DOI 10.1007/s10801-007-0071-6
A geometric model for cluster categories of type D
nRalf Schiffler
Received: 30 August 2006 / Accepted: 27 March 2007 / Published online: 26 April 2007
© Springer Science+Business Media, LLC 2007
Abstract We give a geometric realization of cluster categories of type Dn using a polygon withnvertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
Keywords Cluster category·Triangulated surface·Punctured polygon· Elementary move
1 Introduction
Cluster categories were introduced in [6] and, independently, in [11] for type An, as a means for better understanding of the cluster algebras of Fomin and Zelevin- sky [15,16]. Since then cluster categories have been the subject of many investiga- tions; see, for instance, [1,2,7–10,12–14,21–23,26].
In the approach of [6], the cluster categoryCAis defined as the quotientDbA/Fof the derived categoryDbAof a hereditary algebraAby the endofunctorF =τD−b1
A[1], whereτDbA is the Auslander–Reiten translation, and [1]is the shift. On the other hand, in the approach of [11], which is only valid in typeAn, the cluster category is realized by an ad-hoc method as a category of diagonals of a regular polygon withn+3 vertices. The morphisms between diagonals are constructed geometrically using so-called elementary moves and mesh relations. In that realization, clusters are in one-to-one correspondence with triangulations of the polygon, and mutations are given by flips of diagonals in the triangulation. Recently, Baur and Marsh [5] have generalized this model tom-cluster categories of typeAn.
R. Schiffler (
)Department of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst, MA 01003-9305, USA
e-mail: schiffler@math.umass.edu
In this paper, we give a geometric realization of the cluster categories of typeDn in the spirit of [11]. The polygon with(n+3)vertices has to be replaced by a polygon withnvertices and one puncture in the center, and instead of looking at diagonals, which are straight lines between two vertices, one has to consider homotopy classes of paths between two vertices, which we will call edges. This punctured polygon model has appeared recently in the work of Fomin, Shapiro, and Thurston [17] on the relation between cluster algebras and triangulated surfaces. Let us point out that they work in a vastly more general context, and the punctured polygon is only one example of their theory. We define the cluster category by an ad-hoc method as the category of (tagged) edges inside the punctured polygon. Morphisms are defined us- ing so-called elementary moves and mesh relations, which are generalizations of the elementary moves and mesh relations of [11]. Our main results are the equivalence of the category of tagged edges and the cluster category of [6], see Theorem4.3, and the realization of the dimension of Ext1of tagged edges as the number of crossings between the same tagged edges, see Theorem5.3.
The article is organized as follows. After a brief preliminary section, in which we fix the notation and recall some concepts needed later, Sect.3is devoted to the defi- nition of the categoryCof tagged edges. In Sect.4, we show the equivalence of this category and the cluster category and, in Sect.5, we study Ext1of indecomposable objects inC. As an application, in Sect.6, we describe Auslander–Reiten triangles, tilting objects and exchange relations and show a geometric method to construct the Auslander–Reiten quiver of any cluster tilted algebra of typeDn, using a result of [7].
2 Preliminaries
2.1 Notation
Letkbe an algebraically closed field. IfQis a quiver, we denote byQ0 the set of vertices and byQ1the set of arrows ofQ. The path algebra ofQoverkis denoted bykQ. It is of finite representation type if there is only a finite number of isoclasses of indecomposable modules. By Gabriel’s theoremkQis of finite representation type if and only ifQis a Dynkin quiver, that is, the underlying graph ofQis a Dynkin diagram of typeAn, Dn, orEn[18].
IfA is an algebra, we denote by modAthe category of finitely generated right A-modules and by indAthe full subcategory whose objects are a full set of repre- sentatives of the isoclasses of indecomposableA-modules. LetDbA=Db (modA) denote the derived category of bounded complexes of finitely generatedA-modules.
For further facts about mod(A)andDbA, we refer the reader to [3,4,19,25].
IfAis an additivek-category, then its additive hull⊕Ais defined as follows: the objects of⊕Aare direct sums of objects inA, morphisms⊕iXi→ ⊕jYj are given componentwise by morphismsXi→Yj ofA, and the composition of morphisms is given by matrix multiplication.
2.2 Translation quivers
Following [24], we define a stable translation quiver (Γ , τ ) to be a quiver Γ = (Γ0, Γ1) without loops together with a bijection τ (the translation) such that the
number of arrows fromy→x is equal to the number of arrows fromτ x→y for anyx, y∈Γ0. Given a stable translation quiver(Γ , τ ), a polarization ofΓ is a bijec- tionσ:Γ1→Γ1such thatσ (α):τ x→yfor every arrowα:y→x∈Γ1. IfΓ has no multiple arrows, then there is a unique polarization.
