}y))\text{mbox}andM=\underset{i=1}{\overset{n}{⨁}}k[[x,y]]/((x-{\lambda }_{1}y)\cdots (x-{\lambda }_{i}y))$$

$$D{\mathrm{Hom}}_{\mathcal{C}}(T,T)\simeq {\mathrm{Hom}}_{\mathcal{C}}(T,{\Sigma }^{2}T)={\mathrm{Hom}}_{\mathcal{C}}(T,{\tau }^{2}T)\simeq {\mathrm{Hom}}_{\mathcal{C}}(T,T).$$

$$I_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$N_{-}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Ext}}^1(M_i,M_i)\simeq \underline{\mathrm{Hom}}(M_i,\tau M_i)\simeq \underline{\mathrm{Hom}}(M_i,M_i)\ne 0.$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$C_{+}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$D_{-}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$C_{-}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\{C_{+},D_{-}\}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$B\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$C\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A_2\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$R=k[x_1,x_2,\dots ,x_n{]}_{⟨x_1,x_2,\dots ,x_n⟩}/I$$

$${\dim }_k({\mathrm{Ext}}_R^i(M,N))={\dim }_k({\mathrm{Ext}}_{\hat{R}}^i(\hat{M},\hat{N})).$$

$$C\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

The first author was supported by the DFG project Bu 1866/1-1, the second and last author by a Storforsk grant 167130 from the Norwegian Research Council

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 $\mathcal{C}_{H}$ is the orbit category $\mathcal{D}^{b}(H)/{F}$, where $\mathcal{D}^{b}(H)$ is the bounded derived category of finite dimensional $H$-modules and the functor $F\colon\mathcal{D}^{b}(H)\to\mathcal{D}^{b}(H)$ is $\tau^{-1}[1]=S^{-1}[2]$. Here $\tau$ denotes the translation associated with almost split sequences/triangles and $S$ the Serre functor [BK] on $\mathcal{D}^{b}(H)$. (See [CCS] for an independent definition of a category equivalent to the cluster category when $H$ is of Dynkin type $A_{n}$).

An object $T$ in a cluster category $\mathcal{C}_{H}$ was defined to be a (cluster) tilting object if $\operatorname{Ext}\nolimits_{\mathcal{C}_{H}}^{1}(T,T)=0$, and if $\operatorname{Ext}\nolimits_{\mathcal{C}_{H}}^{1}(X,X\oplus T)=0$, then $X$ is in $\operatorname{add}\nolimits T$. 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 $\operatorname{Ext}\nolimits_{\mathcal{C}_{H}}^{1}(X,T)=0$ implies that $X$ is in $\operatorname{add}\nolimits T$, 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 notion corresponding to the above definition was called maximal 1-orthogonal. For the category $\operatorname{mod}\nolimits(\Lambda)$ of finite dimensional modules over a preprojective algebra of Dynkin type $\Lambda$ over an algebraically closed field $k$, the concept 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 $\underline{\operatorname{mod}\nolimits}(\Lambda)$.

The categories $\mathcal{C}_{H}$ and $\underline{\operatorname{mod}\nolimits}(\Lambda)$ 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 $S=\Sigma^{2}$, where $\Sigma$ is the shift functor in the triangulated category.

For an arbitrary 2-CY triangulated category $\mathcal{C}$ with finite dimensional homomorphism spaces over a field $k$, a cluster tilting object $T$ in $\mathcal{C}$ 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 $\underline{\operatorname{mod}\nolimits}(\Lambda)$, 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 $\operatorname{Hom}\nolimits(T,\text{ })$ 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 $\underline{\operatorname{CM}\nolimits}(R)$ is the stable category of maximal Cohen-Macaulay modules over an odd-dimensional isolated hypersurface singularity, then $\underline{\operatorname{CM}\nolimits}(R)$ is 2-CY. If it contains a cluster tilting object $T$, then the functor $\operatorname{Hom}\nolimits(T,\text{ })$ induces an equivalence between the quotient of $\underline{\operatorname{CM}\nolimits}(R)$ by the ideal of morphisms factoring through $\tau T$ and the category of finite dimensional modules over the endomorphism algebra $B=\operatorname{End}\nolimits(T)$. It is then not hard to see that $B$ is symmetric and the indecomposable nonprojective $B$-modules are $\tau$-periodic of $\tau$-period at most 2. In this article, we study examples of this setup arising from finite, tame and wild $\operatorname{CM}\nolimits$-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 (wild) selfinjective algebras whose stable AR-quiver consists only of tubes of rank one or three were known previously [AR].

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 $\underline{\operatorname{CM}\nolimits}(R)$ 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 deal with cluster tilting and (maximal) rigid objects in the category $\operatorname{CM}\nolimits(R)$, defined in an analogous way.

We also investigate the other Dynkin diagrams, and it is interesting to notice that there are 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 $\mathcal{C}_{H}$ and $\underline{\operatorname{mod}\nolimits}(\Lambda)$ [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 [Kn, So].

We shall construct a large class of one-dimensional hypersurface singularities $R$, where $\operatorname{CM}\nolimits(R)$ or $\underline{\operatorname{CM}\nolimits}(R)$ has a cluster tilting object, including examples coming from simple singularities and minimally elliptic singularities. We classify all rigid objects in $\operatorname{CM}\nolimits(R)$ for these $R$, in particular, we give a bijection between cluster tilting objects in $\operatorname{CM}\nolimits(R)$ 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 $\operatorname{End}\nolimits_{R}(T)$ (called ‘three-dimensional Auslander algebras’) of cluster tilting objects $T$ in $\operatorname{CM}\nolimits(R)$. These algebras have global dimension three, and have 2-CY tilted algebras as stable factors. The functor $\operatorname{Hom}\nolimits_{R}(T,\ ):\operatorname{CM}\nolimits(R)\to\operatorname{mod}\nolimits(\operatorname{End}\nolimits_{R}(T))$ sends cluster tilting objects in $\operatorname{CM}\nolimits(R)$ to tilting modules over $\operatorname{End}\nolimits_{R}(T)$. By comparing cluster tilting mutations in $\operatorname{CM}\nolimits(R)$ and tilting mutation in $\operatorname{CM}\nolimits(\operatorname{End}\nolimits_{R}(T))$, we can apply results on tilting mutation due to Riedtmann-Schofield [RS] and Happel-Unger [HU1, HU2] to get information on cluster tilting objects in $\operatorname{CM}\nolimits(R)$.

We focus on the interplay between cluster tilting theory and birational geometry (see section 5 for definitions). 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 endomorphism 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 they form a class of non-commutative crepant resolutions [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 for existence of crepant resolutions to a criterion for one-dimensional hypersurface singularities for 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 non-commutative crepant resolutions. 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 $\underline{\operatorname{CM}\nolimits}(R)$ 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 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 classify cluster tilting and indecomposable rigid objects for this class. 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 some of 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, GLSh] for classification of singularities.

Our modules are usually right modules, and composition of maps $fg$ means first $g$, then $f$. We call a module basic if it is a direct sum of mutually non-isomorphic indecomposable modules.

Acknowledgment

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

1. Main results

Let $(R,{\mathfrak{m}})$ be a local complete $d$-dimensional commutative noetherian Gorenstein isolated singularity and $R/{\mathfrak{m}}=k\subset R$, where $k$ is an algebraically closed field of characteristic zero. We denote by $\operatorname{CM}\nolimits(R)$ the category of maximal Cohen-Macaulay modules over $R$. Then $\operatorname{CM}\nolimits(R)$ is a Frobenius category (i.e. an exact category with enough projectives and injectives which coincide), and so the stable category $\underline{\operatorname{CM}\nolimits}(R)$ is a $\operatorname{Hom}\nolimits$-finite triangulated category with shift functor $\Sigma=\Omega^{-1}$ [H]. For an integer $n$, we say that $\underline{\operatorname{CM}\nolimits}(R)$ or $\operatorname{CM}\nolimits(R)$ is $n$-CY if there exists a functorial isomorphism

[TABLE]

for any $X,Y\in\operatorname{CM}\nolimits(R)$.

We collect some fundamental results.

• •

We have AR-duality

[TABLE]

with $\tau\simeq\Omega^{2-d}$ [Au]. In particular, $\underline{\operatorname{CM}\nolimits}(R)$ is $(d-1)$-CY.

• •

If $R$ is a hypersurface singularity, then $\Sigma^{2}=\operatorname{id}\nolimits$ [Ei].

Consequently, if $d$ is odd, then $\tau=\Omega$ and $\underline{\operatorname{CM}\nolimits}(R)$ is 2-CY. If $d$ is even, then $\tau=\operatorname{id}\nolimits$ and $\underline{\operatorname{CM}\nolimits}(R)$ is 1-CY, hence any non-free Cohen-Macaulay $R$-module $M$ satisfies $\operatorname{Ext}\nolimits^{1}_{R}(M,M)\neq 0$.

• •

(Knörrer periodicity)

[TABLE]

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

We state some of the definitions, valid more generally, in the context of $\operatorname{CM}\nolimits(R)$ and $\underline{\operatorname{CM}\nolimits}(R)$.

Definition 1.1**.**

Let $\mathcal{C}=\operatorname{CM}\nolimits(R)$ or $\underline{\operatorname{CM}\nolimits}(R)$. We call an object $M\in\mathcal{C}$

• •

rigid* if $\operatorname{Ext}\nolimits^{1}_{R}(M,M)=0$,*

• •

maximal rigid* if it is rigid and any rigid $N\in\mathcal{C}$ satisfying $M\in\operatorname{add}\nolimits N$ satisfies $N\in\operatorname{add}\nolimits M$,*

• •

cluster tilting* if $\operatorname{add}\nolimits M=\{X\in\mathcal{C}\ |\ \operatorname{Ext}\nolimits^{1}_{R}(M,X)=0\}=\{X\in\mathcal{C}\ |\ \operatorname{Ext}\nolimits^{1}_{R}(X,M)=0\}$.*

Cluster tilting objects are maximal rigid, but we show that the converse does not necessarily hold for 2-CY triangulated categories $\underline{\operatorname{CM}\nolimits}(R)$. If $\mathcal{C}$ is 2-CY, then $M\in\mathcal{C}$ is cluster tilting if and only if $\operatorname{add}\nolimits M=\{X\in\mathcal{C}\ |\ \operatorname{Ext}\nolimits^{1}_{R}(M,X)=0\}$.

