Cluster tilting subcategories and torsion pairs in Igusa--Todorov cluster categories of Dynkin type A ∞
aa r X i v : . [ m a t h . R T ] J un CLUSTER TILTING SUBCATEGORIES AND TORSION PAIRS INIGUSA–TODOROV CLUSTER CATEGORIES OF DYNKIN TYPE A ∞ SIRA GRATZ, THORSTEN HOLM, AND PETER JØRGENSEN
Abstract.
We give a combinatorial classification of cluster tilting subcategories and torsionpairs in Igusa–Todorov cluster categories of Dynkin type A ∞ . Introduction
Let C ( A n ) be the cluster category of Dynkin type A n , see [3, sec. 1] and [4]. It is well knownthat C ( A n ) has a combinatorial model by an ( n + 3)-gon P . The indecomposable objects arein bijection with the diagonals of P , and non-vanishing Ext groups correspond to crossingdiagonals.The combinatorial model has two key properties: Cluster tilting subcategories of C ( A n ) cor-respond to triangulations of P , and torsion pairs in C ( A n ) correspond to so-called Ptolemydiagrams in P . The former result is well known and appears to be folklore; the latter is provedin [10, thm. A].The aim of this paper is to prove similar key properties for the cluster categories C ( Z ) ofDynkin type A ∞ , which were introduced by Igusa and Todorov. Subsection A is a primeron C ( Z ) and its combinatorial model by an ∞ -gon, and Subsections B and C state the keyproperties we will prove.Our results generalise the following parts of the literature: • When Z has one, respectively two limit points (see Definition 0.1(iii)), [9, thms.A,B,C], respectively [16, thms. 3.13, 5.7] classified cluster tilting subcategories in C ( Z ),and showed that they form a cluster structure in the sense of [2, sec. II.1]. • When Z has one, respectively two limit points, [17, thm. 3.18], respectively [5, thm.4.4] classified torsion pairs in C ( Z ).Furthermore, our Theorem 0.5 is closely related to [19, thm. 7.17].We would also like to mention that there are a number of papers on the classification of clustertilting subcategories and torsion pairs in more general cluster categories, mainly based oncombinatorial models of Riemann surfaces with marked points on the boundary, see [1], [18],[20] for surface type and [11] for cluster tubes.A. The Igusa–Todorov cluster categories C ( Z ) of Dynkin type A ∞ . To explain thecategories C ( Z ) and their combinatorial models by ∞ -gons, we first state two definitions. Mathematics Subject Classification.
Key words and phrases.
Cyclically ordered set, fountain, infinite polygon, leapfrog, Ptolemy condition,Ptolemy diagram, triangulation.
Definition 0.1 (Admissible subsets of S ) . A subset Z of the circle S is called admissible ifit satisfies the following conditions.(i) Z has infinitely many elements.(ii) Z ⊂ S is a discrete subset, i.e. for each z ∈ Z there is an open neighbourhood of z in S , equipped with its usual topology, containing no other element of Z .(iii) Z satisfies the two-sided limit condition , i.e. each x ∈ S which is the limit of a sequencefrom Z is a limit of both an increasing and a decreasing sequence from Z with respectto the cyclic order.Throughout the paper, Z ⊂ S is a fixed admissible subset. We think of Z as the vertices ofan ∞ -gon, see Figure 1. z z − z + Figure 1.
An example of an admissible subset Z of S , to be thought of as thevertices of an ∞ -gon, see Definition 0.1. The points in Z converge to the limitpoints marked with small circles. Each point z ∈ Z has a predecessor z − and asuccessor z + in Z , see Remark 1.2. Note that the limit points are not elementsof Z since Z is discrete. Definition 0.2 (Diagonals) . A diagonal of Z is a subset X = { x , x } ⊂ Z where x x − , x , x +0 } . If Y = { y , y } is another diagonal, then X and Y cross if x < y < x < y or x < y < x < y . See Definition 1.1 for an explanation of inequalities.If D is the disk bounded by S , then we think of the diagonal X as an isotopy class of non-selfintersecting curves in D between the non-neighbouring vertices x and x , see Figure 2.Two diagonals cross if their representing curves intersect in the interior of D .Starting from Z and an algebraically closed field k , Igusa and Todorov in [12, sec. 2.4] con-structed a cluster category C ( Z ) of Dynkin type A ∞ , which has a similar combinatorial modelto that of C ( A n ). To wit, C ( Z ) is a k -linear Hom-finite Krull–Schmidt 2-Calabi–Yau trian-gulated category; the indecomposable objects are in bijection with the diagonals of Z , andnon-vanishing Ext groups correspond to crossing diagonals. Further properties of C ( Z ) aregiven in Section 2. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 3 a a a a Figure 2.
A set of diagonals of Z with fountains converging to the limit points a , a , a and a leapfrog converging to the limit point a , see Definition 0.4. Suchconvergence must occur in each cluster tilting subcategory of C ( Z ) by Theorem0.5.B. Cluster tilting subcategories of the cluster categories C ( Z ) . Our first main result isa classification of the cluster tilting subcategories of C ( Z ) (see Definition 5.1). Cluster tiltingsubcategories of C ( A n ) correspond to triangulations of a finite polygon P , that is, maximal setsof pairwise non-crossing diagonals of P . By analogy, we expect cluster tilting subcategories of C ( Z ) to correspond to triangulations of the ∞ -gon with vertex set Z .This is, in a sense, true, but there is more to say: The definition of admissible subset permits Z to have a complicated configuration of limit points, and it is crucial how the endpoints ofdiagonals converge to the limit points. Hence the following two definitions. Definition 0.3 (The proper limit points of Z ) . We denote by Z the topological closure of Z in S , and by L ( Z ) = Z \ Z the set of proper limit points of Z . It is disjoint from Z because Z is discrete. Definition 0.4 (Leapfrogs and fountains) . Let X be a set of diagonals of Z . The followingnotions are illustrated by Figure 2. • Given a ∈ L ( Z ), we say that X has a leapfrog converging to a ∈ L ( Z ) if there is asequence { x i , y i } i ∈ Z > of diagonals from X with x i → a from below and y i → a fromabove. (Convergence from below and above is explained in Definition 1.4.) • Given a ∈ L ( Z ), z ∈ Z . We say that X has a right fountain at z converging to a if there is a sequence { z, x i } i ∈ Z > from X with x i → a from below. We say that X has a left fountain at z converging to a if there is a sequence { z, y i } i ∈ Z > from X with y i → a from above.We say that X has a fountain at z converging to a if it has a right fountain and a leftfountain at z converging to a .Here is our first main result. It is closely related to [19, thm. 7.17]. Given a set X of diagonalsof Z , we write E ( X ) for the corresponding set of indecomposable objects of C ( Z ). SIRA GRATZ, THORSTEN HOLM, AND PETER JØRGENSEN x i px ′ i x i q x ′ i x i px ′ i x i q x ′ i Figure 3.
Illustration of conditions PC1 and PC2 from Definition 0.7.
Theorem 0.5 (=Theorem 5.7) . Let X be a set of diagonals of Z . Then add E ( X ) is acluster tilting subcategory if and only if X is a maximal set of pairwise non-crossing diagonals,such that for each a ∈ L ( Z ) , the set X has a fountain or a leapfrog converging to a . One of the salient features of cluster tilting subcategories are their nice combinatorial propertiesencoded in the notion of cluster structure . We thank Adam-Christiaan van Roosmalen forpointing out that the following result follows from [19, thm. 5.6]. We will give a direct proof.
