CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES
IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Abstract. In this article we study Cohen-Macaulay modules over one-dimensional hypersurface sin-
gularities 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 τ2 = id. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve
singularities.
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 CH is the orbit category D
b(H)/F , where Db(H) is the bounded derived
category of finite dimensional H-modules and the functor F : Db(H) → Db(H) is τ−1[1] = S−1[2]. Here
τ denotes the translation associated with almost split sequences/triangles and S the Serre functor [BK]
on Db(H). (See [CCS] for an independent definition of a category equivalent to the cluster category when
H is of Dynkin type An).
An object T in a cluster category CH was defined to be a (cluster) tilting object if Ext
(T, T ) = 0,
and if Ext1CH (X,X ⊕ T ) = 0, then X is in addT . 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 Ext1CH (X,T ) = 0 implies that X is in
addT , 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 mod(Λ) of
finite dimensional modules over a preprojective algebra of Dynkin type Λ 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 mod(Λ).
The categories CH and mod(Λ) are both triangulated categories [Ke, H], with finite dimensional ho-
momorphism 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 = Σ2, where Σ is the shift
functor in the triangulated category.
For an arbitrary 2-CY triangulated category C with finite dimensional homomorphism spaces over
a field k, a cluster tilting object T in C was defined to be an object satisfying the stronger property
discussed above, corresponding to the property of being maximal 1-orthogonal/Ext-configuration [KR].
The corresponding class of algebras, containing the cluster tilted ones, have been called 2-CY tilted. With
this concept many results have been generalised from cluster categories, and from the stable categories
mod(Λ), to this more general setting in [KR], which moreover contains several results which are new also
in the first two cases.
One of the important applications of classical tilting theory has been the construction of derived
equivalences: Given a tilting bundle T on a smooth projective variety X , the total right derived functor
of Hom(T, ) is an equivalence from the bounded derived category of coherent sheaves onX to the bounded
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.
http://arxiv.org/abs/0704.1249v3
2 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
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 CM(R) is the stable category of maximal Cohen-Macaulay modules over an odd-dimensional
isolated hypersurface singularity, then CM(R) is 2-CY. If it contains a cluster tilting object T , then the
functor Hom(T, ) induces an equivalence between the quotient of CM(R) by the ideal of morphisms
factoring through τT and the category of finite dimensional modules over the endomorphism algebra
B = End(T ). It is then not hard to see that B is symmetric and the indecomposable nonprojective
B-modules are τ -periodic of τ -period at most 2. In this article, we study examples of this setup arising
from finite, tame and wild CM-type isolated hypersurface singularities R. The endomorphism algebras
of the cluster tilting objects in the tame case occur in lists in [BS, Er, Sk]. We also obtain a large class
of symmetric finite dimensional algebras where the stable AR-quiver consists only of tubes of rank one
or two. Examples of (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 CM(R) of
maximal Cohen-Macaulay R-modules being our 2-CY category. These singularities are indexed by the
Dynkin diagrams, and in the cases Dn for odd n and E7 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
CM(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 (An, n even, E6, E8), and cases where the maximal rigid objects coincide with
the cluster tilting objects (An, n odd and Dn, 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 CH and
mod(Λ) [BMRRT, BMR2, GLSc]. The results are also valid for any odd-dimensional simple hypersurface
singularity, since the stable categories of Cohen-Macaulay modules are all triangle equivalent [Kn, So].
We shall construct a large class of one-dimensional hypersurface singularities R, where CM(R) or
CM(R) has a cluster tilting object, including examples coming from simple singularities and minimally
elliptic singularities. We classify all rigid objects in CM(R) for these R, in particular, we give a bijection
between cluster tilting objects in CM(R) and elements in a symmetric group. Our method is based
on a higher theory of almost split sequences [I1, I2], and a crucial role is played by the endomorphism
algebras EndR(T ) (called ‘three-dimensional Auslander algebras’) of cluster tilting objects T in CM(R).
These algebras have global dimension three, and have 2-CY tilted algebras as stable factors. The functor
HomR(T, ) : CM(R) → mod(EndR(T )) sends cluster tilting objects in CM(R) to tilting modules over
EndR(T ). By comparing cluster tilting mutations in CM(R) and tilting mutation in CM(EndR(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 CM(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 sin-
gularities 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 sin-
gularities 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
cAn–singularities for existence of crepant resolutions to a criterion for one-dimensional hypersurface sin-
gularities 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.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 3
In section 2 we investigate maximal rigid objects and cluster tilting objects in CM(R) for simple
one-dimensional hypersurface singularities. We decide whether extension spaces are zero or not by using
covering techniques. In section 3 we point out that we could also use the computer program Singular
[GP] to accomplish the same thing. In section 4 we construct 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,m) be a local complete d-dimensional commutative noetherian Gorenstein isolated singularity
and R/m = k ⊂ R, where k is an algebraically closed field of characteristic zero. We denote by CM(R)
the category of maximal Cohen-Macaulay modules over R. Then CM(R) is a Frobenius category (i.e. an
exact category with enough projectives and injectives which coincide), and so the stable category CM(R)
is a Hom-finite triangulated category with shift functor Σ = Ω−1 [H]. For an integer n, we say that
CM(R) or CM(R) is n-CY if there exists a functorial isomorphism
HomR(X,Y ) ≃ DHomR(Y,Σ
for any X,Y ∈ CM(R).
We collect some fundamental results.
• We have AR-duality
HomR(X,Y ) ≃ DExt
R(Y, τX)
with τ ≃ Ω2−d [Au]. In particular, CM(R) is (d− 1)-CY.
• If R is a hypersurface singularity, then Σ2 = id [Ei].
Consequently, if d is odd, then τ = Ω and CM(R) is 2-CY. If d is even, then τ = id and CM(R)
is 1-CY, hence any non-free Cohen-Macaulay R-module M satisfies Ext1R(M,M) 6= 0.
• (Knörrer periodicity)
CM(k[[x0, · · · , xd, y, z]]/(f + yz)) ≃ CM(k[[x0, · · · , xd]]/(f))
for any f ∈ k[[x0, · · · , xd]] [Kn] ([So] in characteristic two).
We state some of the definitions, valid more generally, in the context of CM(R) and CM(R).
Definition 1.1. Let C = CM(R) or CM(R). We call an object M ∈ C
• rigid if Ext1R(M,M) = 0,
• maximal rigid if it is rigid and any rigid N ∈ C satisfying M ∈ addN satisfies N ∈ addM ,
• cluster tilting if addM = {X ∈ C | Ext1R(M,X) = 0} = {X ∈ C | Ext
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 CM(R). If C is 2-CY, then M ∈ C is cluster tilting if and only if
addM = {X ∈ C | Ext1R(M,X) = 0}.
4 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Definition 1.2. Let C = CM(R) (or CM(R)) be 2-CY and M ∈ C a basic cluster tilting object. Take an
indecomposable summand X of M = X⊕N . Then there exist short exact sequences (or triangles) (called
exchange sequences)
→ Y and Y
such that Ni ∈ addN and f0 is a minimal right (addN)-approximation. Then Y ⊕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 f1 is a minimal right (addN)-approximation and gi is a minimal left (addN)-approximation
automatically, so X ⊕N is a cluster tilting mutation of Y ⊕N . It is known that there are no more basic
cluster tilting objects containing N [IY, Th. 5.3].
Let R = k[[x, y, z2, · · · , zd]]/(f) be a simple hypersurface singularity so that in characteristic zero f is
one of the following polynomials,
(An) x
2 + yn+1 + z22 + z
3 + · · · + z
d (n ≥ 1)
(Dn) x
2y + yn−1 + z22 + z
3 + · · · + z
d (n ≥ 4)
(E6) x
3 + y4 + z22 + z
3 + · · · + z
(E7) x
3 + xy3 + z22 + z
3 + · · · + z
(E8) x
3 + y5 + z22 + z
3 + · · · + z
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 ≥ 1 over an algebraically closed
field k of characteristic zero.
(1) Assume that d is even. Then 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 CM(R) are
as follows:
f indec. rigid cluster tilting max. rigid summands of max. rigid
(An) n : odd 2 2 2 1
(An) n : even 0 0 1 0
(Dn) n : odd 2 0 2 1
(Dn) n : even 6 6 6 2
(E6) 0 0 1 0
(E7) 2 0 2 1
(E8) 0 0 1 0
We also consider a minimally elliptic curve singularity Tp,q(λ) (p ≤ q). Assume for simplicity that
our base field k is algebraically closed of characteristic zero. Then these singularities are given by the
equations
xp + yq + λx2y2 = 0,
where 1
and certain values of λ ∈ k have to be excluded. They are of tame Cohen-Macaulay
representation type [D, Kah, DG]. We divide into two cases.
(i) Assume 1
. This case occurs if and only if (p, q) = (3, 6) or (4, 4), and Tp,q(λ) is called
simply elliptic. The corresponding coordinate rings can be written in the form
T3,6(λ) = k[[x, y]]/(y(y − x
2)(y − λx2))
T4,4(λ) = k[[x, y]]/(xy(x − y)(x− λy)),
where in both cases λ ∈ k \ {0, 1}.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 5
(ii) Assume 1
. Then Tp,q(λ) does not depend on the continuous parameter λ, and is called a
cusp singularity. In this case the corresponding coordinate rings can be written in the form
Tp,q = k[[x, y]]/((x
p−2 − y2)(x2 − yq−2)).
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 Tp,q(λ) over an algebraically closed field
k of characteristic zero.
(a) CM(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 CM(R) are as follows:
p, q indec. rigid cluster tilting summands of cluster tilting
p = 3, q : even 6 6 2
p, q : even 14 24 3
We also prove the following general theorem, which includes both Theorem 1.3 (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 ∈ (x, y)) be a one-dimensional reduced hypersurface singularity
over an algebraically closed field k of characteristic zero.
(a) CM(R) has a cluster tilting object if and only if f is a product f = f1 · · · fn with fi /∈ (x, y)
(b) The number of indecomposable rigid objects, basic cluster tilting objects, and indecomposable
summands of basic cluster tilting objects in CM(R) are as follows:
indec. rigid cluster tilting summands of cluster tilting
2n − 2 n! n− 1
The following result gives a bridge between cluster tilting theory and birational geometry. The termi-
nologies are explained in section 5.
Theorem 1.6. Let (R,m) be a three-dimensional isolated cAn–singularity over an algebraically closed
field k of characteristic zero defined by the equation g(x, y) + zt and R′ a one-dimensional singularity
defined by g(x, y). Then the following conditions are equivalent.
(a) Spec(R) has a small resolution.
(b) Spec(R) has a crepant resolution.
(c) (R,m) has a non-commutative crepant resolution.
(d) CM(R) has a cluster tilting object.
(e) CM(R′) has a cluster tilting object.
(f) The number of irreducible power series in the prime decomposition of g(x, y) is n+ 1.
We end this section by giving an application to finite dimensional algebras. A 2-CY tilted algebra is an
endomorphism ring EndC(M) of a cluster tilting object T in a 2-CY triangulated category C. In section
7, we shall show the following result and compute 2-CY tilted algebras associated with minimally elliptic
curve singularities.
Theorem 1.7. Let (R,m) be an odd-dimensional isolated hypersurface singularity and Γ a 2-CY tilted
algebra coming from CM(R). Then we have the following.
(a) Γ is a symmetric algebra.
(b) All components in the stable AR-quiver of infinite type Γ are tubes of rank 1 or 2.
6 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
For example, put
R = k[[x, y]]/((x− λ1y) · · · (x− λny)) and M =
k[[x, y]]/((x− λ1y) · · · (x− λiy))
for distinct elements λi ∈ k. Then M is a cluster tilting object in CM(R) by Theorem 4.1, so Γ =
EndR(M) satisfies the conditions in Theorem 1.7. Since CM(R) has wild Cohen-Macaulay representation
type if n > 4 [DG, Th. 3], we should get a family of examples of finite dimensional symmetric k-algebras
whose stable AR-quiver consists only of tubes of rank 1 or 2, and are of wild representation type.
2. Simple hypersurface singularities
Let R be a one-dimensional simple hypersurface singularity. In this case the AR-quivers are known
for CM(R) [DW], and so also for CM(R). We use the notation from [Y].
In order to locate the indecomposable rigid modules M , that is, the modules M with
Ext1(M,M) = 0, the following lemmas are useful, where part (a) of the first one is proved in [HKR], and
the second one is a direct consequence of [KR] (generalizing [BMR1]).
Lemma 2.1. (a) Let C be an abelian or triangulated k-category with finite dimensional homomor-
phism spaces. Let A
−−−→ B1 ⊕ B2
(g1,g2)
−−−−→ C be a short exact sequence or a triangle, where
A is indecomposable, B1 and B2 nonzero, and (g1, g2) has no nonzero indecomposable summand
which is an isomorphism. Then Hom(A,C) 6= 0 .
(b) Let 0 → A
−→ C → 0 be an almost split sequence in CM(R), where R is an isolated
hypersurface singularity, and B has at least two indecomposable nonprojective summands in a
decomposition of B into a direct sum of indecomposable modules. Then Ext1(C,C) 6= 0.
Proof. (a) See [HKR, Lem. 6.5].
(b) Using (a) together with the above AR-formula and τ2 = id, we obtain DExt1(C,C) ≃
Hom(τ−1C,C) = Hom(τC,C) ≃ Hom(A,C) 6= 0, where D = Homk( , k). �
Lemma 2.2. Let T be a cluster tilting object in the Hom-finite connected 2-CY category C, and Γ =
EndC(T ).
(a) The functor G = HomC(T, ) : C → mod(Γ) induces an equivalence of categories
G : C/add(τT ) → mod(Γ).
(b) The AR-quiver for Γ is as a translation quiver obtained from the AR-quiver for C by removing
the vertices corresponding to the indecomposable summands of τT .
(c) Assume τ2 = id. Then we have the following.
(i) Γ is a symmetric algebra.
(ii) The indecomposable nonprojective Γ-modules have τ-period one or two.
(iii) If C has an infinite number of nonisomorphic indecomposable objects, then all components
in the stable AR-quiver of Γ are tubes of rank one or two.
(d) If C has only a finite number n of nonisomorphic indecomposable objects, and T has t nonisomor-
phic indecomposable summands, then there are n− t nonisomorphic indecomposable Γ-modules.
Proof. For (a) and (b) see [BMR1, KR]. Since C is 2-CY, we have τ = Σ, and a functorial isomorphism
DHomC(T, T ) ≃ HomC(T,Σ
2T ) = HomC(T, τ
2T ) ≃ HomC(T, T ).
This shows that Γ is symmetric. Let C be an indecomposable nonprojective Γ-module. Viewing C as an
object in C we have τ2CC ≃ C, and τC is not a projective Γ-module since C is not removed. Hence we
have τ2ΓC ≃ C. If C has an infinite number of nonisomorphic indecomposable objects, then Γ is of infinite
type. Then each component of the AR-quiver is infinite, and hence is a tube of rank one or two. Finally,
(d) is a direct consequence of (a). �
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 7
We also use that in our cases we have a covering functor Π: k(ZQ) → CM(R), where Q is the
appropriate Dynkin quiver and k(ZQ) is the mesh category of the translation quiver ZQ [Rie, Am], (see
also [I1, Section 4.4] for another explanation using functorial methods).
For the one-dimensional simple hypersurface singularities we have the cases An (n even or odd), Dn
(n odd or even), E6, E7 and E8. We now investigate them case by case.
Proposition 2.3. In the case An (with n even) there are no indecomposable rigid objects.
Proof. We have the stable AR-quiver
oo // · · · //oo In/2
Here, and later, a dotted line between two indecomposable modules means that they are connected
via τ .
Since τIj ≃ Ij for each j, Ext
1(Ij , Ij) 6= 0 for j = 1, · · · , n/2. Hence no Ij is rigid. �
Proposition 2.4. In the case An (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
(n+1)
Proof. For simplicity, we write l = (n− 1)/2. We have the stable AR-quiver
//oo · · · //oo Ml
==zzzzz
aaDDDDD
Since τMi ≃Mi for i = 1, · · · , l, we have
Ext1(Mi,Mi) ≃ Hom(Mi, τMi) ≃ Hom(Mi,Mi) 6= 0.
So only the indecomposable objects N− and N+ could be rigid. We use covering techniques and additive
functions to compute the support of Hom(N−, ), where we refer to [BG] for the meaning of the diagrams
below.
>>}}}
<<yyy
==zzz
· · ·
>>}}}
>>}}}
==zzz
??���
// N− //// Ml
>>~~~
// N+ // Ml
>>~~~
// N− · · ·
??���
>>~~~
>>~~~
BB���
BB���
BB���
BB���
BB���
BB���
// 1 // 1
BB���
// 0 // 1
BB���
// 1 1
// 0 // 0
BB���
BB���
BB���
BB���
We see that Hom(N−, N+) = 0, so Ext
1(N+, N+) = Ext
1(N+, τN−) = 0, and Ext
1(N−, N−) = 0.
Since Ext1(N+, N−) 6= 0, we see that N+ and N− are exactly the maximal rigid objects. Further
Hom(N−,Mi) 6= 0 for all i, so Ext
1(N+,Mi) 6= 0 and Ext
1(N−,Mi) 6= 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 Dn with n odd we have two maximal rigid objects, which both are inde-
composable, and neither one is cluster tilting.
8 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Proof. We have the AR-quiver
// Y1
// Y2 //
M2 //
// · · · // M(n−3)/2
**UUU
X(n−1)/2
jjUUU
ttiiii
A // X1
\\9999999
// N1
]];;;;;;;
// X2
]];;;;;;;
// N2
]];;;;;;;
// · · · // N(n−3)/2
bbDDDDDDDD 44iiii
Using Lemma 2.1, the only candidates for being indecomposable rigid are A and B. We compute the
support of Hom(A, )
AA���
BB���
· · ·
AA���
??���
??���
BB���
Y1 · · ·
CC���
BB���
where B = τA and l = (n− 3)/2. We see that Hom(A,B) = 0, so that Ext1(A,A) = 0. Then A is
clearly maximal rigid. Since Hom(A,M1) = 0, we have Ext
1(A,N1) = 0, so A is not cluster tilting.
Alternatively, we could use that we see that End(A)op ≃ k[x]/(x2), which has two indecomposable
modules, whereas CM(R) has 2n− 3 indecomposable objects. If A was cluster tilting, End(A)op would
have had 2n− 3 − 1 = 2n− 4 indecomposable modules, by Lemma 2.2. �
Proposition 2.6. In the case D2n 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 ·
α // ·
oo αβα = 0 = βαβ
in the case D4, and by ·γ ;;
α // ·
oo with γn−1 = βα, γβ = 0 = αγ and ·
α // ·
oo with (αβ)n−1α = 0 =
(βα)n−1β for 2n > 4.
Proof. We have the AR-quiver
vvnnn
// Y1
· · · // Yn−1
``BBBBBBB
A // X1 //
XX1111111111
YY3333333333
// X2
YY3333333333
// N2
YY3333333333
// · · · // Xn−1
ZZ66666666666
FF













66nnnn
hhQQQQ
By Lemma 2.1, the only possible indecomposable rigid objects are: A, B, C+, C−, D+, D−.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 9
We compute the support of Hom(C+, ):
C+ // Yl
AA���
// D+ // Xl
??���
// C+ // Yl
??���
// D+ // Xl
??���
// C+ // Yl
CC���
· · ·
@@���
??���
Nl−1 Ml−1
>>~~~
@@���
@@���
??���
AA���
>>~~~
>>~~~
>>~~~
BB���
>>}}}
· · ·
AA���
>>}}}
where l = n− 1
1 // 1
BB���
// 0 // 1
BB���
// 1 // 1
BB���
// 0 // 1
BB���
// 1 // 0
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
We see that Hom(C+, D+) = 0, so Ext
1(C+, C+) = 0 = Ext
1(D+, D+). Further, Hom(C+, C−) = 0,
so Ext1(C+, D−) = 0. By symmetry Ext
1(D−, D−) = 0 = Ext
1(C−, C−) and Ext
1(D+, C−) = 0. Also
Ext1(C+, A) = 0, Ext
1(C+, B) 6= 0, so Ext
1(D+, B) = 0, Ext
1(D+, A) 6= 0. Further Ext
1(C+, X) 6= 0 for
X 6= A,D−, C+.