Given a quiver Q, one can construct a stable translation quiverZQas follows:
(ZQ)0=Z×Q0; the number of arrows inQfrom(i, x)to(j, y)equals the number of arrows inQfromx toy ifi=j and equals the number of arrows in Qfromy tox ifj=i+1, and there are no arrows otherwise. The translationτ is defined by τ ((i, x))=(i−1, x).
The path category of(Γ , τ )is the category whose objects are the vertices ofΓ, and givenx, y∈Γ0, thek-space of morphisms fromx toyis given by thek-vector space with basis the set of all paths fromx toy. The composition of morphisms is induced from the usual composition of paths.
The mesh ideal in the path category ofΓ is the ideal generated by the mesh rela- tions
mx=
α:y→x
σ (α)α. (1)
The mesh categoryM(Γ , τ )of(Γ , τ )is the quotient of the path category of(Γ , τ ) by the mesh ideal. In general, the mesh category depends on the polarization. In this article, however, we only consider mesh categories with unique polarizations.
Important examples of translation quivers are the Auslander–Reiten quivers of the derived categories of hereditary algebras of finite representation type. We shall need the following proposition.
Proposition 2.1 LetQbe a Dynkin quiver. Then
1. For any quiverQ of the same Dynkin type asQ, the derived categoriesDbkQ andDbkQare equivalent.
2. The Auslander–Reiten quiver ofDbkQisZQ.
3. The category indDbkQis equivalent to the mesh category ofZQ.
Proof See [20, I.5].
2.3 Cluster categories
LetCDnbe the cluster category of typeDn, see [6]. By definition,CDnis the quotient of the derived categoryDbAof a hereditary algebraAof typeDnby the endofunctor F =τD−b1
A[1], whereτDbAis the Auslander–Reiten translation inDbA, and[1]is the shift. Thus the objectsM˜ ofCDnare the orbitsM˜ =(FiM)i∈Zof objectsM∈DbA and HomCDn(M,˜ N )˜ =
i∈ZHomDbA(M, FiN ). Let us denote the vertices of the
Dynkin diagram ofDnas follows:
n
1 2 . . . (n−3) (n−2)
(n−1) and, for convenience, let us choose the algebraAto be the path algebrakQof the quiver
n
Q= 1 2 . . . (n−3) (n−2)
(n−1).
Since, by Proposition2.1, the Auslander–Reiten quiver ofDbAis the stable transla- tion quiverZQ, the labels of Q0 induce labels on the vertices ofZQas usual (see Fig.1):
(ZQ)0=
(i, j )|i∈Z, j∈Q0
=Z× {1, . . . , n}.
Moreover, we can identify the indecomposable objects of DbA with the vertices ofZQ. LetP1be the indecomposable projective module corresponding to the vertex 1∈Q0. Then, by defining the position ofP1to be(1,1), we have a bijection
pos:indDbA→Z× {1, . . . , n}.
In other terms, for M ∈indDbA, we have pos(M)=(i, j ) if and only if M = τD−ib
APj, wherePj is the indecomposable projectiveA-module at vertexj. The inte- gerj∈ {1, . . . , n}is called the level ofMand is denoted by level(M).
Fig. 1 Labels of the Auslander–Reiten quiver ofDbAifn=6
There is a quiver automorphism ZQ→ZQthat fixes all indecomposable ob- jects M with level(M) < n−1 and exchanges the indecomposable objects at po- sitions (i, n−1)and (i, n) for all i∈Z. IfM is an indecomposable object with level(M)≥n−1, then letM− be its image under this quiver automorphism. The structure of the module category ofAand of its derived category is well known. In particular, we have the following result.
Lemma 2.2 LetM∈indDbA.
1. Ifnis even, thenM[1] =τD−bn+1
A M.
2. Ifnis odd, then M[1] =
τ−n+1
DbA M if level(M)≤n−2, τ−n+1
DbA M− if level(M)∈ {n−1, n}.
Proof It suffices to prove the statement in the case whereM=P is an indecom- posable projectiveA-module. LetνA denote the Nakayama functor of modA. Then P[1] =τD−1b
AνAP. Now the statement follows from [19, Proposition 6.5].
We also define the position of indecomposable objects of the cluster categoryCA= DbA/F. The set modA A[1] is a fundamental domain for the cluster category (hereA[1]denotes the first shift of all indecomposable projectiveA-modules). Then, forX˜ ∈indCA, we define pos(X)˜ to be pos(X), whereXis the unique element in the fundamental domain such thatX˜ is the orbit ofX. Thus pos(X)˜ ∈ {1,2, . . . , n}2.
We recall now a well-known result that uses the position coordinates to de- scribe the dimension of the space of morphisms between indecomposableCA-objects.