Definition 1.2**.**

Let $\mathcal{C}=\operatorname{CM}\nolimits(R)$ (or $\underline{\operatorname{CM}\nolimits}(R)$) be 2-CY and $M\in\mathcal{C}$ a basic cluster tilting object. Take an indecomposable summand $X$ of $M=X\oplus N$. Then there exist short exact sequences (or triangles) (called exchange sequences)

[TABLE]

such that $N_{i}\in\operatorname{add}\nolimits N$ and $f_{0}$ is a minimal right $(\operatorname{add}\nolimits N)$-approximation. Then $Y\oplus N$ is a basic cluster tilting object again called cluster tilting mutation of $M$ [BMRRT, GLSc][IY, Def. 2.5, Th. 5.3]. In this case $f_{1}$ is a minimal right $(\operatorname{add}\nolimits N)$-approximation and $g_{i}$ is a minimal left $(\operatorname{add}\nolimits N)$-approximation automatically, so $X\oplus N$ is a cluster tilting mutation of $Y\oplus N$. It is known that there are no more basic cluster tilting objects containing $N$ [IY, Th. 5.3].

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

[TABLE]

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

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 1.3**.**

Let $R$ be a simple hypersurface singularity of dimension $d\geq 1$ over an algebraically closed field $k$ of characteristic zero.

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

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

[TABLE]

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

[TABLE]

where $\frac{1}{p}+\frac{1}{q}\leq\frac{1}{2}$ and certain values of $\lambda\in k$ have to be excluded. They are of 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 $T_{p,q}(\lambda)$ is called simply elliptic. The corresponding coordinate rings can be written in the form

[TABLE]

and

[TABLE]

where in both cases $\lambda\in k\setminus\{0,1\}$.

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

[TABLE]

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

Theorem 1.4**.**

Let $R$ be a minimally elliptic curve singularity $T_{p,q}(\lambda)$ over an algebraically closed field $k$ of characteristic zero.

• (a)

$\underline{\operatorname{CM}\nolimits}(R)$* 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, basic cluster tilting objects, and indecomposable summands of basic cluster tilting objects in $\underline{\operatorname{CM}\nolimits}(R)$ are as follows:

[TABLE]

We also prove the following general theorem, which includes both Theorem 1.3 (except the assertion on maximal rigid objects) and Theorem 1.4. 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 1.5**.**

Let $R=k[[x,y]]/(f)$ ($f\in(x,y)$) be a one-dimensional reduced hypersurface singularity over an algebraically closed field $k$ of characteristic zero.

• (a)

$\underline{\operatorname{CM}\nolimits}(R)$* has a cluster tilting object if and only if $f$ is a product $f=f_{1}\cdots f_{n}$ with $f_{i}\notin(x,y)^{2}$.*

• (b)

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

[TABLE]

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

Theorem 1.6**.**

Let $(R,{\mathfrak{m}})$ be a three-dimensional isolated $cA_{n}$–singularity over an algebraically closed field $k$ of characteristic zero defined by the equation $g(x,y)+zt$ and $R^{\prime}$ a one-dimensional singularity defined by $g(x,y)$. Then the following conditions are equivalent.

• (a)

$\operatorname{Spec}\nolimits(R)$* has a small resolution.*

• (b)

$\operatorname{Spec}\nolimits(R)$* has a crepant resolution.*

• (c)

$(R,{\mathfrak{m}})$* has a non-commutative crepant resolution.*

• (d)

$\underline{\operatorname{CM}\nolimits}(R)$* has a cluster tilting object.*

• (e)

$\underline{\operatorname{CM}\nolimits}(R^{\prime})$* 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 $\operatorname{End}\nolimits_{\mathcal{C}}(M)$ of a cluster tilting object $T$ in a 2-CY triangulated category $\mathcal{C}$. In section 7, we shall show the following result and compute 2-CY tilted algebras associated with minimally elliptic curve singularities.

Theorem 1.7**.**

Let $(R,{\mathfrak{m}})$ be an odd-dimensional isolated hypersurface singularity and $\Gamma$ a 2-CY tilted algebra coming from $\underline{\operatorname{CM}\nolimits}(R)$. Then we have the following.

• (a)

$\Gamma$* is a symmetric algebra.*

• (b)

All components in the stable AR-quiver of infinite type $\Gamma$ are tubes of rank 1 or 2.

For example, put

[TABLE]

for distinct elements $\lambda_{i}\in k$. Then $M$ is a cluster tilting object in $\operatorname{CM}\nolimits(R)$ by Theorem 4.1, so $\Gamma=\underline{\operatorname{End}\nolimits}_{R}(M)$ satisfies the conditions in Theorem 1.7. Since $\operatorname{CM}\nolimits(R)$ 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 $\operatorname{CM}\nolimits(R)$ [DW], and so also for $\underline{\operatorname{CM}\nolimits}(R)$. We use the notation from [Y].

In order to locate the indecomposable rigid modules $M$, that is, the modules $M$ with

$\operatorname{Ext}\nolimits^{1}(M,M)=0$, 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 $\mathcal{C}$ be an abelian or triangulated $k$-category with finite dimensional homomorphism spaces. Let $A\xrightarrow{\left(\begin{smallmatrix}f_{1}\\ f_{2}\end{smallmatrix}\right)}B_{1}\oplus B_{2}\xrightarrow{(g_{1},g_{2})}C$ 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 $\operatorname{Hom}\nolimits(A,C)\neq 0$ .

• (b)

Let $0\to A\xrightarrow{f}B\xrightarrow{g}C\to 0$ be an almost split sequence in $\operatorname{CM}\nolimits(R)$, 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 $\operatorname{Ext}\nolimits^{1}(C,C)\neq 0$.

Proof.

(a) See [HKR, Lem. 6.5].

(b) Using (a) together with the above AR-formula and $\tau^{2}=\operatorname{id}\nolimits$, we obtain $D\operatorname{Ext}\nolimits^{1}(C,C)\simeq$

$\underline{\operatorname{Hom}\nolimits}(\tau^{-1}C,C)=\underline{\operatorname{Hom}\nolimits}(\tau C,C)\simeq\underline{\operatorname{Hom}\nolimits}(A,C)\neq 0$, where $D=\operatorname{Hom}\nolimits_{k}(\text{ },k)$.∎

Lemma 2.2**.**

Let $T$ be a cluster tilting object in the $\operatorname{Hom}\nolimits$-finite connected 2-CY category $\mathcal{C}$, and $\Gamma=\operatorname{End}\nolimits_{{\mathcal{C}}}(T)$.

• (a)

*The functor $G=\operatorname{Hom}\nolimits_{{\mathcal{C}}}(T,\text{ })\colon{\mathcal{C}}\to\operatorname{mod}\nolimits(\Gamma)$ induces an equivalence of categories *

$\overline{G}\colon{\mathcal{C}}/{\operatorname{add}\nolimits(\tau T)}\to\operatorname{mod}\nolimits(\Gamma)$.

• (b)

The AR-quiver for $\Gamma$ is as a translation quiver obtained from the AR-quiver for ${\mathcal{C}}$ by removing the vertices corresponding to the indecomposable summands of $\tau T$.

• (c)

Assume $\tau^{2}=\operatorname{id}\nolimits$. Then we have the following.

• (i)

$\Gamma$* is a symmetric algebra.*

• (ii)

The indecomposable nonprojective $\Gamma$-modules have $\tau$-period one or two.

• (iii)

If ${\mathcal{C}}$ has an infinite number of nonisomorphic indecomposable objects, then all components in the stable AR-quiver of $\Gamma$ are tubes of rank one or two.

• (d)

If ${\mathcal{C}}$ has only a finite number $n$ of nonisomorphic indecomposable objects, and $T$ has $t$ nonisomorphic indecomposable summands, then there are $n-t$ nonisomorphic indecomposable $\Gamma$-modules.

Proof.

For (a) and (b) see [BMR1, KR]. Since ${\mathcal{C}}$ is 2-CY, we have $\tau=\Sigma$, and a functorial isomorphism

[TABLE]

This shows that $\Gamma$ is symmetric. Let $C$ be an indecomposable nonprojective $\Gamma$-module. Viewing $C$ as an object in ${\mathcal{C}}$ we have $\tau^{2}_{{\mathcal{C}}}C\simeq C$, and $\tau C$ is not a projective $\Gamma$-module since $C$ is not removed. Hence we have $\tau^{2}_{\Gamma}C\simeq C$. If ${\mathcal{C}}$ has an infinite number of nonisomorphic indecomposable objects, then $\Gamma$ 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 $\Pi\colon k(\mathbb{Z}Q)\to\underline{\operatorname{CM}\nolimits}(R)$, where $Q$ is the appropriate Dynkin quiver and $k(\mathbb{Z}Q)$ is the mesh category of the translation quiver $\mathbb{Z}Q$ [Rie, Am], (see also [I1, Section 4.4] for another explanation using functorial methods).

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

[TABLE]

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

Since $\tau I_{j}\simeq I_{j}$ for each $j$, $\operatorname{Ext}\nolimits^{1}(I_{j},I_{j})\neq 0$ for $j=1,\cdots,n/2$. 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.

