Cluster categories of type A ∞ ∞ and triangulations of the infinite strip
aa r X i v : . [ m a t h . R T ] M a y CLUSTER CATEGORIES OF TYPE A ∞∞ ANDTRIANGULATIONS OF THE INFINITE STRIP
SHIPING LIU AND CHARLES PAQUETTE
Abstract.
We first study the (canonical) orbit category of the bounded de-rived category of finite dimensional representations of a quiver with no infinitepath, and we pay more attention on the case where the quiver is of infiniteDynkin type. In particular, its Auslander-Reiten components are explicitlydescribed. When the quiver is of type A ∞ or A ∞∞ , we show that this orbitcategory is a cluster category, that is, its cluster-tilting subcategories form acluster structure as defined in [3]. When the quiver is of type A ∞∞ , we shallgive a geometrical description of the cluster structure of the cluster categoryby using triangulations of the infinite strip in the plane. In particular, we shallshow that the cluster-tilting subcategories are precisely given by compact tri-angulations. Introduction
One of the most important developments of the representation theory of quivers isits interaction with the theory of cluster algebras. Cluster algebras were introducedby Fomin and Zelevinsky in connection with dual canonical bases and total posi-tivity in semi-simple Lie groups; see [6, 7]. The two theories are linked together viathe notion of cluster categories introduced by Buan, Marsh, Reineke, Reiten andTodorov in [4]. In its original definition, a cluster category is the orbit category ofthe bounded derived category of finite dimensional representations of a finite acyclicquiver under an auto-equivalence, which is the composite of the shift functor andthe inverse Auslander-Reiten translation. In such a cluster category, cluster tiltingobjects correspond to clusters of the cluster algebra defined by the same quiver,and replacing an indecomposable direct summand of a cluster tilting object by an-other non-isomorphic indecomposable object correspond to mutation of a clustervariable within a cluster. For cluster categories of type A n , Caldero, Chapoton andSchiffler gave a beautiful geometrical realization in terms of triangulations of the( n + 3)-gon; see [5]. Later on, by replacing cluster tilting objects by cluster tilt-ing subcategories, Buan, Iyama, Reiten and Scott introduced the notion of clusterstructure in a 2-Calabi-Yau triangulated category; see [3]. A canonical example ofa 2-Calabi-Yau category is the orbit category of the bounded derived category of a Mathematics Subject Classification.
Key words and phrases.
Representations of infinite Dynkin quivers; derived categories; 2-Calabi-Yau categories; Auslander-Reiten theory; cluster categories; cluster-tilting subcategories;geometric triangulations.Both authors were supported in part by the Natural Science and Engineering Research Councilof Canada, while the second-named author was also supported in part by the Atlantic Associa-tion for Research in the Mathematical Sciences and by the Department of Mathematics at theUniversity of Connecticut.
Hom-finite hereditary abelian category under the composite of the shift functor andthe inverse Auslander-Reiten translation, when there is such a translation; see [14].In general, cluster tilting subcategories in a 2-Calabi-Yau category do not necessar-ily form a cluster structure. If this is the case, then the 2-Calabi-Yau category willbe called a cluster category . In [9], Holm and Jørgensen studied a cluster categoryof infinite Dynkin type A ∞ , which is the derived category of differential gradedmodules with finite dimensional homology over the polynomial ring viewed as adifferential graded algebra, and whose Auslander-Reiten quiver is of shape ZA ∞ .It was mentioned that this category is equivalent to the canonical orbit categoryof the bounded derived category of finite dimensional representations of a quiverof type A ∞ with the zigzag orientation. More importantly, its cluster structureadmits a geometrical realization in terms of triangulations of the infinity-gon, anatural generalization of the A n case.The aim of this paper is to extend the above-mentioned work of Holm andJørgensen from a representation theoretic point of view. Indeed, let Q be a locallyfinite quiver with no infinite path. It is well known that the category rep( Q ) of finitedimensional representations of Q is a Hom-finite hereditary abelian category suchthat D b (rep( Q )) has almost split triangles and hence admits a Serre functor; see [2,(7.11)]. Therefore, the canonical orbit category C ( Q ) of D b (rep( Q )), as mentionedabove, is a natural candidate for a cluster category of type Q . By making use ofsome results obtained in [2], we shall describe the Auslander-Reiten components of C ( Q ); see (4.1) and prove that the projective representations in rep( Q ) generate acluster-tilting subcategory of C ( Q ); see (4.4). In case Q is of infinite Dynkin type,every indecomposable object of C ( Q ) is rigid, and consequently, a cluster-tiltingsubcategory of C ( Q ) is simply a maximal rigid subcategory which is functoriallyfinite; see (4.7). However, we shall only prove that C ( Q ) is a cluster category incase Q is of types A ∞ or A ∞∞ ; see (4.9). Nevertheless, we conjecture that C ( Q )is a cluster category whenever Q is locally finite without infinite paths. In case Q is of type A ∞∞ , we shall use an infinite strip B ∞ with marked points in the planeas a geometric realization of C ( Q ). We shall parameterize the indecomposableobjects in C ( Q ) by arcs in B ∞ in such a way that two indecomposable objectshave no non-trivial extension between them if and only the corresponding arcs donot cross; see (6.4). Therefore, maximal rigid subcategories of C ( Q ) correspondto triangulations of B ∞ ; see (6.4), and cluster-tilting subcategories correspond tocompact triangulations of B ∞ ; see (6.9). We shall give an easy criterion for atriangulation to be compact; see (5.20). In this way, it yields a complete descriptionof the cluster-tilting subcategories. Moreover, it also enables us to count the numberof connected components of the quiver of a cluster-tilting subcategory; see (6.13).To conclude this introduction, we should mention that triangulations of B ∞ werefirst considered in [11], and further studied in [10] as for a geometrical model of aclass of cluster categories constructed in a different approach.1. Preliminaries
Throughout this paper, k stands for an algebraically closed field. The standardduality for the category of finite dimensional k -vector spaces will be denote by D . Throughout this section, A stands for a Hom-finite Krull-Schmidt additive k -category, whose Jacobson radical will be written as rad( A ). A radical morphism in LUSTER CATEGORIES 3 A is a morphism lying in rad( A ). A short cycle in A is a sequence X / / Y / / X of non-zero radical morphisms between indecomposable objects. Given two collec-tions Θ and Ω of objects of A , we write Hom A ( Θ , Ω ) = 0 if Hom A ( X, Y ) = 0for all objects X ∈ Θ and Y ∈ Ω . We shall say that Θ and Ω are orthogonal ifHom A ( Θ , Ω ) = 0 and Hom A ( Ω , Θ ) = 0 . A morphism f : M → N in A is called right almost split in A if it is not aretraction and every non-retraction morphism g : X → N in A factors through f ;and right minimal if every factorization f = f h implies that h is an automorphism.If f is right minimal and right almost split in A , then it is called a sink morphism for N . In dual situations, one says that f is left almost split, left minimal , and a sourcemorphism for M . One says that A has source (respectively, sink) morphisms ifthere exists a source (respectively, sink) morphism for every indecomposable objectof A . An almost split sequence in A is a sequence of morphisms L f / / M g / / N with M = 0 such that f is a source morphism and a pseudo-kernel of g, and g is asink morphism and a pseudo-cokernel of f . Such an almost split sequence is uniquefor L and for N ; see [16]. Thus, we may write L = τ A N and N = τ − A L , and call τ A the Auslander-Reiten translation for A . If no confusion is possible, τ A will besimply written as τ .For convenience, we state the following well known result; see, for example, [15].1.1. Lemma.