We now compute the support of Hom(A, )
>>~~~
C+ // Yl
??���
D+ // Xl
· · ·
??���
>>~~~
@@���
M1 M1
@@���
>>~~~
??���
>>}}}
· · ·
??~~~
>>~~~
>>}}}
??~~~
10 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
where l = n− 1 and we have an odd number of columns and rows.
BB���
1 // 1
BB���
0 // 0
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
We see that Hom(A,B) = 0, so Ext1(A,A) = 0, hence also Ext1(B,B) = 0. Since Hom(A,D−) = 0, we
have Ext1(A,C−) = 0, hence Ext
1(B,D−) = 0. Since Hom(A,C−) 6= 0, we have Ext
1(A,D−) 6= 0, so
Ext1(B,D+) 6= 0.
It follows that C+ ⊕D−, C− ⊕D+, C+ ⊕A, D+ ⊕B, A⊕ C− and B ⊕D− are maximal rigid.
These are also cluster tilting: We have Hom(A,Xi) 6= 0, Hom(A,Ni) 6= 0, so Ext
1(B,Xi) 6= 0,
Ext1(B,Ni) 6= 0. Similarly, Ext
1(A, Yi) 6= 0, Ext
1(A,Mi) 6= 0. Also Hom(C+, Yi) 6= 0, Hom(C+, Ni) 6= 0,
so Ext1(D+, Yi) 6= 0, Ext
1(D+, Ni) 6= 0. Hence Ext
1(C+, Xi) 6= 0,
Ext1(C+,Mi) 6= 0. So Ext
1(D−, Yi) 6= 0, Ext
1(D−, Ni) 6= 0, Ext
1(C−, Xi) 6= 0, Ext
1(C−,Mi) 6= 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:
{C+, D−}
{B,D−}
rrrrrr
{A,C+}
{B,D+}
{A,C−}
{C−, D+}
rrrrrr
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 E6 there are no indecomposable rigid objects.
Proof. We have the AR-quiver
// M1
::uuuu
ddIIII
// N1
\\8888888
The only candidates for indecomposable rigid objects according to Lemma 2.1 are M1 and N1. We
compute the support of Hom(M1, ).
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 11
>>~~~
??���
· · ·
??���
// M2 // X
>>~~~
// M2 // X
AA���
>>~~~
??���
· · ·
AA���
??���
>>~~~
BB���
BB���
· · ·
BB���
// 1 // 1
BB���
0 // 1
BB���
BB���
BB���
· · ·
BB���
BB���
BB���
We see that Hom(M1, N1) 6= 0, so that Ext
1(M1,M1) 6= 0 and Ext
1(N1, N1) 6= 0. �
Proposition 2.8. In the case E7 there are two maximal rigid objects, which both are indecomposable,
and neither of them is cluster tilting.
Proof. We have the AR-quiver
___ D
// M2 //
// Y3
^^>>>>>>>>
wwppp
// Y1
xxppp
B // N2
^^========
// X2
__????????
// X3
���������������
__>>>>>>>>
// X1
^^========
// N1
__????????
Using Lemma 2.1, we see that the only candidates for indecomposable rigid objects are A, B, M1, N1,
C and D. We first compute the support of Hom(A, ).
??���
X1 · · ·
@@���
// C // X3
??���
AA���
??���
@@���
@@���
??���
· · ·
AA���
@@���
??���
// 1 // 1
// 0 // 1
// 1 // 1
// 0 // 1
// 1 // 1
// 0 // 0
We see that Ext1(A,A) = 0, and so also Ext1(B,B) = 0, so A and B are rigid.
Next we compute the support of Hom(M1, ).
@@���
??���
@@���
· · ·
@@���
// C // X3
??���
// D // Y3
??���
@@���
??���
N2 · · ·
??���
BB���
BB���
BB���
· · ·
BB���
1 // 1
BB���
// 0 // 1
BB���
BB���
BB���
1 · · ·
BB���
We see that Ext1(M1,M1) 6= 0 and Ext
1(N1, N1) 6= 0, so that M1 and N1 are not rigid.
Then we compute the support of Hom(C, ).
12 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
@@���
??���
X1 · · ·
C // X3
@@���
D // Y3
@@���
C // X3
??���
@@���
??���
@@���
??���
· · ·
??���
BB���
BB���
1 · · ·
1 // 1
BB���
0 // 1
BB���
1 // 2 //
BB���
BB���
BB���
BB���
BB���
· · ·
BB���
We see that Ext1(C,C) 6= 0 and Ext1(D,D) 6= 0, so that C and D are not rigid. Hence A and B are
the rigid indecomposable objects, and they are maximal rigid.
Since Ext1(A,C) = 0, we see that A and hence B is not cluster tilting. �
Proposition 2.9. In the case E8 there are no indecomposable rigid objects.
Proof. We have the AR-quiver
__ B2
N2 //
// X1
]];;;;;;;
xxqqq
xxqqq
// C1
// B1
// N1
M2 // C2
^^<<<<<<<
// Y1
]];;;;;;;
��������������
// Y2
]];;;;;;;
// D1
]];;;;;;;
// A1
]];;;;;;;
// M1
]]<<<<<<<
The only candidates for indecomposable rigid objects are M1, N1, M2, N2, A2 and B2, by Lemma 2.1.
We first compute the support of Hom(M1, ):
??���
??���
· · ·
??���
B2 // X1
??���
// A2 // Y1
??���
??���
??���
@@���
??���
??���
@@���
??���
??���
??���
· · ·
@@���
@@���
??���
??���
BB���
BB���
0 · · ·
BB���
// 1 // 1
BB���
// 0 // 1
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
· · ·
BB���
BB���
BB���
BB���
BB���
We see that Ext1(M1,M1) 6= 0, and hence Ext
1(N1, N1) 6= 0.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 13
Next we compute the support of Hom(M2, ):
@@���
??���
??���
· · ·
@@���
// B2 // X1
??���
// A2 // Y1
??���
??���
??���
??���
· · ·
BB���
BB���
BB���
· · ·
BB���
1 // 1
BB���
0 // 1
BB���
BB���
BB���
BB���
· · ·
We see that Ext1(M2,M2) 6= 0, and hence Ext
1(N2, N2) 6= 0.
Finally we compute the support of Hom(A2, ):
??���
??���
D2 · · ·
A2 // Y1
AA���
// B2 // X1
??���
// A2 // Y1 //
??���
??���
??���
??���
??���
??���
B1 · · ·
??���
BB���
BB���
1 · · ·
1 // 1
BB���
// 0 // 1
BB���
// 1 // 2
BB���
BB���
BB���
BB���
BB���
BB���
1 · · ·
BB���
It follows that Ext1(A2, A2) 6= 0, and similarly Ext
1(B2, B2) 6= 0. Hence there are no indecomposable
rigid objects. �
3. Computation with Singular
An alternative way to carry out computations of Ext1–spaces in the stable category of maximal Cohen-
Macaulay modules is to use the computer algebra system Singular, see [GP]. Let
R = k[x1, x2, . . . , xn]〈x1,x2,...,xn〉/I
be a Cohen-Macaulay local ring which is an isolated singularity, and M and N two maximal Cohen-
Macaulay modules. Denote by R̂ the completion of R. Since all the spaces ExtiR(M,N) (i ≥ 1) are finite-
dimensional over k and the functor mod(R) → mod(R̂) is exact, maps the maximal Cohen-Macaulay
modules to maximal Cohen-Macaulay modules and the finite length modules to finite length modules,
we can conclude that
dimk(Ext
R(M,N)) = dimk(Ext
bR(M̂, N̂)).
As an illustration we show how to do this for the case E7.
Proposition 3.1. In the case E7 there are two maximal rigid objects, which both are indecomposable
and neither of them is cluster tilting.
14 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
By [Y] the AR-quiver of CM(R) has the form
// M2 //
// Y3
aaCCCCC
vvmmm
// Y1
vvmmm
B // N2
``BBBBB
// X2
aaDDDDD
// X3
aaCCCCC
FF









aaBBBBB
// N1
aaDDDDD
By Lemma 2.1 only the modules A,B,C,D,M1, N1 can be rigid. Since B = τ(A), D = τ(C), N1 = τ(M1),
the pairs of modules (A,B), (C,D) and (M1, N1) are rigid or not rigid simultaneously. By [Y] we have
the following presentations:
x2+y3
−−−−→ R
−→ R −→ A −→ 0,
( x y
y2 −x)
−−−−−→ R2
x( x y
y2 −x)
−−−−−−→ R2 −→ C −→ 0,
xy2 −x)
−−−−−−→ R2
xy2 −x2)
−−−−−−→ R2 −→M1 −→ 0,
so we can use the computer algebra system Singular in order to compute the Ext1–spaces between these
modules.
> Singular (call the program ‘‘Singular’’)
> LIB ‘‘homolog.lib’’; (call the library of homological algebra)
> ring S = 0,(x,y),ds; (defines the ring S = Q[x, y]〈x,y〉)
> ideal I = x3 + xy3; (defines the ideal x3 + xy3 in S)
> qring R = std(I); (defines the ring Q[x, y]〈x,y〉/I)
> module A = [x];
> module C = [x2, xy2], [xy, -x2];
> module M1 = [x2, xy2], [y, -x2]; (define modules A,C,M1)
> list l = Ext(1,A,A,1);
// dimension of Ext1: -1 (Output: Ext1R(A,A) = 0)
> list l = Ext(1,C,C,1);
// ** redefining l **
// dimension of Ext1: 0 (the Krull dimension of Ext1R(C,C) is 0)
// vdim of Ext1: 2 (dimk(Ext
R(C,C)) = 2)
> list l = Ext(1,M1,M1,1);
// ** redefining l **
// dimension of Ext1: 0
// vdim of Ext1: 10
> list l = Ext(1,A,C,1);
// ** redefining l **
// dimension of Ext1: -1
This computation shows that the modules A and B are rigid, C,D,M1 and N1 are not rigid and since
Ext1R(A,C) = 0, there are no cluster tilting objects in the stable category CM(R).
4. One-dimensional hypersurface singularities
We shall construct a large class of one-dimensional hypersurface singularities having a cluster tilting
object, then classify all cluster tilting objects. Our method is based on the higher theory of almost split
sequences and Auslander algebras studied in [I1, I2]. We also use a relationship between cluster tilting
objects in CM(R) and tilting modules over the endomorphism algebra of a cluster tilting object [I2].
Then we shall compare cluster 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 CM(R) instead of CM(R).
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 15
Let k be an infinite field, S := k[[x, y]] and m := (x, y). We fix f ∈ m and write f = f1 · · · fn for
irreducible formal power series fi ∈ m (1 ≤ i ≤ n). Put
Si := S/(f1 · · · fi) and R := Sn = S/(f).
We assume that R is reduced, so we have (fi) 6= (fj) for any i 6= 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)
i=1 Si is a rigid object in CM(R).
i=1 Si is a cluster tilting object in CM(R) if the following condition (A) is satisfied.
(A) fi /∈ m
2 for any 1 ≤ i ≤ n.
Let Sn be the symmetric group of degree n. For w ∈ Sn and I ⊆ {1, · · · , n}, we put
Swi := S/(fw(1) · · · fw(i)), Mw :=
Swi and SI := S/(
Theorem 4.2. Assume that (A) is satisfied.
(a) There are exactly n! basic cluster tilting objects Mw (w ∈ Sn) and exactly 2
n− 1 indecomposable
rigid objects SI (∅ 6= I ⊆ {1, · · · , n}) in CM(R).
(b) For any w ∈ Sn, there are exactly n! basic Cohen-Macaulay tilting EndR(Mw)-modules
HomR(Mw,Mw′) (w
′ ∈ Sn) of projective dimension at most one. Moreover, all algebras
EndR(Mw) (w ∈ Sn) are derived equivalent.
It is interesting to compare with results in [IR], where two-dimensional (2-Calabi-Yau) algebras Γ are
treated and a bijection between elements in an affine Weyl group and tilting Γ-modules of projective
dimension at most one is given. Here the algebra is one-dimensional, and Weyl groups appear.
Here we consider three examples.
(a) Let R be a curve singularity of type A2n−1 or D2n+2, so
R = S/((x− yn)(x + yn)) or R = S/(y(x− yn)(x+ yn)).
By our theorems, there are exactly 2 or 6 cluster tilting objects and exactly 3 or 7 indecomposable
rigid objects in CM(R), which fits with our computations in section 1.
(b) Let R be a curve singularity of type T3,2q+2(λ) or T2p+2,2q+2(λ), so
R = S/((x− y2)(x − yq)(x + yq)) (R = S/(y(y − x2)(y − λx2)) for q = 2),
R = S/((xp − y)(xp + y)(x− yq)(x + yq)) (R = S/(xy(x− y)(x− λy)) for p = q = 1).
By our theorems, there are exactly 6 or 24 cluster tilting objects and exactly 7 or 15 indecom-
posable rigid objects in CM(R).
(c) Let λi ∈ k (1 ≤ i ≤ n) be mutually distinct elements in k. Put
R := S/((x− λ1y) · · · (x− λny)).
By our theorems, there are exactly n! cluster tilting objects and exactly 2n − 1 indecomposable
rigid objects in CM(R).
First of all, Theorem 4.1(a) follows immediately from the following observation.
Proposition 4.3. For g1, g2 ∈ m and g3 ∈ S, put R := S/(g1g2g3). If g1 and g2 have no common factor,
then Ext1R(S/(g1g3), S/(g1)) = 0 = Ext
R(S/(g1), S/(g1g3)).
Proof. We have a projective resolution
→ R → S/(g1g3) → 0.
Applying HomR( , S/(g1)), we have a complex
S/(g1)
g1g3=0
−→ S/(g1)
→ S/(g1).
This is exact since g1 and g2 have no common factor. Thus we have the former equation, and the other
one can be proved similarly. �
16 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Our plan of proof of Theorem 4.1(b) is the following.
(i) First we shall prove Theorem 4.1 under the following stronger assumption:
(B) m = (f1, f2) = · · · = (fn−1, fn).
(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 CM(R).
(b) gl. dim EndR(M) ≤ 3.
(c) For any X ∈ CM(R), there exists an exact sequence 0 →M1 →M0 → X → 0 with Mi ∈ addM .
(d) For any indecomposable direct summand X of M , there exists an exact sequence 0 → M2
→ X with Mi ∈ addM and a is a right almost split map in addM .
Proof. (a)⇔(b) For d = dimR, take the d-cotilting module T = R and apply [I2, Th. 5.1(3)] for m = d
and n = 2 there.
(a)⇔(c) See [I1, Prop. 2.2.2].
(a)⇒(d) See [I1, Th. 3.3.1].
(d)⇒(b) For any simple EndR(M)-module S, there exists an indecomposable direct summand X of M
such that S is the top of the projective HomR(M,X). Since Ext
R(M,M2) = 0, the sequence in (d) gives a
projective resolution 0 → HomR(M,M2) → HomR(M,M1) → HomR(M,M0) → HomR(M,X) → S → 0.
Thus we have pdS ≤ 3 and gl. dim EndR(M) ≤ 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 addM , 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
HomR(Si, Sj) ≃
(fi+1 · · · fj)/(f1 · · · fj) i < j
S/(f1 · · · fj) i ≥ j.
Lemma 4.5. Let R = S/(f) be a one-dimensional reduced hypersurface singularity, S0 := 0 and 1 ≤ i <
(a) We have exchange sequences (see Definition 1.2)
0 → Si
−−−−→ Si+1 ⊕ Si−1
(1 fi+1)
−−−−−→ S/(f1 · · · fi−1fi+1) → 0,
0 → S/(f1 · · · fi−1fi+1)
(fi1 )
−−→ Si+1 ⊕ Si−1
(−1 fi)
−−−−−→ Si → 0.
(b) If (fi, fi+1) = m, then we have a 2-almost split sequence
0 → Si
−−−−→ Si+1 ⊕ Si−1
fi fifi+1
1 fi+1
−−−−−−−→ Si+1 ⊕ Si−1
(−1 fi)
−−−−−→ Si → 0
in add
i=1 Si.
Proof. (a) Consider the map a := (−1 fi) : Si+1 ⊕ Si−1 → Si. Any morphism from Sj to Si factors
through 1 : Si+1 → Si (respectively, fi : Si−1 → Si) if j > i (respectively, j < i). Thus a is a minimal
right (add
j 6=i Sj)-approximation.
It is easily checked that Ker a = {s ∈ Si+1 | s ∈ fiSi} = (fi)/(f1 · · · fi+1) ≃ S/(f1 · · · fi−1fi+1), where
we denote by s the image of s via the natural surjection Si+1 → Si.
Consider the surjective map b := (1 fi+1) : Si+1 ⊕Si−1 → S/(f1 · · · fi−1fi+1). It is easily checked that
Ker b = {s ∈ Si+1 | s ∈ (fi+1)/(f1 · · · fi−1fi+1)} = (fi+1)/(f1 · · · fi+1) ≃ Si, where we denote by s the
image of s via the natural surjection Si+1 → S/(f1 · · · fi−1fi+1).
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 17
(b) This sequence is exact by (a). Any non-isomorphic endomorphism of Si is multiplication with an
element in m, which is equal to (fi, fi+1) by our assumption. Since fi+1 (respectively, fi) : Si → Si factors
through 1 : Si+1 → Si (respectively, fi : Si−1 → Si), we have that a is a right almost split map. �
Now we choose fn+1 ∈ m such that m = (fn, fn+1), and fn+1 and f1 · · · fn have no common factor.
This is possible by our assumption (A).
Lemma 4.6. We have an exact sequence
0 → Sn−1
−fn+1
−−−−−→ Sn ⊕ Sn−1
(fn+1 fn)
−−−−−−→ Sn
with a minimal right almost split map (fn+1 fn) in add
i=1 Si.
Proof. Consider the map a := (fn+1 fn) : Sn⊕Sn−1 → Sn. Any morphism from Sj (j < n) to Sn factors
through fn : Sn−1 → Sn.
Any non-isomorphic endomorphism of Sn is multiplication with an element in m = (fn+1, fn). Since
fn : Sn → Sn factors through fn : Sn−1 → Sn, we have that a is a right almost split map.
It is easily checked that Ker a = {s ∈ Sn−1 | fns ∈ fn+1Sn} = (fn+1, f1 · · · fn−1)/(f1 · · · fn−1), which
is isomorphic to Sn−1 by the choice of fn+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 CM(R). We use the notation introduced at the beginning of this section.
Lemma 4.7. For w ∈ Sn, we assume that Mw is a cluster tilting object in CM(R). Then, for 1 ≤ i < n
and si = (i i+ 1), we have exchange sequences
0 → Swi → S
i+1 ⊕ S
i−1 → S
i → 0 and 0 → S
i → S
i+1 ⊕ S
i−1 → S
i → 0.
Proof. Without loss of generality, we can assume w = 1. Then the assertion follows from Lemma
4.5(a). �
Immediately, we have the following.
Proposition 4.8. Assume that Mw is a cluster tilting object in CM(R) for some w ∈ Sn.
(a) The cluster tilting mutations of Mw are Mwsi (1 ≤ i < n).
(b) Mw′ is a cluster tilting object in CM(R) for any w
′ ∈ Sn.
Proof. (a) This follows from Lemma 4.7.