For simplicity, we write $l=({n-1})/2$. We have the stable AR-quiver

[TABLE]

Since $\tau M_{i}\simeq M_{i}$ for $i=1,\cdots,l$, we have

[TABLE]

So only the indecomposable objects $N_{-}$ and $N_{+}$ could be rigid. We use covering techniques and additive functions to compute the support of $\underline{\operatorname{Hom}\nolimits}(N_{-},\ )$, where we refer to [BG] for the meaning of the diagrams below.

\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ M_{1}\ }$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ M_{2}\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{M_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ M_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ }$$\textstyle{M_{l-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{l-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{l-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{-}}$$\textstyle{\cdots}$$\textstyle{\ N_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ }$$\textstyle{N_{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{-}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}

We see that $\underline{\operatorname{Hom}\nolimits}(N_{-},N_{+})=0$, so $\operatorname{Ext}\nolimits^{1}(N_{+},N_{+})=\operatorname{Ext}\nolimits^{1}(N_{+},\tau N_{-})=0$, and $\operatorname{Ext}\nolimits^{1}(N_{-},N_{-})=0$. Since $\operatorname{Ext}\nolimits^{1}(N_{+},N_{-})\neq 0$, we see that $N_{+}$ and $N_{-}$ are exactly the maximal rigid objects. Further $\underline{\operatorname{Hom}\nolimits}(N_{-},M_{i})\neq 0$ for all $i$, so $\operatorname{Ext}\nolimits^{1}(N_{+},M_{i})\neq 0$ and $\operatorname{Ext}\nolimits^{1}(N_{-},M_{i})\neq 0$ 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

[TABLE]

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

[TABLE] \textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{Y_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{l+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}          

\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

where $B=\tau A$ and $l={(n-3)}/2$. We see that $\underline{\operatorname{Hom}\nolimits}(A,B)=0$, so that $\operatorname{Ext}\nolimits^{1}(A,A)=0$. Then $A$ is clearly maximal rigid. Since $\underline{\operatorname{Hom}\nolimits}(A,M_{1})=0$, we have $\operatorname{Ext}\nolimits^{1}(A,N_{1})=0$, so $A$ is not cluster tilting. Alternatively, we could use that we see that $\underline{\operatorname{End}\nolimits}(A)^{\operatorname{op}\nolimits}\simeq k[x]/(x^{2})$, which has two indecomposable modules, whereas $\underline{\operatorname{CM}\nolimits}(R)$ has $2n-3$ indecomposable objects. If $A$ was cluster tilting, $\underline{\operatorname{End}\nolimits}(A)^{\operatorname{op}\nolimits}$ 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

\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}

$\alpha\beta\alpha=0=\beta\alpha\beta$ in the case $D_{4}$, and by

\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\scriptstyle{\alpha}$$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}

with $\gamma^{n-1}=\beta\alpha$, $\gamma\beta=0=\alpha\gamma$ and

\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\cdot\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}

with $(\alpha\beta)^{n-1}\alpha=0=(\beta\alpha)^{n-1}\beta$ for $2n>4$.*

Proof.

We have the AR-quiver

[TABLE]

By Lemma 2.1, the only possible indecomposable rigid objects are: $A$, $B$, $C_{+}$, $C_{-}$, $D_{+}$, $D_{-}$.

We compute the support of $\underline{\operatorname{Hom}\nolimits}(C_{+},\ )$:

[TABLE]

where $l=n-1$

[TABLE]

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

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

[TABLE]

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

[TABLE]

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

It follows that $C_{+}\oplus D_{-}$, $C_{-}\oplus D_{+}$, $C_{+}\oplus A$, $D_{+}\oplus B$, $A\oplus C_{-}$ and $B\oplus D_{-}$ are maximal rigid.

These are also cluster tilting: We have $\underline{\operatorname{Hom}\nolimits}(A,X_{i})\neq 0$, $\underline{\operatorname{Hom}\nolimits}(A,N_{i})\neq 0$, so $\operatorname{Ext}\nolimits^{1}(B,X_{i})\neq 0$, $\operatorname{Ext}\nolimits^{1}(B,N_{i})\neq 0$. Similarly, $\operatorname{Ext}\nolimits^{1}(A,Y_{i})\neq 0$, $\operatorname{Ext}\nolimits^{1}(A,M_{i})\neq 0$. Also $\underline{\operatorname{Hom}\nolimits}(C_{+},Y_{i})\neq 0$, $\underline{\operatorname{Hom}\nolimits}(C_{+},N_{i})\neq 0$, so $\operatorname{Ext}\nolimits^{1}(D_{+},Y_{i})\neq 0$, $\operatorname{Ext}\nolimits^{1}(D_{+},N_{i})\neq 0$. Hence $\operatorname{Ext}\nolimits^{1}(C_{+},X_{i})\neq 0$,

$\operatorname{Ext}\nolimits^{1}(C_{+},M_{i})\neq 0$. So $\operatorname{Ext}\nolimits^{1}(D_{-},Y_{i})\neq 0$, $\operatorname{Ext}\nolimits^{1}(D_{-},N_{i})\neq 0$, $\operatorname{Ext}\nolimits^{1}(C_{-},X_{i})\neq 0$, $\operatorname{Ext}\nolimits^{1}(C_{-},M_{i})\neq 0$. 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, Th. 5.3].

The exchange graph is as follows:

[TABLE]

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

[TABLE]

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

\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}}      

\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}

We see that $\underline{\operatorname{Hom}\nolimits}(M_{1},N_{1})\neq 0$, so that $\operatorname{Ext}\nolimits^{1}(M_{1},M_{1})\neq 0$ and $\operatorname{Ext}\nolimits^{1}(N_{1},N_{1})\neq 0$. ∎

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

[TABLE]

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 $\underline{\operatorname{Hom}\nolimits}(A,\ )$.

\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}}$$\textstyle{\cdots}$$\textstyle{Y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}

\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}

We see that $\operatorname{Ext}\nolimits^{1}(A,A)=0$, and so also $\operatorname{Ext}\nolimits^{1}(B,B)=0$, so $A$ and $B$ are rigid.

Next we compute the support of $\underline{\operatorname{Hom}\nolimits}(M_{1},\ )$.

\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{Y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{3}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{2}}$$\textstyle{\cdots}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}  

[TABLE] \textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

We see that $\operatorname{Ext}\nolimits^{1}(M_{1},M_{1})\neq 0$ and $\operatorname{Ext}\nolimits^{1}(N_{1},N_{1})\neq 0$, so that $M_{1}$ and $N_{1}$ are not rigid.

Then we compute the support of $\underline{\operatorname{Hom}\nolimits}(C,\ )$.

[TABLE] \textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}}$$\textstyle{\cdots}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{3}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}}$$\textstyle{N_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B}      

\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}

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

Since $\operatorname{Ext}\nolimits^{1}(A,C)=0$, 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

[TABLE]

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 $\underline{\operatorname{Hom}\nolimits}(M_{1},\ )$:

[TABLE] \textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{2}}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}}  

\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}

We see that $\operatorname{Ext}\nolimits^{1}(M_{1},M_{1})\neq 0$, and hence $\operatorname{Ext}\nolimits^{1}(N_{1},N_{1})\neq 0$.

Next we compute the support of $\underline{\operatorname{Hom}\nolimits}(M_{2},\ )$:

\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{2}}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{M_{1}}      

[TABLE] \textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cdots}$$\textstyle{1}

We see that $\operatorname{Ext}\nolimits^{1}(M_{2},M_{2})\neq 0$, and hence $\operatorname{Ext}\nolimits^{1}(N_{2},N_{2})\neq 0$.

Finally we compute the support of $\underline{\operatorname{Hom}\nolimits}(A_{2},\ )$:

[TABLE] \textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{D_{2}}$$\textstyle{\cdots}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{2}}$$\textstyle{Y_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{2}}$$\textstyle{D_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{C_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{B_{1}}$$\textstyle{\cdots}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}      

[TABLE] \textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{1}$$\textstyle{\cdots}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}

It follows that $\operatorname{Ext}\nolimits^{1}(A_{2},A_{2})\neq 0$, and similarly $\operatorname{Ext}\nolimits^{1}(B_{2},B_{2})\neq 0$. Hence there are no indecomposable rigid objects. ∎

3. Computation with Singular

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

[TABLE]

be a Cohen-Macaulay local ring which is an isolated singularity, and $M$ and $N$ two maximal Cohen-Macaulay modules. Denote by $\widehat{R}$ the completion of $R$. Since all the spaces $\operatorname{Ext}\nolimits^{i}_{R}(M,N)$ ($i\geq 1$) are finite-dimensional over $k$ and the functor $\operatorname{mod}\nolimits(R)\to\operatorname{mod}\nolimits(\widehat{R})$ 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

[TABLE]

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 $\underline{\operatorname{CM}\nolimits}(R)$ has the form

[TABLE]

By Lemma 2.1 only the modules $A,B,C,D,M_{1},N_{1}$ can be rigid. Since $B=\tau(A),D=\tau(C)$, $N_{1}=\tau(M_{1})$, 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:

[TABLE]

[TABLE]

[TABLE]

so we can use the computer algebra system Singular in order to compute the $\operatorname{Ext}\nolimits^{1}$–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 $S=\mathbb{Q}[x,y]_{\langle x,y\rangle}$)

ideal I = x3 + xy3; (defines the ideal $x^{3}+xy^{3}$ in $S$)

qring R = std(I); (defines the ring $\mathbb{Q}[x,y]_{\langle x,y\rangle}/I)$

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 $\operatorname{Ext}\nolimits^{1}$: -1 (Output: $\operatorname{Ext}\nolimits^{1}_{R}(A,A)=0$)

list l = Ext(1,C,C,1);

// ** redefining l **

// dimension of $\operatorname{Ext}\nolimits^{1}$: 0 (the Krull dimension of $\operatorname{Ext}\nolimits^{1}_{R}(C,C)$ is 0)

// vdim of $\operatorname{Ext}\nolimits^{1}$: 2   ($\dim_{k}(\operatorname{Ext}\nolimits^{1}_{R}(C,C))=2$)

list l = Ext(1,M1,M1,1);

// ** redefining l **

// dimension of $\operatorname{Ext}\nolimits^{1}$: 0

// vdim of $\operatorname{Ext}\nolimits^{1}$: 10

list l = Ext(1,A,C,1);

// ** redefining l **

// dimension of $\operatorname{Ext}\nolimits^{1}$: -1

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

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 the higher theory of almost split sequences and Auslander algebras studied in [I1, I2]. We also use a relationship between cluster tilting objects in $\operatorname{CM}\nolimits(R)$ and tilting modules over the endomorphism algebra of a cluster tilting object [I2]. Then we shall compare cluster tilting mutation given in Definition 1.2 with tilting mutation by using results due to Riedtmann-Schofield [RS] and Happel-Unger [HU1, HU2].