Theorem 0.6 (=Theorem 5.9) . The cluster tilting subcategories of C ( Z ) form a cluster struc-ture in the sense of [2, sec. II.1] . To get this from [19, thm. 5.6] requires the existence of a so-called directed cluster tiltingsubcategory of C ( Z ), which can be obtained from Theorem 0.5 by picking a vertex z ∈ Z and letting X be the set of all diagonals from z to non-neighbouring vertices.C. Torsion pairs in the cluster categories C ( Z ) . Our second main result is a classificationof the torsion pairs in C ( Z ) (see Definition 4.1). Recall that torsion pairs in C ( A n ) correspondto so-called Ptolemy diagrams in a finite polygon P , see [10, thm. A]. Again there is an analoguefor C ( Z ), and again, convergence plays a crucial role. Hence the following definition. Definition 0.7 (Conditions PC1 and PC2) . We can impose the following conditions on a set X of diagonals of Z , see Figure 3. The letters “PC” stands for “precovering”.PC1: If there is a sequence { x i , x i } i ∈ Z > from X with x i → p from below and x i → q frombelow with p = q , then there is a sequence { x ′ i , x ′ i } i ∈ Z > from X with x ′ i → p fromabove and x ′ i → q from above.PC2: If there is a sequence { x i , x i } i ∈ Z > from X with x i → p from below and x i → q fromabove with p = q , then there is a sequence { x ′ i , x ′ i } i ∈ Z > from X with x ′ i → p fromabove and x ′ i → q from above.The following combinatorial notion was introduced in [17, def. 0.3]. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 5 x y x y Figure 4.
The Ptolemy condition from Definition 0.8: If the crossing diagonals { x , x } and { y , y } are in X , then so are those of { x , y } , { x , y } , { x , y } , { x , y } which are diagonals. Definition 0.8 (The Ptolemy condition) . Let X be a set of diagonals of Z . We say that X satisfies the Ptolemy condition if, whenever { x , x } ∈ X and { y , y } ∈ X cross, thenthose of { x , y } , { x , y } , { x , y } and { x , y } which are diagonals of Z (i.e. whose verticesare non-neighbouring) also lie in X . See Figure 4.Here is our second main result. Theorem 0.9 (=Theorem 4.7) . Let X be a set of diagonals of Z . Then add E ( X ) is thefirst half of a torsion pair in C ( Z ) if and only if X satisfies conditions PC1, PC2, and thePtolemy condition. Note that the first half of a torsion pair determines the second half, so our result does providea complete classification.
Remark 0.10.
The conjunction of PC1 and PC2 is equivalent to the following condition.PC: If there is a sequence { x i , x i } i ∈ Z > from X with x i → p from below and x i → q with p = q , then there is a sequence { x ′ i , x ′ i } i ∈ Z > from X with x ′ i → p from above and x ′ i → q from above.It is clear that PC implies PC1 and PC2. To see the converse, note that the sequence { x i , x i } i ∈ Z > in PC will either have a subsequence with x i → q from below, and then PC1can be applied, or a subsequence with x i → q from above, and then PC2 can be applied.The paper is organised as follows: Section 1 shows some properties of admissible subsets of S . Section 2 recalls the cluster category C ( Z ) from [12, sec. 2.4]. Section 3 provides a mainingredient for the proof of Theorem 0.9 by showing that add E ( X ) is precovering if and only if X satisfies conditions PC1 and PC2. Section 4 proves Theorem 0.9. Section 5 proves Theorems0.5 and 0.6. SIRA GRATZ, THORSTEN HOLM, AND PETER JØRGENSEN x x x x x x ab Figure 5.
Illustration of Definition 1.1. The elements x , . . . , x of S satisfy x < x < x < x < x < x , and the interval [ a, b ] is marked by a thick arc.1. Admissible subsets of the circle S Definition 1.1 (Cyclically ordered subsets of S ) . The circle S , equipped with its usualtopology and orientation, has a natural structure as a cyclically ordered set.We choose anticlockwise as the positive direction, whence the inequalities x < x < . . .
0. Thus each convergentsequence from Z that is not constant from some step converges to an element of L ( Z ). Fur-thermore, since S is compact, each sequence { z i } i ∈ Z > from Z has a convergent subsequence { z ′ i } i ∈ Z > converging to some point in Z .There is hence a dichotomy: LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 7 • Either the subsequence { z ′ i } i ∈ Z > converges to z ∈ Z , and z ′ i is constant from somestep, • or the subsequence { z ′ i } i ∈ Z > converges to a proper limit point a ∈ L ( Z ) and z ′ i is notconstant from any step.In the latter case, by refining the sequence further if necessary, we can suppose that the sequenceis increasing (i.e. z ′ z ′ . . . z ′ k < a for each k ∈ Z > ) or decreasing (i.e. z ′ > z ′ > . . . > z ′ k > a for each k ∈ Z > ). Definition 1.4 (Convergence from below and above) . Let { z i } i ∈ Z > be a convergent sequencefrom Z . If { z i } i ∈ Z > converges to p ∈ Z , then we write z i → p . • We say that z i → p from below if there is a µ ∈ S \ { p } such that z i ∈ [ µ, p ] from somestep. • We say that z i → p from above if there is a ν ∈ S \ { p } such that z i ∈ [ p, ν ] from somestep.If z i → p with p ∈ Z , then z i = p from some step by Remark 1.3, so z i → p from below and from above. Definition 1.5 (Infimum and supremum) . Let a, b ∈ S . Each non-empty subset P ⊆ [ a, b ] ∩ Z ⊂ S has an infimum and a supremum, and there is a decreasing sequence in P converging toits infimum, denoted by inf [ a,b ] P , and an increasing sequence in P converging to its supremum,denoted by sup [ a,b ] P .Note that the infimum and the supremum are contained in the interval [ a, b ], but not necessarilyin P or in Z . If i = inf [ a,b ] P and s = sup [ a,b ] P then a i p s b for each p ∈ P .Note that any increasing or decreasing sequence in an interval [ a, b ] is convergent to a point inthat interval.Recall that Z contains infinitely many points by Definition 0.1(i). For each z ∈ Z , thesequence { z + n } n > defined iteratively by z +0 = z and z +( k +1) = ( z + k ) + for each k ∈ Z > isan increasing sequence. Moreover, there are infinitely many points of Z in [ z, z − ] whence z z + n < z − . So { z + n } n > is an increasing sequence in [ z, z − ] and it must converge to a limitpoint. Definition 1.6.
The limit point of { z + n } n > will be denoted z + ∞ . Symmetrically, we candefine { z − n } n > and its limit point will be denoted z −∞ . Lemma 1.7.
We have [ z, z + ∞ ] ∩ L ( Z ) = { z + ∞ } and [ z −∞ , z ] ∩ L ( Z ) = { z −∞ } .Proof. We only prove that [ z, z + ∞ ] ∩ L ( Z ) = { z + ∞ } ; the equality [ z −∞ , z ] ∩ L ( Z ) = { z −∞ } is proved symmetrically. The inclusion { z + ∞ } ⊆ [ z, z + ∞ ] ∩ L ( Z ) is clear by definition. Theinclusion [ z, z + ∞ ] ∩ L ( Z ) ⊆ { z + ∞ } amounts to showing that [ z, z + ∞ ) ∩ L ( Z ) = ∅ , whichagain amounts to showing that ( z, z + ∞ ) ∩ L ( Z ) = ∅ , since z ∈ Z , and hence z / ∈ L ( Z ).So suppose for a contradiction that there exists x ∈ ( z, z + ∞ ) ∩ L ( Z ). In particular, x / ∈ Z and there exists a sequence { z i } i ∈ Z > from Z converging to x . By construction we have that SIRA GRATZ, THORSTEN HOLM, AND PETER JØRGENSEN z + ∞ = sup [ z,z − ] { z + n | n > } , so we can find m > z + m < x < z +( m +1) (note that x can not equal any of the z + n since x / ∈ Z ). But since the sequence { z i } i ∈ Z > converges to x , the open neighbourhood ( z + m , z +( m +1) ) of x contains infinitely many entries of the sequence { z i } i ∈ Z > . Since the z i are in Z , this clearly contradicts the definition of z +( m +1) . (cid:3) The Igusa–Todorov cluster categories C ( Z ) of Dynkin type A ∞ Setup 2.1.