(b) This follows from (a) since Sn is generated by si (1 ≤ i < n). �
The following result is also useful.
Lemma 4.9. Let R and R′ be complete local Gorenstein rings with dimR = dimR′ and M a rigid object
in CM(R) which is a generator. Assume that there exists a surjection R′ → R, and we regard CM(R) as
a full subcategory of CM(R′). If R′ ⊕M is a cluster tilting object in CM(R′), then M is a cluster tilting
object in CM(R).
Proof. We use the equivalence (a)⇔(c) in Proposition 4.4, which remains true in any dimension [I1,
Prop. 2.2.2]. For any X ∈ CM(R), take a right (addM)-approximation f : M0 → X of X . Since M is
a generator of R, we have an exact sequence 0 → Y → M0
→ X → 0 with Y ∈ CM(R). Since R′ is a
projective R′-module, f is a right add(R′ ⊕M)-approximation of X . Since R′ ⊕M is a cluster tilting
object in CM(R′), we have Y ∈ add(R′ ⊕M). Since Y ∈ CM(R), we have Y ∈ addM . 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 gi ∈ m (1 ≤ i < n) such that h2i−1 := fi and h2i := gi satisfy
the following conditions:
• (hi) 6= (hj) for any i 6= j.
18 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
• m = (h1, h2) = (h2, h3) = · · · = (h2n−2, h2n−1).
Put R′ := S/(h1 · · ·h2n−1). This is reduced by the first condition.
Since we have already proved Theorem 4.1 under the assumption (B), we have that⊕2n−1
i=1 S/(h1 · · ·hi) is a cluster tilting object in CM(R
′). By Proposition 4.8,⊕2n−1
i=1 S/(hw(1) · · ·hw(i)) is a cluster tilting object in CM(R
′) for any w ∈ S2n−1. In particular,
S/(f1 · · · fi)) ⊕ (
S/(f1 · · · fng1 · · · gi))
is a cluster tilting object in CM(R′). Moreover we have surjections
R′ → · · · → S/(f1 · · · fng1g2) → S/(f1 · · · fng1) → R.
Using Lemma 4.9 repeatedly, we have that
i=1(S/(f1 · · · fi)) is a cluster tilting object in CM(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 EndR(
i=1 Si) is
// · · ·oo // Sn−1oo
where in addition there is a loop at Si (1 ≤ i < n) if and only if (fi, fi+1) 6= m.
(b) We have the quiver of EndR(
i=1 Si) by removing the vertex Sn from the quiver in (a).
Proof. We only have to show (a). We only have to calculate minimal right almost split maps in
i=1 Si. We have a minimal right almost split map Sn⊕Sn−1 → Sn by Lemma 4.6. If (fi, fi+1) = m
(1 ≤ i < n), then we have a minimal right almost split map Si+1 ⊕ Si−1 → Si by Lemma 4.5.
We only have to consider the case (fi, fi+1) 6= m (1 ≤ i < n). Take g ∈ m such that (fi, fi+1, g) = m.
It is easily check (cf proof of Lemma 4.5) that we have a right almost split map
c := (−1 g fi) : Si+1 ⊕ Si ⊕ Si−1 → Si.
Assume that c is not right minimal. Then there exists a right almost split map of the form c′ :
Si+1 ⊕ Si → Si (i > 1), Si ⊕ Si−1 → Si or Si+1 ⊕ Si−1 → Si. For the first case, it is easily checked that
fi : Si−1 → Si does not factor through c
′, 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 Γ 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 Γ-modules T and U , we
write
T ≥ U
if Ext1Γ(T, U) = 0. By tilting theory, we have FacT = {X ∈ mod(Γ) | Ext
Γ(T,X) = 0}. Thus
Ext1Γ(T, U) = 0 is equivalent to FacT ⊃ FacU , and ≥ gives a partial order. On the other hand, we call a
Γ-module T almost complete tilting if pd ΓT ≤ 1, Ext
Γ(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 Γ-module has at most two complements.
(b) T and U are neighbors in the partial order if and only if there exists an almost complete tilting
Γ-module which is a common direct summand of T and U .
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 19
(c) Assume T ≥ U . Then there exists a sequence T = T0 > T1 > T2 > · · · > U satisfying the
following conditions.
(i) Ti and Ti+1 are neighbors.
(ii) Either Ti = U for some i or the sequence is infinite.
(d) T ≥ U if and only if there exists an exact sequence 0 → T → U0 → U1 → 0 with Ui ∈ addU .
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 Γ, we call a Γ-module Cohen-Macaulay if it is a Cohen-Macaulay R-module. As usual, we
denote by CM(Γ) the category of Cohen-Macaulay Γ-modules.
Lemma 4.12. Let Γ be a module-finite algebra over a complete local Gorenstein ring R such that Γ ∈
CM(R), and T and U tilting Γ-modules. Assume U ∈ CM(Γ).
(a) If T ≥ U , then T ∈ CM(Γ).
(b) Let P be a projective Γ-module such that HomR(P,R) is a projective Γ
op-module. Then P ∈
addU .
Proof. (a) By Proposition 4.11(d), there exists an exact sequence 0 → T → U0 → U1 → 0 with Ui ∈
addU . Thus the assertion holds.
(b) We have Ext1Γop(HomR(P,R),HomR(U,R)) = 0. Since we have a duality HomR( , R) : CM(Γ) ↔
CM(Γop), it holds Ext1Γ(U, P ) = 0. There exists an exact sequence 0 → P → U0 → U1 → 0 with
Ui ∈ addU [H, Lem. III.2.3], which must split since Ext
Γ(U, P ) = 0. Thus we have P ∈ addU . �
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 ′ cluster tilting objects in CM(R).
(a) HomR(M,N) is a tilting EndR(M)-module of projective dimension at most one.
(b) If N ′ is a cluster tilting mutation of N , then HomR(M,N
′) is a tilting mutation of HomR(M,N).
Now we shall prove Theorem 4.2. Fix w ∈ Sn and put Γ := EndR(Mw). Since Mw is a generator
of R, the functor HomR(Mw, ) : CM(R) → CM(Γ) is fully faithful. By Theorem 4.1, Mw is a cluster
tilting object in CM(R). By Proposition 4.13(a), HomR(Mw,Mw′) (w
′ ∈ Sn) is a Cohen-Macaulay tilting
Γ-module.
(b) Take any Cohen-Macaulay tilting Γ-module U . Since P := HomR(Mw, R) is a projective Γ-module
such that HomR(P,R) = Mw = HomR(R,Mw) is a projective Γ-module, we have P ∈ addU by Lemma
4.12(b). In particular, by Proposition 4.11(a)(b),
• any Cohen-Macaulay tilting Γ-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 Γ-module of the form HomR(Mw,Mw′) (w
′ ∈ Sn) has precisely
(n− 1) tilting mutations HomR(Mw,Mw′si) (1 ≤ i < n) which are Cohen-Macaulay.
Consequently, any successive tilting mutation of Γ = HomR(Mw,Mw) has the form HomR(Mw,Mw′) for
some w′ ∈ Sn if each step is Cohen-Macaulay.
Using this observation, we shall show that U is isomorphic to HomR(Mw,Mw′) for some w
′. Since
Γ ≥ U , there exists a sequence
Γ = T0 > T1 > T2 > · · · > U
satisfying the conditions in Proposition 4.11(c). By Lemma 4.12(a), each Ti is Cohen-Macaulay. Thus
the above observation implies that each Ti has the form HomR(Mw,Mwi) for some wi ∈ Sn. Moreover,
wi 6= wj for i 6= j. Since Sn is a finite group, the above sequence must be finite. Thus U = Ti holds for
some i, hence the proof is completed.
20 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
(a) Let U be a cluster tilting object in CM(R). Again by Proposition 4.13(a), HomR(Mw, U) is a
Cohen-Macaulay tilting Γ-module. By part (b) which we already proved, HomR(Mw, U) is isomorphic to
HomR(Mw,Mw′) for some w
′ ∈ Sn. Since the functor HomR(Mw, ) : CM(R) → CM(Γ) is fully faithful,
U is isomorphic to Mw′, and the former assertion is proved.
For the latter assertion, we only have to show that any rigid object in CM(R) is a direct summand of
some cluster tilting object in CM(R). This is valid by the following general result in [BIRS, Th. 1.9]. �
Proposition 4.14. Let C be a 2-CY Frobenius category with a cluster tilting object. Then any rigid
object in C is a direct summand of some cluster tilting object in C.
We end this section with the following application to dimension three.
Now let S′′ := k[[x, y, u, v]], fi ∈ m = (x, y) (1 ≤ i ≤ n) and R
′′ := S′′/(f1 · · · fn + uv). For w ∈ Sn
and I ⊆ {1, · · · , n}, we put
Uwi := (u, fw(1) · · · fw(i)) ⊂ R
′′, Mw :=
Uwi and UI := (u,
fi) ⊂ S
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 Mw (w ∈ Sn) and exactly 2
n− 1 indecomposable
rigid objects UI (∅ 6= I ⊂ {1, · · · , n}) in CM(R
(b) There are non-commutative crepant resolutions EndR′′(Mw) (w ∈ Sn) of R
′′, which are derived
equivalent.
Proof. (a) We only have to apply Knörrer periodicity CM(R) → CM(R′′) [Kn, So] as follows:
Since Swi ∈ CM(R) has a projective resolution
−→ R → Swi → 0
for a := fw(1) · · · fw(i) and b := fw(i+1) · · · fw(n), the corresponding object X ∈ CM(R
′′) has a projective
resolution
(u ab −v)
−→ R′′2
(v ab −u)
−→ R′′2 → X → 0.
It is easily checked that X is isomorphic to (u, a) = Uwi .
(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,
k[[x, y, u, v]]/((x− λ1y) · · · (x− λny) + uv)
has a non-commutative crepant resolution for distinct elements λ1, · · · , λn ∈ 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,m) be a three-dimensional complete normal Gorenstein singularity over an algebraically closed
field k of characteristic zero, and let X = Spec(R). A resolution of singularities Y
−→ X is called
• crepant, if ωY ∼= π
∗ωX for canonical sheaves ωX and ωY of X and Y respectively.
• small, if the fibre of the closed point has dimension at most one.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 21
A small resolution is automatically crepant, but the converse is in general not true. However, both types
of resolutions coincide for certain important classes of three-dimensional singularities.
A cDV (compound Du Val) singularity is a three-dimensional singularity given by the equation
f(x, y, z) + tg(x, y, z, t) = 0,
where f(x, y, z) defines a simple surface singularity and g(x, y, z, t) is arbitrary.
A cDV singularity is called cAn if the intersection of f(x, y, z) + tg(x, y, z, t) = 0 with a generic
hyperplane ax+ by + cz + dt = 0 in k4 is an An surface singularity. Generic means that the coefficients
(a, b, c, d) belong to a non-empty Zariski open subset in k4.
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,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 = EndR(M) is Cohen-Macaulay as an R–module,
(iii) gl.dim(A) = 3.
The following result establishes a useful connection.
Theorem 5.3. [V1, Cor. 3.2.11][V2, Th. 6.6.3] Let (R,m) be an isolated cDV singularity. Then there
exists a crepant resolution of X = Spec(R) if and only if there exists a non-commutative one in the sense
of Definition 5.2.
The existence of a non-commutative crepant resolution turns out to be equivalent to the existence of a
cluster tilting object in the triangulated category CM(R).
Theorem 5.4. [I2, Th. 5.2.1][IR, Cor. 8.13] Let (R,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 CM(R).
Proof. For convenience of the reader, we give an outline of the proof (see also Proposition 4.4).
Let us first assume that M is a cluster tilting object in CM(R). Then M is automatically reflexive.
From the exact sequence
0 −→ Ω(M) −→ F −→M −→ 0
we obtain
(1) 0 −→ EndR(M) −→ HomR(F,M) −→ HomR(Ω(M),M) −→ Ext
R(M,M) −→ 0.
Since M is rigid, Ext1R(M,M) = 0. Moreover, depth(HomR(F,M)) = depth(M) = 3 and
depth(HomR(Ω(M),M) ≥ 2, and hence depth(EndR(M)) = 3 and A = End(M) is maximal Cohen-
Macaulay over R.
For the difficult part of this implication, claiming that gl.dim(A) = 3, we refer to [I1, Th. 3.6.2].
For the other direction, let M be a module giving rise to a non-commutative crepant resolution. Then
by [IR, Th. 8.9] there exists another module M ′ giving rise to a non-commutative crepant resolution,
which is maximal Cohen-Macaulay and contains R as a direct summand.
By the assumption, depth(EndR(M
′)) = 3 and we can apply [IR, Lem. 8.5] to the exact sequence (1)
to deduce that Ext1R(M
′,M ′) = 0, so that M ′ is rigid. The difficult part saying that M ′ is cluster tilting
is proven in [I2, Th. 5.2.1]. �
We now summarize the results of this section.
22 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Theorem 5.5. Let (R,m) be an isolated cDV singularity. Then the following are equivalent.
(a) Spec(R) has a small resolution.
(b) Spec(R) has a crepant resolution.
(c) (R,m) has a non-commutative crepant resolution.
(d) CM(R) has a cluster tilting object.
We have an efficient criterion for existence of a small resolution of a cAn–singularity.
Theorem 5.6. [Kat, Th. 1.1] Let X = Spec(R) be an isolated cAn–singularity.
(a) Let Y −→ X be a small resolution. Then the exceptional curve in Y is a chain of n projective
lines and X has the form g(x, y) + uv, where the curve singularity g(x, y) has n + 1 distinct
branches at the origin.
(b) If X has the form g(x, y) + uv, where the curve singularity g(x, y) has n+ 1 distinct branches at
the origin, then X has a small resolution.
Using the criterion of Katz together with Knörrer periodicity, we get additional equivalent conditions
in a special case.
Theorem 5.7. Let (R,m) be an isolated cAn–singularity defined by the equation g(x, y) + zt. Then the
following conditions are equivalent in addition to (a)-(d) in Theorem 5.5.
(e) Let R′ be a one-dimensional singularity defined by g(x, y). Then CM(R′) has a cluster tilting
object.
(f) The number of irreducible power series in the prime decomposition of g(x, y) is n+ 1.
Proof. (a)⇔(f) This follows from Theorem 5.6.
(d)⇔(e) By the Knörrer periodicity there is an equivalence of triangulated categories between the
stable categories CM(R) ∼= CM(R′). For, the equivalence of these stable categories given in [Kn, So] is
induced by an exact functor taking projectives to projectives. �
Theorem 5.8. Assume that the equivalent conditions in Theorem 5.7 are satisfied. Then the following
numbers are equal.
(a) One plus the number of irreducible components of the exceptional curve of a small resolution of
Spec(R).
(b) The number of irreducible power series in the prime decomposition of g(z, t).
(c) The number of simple modules of non-commutative crepant resolutions of (R,m).
(d) One plus the number of non-isomorphic indecomposable summands of basic cluster tilting objects
in CM(R).
Proof. (a) and (b) are equal by Theorem 5.6.
(a) and (c) are equal by [V1, Th. 3.5.6].
(c) and (d) are equal by [IR, Cor. 8.8]. �
6. Application to curve singularities
In this section we apply results in the previous section to some curve singularities to investigate whether
they have 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
Tp,q,2,2(λ) = k[[x, y, u, v]]/(x
p + yq + λx2y2 + uv).
To apply Theorem 5.7 to a curve singularity, we have to know that the corresponding three-dimensional
singularity is cAn. It is given by the following result, where we denote by ord(g) the degree of the lowest
term of a power series g.
Proposition 6.1. We have the following properties of three-dimensional hypersurface singularities:
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 23
(a) An (1 ≤ n) is a cA1–singularity,
(b) Dn (4 ≤ n) and En (n = 6, 7, 8) are cA2–singularities,
(c) T3,q,2,2(λ) (6 ≤ q) is a cA2–singularity,
(d) Tp,q,2,2(λ) (4 ≤ p ≤ q) is a cA3–singularity,
(e) k[[x, y, z, t]]/(x2 + y2 + g(z, t)) (g ∈ k[[z, t]]) is a cAm–singularity if m = ord(g) − 1 ≥ 1.
We shall give a detailed proof at the end of this section. In view of Theorem 5.7 and Proposition 6.1,
we have the following main result in this section.
Theorem 6.2. (a) A simple three-dimensional singularity satisfies the equivalent conditions in The-
orem 5.7 if and only if it is of type An (n is odd) or Dn (n is even).
(b) A Tp,q,2,2(λ)–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 + f1 · · · fn) with irreducible and mutually prime fi ∈ (x, y) ⊂
k[[x, y]] (1 ≤ i ≤ n) satisfies the equivalent conditions in Theorem 5.7 if and only if fi /∈ (x, y)
for any i.
Proof. Each singularity is cAm 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 An–singularity, we have m = 1 and g(x, y) = x
2 + yn+1. So g has two factors if and only if
n is odd.
For a Dn–singularity, we have m = 2 and g(x, y) = (x
2 + yn−2)y. So g has three factors if and only if
n is even.
For an En–singularity, we have m = 2 and g(x, y) = x
3 + y4, x(x2 + y3) or x3 + y5. 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 − x2)(y − λx2) for
(p, q) = (3, 6), and m = 3 and g(x, y) = xy(x − y)(x− λy) for (p, q) = (4, 4). In both cases, g has m+ 1
factors.
Now we consider the cusp case. We have m = 2 for p = 3 and m = 3 for p > 3, and g(x, y) =
(xp−2 − y2)(x2 − yq−2). So g has m+ 1 factors if and only if p = 3 and q is even or if both p and q are
even.
(c) We have m =
i=1 ord(fi) − 1 and g = f1 · · · fn. So g has m+ 1 factors if and only if ord(fi) = 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 An (n is odd) or Dn (n is even). The number of non-isomorphic indecomposable summands
of basic cluster tilting objects in CM(R) is 1 for type An (n is odd) and 2 for type Dn (n is even).
(b) A Tp,q(λ)-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 CM(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]]/(f1 · · · fn) with irreducible and mutually prime fi ∈ (x, y) ⊂ k[[x, y]]
(1 ≤ i ≤ n) has a cluster tilting object if and only if fi /∈ (x, y)
2 for any i. In this case, the
number of non-isomorphic indecomposable summands of basic cluster tilting objects in CM(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[[x1, x2, . . . , xn]] the local ring of
formal power series and m its maximal ideal. We shall need the following standard notions.
Definition 6.4. For f ∈ m2 we denote by J(f) = 〈 ∂f
, . . . , ∂f
〉 its Jacobi ideal. The Milnor number
µ(f) is defined as
µ(f) := dimk(R/J(f)).
24 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
The following lemma is standard (see for example [AGV, GLSh]):
Lemma 6.5. A hypersurface singularity f = 0 is isolated if and only if µ(f) <∞.
Definition 6.6 ([AGV]). Two hypersurface singularities f = 0 and g = 0 are called right equivalent
∼ g) if there exists an algebra automorphism ϕ ∈ Aut(R) such that g = ϕ(f).
Note that f
∼ g implies an isomorphism of k–algebras
R/(f) ∼= R/(g).
The following lemma is straightforward, see for example [GLSh, Lem. 2.10].
Lemma 6.7. Assume f
∼ g, then µ(f) = µ(g).
In what follows, we shall need the next standard result on classification of singularities, see for example
[GLSh, Cor. 2.24].
Theorem 6.8. Let f ∈ m2 be an isolated singularity with Milnor number µ. Then
∼ f + g
for any g ∈ mµ+2.
We shall need the following easy lemma.
Lemma 6.9. Let f = x2 + y2 + p(x, y, z), where
p(x, y, z) = zn + p1(x, y)z
n−1 + · · · + pn(x, y)
is a homogeneous form of degree n ≥ 3. Then
∼ x2 + y2 + zn.