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

Let $k$ be an infinite field, $S:=k[[x,y]]$ and ${\mathfrak{m}}:=(x,y)$. We fix $f\in{\mathfrak{m}}$ and write $f=f_{1}\cdots f_{n}$ for irreducible formal power series $f_{i}\in{\mathfrak{m}}$ ($1\leq i\leq n$). Put

[TABLE]

We assume that $R$ is reduced, so we have $(f_{i})\neq(f_{j})$ for any $i\neq j$, and $R$ is then an isolated singularity.

Our main results in this section are the following, where the part (a) remains true in any dimension.

Theorem 4.1**.**

• (a)

$\bigoplus_{i=1}^{n}S_{i}$* is a rigid object in $\operatorname{CM}\nolimits(R)$.*

• (b)

$\bigoplus_{i=1}^{n}S_{i}$* is a cluster tilting object in $\operatorname{CM}\nolimits(R)$ if the following condition (A) is satisfied.*

(A) $f_{i}\notin{\mathfrak{m}}^{2}$ for any $1\leq i\leq n$.

Let ${\mathfrak{S}}_{n}$ be the symmetric group of degree $n$. For $w\in{\mathfrak{S}}_{n}$ and $I\subseteq\{1,\cdots,n\}$, we put

[TABLE]

Theorem 4.2**.**

Assume that (A) is satisfied.

• (a)

There are exactly $n!$ basic cluster tilting objects $M_{w}$ ($w\in{\mathfrak{S}}_{n}$) and exactly $2^{n}-1$ indecomposable rigid objects $S_{I}$ ($\emptyset\neq I\subseteq\{1,\cdots,n\}$) in $\operatorname{CM}\nolimits(R)$.

• (b)

For any $w\in{\mathfrak{S}}_{n}$, there are exactly $n!$ basic Cohen-Macaulay tilting $\operatorname{End}\nolimits_{R}(M_{w})$-modules

$\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}})$* ($w^{\prime}\in{\mathfrak{S}}_{n}$) of projective dimension at most one. Moreover, all algebras*

$\operatorname{End}\nolimits_{R}(M_{w})$* ($w\in{\mathfrak{S}}_{n}$) are derived equivalent.*

It is interesting to compare with results in [IR], where two-dimensional (2-Calabi-Yau) algebras $\Gamma$ are treated and a bijection between elements in an affine Weyl group and tilting $\Gamma$-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

[TABLE]

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

• (b)

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

[TABLE]

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

• (c)

Let $\lambda_{i}\in k$ ($1\leq i\leq n$) be mutually distinct elements in $k$. Put

[TABLE]

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

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

Proposition 4.3**.**

For $g_{1},g_{2}\in{\mathfrak{m}}$ and $g_{3}\in S$, put $R:=S/(g_{1}g_{2}g_{3})$. If $g_{1}$ and $g_{2}$ have no common factor, then $\operatorname{Ext}\nolimits^{1}_{R}(S/(g_{1}g_{3}),S/(g_{1}))=0=\operatorname{Ext}\nolimits^{1}_{R}(S/(g_{1}),S/(g_{1}g_{3}))$.

Proof.

We have a projective resolution

[TABLE]

Applying $\operatorname{Hom}\nolimits_{R}(\ ,S/(g_{1}))$, we have a complex

[TABLE]

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) ${\mathfrak{m}}=(f_{1},f_{2})=\cdots=(f_{n-1},f_{n})$.

• (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 (i.e. $M$ contains $R$ as a direct summand). Then the following conditions are equivalent.

• (a)

$M$* is a cluster tilting object in $\operatorname{CM}\nolimits(R)$.*

• (b)

$\operatorname{gl.dim}\nolimits\operatorname{End}\nolimits_{R}(M)\leq 3$.

• (c)

For any $X\in\operatorname{CM}\nolimits(R)$, there exists an exact sequence $0\to M_{1}\to M_{0}\to X\to 0$ with $M_{i}\in\operatorname{add}\nolimits M$.

• (d)

For any indecomposable direct summand $X$ of $M$, there exists an exact sequence $0\to M_{2}\stackrel{{\scriptstyle c}}{{\to}}M_{1}\stackrel{{\scriptstyle b}}{{\to}}M_{0}\stackrel{{\scriptstyle a}}{{\to}}X$ with $M_{i}\in\operatorname{add}\nolimits M$ and $a$ is a right almost split map in $\operatorname{add}\nolimits M$.

Proof.

(a)$\Leftrightarrow$(b) For $d=\dim R$, take the $d$-cotilting module $T=R$ and apply [I2, Th. 5.1(3)] for $m=d$ and $n=2$ there.

(a)$\Leftrightarrow$(c) See [I1, Prop. 2.2.2].

(a)$\Rightarrow$(d) See [I1, Th. 3.3.1].

(d)$\Rightarrow$(b) For any simple $\operatorname{End}\nolimits_{R}(M)$-module $S$, there exists an indecomposable direct summand $X$ of $M$ such that $S$ is the top of the projective $\operatorname{Hom}\nolimits_{R}(M,X)$. Since $\operatorname{Ext}\nolimits^{1}_{R}(M,M_{2})=0$, the sequence in (d) gives a projective resolution $0\to\operatorname{Hom}\nolimits_{R}(M,M_{2})\to\operatorname{Hom}\nolimits_{R}(M,M_{1})\to\operatorname{Hom}\nolimits_{R}(M,M_{0})\to\operatorname{Hom}\nolimits_{R}(M,X)\to S\to 0$. Thus we have $\operatorname{pd}\nolimits S\leq 3$ and $\operatorname{gl.dim}\nolimits\operatorname{End}\nolimits_{R}(M)\leq 3$. ∎

The sequence in (d) is called a 2-almost split sequence when $X$ is non-projective and $a$ and $b$ are right minimal. In this case $a$ is surjective, $c$ is a left almost split map in $\operatorname{add}\nolimits M$, and $b$ and $c$ are left minimal. 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 isomorphism

[TABLE]

Lemma 4.5**.**

Let $R=S/(f)$ be a one-dimensional reduced hypersurface singularity, $S_{0}:=0$ and $1\leq i<n$.

• (a)

We have exchange sequences (see Definition 1.2)

[TABLE]

• (b)

If $(f_{i},f_{i+1})=\mathfrak{m}$, then we have a 2-almost split sequence

[TABLE]

in $\operatorname{add}\nolimits\bigoplus_{i=1}^{n}S_{i}$.

Proof.

(a) Consider the map $a:=(-1\ f_{i}):S_{i+1}\oplus S_{i-1}\to S_{i}$. Any morphism from $S_{j}$ to $S_{i}$ factors through $1:S_{i+1}\to S_{i}$ (respectively, $f_{i}:S_{i-1}\to S_{i}$) if $j>i$ (respectively, $j<i$). Thus $a$ is a minimal right $(\operatorname{add}\nolimits\bigoplus_{j\neq i}S_{j})$-approximation.

It is easily checked that $\operatorname{Ker}\nolimits a=\{s\in S_{i+1}\ |\ \overline{s}\in f_{i}S_{i}\}=(f_{i})/(f_{1}\cdots f_{i+1})\simeq S/(f_{1}\cdots f_{i-1}f_{i+1})$, where we denote by $\overline{s}$ the image of $s$ via the natural surjection $S_{i+1}\to S_{i}$.

Consider the surjective map $b:=(1\ f_{i+1}):S_{i+1}\oplus S_{i-1}\to S/(f_{1}\cdots f_{i-1}f_{i+1})$. It is easily checked that $\operatorname{Ker}\nolimits b=\{s\in S_{i+1}\ |\ \overline{s}\in(f_{i+1})/(f_{1}\cdots f_{i-1}f_{i+1})\}=(f_{i+1})/(f_{1}\cdots f_{i+1})\simeq S_{i}$, where we denote by $\overline{s}$ the image of $s$ via the natural surjection $S_{i+1}\to S/(f_{1}\cdots f_{i-1}f_{i+1})$.

(b) This sequence is exact by (a). Any non-isomorphic endomorphism of $S_{i}$ is multiplication with an element in ${\mathfrak{m}}$, which is equal to $(f_{i},f_{i+1})$ by our assumption. Since $f_{i+1}$ (respectively, $f_{i}$) $:S_{i}\to S_{i}$ factors through $1:S_{i+1}\to S_{i}$ (respectively, $f_{i}:S_{i-1}\to S_{i}$), we have that $a$ is a right almost split map. ∎

Now we choose $f_{n+1}\in{\mathfrak{m}}$ such that ${\mathfrak{m}}=(f_{n},f_{n+1})$, and $f_{n+1}$ and $f_{1}\cdots f_{n}$ have no common factor. This is possible by our assumption (A).

Lemma 4.6**.**

We have an exact sequence

[TABLE]

with a minimal right almost split map $(f_{n+1}\ f_{n})$ in $\operatorname{add}\nolimits\bigoplus_{i=1}^{n}S_{i}$.

Proof.