Let M, N ∈indCA be such that pos(M)=(1, m) and pos(N )=(i, j ). The next proposition follows from the structure of the mesh category.
Proposition 2.3 The dimension of HomCA(M, N )is 0,1, or 2 and it can be charac- terized as follows:
1. Ifm≤n−2, then dim HomCA(M, N ) =0 if and only if
1≤i≤m & i+j≥m+1,
or m+1≤i≤n−1 & n≤i+j≤n+m−1.
2. Ifm∈ {n−1, n}, definem to be the unique element in{n−1, n} \ {m}. Then dim HomCA(M, N ) =0 if and only if
⎧⎨
⎩
2≤i≤n−1 & i+j≥n & j ≤n−2, or 1≤i≤n−1 & j=m & iis odd, or 1≤i≤n−1 & j=m & iis even.
Moreover, dim HomCA(M, N )=2 if and only if
2≤m≤n−2 & 2≤i≤m & 2≤j≤n−2 & i+j ≥n.
To illustrate this statement, we give an example wheren=6 andM is on posi- tion(1,3). The following picture shows the Auslander–Reiten quiver ofCA(without arrows) and where the label on any vertexNis dim HomCA(M, N ). If this number is zero, we write a dot instead of 0. The two underlined vertices should be identified, they indicate the position ofM.
. 1 1 1 . . . 1 1 1 .
. . 1 1 2 1 2 1 1 . . . . . 1 1 2 1 2 1 1 . .
1 1 2 1 1 . 1 1 2 1 1
. 1 1 1 1 . . 1 1 1 1 .
. 1 . 1 . . . 1 . 1 .
3 The category of tagged edges
3.1 Tagged edges
Consider a regular polygon withn≥3 vertices and one puncture in its center.
Ifa =bare any two vertices on the boundary, then letδa,bdenote a path along the boundary froma tobin counterclockwise direction which does not run through the same point twice. Letδa,adenote a path along the boundary fromatoain counter- clockwise direction which goes around the polygon exactly once and such thatais the only point through whichδa,aruns twice. Fora =b, let|δa,b|be the number of vertices on the pathδa,b(includingaandb), and let|δa,a| =n+1. Two verticesa, b are called neighbors if|δa,b| =2 or|δa,b| =n, andbis the counterclockwise neighbor ofaif|δa,b| =2.
An edge is a triple(a, α, b)wherea andbare vertices of the punctured polygon andαis a path fromatobsuch that
(E1) αis homotopic toδa,b.
(E2) Except for its starting pointaand its endpointb, the pathαlies in the interior of the punctured polygon.
(E3) αdoes not cross itself, that is, there is no point in the interior of the punctured polygon through whichαruns twice.
(E4) |δa,b| ≥3.
Note that condition (E1) implies that ifa=bthenαis not homotopic to the constant path, and condition (E4) means thatbis not the counterclockwise neighbor ofa.
Two edges(a, α, b), (c, β, d)are equivalent ifa=c, b=d, andαis homotopic toβ. LetEbe the set of equivalence classes of edges. Since the homotopy class of an edge is already fixed by condition (E1), an element ofEis uniquely determined by an ordered pair of vertices(a, b). We will therefore use the notationMa,bfor the equivalence class of edges(a, α, b)inE.
Define the set of tagged edgesEby E=
Ma,b|Ma,b∈E, = ±1 and =1 ifa =b .
Ifa =b, we will often drop the exponent and writeMa,b instead ofMa,b1 . In other terms, for any ordered pair(a, b)of vertices witha =bandbnot the counterclock- wise neighbor ofa, there is exactly one tagged edgeMa,b∈E, and, for any vertexa, there are exactly two tagged edgesMa,a−1andMa,a1 . A simple count shows that there are (n−n!2)!−n+2n=n2elements inE. These tagged edges will correspond to the indecomposable objects in the cluster category.
Remark 3.1 Our definition of (tagged) edges is for the punctured polygon only. It is an adapted version of the much more general definitions of (tagged) arcs in [17].
Remark 3.2 In pictures, the tagged edgesMa,athat start and end at the same vertexa will not be represented as loops around the puncture but as lines from the vertexato the puncture. If = −1, the line will have a tag on it and if =1, there will be no tag. The reason for this is the definition of the crossing number in the next section.