Proof. Write p(x, y, z) = zn + xu + yv for some homogeneous forms u and v of degree n− 1. Then
x2 + y2 + zn + xu+ yv = (x+ u/2)2 + (y + v/2)2 + zn − (u2 + v2)/4.
After a change of variables x 7→ x+ u/2, y 7→ y + v/2 and z 7→ z we reduce f to the form
f = x2 + y2 + zn + h,
where h ∈ m2(n−1) ⊂ mn+1. Note that µ(x2 + y2 + zn) = n− 1, hence by Theorem 6.8 we have
∼ x2 + y2 + zn.
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 = αx+ βy + γz, α, β, γ ∈ k. We put
g(z, t) = a0z
m+1 + a1z
mt+ · · · + am+1t
m+1 + (higher terms).
Then the intersection of H with the singularity defined by the equation x2 + y2 + g(z, t) is given by the
equation f = h+ (higher terms), where
h = x2 + y2 + a0z
m+1 + a1z
m(αx+ βy + γz) + · · · + am+1(αx+ βy + γz)
Now we consider the case m = 1. We have h
∼ x2 + y2 + z2 since any quadratic form can be
diagonalized using linear transformations. By Lemma 6.7, we have µ(h) = µ(x2 + y2 + z2) = 1. Hence
∼ x2 + y2 + z2 by Theorem 6.8.
Next we consider the case m ≥ 2. Assume α ∈ k satisfies a0 + a1α+ · · ·+ am+1α
m+1 6= 0. By Lemma
6.9, we have h
∼ x2 + y2 + zm+1. By Lemma 6.7, we have µ(h) = µ(x2 + y2 + zm+1) = m. Hence
∼ x2 + y2 + zm+1 by Theorem 6.8.
Consequently, x2 + y2 + g(z, t) is cAm. �
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 25
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,m) be an odd-dimensional isolated hypersurface singularity and Γ a 2-CY tilted
algebra coming from CM(R). Then we have the following.
(a) Γ is a symmetric algebra.
(b) All components in the stable AR-quiver of infinite type Γ are tubes of rank 1 or 2.
We now start with our computations of 2-CY tilted algebras coming from minimally elliptic singular-
ities. We first introduce and investigate two classes of algebras, and then show that they are isomorphic
to 2-CY tilted algebras coming from minimally elliptic singularities.
For a quiver Q with finitely many vertices and arrows we define the radical completion k̂Q of the path
algebra kQ by the formula
k̂Q = lim
kQ/ radn(kQ).
The reason we deal with completion is the following: Let Q be a finite quiver, J the ideal of k̂Q
generated by the arrows and I ⊆ J2 a complete ideal such that Λ = k̂Q/I is finite-dimensional.
Lemma 7.2. The ideal I is generated in k̂Q by a minimal system of relations, that is, a set of elements
ρ1, · · · , ρn of I whose images form a k-basis of I/IJ + JI.
The lemma is shown by a standard argument (cf [BMR3, Section 3]). Its analogue for the non complete
path algebra is not always true. For example, for the algebra Λ = B2,2(λ) defined below, the elements
ρ1, · · · , ρn listed as generators for I form a minimal system of relations. So they generate I in k̂Q. They
also yield a k-basis of I ′/I ′J + JI ′
−→ I/IJ + JI, where I ′ = I ∩ kQ and J ′ = J ∩ kQ. But they do not
generate the ideal I ′ of kQ since, as one can show, the quotient kQ/〈ρ1, · · · , ρn〉 is infinite-dimensional.
On the other hand, the ideal I ′ is generated by the preimage ρ1, · · · , ρn of a basis of I
′/I ′J ′ + J ′I ′ if
the quotient kQ/〈ρ1, · · · , ρn〉 is finite-dimensional, since then the ideal 〈ρ1, · · · , ρn〉 contains a power of
J ′. This happens for example for the algebra A2(λ) as defined below, cf. also [Sk, 5.9] and [BS, Th. 1].
We know that for all vertices i, j of Q, we have
dimk ei(I/IJ + JI)ej = dimk Ext
Λ(Si, Sj)
where Si and Sj denote the simple Λ-modules corresponding to the vertices i and j [B]. When Λ is 2-CY
tilted, then
dim Ext1Λ(Sj , Si) ≥ dim Ext
Λ(Si, Sj)
(see [BMR3, KR]). Thus the number of arrows in Q is an upper bound on the number of elements in a
minimal system of relations.
Definition 7.3. (1) For q ≥ 2 and λ ∈ k∗ we write Aq(λ) = k̂Q/I, where
Q = ·ϕ
## α // ·
I = 〈ψα− αϕ, βψ − ϕβ, ϕ2 − βα, ψq − λαβ〉.
If q = 2, then we additionally assume λ 6= 1. (It can be shown that for q ≥ 3 we have Aq(λ) ∼= Aq(1), so
we drop the parameter λ in this case.)
(2) For p, q ≥ 1 and λ ∈ k∗ we write Bp,q(λ) = k̂Q/I, where
Q = ·ϕ
## α // ·
γ // ·
26 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
I = 〈βα− ϕp, γδ − λψq, αϕ− δγα, ϕβ − βδγ, δψ − αβδ, ψγ − γαβ〉.
For p = q = 1 we additionally assume λ 6= 1.
When p = q = 1, the generators ϕ and ψ can be excluded and B1,1(λ) is given by the completion of
the path algebra of the quiver
Q = ·
α // ·
γ // ·
modulo the relations
I = 〈αβα − δγα, αβδ − λδγδ, γαβ − λγδγ, βδγ − βαβ〉.
For (p, q) 6= (1, 1) we have Bp,q(λ) ∼= Bp,q(1). In particular, for p = 1 and q ≥ 2 the algebra is isomorphic
to k̂Q/I, where
Q = ·
α // ·
γ // ·
I = 〈γδ − ψq, αβα − δγα, βαβ − βδγ, δψ − αβδ, ψγ − γαβ〉.
It turns out that the algebras Aq(λ) and Bp,q(λ) are finite dimensional. In order to show this it suffices
to check that all oriented cycles in k̂Q/I are nilpotent.
Lemma 7.4. In the algebra Aq(λ) the following zero relations hold:
αβα = 0, βαβ = 0, αϕ2 = ψ2α = 0, ϕ2β = βψ2 = 0, ϕ4 = 0, ψq+2 = 0.
Proof. We have to consider separately the cases q = 2 and q ≥ 3.
Let q = 2, then we assumed λ 6= 1. We have
αβα = αϕ2 = ψ2α = λ−1αβα,
hence αβα = 0. In a similar way we obtain βαβ = 0. Then αϕ2 = αβα = 0, ϕ2 = βαβα = 0 and the
remaining zero relations follow analogously.
Let q ≥ 3. Then
ψqα = αβα = αϕ2 = ψ2α,
so (1 − ψq−2)ψ2α = 0 and hence
ψ2α = αβα = 0
in k̂Q/I. The remaining zero relations follow similarly. �
Lemma 7.5. We have the following relations in Bp,q(λ):
ϕp+2 = 0, ψq+2 = 0, γαϕ = ψγα = 0, ϕβδ = βδψ = 0.
Moreover, αβ · δγ = δγ · αβ. For q ≥ p ≥ 2 we have
(αβ)2 = (δγ)2 = 0,
for q > p = 1 we have
(αβ)3 = 0, (δγ)2 = 0, (αβ)2 · (δγ) = 0
and for p = q = 1
(αβ)3 = (γδ)3 = 0, (αβ)2 = αβ · δγ = λ(δγ)2.
The proof is completely parallel to the proof of the previous lemma and is therefore skipped. �
The main result of this section is the following
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 27
Theorem 7.6. (a) Let R be a T3,2q+2(λ)–singularity, where q ≥ 2 and λ ∈ k
∗. Then in the triangulated
category CM(R) there exists a cluster tilting object with the corresponding 2-CY-tilted algebra isomorphic
to Aq(λ).
(b)For R = T2p+2,2q+2(λ) the category CM(R) has a cluster tilting object with endomorphism algebra
isomorphic to Bp,q(λ).
Proof. (a) We consider first the case of T3,2q+2(λ).
The coordinate ring of T3,6(λ) is isomorphic to
R = k[[x, y]]/(y(y − x2)(y − λx2)),
where λ 6= 0, 1. Consider Cohen-Macaulay modules M and N given by the two-periodic free resolutions
M = (R
−−−→ R
y(y−λx2)
−−−−−−→ R),
N = (R
y(y−x2)
−−−−−→ R
y−λx2
−−−−→ R).
Then M ⊕N is cluster tilting by Theorem 4.1 or Corollary 6.3. In order to compute the endomorphism
algebra End(M ⊕N), note that
End(M) ∼= k[ϕ]/〈ϕ
where ϕ = (x, x) is an endomorphism of M viewed as a two-periodic map of a free resolution. In End(M)
we have (y, y) = (x, x)2 = ϕ2. Similarly,
End(N) ∼= k[ψ]/〈ψ
4〉, ψ = (x, x), (y, y) = λ(x, x)2 = λψ2,
Hom(M,N) = k2 = 〈(1, y), (x, xy)〉, Hom(N,M) = k2 = 〈(y, 1), (xy, x)〉.
The isomorphism A2(λ) −→ End(M ⊕N) is given by
ϕ 7→ (x, x), ψ 7→ (x, x), α 7→ (1, y), β 7→ (y, 1).
Assume now q ≥ 3 and R = T3,2q+2. By [AGV] we may write
R = k[[x, y]]/((x− y2)(x2 − y2q)).
Consider the Cohen-Macaulay module M ⊕N , where
M = (R
−−−→ R
x2−y2q
−−−−−→ R),
N = (R
(x−y2)(x+yq)
−−−−−−−−−→ R
−−−→ R).
Again, by a straightforward calculation
End(M) ∼= k[ϕ]/〈ϕ4〉, ϕ = (y, y), End(N) ∼= k[ψ]/〈ψq+2〉, ψ = (y, y)
Hom(M,N) = k2 = 〈(1, x+ yq), (y, y(x+ yq))〉,
Hom(M,N) = k2 = 〈(x+ yq, 1), (y(x+ yq), y)〉.
If q ≥ 4 then End(M ⊕N) is isomorphic to k̂Q/I, where
Q = ·ϕ
## α // ·
and the relations are
βα = ϕ2, αβ = 2ψq, αϕ = ψα, ϕβ = βψ
ϕ = (y, y), ψ = (y, y), α = (1, x+ yq), β = (x+ yq, 1).
By rescaling all generators α 7→ 2aα, β 7→ 2bβ, ϕ 7→ 2fϕ, ψ 7→ 2gψ for properly chosen a, b, f, g ∈ Q one
can easily show End(M ⊕N) ∼= Aq.
28 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
The case q = 3 has to be considered separately, since this time the relations are
βα = ϕ2 + ϕ3, αβ = 2ψq, αϕ = ψα, ϕβ = βψ.
We claim that there exist invertible power series u(t), v(t), w(t), z(t) ∈ k[[t]] such that the new generators
ϕ′ = u(ϕ)ϕ, ψ′ = v(ψ)ψ, α′ = αw(ϕ) = w(ψ)α, β′ = βz(ψ) = z(ϕ)β
satisfy precisely the relations of the algebra A3. This is fulfilled provided we have the following equations
in k[[t]]: 
zw = u2(1 + tu)
zw = 2v3
uw = vw
uz = vz.
This system is equivalent to
u(t) = v(t) = (2 − t)−1 =
)2 + . . . )
and hence the statement is proven.
The case of T2p+2,2q+2(λ) is essentially similar. For p = q = 1 we have
R = k[[x, y]]/(xy(x − y)(x− λy)).
Take 
M = (R
−−−→ R
xy(x−λy)
−−−−−−→ R),
N = (R
x(x−y)
−−−−−→ R
y(x−λy)
−−−−−→ R),
K = (R
xy(x−y)
−−−−−→ R
−−−→ R).
By Theorem 4.1 or Corollary 6.3, M ⊕N ⊕K is cluster tilting. Moreover, B1,1(λ) ≃ End(M ⊕N ⊕K).
Let now
R = k[[x, y]]/((xp − y)(xp + y)(yq − x)(yq + x)),
where (p, q) 6= (1, 1) and 
M = (R
−−−→ R
(yq+x)(yq−x)(xp+y)
−−−−−−−−−−−−−−→ R),
N = (R
(xp−y)(xp+y)
−−−−−−−−−→ R
(yq−x)(yq+x)
−−−−−−−−−→ R),
K = (R
(xp−y)(xp+y)(yq+x)
−−−−−−−−−−−−−−→ R
−−−→ R).
By Theorem 4.1 or Corollary 6.3, M ⊕N ⊕K is cluster tilting, and by a similar case-by-case analysis it
can be verified that End(M ⊕N ⊕K) ∼= Bp,q. �
We have seen that the algebras Aq(λ) and Bp,q(λ) are symmetric, and the indecomposable nonpro-
jective modules have τ -period at most 2, hence Ω-period dividing 4 since τ = Ω2 in this case. A direct
computation shows that the Cartan matrix is nonsingular. Note that these algebras appear in Erd-
mann’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
Bia lkowski-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 Coh(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
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 29
singularity. And the canonical algebras are obtained as endomorphism algebras of certain tilting objects
in Coh(X) which are vector bundles.
Note that it is known from work of Dieterich [D], Kahn [Kah], Drozd and Greuel [DG] that mini-
mally 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 Tp,q(λ), see [DG].
Moreover, simply elliptic singularities are tame of polynomial growth and cusp singularities are tame of
exponential growth. Furthermore, the Auslander-Reiten quiver of the corresponding stable categories of
maximal Cohen-Macaulay modules consists of tubes of rank one or two, see [Kah, Th. 3.1] and [DGK,
Cor. 7.2].
It should follow from the tameness of CM(T3,p(λ)) and CM(Tp,q(λ)) that the associated 2-CY tilted
algebras are tame.
We point out that in the wild case we can obtain symmetric 2-CY tilted algebras where the stable
AR-quiver consists of tubes of rank one and two, and most of them should be wild. It was previously
known that there are examples of wild selfinjective algebras whose AR-quivers consist of tubes of rank
one or three [AR].
8. Appendix: 2-CY triangulated categories of finite type
In this section, we consider a more general situation than in section 2. Let k be an algebraically closed
field and C a k-linear connected 2-Calabi-Yau triangulated category with only finitely many indecompos-
able objects. We show that it follows from the shape of the AR quiver of C whether cluster tilting objects
(respectively, non-zero rigid objects) exist in C or not. Let us start with giving the possible shapes of
the AR quiver of C. Recall that a subgroup G of Aut(Z∆) is called weakly admissible if x and gx do not
have a common direct successor for any vertex x in Z∆ and g ∈ G\{1} [XZ, Am].
Proposition 8.1. The AR quiver of C is Z∆/G for a Dynkin diagram ∆ and a weakly admissible
subgroup G of Aut(Z∆) which contains F ∈ Aut(Z∆) defined by the list below. Moreover, G is generated
by a single element g ∈ Aut(Z∆) in the list below.
∆ Aut(Z∆) F g
(An) n : odd Z× Z/2Z (
, 1) (k, 1) (k|n+3
, n+3
is odd)
(An) n : even Z n+ 3 k (k|n+ 3)
(Dn) n : odd Z× Z/2Z (n, 1) (k, 1) (k|n)
(D4) Z× S3 (4, 0) (k, σ) (k|4, σ
k = 1)
(Dn) n : even, n > 4 Z× Z/2Z (n, 0) (k, 0) (k|n) or (k, 1) (k|n,
is even)
(E6) Z× Z/2Z (7, 1) (1, 1) or (7, 1)
(E7) Z 10 1, 2, 5 or 10
(E8) Z 16 1, 2, 4, 8 or 16
In each case, elements in the torsion part of Aut(Z∆) are induced by the automorphism of ∆. The
torsionfree part of Aut(Z∆) is generated by τ except the case (An) with even n, in which case it is
generated by the square root of τ .
Proof. By [XZ] (see also [Am, 4.0.4]), the AR quiver of C is Z∆/G for a Dynkin diagram ∆ and a
weakly admissible subgroup G of Aut(Z∆). Since 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 ∈ 〈g〉, we have the above list. �
Note that, by a result of Keller [Ke], the translation quiver Z∆/G for any Dynkin diagram ∆ and
any weakly admissible group G of Aut(Z∆) is realized as the AR quiver of a triangulated orbit category
Db(H)/g for a hereditary algebra H of type ∆ and some autofunctor g of Db(H).
30 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Theorem 8.2. (1) C has a cluster tilting object if and only if the AR quiver of C is Z∆/g for a Dynkin
diagram ∆ and g ∈ Aut(Z∆) in the list below.
∆ Aut(Z∆) g
(An) n : odd Z× Z/2Z (
, 1) (3|n) or (n+3
(An) n : even Z
(3|n) or n+ 3
(Dn) n : odd Z× Z/2Z (k, 1) (k|n)
(D4) Z× S3 (k, σ) (k|4, σ
k = 1, (k, σ) 6= (1, 1))
(Dn) n : even, n > 4 Z× Z/2Z (k, k) (k|n)
(E6) Z× Z/2Z (7, 1)
(E7) Z 10
(E8) Z 8 or 16
(2) C does not have a non-zero rigid object if and only if the AR quiver of C is Z∆/g for a Dynkin
diagram ∆ and g ∈ Aut(Z∆) in the list below.
∆ Aut(Z∆) g
(An) n : odd Z× Z/2Z −
(An) n : even Z 1
(Dn) n : odd Z× Z/2Z −
(D4) Z× S3 (1, 1)
(Dn) n : even, n > 4 Z× Z/2Z (1, 0)
(E6) Z× Z/2Z (1, 1)
(E7) Z 1
(E8) Z 1 or 2
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 C has a cluster tilting object. For example,
consider the (Dn) case here. Fix a vertex x ∈ Z∆ corresponding to an end point of ∆ which is adjacent
to the branch vertex of ∆. Then the subset {(1, 1)lx | l ∈ Z} of Z∆ is stable under the action of g, and
gives a cluster tilting object of C.
Conversely, assume that C has a cluster tilting object. Then one can check that g is on the list. For
example, consider the (An) 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 2kπ
-radian. Since the center of the regular (n + 3)-polygon is
contained in some triangle or its edge, we have 2kπ
= 2π, 4π
, π or 2π
. Since k|n+ 3 and n is even, we
have k = n+ 3 or n+3
(2) If g is on the list above, then one can easily check that 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 ∆ is rigid. �
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[Sk] A. Skowroński, Selfinjective algebras: finite and tame type, Trends in representation theory of algebras and related
topics, 169–238, Contemp. Math., 406, Amer. Math. Soc., Providence, RI, 2006.
[So] Ø. Solberg, Hypersurface singularities of finite Cohen-Macaulay type, Proc. London Math. Soc. 58 (1989) 258–280.
[V1] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455.
[V2] M. Van den Bergh, Non-commutative crepant resolutions, The legacy of Niels Henrik Abel, 749–770, Springer, Berlin,
2004.
[XZ] J. Xiao, B. Zhu, Locally finite triangulated categories, J. Algebra 290 (2005), no. 2, 473–490.
[Y] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, LMS 146, 1990.
Johannes-Gutenberg Universität Mainz, Fachbereich Physik, Mathematik und Informatik, Institut für Math-
ematik, 55099 Mainz, Germany
E-mail address: burban@mathematik.uni-mainz.de
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
E-mail address: iyama@math.nagoya-u.ac.jp
UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2 place Jussieu, 75251 Paris
Cedex 05, France
E-mail address: keller@math.jussieu.fr
Institutt for matematiske fag, Norges Teknisk-naturvitenskapelige universitet, N-7491, Trondheim, Norway
E-mail address: idunr@math.ntnu.no
	Introduction
	Acknowledgment
	1. Main results
	2. Simple hypersurface singularities
	3. Computation with Singular
	4. One-dimensional hypersurface singularities
	5. Link with birational geometry
	6. Application to curve singularities
	7. Examples of 2-CY tilted algebras
	8. Appendix: 2-CY triangulated categories of finite type
	References

  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.