Consider the map $a:=(f_{n+1}\ f_{n}):S_{n}\oplus S_{n-1}\to S_{n}$. Any morphism from $S_{j}$ ($j<n$) to $S_{n}$ factors through $f_{n}:S_{n-1}\to S_{n}$.

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

It is easily checked that $\operatorname{Ker}\nolimits a=\{s\in S_{n-1}\ |\ f_{n}s\in f_{n+1}S_{n}\}=(f_{n+1},f_{1}\cdots f_{n-1})/(f_{1}\cdots f_{n-1})$, which is isomorphic to $S_{n-1}$ by the choice of $f_{n+1}$. In particular, $a$ is right minimal. ∎

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. Let us consider cluster tilting mutation in $\operatorname{CM}\nolimits(R)$. We use the notation introduced at the beginning of this section.

Lemma 4.7**.**

For $w\in{\mathfrak{S}}_{n}$, we assume that $M_{w}$ is a cluster tilting object in $\operatorname{CM}\nolimits(R)$. Then, for $1\leq i<n$ and $s_{i}=(i\ i+1)$, we have exchange sequences

[TABLE]

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 4.8**.**

Assume that $M_{w}$ is a cluster tilting object in $\operatorname{CM}\nolimits(R)$ for some $w\in{\mathfrak{S}}_{n}$.

• (a)

The cluster tilting mutations of $M_{w}$ are $M_{ws_{i}}$ ($1\leq i<n$).

• (b)

$M_{w^{\prime}}$* is a cluster tilting object in $\operatorname{CM}\nolimits(R)$ for any $w^{\prime}\in{\mathfrak{S}}_{n}$.*

Proof.

(a) This follows from Lemma 4.7.

(b) This follows from (a) since ${\mathfrak{S}}_{n}$ is generated by $s_{i}$ ($1\leq i<n$). ∎

The following result is also useful.

Lemma 4.9**.**

Let $R$ and $R^{\prime}$ be complete local Gorenstein rings with $\dim R=\dim R^{\prime}$ and $M$ a rigid object in $\operatorname{CM}\nolimits(R)$ which is a generator. Assume that there exists a surjection $R^{\prime}\to R$, and we regard $\operatorname{CM}\nolimits(R)$ as a full subcategory of $\operatorname{CM}\nolimits(R^{\prime})$. If $R^{\prime}\oplus M$ is a cluster tilting object in $\operatorname{CM}\nolimits(R^{\prime})$, then $M$ is a cluster tilting object in $\operatorname{CM}\nolimits(R)$.

Proof.

We use the equivalence (a)$\Leftrightarrow$(c) in Proposition 4.4, which remains true in any dimension [I1, Prop. 2.2.2]. For any $X\in\operatorname{CM}\nolimits(R)$, take a right $(\operatorname{add}\nolimits M)$-approximation $f:M_{0}\to X$ of $X$. Since $M$ is a generator of $R$, we have an exact sequence $0\to Y\to M_{0}\stackrel{{\scriptstyle f}}{{\to}}X\to 0$ with $Y\in\operatorname{CM}\nolimits(R)$. Since $R^{\prime}$ is a projective $R^{\prime}$-module, $f$ is a right $\operatorname{add}\nolimits(R^{\prime}\oplus M)$-approximation of $X$. Since $R^{\prime}\oplus M$ is a cluster tilting object in $\operatorname{CM}\nolimits(R^{\prime})$, we have $Y\in\operatorname{add}\nolimits(R^{\prime}\oplus M)$. Since $Y\in\operatorname{CM}\nolimits(R)$, we have $Y\in\operatorname{add}\nolimits M$. Thus $M$ satisfies condition (c) in Proposition 4.4.∎

Now we shall prove Theorem 4.1. Since $k$ is an infinite field and the assumption (A) is satisfied, we can take irreducible formal power series $g_{i}\in{\mathfrak{m}}$ ($1\leq i<n$) such that $h_{2i-1}:=f_{i}$ and $h_{2i}:=g_{i}$ satisfy the following conditions:

• •

$(h_{i})\neq(h_{j})$ for any $i\neq j$.

• •

${\mathfrak{m}}=(h_{1},h_{2})=(h_{2},h_{3})=\cdots=(h_{2n-2},h_{2n-1})$.

Put $R^{\prime}:=S/(h_{1}\cdots h_{2n-1})$. This is reduced by the first condition.

Since we have already proved Theorem 4.1 under the assumption (B), we have that

$\bigoplus_{i=1}^{2n-1}S/(h_{1}\cdots h_{i})$ is a cluster tilting object in $\operatorname{CM}\nolimits(R^{\prime})$. By Proposition 4.8,

$\bigoplus_{i=1}^{2n-1}S/(h_{w(1)}\cdots h_{w(i)})$ is a cluster tilting object in $\operatorname{CM}\nolimits(R^{\prime})$ for any $w\in{\mathfrak{S}}_{2n-1}$. In particular,

[TABLE]

is a cluster tilting object in $\operatorname{CM}\nolimits(R^{\prime})$. Moreover we have surjections

[TABLE]

Using Lemma 4.9 repeatedly, we have that $\bigoplus_{i=1}^{n}(S/(f_{1}\cdots f_{i}))$ is a cluster tilting object in $\operatorname{CM}\nolimits(R)$. Thus we have proved Theorem 4.1.∎

Before proving Theorem 4.2, we give the following description of the quiver of the endomorphism algebras.

Proposition 4.10**.**

Assume that (A) is satisfied.

• (a)

The quiver of $\operatorname{End}\nolimits_{R}(\bigoplus_{i=1}^{n}S_{i})$ is

[TABLE]

where in addition there is a loop at $S_{i}$ ($1\leq i<n$) if and only if $(f_{i},f_{i+1})\neq\mathfrak{m}$.

• (b)

We have the quiver of $\underline{\operatorname{End}\nolimits}_{R}(\bigoplus_{i=1}^{n}S_{i})$ by removing the vertex $S_{n}$ from the quiver in (a).

Proof.

We only have to show (a). We only have to calculate minimal right almost split maps in $\operatorname{add}\nolimits\bigoplus_{i=1}^{n}S_{i}$. We have a minimal right almost split map $S_{n}\oplus S_{n-1}\to S_{n}$ by Lemma 4.6. If $(f_{i},f_{i+1})=\mathfrak{m}$ ($1\leq i<n$), then we have a minimal right almost split map $S_{i+1}\oplus S_{i-1}\to S_{i}$ by Lemma 4.5.

We only have to consider the case $(f_{i},f_{i+1})\neq\mathfrak{m}$ ($1\leq i<n$). Take $g\in\mathfrak{m}$ such that $(f_{i},f_{i+1},g)=\mathfrak{m}$. It is easily check (cf proof of Lemma 4.5) that we have a right almost split map

[TABLE]

Assume that $c$ is not right minimal. Then there exists a right almost split map of the form $c^{\prime}:S_{i+1}\oplus S_{i}\to S_{i}$ ($i>1$), $S_{i}\oplus S_{i-1}\to S_{i}$ or $S_{i+1}\oplus S_{i-1}\to S_{i}$. For the first case, it is easily checked that $f_{i}:S_{i-1}\to S_{i}$ does not factor through $c^{\prime}$, a contradiction. Similarly we have the contradiction for the remaining cases. Thus $c$ is the minimal right almost split map. ∎

In the rest we shall show Theorem 4.2. We recall results on tilting mutation due to Riedtmann-Schofield [RS] and Happel-Unger [HU1, HU2]. For simplicity, a tilting module means a tilting module of projective dimension at most one.

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

[TABLE]

if $\operatorname{Ext}\nolimits^{1}_{\Gamma}(T,U)=0$. By tilting theory, we have $\operatorname{Fac}\nolimits T=\{X\in\operatorname{mod}\nolimits(\Gamma)\ |\ \operatorname{Ext}\nolimits^{1}_{\Gamma}(T,X)=0\}$. Thus $\operatorname{Ext}\nolimits^{1}_{\Gamma}(T,U)=0$ is equivalent to $\operatorname{Fac}\nolimits T\supset\operatorname{Fac}\nolimits U$, and $\geq$ gives a partial order. On the other hand, we call a $\Gamma$-module $T$ almost complete tilting if $\operatorname{pd}\nolimits{}_{\Gamma}T\leq 1$, $\operatorname{Ext}\nolimits^{1}_{\Gamma}(T,T)=0$ and $T$ has exactly $(n-1)$ non-isomorphic indecomposable direct summands.

We collect some basic results.

Proposition 4.11**.**

• (a)

Any almost complete tilting $\Gamma$-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 $\Gamma$-module which is a common direct summand of $T$ and $U$.*

• (c)

Assume $T\geq U$. Then there exists a sequence $T=T_{0}>T_{1}>T_{2}>\cdots>U$ 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.

• (d)

$T\geq U$* if and only if there exists an exact sequence $0\to T\to U_{0}\to U_{1}\to 0$ with $U_{i}\in\operatorname{add}\nolimits U$.*

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. For a module-finite $R$-algebra $\Gamma$, we call a $\Gamma$-module Cohen-Macaulay if it is a Cohen-Macaulay $R$-module. As usual, we denote by $\operatorname{CM}\nolimits(\Gamma)$ the category of Cohen-Macaulay $\Gamma$-modules.

Lemma 4.12**.**

Let $\Gamma$ be a module-finite algebra over a complete local Gorenstein ring $R$ such that $\Gamma\in\operatorname{CM}\nolimits(R)$, and $T$ and $U$ tilting $\Gamma$-modules. Assume $U\in\operatorname{CM}\nolimits(\Gamma)$.

• (a)

If $T\geq U$, then $T\in\operatorname{CM}\nolimits(\Gamma)$.

• (b)

Let $P$ be a projective $\Gamma$-module such that $\operatorname{Hom}\nolimits_{R}(P,R)$ is a projective $\Gamma^{\operatorname{op}\nolimits}$-module. Then $P\in\operatorname{add}\nolimits U$.