3.2 Crossing numbers
LetM=Ma,b andN=Mc,d be in E, and let0denote the interior of the punc- tured polygon. The crossing numbere(M, N )ofMandNis the minimal number of intersections ofMa,bandMc,din0. More precisely,
1. Ifa =bandc =d, then e(M, N )=min
Card
α∩β∩0
|(a, α, b)∈Ma,b, (c, β, d)∈Mc,d . 2. Ifa=bandc =d, letαbe the straight line fromato the puncture. Then
e(M, N )=min Card
α∩β∩0
|(c, β, d)∈Mc,d . 3. Ifa =bandc=d, letβbe the straight line fromcto the puncture. Then
e(M, N )=min Card
α∩β∩0
|(a, α, b)∈Ma,b . 4. Ifa=bandc=d, that is,M=Ma,aandN=Mc,c , then
e(M, N )=
1 ifa =cand = , 0 otherwise.
Fig. 2 Examples of elementary moves
Note thate(M, M)=0. We say thatMcrossesN ife(M, N ) =0. The next lemma immediately follows from the construction.
Lemma 3.3 For anyM, N∈indC, we havee(M, N )∈ {0,1,2}. 3.3 Elementary moves
Generalizing a concept from [11], we will now define elementary moves, which will correspond to irreducible morphisms in the cluster category.
An elementary move sends a tagged edge Ma,b ∈E to another tagged edge Ma,b∈E (i.e., it is an ordered pair(Ma,b, Ma,b)of tagged edges) satisfying cer- tain conditions which we will define in four separate cases according to the relative position of a andb. Let c(respectivelyd) be the counterclockwise neighbor ofa (respectivelyb).
1. If|δa,b| =3, then there is precisely one elementary moveMa,b→Ma,d.
2. If 4≤ |δa,b| ≤n−1, then there are precisely two elementary movesMa,b→Mc,b andMa,b→Ma,d.
3. If|δa,b| =n, thend=a,and there are precisely three elementary movesMa,b→ Mc,b,Ma,b→Ma,a1 , andMa,b→Ma,a−1.
4. If|δa,b| =n+1, thena=b, and there is precisely one elementary moveMa,a→ Mc,a.
Some examples of elementary moves are shown in Fig.2.
3.4 A triangulation of typeDn
A triangulation of the punctured polygon is a maximal set of noncrossing tagged edges.
Lemma 3.4 Any triangulation of the punctured polygon hasnelements.
Proof We prove the lemma by induction onn. If n=3, there exist, up to rotation, symmetry, and changing tags, three different triangulations (each having 3 tagged edges):
Suppose now thatn >3, and letT be a triangulation. If T contains an edge of the formM=Ma,bwitha =b, then this edge cuts the punctured polygon into two parts. Let pbe the number of vertices of the punctured polygon that lie in the part containing the puncture. Thenn−p+2 vertices lie in the other part, sincea andb lie in the both. The setT\ {M}defines triangulations on both parts. Ifp≥3, then, by induction, the number of edges inT\ {M}that lie in the part containing the puncture is equal top. Ifp=2, then this number is also equal top(in this case, the 2 edges are either(Ma,a, Ma,a− ),(Ma,a, Mc,c), or(Mc,c, Mc,c− )). Thus, in all cases, the number of edges inT \ {M}that lie in the part containing the puncture is equal top. On the other hand, the number of edges inT \ {M}that lie in other part equals the number of edges in a triangulation of an(n−p+2)-polygon, and this number is equal to n−p−1. So the cardinality ofT\ {M}isn−1, henceT hasnelements.
IfT does not contain an edge of the formMa,bwitha =b, thenT contains exactly one elementMa,afor each vertexa, and all edges have the same tag =1 or = −1.
Clearly,T hasnelements.
We now construct a particular triangulation of the punctured polygon. Leta1be a vertex on the boundary of the punctured polygon. Denote bya2the counterclock- wise neighbor ofa1and, recursively, letakbe the counterclockwise neighbor ofak−1 for anyksuch that 2≤k≤n. Then the triangulationT =T (a1)is the set of tagged edges
T = Ma1
1,a1, Ma−1
1,a1
∪ {Ma1,ak|3≤k≤n}. Here is an example forT in the casen=8.
From the construction it is clear that the elementary moves between the elements ofT are precisely the elementary movesMa1,ak→Ma1,ak+1 for allk with 3≤k≤ n−1 andMa1,an→Ma
1,a1 with = ±1. We can associate a quiver QT toT as follows: The vertices ofQT are the elements ofT, and the arrows are the elementary moves between the elements ofT. By the above observation,QT is the following
quiver of typeDn(forn=3,setD3=A3):
Ma1
1,a1
Ma1,a3 Ma1,a4 · · · Ma1,an
Ma−1
1,a1.
Note that QT is isomorphic to the quiverQ that we have chosen to construct the cluster category in Sect.2.3.
3.5 Translation
We define the translationτto be the following bijectionτ :E→E: LetMa,b∈E, and leta(respectivelyb) be the clockwise neighbor ofa(respectivelyb).