In the rest of the paper, k is an algebraically closed field.Igusa and Todorov [12] constructed a cluster category C ( Z ). They proved in [12, sec. 2.4] thatit has the following properties.(i) C ( Z ) is a k -linear Hom-finite Krull–Schmidt triangulated category.(ii) C ( Z ) is 2-Calabi–Yau, that is, there are natural isomorphismsExt C ( Z ) ( X, Y ) ∼ = D Ext C ( Z ) ( Y, X )where D( − ) = Hom k ( − , k ).(iii) If X = { x , x } is a diagonal of Z , then there is an indecomposable object E ( X ) = E ( x , x ) in C ( Z ), and this induces a bijection from diagonals of Z to isomorphismclasses of indecomposable objects of C ( Z ).(iv) The suspension functor acts on the indecomposable objects E ( X ) byΣ( E ( x , x )) = E ( x − , x − ) . (v) We have Ext C ( Z ) ( E ( X ) , E ( Y )) ∼ = ( k if X and Y cross,0 otherwise.(vi) Since Hom C ( Z ) ( E ( X ) , E ( Y )) ∼ = Ext C ( Z ) ( E ( X ) , Σ − E ( Y )), it follows from (iv) and (v)that Hom C ( Z ) ( E ( X ) , E ( Y )) is isomorphic to ( k if we can write X = { x , x } and Y = { y , y } with x y x −− < x y x −− ,0 otherwise.(vii) In part (vi), if X = { x , x } and Y = { y , y } with x y x −− < x y x −− ,then a morphism E ( X ) → E ( Y ) factors through E ( S ) if and only if we can write S = { s , s } with x s y and x s y .In part (v) observe that non-vanishing of Ext is symmetric in the two arguments, as indeed itmust be by the 2-Calabi–Yau property from (ii). Figure 6 provides an illustration of morphismsbetween indecomposable objects.3. Precovering subcategories of the cluster categories C ( Z )This section provides the following main ingredient for the proof of Theorem 0.9. Theorem 3.1.
Let X be a set of diagonals of Z . Then add E ( X ) is a precovering subcategoryof C ( Z ) if and only if X satisfies conditions PC1 and PC2 from Definition 0.7. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 9 x y x y z Figure 6.
The non-zero morphism spaces between the indecomposable objectscorresponding to the pictured diagonals are precisely Hom( E ( x , x ) , E ( y , y )),Hom( E ( y , y ) , E ( x , x )) and Hom( E ( x , z ) , E ( x , x )), as well as the endomor-phism spaces of each of the three indecomposable objects. All other morphismspaces between these three objects are zero. See Section 2(vi).The proof can be found at the end of the section. First we require some preparation, not leastthe following definition due to [7, sec. 1]. Definition 3.2 (Precovers) . Let T be a category, X ⊆ T a full subcategory.(i) Let t ∈ T be an object. An object x ∈ X together with a morphism f : x → t is calledan X -precover of t if each morphism g : x ′ → t with x ′ ∈ X factors through f . Thatis, there exists a morphism h : x ′ → x such that g = f ◦ h . x f (cid:15) (cid:15) x ′ g / / h ? ? ⑦⑦⑦⑦ t (ii) The subcategory X ⊆ T is called precovering if each object t ∈ T has an X -precover. Definition 3.3.
Let T be an additive category. An additive subcategory X of T is a fullsubcategory of T closed under isomorphisms, finite direct sums, and direct summands. Remark 3.4.
Since C ( Z ) is Krull-Schmidt, its additive subcategories are determined by theindecomposable objects they contain. Thus, there is a one-to-one correspondence betweenadditive subcategories of C ( Z ) and sets of diagonals of Z .Given a set of diagonals X we write E ( X ) for the corresponding set of indecomposable objectsof C ( Z ). The corresponding additive subcategory of C ( Z ) is given by add E ( X ). Lemma 3.5.
Let D ⊆ C ( Z ) be an additive subcategory, e ∈ C ( Z ) an indecomposable object,and δ : d ⊕ . . . ⊕ d n → e a morphism in C ( Z ) with d i ∈ D indecomposable for each i ∈ { , . . . , n } . We can write δ = ( δ , . . . , δ n ) , and δ is a D -precover of e if and only if each morphism ϕ : d → e with d ∈ D indecomposable factors through at least one of the δ i .Proof. It is clear that if each morphism ϕ : d → e with d ∈ D indecomposable factors throughat least one of the δ i , then it also factors through δ which is hence a D -precover.Conversely, assume that δ is a D -precover. Let ϕ : d → e be a morphism in C ( Z ) with d ∈ D indecomposable. If ϕ = 0, then ϕ factors trivially through each δ i and we are done. If ϕ = 0,then choose a morphism ϕ ′ : d → d ⊕ . . . ⊕ d n with ϕ = δ ◦ ϕ ′ . Writing ϕ ′ in components ϕ ′ i , thismeans ϕ = δ ϕ ′ + . . . + δ n ϕ ′ n . Because ϕ = 0 there exists an i ∈ { , . . . , n } such that δ i ◦ ϕ ′ i = 0.Now ϕ and δ i ◦ ϕ ′ i are non-zero elements of Hom C ( Z ) ( d, e ) which must be a one-dimensional k -vector space by Section 2(vi). Hence ϕ = α δ i ◦ ϕ ′ i for some α ∈ k , so ϕ factors through δ i . (cid:3) Lemma 3.6.
Let X be a set of diagonals of Z . Then add E ( X ) is a precovering subcategoryof C ( Z ) if and only if X satisfies the following condition:For each diagonal Y = { y , y } of Z there is a finite set of diagonals X , . . . , X l ∈ X , suchthat for each X = { x , x } ∈ X with x y x −− < x y x −− there is an i ∈ { , . . . , l } with X i = { x i , x i } and x x i y and x x i y . Proof.
This is immediate by combining Section 2(vii) with Lemma 3.5. (cid:3)
Proposition 3.7.
Let X be a set of diagonals of Z . If add E ( X ) is a precovering subcategoryof C ( Z ) then X satisfies conditions PC1 and PC2.Proof. Let X be a set of diagonals such that add E ( X ) is precovering. We show that X satisfies condition PC1. The fact that X satisfies condition PC2 follows by an analogousargument. Hence let X i = { x i , x i } i ∈ Z > be a sequence from X with x i → p from below and x i → q from below with p = q .If p, q ∈ Z , we have x i = p and x i = q from some step, whence { p, q } ∈ X and condition PC1is clearly satisfied with x ′ j = p and x ′ j = q for each j ∈ Z > .We can thus assume that p ∈ L ( Z ) or q ∈ L ( Z ).Then by passing to a subsequence we may assume x i p x i −− < x i q x i −− for each i ∈ Z > . Let Y = { y , y } be a diagonal of Z with p y x i −− and q y x i −− for each i ∈ Z > , see Figure 7. Note that such diagonals exist; in fact since Z satisfies the two-sided limit condition (see Definition 0.1), we can even find an entire sequence of such diagonalswith endpoints converging to p and q (at least one of which lies in L ( Z )) from above.Then for each i ∈ Z > we have x i y x i −− and x i y x i −− . LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 11 qp y y x i x i Figure 7.