Proof.

(a) By Proposition 4.11(d), there exists an exact sequence $0\to T\to U_{0}\to U_{1}\to 0$ with $U_{i}\in\operatorname{add}\nolimits U$. Thus the assertion holds.

(b) We have $\operatorname{Ext}\nolimits^{1}_{\Gamma^{\operatorname{op}\nolimits}}(\operatorname{Hom}\nolimits_{R}(P,R),\operatorname{Hom}\nolimits_{R}(U,R))=0$. Since we have a duality $\operatorname{Hom}\nolimits_{R}(\ ,R):\operatorname{CM}\nolimits(\Gamma)\leftrightarrow\operatorname{CM}\nolimits(\Gamma^{\operatorname{op}\nolimits})$, it holds $\operatorname{Ext}\nolimits^{1}_{\Gamma}(U,P)=0$. There exists an exact sequence $0\to P\to U_{0}\to U_{1}\to 0$ with $U_{i}\in\operatorname{add}\nolimits U$ [H, Lem. III.2.3], which must split since $\operatorname{Ext}\nolimits^{1}_{\Gamma}(U,P)=0$. Thus we have $P\in\operatorname{add}\nolimits U$. ∎

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 4.13**.**

Let $R=S/(f)$ be a one-dimensional reduced hypersurface singularity and $M$, $N$ and $N^{\prime}$ cluster tilting objects in $\operatorname{CM}\nolimits(R)$.

• (a)

$\operatorname{Hom}\nolimits_{R}(M,N)$* is a tilting $\operatorname{End}\nolimits_{R}(M)$-module of projective dimension at most one.*

• (b)

If $N^{\prime}$ is a cluster tilting mutation of $N$, then $\operatorname{Hom}\nolimits_{R}(M,N^{\prime})$ is a tilting mutation of $\operatorname{Hom}\nolimits_{R}(M,N)$.

Now we shall prove Theorem 4.2. Fix $w\in{\mathfrak{S}}_{n}$ and put $\Gamma:=\operatorname{End}\nolimits_{R}(M_{w})$. Since $M_{w}$ is a generator of $R$, the functor $\operatorname{Hom}\nolimits_{R}(M_{w},\ ):\operatorname{CM}\nolimits(R)\to\operatorname{CM}\nolimits(\Gamma)$ is fully faithful. By Theorem 4.1, $M_{w}$ is a cluster tilting object in $\operatorname{CM}\nolimits(R)$. By Proposition 4.13(a), $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}})$ ($w^{\prime}\in{\mathfrak{S}}_{n}$) is a Cohen-Macaulay tilting $\Gamma$-module.

(b) Take any Cohen-Macaulay tilting $\Gamma$-module $U$. Since $P:=\operatorname{Hom}\nolimits_{R}(M_{w},R)$ is a projective $\Gamma$-module such that $\operatorname{Hom}\nolimits_{R}(P,R)=M_{w}=\operatorname{Hom}\nolimits_{R}(R,M_{w})$ is a projective $\Gamma$-module, we have $P\in\operatorname{add}\nolimits U$ by Lemma 4.12(b). In particular, by Proposition 4.11(a)(b),

• •

any Cohen-Macaulay tilting $\Gamma$-module has at most $(n-1)$ tilting mutations which are Cohen-Macaulay.

Conversely, by Proposition 4.8 and Proposition 4.13(b),

• •

any Cohen-Macaulay tilting $\Gamma$-module of the form $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}})$ ($w^{\prime}\in{\mathfrak{S}}_{n}$) has precisely $(n-1)$ tilting mutations $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}s_{i}})$ ($1\leq i<n$) which are Cohen-Macaulay.

Consequently, any successive tilting mutation of $\Gamma=\operatorname{Hom}\nolimits_{R}(M_{w},M_{w})$ has the form $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}})$ for some $w^{\prime}\in{\mathfrak{S}}_{n}$ if each step is Cohen-Macaulay.

Using this observation, we shall show that $U$ is isomorphic to $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}})$ for some $w^{\prime}$. Since $\Gamma\geq U$, there exists a sequence

[TABLE]

satisfying the conditions in Proposition 4.11(c). By Lemma 4.12(a), each $T_{i}$ is Cohen-Macaulay. Thus the above observation implies that each $T_{i}$ has the form $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w_{i}})$ for some $w_{i}\in{\mathfrak{S}}_{n}$. Moreover, $w_{i}\neq w_{j}$ for $i\neq j$. Since ${\mathfrak{S}}_{n}$ 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 $\operatorname{CM}\nolimits(R)$. Again by Proposition 4.13(a), $\operatorname{Hom}\nolimits_{R}(M_{w},U)$ is a Cohen-Macaulay tilting $\Gamma$-module. By part (b) which we already proved, $\operatorname{Hom}\nolimits_{R}(M_{w},U)$ is isomorphic to $\operatorname{Hom}\nolimits_{R}(M_{w},M_{w^{\prime}})$ for some $w^{\prime}\in{\mathfrak{S}}_{n}$. Since the functor $\operatorname{Hom}\nolimits_{R}(M_{w},\ ):\operatorname{CM}\nolimits(R)\to\operatorname{CM}\nolimits(\Gamma)$ is fully faithful, $U$ is isomorphic to $M_{w^{\prime}}$, and the former assertion is proved.

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

Proposition 4.14**.**

Let $\mathcal{C}$ be a 2-CY Frobenius category with a cluster tilting object. Then any rigid object in $\mathcal{C}$ is a direct summand of some cluster tilting object in $\mathcal{C}$.

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

Now let $S^{\prime\prime}:=k[[x,y,u,v]]$, $f_{i}\in{\mathfrak{m}}=(x,y)$ ($1\leq i\leq n$) and $R^{\prime\prime}:=S^{\prime\prime}/(f_{1}\cdots f_{n}+uv)$. For $w\in{\mathfrak{S}}_{n}$ and $I\subseteq\{1,\cdots,n\}$, we put

[TABLE]

We have the following result (see 5.2 for definition).

Corollary 4.15**.**

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

• (a)

There are exactly $n!$ basic cluster tilting objects $M_{w}$ ($w\in{\mathfrak{S}}_{n}$) and exactly $2^{n}-1$ indecomposable rigid objects $U_{I}$ ($\emptyset\neq I\subset\{1,\cdots,n\}$) in $\operatorname{CM}\nolimits(R^{\prime\prime})$.

• (b)

There are non-commutative crepant resolutions $\operatorname{End}\nolimits_{R^{\prime\prime}}(M_{w})$ ($w\in{\mathfrak{S}}_{n}$) of $R^{\prime\prime}$, which are derived equivalent.

Proof.

(a) We only have to apply Knörrer periodicity $\underline{\operatorname{CM}\nolimits}(R)\to\underline{\operatorname{CM}\nolimits}(R^{\prime\prime})$ [Kn, So] as follows:

Since $S^{w}_{i}\in\operatorname{CM}\nolimits(R)$ has a projective resolution

[TABLE]

for $a:=f_{w(1)}\cdots f_{w(i)}$ and $b:=f_{w(i+1)}\cdots f_{w(n)}$, the corresponding object $X\in\operatorname{CM}\nolimits(R^{\prime\prime})$ has a projective resolution

[TABLE]

It is easily checked that $X$ is isomorphic to $(u,a)=U^{w}_{i}$.

(b) Any cluster tilting object gives a non-commutative crepant resolution by [I2, Th. 5.2.1]. They are derived equivalent by [I2, Cor. 5.3.3]. ∎

For example,

[TABLE]

has a non-commutative crepant resolution for distinct elements $\lambda_{1},\cdots,\lambda_{n}\in k$.

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 gives at the same time alternative proofs for geometric results, using cluster tilting objects. The aim of this section is to establish a 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 $(R,{\mathfrak{m}})$ be a three-dimensional complete normal Gorenstein singularity over an algebraically closed field $k$ of characteristic zero, and let $X=\operatorname{Spec}\nolimits(R)$. A resolution of singularities $Y\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}X$ is called

• •

crepant, if $\omega_{Y}\cong\pi^{*}\omega_{X}$ for canonical sheaves $\omega_{X}$ and $\omega_{Y}$ of $X$ and $Y$ respectively.

• •

small, if the fibre of the closed point has dimension at most 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

[TABLE]

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

A cDV singularity is called $cA_{n}$ if the intersection of $f(x,y,z)+tg(x,y,z,t)=0$ with a generic hyperplane $ax+by+cz+dt=0$ in $k^{4}$ is an $A_{n}$ surface singularity. Generic means that the coefficients $(a,b,c,d)$ belong to a non-empty Zariski open subset in $k^{4}$.

Theorem 5.1**.**

[Re, Cor. 1.12, Th. 1.14]** Let $X$ be a three-dimensional Gorenstein 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.

Since any isolated cDV singularity is terminal [Re], we can apply Van den Bergh’s results on non-commutative crepant resolutions defined as follows.

Definition 5.2**.**

[V2, Def. 4.1]** Let $(R,{\mathfrak{m}})$ be a three-dimensional normal Gorenstein domain. An $R$-module $M$ gives rise to a non-commutative crepant resolution if

• (i)

$M$* is reflexive,*

• (ii)

$A=\operatorname{End}\nolimits_{R}(M)$* is Cohen-Macaulay as an $R$–module,*

• (iii)

$\mathrm{gl.dim}(A)=3$.

The following result establishes a useful connection.

Theorem 5.3**.**

[V1, Cor. 3.2.11]**[V2, Th. 6.6.3]** Let $(R,{\mathfrak{m}})$ be an isolated cDV singularity. Then there exists a crepant resolution of $X=\operatorname{Spec}\nolimits(R)$ 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 $\underline{\operatorname{CM}\nolimits}(R)$.

