Some cluster tilting modules for weighted surface algebras
aa r X i v : . [ m a t h . R T ] J a n SOME CLUSTER TILTING MODULES FOR WEIGHTED SURFACEALGEBRAS
KARIN ERDMANN
Abstract.
Non-singular weighted surface algebras satisfy the necessary condition found in [6]for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiveris bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is3-cluster tilting precisely for non-singular triangular or spherical algebras, but not for any otherweighted surface algebra with bipartite Gabriel quiver.
Keywords:
Symmetric algebra, Surface algebra, cluster tilting modules. Introduction
A module M of a finite-dimensional algebra A is an n -cluster tilting module (or maximal ( n − M ) = { N | Ext i ( M, N ) = 0 for 1 ≤ i ≤ n − } = { N | Ext i ( N, M ) = 0 for 1 ≤ i ≤ n − } , (see [12], [13]). We would like to know whether non-singular weighted surface algebras have clustertilting modules. Weighted surface algebras are a class of tame symmetric algebras, periodic asbimodules, of period 4 (see [7] and [8], [9]). This means that they satisfy the necessary conditionfound in [6], requiring that all non-projective modules should have bounded periodic resolutions.As observed in [6] if such an algebra has an n -cluster tilting module then the only option is n = 3.Here we study weighted surface algebras which have a bipartite Gabriel quiver, which means thatin the presentation as in [8] (see also [9]) it has many virtual arrows. We introduce a module M ,defined in the same way for each of the algebras, which satisfies Ext ( M, M ) = 0 and Ext ( M, M ) =0. We show that it is 3-cluster tilting when Λ is either a triangle algebra T ( λ ), or a spherical algebra S ( λ ) (see § λ ∈ K and λ = 0 ,
1. We also show that for any other weightedsurface algebra whose Gabriel quiver is bipartite, M cannot be a direct summand of a 3-clustertilting module.The algebra T ( λ ) occurs in various places in the literature. It is an algebra with k = 1 in thefamily Q (3 A ) of algebras of quaternion type, in [5]. Furthermore, it occurs with the name B , ( λ )in [3]. As well, it occurs in [1] with the name A ( λ ). In [8] it is called the triangle algebra T ( λ )(in Example 3.4). Similarly the spherical algebra S ( λ ) was introduced in [8] (Example 3.6).Spherical algebras are a special case of the family of algebras which come from the triangulation T ( n ) of the sphere as defined in Example 7.5 of [10]. We call these algebras n -spherical; when n = 2 they are the same as the spherical algebras. One may also observe that the spherical algebrawith n = 1 with the multiplicities 2 , , T ( λ ), allowing a multiplicity k > Q (3 A ) k in the labelling of [11]. When the characteristic of K is 2 these occur as the basic algebras of blocks of finite groups. Very recently B.B¨ohmler and R. Marcinczik proved using computer calculations that for k = 2, it has a 3-clustertilting module (see [2]).Much of this note was written five years ago, when talking to Idun Reiten about [3], and it wasextended first when spherical algebras had been discovered, and then again, inspired by an emailfrom R. Marcinczik (for which I am grateful).2. Preliminaries
Throughout K is an algebraically closed field, of arbitrary characteristic. Assume Λ is a finite-dimensional symmetric K -algebra. We recall some identities for the stable category modΛ.(1) D Ext ( M, N ) ∼ = Hom( τ − N, M ), and in this case τ ∼ = Ω .(2) Ext i ( U, V ) ∼ = Hom(Ω i U, V ) ∼ = Hom( U, Ω − i V ).This implies that dim Ext i ( M, N ) = dim Ext ( N, Ω i +2 M ). The algebras we consider have theproperty that all non-projective indecomposable right Λ-modules are Ω periodic of periods dividing4. This gives us the following, we refer to this as ext symmetry. Corollary 2.1.
Assume Λ is symmetric and all modules have Ω -period dividing . Then for all M, N we have dim Ext ( M, N ) = dim Ext ( N, M ) as vector spaces. This simplifies the search for 3-cluster tilting modules. If we know that Ext ( N, X ) = 0 andExt ( X, N ) = 0 then automatically Ext ( X, N ) = 0 and Ext ( N, X ) = 0.3.
The algebras
Weighted surface algebras.