1. Ifa =b,thenτ Ma,b=Ma,b.
2. Ifa=b,thenτ Ma,a=Ma−,a for = ±1.
The next lemma immediately follows from the definition ofτ. Lemma 3.5
(1) Ifnis even, thenτn=id.
(2) Ifnis odd, thenτnMa,b=
Ma,b ifa =b,
Ma,a− ifa=b.
We will need the following lemma when we define the category of tagged edges.
Lemma 3.6 LetMa,bλ , Mc,d∈E. Then there is an elementary moveMa,bλ →Mc,d if and only if there is an elementary moveτ Mc,d→Ma,bλ .
Proof LetMc,d=τ Mc,d. Suppose that there is an elementary moveMa,bλ →Mc,d. Then eithera=corb=d.
1. Ifa =c, then b=d and |δc,d| = |δc,d| = |δa,b| +1. In particular, a =b and
|δc,d| ≥4. Now, if 4 ≤ |δc,d| ≤n, then c =d and, by 3.3.2, 3.3.3, there is an elementary move τ Mc,d=Mc,b →Mc,b=Ma,b. On the other hand, if
|δc,d| =n+1,thenc=d=bandais the counterclockwise neighbor ofb,and thus, by3.3.4, there is an elementary moveτ Mc,d=Mb,b− →Ma,b.
2. Ifb=d,thena=cand|δc,d| = |δc,d| = |δa,b| −1. In particular,|δc,d| ≤n.
Now, if 3≤ |δc,d| ≤n−1, then, by 3.3.1,3.3.2, there is an elementary move τ Mc,d=Ma,d →Ma,b. On the other hand, if|δc,d| =n,thena=b=d andb is the counterclockwise neighbor ofd, and thus, by3.3.3, there is an elementary moveτ Mc,d=Ma,d→Ma,bλ forλ= ±1.
Thus, there is an elementary moveτ Mc,d→Ma,bλ . By symmetry, the other implica-
tion also holds.
3.6 The categoryC
We will now define ak-linear additive categoryCas follows. The objects are direct sums of tagged edges inE. By additivity, it suffices to define morphisms between tagged edges. The space of morphisms from a tagged edge Mto a tagged edge N is a quotient of the vector space overkspanned by sequences of elementary moves starting atMand ending atN. The subspace which defines the quotient is spanned by the so-called mesh relations.
For any tagged edgeX∈E, we define the mesh relation mX=
α
(α)α,
where the sum is over all elementary movesY→α Xthat send a tagged edgeY toX, andτ X→(α)Y denotes the elementary move given by Lemma3.6. In other terms, if there is only oneY ∈Esuch that there is an elementary moveY→α X,then the com- position of morphismsτ X(α)→Y →α Xis zero, and if there are severalY1, . . . , Yp∈E with elementary moves Yi →αi X, then the sum of all compositions of morphisms τ X(α→i)Yi
αi
→X,i=1, . . . , n, is zero.
More generally, a mesh relation is an equality between sequences of elementary moves which differ precisely by such a relationmX.
We can now define the set of morphisms from a tagged edgeMto a tagged edgeN to be the quotient of the vector space over k spanned by sequences of elementary moves fromMtoNby the subspace generated by mesh relations.
4 Equivalence of categories
In this section, we will prove the equivalence of the categoryCand the cluster cate- goryCA.
4.1 Translation quiver
First we construct a stable translation quiver in our situation. Let Γ =(Γ0, Γ1)be the following quiver: The set of vertices is the setZ×E,whereE is the set of tagged edges of Sect.3.1. In order to define the set of arrows, let us fix a vertexa1 on the boundary of the punctured polygon, and letT =T (a1)be the triangulation of Sect. 3.4. GivenM, N∈E, there is an arrow(i, M)→(i, N )inΓ1if there is an elementary moveM→N and eitherN /∈T orM andN are both inT, and there is an arrow(i, M)→(i+1, N )inΓ1if there is an elementary moveM→N and N∈T andM /∈T. Note thatΓ has no loops and no multiple arrows.
The translationτ:E→Edefined in Sect.3.5induces a bijectionτΓ onΓ0by τΓ(i, M)=
(i, τ M) ifM /∈T , (i−1, τ M) ifM∈T.
Then from Lemma 3.6 it follows that (Γ , τΓ) is a stable translation quiver. Let M(Γ , τΓ)denote the mesh category of(Γ , τΓ).
Now consider the vertex subset 0×T ⊂Γ0. This set contains exactly one element of eachτΓ-orbit ofΓ. Moreover, the full subquiver induced by 0×T is the quiver Q=QT of Sects.2.3and3.4, a quiver of typeDn, and from Sect.3.3it follows that Γ is the translation quiverZQ. LetAbe the path algebra ofQ, as before, and let us use the notation( )−of Sect.2.3. Then, by Proposition2.1, we have the following result.