Illustration of the proof of Proposition 3.7 .By assumption, add E ( X ) is a precovering subcategory of C ( Z ). So by Lemma 3.6 there mustexist finitely many diagonals U j = { u j , u j } ∈ X for j ∈ { , . . . , l } , such that for each i ∈ Z > there is a j ∈ { , . . . , l } with x i u j y and x i u j y . There must be a j ∈ { , . . . , l } which works for infinitely many values of i ∈ Z > , i.e. there is adiagonal V = { v , v } ∈ X such that for infinitely many values of i ∈ Z > we have x i v y and x i v y . Since they hold for infinitely many i ∈ Z > , the first of these inequalities forces p v y ,while the second forces q v y . As mentioned above, since Z satisfies the two-sided limitcondition, we can pick a sequence of diagonals Y j = { y j , y j } of Z with y j → p from above and y j → q from above and such that p y j x i −− and q y j x i −− for all i, j ∈ Z > (note that if p ∈ Z , respectively q ∈ Z , we can pick y j = p for each j ∈ Z > ,respectively y j = q for each j ∈ Z > ). Applying the above argument for each of the diagonals Y j in this sequence, we find a sequence { v j , v j } ∈ X with v j → p from above and v j → q fromabove. Thus condition PC1 holds. (cid:3) Remark 3.8.
Either of conditions PC1 and PC2 implies the following condition: Suppose X has a right fountain at z ∈ Z converging to a ∈ L ( Z ), that is, a sequence { z, x i } i ∈ Z > with x i → a from below. Then X has a fountain at z converging to a .Namely, if condition PC1 holds, then there is a sequence { x ′ i , x ′ i } i ∈ Z > from X with x ′ i → z from above and x ′ i → a from above. Since Z is discrete, x ′ i = z from some step (see Remark1.3), so X has a left fountain at z converging to a .If condition PC2 holds, the analogous argument works with z in the role of q and a in the roleof p in the definition of condition PC2. y ++1 x x ′ x ′′ s t y y ++0 t x x ′ x ′′ s y Figure 8.
Illustration of Definition 3.9. The set W ( X , Y, t , t ) has elements x , x ′ , and x ′′ among others, and supremum s , where { x , x } , { x ′ , x ′ } , { x ′′ , x ′′ } , { s , s } are diagonals in X . Definition 3.9.
Let X be a set of diagonals of Z , let Y = { y , y } be in X , and let t ∈ [ y ++1 , y ] ∩ Z and t ∈ [ y ++0 , y ] ∩ Z . We write W ( X , Y, t , t ) = (cid:8) x ∈ [ y ++1 , t ] ∩ Z (cid:12)(cid:12) ∃ { x , x } ∈ X with x ∈ [ t , y ] (cid:9) . For u ∈ [ y ++1 , y ] ∩ Z we write W ( X , Y, u , t ) = (cid:8) x ∈ [ t , y ] ∩ Z (cid:12)(cid:12) { u , x } ∈ X (cid:9) . The set W ( X , Y, t , t ) consists of the end points in [ y ++1 , t ] of diagonals in X between thetwo intervals shown in Figure 8. The set W ( X , Y, u , t ) consists of end points in [ t , y ] ofdiagonals of X with other end point u . Lemma 3.10.
Let X be a set of diagonals of Z satisfying conditions PC1 and PC2, let Y = { y , y } be in X , and let t , t and u be as in Definition 3.9. Then the following holds.(i) If the set W := W ( X , Y, t , t ) is non-empty, then s := sup [ y ++1 ,t ] W ∈ Z .(ii) If the set W := W ( X , Y, u , t ) is non-empty, then s := sup [ t ,y ] W ∈ Z .Proof. We start by showing (i). Suppose s / ∈ Z , in particular s = y ++1 and s = t , so s ∈ ( y ++1 , t ). There is a sequence { x i , x i } from X with x i ∈ [ y ++1 , t ], x i ∈ [ t , y ] for each i ∈ Z > and x i → s from below. Passing to a subsequence we can assume x i → ˜ s from belowor above for some ˜ s ∈ [ t , y ]. Note that since y ++1 < s < t y < y ++0 t ˜ s y , we have s = ˜ s . So conditions PC1 and PC2 imply that there is a sequence { x ′ i , x ′ i } from X with x ′ i → s and x ′ i → ˜ s both from above. So for some i ∈ Z > we have s < x ′ i t and ˜ s x ′ i y . In particular, x ′ i ∈ [ y ++1 , t ] and x ′ i ∈ [ t , y ] for these i , so x ′ i ∈ W and the first of the aboveinequalities violates the definition of s as a supremum. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 13 y y +0 = s = s y ++0 s s s y y +1 y ++1 s s s Figure 9.
Illustration of the proof of Theorem 3.1.We now show (ii). Suppose s / ∈ Z , in particular s = t and s = y , so s ∈ ( t , y ). Thereis a sequence { u , x i } from X with x i ∈ [ t , y ] for each i ∈ Z > and x i → s from below. Bycondition PC1 (or PC2) and Remark 3.8 there is a sequence { u , x ′ i } from X with x ′ i → s from above. However, then we obtain s < x ′ i y from some step, violating the definition of s as a supremum. (cid:3) We can now prove Theorem 3.1.
Proof.
If add E ( X ) is precovering, then X satisfies conditions PC1 and PC2 by Proposition3.7.Conversely, assume that X satisfies conditions PC1 and PC2. Let Y = { y , y } be an arbitrarydiagonal of Z . According to Lemma 3.6 we have to show that Y satisfies the following condition:( ∗ ) There exists a finite set of diagonals S = { X , . . . , X l } ⊆ X , such that for eachdiagonal X = { x , x } ∈ X with x y x −− and x y x −− there is an i ∈ { , . . . , l } with X i = { x i , x i } and x x i y and x x i y . We are going to construct inductively a sequence S of diagonals from X , see Figure 9.Set s = s = y +0 . For l >
1, if y ++1 s l − y +0 and y +0 s l − y have already been defined, then we proceed as follows: • If s l − = y ++1 or s l − = y , then we terminate. (Note that for l = 1 this can not happensince { y , y } is a diagonal, i.e. y and y are not neighbouring vertices of Z .) • If s l − = y ++1 and s l − = y , then y ++1 ( s l − ) − y and y ++0 ( s l − ) + y and we set t = ( s l − ) − , t = ( s l − ) + . If W ( X , Y, t , t ) = ∅ then we terminate. (Note that if this happens for l = 1 thenthere are no relevant diagonals X as in condition ( ∗ ), thus ( ∗ ) is trivially satisfied.)If W ( X , Y, t , t ) = ∅ then we set s l = sup [ y ++1 ,t ] W ( X , Y, t , t ) . (3.1)This supremum lies in Z by Lemma 3.10. We then set u = s l and consider the set W ( X , Y, u , t ). It is non-empty since W ( X , Y, t , t ) = ∅ , and we set s l = sup [ t ,y ] W ( X , Y, u , t ) . Note that by construction we have y ++1 . . . < s < s < s y , (3.2) y ++0 s < s < s < . . . y , (3.3)and { s l , s l } ∈ X by Lemma 3.10 for all l > Y of Z . Suppose by contradiction that for some diagonal Y = { y , y } of Z , our construction doesnot terminate. Then by the inequalities (3.2) and (3.3) there must exist a ∈ ( y ++1 , y ) ∩ L ( Z )and b ∈ ( y ++0 , y ) ∩ L ( Z ) such that s l → a from above and s l → b from below. By conditionPC2 for X , there is a sequence { s ′ m , s ′ m } from X such that s ′ m → a from above and s ′ m → b from above. Moreover, there exist m, l ∈ Z > such that s l < s ′ m s l − and b < s ′ m < y . If wehave s ′ m = s l − then { s l − , s ′ m } ∈ X contradicts the definition of s l − as a supremum. Else,since we now have ( s l − ) + < b < s ′ m < y , the diagonal { s ′ m , s ′ m } ∈ X violates the definitionof s l as a supremum.So we have shown that our construction terminates after finitely many steps. By the aboveremarks on the case l = 1 (i.e. that if the construction terminates without defining s and s then condition ( ∗ ) is trivially satisfied) we can assume that the construction provides anon-empty finite set S = (cid:8) { s l , s l } (cid:12)(cid:12) l N (cid:9) of diagonals from X , for some N ∈ N .We now finally show that the set S has the desired property from condition ( ∗ ). Let X = { x , x } ∈ X with x y x −− and x y x −− , i.e. x ∈ [ y ++1 , y ] and x ∈ [ y ++0 , y ].We distinguish two cases. Assume first that there is an l > s l < x s l − . Notethat then l > l = 1 this would violate the definition of s as supremum. Recall fromequation (3.1) that s l = sup [ y ++1 , ( s l − ) − ] W ( X , Y, ( s l − ) − , ( s l − ) + ) , so s l < x s l − implies that there is no diagonal { x , v } ∈ X with v ∈ [( s l − ) + , y ]. Thatis, we must have y ++0 x s l − . We get that x s l − y and x s l − y , so we aredone in this case.Assume now that there is no l > s l < x s l − . This means that x ∈ [ y ++1 , s N ].Since s N +10 has not been defined in our construction and by the choice of s N as supremum wemust have x ∈ [ y ++0 , s N ]. In other words, x s N y and x s N y , and hence condition( ∗ ) is also satisfied in this case. (cid:3) LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 15 sv p ′ tpwqq ′ Figure 10.