We review the definition from [9], for details see [7], [8], [9].Assume Q is a finite quiver. Denote by KQ the path algebra of Q over K . We will consideralgebras of the form A = KQ/I where I is an ideal of KQ which contains all paths of length ≥ m for some m >>
0, so that the algebra is finite-dimensional and basic. The Gabriel quiver Q A of A is then the full subquiver of Q obtained from Q by removing all arrows α with α + I ∈ R Q + I .A quiver Q is 2 -regular if for each vertex i ∈ Q there are precisely two arrows starting at i andtwo arrows ending at i . Such a quiver has an involution on the arrows, α ¯ α , such that for eacharrow α , the arrow ¯ α is the arrow = α such that s ( α ) = s (¯ α ).A triangulation quiver is a pair ( Q, f ) where Q is a (finite) connected 2-regular quiver, withat least two vertices, and where f is a fixed permutation of the arrows such that t ( α ) = s ( f ( α ))for each arrow α , and such that f is the identity. The permutation f uniquely determines apermutation g of the arrows, defined by g ( α ) := f ( α ) for any arrow α . We assume throughoutthat ( Q, f ) is a triangulation quiver. To give the presentations of the algebras in question, we usethe following notation. For each arrow α , we fix m α ∈ N ∗ a weight, constant on g -cycles, and c α ∈ K ∗ a parameter, constant on g -cycles, and define n α := the length of the g -cycle of α , B α := αg ( α ) . . . g m α n α − ( α ) the path along the g -cycle of α of length m α n α ,A α := αg ( α ) . . . g m α n α − ( α ) the path along the g -cycle of α of length m α n α − OME CLUSTER TILTING MODULES FOR WEIGHTED SURFACE ALGEBRAS 3
Definition 3.1.
We say that an arrow α of Q is virtual if m α n α = 2, that is A α has length 1. Notethat this condition is preserved under the permutation g , and that virtual arrows form g -orbits ofsizes 1 or 2.We assume that the following conditions hold.(1) m α n α ≥ α , and(2) m α n α ≥ α such that ¯ α is virtual and ¯ α is not a loop, and m α n α ≥ α such that ¯ α is virtual and ¯ α is a loop.Condition (1) is a general assumption, and (2) is needed to eliminate two small algebras (see [8]).We also assume that Q has at least three vertices. With this, the definition of a weighted surfacealgebra (as revised in [9]) is as follows. Definition 3.2.
The algebra
Λ = Λ(
Q, f, m • , c • ) = KQ/I is a weighted surface algebra if ( Q, f ) is a triangulation quiver, with | Q | ≥ , and I = I ( Q, f, m • , c • ) is the ideal of KQ generated by: (1) αf ( α ) − c ¯ α A ¯ α for all arrows α of Q , (2) αf ( α ) g ( f ( α )) for all arrows α of Q unless f ( α ) is virtual, or unless f (¯ α ) is virtual and m ¯ α = 1 , n ¯ α = 3 . (3) αg ( α ) f ( g ( α )) for all arrows α of Q unless f ( α ) is virtual, or unless f ( α ) is virtual and m f ( α ) = 1 , n f ( α ) = 3 . The Gabriel quiver Q Λ is the subquiver of Q obtained by removing all virtual arrows.We recall a few properties.(1) Any such algebra is symmetric and tame.(2) The dimension of e i Λ is equal to m α n α + m ¯ α n ¯ α where α, ¯ α are the arrows starting at i .The relations also imply that c α B α = c ¯ α B ¯ α in Λ. One can show that this spans the socle of e i Λ.We wish to define a module M such that Ext ( M, M ) = 0 and Ext ( M, M ) = 0, as a candidateto be 3-cluster tilting. This can be done for a weighted surface algebra whose quiver is bipartite;this requires that each triangle of f must contain a virtual arrow. Such a quiver can be thoughtof made up of three building blocks, first a quiver of the form a b d a b d a · · · a n b n d n a n +1 ξ δ γ η σ ̺ ξ δ γ η σ ̺ ξ n δ n γ n η n σ n ̺ n ,where the shaded triangles define the f -orbits.Next, quivers of the form 1 ε $ $ α / / β o o or 2 ′ γ / / δ o o ε ′ d d K. ERDMANN
We describe the quivers and algebras we consider. We always take the multiplicities at 2-cyclesof g equal to 1, and at loops we take multiplicity 2. That is, all arrows in 2-cycles and loops arevirtual and not part of the Gabriel quiver.3.2. Algebras with Gabriel quiver A . We take the quiver Q obtained by glueing the secondand the third type above, identifying vertex 2 with vertex 2 ′ .1 ε $ $ α / / β o o γ / / δ o o ε ′ d d The permutation g is of the form ( α γ δ β )( ε )( ε ′ ). Let m α = k ≥ k = 1 is special, this gives the triangular algebra, called T ( λ ) in [8], here λ = 1.With suitable choice of c • , the presentation of the weighted surface algebra induces the (Gabriel)presentation of T ( λ ) αβα = αγδ, δβα = λ ( δγδ ) , βαβ = γδβ, βαγ = λ ( γδγ ) ,αβαγ = 0 , βαβαβ = 0 , δγδβ = 0 , γδγδγ = 0 ,αβαβα = 0 , δγδγδ = 0 , δβαβ = 0 . One can show that T ( λ ) and T ( µ ) are not isomorphic for λ = µ . The weighted surface algebraswith the same quiver and ε, ε ′ virtual loops have Gabriel quiver denoted by Q (3 A ) k in [11] (whichis the algebra with parameter k in the the family Q (3 A ) of [5]).3.3. Spherical algebras.