Given any morphism f : X → Y in A , there exist decompositions (1) f = ( f , f ) : X = X ⊕ X → Y , where f is right minimal and f = 0; and (2) f = (cid:0) g g (cid:1) : X → Y ⊕ Y , where g is left minimal and g = 0 . The
Auslander-Reiten quiver Γ A of A is a translation quiver endowed with theAuslander-Reiten translation τ A , whose underlying quiver is defined as follows;see, for example, [16]. The vertex set is a complete set of representatives of theisomorphism classes of indecomposable objects in A . The number of arrows froma vertex X to a vertex Y is the k -dimension ofirr( X, Y ) := rad A ( X, Y ) / rad A ( X, Y ) . Let Γ be a connected component of Γ A . One defines the path category k Γ , aswell as the mesh category k ( Γ ), of Γ over k ; see, for example, [20]. If u ∈ k Γ , thenits image in k ( Γ ) will be denoted by ¯ u . Recall that Γ is called standard if k ( Γ ) isequivalent to the full subcategory A ( Γ ) of A generated by the objects lying in Γ ;see [18, 20].1.2. Lemma.
Let Γ be a connected component of Γ A of shape ZA ∞ or ZA ∞∞ . If p, q are two parallel paths in Γ , then ¯ p = ± ¯ q in k ( Γ ) .Proof. Let p, q be parallel paths from an object X to an object Y . Observing thatparallel paths in Γ have the same length, we shall proceed by induction on thelength of p . The lemma holds trivially if p is of length zero. Suppose that p is ofpositive length n . If p, q have the same terminal arrow, then the lemma followsimmediately from the induction hypothesis. Otherwise, p = α p and q = α q ,where p , q are paths, and α , α are two distinct arrows in Γ . If p is sectional,then it is the only path in Γ from X to Y , a contradiction. Thus, p is not sectional, SHIPING LIU AND CHARLES PAQUETTE and hence, Γ has a path u : X τ Y . Let β , β be the two arrows starting in τ Y such that α β + α β is a mesh relation in k Γ . By the induction hypothesis,¯ p = ± ¯ β ¯ u and ¯ q = ± ¯ β ¯ u . This yields¯ p = ¯ α ¯ p = ± ¯ α ¯ β ¯ u = ± ¯ α ¯ β ¯ u = ± ¯ α ¯ q = ± ¯ q. The proof of the lemma is completed.Let Γ be a connected component of Γ A of shape ZA ∞ or ZA ∞∞ . For each object X ∈ Γ , we define the forward rectangle R X of X to be the full subquiver of Γ generated by its successors Y such that, for any path p : X Y and anyfactorization p = vu with paths u : X Z and v : Z Y , either u is sectional, orelse, Z has two distinct immediate predecessors. The backward rectangle R X in Γ of X is defined in a dual manner.1.3. Proposition.
Let A be a Hom-finite Krull-Schmidt additive k -category, andlet Γ be a standard component of Γ A of shape ZA ∞ or ZA ∞∞ . If X, Y ∈ Γ , then
Hom A ( X, Y ) = 0 if and only if Y ∈ R X if and only if X ∈ R Y . In this case,moreover,
Hom A ( X, Y ) is one-dimensional over k .Proof. By hypothesis, there exists an equivalence ϕ : k ( Γ ) → A ( Γ ), which actsidentically on the objects. For each arrow α : M → N ∈ Γ , write f α = ϕ (¯ α ), whichis irreducible in A ; see [18, (1.3)]. Since ¯ α forms a k -basis for Hom k ( Γ ) ( M, N ), theset { f α } is a k -basis for Hom A ( M, N ). More generally, for a path p in Γ , we write f p = ϕ (¯ p ).Let X, Y ∈ Γ . First of all, it is clear that Y ∈ R X if and only if X ∈ R Y . Hence,we need only to consider forward rectangles. Suppose that Y R X . Then either Y is not a successor of X in Γ , or else, Γ has a path p : X Y which factors througha monomial mesh relation. In the first case, it is clear that Hom k ( Γ ) ( X, Y ) = 0. Inthe second case, ¯ p = 0, and by Lemma 1.2, ¯ q = 0 for all paths q : X Y . Thus,Hom k ( Γ ) ( X, Y ) = 0. As a consequence, Hom A ( X, Y ) = 0.Suppose now that Y ∈ R X . Let p : X Y be a path in Γ . It suffices to provethat { f p } is a k -basis for Hom A ( X, Y ). For this purpose, we proceed by inductionon the length of p . If p is trivial, then the claim is evident. Otherwise, p = αq ,where q : X U is a path and α : U → Y is an arrow. Observe that U ∈ R X . Bythe induction hypothesis, { f q } is a k -basis for Hom A ( X, U ) = 0. If p is sectional,then ¯ p = 0, and p is the only path in Γ from X to Y . Therefore, { ¯ p } is a k -basisfor Hom k ( Γ ) ( X, Y ), and consequently, { f p } is a k -basis for Hom A ( X, Y ). Assume p is not sectional. Then Γ has a path w : X τ Y . By the induction hypothesis, { f w } is a k -basis for Hom A ( X, τ Y ) . Moreover, by definition, there exists a binomialmesh relation αγ + βδ from τ Y to Y , where γ : τ Y → U , and β : V → Y , and δ : τ Y → V are arrows in Γ . Then A has an almost split sequence( ∗ ) τ Y ( fγfδ ) / / U ⊕ V ( f α ,f β ) / / Y. Suppose to the contrary that f p = 0. That is, f α f q = 0. In view of the pseudo-exactness of ( ∗ ), A has a morphism u : X → τ Y such that f q = f γ u and f δ u = 0.Since f q = 0, we see that u = 0. Again by the induction hypothesis, { f w } is a k -basis for Hom A ( X, τ Y ) and { f δw } is a k -basis for Hom A ( X, V ). In particular, u = λf w for some λ ∈ k ∗ . However, this gives rise to λf δw = f δ ( λf w ) = f δ u = 0,and hence f δw = 0, a contradiction. Therefore, f p = 0. As a consequence, ¯ p = 0. LUSTER CATEGORIES 5
Applying Lemma 1.2, we deduce easily that { ¯ p } is a k -basis for Hom k ( Γ ) ( X, Y ),and therefore, { f p } is a k -basis for Hom A ( X, Y ). The proof of the proposition iscompleted.If Γ is a translation quiver of shape ZA ∞∞ , then the forward rectangle of a vertexis the full subquiver generated by its successors. Thus, the following statement isan immediate consequence of Proposition 1.3.1.4. Corollary.