Theorem 5.4**.**

[I2, Th. 5.2.1]**[IR, Cor. 8.13]** Let $(R,{\mathfrak{m}})$ be a three-dimensional normal Gorenstein domain which is an isolated singularity. 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 $\underline{\operatorname{CM}\nolimits}(R)$.

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 $\underline{\operatorname{CM}\nolimits}(R)$. Then $M$ is automatically reflexive. From the exact sequence

[TABLE]

we obtain

[TABLE]

Since $M$ is rigid, $\operatorname{Ext}\nolimits^{1}_{R}(M,M)=0$. Moreover, $\operatorname{depth}\nolimits(\operatorname{Hom}\nolimits_{R}(F,M))=\operatorname{depth}\nolimits(M)=3$ and

$\operatorname{depth}\nolimits(\operatorname{Hom}\nolimits_{R}(\Omega(M),M)\geq 2$, and hence $\operatorname{depth}\nolimits(\operatorname{End}\nolimits_{R}(M))=3$ and $A=\operatorname{End}\nolimits(M)$ is maximal Cohen-Macaulay over $R$.

For the difficult part of this implication, claiming that $\mathrm{gl.dim}(A)=3$, 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 $M^{\prime}$ giving rise to a non-commutative crepant resolution, which is maximal Cohen-Macaulay and contains $R$ as a direct summand.

By the assumption, $\operatorname{depth}\nolimits(\operatorname{End}\nolimits_{R}(M^{\prime}))=3$ and we can apply [IR, Lem. 8.5] to the exact sequence (1) to deduce that $\operatorname{Ext}\nolimits^{1}_{R}(M^{\prime},M^{\prime})=0$, so that $M^{\prime}$ is rigid. The difficult part saying that $M^{\prime}$ is cluster tilting is proven in [I2, Th. 5.2.1]. ∎

We now summarize the results of this section.

Theorem 5.5**.**

Let $(R,{\mathfrak{m}})$ be an isolated cDV singularity. Then the following are equivalent.

• (a)

$\operatorname{Spec}\nolimits(R)$* has a small resolution.*

• (b)

$\operatorname{Spec}\nolimits(R)$* has a crepant resolution.*

• (c)

$(R,{\mathfrak{m}})$* has a non-commutative crepant resolution.*

• (d)

$\underline{\operatorname{CM}\nolimits}(R)$* 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 $X=\operatorname{Spec}\nolimits(R)$ be an isolated $cA_{n}$–singularity.

• (a)

Let $Y\longrightarrow X$ 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 $(R,{\mathfrak{m}})$ be an 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 $R^{\prime}$ be a one-dimensional singularity defined by $g(x,y)$. Then $\underline{\operatorname{CM}\nolimits}(R^{\prime})$ 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)$\Leftrightarrow$(f) This follows from Theorem 5.6.

(d)$\Leftrightarrow$(e) By the Knörrer periodicity there is an equivalence of triangulated categories between the stable categories $\underline{\operatorname{CM}\nolimits}(R)\cong\underline{\operatorname{CM}\nolimits}(R^{\prime})$. 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 $\operatorname{Spec}\nolimits(R)$.

• (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 $(R,{\mathfrak{m}})$.

• (d)

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

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 some 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

[TABLE]

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 $\operatorname{ord}\nolimits(g)$ 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)

$A_{n}$* ($1\leq n$) is a $cA_{1}$–singularity,*

• (b)

$D_{n}$* ($4\leq n$) and $E_{n}$ ($n=6,7,8$) are $cA_{2}$–singularities,*

• (c)

$T_{3,q,2,2}(\lambda)$* ($6\leq q$) is a $cA_{2}$–singularity,*

• (d)

$T_{p,q,2,2}(\lambda)$* ($4\leq p\leq q$) is a $cA_{3}$–singularity,*

• (e)

$k[[x,y,z,t]]/(x^{2}+y^{2}+g(z,t))$* ($g\in k[[z,t]]$) is a $cA_{m}$–singularity if $m=\operatorname{ord}\nolimits(g)-1\geq 1$.*

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 5.7 if and only if it is of type $A_{n}$ ($n$* is odd) or $D_{n}$ ($n$ is even).*

• (b)

A $T_{p,q,2,2}(\lambda)$–singularity satisfies the equivalent conditions in Theorem 5.7 if and only if $p=3$ and $q$ is even or if both $p$ and $q$ are even.

• (c)

A singularity $k[[x,y,u,v]]/(uv+f_{1}\cdots f_{n})$ with irreducible and mutually prime $f_{i}\in(x,y)\subset k[[x,y]]$ ($1\leq i\leq n$) satisfies the equivalent conditions in Theorem 5.7 if and only if $f_{i}\notin(x,y)^{2}$ 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 5.7, 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 $g(x,y)=y(y-x^{2})(y-\lambda x^{2})$ for $(p,q)=(3,6)$, and $m=3$ and $g(x,y)=xy(x-y)(x-\lambda y)$ 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 $m=\sum_{i=1}^{n}\operatorname{ord}\nolimits(f_{i})-1$ and $g=f_{1}\cdots f_{n}$. So $g$ has $m+1$ factors if and only if $\operatorname{ord}\nolimits(f_{i})=1$ for any $i$. ∎

Immediately we have the following conclusion.

Corollary 6.3**.**

• (a)

A simple curve singularity $R$ 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 basic cluster tilting objects in $\underline{\operatorname{CM}\nolimits}(R)$ is $1$ for type $A_{n}$ ($n$ is odd) and $2$ for type $D_{n}$ ($n$ is even).*

• (b)

A $T_{p,q}(\lambda)$-singularity $R$ 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 basic cluster tilting objects in $\underline{\operatorname{CM}\nolimits}(R)$ is $2$ if $p=3$ and $q$ is even, and $3$ if both $p$ and $p$ are even.

• (c)

A singularity $R=k[[x,y]]/(f_{1}\cdots f_{n})$ with irreducible and mutually prime $f_{i}\in(x,y)\subset k[[x,y]]$ ($1\leq i\leq n$) has a cluster tilting object if and only if $f_{i}\notin(x,y)^{2}$ for any $i$. In this case, the number of non-isomorphic indecomposable summands of basic cluster tilting objects in $\underline{\operatorname{CM}\nolimits}(R)$ is $n-1$.

In view of Theorem 4.2, we have completed the proof of Theorem 1.5.

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

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

Definition 6.4**.**

For $f\in{\mathfrak{m}}^{2}$ we denote by $J(f)=\langle\frac{\partial f}{\partial x_{1}},\dots,\frac{\partial f}{\partial x_{n}}\rangle$ its Jacobi ideal. The Milnor number $\mu(f)$ is defined as

[TABLE]

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

Lemma 6.5**.**

A hypersurface singularity $f=0$ is isolated if and only if $\mu(f)<\infty$.

Definition 6.6** ([AGV]).**

Two hypersurface singularities $f=0$ and $g=0$ are called right equivalent ($f\stackrel{{\scriptstyle r}}{{\sim}}g$) if there exists an algebra automorphism $\varphi\in Aut(R)$ such that $g=\varphi(f)$.

Note that $f\stackrel{{\scriptstyle r}}{{\sim}}g$ implies an isomorphism of $k$–algebras

[TABLE]

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

Lemma 6.7**.**

Assume $f\stackrel{{\scriptstyle r}}{{\sim}}g$, then $\mu(f)=\mu(g)$.

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 $f\in{\mathfrak{m}}^{2}$ be an isolated singularity with Milnor number $\mu$. Then

[TABLE]

for any $g\in{\mathfrak{m}}^{\mu+2}$.

We shall need the following easy lemma.

Lemma 6.9**.**

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

[TABLE]

is a homogeneous form of degree $n\geq 3$. Then

[TABLE]

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

[TABLE]

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

[TABLE]

where $h\in{\mathfrak{m}}^{2(n-1)}\subset{\mathfrak{m}}^{n+1}$. Note that $\mu(x^{2}+y^{2}+z^{n})=n-1$, hence by Theorem 6.8 we have

[TABLE]

∎

Now we are ready to give a proof of Proposition 6.1. We only have to show the assertion (e) since the other cases are special cases of this. We denote by $H$ the hyperplane in a four-dimensional space defined by the equation $t=\alpha x+\beta y+\gamma z$, $\alpha,\beta,\gamma\in k$. We put

[TABLE]

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

[TABLE]

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

Next we consider the case $m\geq 2$. Assume $\alpha\in k$ satisfies $a_{0}+a_{1}\alpha+\cdots+a_{m+1}\alpha^{m+1}\neq 0$. By Lemma 6.9, we have $h\stackrel{{\scriptstyle r}}{{\sim}}x^{2}+y^{2}+z^{m+1}$. By Lemma 6.7, we have $\mu(h)=\mu(x^{2}+y^{2}+z^{m+1})=m$. Hence $f\stackrel{{\scriptstyle r}}{{\sim}}h\stackrel{{\scriptstyle r}}{{\sim}}x^{2}+y^{2}+z^{m+1}$ 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, Sk].

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

Theorem 7.1**.**

Let $(R,{\mathfrak{m}})$ be an odd-dimensional isolated hypersurface singularity and $\Gamma$ a 2-CY tilted algebra coming from $\underline{\operatorname{CM}\nolimits}(R)$. Then we have the following.

• (a)

$\Gamma$* is a symmetric algebra.*

• (b)

All components in the stable AR-quiver of infinite type $\Gamma$ 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 $\widehat{kQ}$ of the path algebra $kQ$ by the formula

[TABLE]

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

Lemma 7.2**.**

The ideal $I$ is generated in $\widehat{kQ}$ by a minimal system of relations, that is, a set of elements $\rho_{1},\cdots,\rho_{n}$ 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 $\Lambda=B_{2,2}(\lambda)$ defined below, the elements $\rho_{1},\cdots,\rho_{n}$ listed as generators for $I$ form a minimal system of relations. So they generate $I$ in $\widehat{kQ}$. They also yield a $k$-basis of $I^{\prime}/{I^{\prime}J+JI^{\prime}}\xrightarrow{\sim}I/{IJ+JI}$, where $I^{\prime}=I\cap kQ$ and $J^{\prime}=J\cap kQ$. But they do not generate the ideal $I^{\prime}$ of $kQ$ since, as one can show, the quotient $kQ/\langle\rho_{1},\cdots,\rho_{n}\rangle$ is infinite-dimensional.