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 CH is the orbit category D
b(H)/F , where Db(H) is the bounded derived
category of finite dimensional H-modules and the functor F : Db(H) → Db(H) is τ−1[1] = S−1[2]. Here
τ denotes the translation associated with almost split sequences/triangles and S the Serre functor [BK]
on Db(H). (See [CCS] for an independent definition of a category equivalent to the cluster category when
H is of Dynkin type An).
An object T in a cluster category CH was defined to be a (cluster) tilting object if Ext
(T, T ) = 0,
and if Ext1CH (X,X ⊕ T ) = 0, then X is in addT . 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 Ext1CH (X,T ) = 0 implies that X is in
addT , 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 mod(Λ) of
finite dimensional modules over a preprojective algebra of Dynkin type Λ 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 mod(Λ).
The categories CH and mod(Λ) are both triangulated categories [Ke, H], with finite dimensional ho-
momorphism 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 = Σ2, where Σ is the shift
functor in the triangulated category.
For an arbitrary 2-CY triangulated category C with finite dimensional homomorphism spaces over
a field k, a cluster tilting object T in C was defined to be an object satisfying the stronger property
discussed above, corresponding to the property of being maximal 1-orthogonal/Ext-configuration [KR].
The corresponding class of algebras, containing the cluster tilted ones, have been called 2-CY tilted. With
this concept many results have been generalised from cluster categories, and from the stable categories
mod(Λ), to this more general setting in [KR], which moreover contains several results which are new also
in the first two cases.
One of the important applications of classical tilting theory has been the construction of derived
equivalences: Given a tilting bundle T on a smooth projective variety X , the total right derived functor
of Hom(T, ) is an equivalence from the bounded derived category of coherent sheaves onX to the bounded
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.
http://arxiv.org/abs/0704.1249v3
2 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
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 CM(R) is the stable category of maximal Cohen-Macaulay modules over an odd-dimensional
isolated hypersurface singularity, then CM(R) is 2-CY. If it contains a cluster tilting object T , then the
functor Hom(T, ) induces an equivalence between the quotient of CM(R) by the ideal of morphisms
factoring through τT and the category of finite dimensional modules over the endomorphism algebra
B = End(T ). It is then not hard to see that B is symmetric and the indecomposable nonprojective
B-modules are τ -periodic of τ -period at most 2. In this article, we study examples of this setup arising
from finite, tame and wild CM-type isolated hypersurface singularities R. The endomorphism algebras
of the cluster tilting objects in the tame case occur in lists in [BS, Er, Sk]. We also obtain a large class
of symmetric finite dimensional algebras where the stable AR-quiver consists only of tubes of rank one
or two. Examples of (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 CM(R) of
maximal Cohen-Macaulay R-modules being our 2-CY category. These singularities are indexed by the
Dynkin diagrams, and in the cases Dn for odd n and E7 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
CM(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 (An, n even, E6, E8), and cases where the maximal rigid objects coincide with
the cluster tilting objects (An, n odd and Dn, 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 CH and
mod(Λ) [BMRRT, BMR2, GLSc]. The results are also valid for any odd-dimensional simple hypersurface
singularity, since the stable categories of Cohen-Macaulay modules are all triangle equivalent [Kn, So].
We shall construct a large class of one-dimensional hypersurface singularities R, where CM(R) or
CM(R) has a cluster tilting object, including examples coming from simple singularities and minimally
elliptic singularities. We classify all rigid objects in CM(R) for these R, in particular, we give a bijection
between cluster tilting objects in CM(R) and elements in a symmetric group. Our method is based
on a higher theory of almost split sequences [I1, I2], and a crucial role is played by the endomorphism
algebras EndR(T ) (called ‘three-dimensional Auslander algebras’) of cluster tilting objects T in CM(R).
These algebras have global dimension three, and have 2-CY tilted algebras as stable factors. The functor
HomR(T, ) : CM(R) → mod(EndR(T )) sends cluster tilting objects in CM(R) to tilting modules over
EndR(T ). By comparing cluster tilting mutations in CM(R) and tilting mutation in CM(EndR(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 CM(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 sin-
gularities 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 sin-
gularities 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
cAn–singularities for existence of crepant resolutions to a criterion for one-dimensional hypersurface sin-
gularities 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.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 3
In section 2 we investigate maximal rigid objects and cluster tilting objects in CM(R) for simple
one-dimensional hypersurface singularities. We decide whether extension spaces are zero or not by using
covering techniques. In section 3 we point out that we could also use the computer program Singular
[GP] to accomplish the same thing. In section 4 we construct 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,m) be a local complete d-dimensional commutative noetherian Gorenstein isolated singularity
and R/m = k ⊂ R, where k is an algebraically closed field of characteristic zero. We denote by CM(R)
the category of maximal Cohen-Macaulay modules over R. Then CM(R) is a Frobenius category (i.e. an
exact category with enough projectives and injectives which coincide), and so the stable category CM(R)
is a Hom-finite triangulated category with shift functor Σ = Ω−1 [H]. For an integer n, we say that
CM(R) or CM(R) is n-CY if there exists a functorial isomorphism
HomR(X,Y ) ≃ DHomR(Y,Σ
for any X,Y ∈ CM(R).
We collect some fundamental results.
• We have AR-duality
HomR(X,Y ) ≃ DExt
R(Y, τX)
with τ ≃ Ω2−d [Au]. In particular, CM(R) is (d− 1)-CY.
• If R is a hypersurface singularity, then Σ2 = id [Ei].
Consequently, if d is odd, then τ = Ω and CM(R) is 2-CY. If d is even, then τ = id and CM(R)
is 1-CY, hence any non-free Cohen-Macaulay R-module M satisfies Ext1R(M,M) 6= 0.
• (Knörrer periodicity)
CM(k[[x0, · · · , xd, y, z]]/(f + yz)) ≃ CM(k[[x0, · · · , xd]]/(f))
for any f ∈ k[[x0, · · · , xd]] [Kn] ([So] in characteristic two).
We state some of the definitions, valid more generally, in the context of CM(R) and CM(R).
Definition 1.1. Let C = CM(R) or CM(R). We call an object M ∈ C
• rigid if Ext1R(M,M) = 0,
• maximal rigid if it is rigid and any rigid N ∈ C satisfying M ∈ addN satisfies N ∈ addM ,
• cluster tilting if addM = {X ∈ C | Ext1R(M,X) = 0} = {X ∈ C | Ext
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 CM(R). If C is 2-CY, then M ∈ C is cluster tilting if and only if
addM = {X ∈ C | Ext1R(M,X) = 0}.
4 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Definition 1.2. Let C = CM(R) (or CM(R)) be 2-CY and M ∈ C a basic cluster tilting object. Take an
indecomposable summand X of M = X⊕N . Then there exist short exact sequences (or triangles) (called
exchange sequences)
→ Y and Y
such that Ni ∈ addN and f0 is a minimal right (addN)-approximation. Then Y ⊕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 f1 is a minimal right (addN)-approximation and gi is a minimal left (addN)-approximation
automatically, so X ⊕N is a cluster tilting mutation of Y ⊕N . It is known that there are no more basic
cluster tilting objects containing N [IY, Th. 5.3].
Let R = k[[x, y, z2, · · · , zd]]/(f) be a simple hypersurface singularity so that in characteristic zero f is
one of the following polynomials,
(An) x
2 + yn+1 + z22 + z
3 + · · · + z
d (n ≥ 1)
(Dn) x
2y + yn−1 + z22 + z
3 + · · · + z
d (n ≥ 4)
(E6) x
3 + y4 + z22 + z
3 + · · · + z
(E7) x
3 + xy3 + z22 + z
3 + · · · + z
(E8) x
3 + y5 + z22 + z
3 + · · · + z
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 ≥ 1 over an algebraically closed
field k of characteristic zero.
(1) Assume that d is even. Then 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 CM(R) are
as follows:
f indec. rigid cluster tilting max. rigid summands of max. rigid
(An) n : odd 2 2 2 1
(An) n : even 0 0 1 0
(Dn) n : odd 2 0 2 1
(Dn) n : even 6 6 6 2
(E6) 0 0 1 0
(E7) 2 0 2 1
(E8) 0 0 1 0
We also consider a minimally elliptic curve singularity Tp,q(λ) (p ≤ q). Assume for simplicity that
our base field k is algebraically closed of characteristic zero. Then these singularities are given by the
equations
xp + yq + λx2y2 = 0,
where 1
and certain values of λ ∈ k have to be excluded. They are of tame Cohen-Macaulay
representation type [D, Kah, DG]. We divide into two cases.
(i) Assume 1
. This case occurs if and only if (p, q) = (3, 6) or (4, 4), and Tp,q(λ) is called
simply elliptic. The corresponding coordinate rings can be written in the form
T3,6(λ) = k[[x, y]]/(y(y − x
2)(y − λx2))
T4,4(λ) = k[[x, y]]/(xy(x − y)(x− λy)),
where in both cases λ ∈ k \ {0, 1}.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 5
(ii) Assume 1
. Then Tp,q(λ) does not depend on the continuous parameter λ, and is called a
cusp singularity. In this case the corresponding coordinate rings can be written in the form
Tp,q = k[[x, y]]/((x
p−2 − y2)(x2 − yq−2)).
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 Tp,q(λ) over an algebraically closed field
k of characteristic zero.
(a) CM(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 CM(R) are as follows:
p, q indec. rigid cluster tilting summands of cluster tilting
p = 3, q : even 6 6 2
p, q : even 14 24 3
We also prove the following general theorem, which includes both Theorem 1.3 (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 ∈ (x, y)) be a one-dimensional reduced hypersurface singularity
over an algebraically closed field k of characteristic zero.
(a) CM(R) has a cluster tilting object if and only if f is a product f = f1 · · · fn with fi /∈ (x, y)
(b) The number of indecomposable rigid objects, basic cluster tilting objects, and indecomposable
summands of basic cluster tilting objects in CM(R) are as follows:
indec. rigid cluster tilting summands of cluster tilting
2n − 2 n! n− 1
The following result gives a bridge between cluster tilting theory and birational geometry. The termi-
nologies are explained in section 5.
Theorem 1.6. Let (R,m) be a three-dimensional isolated cAn–singularity over an algebraically closed
field k of characteristic zero defined by the equation g(x, y) + zt and R′ a one-dimensional singularity
defined by g(x, y). Then the following conditions are equivalent.
(a) Spec(R) has a small resolution.
(b) Spec(R) has a crepant resolution.
(c) (R,m) has a non-commutative crepant resolution.
(d) CM(R) has a cluster tilting object.
(e) CM(R′) has a cluster tilting object.
(f) The number of irreducible power series in the prime decomposition of g(x, y) is n+ 1.
We end this section by giving an application to finite dimensional algebras. A 2-CY tilted algebra is an
endomorphism ring EndC(M) of a cluster tilting object T in a 2-CY triangulated category C. In section
7, we shall show the following result and compute 2-CY tilted algebras associated with minimally elliptic
curve singularities.
Theorem 1.7. Let (R,m) be an odd-dimensional isolated hypersurface singularity and Γ a 2-CY tilted
algebra coming from CM(R). Then we have the following.
(a) Γ is a symmetric algebra.
(b) All components in the stable AR-quiver of infinite type Γ are tubes of rank 1 or 2.
6 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
For example, put
R = k[[x, y]]/((x− λ1y) · · · (x− λny)) and M =
k[[x, y]]/((x− λ1y) · · · (x− λiy))
for distinct elements λi ∈ k. Then M is a cluster tilting object in CM(R) by Theorem 4.1, so Γ =
EndR(M) satisfies the conditions in Theorem 1.7. Since CM(R) has wild Cohen-Macaulay representation
type if n > 4 [DG, Th. 3], we should get a family of examples of finite dimensional symmetric k-algebras
whose stable AR-quiver consists only of tubes of rank 1 or 2, and are of wild representation type.
2. Simple hypersurface singularities
Let R be a one-dimensional simple hypersurface singularity. In this case the AR-quivers are known
for CM(R) [DW], and so also for CM(R). We use the notation from [Y].
In order to locate the indecomposable rigid modules M , that is, the modules M with
Ext1(M,M) = 0, the following lemmas are useful, where part (a) of the first one is proved in [HKR], and
the second one is a direct consequence of [KR] (generalizing [BMR1]).
Lemma 2.1. (a) Let C be an abelian or triangulated k-category with finite dimensional homomor-
phism spaces. Let A
−−−→ B1 ⊕ B2
(g1,g2)
−−−−→ C be a short exact sequence or a triangle, where
A is indecomposable, B1 and B2 nonzero, and (g1, g2) has no nonzero indecomposable summand
which is an isomorphism. Then Hom(A,C) 6= 0 .
(b) Let 0 → A
−→ C → 0 be an almost split sequence in CM(R), where R is an isolated
hypersurface singularity, and B has at least two indecomposable nonprojective summands in a
decomposition of B into a direct sum of indecomposable modules. Then Ext1(C,C) 6= 0.
Proof. (a) See [HKR, Lem. 6.5].
(b) Using (a) together with the above AR-formula and τ2 = id, we obtain DExt1(C,C) ≃
Hom(τ−1C,C) = Hom(τC,C) ≃ Hom(A,C) 6= 0, where D = Homk( , k). �
Lemma 2.2. Let T be a cluster tilting object in the Hom-finite connected 2-CY category C, and Γ =
EndC(T ).
(a) The functor G = HomC(T, ) : C → mod(Γ) induces an equivalence of categories
G : C/add(τT ) → mod(Γ).
(b) The AR-quiver for Γ is as a translation quiver obtained from the AR-quiver for C by removing
the vertices corresponding to the indecomposable summands of τT .
(c) Assume τ2 = id. Then we have the following.
(i) Γ is a symmetric algebra.
(ii) The indecomposable nonprojective Γ-modules have τ-period one or two.
(iii) If C has an infinite number of nonisomorphic indecomposable objects, then all components
in the stable AR-quiver of Γ are tubes of rank one or two.
(d) If C has only a finite number n of nonisomorphic indecomposable objects, and T has t nonisomor-
phic indecomposable summands, then there are n− t nonisomorphic indecomposable Γ-modules.
Proof. For (a) and (b) see [BMR1, KR]. Since C is 2-CY, we have τ = Σ, and a functorial isomorphism
DHomC(T, T ) ≃ HomC(T,Σ
2T ) = HomC(T, τ
2T ) ≃ HomC(T, T ).
This shows that Γ is symmetric. Let C be an indecomposable nonprojective Γ-module. Viewing C as an
object in C we have τ2CC ≃ C, and τC is not a projective Γ-module since C is not removed. Hence we
have τ2ΓC ≃ C. If C has an infinite number of nonisomorphic indecomposable objects, then Γ is of infinite
type. Then each component of the AR-quiver is infinite, and hence is a tube of rank one or two. Finally,
(d) is a direct consequence of (a). �
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 7
We also use that in our cases we have a covering functor Π: k(ZQ) → CM(R), where Q is the
appropriate Dynkin quiver and k(ZQ) is the mesh category of the translation quiver ZQ [Rie, Am], (see
also [I1, Section 4.4] for another explanation using functorial methods).
For the one-dimensional simple hypersurface singularities we have the cases An (n even or odd), Dn
(n odd or even), E6, E7 and E8. We now investigate them case by case.
Proposition 2.3. In the case An (with n even) there are no indecomposable rigid objects.
Proof. We have the stable AR-quiver
oo // · · · //oo In/2
Here, and later, a dotted line between two indecomposable modules means that they are connected
via τ .
Since τIj ≃ Ij for each j, Ext
1(Ij , Ij) 6= 0 for j = 1, · · · , n/2. Hence no Ij is rigid. �
Proposition 2.4. In the case An (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
(n+1)
Proof. For simplicity, we write l = (n− 1)/2. We have the stable AR-quiver
//oo · · · //oo Ml
==zzzzz
aaDDDDD
Since τMi ≃Mi for i = 1, · · · , l, we have
Ext1(Mi,Mi) ≃ Hom(Mi, τMi) ≃ Hom(Mi,Mi) 6= 0.
So only the indecomposable objects N− and N+ could be rigid. We use covering techniques and additive
functions to compute the support of Hom(N−, ), where we refer to [BG] for the meaning of the diagrams
below.
>>}}}
<<yyy
==zzz
· · ·
>>}}}
>>}}}
==zzz
??���
// N− //// Ml
>>~~~
// N+ // Ml
>>~~~
// N− · · ·
??���
>>~~~
>>~~~
BB���
BB���
BB���
BB���
BB���
BB���
// 1 // 1
BB���
// 0 // 1
BB���
// 1 1
// 0 // 0
BB���
BB���
BB���
BB���
We see that Hom(N−, N+) = 0, so Ext
1(N+, N+) = Ext
1(N+, τN−) = 0, and Ext
1(N−, N−) = 0.
Since Ext1(N+, N−) 6= 0, we see that N+ and N− are exactly the maximal rigid objects. Further
Hom(N−,Mi) 6= 0 for all i, so Ext
1(N+,Mi) 6= 0 and Ext
1(N−,Mi) 6= 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 Dn with n odd we have two maximal rigid objects, which both are inde-
composable, and neither one is cluster tilting.
8 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Proof. We have the AR-quiver
// Y1
// Y2 //
M2 //
// · · · // M(n−3)/2
**UUU
X(n−1)/2
jjUUU
ttiiii
A // X1
\\9999999
// N1
]];;;;;;;
// X2
]];;;;;;;
// N2
]];;;;;;;
// · · · // N(n−3)/2
bbDDDDDDDD 44iiii
Using Lemma 2.1, the only candidates for being indecomposable rigid are A and B. We compute the
support of Hom(A, )
AA���
BB���
· · ·
AA���
??���
??���
BB���
Y1 · · ·
CC���
BB���
where B = τA and l = (n− 3)/2. We see that Hom(A,B) = 0, so that Ext1(A,A) = 0. Then A is
clearly maximal rigid. Since Hom(A,M1) = 0, we have Ext
1(A,N1) = 0, so A is not cluster tilting.
Alternatively, we could use that we see that End(A)op ≃ k[x]/(x2), which has two indecomposable
modules, whereas CM(R) has 2n− 3 indecomposable objects. If A was cluster tilting, End(A)op would
have had 2n− 3 − 1 = 2n− 4 indecomposable modules, by Lemma 2.2. �
Proposition 2.6. In the case D2n 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 ·
α // ·
oo αβα = 0 = βαβ
in the case D4, and by ·γ ;;
α // ·
oo with γn−1 = βα, γβ = 0 = αγ and ·
α // ·
oo with (αβ)n−1α = 0 =
(βα)n−1β for 2n > 4.
Proof. We have the AR-quiver
vvnnn
// Y1
· · · // Yn−1
``BBBBBBB
A // X1 //
XX1111111111
YY3333333333
// X2
YY3333333333
// N2
YY3333333333
// · · · // Xn−1
ZZ66666666666
FF













66nnnn
hhQQQQ
By Lemma 2.1, the only possible indecomposable rigid objects are: A, B, C+, C−, D+, D−.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 9
We compute the support of Hom(C+, ):
C+ // Yl
AA���
// D+ // Xl
??���
// C+ // Yl
??���
// D+ // Xl
??���
// C+ // Yl
CC���
· · ·
@@���
??���
Nl−1 Ml−1
>>~~~
@@���
@@���
??���
AA���
>>~~~
>>~~~
>>~~~
BB���
>>}}}
· · ·
AA���
>>}}}
where l = n− 1
1 // 1
BB���
// 0 // 1
BB���
// 1 // 1
BB���
// 0 // 1
BB���
// 1 // 0
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
We see that Hom(C+, D+) = 0, so Ext
1(C+, C+) = 0 = Ext
1(D+, D+). Further, Hom(C+, C−) = 0,
so Ext1(C+, D−) = 0. By symmetry Ext
1(D−, D−) = 0 = Ext
1(C−, C−) and Ext
1(D+, C−) = 0. Also
Ext1(C+, A) = 0, Ext
1(C+, B) 6= 0, so Ext
1(D+, B) = 0, Ext
1(D+, A) 6= 0. Further Ext
1(C+, X) 6= 0 for
X 6= A,D−, C+.
