The ubiquity of generalized cluster categories

Abstract

Associated with a finite-dimensional algebra of global dimension at most 2, a generalized cluster category was introduced in Amiot (2009) [1]. It was shown to be triangulated, and 2-Calabi–Yau when it is Hom-finite. By definition, the cluster categories of Buan et al. (2006) [4] are a special case. In this paper we show that a large class of 2-Calabi–Yau triangulated categories, including those associated with elements in Coxeter groups from Buan et al. (2009) [7], are triangle equivalent to generalized cluster categories. This was already shown for some special elements in Amiot (2009) [1].

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arXiv:0911.4819v1 [math.RT] 25 Nov 2009
THE UBIQUITY OF GENERALIZED CLUSTER CATEGORIES
CLAIRE AMIOT, IDUN REITEN, AND GORDANA TODOROV
Abstract. Associated with some finite dimensional algebras of global dimension at most 2,
a generalized cluster category was introduced in [Ami08], which was shown to be triangulated
and 2-Calabi-Yau when it is Hom-finite. By definition, the cluster categories of [BMR+06]
are a special case.In this paper we show that a large class of 2-Calabi-Yau triangulated
categories, including those associated with elements in Coxeter groups from [BIRS09a], are
triangle equivalent to generalized cluster categories. This was already shown for some special
elements in [Ami08].
Contents
Introduction
Notations
1. Background
1.1. Cluster-tilting objects
1.2.Generalized cluster categories
1.3.Jacobian algebras and generalizations
2.A useful triangle
2.1.Description of projective and injective resolutions in mod(¯Bop⊗ B)
2.2.Grading on the quiver
3.Main Theorem
3.1.Global dimension of¯A
3.2.Construction of a triangle functor
3.3.Proof of the equivalence
4. Application to 2-Calabi-Yau categories associated with elements in the Coxeter
group
4.1. Results of [BIRS09a] and [BIRS09b]
4.2. Description of the grading
4.3. Meaning of the grading
5. Examples
5.1.Canonical cluster-tilting object associated to a word
5.2.Example which is not associated with a word
References
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all authors were supported by the Storforsk-grant 167130 from the Norwegian Research Council.
the third author was also supported by the NSA-grant MSPF-08G-228.
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2CLAIRE AMIOT, IDUN REITEN, AND GORDANA TODOROV
Introduction
Throughout the paper k is an algebraically closed field. Let Q be a finite quiver without
oriented cycles. In [BMR+06], the cluster category CQwas defined to be the orbit category
Db(kQ)/τ−[1] where τ is the Auslander-Reiten translation in the bounded derived category
Db(kQ). The category CQ is Hom-finite and triangulated [Kel05], and 2-Calabi-Yau (2-CY
for short), that is, there is a functorial isomorphism DHomCQ(X,Y ) ≃ HomCQ(Y,X[2]), where
D = Homk(−,k). A theory for a special kind of objects, called cluster-tilting objects, was
developed in [BMR+06]. This work was motivated via [MRZ03] by the Fomin-Zelevinsky theory
of cluster algebras [FZ02], where the cluster-tilting objects are the analogs of clusters.
Another category where a similar theory was developed is the category modΛ of finite dimen-
sional modules over a preprojective algebra of Dynkin type [GLS07a], [GLS06]. This category
is Hom-finite and Frobenius. Moreover it is stably 2-CY, that is, its stable category (which is
triangulated) is 2-CY.
Much of the work on cluster categories has been generalized to the setting of 2-CY triangu-
lated categories with cluster-tilting objects, and new results have been proved in the general
setting ( [IY08], [KR08], [KR07], and others). It is of interest to investigate such categories,
both for developing new theory and for providing applications to new classes of cluster algebras.
In particular, it is of interest to find new classes of 2-CY triangulated categories with cluster-
tilting objects. An important class is the stable categories Ewof the Frobenius categories Ew
associated with reduced words w in Coxeter groups [BIRS09a], [GLS08], [GLS07b]. This class
contains both classes of examples discussed above as special cases (see [BIRS09a], [GLS07b]).