We have the algebras whose quiver is given by the first building blockwhere we identify a = a n +1 , for n ≥
2. The case n = 2 gives the algebra S ( λ ), called sphericalalgebra, introduced in [8], Example 3.6, as follows.12 3 45 6 αξ δη βν ̺ εσ µωγ where the four shaded triangles denote the f -orbits. We take all multiplicities equal to 1, thepresentation induced by the weighted surface algebra presentation is, with suitable choice of c • , αβν = ̺ων, βνδ = λβγσ, νδα = λγσα, δαβ = δ̺ω,γσ̺ = νδ̺, σ̺ω = λσαβ, ̺ωγ = λαβγ, ωγσ = ωνδ,αβνδα = 0 , βνδ̺ = 0 , νδαβν = 0 , δαβγ = 0 ,γσ̺ωγ = 0 , σ̺ων = 0 , ̺ωγσ̺ = 0 , ωγσα = 0 ,βγσ̺ = 0 , σαβν = 0 , δ̺ωγ = 0 , ωνδα = 0 ,βνδαβ = 0 , δαβνδ = 0 , σ̺ωγσ = 0 , ωγσ̺ω = 0 . OME CLUSTER TILTING MODULES FOR WEIGHTED SURFACE ALGEBRAS 5
The n -spherical algebra. When n ≥
2, the permutation g is of the form n Y i =1 ( ξ i η i ) · ( γ σ γ σ . . . γ n σ n ) · ( ̺ n δ n ̺ n − δ n − . . . ̺ δ ) . We take the multiplicities for the 2 n -cycles to be m, m ′ ≥
1, and write c, c ′ for the parameters atthese cycles.3.5. A mixed algebra.
We can glue together the three building blocks by identifying 2 = a ,and 2 ′ = a n +1 . In this case, the permutation g is the product of one large cycle with n cycles oflength 2, and two loops: n Y i =1 ( ξ i η i ) · ( γ σ γ σ . . . γ n σ n γ δ ̺ n δ n ̺ n − δ n − . . . ̺ δ β α )( ε )( ε ′ ) . We take again the multiplicities equal to 1 on 2-cycles of g , or m γ = m , and the parameterfunction with value 1 on each virtual arrow. These algebras were not studied in previous papersbut they fit into the same scheme.4. Construction of the module M Let Λ be one of the algebras as described above. Let Γ ⊂ Q be the set of vertices which arenot adjacent to a virtual arrow. Definition 4.1.
Let M be the (right) Λ -module M := Λ ⊕ [ M i ∈ Γ S i ] ⊕ [ M ν Γ Ω ( S ν )]In the following we write down the details for the case of the n -spherical algebra, for the otheralgebras they are essentially the same. In this case Γ = { a i | ≤ i ≤ n } .4.1. The Ω -translates of the simple modules. For the algebra in question, the dimensions ofthe indecomposable projectives are:dim P a i = 2 n ( m + m ′ ) , dim P b i = 2 nm + 2 , dim P d i = 2 nm ′ + 2 . Let a i ∈ Γ. The structure of Ω ± ( S a i ) can be seen from the presentation of the algebra. Themodule Ω ( S a i ) has dimension 5, the Loewy structure is b i − d i a i b i d i − That is, the module has a ’simple waist’. Now let ν ∈ Q \ Γ we set U ν := Ω ( S ν ). ThenΩ( U ν ) = Ω − ( S ν ) and Ω − ( U ν ) = Ω( S ν ), their structure can also be seen from the presentation.We describe U ν . Lemma 4.2.