Let A be a Hom-finite Krull-Schmidt additive k -category, and letΓ be a standard component of Γ A of shape ZA ∞∞ . For any objects
X, Y ∈ Γ , wehave
Hom A ( X, Y ) = 0 if and only if Y is a successor of X in Γ . Let D be a full subcategory of A . Recall that D is covariantly finite in A providedthat every object X of A admits a left D -approximation , that is, a morphism f : X → M in A such that every morphism g : X → N with N ∈ D factorsthrough f ; and contravariantly finite in A provided that every object X of A admits a right D -approximation , that is, a morphism f : M → X such that everymorphism g : N → X with N ∈ D factors through f ; and functorially finite in A ifit is is both covariantly and contravariantly finite in A . Now, we shall say that D is covariantly bounded in A if, for any object X ∈ A , there exists at most finitelymany non-isomorphic indecomposable objects M ∈ D such that Hom A ( X, M ) = 0;and dually, D is contravariantly bounded in A if, for any object X ∈ A , there existsat most finitely many non-isomorphic indecomposable objects M ∈ D such thatHom A ( M, X ) = 0.1.5. Lemma.
Let D be a full subcategory of A . If D is covariantly ( respectively,contravariantly ) bounded in A , then it is covariantly ( respectively, contravariantly ) finite in A .Proof. We shall prove only one part of the statement. Suppose that D is con-travariantly bounded in A . Let X be an object in A . Let M , . . . , M n be the non-isomorphic indecomposable objects in D such that Hom A ( M i , X ) = 0, i = 1 , . . . , n. For each 1 ≤ i ≤ n , choose a k -basis { f i , . . . , f i,d i } of Hom A ( M i , X ) and set f i = ( f i , . . . , f i,d i ) : M d i i → X , where M d i i denotes the direct sum of d i copies of M i . Since A is Krull-Schmidt, it is easy to see that f = ( f , · · · , f n ) : M d ⊕ · · · ⊕ M d n n → X is a right D -approximation for X . The proof of the lemma is completed.For the rest of this section we assume, in addition, that A is a triangulatedcategory, whose shift functor is denoted by [1]. A Serre functor for A is an auto-equivalence S of A such that, for any objects X, Y ∈ A , there exists a naturalisomorphism Hom A ( X, Y ) → D Hom A ( Y, S ( X )) . If such a Serre functor S exists, then A has almost split triangles and its Ausander-Reiten translation is given by the composite of the shift by − S ; see [19]. Now, A is called 2- Calabi-Yau if the shift by 2 functor is a Serre functor;see [4]. The following easy observation is important for our investigation.
SHIPING LIU AND CHARLES PAQUETTE
Lemma.
Let A be a 2-Calabi-Yau triangulated k -category. For any objects X, Y in A , there exists a natural isomorphism Hom A ( X, Y [1]) ∼ = D Hom A ( Y, X [1]) . We say that a full subcategory T of A is strictly additive provided that T isclosed under isomorphisms, finite direct sums, and taking summands. We recallthe following definition from [3], which is our main objective of study.1.7. Definition.
Let A be a 2-Calabi-Yau triangulated category. A strictly addi-tive category T of A is called weakly cluster-tilting provided that, for any X ∈ A ,Hom A ( T , X [1]) = 0 if and only if X ∈ T ; and cluster-tilting provided that T isweakly cluster-tilting and functorially finite in A .Let T be a strictly additive subcategory of A . In particular, T is a Krull-Schmidt additive category. By definition, the quiver of T , denoted by Q T , is theunderlying quiver of its Auslander-Reiten quiver. For each indecomposable object M of T , we shall denote by T M the full additive subcategory of T generated bythe indecomposable objects not isomorphic to M . Observe that T M is also strictlyadditive.1.8. Proposition.
Let A be a -Calabi-Yau triangulated category. If T is a cluster-tilting subcategory of A , then it has source morphisms and sink morphisms ; andconsequently, its quiver Q T is locally finite.Proof. Let T be a cluster-tilting subcategory of A . Suppose that M is an inde-composable object in T . Then T M is functorially finite in A ; see [13, (4.1)]. Let f : X → M be a right T M -approximation of M . By Lemma 1.1(1), f restricts to aright minimal morphism g : Y → M , where Y is a direct summand of X . Observethat g is a minimal right T M -approximation of M .If rad(End A ( M )) = 0, then g is right almost split, and hence, a sink morphismfor M in T . Otherwise, pick a k -basis { h , . . . , h m } of rad(End A ( M )) and set h = ( h , · · · , h m ) : M m → M . We see that every radical endomorphism M → M factors through h . As a consequence, u = ( g, h ) : Y ⊕ M m → M is right almostsplit in T . By Lemma 1.1(1), u restricts to a right minimal morphism v : Z → M ,where Z is a direct summand of Y ⊕ M m . Note that v is also right almost split,and hence, a sink morphism for M in T . Dually, we may show that M admits asource morphism in T . The proof of the proposition is completed.For convenience, we reformulate the notion of a cluster structure without coeffi-cients of a 2-Calabi-Yau triangulated category, originally introduced in [3].1.9. Definition.
Let A be a 2-Calabi-Yau triangulated category. A non-emptycollection C of strictly additive subcategories of A is called a cluster structure pro-vided that, for any T ∈ C and any indecomposable object M ∈ T , the followingconditions are verified.(1) The quiver of T contains no oriented cycle of length one or two.(2) There exists a unique (up to isomorphism) indecomposable object M ∗ ∈ A with M ∗ = M such that the additive subcategory of A generated by T M and M ∗ , written as µ M ( T ), belongs to C . LUSTER CATEGORIES 7 (3) The quiver of µ M ( T ) is obtained from the quiver of T by the Fomin-Zelevinskymutation at M as described in [7, (1.1)]; see also [3, Section II].(4) There exist two exact triangles in A as follows : M f / / N g / / M ∗ / / M [1] and M ∗ u / / L v / / M / / M ∗ [1] , where f, u are minimal left T M -approximations, and g, v are minimal right T M -approximations in A .The following notion is our main objective of study.1.10. Definition.