A new class of triangulated 2-CY categories was introduced in [Ami08]. They are generalized
cluster categories C ¯ Aassociated with algebras¯A of global dimension at most 2, rather than
global dimension 1. In this case the orbit category Db(¯A)/τ−[1] is not necessarily triangulated,
so C ¯ Ais defined to be its triangulated hull. If C ¯ Ais Hom-finite, then it is triangulated 2-CY
and¯A is a cluster-tilting object in C ¯ A.
A natural question is how the generalized cluster categories are related to the previous classes
of triangulated 2-CY categories. It was already shown in [Ami08] that some classes of categories
Eware equivalent to generalized cluster categories, including CQ and modΛ. This result is
extended to the case of c-sortable words in [AIRT09] with a similar choice for¯A. One of the
main results in this paper is: Each category Ewassociated with a reduced word is equivalent to
a generalized cluster category C ¯ Afor some algebra¯A of global dimension at most 2 (Theorem
4.4).
Actually, we prove our main result in a more general setting: We start with a Frobenius
category E which we assume to be Hom-finite, stably 2-CY and with a cluster-tilting object T.
We assume that the endomorphism algebra EndE(T) is Jacobian and has a grading with certain
properties. From these data we construct an algebra¯A of global dimension ≤ 2 and a triangle
equivalence C ¯ A≃ E (Theorem 3.1) sending the canonical cluster-tilting object¯A of C ¯ Ato the
cluster-tilting object T. The algebra¯A is constructed as the degree zero part of EndE(T), and
we show EndE(T) ≃ EndC¯
equivalence. It is however not known in general if 2-CY categories are equivalent when they
have cluster-tilting objects whose endomorphism algebras are isomorphic. The only general
result known of this type is that if the quiver Q of EndC(T) has no oriented cycles, where T
is a cluster-tilting object in an algebraic 2-CY category, then C is triangle equivalent to the
cluster category CQ[KR08]. A crucial step in this paper for proving the equivalence C ¯ A≃ E is
the construction of a triangle functor C ¯ A→ E sending¯A to T. This is done by constructing a
A(¯A) (Proposition 3.5), which is an important step in the proof of the

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THE UBIQUITY OF GENERALIZED CLUSTER CATEGORIES3
triangle functor Db(¯A) → E with the strong use of the Frobenius structure of E and then using
the universal property of the projection π ¯ A: Db(¯A) → C ¯ Afrom [Kel05] and [Ami08]. It will be
important to also deal with the endomorphism algebra EndE(T) of the cluster-tilting object T
in the Frobenius category, which we assume to be a graded frozen Jacobian algebra (see section
1 for definitions) with potential homogeneous of degree 1. Theorem 3.1 applies in particular to
the categories Ewassociated to any element of the Coxeter group (see section 4).
The paper is organized as follows. In section 1 we recall some background material on
cluster-tilting objects in 2-CY categories, on generalized cluster categories from [Ami08] and on
Jacobian algebras from [DWZ08], together with the generalization to frozen Jacobian algebras
from [BIRS09b].
In section 2 we construct a special triangle (Proposition 2.8) which is useful for our construc-
tion of a functor from C ¯ Ato E.
Section 3 is devoted to the proof of the equivalence from C ¯ Ato E (Theorem 3.1) . We first
show that the global dimension of¯A is at most 2. Then we construct our triangle functor
from C ¯ Ato E using the special triangle from section 2, together with a universal property from
[Ami08]. Finally we show that our functor is an equivalence by using a criterion from [KR08].
In section 4 we apply the main theorem to prove that for any reduced word w, the 2-CY
triangulated category Ewis triangle equivalent to some generalized cluster category, which was
our original motivation (Theorem 4.4).
In section 5 we give two examples to illustrate our theorems. The first one is an application
of Theorem 4.4. In the second one we use Theorem 3.1 to construct a triangle equivalence from
a generalized cluster category C ¯ Ato a category Ewwhich sends the canonical cluster-tilting
object¯A to a cluster-tilting object T of Ew, where T is not associated to a reduced expression
of w.