Proposition 4.1 There is an equivalence of categories ϕ:M(Γ , τΓ)−→∼ indDbA such that
1. ϕmaps the elements of 0×T to the projectiveA-modules.
2. ϕ◦τΓ =τDbA◦ϕ.
3. If level(ϕ(, Ma,b))=j,then
j = |δa,b| −2 ifa =b and j∈ {n−1, n} ifa=b.
4. IfM=(, Ma,a)andM−=(, Ma,a− ),then ϕ(M)−
=ϕ(M−).
The equivalenceϕinduces an equivalence of categoriesϕ: ⊕M(Γ , τΓ)→∼ DbA, where⊕M(Γ , τΓ)denotes the additive hull ofM(Γ , τΓ). In particular,⊕M(Γ , τΓ) has a triangulated structure such thatϕis an equivalence of triangulated categories.
Letρbe the endofunctor of⊕M(Γ , τΓ)given by the full counterclockwise rotation of the punctured polygon. Thus, if(i, M)∈M(Γ , τΓ),thenρ(i, M)=(i+1, M).
The next result follows from Lemma3.5.
Lemma 4.2 (1) Ifnis even, thenρ=τ−n
Γ .
(2) Ifnis odd andM=(, Ma,b)witha =b,thenρ(M)=τ−n
Γ (M).
(3) Ifnis odd andM=(, Ma,a)andM−=(, Ma,a− ), thenρ(M)=τ−n
Γ (M−).
4.2 Main result
The category of tagged edges C of Sect. 3.6 is the quotient category of the trian- gulated category ⊕M(Γ , τΓ) by the endofunctor ρ. Hence the objects of C are the ρ-orbits M˜ =(ρiM)i∈Z of objects M in ⊕M(Γ , τΓ), and HomC(M,˜ N )˜ =
⊕iHom(M, ρiM), where the Hom-spaces on the right are taken in the category
⊕M(Γ , τΓ). We are now able to show our main result.
Theorem 4.3 There is an equivalence of categories
ϕ:C−→CA
between the category of tagged edgesCand the cluster categoryCAwhich sends the ρ-orbit of the triangulation 0×T to theF-orbit of the projectiveA-modules.
Proof Let ϕ : ⊕M(Γ , τΓ)→DbA be the equivalence of Sect. 4.1. Since C =
⊕M(Γ , τΓ)/ρandCA=DbA/F, we only have to show thatϕ(ρ(M))=F (ϕ(M)) for anyM∈M(Γ , τΓ).
Ifnis even, thenF ϕ=τD−1b
A[1]ϕ=τD−nb
Aϕ by Lemma2.2. On the other hand, ϕ ρ=ϕ τ−n
Γ by Lemma4.2, and then the statement follows from Proposition4.1.
Suppose now that n is odd. Let M=(, Ma,b)∈M(Γ , τΓ). If a =b, then level(ϕ(M))= |δa,b| −2≤n−2 by Proposition4.1, and, therefore, F (ϕ(M))= τD−nb
Aϕ(M)by Lemma2.2. On the other hand, Lemma4.2implies thatϕ ρ(M)= ϕ τ−n
Γ (M), and the statement follows from Proposition 4.1. Finally, suppose that a=b, that is, M=(, Ma,a)and level(ϕ(M))∈ {n−1, n}. LetM−=(, Ma,a− );
then, by Proposition 4.1, we have (ϕ(M))− = ϕ(M−) and thus F (ϕ(M)) = τD−nb
Aϕ(M−)by Lemma2.2. On the other hand, Lemma4.2implies thatϕ ρ(M)= ϕ τ−n
Γ (M−), and again the statement follows from Proposition4.1.
Remark 4.4 It has been shown in [21] that the cluster categoryCAis triangulated. The shift functor[1]of this triangulated structure is induced by the shift functor ofDbA and[1] =τCA, by construction.
Thus the category of tagged edges is also triangulated and its shift functor is equal toτ. In particular, we can define the Ext1of two objectsM, N∈indCas
Ext1C(M, N )=HomC(M, τ N ).
We study Ext1in the next section.
5 Dimension of Ext1
We want to translate the statement of Proposition2.3in the categoryC. For an element N∈indC, let pos(N )=(i, j )be the position of the corresponding indecomposable objectϕ(N )inCAunder the equivalenceϕof Theorem4.3. For two verticesa, bon the boundary of the punctured polygon, define the closed interval[a, b]to be the set of all vertices that lie on the counterclockwise path fromatobon the boundary. The open interval]a, b[is[a, b]\{a, b}. Recall thate(M, N )denotes the crossing number of tagged edgesM, N (see Sect.3.2). The various cases of the following lemma are illustrated in Figs.3and4.