On the other hand, the ideal $I^{\prime}$ is generated by the preimage $\rho_{1},\cdots,\rho_{n}$ of a basis of $I^{\prime}/{I^{\prime}J^{\prime}+J^{\prime}I^{\prime}}$ if the quotient $kQ/\langle\rho_{1},\cdots,\rho_{n}\rangle$ is finite-dimensional, since then the ideal $\langle\rho_{1},\cdots,\rho_{n}\rangle$ contains a power of $J^{\prime}$. This happens for example for the algebra $A_{2}(\lambda)$ as defined below, cf. also [Sk, 5.9] and [BS, Th. 1].

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

[TABLE]

where $S_{i}$ and $S_{j}$ denote the simple $\Lambda$-modules corresponding to the vertices $i$ and $j$ [B]. When $\Lambda$ is 2-CY tilted, then

[TABLE]

(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 $q\geq 2$ and $\lambda\in k^{*}$ we write $A_{q}(\lambda)=\widehat{kQ}/I$, where

[TABLE]

and

[TABLE]

If $q=2$, then we additionally assume $\lambda\neq 1$. (It can be shown that for $q\geq 3$ we have $A_{q}(\lambda)\cong A_{q}(1)$, so we drop the parameter $\lambda$ in this case.)

(2) For $p,q\geq 1$ and $\lambda\in k^{*}$ we write $B_{p,q}(\lambda)=\widehat{kQ}/I$, where

[TABLE]

and

[TABLE]

For $p=q=1$ we additionally assume $\lambda\neq 1$.

When $p=q=1$, the generators $\varphi$ and $\psi$ can be excluded and $B_{1,1}(\lambda)$ is given by the completion of the path algebra of the quiver

[TABLE]

modulo the relations

[TABLE]

For $(p,q)\neq(1,1)$ we have $B_{p,q}(\lambda)\cong B_{p,q}(1)$. In particular, for $p=1$ and $q\geq 2$ the algebra is isomorphic to $\widehat{kQ}/I$, where

[TABLE]

and

[TABLE]

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

Lemma 7.4**.**

In the algebra $A_{q}(\lambda)$ the following zero relations hold:

[TABLE]

Proof. We have to consider separately the cases $q=2$ and $q\geq 3$.

Let $q=2$, then we assumed $\lambda\neq 1$. We have

[TABLE]

hence $\alpha\beta\alpha=0.$ In a similar way we obtain $\beta\alpha\beta=0$. Then $\alpha\varphi^{2}=\alpha\beta\alpha=0$, $\varphi^{2}=\beta\alpha\beta\alpha=0$ and the remaining zero relations follow analogously.

Let $q\geq 3$. Then

[TABLE]

so $(1-\psi^{q-2})\psi^{2}\alpha=0$ and hence

[TABLE]

in $\widehat{kQ}/I$. The remaining zero relations follow similarly. ∎

Lemma 7.5**.**

We have the following relations in $B_{p,q}(\lambda)$:

[TABLE]

Moreover, $\alpha\beta\cdot\delta\gamma=\delta\gamma\cdot\alpha\beta$. For $q\geq p\geq 2$ we have

[TABLE]

for $q>p=1$ we have

[TABLE]

and for $p=q=1$

[TABLE]

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 $T_{3,2q+2}(\lambda)$–singularity, where $q\geq 2$ and $\lambda\in k^{*}$. Then in the triangulated category $\underline{\operatorname{CM}\nolimits}(R)$ there exists a cluster tilting object with the corresponding 2-CY-tilted algebra isomorphic to $A_{q}(\lambda)$.

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

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

The coordinate ring of $T_{3,6}(\lambda)$ is isomorphic to

[TABLE]

where $\lambda\neq 0,1$. Consider Cohen-Macaulay modules $M$ and $N$ given by the two-periodic free resolutions

[TABLE]

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

[TABLE]

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

[TABLE]

and

[TABLE]

The isomorphism $A_{2}(\lambda)\longrightarrow\underline{\operatorname{End}\nolimits}(M\oplus N)$ is given by

[TABLE]

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

[TABLE]

Consider the Cohen-Macaulay module $M\oplus N$, where

[TABLE]

Again, by a straightforward calculation

[TABLE]

and

[TABLE]

[TABLE]

If $q\geq 4$ then $\underline{\operatorname{End}\nolimits}(M\oplus N)$ is isomorphic to $\widehat{kQ}/I$, where

[TABLE]

and the relations are

[TABLE]

for

[TABLE]

By rescaling all generators $\alpha\mapsto 2^{a}\alpha,\beta\mapsto 2^{b}\beta,\varphi\mapsto 2^{f}\varphi,\psi\mapsto 2^{g}\psi$ for properly chosen $a,b,f,g\in\mathbb{Q}$ one can easily show $\underline{\operatorname{End}\nolimits}(M\oplus N)\cong A_{q}$.

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

[TABLE]

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

[TABLE]

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

[TABLE]

This system is equivalent to

[TABLE]

and hence the statement is proven.

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

[TABLE]

Take

[TABLE]

By Theorem 4.1 or Corollary 6.3, $M\oplus N\oplus K$ is cluster tilting. Moreover, $B_{1,1}(\lambda)\simeq\underline{\operatorname{End}\nolimits}(M\oplus N\oplus K)$.

Let now

[TABLE]

where $(p,q)\neq(1,1)$ and

[TABLE]

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

We have seen that the algebras $A_{q}(\lambda)$ and $B_{p,q}(\lambda)$ are symmetric, and the indecomposable nonprojective modules have $\tau$-period at most 2, hence $\Omega$-period dividing 4 since $\tau=\Omega^{2}$ 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 ${\rm Coh}(\mathbb{X})$ 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 ${\rm Coh}(\mathbb{X})$ 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 $T_{p,q}(\lambda)$, 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 follow from the tameness of $\operatorname{CM}\nolimits(T_{3,p}(\lambda))$ and $\operatorname{CM}\nolimits(T_{p,q}(\lambda))$ 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 a more general situation than in section 2. Let $k$ be an algebraically closed field and $\mathcal{C}$ 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 $\mathcal{C}$ whether cluster tilting objects (respectively, non-zero rigid objects) exist in $\mathcal{C}$ or not. Let us start with giving the possible shapes of the AR quiver of $\mathcal{C}$. Recall that a subgroup $G$ of ${\rm Aut}({\bf Z}\Delta)$ is called weakly admissible if $x$ and $gx$ do not have a common direct successor for any vertex $x$ in ${\bf Z}\Delta$ and $g\in G\backslash\{1\}$ [XZ, Am].

Proposition 8.1**.**

The AR quiver of $\mathcal{C}$ is ${\bf Z}\Delta/G$ for a Dynkin diagram $\Delta$ and a weakly admissible subgroup $G$ of ${\rm Aut}({\bf Z}\Delta)$ which contains $F\in{\rm Aut}({\bf Z}\Delta)$ defined by the list below. Moreover, $G$ is generated by a single element $g\in{\rm Aut}({\bf Z}\Delta)$ in the list below.

[TABLE]

In each case, elements in the torsion part of ${\rm Aut}({\bf Z}\Delta)$ are induced by the automorphism of $\Delta$. The torsionfree part of ${\rm Aut}({\bf Z}\Delta)$ is generated by $\tau$ except the case $(A_{n})$ with even $n$, in which case it is generated by the square root of $\tau$.

Proof.

By [XZ] (see also [Am, 4.0.4]), the AR quiver of $\mathcal{C}$ is ${\bf Z}\Delta/G$ for a Dynkin diagram $\Delta$ and a weakly admissible subgroup $G$ of ${\rm Aut}({\bf Z}\Delta)$. Since $\mathcal{C}$ is 2-Calabi-Yau, $G$ contains $F$. By [Am, 2.2.1], $G$ is generated by a single element $g$. By the condition $F\in\langle g\rangle$, we have the above list. ∎

Note that, by a result of Keller [Ke], the translation quiver ${\bf Z}\Delta/G$ for any Dynkin diagram $\Delta$ and any weakly admissible group $G$ of ${\rm Aut}({\bf Z}\Delta)$ is realized as the AR quiver of a triangulated orbit category $\mathcal{D}^{b}(H)/g$ for a hereditary algebra $H$ of type $\Delta$ and some autofunctor $g$ of $\mathcal{D}^{b}(H)$.

Theorem 8.2**.**

(1) $\mathcal{C}$ has a cluster tilting object if and only if the AR quiver of $\mathcal{C}$ is ${\bf Z}\Delta/g$ for a Dynkin diagram $\Delta$ and $g\in{\rm Aut}({\bf Z}\Delta)$ in the list below.

[TABLE]

(2) $\mathcal{C}$ does not have a non-zero rigid object if and only if the AR quiver of $\mathcal{C}$ is ${\bf Z}\Delta/g$ for a Dynkin diagram $\Delta$ and $g\in{\rm Aut}({\bf Z}\Delta)$ in the list below.

[TABLE]

Proof.

Our method is based on the computation of additive functions in section 2. We refer to [I1, Section 4.4] for detailed explanation.

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

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

(2) If $g$ is on the list above, then one can easily check that $\mathcal{C}$ does not have non-zero rigid objects. Conversely, if $g$ is not on the list, then one can easily check that at least one indecomposable object which corresponds to an end point of $\Delta$ is rigid. ∎