Notations. By triangulated category we mean k-linear triangulated category satisfying the
Krull-Schmidt property. For all triangulated categories we will denote the shift functor by [1].
By Frobenius category we mean an exact k-category with enough projectives and injectives
and where projectives and injectives coincide. For a finite-dimensional k-algebra A, we denote
by modA the category of finite-dimensional right A-modules. Let D be the usual duality
Homk(?,k). The tensor product −⊗−, when not specified, will be over the ground field k. For
a quiver Q we will denote by Q0its set of vertices, by Q1its set of arrows, by s the source map
and by t the target map.
1. Background
In this section we collect some background material relevant for this paper.
1.1. Cluster-tilting objects. Let C be a k-category which is Hom-finite, that is, has finite
dimensional homomorphism spaces. Assume that C is either Frobenius stably 2-CY (that is, its
stable category is 2-CY) or triangulated 2-CY. Then an object T in C is said to be cluster-tilting
if
(i) T is rigid, i.e. Ext1
(ii) Ext1
Note that when C is Frobenius stably 2-CY, then any indecomposable projective-injective
module is a summand of every cluster-tilting object. The finite dimensional algebras EndC(T),
where C is triangulated 2-CY, are called 2-CY-tilted algebras.
C(T,T) = 0, and
C(T,X) = 0 implies that X is a summand of a finite direct sum of copies of T.

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4CLAIRE AMIOT, IDUN REITEN, AND GORDANA TODOROV
Assume that T = T1⊕ ... ⊕ Tnis a cluster-tilting object in a triangulated 2-CY category
C, where the Tiare indecomposable and pairwise non isomorphic. Then for each i = 1,...,n
there is a unique indecomposable object T∗
a cluster-tilting object [BMR+06],[IY08]. The new object T∗is called the mutation of T at Ti.
If T = T1⊕ ... ⊕ Tnis a cluster-tilting object in a Frobenius stably 2-CY category, we can
only mutate the Tiwhich are not projective-injective.
When T∗
inot isomorphic to Ti, such that T∗= T/Ti⊕ T∗
iis
iis defined, there are exchange sequences if C is Frobenius
0
??T∗
i
f
??B
g
??Ti
??0
and
0
??Ti
f′
??B′
g′
??T∗
i
??0
or exchange triangles if C is triangulated
T∗
i
f
??B
g
??Ti
??T∗
i[1]
and
Ti
f′
??B′
g′
??T∗
i
??Ti[1]
where f, f′are minimal left add(T/Ti)-approximations and g, g′are minimal right add(T/Ti)-
approximations. These sequences (or triangles) play an important role in the categorification
of cluster algebras.
There is also a related kind of sequences investigated in [IY08].
Proposition 1.1. [Iyama-Yoshino] Let C be a Hom-finite Frobenius stably 2-CY category with
a cluster-tilting object T = T1⊕...⊕Tn. For each i = 1,...,n, if Tiis not projective-injective,
there are exact sequences
0
??T+
i
f
??E
g
??Ti
??0
and
0
??Ti
f′
??E′
g′
??T+
i
??0
for some indecomposable object T+
(resp. in add(T/Ti⊕T+
T+
i in C, such that g (resp. g′) is right almost split in add(T)
i)) and f′(resp. f) is left almost split in add(T) (resp. in add(T/Ti⊕
i)).
The induced sequence 0
sequence associated with Ti.
There is the corresponding result when C is triangulated. For any direct summand Tiof a
cluster-tilting object T, there are triangles
??Ti
f′
??E′
fg′
??E
g
??Ti
??0 is called the 2-almost split
T+
i
f
??E
g
??Ti
??T+
i[1]
and
Ti
f′
??E′
g′
??T+
i
??Ti[1] )
where the maps f,g,f′and g′are almost split. For cluster categories, it was shown in [BMR+06]
that these triangles coincide with the exchange triangles. More generally, they clearly coincide
if and only if there are no loops in the quiver of EndC(T).