We now compute the support of Hom(A, )
>>~~~
C+ // Yl
??���
D+ // Xl
· · ·
??���
>>~~~
@@���
M1 M1
@@���
>>~~~
??���
>>}}}
· · ·
??~~~
>>~~~
>>}}}
??~~~
10 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
where l = n− 1 and we have an odd number of columns and rows.
BB���
1 // 1
BB���
0 // 0
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
We see that Hom(A,B) = 0, so Ext1(A,A) = 0, hence also Ext1(B,B) = 0. Since Hom(A,D−) = 0, we
have Ext1(A,C−) = 0, hence Ext
1(B,D−) = 0. Since Hom(A,C−) 6= 0, we have Ext
1(A,D−) 6= 0, so
Ext1(B,D+) 6= 0.
It follows that C+ ⊕D−, C− ⊕D+, C+ ⊕A, D+ ⊕B, A⊕ C− and B ⊕D− are maximal rigid.
These are also cluster tilting: We have Hom(A,Xi) 6= 0, Hom(A,Ni) 6= 0, so Ext
1(B,Xi) 6= 0,
Ext1(B,Ni) 6= 0. Similarly, Ext
1(A, Yi) 6= 0, Ext
1(A,Mi) 6= 0. Also Hom(C+, Yi) 6= 0, Hom(C+, Ni) 6= 0,
so Ext1(D+, Yi) 6= 0, Ext
1(D+, Ni) 6= 0. Hence Ext
1(C+, Xi) 6= 0,
Ext1(C+,Mi) 6= 0. So Ext
1(D−, Yi) 6= 0, Ext
1(D−, Ni) 6= 0, Ext
1(C−, Xi) 6= 0, Ext
1(C−,Mi) 6= 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:
{C+, D−}
{B,D−}
rrrrrr
{A,C+}
{B,D+}
{A,C−}
{C−, D+}
rrrrrr
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 E6 there are no indecomposable rigid objects.
Proof. We have the AR-quiver
// M1
::uuuu
ddIIII
// N1
\\8888888
The only candidates for indecomposable rigid objects according to Lemma 2.1 are M1 and N1. We
compute the support of Hom(M1, ).
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 11
>>~~~
??���
· · ·
??���
// M2 // X
>>~~~
// M2 // X
AA���
>>~~~
??���
· · ·
AA���
??���
>>~~~
BB���
BB���
· · ·
BB���
// 1 // 1
BB���
0 // 1
BB���
BB���
BB���
· · ·
BB���
BB���
BB���
We see that Hom(M1, N1) 6= 0, so that Ext
1(M1,M1) 6= 0 and Ext
1(N1, N1) 6= 0. �
Proposition 2.8. In the case E7 there are two maximal rigid objects, which both are indecomposable,
and neither of them is cluster tilting.
Proof. We have the AR-quiver
___ D
// M2 //
// Y3
^^>>>>>>>>
wwppp
// Y1
xxppp
B // N2
^^========
// X2
__????????
// X3
���������������
__>>>>>>>>
// X1
^^========
// N1
__????????
Using Lemma 2.1, we see that the only candidates for indecomposable rigid objects are A, B, M1, N1,
C and D. We first compute the support of Hom(A, ).
??���
X1 · · ·
@@���
// C // X3
??���
AA���
??���
@@���
@@���
??���
· · ·
AA���
@@���
??���
// 1 // 1
// 0 // 1
// 1 // 1
// 0 // 1
// 1 // 1
// 0 // 0
We see that Ext1(A,A) = 0, and so also Ext1(B,B) = 0, so A and B are rigid.
Next we compute the support of Hom(M1, ).
@@���
??���
@@���
· · ·
@@���
// C // X3
??���
// D // Y3
??���
@@���
??���
N2 · · ·
??���
BB���
BB���
BB���
· · ·
BB���
1 // 1
BB���
// 0 // 1
BB���
BB���
BB���
1 · · ·
BB���
We see that Ext1(M1,M1) 6= 0 and Ext
1(N1, N1) 6= 0, so that M1 and N1 are not rigid.
Then we compute the support of Hom(C, ).
12 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
@@���
??���
X1 · · ·
C // X3
@@���
D // Y3
@@���
C // X3
??���
@@���
??���
@@���
??���
· · ·
??���
BB���
BB���
1 · · ·
1 // 1
BB���
0 // 1
BB���
1 // 2 //
BB���
BB���
BB���
BB���
BB���
· · ·
BB���
We see that Ext1(C,C) 6= 0 and Ext1(D,D) 6= 0, so that C and D are not rigid. Hence A and B are
the rigid indecomposable objects, and they are maximal rigid.
Since Ext1(A,C) = 0, we see that A and hence B is not cluster tilting. �
Proposition 2.9. In the case E8 there are no indecomposable rigid objects.
Proof. We have the AR-quiver
__ B2
N2 //
// X1
]];;;;;;;
xxqqq
xxqqq
// C1
// B1
// N1
M2 // C2
^^<<<<<<<
// Y1
]];;;;;;;
��������������
// Y2
]];;;;;;;
// D1
]];;;;;;;
// A1
]];;;;;;;
// M1
]]<<<<<<<
The only candidates for indecomposable rigid objects are M1, N1, M2, N2, A2 and B2, by Lemma 2.1.
We first compute the support of Hom(M1, ):
??���
??���
· · ·
??���
B2 // X1
??���
// A2 // Y1
??���
??���
??���
@@���
??���
??���
@@���
??���
??���
??���
· · ·
@@���
@@���
??���
??���
BB���
BB���
0 · · ·
BB���
// 1 // 1
BB���
// 0 // 1
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
BB���
· · ·
BB���
BB���
BB���
BB���
BB���
We see that Ext1(M1,M1) 6= 0, and hence Ext
1(N1, N1) 6= 0.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 13
Next we compute the support of Hom(M2, ):
@@���
??���
??���
· · ·
@@���
// B2 // X1
??���
// A2 // Y1
??���
??���
??���
??���
· · ·
BB���
BB���
BB���
· · ·
BB���
1 // 1
BB���
0 // 1
BB���
BB���
BB���
BB���
· · ·
We see that Ext1(M2,M2) 6= 0, and hence Ext
1(N2, N2) 6= 0.
Finally we compute the support of Hom(A2, ):
??���
??���
D2 · · ·
A2 // Y1
AA���
// B2 // X1
??���
// A2 // Y1 //
??���
??���
??���
??���
??���
??���
B1 · · ·
??���
BB���
BB���
1 · · ·
1 // 1
BB���
// 0 // 1
BB���
// 1 // 2
BB���
BB���
BB���
BB���
BB���
BB���
1 · · ·
BB���
It follows that Ext1(A2, A2) 6= 0, and similarly Ext
1(B2, B2) 6= 0. Hence there are no indecomposable
rigid objects. �
3. Computation with Singular
An alternative way to carry out computations of Ext1–spaces in the stable category of maximal Cohen-
Macaulay modules is to use the computer algebra system Singular, see [GP]. Let
R = k[x1, x2, . . . , xn]〈x1,x2,...,xn〉/I
be a Cohen-Macaulay local ring which is an isolated singularity, and M and N two maximal Cohen-
Macaulay modules. Denote by R̂ the completion of R. Since all the spaces ExtiR(M,N) (i ≥ 1) are finite-
dimensional over k and the functor mod(R) → mod(R̂) is exact, maps the maximal Cohen-Macaulay
modules to maximal Cohen-Macaulay modules and the finite length modules to finite length modules,
we can conclude that
dimk(Ext
R(M,N)) = dimk(Ext
bR(M̂, N̂)).
As an illustration we show how to do this for the case E7.
Proposition 3.1. In the case E7 there are two maximal rigid objects, which both are indecomposable
and neither of them is cluster tilting.
14 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
By [Y] the AR-quiver of CM(R) has the form
// M2 //
// Y3
aaCCCCC
vvmmm
// Y1
vvmmm
B // N2
``BBBBB
// X2
aaDDDDD
// X3
aaCCCCC
FF









aaBBBBB
// N1
aaDDDDD
By Lemma 2.1 only the modules A,B,C,D,M1, N1 can be rigid. Since B = τ(A), D = τ(C), N1 = τ(M1),
the pairs of modules (A,B), (C,D) and (M1, N1) are rigid or not rigid simultaneously. By [Y] we have
the following presentations:
x2+y3
−−−−→ R
−→ R −→ A −→ 0,
( x y
y2 −x)
−−−−−→ R2
x( x y
y2 −x)
−−−−−−→ R2 −→ C −→ 0,
xy2 −x)
−−−−−−→ R2
xy2 −x2)
−−−−−−→ R2 −→M1 −→ 0,
so we can use the computer algebra system Singular in order to compute the Ext1–spaces between these
modules.
> Singular (call the program ‘‘Singular’’)
> LIB ‘‘homolog.lib’’; (call the library of homological algebra)
> ring S = 0,(x,y),ds; (defines the ring S = Q[x, y]〈x,y〉)
> ideal I = x3 + xy3; (defines the ideal x3 + xy3 in S)
> qring R = std(I); (defines the ring Q[x, y]〈x,y〉/I)
> module A = [x];
> module C = [x2, xy2], [xy, -x2];
> module M1 = [x2, xy2], [y, -x2]; (define modules A,C,M1)
> list l = Ext(1,A,A,1);
// dimension of Ext1: -1 (Output: Ext1R(A,A) = 0)
> list l = Ext(1,C,C,1);
// ** redefining l **
// dimension of Ext1: 0 (the Krull dimension of Ext1R(C,C) is 0)
// vdim of Ext1: 2 (dimk(Ext
R(C,C)) = 2)
> list l = Ext(1,M1,M1,1);
// ** redefining l **
// dimension of Ext1: 0
// vdim of Ext1: 10
> list l = Ext(1,A,C,1);
// ** redefining l **
// dimension of Ext1: -1
This computation shows that the modules A and B are rigid, C,D,M1 and N1 are not rigid and since
Ext1R(A,C) = 0, there are no cluster tilting objects in the stable category CM(R).
4. One-dimensional hypersurface singularities
We shall construct a large class of one-dimensional hypersurface singularities having a cluster tilting
object, then classify all cluster tilting objects. Our method is based on the higher theory of almost split
sequences and Auslander algebras studied in [I1, I2]. We also use a relationship between cluster tilting
objects in CM(R) and tilting modules over the endomorphism algebra of a cluster tilting object [I2].
Then we shall compare cluster 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 CM(R) instead of CM(R).
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 15
Let k be an infinite field, S := k[[x, y]] and m := (x, y). We fix f ∈ m and write f = f1 · · · fn for
irreducible formal power series fi ∈ m (1 ≤ i ≤ n). Put
Si := S/(f1 · · · fi) and R := Sn = S/(f).
We assume that R is reduced, so we have (fi) 6= (fj) for any i 6= 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)
i=1 Si is a rigid object in CM(R).
i=1 Si is a cluster tilting object in CM(R) if the following condition (A) is satisfied.
(A) fi /∈ m
2 for any 1 ≤ i ≤ n.
Let Sn be the symmetric group of degree n. For w ∈ Sn and I ⊆ {1, · · · , n}, we put
Swi := S/(fw(1) · · · fw(i)), Mw :=
Swi and SI := S/(
Theorem 4.2. Assume that (A) is satisfied.
(a) There are exactly n! basic cluster tilting objects Mw (w ∈ Sn) and exactly 2
n− 1 indecomposable
rigid objects SI (∅ 6= I ⊆ {1, · · · , n}) in CM(R).
(b) For any w ∈ Sn, there are exactly n! basic Cohen-Macaulay tilting EndR(Mw)-modules
HomR(Mw,Mw′) (w
′ ∈ Sn) of projective dimension at most one. Moreover, all algebras
EndR(Mw) (w ∈ Sn) are derived equivalent.
It is interesting to compare with results in [IR], where two-dimensional (2-Calabi-Yau) algebras Γ are
treated and a bijection between elements in an affine Weyl group and tilting Γ-modules of projective
dimension at most one is given. Here the algebra is one-dimensional, and Weyl groups appear.
Here we consider three examples.
(a) Let R be a curve singularity of type A2n−1 or D2n+2, so
R = S/((x− yn)(x + yn)) or R = S/(y(x− yn)(x+ yn)).
By our theorems, there are exactly 2 or 6 cluster tilting objects and exactly 3 or 7 indecomposable
rigid objects in CM(R), which fits with our computations in section 1.
(b) Let R be a curve singularity of type T3,2q+2(λ) or T2p+2,2q+2(λ), so
R = S/((x− y2)(x − yq)(x + yq)) (R = S/(y(y − x2)(y − λx2)) for q = 2),
R = S/((xp − y)(xp + y)(x− yq)(x + yq)) (R = S/(xy(x− y)(x− λy)) for p = q = 1).
By our theorems, there are exactly 6 or 24 cluster tilting objects and exactly 7 or 15 indecom-
posable rigid objects in CM(R).
(c) Let λi ∈ k (1 ≤ i ≤ n) be mutually distinct elements in k. Put
R := S/((x− λ1y) · · · (x− λny)).
By our theorems, there are exactly n! cluster tilting objects and exactly 2n − 1 indecomposable
rigid objects in CM(R).
First of all, Theorem 4.1(a) follows immediately from the following observation.
Proposition 4.3. For g1, g2 ∈ m and g3 ∈ S, put R := S/(g1g2g3). If g1 and g2 have no common factor,
then Ext1R(S/(g1g3), S/(g1)) = 0 = Ext
R(S/(g1), S/(g1g3)).
Proof. We have a projective resolution
→ R → S/(g1g3) → 0.
Applying HomR( , S/(g1)), we have a complex
S/(g1)
g1g3=0
−→ S/(g1)
→ S/(g1).
This is exact since g1 and g2 have no common factor. Thus we have the former equation, and the other
one can be proved similarly. �
16 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Our plan of proof of Theorem 4.1(b) is the following.
(i) First we shall prove Theorem 4.1 under the following stronger assumption:
(B) m = (f1, f2) = · · · = (fn−1, fn).
(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 CM(R).
(b) gl. dim EndR(M) ≤ 3.
(c) For any X ∈ CM(R), there exists an exact sequence 0 →M1 →M0 → X → 0 with Mi ∈ addM .
(d) For any indecomposable direct summand X of M , there exists an exact sequence 0 → M2
→ X with Mi ∈ addM and a is a right almost split map in addM .
Proof. (a)⇔(b) For d = dimR, take the d-cotilting module T = R and apply [I2, Th. 5.1(3)] for m = d
and n = 2 there.
(a)⇔(c) See [I1, Prop. 2.2.2].
(a)⇒(d) See [I1, Th. 3.3.1].
(d)⇒(b) For any simple EndR(M)-module S, there exists an indecomposable direct summand X of M
such that S is the top of the projective HomR(M,X). Since Ext
R(M,M2) = 0, the sequence in (d) gives a
projective resolution 0 → HomR(M,M2) → HomR(M,M1) → HomR(M,M0) → HomR(M,X) → S → 0.
Thus we have pdS ≤ 3 and gl. dim EndR(M) ≤ 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 addM , 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
HomR(Si, Sj) ≃
(fi+1 · · · fj)/(f1 · · · fj) i < j
S/(f1 · · · fj) i ≥ j.
Lemma 4.5. Let R = S/(f) be a one-dimensional reduced hypersurface singularity, S0 := 0 and 1 ≤ i <
(a) We have exchange sequences (see Definition 1.2)
0 → Si
−−−−→ Si+1 ⊕ Si−1
(1 fi+1)
−−−−−→ S/(f1 · · · fi−1fi+1) → 0,
0 → S/(f1 · · · fi−1fi+1)
(fi1 )
−−→ Si+1 ⊕ Si−1
(−1 fi)
−−−−−→ Si → 0.
(b) If (fi, fi+1) = m, then we have a 2-almost split sequence
0 → Si
−−−−→ Si+1 ⊕ Si−1
fi fifi+1
1 fi+1
−−−−−−−→ Si+1 ⊕ Si−1
(−1 fi)
−−−−−→ Si → 0
in add
i=1 Si.
Proof. (a) Consider the map a := (−1 fi) : Si+1 ⊕ Si−1 → Si. Any morphism from Sj to Si factors
through 1 : Si+1 → Si (respectively, fi : Si−1 → Si) if j > i (respectively, j < i). Thus a is a minimal
right (add
j 6=i Sj)-approximation.
It is easily checked that Ker a = {s ∈ Si+1 | s ∈ fiSi} = (fi)/(f1 · · · fi+1) ≃ S/(f1 · · · fi−1fi+1), where
we denote by s the image of s via the natural surjection Si+1 → Si.
Consider the surjective map b := (1 fi+1) : Si+1 ⊕Si−1 → S/(f1 · · · fi−1fi+1). It is easily checked that
Ker b = {s ∈ Si+1 | s ∈ (fi+1)/(f1 · · · fi−1fi+1)} = (fi+1)/(f1 · · · fi+1) ≃ Si, where we denote by s the
image of s via the natural surjection Si+1 → S/(f1 · · · fi−1fi+1).
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 17
(b) This sequence is exact by (a). Any non-isomorphic endomorphism of Si is multiplication with an
element in m, which is equal to (fi, fi+1) by our assumption. Since fi+1 (respectively, fi) : Si → Si factors
through 1 : Si+1 → Si (respectively, fi : Si−1 → Si), we have that a is a right almost split map. �
Now we choose fn+1 ∈ m such that m = (fn, fn+1), and fn+1 and f1 · · · fn have no common factor.
This is possible by our assumption (A).
Lemma 4.6. We have an exact sequence
0 → Sn−1
−fn+1
−−−−−→ Sn ⊕ Sn−1
(fn+1 fn)
−−−−−−→ Sn
with a minimal right almost split map (fn+1 fn) in add
i=1 Si.
Proof. Consider the map a := (fn+1 fn) : Sn⊕Sn−1 → Sn. Any morphism from Sj (j < n) to Sn factors
through fn : Sn−1 → Sn.
Any non-isomorphic endomorphism of Sn is multiplication with an element in m = (fn+1, fn). Since
fn : Sn → Sn factors through fn : Sn−1 → Sn, we have that a is a right almost split map.
It is easily checked that Ker a = {s ∈ Sn−1 | fns ∈ fn+1Sn} = (fn+1, f1 · · · fn−1)/(f1 · · · fn−1), which
is isomorphic to Sn−1 by the choice of fn+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 CM(R). We use the notation introduced at the beginning of this section.
Lemma 4.7. For w ∈ Sn, we assume that Mw is a cluster tilting object in CM(R). Then, for 1 ≤ i < n
and si = (i i+ 1), we have exchange sequences
0 → Swi → S
i+1 ⊕ S
i−1 → S
i → 0 and 0 → S
i → S
i+1 ⊕ S
i−1 → S
i → 0.
Proof. Without loss of generality, we can assume w = 1. Then the assertion follows from Lemma
4.5(a). �
Immediately, we have the following.
Proposition 4.8. Assume that Mw is a cluster tilting object in CM(R) for some w ∈ Sn.
(a) The cluster tilting mutations of Mw are Mwsi (1 ≤ i < n).