Let A be a 2-Calabi-Yau triangulated category. We shall call A a cluster category if its cluster-tilting subcategories form a cluster structure.2. Representations of quivers
Throughout this section, let Q stand for a connected locally finite quiver with noinfinite path, whose vertex set is written as Q . By K¨onig’s Lemma, the number ofpaths between any two pre-fixed vertices is finite. Thus, Q is strongly locally finite,and hence, the representation theory of Q obtained in [2] applies in this case. Foreach x ∈ Q , let P x , I x , and S x be the indecomposable projective representation,the indecomposable injective representation, and the simple representation at x ,respectively, which are defined in a canonical way; see, for example, [2, Section 1].Let rep( Q ) denote the category of finite dimensional k -linear representations of Q .This is a Hom-finite hereditary abelian category, having almost split sequences; see[2, 18]. The Auslander-Reiten quiver Γ rep( Q ) of rep( Q ) is chosen to contain therepresentations P x , I x and S x with x ∈ Q , and its Auslander-Reiten translation iswritten as τ Q . The following result is implicitly stated in the proof of Theorem 2.8in [2].2.1. Lemma.
Let Q be a connected locally finite quiver with no infinite path. If M, N are representations lying in Γ rep( Q ) , then D Hom rep( Q ) ( N, τ Q M ) ∼ = Ext Q ) ( M, N ) ∼ = D Hom rep( Q ) ( τ − Q N, M ) . Recall that Γ rep( Q ) has a unique preprojective component P containing all the P x with x ∈ Q , a unique preinjective component I containing all the I x with x ∈ Q , and possibly some other regular components which contain none of the P x and I x with x ∈ Q ; see [2].2.2. Theorem.
Let Q be a connected locally finite quiver with no infinite path. (1) The preprojective component P of Γ rep( Q ) is standard of shape N Q op . (2) The preinjective component I of Γ rep( Q ) is standard of shape N − Q op with Hom rep( Q ) ( I , P ) = 0 . (3) If R is a regular component of Γ rep( Q ) , then it is of shape ZA ∞ such that Hom rep( Q ) ( I , R ) = 0 and Hom rep( Q ) ( R , P ) = 0 . Proof.
Let f : M → N be a non-zero morphism in rep( Q ), where M, N ∈ Γ rep( Q ) .Suppose that M is preinjective. Then there exists some r ≥ τ − r M = I x for some x ∈ Q . If N is not preinjective, then τ − r N is defined and not injective. Onthe other hand, by Lemma 2.1, we have a non-zero morphism g : τ − r M → τ − r N . SHIPING LIU AND CHARLES PAQUETTE
Since rep( Q ) is hereditary, τ − r N is injective, a contradiction. Dually, if N ispreprojective, then so is M . The rest of the theorem has already been establishedin [18]. The proof of the theorem is completed.Let Γ be a regular component of Γ rep( Q ) , and let X be a representation lying in Γ . One says that X is quasi-simple if it has only one immediate predecessor in Γ .In general, since Γ is of shape ZA ∞ , it has a unique sectional path X = X n / / X n − / / · · · / / X with X being quasi-simple. One defines then the quasi-length ℓ ( X ) of X to be n .Let Q be a quiver of type A ∞∞ with no infinite path. A vertex a lying on apath p is called a middle point of p if a is neither the starting point nor the endingpoint. A string in Q is a finite reduced walk w , to which one associates a stringrepresentation M ( w ); see [2, Section 5]. Let a i , b i , i ∈ Z , be the source vertices andthe sink vertices, respectively, in Q such that there exist paths p i : a i b i and q i : a i b i − in Q , for i ∈ Z . Let Q R be the union of the paths p i with i ∈ Z andthe trivial paths ε a with a being a middle point of q j for some j ∈ Z . Dually, let Q L be the union of the paths q i with i ∈ Z and the trivial paths ε b with b beinga middle point of p j for some j ∈ Z . These notations will allow us to describe thequasi-simple regular representations as follows; see [2, (5.15)].2.3. Lemma.
Let Q be a quiver of type A ∞∞ . If Q contains no infinite path, thenΓ rep( Q ) has exactly two regular components R R and R L such that the quasi-simplerepresentations in R R are the string representations M ( p ) with p ∈ Q R , and thosein R L are the string representations M ( q ) with q ∈ Q L . In the infinite Dynkin case, we will have an explicit description of all Auslander-Reiten components of rep( Q ).2.4. Theorem.
Let Q be an infinite Dynkin quiver. If Q has no infinite path, thenthe connected components of Γ rep( Q ) are all standard and consist of the preprojectivecomponent P , the preinjective component I , and r regular components, where (1) r = 0 in case Q is of type A ∞ ;(2) r = 1 in case Q is of type D ∞ ;(3) r = 2 in case Q is of type A ∞∞ ; and in this case, the two regular componentsare orthogonal.Proof. We need only to prove the second part of Statement (3), since all other partsare known; see [18] and [2, (5.16), (5.22)]. Assume now that Q is of type A ∞∞ withno infinite path. By Lemma 2.3, Γ rep( Q ) has exactly two regular components R R and R L , both are of shape ZA ∞ . Suppose that rep( Q ) has a non-zero morphism f : M → N with M ∈ R R and N ∈ R L . We may assume that m = ℓ ( M ) + ℓ ( N )is minimal with respect to this property. Suppose that ℓ ( N ) > . Then rep( Q ) hasa short exact sequence 0 / / X u / / N v / / Y / / , where X, Y ∈ R L with ℓ ( X ) = ℓ ( N ) − ℓ ( Y ) = 1. By the minimality of m , wehave vf = 0, and hence, f = uw for some non-zero morphism w : M → X , whichcontradicts the minimality of m . Therefore, ℓ ( N ) = 1, and dually, ℓ ( M ) = 1. Thatis, M, N are quasi-simple such that their supports intersect.
LUSTER CATEGORIES 9
Let a i , b i , i ∈ Z , be the source vertices and the sink vertices, respectively suchthat Q has paths p i : a i b i and q i : a i b i − in Q , for i ∈ Z . By Lemma2.3, M = M ( p r ) with r ∈ Z or M = M ( ε a ), where a is a middle point of some q s with s ∈ Z . In the first case, N = M ( q i ) with r ≤ i ≤ r + 1 or N = M ( ε b )with b a middle point of p r . Since Hom rep( Q ) ( M, N ) = 0, the top of M , that isthe simple representation S a r , appears as a composition factor of N . Therefore, N = M ( q r ). Hence, the socle of N , that is, the simple representation S b r − , isa composition factor of M , which is absurd. In the second case, M = S a and N = M ( q s ). Since a is a middle point of q s , we see that Hom rep( Q ) ( S a , M ( q s )) = 0,a contradiction. This shows that Hom rep( Q ) ( R R , R L ) = 0 . Similarly, one can showthat Hom rep( Q ) ( R L , R R ) = 0 . The proof of the theorem is completed.In case Q is of type A ∞ or A ∞∞ , we shall be able to obtain more information onthe morphisms between the indecomposable representations.2.5. Lemma.