Using the existence of 2-almost split sequences, we can construct minimal projective and
injective resolutions of simple modules over EndC(T).
Proposition 1.2. Let C be a Hom-finite Frobenius stably 2-CY category with a cluster-tilting
object T = T1⊕ ... ⊕ Tn. Let B := EndC(T) be the endomorphism algebra of T, and let Q be
the quiver of B ≃ kQ/I. For each i = 1,...,n, such that Tiis not projective-injective, denote
by Sithe simple B-module HomC(T,Ti)/Rad(HomC(T,Ti)). Then we have exact sequences in
modB:
(b)???
0
??eiB
b∈Q1,s(b)=iet(b)B
(rab)???
a∈Q1,t(a)=ies(a)B
(a)
??eiB
??Si
??0

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THE UBIQUITY OF GENERALIZED CLUSTER CATEGORIES5
0
??Si
??D(Bei)
(b)???
b∈Q1,s(b)=iD(Bet(b))
(r′
ab)???
a∈Q1,t(a)=iD(Bes(a))
(a)??D(Bei)
??0 ,
where the sets {arab| a,b ∈ Q1,t(a) = s(b) = i,t(b) = j} and {r′
are bases of eiIej.
abb| s(b) = t(a) = i,s(a) = j}
Proof. Applying the functor HomC(T,−) to the 2-almost-split sequence
0
??Ti
f′
??E′
fg′
??E
g
??Ti
??0
we get the following exact sequence of B-modules
0
??HomC(T,Ti)
??HomC(T,E′)
??HomC(T,E)
??HomC(T,Ti)
??Si
??0
which is a minimal projective resolution of the simple B-module Si.
Let Q be the quiver of B, and B ≃ kQ/I. Since g is right almost split in add(T), we have
?
a∈Q1| t(a)=i
E ≃Ts(a)
and
HomC(T,g) ≃ (a){a∈Q1| t(a)=i}.
Since f′is left almost split in add(T), we have
E′≃
?
b∈Q1| s(b)=i
Tt(b)
and
HomC(T,f′) ≃ (b){b∈Q1| s(b)=i}.
For a,b ∈ Q1with t(a) = i and s(b) = i, let rab: et(b)B → es(a)B be the map induced by
fg′:?
induces a minimal projective resolution of the simple Si, the set {arab| a,b ∈ Q1,t(a) = s(b) =
i,t(b) = j} is a basis of the set of relations eiIej.
To get the other sequence of the proposition, we apply the functor DHomC(−,T) to the
2-almost split sequence associated to Ti, and we proceed similarly.
b∈Q1| s(b)=iTt(b)→?
a∈Q1| t(a)=iTs(a). Since the 2-almost split sequence associated to Ti
?
1.2. Generalized cluster categories. Let Λ be a finite dimensional k-algebra of global di-
mension ≤ 2. We denote by Db(Λ) the bounded derived category of finitely generated Λ-
modules. It has a Serre functor that we denote by S, which coincides with τ[1].
The category CΛhas been defined in [Ami08] as the triangulated hull of the orbit category
Db(Λ)/S[−2]. There is a triangle functor
πΛ: Db(Λ)
?? ??Db(Λ)/S[−2]??
??CΛ.
When the endomorphism algebra EndCΛ(πΛ(Λ)) is finite dimensional, the category CΛis called
the generalized cluster category and we have the following result:
Theorem 1.3 (Theorem 4.10 of [Ami08]). Let Λ be a finite dimensional algebra of global
dimension ≤ 2, and assume that the endomorphism algebra EndCΛ(πΛ(Λ)) is finite dimensional.
Then CΛis a Hom-finite, 2-Calabi-Yau category and πΛ(Λ) ∈ CΛis a cluster-tilting object.
There is the following criterion for constructing triangle functors from a generalized cluster
category CΛto some stable category E. It can be deduced from the universal property of πΛ
given in subsection 4.1 of [Ami08] (see also section 9 of [Kel05] or appendix of [IO09] for more
details). The next proposition is a key-step in the process of constructing the equivalence of
the main theorem of this paper.