(b) Mw′ is a cluster tilting object in CM(R) for any w
′ ∈ Sn.
Proof. (a) This follows from Lemma 4.7.
(b) This follows from (a) since Sn is generated by si (1 ≤ i < n). �
The following result is also useful.
Lemma 4.9. Let R and R′ be complete local Gorenstein rings with dimR = dimR′ and M a rigid object
in CM(R) which is a generator. Assume that there exists a surjection R′ → R, and we regard CM(R) as
a full subcategory of CM(R′). If R′ ⊕M is a cluster tilting object in CM(R′), then M is a cluster tilting
object in CM(R).
Proof. We use the equivalence (a)⇔(c) in Proposition 4.4, which remains true in any dimension [I1,
Prop. 2.2.2]. For any X ∈ CM(R), take a right (addM)-approximation f : M0 → X of X . Since M is
a generator of R, we have an exact sequence 0 → Y → M0
→ X → 0 with Y ∈ CM(R). Since R′ is a
projective R′-module, f is a right add(R′ ⊕M)-approximation of X . Since R′ ⊕M is a cluster tilting
object in CM(R′), we have Y ∈ add(R′ ⊕M). Since Y ∈ CM(R), we have Y ∈ addM . 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 gi ∈ m (1 ≤ i < n) such that h2i−1 := fi and h2i := gi satisfy
the following conditions:
• (hi) 6= (hj) for any i 6= j.
18 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
• m = (h1, h2) = (h2, h3) = · · · = (h2n−2, h2n−1).
Put R′ := S/(h1 · · ·h2n−1). This is reduced by the first condition.
Since we have already proved Theorem 4.1 under the assumption (B), we have that⊕2n−1
i=1 S/(h1 · · ·hi) is a cluster tilting object in CM(R
′). By Proposition 4.8,⊕2n−1
i=1 S/(hw(1) · · ·hw(i)) is a cluster tilting object in CM(R
′) for any w ∈ S2n−1. In particular,
S/(f1 · · · fi)) ⊕ (
S/(f1 · · · fng1 · · · gi))
is a cluster tilting object in CM(R′). Moreover we have surjections
R′ → · · · → S/(f1 · · · fng1g2) → S/(f1 · · · fng1) → R.
Using Lemma 4.9 repeatedly, we have that
i=1(S/(f1 · · · fi)) is a cluster tilting object in CM(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 EndR(
i=1 Si) is
// · · ·oo // Sn−1oo
where in addition there is a loop at Si (1 ≤ i < n) if and only if (fi, fi+1) 6= m.
(b) We have the quiver of EndR(
i=1 Si) by removing the vertex Sn from the quiver in (a).
Proof. We only have to show (a). We only have to calculate minimal right almost split maps in
i=1 Si. We have a minimal right almost split map Sn⊕Sn−1 → Sn by Lemma 4.6. If (fi, fi+1) = m
(1 ≤ i < n), then we have a minimal right almost split map Si+1 ⊕ Si−1 → Si by Lemma 4.5.
We only have to consider the case (fi, fi+1) 6= m (1 ≤ i < n). Take g ∈ m such that (fi, fi+1, g) = m.
It is easily check (cf proof of Lemma 4.5) that we have a right almost split map
c := (−1 g fi) : Si+1 ⊕ Si ⊕ Si−1 → Si.
Assume that c is not right minimal. Then there exists a right almost split map of the form c′ :
Si+1 ⊕ Si → Si (i > 1), Si ⊕ Si−1 → Si or Si+1 ⊕ Si−1 → Si. For the first case, it is easily checked that
fi : Si−1 → Si does not factor through c
′, 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 Γ 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 Γ-modules T and U , we
write
T ≥ U
if Ext1Γ(T, U) = 0. By tilting theory, we have FacT = {X ∈ mod(Γ) | Ext
Γ(T,X) = 0}. Thus
Ext1Γ(T, U) = 0 is equivalent to FacT ⊃ FacU , and ≥ gives a partial order. On the other hand, we call a
Γ-module T almost complete tilting if pd ΓT ≤ 1, Ext
Γ(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 Γ-module has at most two complements.
(b) T and U are neighbors in the partial order if and only if there exists an almost complete tilting
Γ-module which is a common direct summand of T and U .
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 19
(c) Assume T ≥ U . Then there exists a sequence T = T0 > T1 > T2 > · · · > U satisfying the
following conditions.
(i) Ti and Ti+1 are neighbors.
(ii) Either Ti = U for some i or the sequence is infinite.
(d) T ≥ U if and only if there exists an exact sequence 0 → T → U0 → U1 → 0 with Ui ∈ addU .
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 Γ, we call a Γ-module Cohen-Macaulay if it is a Cohen-Macaulay R-module. As usual, we
denote by CM(Γ) the category of Cohen-Macaulay Γ-modules.
Lemma 4.12. Let Γ be a module-finite algebra over a complete local Gorenstein ring R such that Γ ∈
CM(R), and T and U tilting Γ-modules. Assume U ∈ CM(Γ).
(a) If T ≥ U , then T ∈ CM(Γ).
(b) Let P be a projective Γ-module such that HomR(P,R) is a projective Γ
op-module. Then P ∈
addU .
Proof. (a) By Proposition 4.11(d), there exists an exact sequence 0 → T → U0 → U1 → 0 with Ui ∈
addU . Thus the assertion holds.
(b) We have Ext1Γop(HomR(P,R),HomR(U,R)) = 0. Since we have a duality HomR( , R) : CM(Γ) ↔
CM(Γop), it holds Ext1Γ(U, P ) = 0. There exists an exact sequence 0 → P → U0 → U1 → 0 with
Ui ∈ addU [H, Lem. III.2.3], which must split since Ext
Γ(U, P ) = 0. Thus we have P ∈ addU . �
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 ′ cluster tilting objects in CM(R).
(a) HomR(M,N) is a tilting EndR(M)-module of projective dimension at most one.
(b) If N ′ is a cluster tilting mutation of N , then HomR(M,N
′) is a tilting mutation of HomR(M,N).
Now we shall prove Theorem 4.2. Fix w ∈ Sn and put Γ := EndR(Mw). Since Mw is a generator
of R, the functor HomR(Mw, ) : CM(R) → CM(Γ) is fully faithful. By Theorem 4.1, Mw is a cluster
tilting object in CM(R). By Proposition 4.13(a), HomR(Mw,Mw′) (w
′ ∈ Sn) is a Cohen-Macaulay tilting
Γ-module.
(b) Take any Cohen-Macaulay tilting Γ-module U . Since P := HomR(Mw, R) is a projective Γ-module
such that HomR(P,R) = Mw = HomR(R,Mw) is a projective Γ-module, we have P ∈ addU by Lemma
4.12(b). In particular, by Proposition 4.11(a)(b),
• any Cohen-Macaulay tilting Γ-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 Γ-module of the form HomR(Mw,Mw′) (w
′ ∈ Sn) has precisely
(n− 1) tilting mutations HomR(Mw,Mw′si) (1 ≤ i < n) which are Cohen-Macaulay.
Consequently, any successive tilting mutation of Γ = HomR(Mw,Mw) has the form HomR(Mw,Mw′) for
some w′ ∈ Sn if each step is Cohen-Macaulay.
Using this observation, we shall show that U is isomorphic to HomR(Mw,Mw′) for some w
′. Since
Γ ≥ U , there exists a sequence
Γ = T0 > T1 > T2 > · · · > U
satisfying the conditions in Proposition 4.11(c). By Lemma 4.12(a), each Ti is Cohen-Macaulay. Thus
the above observation implies that each Ti has the form HomR(Mw,Mwi) for some wi ∈ Sn. Moreover,
wi 6= wj for i 6= j. Since Sn is a finite group, the above sequence must be finite. Thus U = Ti holds for
some i, hence the proof is completed.
20 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
(a) Let U be a cluster tilting object in CM(R). Again by Proposition 4.13(a), HomR(Mw, U) is a
Cohen-Macaulay tilting Γ-module. By part (b) which we already proved, HomR(Mw, U) is isomorphic to
HomR(Mw,Mw′) for some w
′ ∈ Sn. Since the functor HomR(Mw, ) : CM(R) → CM(Γ) is fully faithful,
U is isomorphic to Mw′, and the former assertion is proved.
For the latter assertion, we only have to show that any rigid object in CM(R) is a direct summand of
some cluster tilting object in CM(R). This is valid by the following general result in [BIRS, Th. 1.9]. �
Proposition 4.14. Let C be a 2-CY Frobenius category with a cluster tilting object. Then any rigid
object in C is a direct summand of some cluster tilting object in C.
We end this section with the following application to dimension three.
Now let S′′ := k[[x, y, u, v]], fi ∈ m = (x, y) (1 ≤ i ≤ n) and R
′′ := S′′/(f1 · · · fn + uv). For w ∈ Sn
and I ⊆ {1, · · · , n}, we put
Uwi := (u, fw(1) · · · fw(i)) ⊂ R
′′, Mw :=
Uwi and UI := (u,
fi) ⊂ S
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 Mw (w ∈ Sn) and exactly 2
n− 1 indecomposable
rigid objects UI (∅ 6= I ⊂ {1, · · · , n}) in CM(R
(b) There are non-commutative crepant resolutions EndR′′(Mw) (w ∈ Sn) of R
′′, which are derived
equivalent.
Proof. (a) We only have to apply Knörrer periodicity CM(R) → CM(R′′) [Kn, So] as follows:
Since Swi ∈ CM(R) has a projective resolution
−→ R → Swi → 0
for a := fw(1) · · · fw(i) and b := fw(i+1) · · · fw(n), the corresponding object X ∈ CM(R
′′) has a projective
resolution
(u ab −v)
−→ R′′2
(v ab −u)
−→ R′′2 → X → 0.
It is easily checked that X is isomorphic to (u, a) = Uwi .
(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,
k[[x, y, u, v]]/((x− λ1y) · · · (x− λny) + uv)
has a non-commutative crepant resolution for distinct elements λ1, · · · , λn ∈ 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,m) be a three-dimensional complete normal Gorenstein singularity over an algebraically closed
field k of characteristic zero, and let X = Spec(R). A resolution of singularities Y
−→ X is called
• crepant, if ωY ∼= π
∗ωX for canonical sheaves ωX and ωY of X and Y respectively.
• small, if the fibre of the closed point has dimension at most one.
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 21
A small resolution is automatically crepant, but the converse is in general not true. However, both types
of resolutions coincide for certain important classes of three-dimensional singularities.
A cDV (compound Du Val) singularity is a three-dimensional singularity given by the equation
f(x, y, z) + tg(x, y, z, t) = 0,
where f(x, y, z) defines a simple surface singularity and g(x, y, z, t) is arbitrary.
A cDV singularity is called cAn if the intersection of f(x, y, z) + tg(x, y, z, t) = 0 with a generic
hyperplane ax+ by + cz + dt = 0 in k4 is an An surface singularity. Generic means that the coefficients
(a, b, c, d) belong to a non-empty Zariski open subset in k4.
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,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 = EndR(M) is Cohen-Macaulay as an R–module,
(iii) gl.dim(A) = 3.
The following result establishes a useful connection.
Theorem 5.3. [V1, Cor. 3.2.11][V2, Th. 6.6.3] Let (R,m) be an isolated cDV singularity. Then there
exists a crepant resolution of X = Spec(R) if and only if there exists a non-commutative one in the sense
of Definition 5.2.
The existence of a non-commutative crepant resolution turns out to be equivalent to the existence of a
cluster tilting object in the triangulated category CM(R).
Theorem 5.4. [I2, Th. 5.2.1][IR, Cor. 8.13] Let (R,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 CM(R).
Proof. For convenience of the reader, we give an outline of the proof (see also Proposition 4.4).
Let us first assume that M is a cluster tilting object in CM(R). Then M is automatically reflexive.
From the exact sequence
0 −→ Ω(M) −→ F −→M −→ 0
we obtain
(1) 0 −→ EndR(M) −→ HomR(F,M) −→ HomR(Ω(M),M) −→ Ext
R(M,M) −→ 0.
Since M is rigid, Ext1R(M,M) = 0. Moreover, depth(HomR(F,M)) = depth(M) = 3 and
depth(HomR(Ω(M),M) ≥ 2, and hence depth(EndR(M)) = 3 and A = End(M) is maximal Cohen-
Macaulay over R.
For the difficult part of this implication, claiming that gl.dim(A) = 3, we refer to [I1, Th. 3.6.2].
For the other direction, let M be a module giving rise to a non-commutative crepant resolution. Then
by [IR, Th. 8.9] there exists another module M ′ giving rise to a non-commutative crepant resolution,
which is maximal Cohen-Macaulay and contains R as a direct summand.
By the assumption, depth(EndR(M
′)) = 3 and we can apply [IR, Lem. 8.5] to the exact sequence (1)
to deduce that Ext1R(M
′,M ′) = 0, so that M ′ is rigid. The difficult part saying that M ′ is cluster tilting
is proven in [I2, Th. 5.2.1]. �
We now summarize the results of this section.
22 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Theorem 5.5. Let (R,m) be an isolated cDV singularity. Then the following are equivalent.
(a) Spec(R) has a small resolution.
(b) Spec(R) has a crepant resolution.
(c) (R,m) has a non-commutative crepant resolution.
(d) CM(R) has a cluster tilting object.
We have an efficient criterion for existence of a small resolution of a cAn–singularity.
Theorem 5.6. [Kat, Th. 1.1] Let X = Spec(R) be an isolated cAn–singularity.
(a) Let Y −→ X be a small resolution. Then the exceptional curve in Y is a chain of n projective
lines and X has the form g(x, y) + uv, where the curve singularity g(x, y) has n + 1 distinct
branches at the origin.
(b) If X has the form g(x, y) + uv, where the curve singularity g(x, y) has n+ 1 distinct branches at
the origin, then X has a small resolution.
Using the criterion of Katz together with Knörrer periodicity, we get additional equivalent conditions
in a special case.
Theorem 5.7. Let (R,m) be an isolated cAn–singularity defined by the equation g(x, y) + zt. Then the
following conditions are equivalent in addition to (a)-(d) in Theorem 5.5.
(e) Let R′ be a one-dimensional singularity defined by g(x, y). Then CM(R′) has a cluster tilting
object.
(f) The number of irreducible power series in the prime decomposition of g(x, y) is n+ 1.
Proof. (a)⇔(f) This follows from Theorem 5.6.
(d)⇔(e) By the Knörrer periodicity there is an equivalence of triangulated categories between the
stable categories CM(R) ∼= CM(R′). For, the equivalence of these stable categories given in [Kn, So] is
induced by an exact functor taking projectives to projectives. �
Theorem 5.8. Assume that the equivalent conditions in Theorem 5.7 are satisfied. Then the following
numbers are equal.
(a) One plus the number of irreducible components of the exceptional curve of a small resolution of
Spec(R).
(b) The number of irreducible power series in the prime decomposition of g(z, t).
(c) The number of simple modules of non-commutative crepant resolutions of (R,m).
(d) One plus the number of non-isomorphic indecomposable summands of basic cluster tilting objects
in CM(R).
Proof. (a) and (b) are equal by Theorem 5.6.
(a) and (c) are equal by [V1, Th. 3.5.6].
(c) and (d) are equal by [IR, Cor. 8.8]. �
6. Application to curve singularities
In this section we apply results in the previous section to some curve singularities to investigate whether
they have 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
Tp,q,2,2(λ) = k[[x, y, u, v]]/(x
p + yq + λx2y2 + uv).
To apply Theorem 5.7 to a curve singularity, we have to know that the corresponding three-dimensional
singularity is cAn. It is given by the following result, where we denote by ord(g) the degree of the lowest
term of a power series g.
Proposition 6.1. We have the following properties of three-dimensional hypersurface singularities:
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 23
(a) An (1 ≤ n) is a cA1–singularity,
(b) Dn (4 ≤ n) and En (n = 6, 7, 8) are cA2–singularities,
(c) T3,q,2,2(λ) (6 ≤ q) is a cA2–singularity,
(d) Tp,q,2,2(λ) (4 ≤ p ≤ q) is a cA3–singularity,
(e) k[[x, y, z, t]]/(x2 + y2 + g(z, t)) (g ∈ k[[z, t]]) is a cAm–singularity if m = ord(g) − 1 ≥ 1.
We shall give a detailed proof at the end of this section. In view of Theorem 5.7 and Proposition 6.1,
we have the following main result in this section.
Theorem 6.2. (a) A simple three-dimensional singularity satisfies the equivalent conditions in The-
orem 5.7 if and only if it is of type An (n is odd) or Dn (n is even).
(b) A Tp,q,2,2(λ)–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 + f1 · · · fn) with irreducible and mutually prime fi ∈ (x, y) ⊂
k[[x, y]] (1 ≤ i ≤ n) satisfies the equivalent conditions in Theorem 5.7 if and only if fi /∈ (x, y)
for any i.
Proof. Each singularity is cAm 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 An–singularity, we have m = 1 and g(x, y) = x
2 + yn+1. So g has two factors if and only if
n is odd.
For a Dn–singularity, we have m = 2 and g(x, y) = (x
2 + yn−2)y. So g has three factors if and only if
n is even.
For an En–singularity, we have m = 2 and g(x, y) = x
3 + y4, x(x2 + y3) or x3 + y5. 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 − x2)(y − λx2) for
(p, q) = (3, 6), and m = 3 and g(x, y) = xy(x − y)(x− λy) for (p, q) = (4, 4). In both cases, g has m+ 1
factors.
Now we consider the cusp case. We have m = 2 for p = 3 and m = 3 for p > 3, and g(x, y) =
(xp−2 − y2)(x2 − yq−2). So g has m+ 1 factors if and only if p = 3 and q is even or if both p and q are
even.
(c) We have m =
i=1 ord(fi) − 1 and g = f1 · · · fn. So g has m+ 1 factors if and only if ord(fi) = 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 An (n is odd) or Dn (n is even). The number of non-isomorphic indecomposable summands
of basic cluster tilting objects in CM(R) is 1 for type An (n is odd) and 2 for type Dn (n is even).
(b) A Tp,q(λ)-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 CM(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]]/(f1 · · · fn) with irreducible and mutually prime fi ∈ (x, y) ⊂ k[[x, y]]
(1 ≤ i ≤ n) has a cluster tilting object if and only if fi /∈ (x, y)
2 for any i. In this case, the
number of non-isomorphic indecomposable summands of basic cluster tilting objects in CM(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[[x1, x2, . . . , xn]] the local ring of
formal power series and m its maximal ideal. We shall need the following standard notions.
Definition 6.4. For f ∈ m2 we denote by J(f) = 〈 ∂f
, . . . , ∂f
〉 its Jacobi ideal. The Milnor number
µ(f) is defined as
µ(f) := dimk(R/J(f)).
24 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
The following lemma is standard (see for example [AGV, GLSh]):
Lemma 6.5. A hypersurface singularity f = 0 is isolated if and only if µ(f) <∞.
Definition 6.6 ([AGV]). Two hypersurface singularities f = 0 and g = 0 are called right equivalent
∼ g) if there exists an algebra automorphism ϕ ∈ Aut(R) such that g = ϕ(f).
Note that f
∼ g implies an isomorphism of k–algebras
R/(f) ∼= R/(g).
The following lemma is straightforward, see for example [GLSh, Lem. 2.10].
Lemma 6.7. Assume f
∼ g, then µ(f) = µ(g).
In what follows, we shall need the next standard result on classification of singularities, see for example
[GLSh, Cor. 2.24].
Theorem 6.8. Let f ∈ m2 be an isolated singularity with Milnor number µ. Then
∼ f + g
for any g ∈ mµ+2.
We shall need the following easy lemma.
Lemma 6.9. Let f = x2 + y2 + p(x, y, z), where
p(x, y, z) = zn + p1(x, y)z
n−1 + · · · + pn(x, y)
is a homogeneous form of degree n ≥ 3. Then
∼ x2 + y2 + zn.