Let Q be a quiver of type A ∞ or A ∞∞ , containing no infinite path. If X, Y ∈ Γ rep( Q ) , then Hom rep( Q ) ( X, Y ) is at most one-dimensional over k .Proof. Let
X, Y ∈ Γ rep( Q ) be such that Hom rep( Q ) ( X, Y ) = 0. Assume first that X is preprojective, that is X = τ − n P x for some n ≥ x ∈ Q . If τ n Y = 0, then Y = τ − m P y for some 0 ≤ m < n . By Lemma 2.1,Hom rep( Q ) ( τ m − n P x , P y ) ∼ = Hom rep( Q ) ( X, Y ) = 0 . Then τ m − n P x is projective; see [2, (4.3)], which is absurd. Now, applying Lemma2.1 again, we obtain Hom rep( Q ) ( X, Y ) ∼ = Hom rep( Q ) ( P x , τ n Y ) , which is one-dimensional over k because τ n Y is a string representation; see [2,(5.9)]. Dually, the result holds if Y is preinjective.Assume now that X is regular. By Theorem 2.2, Y is regular or preinjective. Weonly need to consider the case where Y is regular. By Theorem 2.4, X, Y belongto a connected component of Γ rep( Q ) , which is standard of shape ZA ∞ . Thus, theresult follows from Proposition 1.3. Finally, if X is preinjective, then so is Y byTheorem 2.2, and hence, the statement holds. The proof of the lemma is completed.Let Γ be a connected component of Γ rep( Q ) of shape ZA ∞ , containing a quasi-simple representation S . Observe that Γ has a unique ray starting in S , written as( S → ), and a unique co-ray ending in S written as ( → S ). We denote by W ( S ) thefull subquiver of Γ generated by the representations X for which there exist paths M X N , where M ∈ ( → S ) and N ∈ ( S → ), and call it the infinite wing with wing vertex S ; compare [20].2.6. Proposition.
Let Q be a quiver of type A ∞∞ , containing no infinite path. If X ∈ Γ rep( Q ) is preprojective, then each regular component R of Γ rep( Q ) has a uniquequasi-simple S X such that, for any Y ∈ R , we have Hom rep( Q ) ( X, Y ) = 0 if andonly if Y ∈ W ( S X ); and any non-zero morphism f : X → Y factors through arepresentation lying on the co-ray ( → S X ) ending in S X .Proof. Let a i , b i , i ∈ Z , be the source vertices and the sink vertices, respectivelysuch that Q has paths p i : a i b i and q i : a i b i − , i ∈ Z . By Lemma 2.3, Γ rep( Q ) has exactly two regular components R R and R L . We shall consider onlythe case where R = R R . Then the quasi-simple representations in R are the string representations M ( p ) with p ∈ Q R , where Q R denotes the union of the p i with i ∈ Z , and the trivial paths ε a , where a ranges over the set of middle points of the q j with j ∈ Z . Let X be a preprojective representation in Γ rep( Q ) . Since R is τ Q -stable, in viewof Lemma 2.1, we may assume that X = P x , for some x ∈ Q . Since x appearsin exactly one of the paths in Q R , we have a unique quasi-simple representation S X ∈ R such that Hom rep( Q ) ( P x , S X ) = 0. Observe that each almost split sequence0 / / U / / V / / W / / W ∈ R yields an exact sequence( ∗ ) 0 / / Hom( P x , U ) / / Hom( P x , V ) / / Hom( P x , W ) / / . Let Y ∈ R be of quasi-length n . By Lemma 2.5, Hom rep( Q ) ( P x , Y ) is at most one-dimensional over k . We claim that Hom rep( Q ) ( P x , Y ) = 0 if and only if Y ∈ W ( S X ).If n = 1, then Y ∈ W ( S X ) if and only if Y = S X , and hence, the claim holds.Assume that n >
1. Consider first the case where Y
6∈ W ( S X ). Then rep( Q ) hasan almost split sequence 0 / / U / / V / / W / / , where U, W belong to
R\W ( S X ) and are of quasi-length n −
1, and Y is a di-rect summand of V . By the induction hypothesis, Hom rep( Q ) ( P x , U ) = 0 andHom rep( Q ) ( P x , W ) = 0 . In view of the sequence ( ∗ ), we have Hom rep( Q ) ( P x , Y ) = 0.Consider now the case where Y ∈ W ( S X ). Then Y = τ i M for some 0 ≤ i ≤ n − M ∈ ( S X → ) . We denote by N the immediate predecessor of M in ( S X → ).Suppose first that i = 0, that is, Y = M . Since there exists a monomorphism from S X to Y , we see that Hom rep( Q ) ( P x , Y ) = 0. Suppose that 0 < i ≤ n −
1. Then,there exists an almost split sequence0 / / τ i N ( u u ) / / Z ⊕ Y ( v ,v ) / / τ i − N / / , where Z, τ i N, τ i − N ∈ W ( S X ) are of quasi-length n − n −
2. By the induc-tion hypothesis on n , Hom( P x , τ i N ), Hom( P x , Z ) and Hom( P x , τ i − N ) are one-dimensional over k , and by the exactness of the sequence ( ∗ ), so is Hom( P x , Y ).This establishes our claim.Finally, let f : P x → Y be a non-zero morphism. By our claim, Y ∈ W ( S X ).Then there exists a unique representation Y ′ ∈ ( → S X ) which is a sectional prede-cessor of Y in R . Observe, moreover, that there exists a monomorphism g : Y ′ → Y .This yields a monomorphism Hom( P x , g ) : Hom rep( Q ) ( P x , Y ′ ) → Hom rep( Q ) ( P x , Y ).Since Hom rep( Q ) ( P x , Y ′ ) and Hom rep( Q ) ( P x , Y ) are one-dimensional, Hom( P x , g ) isan isomorphism. In particular, f factors through g . The proof of the propositionis completed. 3. Derived categories
Throughout this section, Q stands for a locally finite quiver with no infinite path.We shall study the bounded derived category D b (rep( Q )) of rep( Q ). It is wellknown that D b (rep( Q )) is a Hom-finite Krull-Schmidt triangulated k -category ha-ving almost split triangles; see [2, (7.11)], whose Auslander-Reiten translation willbe written as τ D . In particular, D b (rep( Q )) admits a Serre functor S such that S ◦ [ −
1] = τ D ; see [19]. This fact is expressed in the following result. LUSTER CATEGORIES 11
Lemma.