The module U b i is uniserial of length nm ′ − , with composition series U b i = U ( a i , d i − , a i − , d i − , a i − , . . . , a i +1 ) The module U d i is uniserial of length nm − , with composition series U d i = U ( a i +1 , b i +1 , a i +2 , b i +2 , . . . , a i ) (taking indices modulo n and writing a i , d i , b i meaning the corresponding simple module). K. ERDMANN
Proof
We compute U b , that is Ω ( S b ) = Ω( S b ) = { x ∈ P a | σ x = 0 } ⊂ P a . From therelations for the algebra, we have σ ̺ δ = cA σ = cσ A ′ σ Hence σ ψ = 0 if we set ψ = ψ ̺ δ := ̺ δ − cA ′ σ . One exhibits a basis for ψ Λ, showing that it has the same dimension as Ω ( S b ), hence we haveequality. The submodule structure follows directly. The case U d i is similar. (cid:3) Proposition 4.3.
We have
Ext ( M, M ) = 0 and
Ext ( M, M ) = 0 .Proof
By ext symmetry, it suffices to show that for any non-projective indecomposable summand X of M we have Ext ( M, X ) = 0 and Ext ( X, M ) = 0. For this, we use the following short exactsequences:Let a i ∈ Γ,(1) 0 → Ω( S a i ) → P a i → S a i → , → Ω ( S a i ) → P c i ⊕ P d i → Ω( S a i ) → ν not in Γ, let ν = c i (2) 0 → Ω − S b i → P a i → U b i → , → S b i → P b i → Ω − ( S b i ) → ν = d i , then(3) 0 → Ω − ( S d i ) → P a i +1 → U d i → , → S d i → P d i → Ω − ( S d i ) → A ( − , X ) to the above exact sequences.(I) Assume X = S a j for some j . We know from the quiver that Ext ( S a i , S a j ) = 0 already.To show that Ext ( S a i , S a j ) = 0 we apply the functor to the second sequence in (1). From thestructure of Ω ( S a i ) we see directly that Hom(Ω ( S a i ) , S a j ) = 0 and hence Ext ( S a i , S a j ) = 0.We have Ext ( U b i , X ) = 0 since Hom(Ω − ( S b i ) , S a j ) = 0 Furthermore Ext ( U b i , X ) = 0 sinceHom( S b i , S a j ) = 0. Similarly one shows Ext ( U d i , X ) = 0 and Ext ( U d i , X ) = 0.(II) Now assume X = U b j for some j . First, by dimension shift Ext ( U ν , U b j ) ∼ = Ext ( S ν , S b j ) =0 for any ν of valency 1, from the quiver. Next, consider Ext ( U ν , X ), by applying the functorHom( − , X ) to the second exact sequence in (2). We have Hom( S µ , U b j ) = 0 (the socle of U b i isalways some S a ), and hence Ext ( U ν , X ) = 0.Now consider Ext t ( S a i , X ) for t = 1 ,
2. By the ext symmetry, it is isomorphic to Ext t ( X, S a i )for t = 2 ,
1. By part (I) we know that it is zero.The proof for X = U d i is analogous. (cid:3) Remark 4.4.
For possible later use, we write down sequences which may be used to showExt ( X, M ) = 0 and Ext ( X, M ) = 0: Let a i ∈ Γ,(1 ∗ ) 0 → S a i → P a i → Ω − ( S a i ) → , → Ω − ( S a i ) → P b i ⊕ P d i → Ω − ( S a i ) → ν not in Γ, let ν = c i (2 ∗ ) 0 → U b i → P a i +1 → Ω( S b i ) → , → Ω( S b i ) → P b i → S b i → ν = d i , then(3 ∗ ) 0 → U d i → P a i → Ω( S d i ) → , → Ω( S d i ) → P d i → S d i → . OME CLUSTER TILTING MODULES FOR WEIGHTED SURFACE ALGEBRAS 7 Ext vanishing and 3-cluster tilting
We would like to determine when M is 3-cluster tilting. Hence take X indecomposable and notprojective, and assume Ext ( M, X ) = 0 = Ext ( M, X ) . By ext symmetry, we get for free that Ext ( X, M ) = 0 = Ext ( X, M ) . The aim is to show that X is in add( M ), or if not, to identify X . Lemma 5.1.