Illustration of Lemma 4.3.4.
Torsion pairs in the cluster categories C ( Z )This section proves Theorem 0.9 from the introduction (=Theorem 4.7). To set the scene, recallthe definition of torsion pairs in triangulated categories, due to Iyama and Yoshino [14, def.2.2], following the lead of Dickson [6, p. 224] from the abelian case. Definition 4.1 (Torsion pairs in triangulated categories) . Let T be a triangulated categorywith suspension functor Σ. A pair ( X, Y ) of full subcategories of T is called a torsion pair ifit satisfies the following two axioms.(T1) Hom T ( x, y ) = 0 for all x ∈ X , y ∈ Y .(T2) For each t ∈ T there exist x ∈ X and y ∈ Y and a distinguished triangle x → t → y → Σ x. Lemma 4.2.
Let X be a set of diagonals of Z satisfying condition PC1 or condition PC2and let s, t ∈ Z . If the set U ([ s, t ]) = { z ∈ [ s, t ] ∩ Z | { s, z } ∈ X } is non-empty then its supremum u = sup [ s,t ] U ([ s, t ]) lies in Z .Proof. Assume by contradiction that the supremum u does not lie in Z . Then there is asequence { s, z i } i ∈ Z > from X with z i → u from below. Since X satisfies condition PC1 orcondition PC2, by Remark 3.8 there is a sequence { s, z ′ i } i ∈ Z > from X with z ′ i → u fromabove. Since u / ∈ Z we have u = t and thus u < z ′ i < t for some i ∈ Z > . Then { s, z ′ i } ∈ X violates the definition of u as a supremum. (cid:3) Lemma 4.3.
Let X be a set of diagonals of Z satisfying conditions PC1 and PC2 and thePtolemy condition. Let s ∈ Z and v ∈ L ( Z ) be given, and assume that there exists t ∈ ( s, v ) ∩ Z such that the following condition is satisfied, see Figure 10:For each w ∈ ( t, v ) ∩ Z there exists a diagonal { p, q } ∈ X with s < p < w < q < v. (4.1) Then for each w ∈ ( t, v ) ∩ Z there exists a diagonal { p ′ , q ′ } ∈ X with s < p ′ t < w < q ′ < v. Proof.
Consider the set V = (cid:8) w ∈ ( t, v ) ∩ Z (cid:12)(cid:12) ∄ { p ′ , q ′ } ∈ X with s < p ′ t < w < q ′ < v (cid:9) and suppose that V = ∅ , setting ˜ w = inf [ t,v ] V . We aim for a contradiction.Assume first that ˜ w ∈ Z . In particular, this implies ˜ w ∈ V . By condition (4.1) there exists adiagonal { p, q } ∈ X with s < p < ˜ w < q < v. Since ˜ w ∈ V we must have s < t < p < ˜ w < q < v. (4.2)Therefore ˜ w lies in ( t + , v ) and thus ˜ w − ∈ ( t, v ). Now, because ˜ w is the infimum of V , we have˜ w − / ∈ V and thus we can find { p ′ , q ′ } ∈ X with s < p ′ t < ˜ w − < q ′ < v .This implies s < p ′ t < ˜ w q ′ < v (4.3)and because ˜ w ∈ V we must have q ′ = ˜ w . Combining (4.2) and (4.3) yields s < p ′ t < p < ˜ w = q ′ < q < v, implying that { p ′ , q ′ } ∈ X and { p, q } ∈ X cross. The Ptolemy condition implies that thediagonal { p ′ , q } is in X and we have s < p ′ t < ˜ w < q < v. This contradicts ˜ w ∈ V .Assume now that ˜ w ∈ L ( Z ). We can pick a sequence { w i } i ∈ Z > from V converging to ˜ w from above. Since Z satisfies the two-sided limit condition (cf. Definition 0.1), we can pick asequence { z i } i ∈ Z > from ( t, ˜ w ) ∩ Z converging to ˜ w from below.Because ˜ w is the infimum of V , we have z i / ∈ V for each i ∈ Z > . Thus for each i ∈ Z > thereis a diagonal { x i , x i } ∈ X with s < x i t < z i < x i < v. The last inequality can even be written x i < ˜ w < v for each i ∈ Z > : If we had ˜ w < x i < v foran i ∈ Z > there would be a j ∈ Z > (in fact, infinitely many) with ˜ w < w j < x i which wouldyield s < x i t < ˜ w < w j < x i < v contradicting the fact that w j ∈ V .Having z i < x i < ˜ w for each i ∈ Z > and z i → ˜ w from below forces x i → ˜ w from below. Wehave x i ∈ [ s + , t ] and passing to a subsequence we can assume x i → c from below or above forsome c ∈ [ s + , t ] ∩ Z . Since ˜ w ∈ ( t, v ) ∩ L ( Z ) we have c = ˜ w . By assumption, the set X satisfies conditions PC1 and PC2 and thus there is a sequence { x ′ i , x ′ i } i ∈ Z > of diagonals from X with x ′ i → c from above and x ′ i → ˜ w from above. We can pick i, j ∈ Z > such that s < x ′ i t < ˜ w < w j < x ′ i < v contradicting the fact that w j ∈ V . (cid:3) Definition 4.4.
Let X be a set of diagonals of Z . Then we setnc X = { Y diagonal of Z | Y crosses no X ∈ X } . We write nc X = nc(nc X ). The letters “nc” stand for “non-crossing”. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 17
Lemma 4.5.