Let Q be a locally finite quiver with no infinite path. If X, Y areindecomposable objects in D b (rep( Q )) , then Hom D b (rep( Q )) ( Y, τ D X ) ∼ = D Hom D b (rep( Q ) ( X, Y [1]) . As usual, we shall regard rep( Q ) as a full subcategory of D b (rep( Q )) by identify-ing a representation X with the stalk complex X [0] concentrated in degree 0. TheAuslander-Reiten quiver Γ D b (rep( Q )) of D b (rep( Q )) has a connecting component C ,which is obtained by gluing the preprojective component P of Γ rep( Q ) with the shiftby − I in such a way that each arrow x → y in Q induces a unique arrow I x [ − → P y in C ; see [2, Section 7], and compare [8]. Forconvenience, we quote the following result from [2, Section 7].3.2. Proposition.
Let Q be an infinite connected quiver, which is locally finite andcontains no infinite path. (1) The connecting component C of Γ D b (rep( Q )) is standard of shape Z Q op . (2) The connected components of Γ D b (rep( Q )) are the shifts of C and the shifts ofthe regular components of Γ rep( Q ) . Specializing to the infinite Dynkin case, we will have a better description of themorphisms between indecomposable objects of D b (rep( Q )) . Theorem.
Let Q be an infinite Dynkin quiver with no infinite path. (1) All the connected components of Γ D b (rep( Q )) are standard. (2) If R , S are distinct regular components of Γ rep( Q ) , then R [ i ] , S [ j ] with i, j ∈ Z are orthogonal in D b (rep( Q )) . (3) If M, N are objects lying in different connected components of Γ D b (rep( Q )) , then Hom D b (rep( Q )) ( M, N ) = 0 or Hom D b (rep( Q )) ( N, M ) = 0 . (4) There exists no short cycle in D b (rep( Q )) . Proof.
Statement (1) follows from Propositions 2.4 and 3.2. For Statement (2), let R , S be two distinct regular components of Γ rep( Q ) with M ∈ R and N ∈ S . Sincerep( Q ) is hereditary, Hom D b (rep( Q )) ( M, N [ j ]) = 0 for j = 0 , . By Theorem2.4,Hom D b (rep( Q )) ( M, N ) = 0, and by Lemma 2.1,Hom D b (rep( Q )) ( M, N [1]) ∼ = Ext Q ) ( M, N ) ∼ = D Hom rep( Q ) ( N, τ Q M ) = 0 . This shows that Hom D b (rep( Q )) ( R , S [ j ]) = 0, for all j ∈ Z . By symmetry, we haveHom D b (rep( Q )) ( S , R [ j ]) = 0 , for all j ∈ Z . As a consequence, R [ i ] and S [ j ] areorthogonal for all i, j ∈ Z .Suppose now that there exist distinct components Γ , Ω of Γ D b (rep( Q )) such thatHom D b (rep( Q )) ( Γ , Ω ) = 0 and Hom D b (rep( Q )) ( Ω , Γ ) = 0 . Since rep( Q ) is hereditary,making use of Statement (2), we may assume that Γ is the connecting component C of Γ D b (rep( Q )) . Suppose that Ω = C [ i ] for some i = 0. Since rep( Q ) is hereditary, i ∈ {− , , } , and since Hom rep( Q ) ( I , P ) = 0 , we have i ∈ { , } . This yieldsHom D b (rep( Q )) ( C , C [ − i ]) ∼ = Hom D b (rep( Q )) ( C [ i ] , C ) = Hom D b (rep( Q )) ( Ω , Γ ) = 0with − i ∈ {− , − } , a contradiction. Therefore, Ω = R [ j ] with R some regularcomponent of Γ rep( Q ) and j ∈ Z . Since rep( Q ) is hereditary, j ∈ {− , , } . Since Hom rep( Q ) ( I , R ) = 0, we have j ∈ { , } ; since Hom rep( Q ) ( R , P ) = 0, we concludethat j = 1. Now, if X ∈ P and Y ∈ R , we haveHom D b (rep( Q )) ( X, Y [1]) ∼ = Ext Q ) ( X, Y ) ∼ = D Hom rep( Q ) ( Y, τ Q X ) = 0 . As a consequence, Hom D b (rep( Q )) ( Γ , Ω ) = Hom D b (rep( Q )) ( C , R [1]) = 0 , a contradic-tion. Statement (3) is established.Finally, suppose that D b (rep( Q )) admits a short cycle X f / / Y g / / X, where X, Y ∈ Γ D b (rep( Q )) . By Statement (2),
X, Y lie in a connected component of Γ D b (rep( Q )) . This is absurd, since all the connected components of Γ D b (rep( Q )) arestandard without oriented cycles. The proof of the theorem is completed.4. Cluster Categories
Throughout this section, let Q be a locally finite quiver with no infinite path. Inparticular, the bounded derived category D b (rep( Q )) has almost split triangles,whose Auslander-Reiten translation is written as τ D . Setting F = τ − D ◦ [1], oneobtains the canonical orbit category C ( Q ) = D b (rep( Q )) /F. Recall that the objects of C ( Q ) are those of D b (rep( Q )); and for any objects X, Y , the morphisms are given byHom C ( Q ) ( X, Y ) = ⊕ i ∈ Z Hom D b (rep( Q )) ( X, F i Y ) . The composition of morphisms is given by( g i ) i ∈ Z ◦ ( f i ) i ∈ Z = ( h i ) i ∈ Z , where h i = P p + q = i F p ( g q ) ◦ f p . There exists a canonical projection functor π : D b (rep( Q )) → C ( Q ) , which acts identically on the objects, and sends a morphism f : X → Y to( f i ) i ∈ Z ∈ ⊕ i ∈ Z Hom D b (rep( Q )) ( X, F i Y ) = Hom C ( Q ) ( X, Y ) , where f i = f if i = 0; and otherwise, f i = 0. In particular, π is faithful.4.1. Theorem.