The socle and the top of X belong to add( ⊕ i S a i ) .Proof Let ν be a vertex = a i for any i . Apply the functor Hom( − , X ) to the second sequenceof (2), this gives the exact sequence0 → Hom(Ω − ( S b i ) , X ) → Hom( P b i , X ) → Hom( S b i , X ) → P b i → X must map the socle to zero, otherwise it would be split. Hence itlies in Hom(Ω − ( S b i ) , X ) and therefore the first two terms are isomorphic. Hence the last term iszero, as required. To show that also Hom( X, S ν ) = 0 we use a sequence from (2 ∗ ). (cid:3) Lemma 5.2.
We have
Hom(Ω( X ) , S a i ) = 0 and Hom( S a i , Ω − ( X )) = 0 .Proof Since Ext ( X, S a i ) = 0, from a minimal projective cover of X we obtain the exactsequence 0 → Hom(
X, S a i ) → Hom( P X , S a i ) → Hom(Ω( X ) , S a i ) → − ( X ) , S a i ) = 0. (cid:3) Let X be the category of A -modules which have socle and top in add( S a i ). This category isequivalent to mod − e Λ e . where e is the idempotent e := P i e a i . An equivalence is given by thefunctor V V e , with inverse the composite of ( − ) ⊗ e Λ e ( e Λ) follows by factoring out the largest A -submodule V ′ with V ′ e = 0 (see for example [4]).We may write down quiver and presentation of the algebra e Λ e . The arrows are x i := γ i σ i and y i := ̺ i δ i , for 1 ≤ i ≤ n where x i : a i a i +1 and y i : a i +1 → a i . From the relations for Λ wesee We claim that x i y i = 0 and y i x i − = 0. That is, e Λ e is special biserial. Moreover, for any i ,the longest non-zero monomial x i x i +1 . . . is up to a scalar equal to the longest non-zero monomial y i − y i − . . . , and this gives the socle relations. Lemma 5.3.
The module X has simple socle and top.Proof The module X , and as well, all projectives (injectives) P a i belong to the category X ,and hence we may fix an injective hull, or projective cover, of X by identifying with the image ofa suitable injective hull, or projective cover, of Xe , in mod e Λ e . The indecomposable e Λ e -modulesare ’strings’ or ’bands’, and their injective hulls or projective covers may be written down explicitly.Assume the socle of X is not simple, then consider the injective hull I X , it has at least twoindecomposable summands, say it is ⊕ i ∈ R P a i . We may assume, with the above convention, andtaking X → I X as inclusion, that X has a generator ω = ( ω , ω , , . . . ) such that ωA has socle oflength two, and moreover, that ωx j = ( ω x j , , . . . ) and ωy j − = (0 , ω y j − , , . . . ) and ωx r = 0, ωy s = 0 for all other generators x r , y s of e Λ e . This implies then that ωJ ⊆ X where J isthe radical of Λ. Now consider π : I X → Ω( X ). The element π ( ω , , . . . ) is non-zero (since K. ERDMANN ω is a generator for X ). Furthermore, [ π ( ω , , . . . )] J = π [( ω , J ] = 0 since ( ω , , . . . ) J iscontained in X . Now π ( ω ,
0) = π ( ω , e , (since top X is in add( ⊕ S a i ). Hence for some i we haveHom( S a i , Ω − ( X )) = 0. This contradicts the previous Lemma.Similarly by exploiting a projective cover, one shows that the top of X must be simple. (cid:3) Proposition 5.4.