Proof. Write p(x, y, z) = zn + xu + yv for some homogeneous forms u and v of degree n− 1. Then
x2 + y2 + zn + xu+ yv = (x+ u/2)2 + (y + v/2)2 + zn − (u2 + v2)/4.
After a change of variables x 7→ x+ u/2, y 7→ y + v/2 and z 7→ z we reduce f to the form
f = x2 + y2 + zn + h,
where h ∈ m2(n−1) ⊂ mn+1. Note that µ(x2 + y2 + zn) = n− 1, hence by Theorem 6.8 we have
∼ x2 + y2 + zn.
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 = αx+ βy + γz, α, β, γ ∈ k. We put
g(z, t) = a0z
m+1 + a1z
mt+ · · · + am+1t
m+1 + (higher terms).
Then the intersection of H with the singularity defined by the equation x2 + y2 + g(z, t) is given by the
equation f = h+ (higher terms), where
h = x2 + y2 + a0z
m+1 + a1z
m(αx+ βy + γz) + · · · + am+1(αx+ βy + γz)
Now we consider the case m = 1. We have h
∼ x2 + y2 + z2 since any quadratic form can be
diagonalized using linear transformations. By Lemma 6.7, we have µ(h) = µ(x2 + y2 + z2) = 1. Hence
∼ x2 + y2 + z2 by Theorem 6.8.
Next we consider the case m ≥ 2. Assume α ∈ k satisfies a0 + a1α+ · · ·+ am+1α
m+1 6= 0. By Lemma
6.9, we have h
∼ x2 + y2 + zm+1. By Lemma 6.7, we have µ(h) = µ(x2 + y2 + zm+1) = m. Hence
∼ x2 + y2 + zm+1 by Theorem 6.8.
Consequently, x2 + y2 + g(z, t) is cAm. �
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 25
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,m) be an odd-dimensional isolated hypersurface singularity and Γ a 2-CY tilted
algebra coming from CM(R). Then we have the following.
(a) Γ is a symmetric algebra.
(b) All components in the stable AR-quiver of infinite type Γ are tubes of rank 1 or 2.
We now start with our computations of 2-CY tilted algebras coming from minimally elliptic singular-
ities. We first introduce and investigate two classes of algebras, and then show that they are isomorphic
to 2-CY tilted algebras coming from minimally elliptic singularities.
For a quiver Q with finitely many vertices and arrows we define the radical completion k̂Q of the path
algebra kQ by the formula
k̂Q = lim
kQ/ radn(kQ).
The reason we deal with completion is the following: Let Q be a finite quiver, J the ideal of k̂Q
generated by the arrows and I ⊆ J2 a complete ideal such that Λ = k̂Q/I is finite-dimensional.
Lemma 7.2. The ideal I is generated in k̂Q by a minimal system of relations, that is, a set of elements
ρ1, · · · , ρn of I whose images form a k-basis of I/IJ + JI.
The lemma is shown by a standard argument (cf [BMR3, Section 3]). Its analogue for the non complete
path algebra is not always true. For example, for the algebra Λ = B2,2(λ) defined below, the elements
ρ1, · · · , ρn listed as generators for I form a minimal system of relations. So they generate I in k̂Q. They
also yield a k-basis of I ′/I ′J + JI ′
−→ I/IJ + JI, where I ′ = I ∩ kQ and J ′ = J ∩ kQ. But they do not
generate the ideal I ′ of kQ since, as one can show, the quotient kQ/〈ρ1, · · · , ρn〉 is infinite-dimensional.
On the other hand, the ideal I ′ is generated by the preimage ρ1, · · · , ρn of a basis of I
′/I ′J ′ + J ′I ′ if
the quotient kQ/〈ρ1, · · · , ρn〉 is finite-dimensional, since then the ideal 〈ρ1, · · · , ρn〉 contains a power of
J ′. This happens for example for the algebra A2(λ) as defined below, cf. also [Sk, 5.9] and [BS, Th. 1].
We know that for all vertices i, j of Q, we have
dimk ei(I/IJ + JI)ej = dimk Ext
Λ(Si, Sj)
where Si and Sj denote the simple Λ-modules corresponding to the vertices i and j [B]. When Λ is 2-CY
tilted, then
dim Ext1Λ(Sj , Si) ≥ dim Ext
Λ(Si, Sj)
(see [BMR3, KR]). Thus the number of arrows in Q is an upper bound on the number of elements in a
minimal system of relations.
Definition 7.3. (1) For q ≥ 2 and λ ∈ k∗ we write Aq(λ) = k̂Q/I, where
Q = ·ϕ
## α // ·
I = 〈ψα− αϕ, βψ − ϕβ, ϕ2 − βα, ψq − λαβ〉.
If q = 2, then we additionally assume λ 6= 1. (It can be shown that for q ≥ 3 we have Aq(λ) ∼= Aq(1), so
we drop the parameter λ in this case.)
(2) For p, q ≥ 1 and λ ∈ k∗ we write Bp,q(λ) = k̂Q/I, where
Q = ·ϕ
## α // ·
γ // ·
26 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
I = 〈βα− ϕp, γδ − λψq, αϕ− δγα, ϕβ − βδγ, δψ − αβδ, ψγ − γαβ〉.
For p = q = 1 we additionally assume λ 6= 1.
When p = q = 1, the generators ϕ and ψ can be excluded and B1,1(λ) is given by the completion of
the path algebra of the quiver
Q = ·
α // ·
γ // ·
modulo the relations
I = 〈αβα − δγα, αβδ − λδγδ, γαβ − λγδγ, βδγ − βαβ〉.
For (p, q) 6= (1, 1) we have Bp,q(λ) ∼= Bp,q(1). In particular, for p = 1 and q ≥ 2 the algebra is isomorphic
to k̂Q/I, where
Q = ·
α // ·
γ // ·
I = 〈γδ − ψq, αβα − δγα, βαβ − βδγ, δψ − αβδ, ψγ − γαβ〉.
It turns out that the algebras Aq(λ) and Bp,q(λ) are finite dimensional. In order to show this it suffices
to check that all oriented cycles in k̂Q/I are nilpotent.
Lemma 7.4. In the algebra Aq(λ) the following zero relations hold:
αβα = 0, βαβ = 0, αϕ2 = ψ2α = 0, ϕ2β = βψ2 = 0, ϕ4 = 0, ψq+2 = 0.
Proof. We have to consider separately the cases q = 2 and q ≥ 3.
Let q = 2, then we assumed λ 6= 1. We have
αβα = αϕ2 = ψ2α = λ−1αβα,
hence αβα = 0. In a similar way we obtain βαβ = 0. Then αϕ2 = αβα = 0, ϕ2 = βαβα = 0 and the
remaining zero relations follow analogously.
Let q ≥ 3. Then
ψqα = αβα = αϕ2 = ψ2α,
so (1 − ψq−2)ψ2α = 0 and hence
ψ2α = αβα = 0
in k̂Q/I. The remaining zero relations follow similarly. �
Lemma 7.5. We have the following relations in Bp,q(λ):
ϕp+2 = 0, ψq+2 = 0, γαϕ = ψγα = 0, ϕβδ = βδψ = 0.
Moreover, αβ · δγ = δγ · αβ. For q ≥ p ≥ 2 we have
(αβ)2 = (δγ)2 = 0,
for q > p = 1 we have
(αβ)3 = 0, (δγ)2 = 0, (αβ)2 · (δγ) = 0
and for p = q = 1
(αβ)3 = (γδ)3 = 0, (αβ)2 = αβ · δγ = λ(δγ)2.
The proof is completely parallel to the proof of the previous lemma and is therefore skipped. �
The main result of this section is the following
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 27
Theorem 7.6. (a) Let R be a T3,2q+2(λ)–singularity, where q ≥ 2 and λ ∈ k
∗. Then in the triangulated
category CM(R) there exists a cluster tilting object with the corresponding 2-CY-tilted algebra isomorphic
to Aq(λ).
(b)For R = T2p+2,2q+2(λ) the category CM(R) has a cluster tilting object with endomorphism algebra
isomorphic to Bp,q(λ).
Proof. (a) We consider first the case of T3,2q+2(λ).
The coordinate ring of T3,6(λ) is isomorphic to
R = k[[x, y]]/(y(y − x2)(y − λx2)),
where λ 6= 0, 1. Consider Cohen-Macaulay modules M and N given by the two-periodic free resolutions
M = (R
−−−→ R
y(y−λx2)
−−−−−−→ R),
N = (R
y(y−x2)
−−−−−→ R
y−λx2
−−−−→ R).
Then M ⊕N is cluster tilting by Theorem 4.1 or Corollary 6.3. In order to compute the endomorphism
algebra End(M ⊕N), note that
End(M) ∼= k[ϕ]/〈ϕ
where ϕ = (x, x) is an endomorphism of M viewed as a two-periodic map of a free resolution. In End(M)
we have (y, y) = (x, x)2 = ϕ2. Similarly,
End(N) ∼= k[ψ]/〈ψ
4〉, ψ = (x, x), (y, y) = λ(x, x)2 = λψ2,
Hom(M,N) = k2 = 〈(1, y), (x, xy)〉, Hom(N,M) = k2 = 〈(y, 1), (xy, x)〉.
The isomorphism A2(λ) −→ End(M ⊕N) is given by
ϕ 7→ (x, x), ψ 7→ (x, x), α 7→ (1, y), β 7→ (y, 1).
Assume now q ≥ 3 and R = T3,2q+2. By [AGV] we may write
R = k[[x, y]]/((x− y2)(x2 − y2q)).
Consider the Cohen-Macaulay module M ⊕N , where
M = (R
−−−→ R
x2−y2q
−−−−−→ R),
N = (R
(x−y2)(x+yq)
−−−−−−−−−→ R
−−−→ R).
Again, by a straightforward calculation
End(M) ∼= k[ϕ]/〈ϕ4〉, ϕ = (y, y), End(N) ∼= k[ψ]/〈ψq+2〉, ψ = (y, y)
Hom(M,N) = k2 = 〈(1, x+ yq), (y, y(x+ yq))〉,
Hom(M,N) = k2 = 〈(x+ yq, 1), (y(x+ yq), y)〉.
If q ≥ 4 then End(M ⊕N) is isomorphic to k̂Q/I, where
Q = ·ϕ
## α // ·
and the relations are
βα = ϕ2, αβ = 2ψq, αϕ = ψα, ϕβ = βψ
ϕ = (y, y), ψ = (y, y), α = (1, x+ yq), β = (x+ yq, 1).
By rescaling all generators α 7→ 2aα, β 7→ 2bβ, ϕ 7→ 2fϕ, ψ 7→ 2gψ for properly chosen a, b, f, g ∈ Q one
can easily show End(M ⊕N) ∼= Aq.
28 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
The case q = 3 has to be considered separately, since this time the relations are
βα = ϕ2 + ϕ3, αβ = 2ψq, αϕ = ψα, ϕβ = βψ.
We claim that there exist invertible power series u(t), v(t), w(t), z(t) ∈ k[[t]] such that the new generators
ϕ′ = u(ϕ)ϕ, ψ′ = v(ψ)ψ, α′ = αw(ϕ) = w(ψ)α, β′ = βz(ψ) = z(ϕ)β
satisfy precisely the relations of the algebra A3. This is fulfilled provided we have the following equations
in k[[t]]: 
zw = u2(1 + tu)
zw = 2v3
uw = vw
uz = vz.
This system is equivalent to
u(t) = v(t) = (2 − t)−1 =
)2 + . . . )
and hence the statement is proven.
The case of T2p+2,2q+2(λ) is essentially similar. For p = q = 1 we have
R = k[[x, y]]/(xy(x − y)(x− λy)).
Take 
M = (R
−−−→ R
xy(x−λy)
−−−−−−→ R),
N = (R
x(x−y)
−−−−−→ R
y(x−λy)
−−−−−→ R),
K = (R
xy(x−y)
−−−−−→ R
−−−→ R).
By Theorem 4.1 or Corollary 6.3, M ⊕N ⊕K is cluster tilting. Moreover, B1,1(λ) ≃ End(M ⊕N ⊕K).
Let now
R = k[[x, y]]/((xp − y)(xp + y)(yq − x)(yq + x)),
where (p, q) 6= (1, 1) and 
M = (R
−−−→ R
(yq+x)(yq−x)(xp+y)
−−−−−−−−−−−−−−→ R),
N = (R
(xp−y)(xp+y)
−−−−−−−−−→ R
(yq−x)(yq+x)
−−−−−−−−−→ R),
K = (R
(xp−y)(xp+y)(yq+x)
−−−−−−−−−−−−−−→ R
−−−→ R).
By Theorem 4.1 or Corollary 6.3, M ⊕N ⊕K is cluster tilting, and by a similar case-by-case analysis it
can be verified that End(M ⊕N ⊕K) ∼= Bp,q. �
We have seen that the algebras Aq(λ) and Bp,q(λ) are symmetric, and the indecomposable nonpro-
jective modules have τ -period at most 2, hence Ω-period dividing 4 since τ = Ω2 in this case. A direct
computation shows that the Cartan matrix is nonsingular. Note that these algebras appear in Erd-
mann’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
Bia lkowski-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 Coh(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
CLUSTER TILTING FOR ONE-DIMENSIONAL HYPERSURFACE SINGULARITIES 29
singularity. And the canonical algebras are obtained as endomorphism algebras of certain tilting objects
in Coh(X) which are vector bundles.
Note that it is known from work of Dieterich [D], Kahn [Kah], Drozd and Greuel [DG] that mini-
mally 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 Tp,q(λ), see [DG].
Moreover, simply elliptic singularities are tame of polynomial growth and cusp singularities are tame of
exponential growth. Furthermore, the Auslander-Reiten quiver of the corresponding stable categories of
maximal Cohen-Macaulay modules consists of tubes of rank one or two, see [Kah, Th. 3.1] and [DGK,
Cor. 7.2].
It should follow from the tameness of CM(T3,p(λ)) and CM(Tp,q(λ)) that the associated 2-CY tilted
algebras are tame.
We point out that in the wild case we can obtain symmetric 2-CY tilted algebras where the stable
AR-quiver consists of tubes of rank one and two, and most of them should be wild. It was previously
known that there are examples of wild selfinjective algebras whose AR-quivers consist of tubes of rank
one or three [AR].
8. Appendix: 2-CY triangulated categories of finite type
In this section, we consider a more general situation than in section 2. Let k be an algebraically closed
field and C a k-linear connected 2-Calabi-Yau triangulated category with only finitely many indecompos-
able objects. We show that it follows from the shape of the AR quiver of C whether cluster tilting objects
(respectively, non-zero rigid objects) exist in C or not. Let us start with giving the possible shapes of
the AR quiver of C. Recall that a subgroup G of Aut(Z∆) is called weakly admissible if x and gx do not
have a common direct successor for any vertex x in Z∆ and g ∈ G\{1} [XZ, Am].
Proposition 8.1. The AR quiver of C is Z∆/G for a Dynkin diagram ∆ and a weakly admissible
subgroup G of Aut(Z∆) which contains F ∈ Aut(Z∆) defined by the list below. Moreover, G is generated
by a single element g ∈ Aut(Z∆) in the list below.
∆ Aut(Z∆) F g
(An) n : odd Z× Z/2Z (
, 1) (k, 1) (k|n+3
, n+3
is odd)
(An) n : even Z n+ 3 k (k|n+ 3)
(Dn) n : odd Z× Z/2Z (n, 1) (k, 1) (k|n)
(D4) Z× S3 (4, 0) (k, σ) (k|4, σ
k = 1)
(Dn) n : even, n > 4 Z× Z/2Z (n, 0) (k, 0) (k|n) or (k, 1) (k|n,
is even)
(E6) Z× Z/2Z (7, 1) (1, 1) or (7, 1)
(E7) Z 10 1, 2, 5 or 10
(E8) Z 16 1, 2, 4, 8 or 16
In each case, elements in the torsion part of Aut(Z∆) are induced by the automorphism of ∆. The
torsionfree part of Aut(Z∆) is generated by τ except the case (An) with even n, in which case it is
generated by the square root of τ .
Proof. By [XZ] (see also [Am, 4.0.4]), the AR quiver of C is Z∆/G for a Dynkin diagram ∆ and a
weakly admissible subgroup G of Aut(Z∆). Since 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 ∈ 〈g〉, we have the above list. �
Note that, by a result of Keller [Ke], the translation quiver Z∆/G for any Dynkin diagram ∆ and
any weakly admissible group G of Aut(Z∆) is realized as the AR quiver of a triangulated orbit category
Db(H)/g for a hereditary algebra H of type ∆ and some autofunctor g of Db(H).
30 IGOR BURBAN, OSAMU IYAMA, BERNHARD KELLER, AND IDUN REITEN
Theorem 8.2. (1) C has a cluster tilting object if and only if the AR quiver of C is Z∆/g for a Dynkin
diagram ∆ and g ∈ Aut(Z∆) in the list below.
∆ Aut(Z∆) g
(An) n : odd Z× Z/2Z (
, 1) (3|n) or (n+3
(An) n : even Z
(3|n) or n+ 3
(Dn) n : odd Z× Z/2Z (k, 1) (k|n)
(D4) Z× S3 (k, σ) (k|4, σ
k = 1, (k, σ) 6= (1, 1))
(Dn) n : even, n > 4 Z× Z/2Z (k, k) (k|n)
(E6) Z× Z/2Z (7, 1)
(E7) Z 10
(E8) Z 8 or 16
(2) C does not have a non-zero rigid object if and only if the AR quiver of C is Z∆/g for a Dynkin
diagram ∆ and g ∈ Aut(Z∆) in the list below.
∆ Aut(Z∆) g
(An) n : odd Z× Z/2Z −
(An) n : even Z 1
(Dn) n : odd Z× Z/2Z −
(D4) Z× S3 (1, 1)
(Dn) n : even, n > 4 Z× Z/2Z (1, 0)
(E6) Z× Z/2Z (1, 1)
(E7) Z 1
(E8) Z 1 or 2
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 C has a cluster tilting object. For example,
consider the (Dn) case here. Fix a vertex x ∈ Z∆ corresponding to an end point of ∆ which is adjacent
to the branch vertex of ∆. Then the subset {(1, 1)lx | l ∈ Z} of Z∆ is stable under the action of g, and
gives a cluster tilting object of C.
Conversely, assume that C has a cluster tilting object. Then one can check that g is on the list. For
example, consider the (An) 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 2kπ
-radian. Since the center of the regular (n + 3)-polygon is
contained in some triangle or its edge, we have 2kπ
= 2π, 4π
, π or 2π
. Since k|n+ 3 and n is even, we
have k = n+ 3 or n+3
(2) If g is on the list above, then one can easily check that 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 ∆ is rigid. �
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Johannes-Gutenberg Universität Mainz, Fachbereich Physik, Mathematik und Informatik, Institut für Math-
ematik, 55099 Mainz, Germany
E-mail address: burban@mathematik.uni-mainz.de
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan
E-mail address: iyama@math.nagoya-u.ac.jp
UFR de Mathématiques, UMR 7586 du CNRS, Case 7012, Université Paris 7, 2 place Jussieu, 75251 Paris
Cedex 05, France
E-mail address: keller@math.jussieu.fr
Institutt for matematiske fag, Norges Teknisk-naturvitenskapelige universitet, N-7491, Trondheim, Norway
E-mail address: idunr@math.ntnu.no
	Introduction
	Acknowledgment
	1. Main results
	2. Simple hypersurface singularities
	3. Computation with Singular
	4. One-dimensional hypersurface singularities
	5. Link with birational geometry
	6. Application to curve singularities
	7. Examples of 2-CY tilted algebras
	8. Appendix: 2-CY triangulated categories of finite type
	References