Let Q be an infinite connected quiver, which is locally finite andcontains no infinite path. (1) The canonical orbit category C ( Q ) is a Hom-finite Krull-Schmidt 2-Calabi-Yautriangulated k -category. (2) The canonical projection π : D b (rep( Q )) → C ( Q ) is a faithful triangle-functor,sending Auslander-Reiten triangles to Auslander-Reiten triangles. (3) If Γ is a connected component of Γ D b (rep( Q )) , then π ( Γ ) is a connected compo-nent of Γ C ( Q ) such that π ( Γ ) ∼ = Γ as translation quivers. (4)
The connected components of Γ C ( Q ) are π ( Γ ) , where Γ is either the connectingcomponent of Γ D b (rep( Q )) or a regular component of Γ rep( Q ) .Proof. Firstly, by a well known result of Keller stated in [14, Section 9], C ( Q ) isa 2-Calabi-Yau triangulated k -category such that the canonical projection functor π : D b (rep( Q )) → C ( Q ) is exact and faithful. Moreover, since rep( Q ) is Hom-finiteand hereditary, C ( Q ) is Hom-finite.We denote by D a skeleton of D b (rep( Q )), containing the objects in Γ D b (rep( Q )) . LUSTER CATEGORIES 13
Then D is a triangulated category such that the inclusion functor D → D b (rep( Q ))is a triangle-equivalence and Γ D = Γ D b (rep( Q )) . Observe that the Auslander-Reitentranslation τ D for D b (rep( Q )) induces an automorphism of D , which is denotedagain by τ D . Setting F = τ − D ◦ [1], we obtain a group G = { F n | n ∈ Z } ofautomorphisms of D , whose action on D is free and locally bounded. Now, theimage C of D under the canonical projection π : D b (rep( Q )) → C ( Q ) is a densefull subcategory of C ( Q ). Restricting π : D b (rep( Q )) → C ( Q ), we obtain a trianglefunctor D → C which, by abuse of notation, is denoted again by π . For X ∈ D and n ∈ Z , we define δ n,X = ( δ n,i ) i ∈ Z ∈ ⊕ i ∈ Z Hom D ( F n X, F i X ) = Hom C ( F n X, X ) , where δ n,i = 1 I F n X if i = n ; otherwise, δ n,i = 0. It is easy to see that δ n,X is anisomorphism which is natural in X such that δ n,X ◦ δ m,F n X = δ n + m,X , for m, n ∈ Z . This yields functorial isomorphisms δ n : π ◦ F n → π , n ∈ Z , such that δ = ( δ n ) n ∈ Z is a G -stabilizer for π . It not hard to verify that the map π X,Y : ⊕ i ∈ Z Hom D ( X, F i Y ) → Hom C ( X, Y ) : ( f i ) i ∈ Z P i ∈ Z δ i,Y ◦ π ( f i )is the identity map. Hence, π is a G -precovering. Since rep( Q ) is Hom-finiteand abelian, it is well known that D b (rep( Q )) is Hom-finie and Krull-Schmidt. Inparticular, the endomorphism algebra of an indecomposable object in D b (rep( Q ))is local with a nilpotent radical. By Lemma 2.9 stated in [1], the functor π satisfiesConditions (2) and (3) stated in [1, (2.8)]. In particular, C is Krull-Schmidt.Moreover, since π is dense, it is a Galois G -covering; see [1, (2.8)].In view of Proposition 3.5 stated in [1], we see that the exact functor π : D → C sends Auslander-Reiten triangles to Auslander-Reiten triangles, and hence State-ment (2) holds. For proving Statement (3), observe that Γ C ( Q ) = Γ C . Let Γ beconnected component of Γ D . By Theorem 4.7 stated in [1], π ( Γ ) is a connectedcomponent of Γ C such that π restricts to Galois G Γ -covering π Γ : Γ → π ( Γ ),where G Γ = { F n | F n ( Γ ) = Γ } . Since Q is infinite, F n ( Γ ) = Γ for each n = 0 . That is, G Γ is trivial, and hence, π Γ is an isomorphism of translation quivers.Finally, since π is dense, Γ C consists of the connected components π ( Θ ) with Θ ranging over the connected components of Γ D . Now, if Θ is such a component,then Θ = F n ( Γ ), where n ∈ Z and Γ is the connecting component of Γ D ora connected component of Γ rep( Q ) . This yields π ( Θ ) = π ( Γ ). The proof of thetheorem is completed. Remark. (1) As seen in the proof of Theorem 4.1, the objects of D b (rep( Q )) lyingin the connecting component of Γ D b (rep( Q )) or a regular components of Γ rep( Q ) forma fundamental domain for C ( Q ), denoted by F ( Q ), that is every indecomposableobject in C ( Q ) is isomorphic to a unique object in F ( Q ).(2) By abuse of language and notation, we shall identify the connecting compo-nent C of Γ D b (rep( Q )) with π ( C ) and call it the connecting component of Γ C ( Q ) , andidentify a regular component R of Γ rep( Q ) with π ( R ) and call it a regular compo-nent of Γ C ( Q ) . Note, however, that a standard component of Γ D b (rep( Q )) is neverstandard as a connected component of Γ C ( Q ) .For our purpose, we shall need the following explicit description of the morphismsin C ( Q ) between the objects in the fundamental domain. Lemma.
Let Q be a locally finite quiver with no infinite path, and let X, Y berepresentations lying in Γ rep( Q ) . (1) Hom C ( Q ) ( X, Y ) ∼ = Hom D b (rep( Q )) ( X, Y ) ⊕ D Hom D b (rep( Q )) ( Y, τ D X ) . (2) If Y is preinjective and X is not, then Hom C ( Q ) ( X, Y [ − ∼ = Hom D b (rep( Q )) ( X, τ − D Y ) . (3) If X is preinjective and Y is not, then Hom C ( Q ) ( X [ − , Y ) ∼ = D Hom D b (rep( Q )) ( Y, τ D X ) . Proof. (1) By definition, we haveHom C ( Q ) ( X, Y ) = ⊕ i ∈ Z Hom D b (rep( Q )) ( X, F i Y ) . Since rep( Q ) is hereditary, Hom D b (rep( Q )) ( X, F i Y ) = 0 , for i = 0 ,
1. Since τ D is anauto-equivalence and S = τ D ◦ F is a Serre functor, we haveHom D b (rep( Q )) ( X, F Y ) ∼ = Hom D b (rep( Q )) ( τ D X, S ( Y )) ∼ = D Hom D b (rep( Q )) ( Y, τ D X ) . (2) Firstly, we have Hom C ( Q ) ( X, Y [ − ∼ = Hom C ( Q ) ( X, τ − D Y ) . Suppose that Y is preinjective and X is not. We start with the case where Y is not injective. Then τ − D Y = τ − Q Y is a preinjective representation in Γ rep( Q ) . Applying Statement (1),we obtainHom C ( Q ) ( X, τ − D Y ) ∼ = Hom D b (rep( Q )) ( X, τ − D Y ) ⊕ D Hom D b (rep( Q )) ( τ − D Y, τ D X ) . Observe that τ D X is a shift by − D b (rep( Q )) ( τ − D Y, τ D X ) = 0. Statement(2) holds in this case.Assume now that Y = I x for some x ∈ Q . Then τ − D Y = P x [1]. By definition, F i P x [1] = τ − i D P x [ i + 1], for all i ∈ Z . If i ≥
1, then Hom D b (rep( Q )) ( X, F i P x [1]) = 0.If i ≤ −
1, then τ − i D P x [ i + 1] = N [ i ] for some preinjective representation N , andhence, Hom D b (rep( Q )) ( X, F i P x [1]) = 0. This yieldsHom C ( Q ) ( X, τ − D Y ) = Hom D b (rep( Q )) ( X, F P [1]) = Hom D b (rep( Q )) ( X, τ − D Y ) . (3) Suppose that X is preinjective and Y is not. Then τ − C Y = τ − D Y ∈ Γ rep( Q ) . Since C ( Q ) is 2-Calabi-Yau and τ C = [1], we haveHom C ( Q ) ( X [ − , Y ) ∼ = D Hom C ( Q ) ( Y, X [1]) ∼ = D Hom C ( Q ) ( τ − C Y, X [ − ∼ = D Hom D b (rep( Q )) ( τ − D Y, τ − D X ) ∼ = D Hom D b (rep( Q )) ( Y, τ D X ) , where the third isomorphism follows from Statement (2). The proof of the lemmais completed.4.3. Corollary.