Let X be a set of diagonals of Z . If nc X = X , then X satisfies the Ptolemycondition.Proof. Assume { x , x } ∈ X and { y , y } ∈ X cross. According to Definition 0.2 this meansthat we can label the vertices so that x < y < x < y . Consider those of { x , y } , { y , x } , { x , y } and { y , x } which are diagonals of Z . Clearly, any diagonal U of Z crossing one ofthese diagonals must also cross one of { x , x } ∈ X and { y , y } ∈ X , i.e. U nc X . Itfollows that those of { x , y } , { y , x } , { x , y } and { y , x } which are diagonals of Z lie innc X . But by assumption nc X = X , so X satisfies the Ptolemy condition. (cid:3) Lemma 4.6.
Let X be a set of diagonals of Z satisfying conditions PC1 and PC2. If X satisfies the Ptolemy condition, then nc X = X .Proof. The inclusion X ⊆ nc X follows immediately from Definition 4.4 (and does not needany of the assumptions on X ).For the inclusion nc X ⊆ X , let { s, t } ∈ nc X be given. Our proof will be divided intocases and subcases. For each one we will show either that { s, t } ∈ X , or that we can deducea contradiction. Case A: There does not exist z ∈ ( s, t ] ∩ Z such that { s, z } ∈ X . We will show that thisassumption leads to a contradiction.Observe that { s, t } ∈ nc X implies { s − , s + } / ∈ nc X , so there exists a z ∈ Z such that { s, z } ∈ X . By assumption we have z / ∈ ( s, t ], so the set V = (cid:8) z ∈ ( t, s ) ∩ Z (cid:12)(cid:12) { s, z } ∈ X (cid:9) is non-empty. Set v = inf ( t,s ) V . We claim that v ∈ L ( Z ). Assume for a contradiction that v ∈ Z . Then we have { s, v } ∈ X . It follows from the assumption in Case A that { s, t } / ∈ X ,so v ∈ [ t + , s −− ]. Then { s + , v } crosses { s, t } ∈ nc X , whence { s + , v } / ∈ nc X . Thus there isa diagonal { p, q } ∈ X crossing { s + , v } . However, this diagonal can not have s as one of itsendpoints, due to the assumption in Case A and the definition of v as infimum. So we candeduce that the diagonal { p, q } ∈ X crosses the diagonal { s, v } ∈ X ; in particular, one ofthe endpoints, say p , lies in ( s, v ). But then the Ptolemy condition yields that { s, p } ∈ X ,contradicting the assumption in Case A and the definition of v as an infimum.We thus have shown that v ∈ L ( Z ) with t < v < s . From the definition of v as infimum theremust exist a sequence of diagonals { s, v i } i ∈ Z > from X with v i ∈ ( v, s ) and v i → v convergingfrom above. Since Z satisfies the two-sided limit condition (see Definition 0.1), there is also asequence of points in Z converging to v from below; in particular, ( t, v ) ∩ Z is non-empty.For each such w ∈ ( t, v ) ∩ Z we have s + < t < w < v < s , so { s, t } ∈ nc X crosses { s + , w } whence { s + , w } / ∈ nc X . So there is a diagonal { p, q } ∈ X crossing { s + , w } . This diagonalcannot have s as one of its endpoints because of the assumption in Case A and the definition of v as infimum. So we can assume p ∈ [ s ++ , w − ] and q ∈ [ w + , s ). If v < q < s then there existsan i ∈ Z > such that { s, v i } ∈ X and { p, q } ∈ X cross; by the Ptolemy condition it followsthat { s, p } ∈ X , contradicting our assumption in Case A and the definition of v as infimum.Since this argument worked for each w ∈ ( t, v ) ∩ Z , we can apply Lemma 4.3. Thus for each w ∈ ( t, v ) ∩ Z there exists a diagonal { p ′ , q ′ } ∈ X with s + < p ′ t < w < q ′ < v. (4.4) As already mentioned above, the two-sided limit condition yields a sequence { w i } i ∈ Z > with w i → v from below. By (4.4) we can find a sequence { p ′ i , q ′ i } i ∈ Z > of diagonals from X with p ′ i ∈ [ s ++ , t ] and q ′ i ∈ ( w i , v ) for each i ∈ Z > . It is clear that q ′ i → v from below and bycompactness and passing to a subsequence we can assume p ′ i → r from below or above forsome r ∈ [ s ++ , t ]. By conditions PC1 and PC2 there is also a sequence { p ′′ i , q ′′ i } i ∈ Z > from X with p ′′ i → r and q ′′ j → v from above. This implies that there must exist j, l ∈ Z > such that { s, v l } ∈ X and { p ′′ j , q ′′ j } ∈ X cross (more precisely, for each j there are infinitely many l suchthat { s, v l } ∈ X and { p ′′ j , q ′′ j } ∈ X cross). But then the Ptolemy condition gives { s, p ′′ j } ∈ X ,contradicting the assumption in Case A.Therefore we have now shown that Case A cannot occur. Case B: There exists a z ∈ ( s, t ] ∩ Z such that { s, z } ∈ X . Then the set U ([ s, t ]) from Lemma4.2 is non-empty, and by Lemma 4.2 its supremum u = sup [ s,t ] U ([ s, t ]) lies in Z . Subcase B1: We have u = t . Then { s, t } = { s, u } ∈ X and we are done. Subcase B2: We have u ∈ ( s, t ) . We will show that this assumption also leads to a contradic-tion.Again, consider the set V = (cid:8) y ∈ ( t, s ) ∩ Z (cid:12)(cid:12) { s, y } ∈ X (cid:9) . If V = ∅ then a symmetric version of the assumption in Case A is satisfied; so we can deducea contradiction exactly as in Case A. So we can assume that V = ∅ . Set v = inf ( t,s ) V .First suppose v ∈ Z . Then { s, v } ∈ X and t < v < s . Since s < u < t we have that { u, v } is a diagonal of Z which crosses { s, t } . Since { s, t } ∈ nc X , this means that { u, v } / ∈ nc X .So there is a diagonal { p, q } ∈ X which crosses { u, v } and we can assume p ∈ [ u + , v − ] and q ∈ [ v + , u − ].Note that q = s is impossible due to the definition of u as supremum and of v as infimum,respectively. Thus we have q = s ; but then the diagonal { p, q } ∈ X crosses { s, u } ∈ X or { s, v } ∈ X . In either case, the Ptolemy condition implies that { s, p } ∈ X , again contradictingthe choice of u and v as supremum and infimum, because p ∈ [ u + , v − ].Therefore we suppose now that v / ∈ Z , so we have v ∈ L ( Z ) ∩ ( t, s ) as sketched in Figure11. Note that indeed there is a sequence of diagonals { s, v i } from X with v i → v from above,since v = inf ( t,s ) V by definition. For each w ∈ ( t, v ) ∩ Z we have u < t < w < v < s ,so { s, t } ∈ nc X crosses { u, w } whence { u, w } / ∈ nc X . So there is a diagonal { p, q } ∈ X crossing { u, w } and we can suppose p ∈ [ u + , w − ] and q ∈ [ w + , u − ].We claim that q ( v, u − ]. Note that q = s is impossible due to the definition of u as supremumand of v as infimum, respectively. Further, if q ∈ ( v, u − ] then { p, q } ∈ X crosses { s, u } ∈ X or one (actually, infinitely many) of { s, v i } ∈ X . In any case, the Ptolemy condition forces { s, p } ∈ X , a contradiction to the choice of u as supremum or of v as infimum.So we have shown that q ∈ [ w + , v ). To sum up, we have t ∈ ( u, v ) ∩ Z with u ∈ Z and v ∈ L ( Z ) and for each w ∈ ( t, v ) ∩ Z there exists { p, q } ∈ X with u < p < w < q < v. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 19 svutw ···
Figure 11.