Lemma 5.1 Let Ma,b∈indC with a =b be such that pos(Ma,b)=(1, m). Let Mx,y∈indCbe arbitrary, and denote pos(Mx,y)by(i, j ). Then
1. e(τ Ma,b, Mx,y) =0 if and only if one of the following conditions hold:
1≤i≤m & m+1≤i+j; m+1≤i≤n−1 & n≤i+j ≤n+m−1.
2. e(τ Ma,b, Mx,y)=2 if and only if
m≥2 & 2≤i≤m & n≤i+j & 2≤j≤n−2.
Fig. 3 Proof of Lemma5.1.1 Fig. 4 Lemma5.1.2
Proof LetMa,bbe as in the lemma. By Proposition4.1, we have 1≤m= |δa,b|−2 ≤ n−2. We use the notationτ Ma,b=Ma,b andτ2Ma,b=Ma,b. LetMx,y∈indC be such that pos(Mx,y)=(i, j ). Thene(τ Ma,b, Mx,y) =0 if and only if one of the following conditions hold (see Fig.3):
x∈ [a, b] & y∈ [b, x], (1) x∈ [b, a] & y∈ [a, b]. (2) In Fig.3, the two drawings on the left both correspond to the case (1) and the drawing on the right to the case (2). We will now rewrite these conditions in terms ofiandj. First note thati= |δa,x|. Moreover, ifx =y, then Proposition4.1implies thatj =
|δx,y| −2; and ifx=y,thenn−1≤j ≤n. Thus conditions (1) and (2) become 1≤i≤ |δa,b| & |δa,b| −1≤i+j≤ |δa,x| +n, (3)
|δa,b| ≤i≤ |δa,a| & |δa,a| −1≤i+j≤ |δa,a| + |δa,b| −2, (4) where (1)⇔(3) and (2)⇔(4). Note that the upper bound fori+j in (3) is always satisfied, since j ≤nandi= |δa,x|. Now the first statement of the lemma follows simply by counting the vertices on the various boundary paths that appear in (3) and (4).
Fig. 5 Two cases of Lemma5.2
On the other hand,e(τ Ma,b, Mx,y)=2 if and only if x∈ ]a, b[ & y∈ ]a, x[
(see Fig.4). Note thatx cannot be equal toa,since otherwisea=x=y and then, by the definition of the crossing number,e(τ Ma,b, Mx,y)≤1. In particular, we have m= |δa,b| −2≥2. Again, using the fact thati= |δa,x|andj= |δx,y| −2,we see thate(τ Ma,b, Mx,y)=2 if and only if
m≥2 & 2≤i≤ |δa,b| −1 & |δa,a| −1≤i+j & 2≤j≤n−2.
Now, the second statement of the lemma follows, since|δa,b| −1=m.
The two cases of the following lemma are illustrated in Fig.5.
Lemma 5.2 LetMa,a∈indCbe such that pos(Ma,a)=(1, m)withm∈ {n−1, n}.
Definemto be the unique element in{n−1, n} \ {m}. LetMx,y ∈indCbe arbitrary, and denote pos(Mx,y )by(i, j ). Then
1. e(τ Ma,a, Mx,y )=1 if and only if one of the following conditions hold:
2≤i≤n−1 & i+j≥n & j≤n−2, 1≤i≤n−1 & j=
m ifiis odd, m ifiis even.
2. e(τ Ma,a, Mx,y )=0 otherwise.
Proof LetMa,abe as in the lemma, and letτ Ma,a=Ma−,a andτ2Ma,a=Ma,a. LetMx,y ∈indC be such that pos(Mx,y )=(i, j ). By the definition of the crossing number we have thate(τ Ma,a, Mx,y )=1 if and only if one of the following condi- tions hold (see Fig.5):
x∈ ]a, a] & y∈ [a, x[, x=y =a & = .
We can rewrite these conditions in terms ofiandj as follows:
2≤i≤ |δa,a| & |δa,a| −1≤i+j & j ≤ |δx,x| −3, 1≤i≤n−1 & j=m ifiis odd,
m ifiis even.
Counting the vertices on the boundary paths yields the lemma.
Theorem 5.3 LetM, N∈indC. Then the dimension of Ext1C(M, N )is equal to the crossing numbere(M, N )ofMandN.
Remark 5.4 Note that the analogue of Theorem5.3also holds in typeAn; see [11].