The module X is uniserial.Proof If X is not uniserial then Xe is not uniserial (using the structure of the projectives inthis case). Then Xe is a ’band module’. This means that X contains a submodule isomorphic tothe second socle of some P a j . That is Hom(Ω ( S a j ) , X ) = 0.Applying Hom( − , X ) to the exact sequence0 → Ω ( S a j ) ι → P := P b j ⊕ P d j − −→ Ω( S a j ) → θ : Ω ( S a j ) → X factors through ι ,say θ = ψ ◦ ι . The kernel of θ is the socle of Ω ( S a j ) which also is the socle of P . We factor outthese socles, then for the induced maps we have¯ θ = ¯ ψ ◦ ¯ ι. Now, the map ¯ ψ on the socle of ¯ P is non-zero on each component. It follows that the image of ¯ ψ has Loewy length equal to the Loewy length of P/ soc P .Note that all modules P a i have the same Loewy length ℓ say. As well P b j ⊕ P d j − has Loewylength ℓ . Hence the Loewy length of P/ soc P is ℓ −
1. The image of ¯ ψ is contained in the radicalof X , which is the unique maximal submodule. It follows that the Loewy length of X is ℓ . Butthis means that X must be projective, a contradiction. This shows that Xe is uniserial, and thenfrom the structure of the projectives, also X is uniserial. (cid:3) We summarize. We have shown that if X is indecomposable and not projective such thatExt ( M, X ) = 0 = Ext ( M, X ) then( ∗ ) X is uniserial, and soc X and top X are in add( ⊕ i S a i ). That is, X is a subquotient of some U b i or U d j .We show now that if X is any module satifying ( ∗ ) then Ext ( M, X ) = 0 and Ext ( M, X ) = 0.
Lemma 5.5.
Let X = U ( a j , b j , a j +1 , b j +1 , . . . a l ) , a subquotient of some U ν . Then Ext ( M, X ) = 0 and
Ext ( M, X ) = 0 .Proof
We use the sequences in the proof of Proposition 4.3. We apply the functor ( − , X ) :=Hom( − , X ) to the exact sequences in (1). We start with the second, this gives0 → (Ω( S a i ) , X ) −→ ( P b i ⊕ P d i − , X ) −→ (Ω ( S a i ) , X ) → Ext (Ω( S a i ) , X ) → ( S a i ) , X ) = 0 ( X is uniserial). Hence the ext space is zero. Moreover, itfollows that the first two terms are isomorphic, which we can use for the first sequence:0 → ( S a i , X ) −→ Xe a i −→ Xe b i ⊕ Xe d i → Ext ( S a i , X ) → Xe d i = 0. Note that in the composition series we have length two subquotients a r , b r ,except that for l = r we have an extra copy of a l . Hence if i = l then the first term is K , and | Xe a l | = 1 + | Xe b l | and ext is zero. Suppose i = ℓ , then the first term is zero and the second andthird are isomorphic. Again ext is zero. OME CLUSTER TILTING MODULES FOR WEIGHTED SURFACE ALGEBRAS 9
Next, we apply ( − , X ) to the sequences in (3). Since S d i does not occur in X , the functor takesthe second sequence to zero. From the first sequence we get0 → Hom( U d i , X ) → Xe a i +1 → → Ext ( U d i , X ) → − , X ) applied to sequences in (2). The second sequence gives Hom(Ω − ( S b i ) , X ) ∼ = Xe b i and Ext (Ω − ( S b i ) , X ) = 0. Consider the first sequence, this gives0 → Hom( U b i , X ) → Xe a i → Xe b i → Ext ( U b i , X ) → S a i ) of U b i is not the same as the socle of X then the hom space is zero and thesecond and third term are isomorphic, and ext is zero. Supoose i = l , then the first term is K , and | Xe a i | = 1 + | Xe b i | and again the ext space is zero. (cid:3) Corollary 5.6.
Assume Λ is the triangle algebra, or the spherical algebra. Then M is 3-clustertilting.Proof For these algebras, all indecomposables satisfying ( ∗ ) are in add( M ). (cid:3) Consider an n -spherical algebra for n ≥ m = m ′ = 1. Then the (finite) set of modules X satifying ( ∗ ) contains all modules of the form U ( a i , b i , a i +1 ) , U ( a i , d i − , a i − ) . To have a 3-cluster tilting module with M as a summand, we would need to take f M = M ⊕V where V is the direct sum of all modules satisfying ( ∗ ). However, f M has self-extensions. For examplethere is a non-split exact sequence0 → U ( a , b , a ) → S a ⊕ U ( a , b , a , b , a ) → U ( a , b , a ) → M cannot be extended to a 3-cluster tilting module for the n -spherical algebra when n ≥ k ≥
2. In this case the list of uniserialmodules X which are subquotients of U and U contains the modules U (2 , , , U (2 , , f M = M ⊕ V where V is the direct sum of all indecomposable modules satisfying ( ∗ ). This isnot a 3-cluster tilting module since it has self-extensions: we have the non-split exact sequence0 → U (2 , , → S ⊕ U (2 , , , , → U (2 , , → References [1]
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Mathematical Institute, Oxford University, ROQ, Oxford OX2 6GG, United Kingdom
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