Illustration of the proof of Lemma 4.6.Lemma 4.3 implies that for each w ∈ ( t, v ) ∩ Z there exists a diagonal { p ′ , q ′ } ∈ X with u < p ′ t < w < q ′ < v. (4.5)Let { w i } i ∈ Z > be a sequence in ( t, v ) with w i → v from below. By (4.5) we can find a sequence { p ′ i , q ′ i } i ∈ Z > of diagonals from X with u < p ′ i t < w i < q ′ i < v, for each i ∈ Z > . It is clear that q ′ i → v from below and by compactness and passing to asuitable subsequence we can assume p ′ i → r from below or above for some r ∈ [ u + , t ].By conditions PC1 and PC2 there is also a sequence { p ′′ i , q ′′ i } ∈ X with p ′′ i → r from aboveand q ′′ i → v from above. By passing to a subsequence we can assume that p ′′ i ∈ [ r, t ) ⊆ [ u + , t ]for each i ∈ Z > .But then it is clear that there must exist j, l ∈ Z > such that { p ′′ j , q ′′ j } ∈ X crosses { s, v l } ∈ X (see Figure 11). Then the Ptolemy condition yields that { s, p ′′ j } ∈ X , contradicting thedefinition of u as a supremum.Therefore we have finally shown that Subcase B2 cannot occur. (cid:3) The following notation will be useful: If X ⊆ T is an additive subcategory then we writeHom T ( X, y ) = 0 when Hom T ( x, y ) = 0 for each x ∈ X , and Hom T ( y, X ) = 0 when Hom T ( y, x ) =0 for each x ∈ X . We set X ⊥ = { y ∈ T | Hom T ( X, y ) = 0 } , ⊥ X = { y ∈ T | Hom T ( y, X ) = 0 } . The following is Theorem 0.9 from the introduction.
Theorem 4.7.
Let X be a set of diagonals of Z . Then add E ( X ) is the first half of a torsionpair in C ( Z ) if and only if X satisfies conditions PC1, PC2, and the Ptolemy condition.Proof. If Y is a diagonal of Z , then by Section 2(v) we have Y ∈ nc X if and only ifExt C ( Z ) (add E ( X ) , E ( Y )) = 0 , if and only if E ( Y ) ∈ (Σ − add E ( X )) ⊥ . Symmetrically (recall that C ( Z ) is 2-Calabi-Yau), Y ∈ nc X if and only if Ext C ( Z ) ( E ( Y ) , add E ( X )) = 0 , if and only if E ( Y ) ∈ ⊥ (Σ add E ( X )). Thus, X = nc ( X ) if and only ifadd E ( X ) = ⊥ ((add E ( X )) ⊥ ) . Now, by [14, Proposition 2.3], the subcategory add( E ( X )) is the first half of a torsion pair ifand only if add( E ( X )) is precovering and add E ( X ) = ⊥ ((add E ( X )) ⊥ ), which by the aboveis the case if and only if add( E ( X )) is precovering and X = nc X . By Theorem 3.1 andLemmas 4.5 and 4.6, this is equivalent to X satisfying conditions PC1, PC2, and the Ptolemycondition. (cid:3) Cluster tilting subcategories of the cluster categories C ( Z )This section proves Theorems 0.5 and 0.6 from the introduction (=Theorems 5.7 and 5.9). Toset the scene, recall the definition of cluster tilting subcategories of triangulated categories dueto Iyama [13, def. 1.1]. Definition 5.1.
Let T be a triangulated category. A full subcategory X ⊆ T is called weaklycluster tilting if X = (Σ − X ) ⊥ = ⊥ (Σ X ).A subcategory Y ⊆ T is called cluster tilting if it is weakly cluster tilting and functoriallyfinite, i.e. it is precovering (see Definition 3.2) and preenveloping (for each t ∈ T there is amorphism f : t → y with y ∈ Y such that each morphism t → y ′ with y ′ ∈ Y factors through f ). Remark 5.2.
By [15, Lemma 3.2(3)] a full subcategory Y ⊆ T is cluster tilting if and only ifit is weakly cluster tilting and precovering. So we will not need to consider the preenvelopingproperty. Lemma 5.3.
Let X be a set of diagonals of Z satisfying condition PC1 or condition PC2.For z ∈ Z and a ∈ L ( Z ) , define U = (cid:8) u ∈ [ z, a ) ∩ Z (cid:12)(cid:12) { z, u } ∈ X (cid:9) . Then one of the following happens:(i) X has a fountain at z converging to a .(ii) U = ∅ .(iii) s = sup [ z,a ] U ∈ Z .Proof. Assume that (ii) and (iii) do not hold. Then there exists a right fountain at z convergingto the supremum s ∈ L ( Z ). By Remark 3.8 there is even a fountain at z converging to s . Butby definition of s as supremum over the interval [ z, a ] we must have s = a , i.e. (i) holds. (cid:3) Proposition 5.4.
Let X be a maximal set of pairwise non-crossing diagonals of Z , andsuppose that X satisfies condition PC2. For each a ∈ L ( Z ) , the set X has a fountain or aleapfrog converging to a . LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 21
Proof.
Assume that X does not have a fountain converging to a . We will show that it has aleapfrog converging to a .Pick any diagonal { x, y } ∈ X . By switching x and y if necessary we can assume x < y < a .By assumption, X does not have a fountain at x converging to a . Thus, by Lemma 5.3 thereis a maximal s ∈ [ x, a ] ∩ Z such that { x, s } ∈ X .We consider the successor s +1 ∈ Z (this exists since a is a limit point, i.e there are infinitelymany elements of Z in the interval [ s , a )). The diagonal { x, s +1 } is not in X (by maximalityof s ). On the other hand, X is maximal non-crossing, thus { x, s +1 } must be crossed by adiagonal from X . However, this diagonal from X cannot cross { x, s } ∈ X (since X isnon-crossing), so it must have s as one of its endpoints, say { s , x } ∈ X crosses { x, s +1 } .There are now two possibilities, namely x ∈ ( a, x ) ∩ Z or x ∈ ( s +1 , a ) ∩ Z . We claim that,without loss of generality, we can assume x ∈ ( a, x ) . (5.1)Assume to the contrary that x ∈ ( s +1 , a ) ∩ Z . Then we apply Lemma 5.3 to the interval [ s , a ](by assumption there is no fountain at s converging to a ) and hence we can suppose that x is maximal in ( s +1 , a ) ∩ Z with the property that { s , x } ∈ X . Now consider the diagonal { x, x } ; it is not in X (by maximality of s ). Since X is maximal non-crossing, there existsa diagonal in X crossing { x, x } . But this diagonal is not allowed to cross { x, s } ∈ X or { s , x } ∈ X ; so this diagonal must have s as one of its endpoints. Now, by definition of x as maximum, the other endpoint of this diagonal is in the interval ( a, x ). This finishes theargument for (5.1). Thus there is a diagonal { s , x } ∈ X with x ∈ ( a, x ).Now we repeat the above argument starting with the diagonal { x , s } instead of { x, y } . Thenwe obtain a diagonal { x , s } ∈ X where s ∈ ( s , a ) and x ∈ ( a, x ).Inductively, we obtain two infinite sequences ( s i ) i ∈ Z > and ( x i ) i ∈ Z > of points in Z such that x < s < s < s . . . < a and a < . . . < x < x < x < x . Moreover, there exists acorresponding sequence of diagonals { x i , s i } i ∈ Z > in X .The strictly increasing sequence ( s i ) i ∈ Z > must converge from below to some limit point b ∈ L ( Z ), and similarly the strictly decreasing sequence ( x i ) i ∈ Z > must converge from above tosome limit point c ∈ L ( Z ).If b = a = c then the diagonals { x i , s i } i ∈ Z > show that X has a leapfrog converging to a , andwe are done.Otherwise, condition PC2 (which requires two different limit points), applied to the diagonals { x i , s i } i ∈ Z > , yields a sequence { y i , y } of diagonals from X such that y i → b from above and y i → c from above. But then some diagonals of this sequence obviously cross some of thediagonals { x i , s i } , a contradiction to X being non-crossing. (cid:3) The following observation follows easily from the definitions of leapfrog and fountain, see Defi-nition 0.4.