Proof We can assume without loss of generality that pos(M)=(1, m) for some m∈ {1, . . . , n}. Let(i, j )denote pos(N ). Because of the equivalence of categories of Theorem 4.3, we can calculate the dimension of HomC(M, N ) in terms of the positions pos(M),pos(N ),using the formulas in Proposition2.3. Comparing this re- sult with the description of e(τ M, N )in Lemmas 3.3,5.1, and5.2, we conclude that the dimension of HomC(M, N ) is equal to e(τ M, N ) for all M, N ∈indC.
Hencee(M, N )=dim HomC(τ−1M, N )=dim HomC(M, τ N ). On the other hand, Ext1C(M, N )=HomC(M, τ N )by definition, and the result follows.
Remark 5.5 Since the crossing number is symmetric, this theorem illustrates the sym- metry of Ext1in the cluster category, that is, for anyM, N∈indC, we have
dim Ext1C(M, N )=dim Ext1C(N, M).
6 Applications
6.1 Auslander–Reiten-triangles
By [21] and Theorem 4.3, C is a triangulated category with Auslander–Reiten- triangles
τ M→L→M→τ M[1] =τ2M.
We have describedτ already; let us now describeL.
1. Suppose thatτ M=Ma,bwitha =b. ThenM=Mc,d, wherec(respectivelyd) is the counterclockwise neighbor ofa(respectivelyb). In particular,c =b,since
|δa,b| ≥3.
Ifa =d, thenLis the direct sum ofMc,bandMa,d,and ifa=d,thenLis the direct sum ofMc,b,Ma,a1 andMa,a−1.
2. Suppose that τ M =Ma,a− . Then M=Mc,c, where c is the counterclockwise neighbor ofa. ThenLis indecomposable,L=Mc,a.
Note that there are irreducible morphismsτ M→LandL→M, andτ M→L→M is a mesh.
Remark 6.1 Let us point out that, in typeAn, all Auslander–Reiten triangles are of type 1 witha =b; see [11].
6.2 Tilting objects and exchange pairs
A tilting objectT in the categoryCis a maximal set of noncrossing tagged edges, that is,T is a triangulation of the punctured polygon. LetM, N∈E. If e(M, N )=1, then, by [6, Theorem 7.5],M, Nform an exchange pair, that is, there exist two trian- gulationsT andTsuch thatM∈T,N∈T,andT \ {M} =T\ {N}. The edgesM andN are the “diagonals” in a generalized quadrilateral inT \ {M}; see Fig.6. The triangulationTis obtained from the triangulationT by “flipping” the diagonalMto the diagonalN.
LetxM, xN be the corresponding cluster variables in the cluster algebra; then the exchange relation is given by
xMxN=
i
xLi+
i
xL i,
Fig. 6 Two examples of exchange pairs
Fig. 7 Ext1of dimension 2
where the products correspond to “opposite” sides in the generalized quadrilateral and can have one, two, or three factors; see Fig.6.
6.3 Ext1of higher dimension
We include an example of two indecomposablesM, N∈indCfor which the dimen- sion of Ext1C(M, N )is equal to two; see Fig.7. In the same figure, we illustrate inde- composable objectsA1, A2, A3, B2, B3, C1, C2,andD1that appear in the following triangles:
N→A1⊕A2⊕A3→M→N[1], N→A1⊕B2⊕B3→M→N[1], N→C1⊕C2→M→N[1], N→D1→M→N[1].
6.4 Cluster-tilted algebra
Let T be any triangulation of the punctured polygon. Then the endomorphism al- gebra EndC(T )op is called a cluster-tilted algebra. By a result of [7], the functor HomC(τ−1T ,−)induces an equivalence of categoriesϕT :C/T →mod EndC(T )op. Labeling the edges inT byT1, T2, . . . , Tn, we get the dimension vector dimϕT(Ma,b) of a moduleϕT(Ma,b)by
dimϕT Ma,b
i =dim HomC
τ−1Ti, Ma,b
=dim Ext1C Ma,b, Ti
=e Ma,b, Ti
, (5)
where the last equation holds by Theorem5.3.
We illustrate this in an example. Letn=4,and letT be the triangulation
Then the cluster-tilted algebra EndC(T )opis the quotient of the path algebra of the quiver
by the ideal generated by the pathsα β γ,β γ δ,γ δ α,andδ α β. The Auslander–
Reiten quiver of the categoryCis
For the four tagged edges of the triangulationT, we have drawn the borders of the punctured polygons in the above picture as dotted lines. Deleting these positions and
using (5), we obtain the Auslander–Reiten quiver of EndC(T )op
1 4 3 2 1
· · · 4 1 2
4 1
3 4
2 3 4
2 3
1 2
4 1 2
· · ·
3 4 1
1 2 3 where modules are represented by their Loewy series.
Acknowledgements The author thanks Philippe Caldero, Frederic Chapoton, and Dylan Thurston for interesting discussions on the subject.
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