Lemma 5.5.
Let X be a set of pairwise non-crossing diagonals of Z and let a ∈ L ( Z ) .(i) Suppose X has a leapfrog converging to a . Then there cannot be a sequence { x i , y i } i ∈ Z > of diagonals in X such that ( x i ) i ∈ Z > converges to a and ( y i ) i ∈ Z > converges to p forsome p ∈ Z with p = a . (ii) Suppose X has a fountain at z ∈ Z converging to a . Then there cannot be a sequence { x i , y i } i ∈ Z > of diagonals in X such that ( x i ) i ∈ Z > converges to a and ( y i ) i ∈ Z > convergesto p for some p ∈ Z with p = z . Proposition 5.6.
Let X be a set of pairwise non-crossing diagonals of Z . Suppose that foreach a ∈ L ( Z ) there is either a fountain or a leapfrog in X converging to a . Then X satisfiesconditions PC1 and PC2.Proof. According to the definition of the conditions PC1 and PC2 (cf. Definition 3.2), let { x i , x i } i ∈ Z > be a sequence of diagonals from X with x i → p from below and x i → q frombelow or above and p = q .If p, q ∈ Z , then { x i , x i } i ∈ Z > is eventually constant and both conditions PC1 and PC2 aretrivially satisfied with x ′ i = x i and x ′ i = x i .If p ∈ L ( Z ) then by Lemma 5.5(i), X cannot have a leapfrog converging to p , so by assumption X must have a fountain at some z ∈ Z converging to p . By Lemma 5.5(ii) this forces q = z . Therefore X has a fountain at z = q converging to p , so there certainly is a sequence { x ′ i , x ′ i } i ∈ Z > from X with x ′ i → p and x ′ i → z = q from above: we can even chose x ′ i = z = q for each i ∈ Z > .If q ∈ L ( Z ) then an analogous argument works. (cid:3) The following is Theorem 0.5 from the introduction.
Theorem 5.7.
Let X be a set of diagonals of Z . Then add E ( X ) is a cluster tilting subca-tegory if and only if X is a maximal set of pairwise non-crossing diagonals, such that for each a ∈ L ( Z ) , the set X has a fountain or a leapfrog converging to a .Proof. By Remark 5.2, the subcategory add E ( X ) is cluster tilting if and only if it is weaklycluster tilting and precovering.It is straightforward from the description of the Ext spaces in Section 2(v) that add E ( X ) isweakly cluster tilting if and only if X is a maximal set of pairwise non-crossing diagonals.Recall from Theorem 3.1 that add E ( X ) is a precovering subcategory of C ( Z ) if and only if X satisfies conditions PC1 and PC2.So it remains to show that if X is a maximal set of pairwise non-crossing diagonals, then X satisfies conditions PC1 and PC2 if and only if for each a ∈ L ( Z ), there is a leapfrog or afountain in X converging to a . But these two implications have been shown in Propositions5.4 and 5.6, respectively. (cid:3) Remark 5.8. If Z has precisely one limit point, then the assertion of Theorem 5.7 was alreadyestablished in [9, Theorem B]. In fact, the condition of being locally finite appearing there isequivalent to the existence of a leapfrog converging to the unique limit point.Figure 12 shows an example of a maximal set of pairwise non-crossing diagonals X of Z for which the corresponding subcategory add E ( X ) is not cluster tilting (only weakly clustertilting). In fact, neither limit point has a fountain or a leapfrog converging to it. LUSTER TILTING SUBCATEGORIES OF IGUSA–TODOROV CLUSTER CATEGORIES 23
Figure 12.
A maximal pairwise non-crossing set of diagonals of Z correspond-ing to a subcategory of C ( Z ) which is not cluster tilting. Neither limit pointhas a fountain or a leapfrog converging to it.Note that X satisfies condition PC1 (because no sequence of diagonals from X satisfies theassumption in PC1), but not condition PC2. This shows that the conclusion of Proposition 5.4would not be true if only condition PC1 was assumed.The following is Theorem 0.6 from the introduction. Theorem 5.9.
The cluster tilting subcategories of C ( Z ) form a cluster structure in the senseof [2, sec. II.1] .Proof. It is enough to verify the conditions in [2, thm. II.1.6].The first condition is that C ( Z ) has a cluster tilting subcategory. This follows from Theorem5.7.The second condition is that if T ⊆ C ( Z ) is a cluster tilting subcategory, then the quiver of T has no loops or 2-cycles. Recall that up to isomorphism, each indecomposable object of T has the form E ( X ) by Section 2(iii).The space Hom C ( Z ) ( E ( X ) , E ( X )) is 1-dimensional over the ground field k by Section 2(vi), soeach non-zero morphism E ( X ) → E ( X ) is invertible whence the quiver of T has no loops.Let E ( X ) = E ( Y ) be indecomposable objects in T and assume Hom C ( Z ) ( E ( X ) , E ( Y )) = 0.By Section 2(vi) we can write X = { x , x } and Y = { y , y } with x y x −− < x y x −− . (5.2)If x = y and x = y then X and Y would cross, contradicting Ext C ( Z ) ( E ( X ) , E ( Y )) = 0which holds since E ( X ) , E ( Y ) ∈ T . Without loss of generality we can suppose x = y and x = y . (5.3)Suppose we had Hom C ( Z ) ( E ( Y ) , E ( X )) = 0. By Section 2(vi) again we would have y x y −− < y x y −− or y x y −− < y x y −− . But each is incompatible with the combination of (5.2) and (5.3), so Hom C ( Z ) ( E ( Y ) , E ( X )) =0. Hence there is no 2-cycle between E ( X ) and E ( Y ) in the quiver of T . (cid:3) Remark 5.10.
For most admissible sets Z , the cluster structure in Theorem 5.9 is differ-ent from the one in [12, Theorem 2.4.1], where the clusters are not necessarily cluster tiltingsubcategories.Namely, the convergence condition in [12, Theorem 2.4.1] only asks that for each right (respec-tively left) fountain at a point z ∈ Z converging to a limit point a ∈ L ( Z ), there be a left(respectively right) fountain at z converging to the same limit point a (cf. [12, Definition 2.4.6]).In fact, the clusters in [12, Theorem 2.4.1] coincide with cluster tilting subcategories if andonly if Z is finite or has exactly one limit point. Figure 12 yields an example of a clusterin the sense of [12, Theorem 2.4.1] (there is no right or left fountain, so the condition in [12,Definition 2.4.6] is empty) which does not correspond to a cluster tilting subcategory. Acknowledgements.
We thank Charles Paquette, Adam-Christiaan van Roosmalen, and BinZhu for illuminating comments on a preliminary version, and the referee for a careful readingand several useful suggestions which have improved the presentation.This project was supported by grant HO 1880/5-1 under the research priority programme SPP1388 “Darstellungstheorie” of the DFG, and by grant EP/P016014/1 “Higher DimensionalHomological Algebra” from the EPSRC.
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