Let Q be a locally finite quiver with no infinite path, and let X, Y be representations in Γ rep( Q ) . If X is preprojective and Y is regular, then Hom C ( Q ) ( X, Y ) ∼ = Hom rep( Q ) ( X, Y ) .Proof. Suppose that X is preprojective and Y is regular. Then τ D X is eithera preprojective representation or the shift by − D b (rep( Q )) ( Y, τ D X ) = 0 . Thus, by Lemma 4.2(1),Hom C ( Q ) ( X, Y ) ∼ = Hom rep( Q ) ( X, Y ). The proof of the corollary is completed.
LUSTER CATEGORIES 15
The following result says that C ( Q ) has at least one cluster-tilting subcategory.4.4. Lemma.
Let Q be a locally finite quiver with no infinite path. The strictlyadditive subcategory P of C ( Q ) generated by the representations P x with x ∈ Q ,is a cluster-tilting subcategory.Proof. Observe that the Auslander-Reiten translation τ C for C ( Q ) coincides withthe shift functor. For any x, y ∈ Q , making use of Lemma 4.2(2), we obtainHom C ( Q ) ( P x , P y [1]) = Hom C ( Q ) ( P x , τ C P y ) = Hom C ( Q ) ( P x , I y [ − ∼ = Hom D b (rep( Q )) ( P x , τ − D I y ) = Hom D b (rep( Q )) ( P x , P y [1])= Ext Q ) ( P x , P y ) = 0 . Let X be an indecomposable object in the fundamental domain of C ( Q ), butnot in P . Assume first that X ∈ Γ rep( Q ) . Then, τ C X = τ D X = τ Q X ∈ Γ rep( Q ) .Choosing a vertex x in the support of τ Q X , in view of Lemma 4.2(1), we obtainHom C ( Q ) ( P x , X [1]) = Hom C ( Q ) ( P x , τ C X ) = Hom C ( Q ) ( P x , τ Q X ) ∼ = Hom rep( Q ) ( P x , τ Q X ) ⊕ D Hom D b (rep( Q )) ( τ Q X, τ D P x ) = 0 . Assume now that X = Y [ − Y is a preinjective representation in Γ rep( Q ) .If y is a vertex in the support of Y , thenHom C ( Q ) ( P y , X [1]) = Hom C ( Q ) ( P y , Y ) ∼ = Hom D b (rep( Q )) ( P y , Y ) ⊕ D Hom D b (rep( Q )) ( Y, τ D P y ) = 0 . This shows that P is weakly cluster-tilting. Next, we need to show that P is functorially finite in C ( Q ). For this end, let Z be an indecomposable objectin the fundamental domain of C ( Q ). We claim that both Hom C ( Q ) ( Z, − ) andHom C ( Q ) ( − , Z ) vanish on P for all but finitely many objects. Start with the casewhere Z ∈ Γ rep( Q ) . Let x ∈ Q be such that the support of P x does not intersectthe support of Z ⊕ τ Q Z . By Lemma 4.2(1), we haveHom C ( Q ) ( P x , Z ) ∼ = Hom D b (rep( Q )) ( P x , Z ) ⊕ D Hom D b (rep( Q )) ( Z, τ D P x ) ∼ = Hom rep( Q ) ( P x , Z ) ⊕ D Hom D b (rep( Q )) ( Z, ( τ Q I x )[ − C ( Q ) ( Z, P x ) ∼ = Hom D b (rep( Q )) ( Z, P x ) ⊕ D Hom D b (rep( Q )) ( P x , τ D Z ) ∼ = Hom rep( Q ) ( Z, P x ) ⊕ D Hom D b (rep( Q )) ( P x , τ D Z )= D Hom D b (rep( Q )) ( P x , τ D Z ) , where the last equation follows from the hypothesis on x . If Z = τ − i Q P y for some y ∈ Q and 0 ≤ i ≤
1, thenHom D b (rep( Q )) ( P x , τ D Z ) = Hom D b (rep( Q )) ( P x , ( τ − i Q I y )[ − . Otherwise, τ D Z = τ Q Z ∈ Γ rep( Q ) , and by the hypothesis on x , we obtainHom D b (rep( Q )) ( P x , τ D Z ) ∼ = Hom rep( Q ) ( P x , τ Q Z ) = 0 . Since Q has no infinite path, our claim holds in this case. Next, suppose that Z = N [ − N ∈ Γ rep( Q ) . Let z ∈ Q be not in the support of τ Q N ⊕ τ − Q N . In view of Lemma 4.2(3), we obtainHom C ( Q ) ( Z, P z ) ∼ = D Hom D b (rep( Q )) ( P z , τ D N ) ∼ = D Hom rep( Q ) ( P z , τ Q N ) = 0;and by Lemma 4.2(2), we haveHom C ( Q ) ( P z , Z ) ∼ = Hom D b (rep( Q )) ( P z , τ − D N ) . If N = I b for some b ∈ Q , thenHom D b (rep( Q )) ( P z , τ − D N ) = Hom D b (rep( Q )) ( P z , P b [1]) = 0 . Otherwise, by the hypothesis on z , we obtainHom D b (rep( Q )) ( P z , τ − D N ) = Hom rep( Q ) ( P z , τ − Q N ) = 0 . Since N ⊕ τ − N is finite dimensional, our claim holds in this case. As a conse-quence, P is covariantly and contrvariantly bounded in C ( Q ), and by Lemma 1.5,it is functorially finite. The proof of the lemma is completed.For the rest of this section, we shall concentrate on the infinite Dynkin case.4.5. Proposition.
Let Q be an infinite Dynkin quiver with no infinite path. Theconnected components of Γ C ( Q ) consist of the connecting component of shape Z Q op and r regular components of shape ZA ∞ , where (1) r = 0 if Q is of type A ∞ ;(2) r = 1 if Q is of type D ∞ ;(3) r = 2 if Q is of type A ∞∞ ; and in this case, the two regular components areorthogonal.Proof. We need only to show the second part of Statement (3), since the other partsfollow from Theorem 4.1 and some results stated in [2, (5.16),(5.17),(5.22)]. Forthis purpose, suppose that Q is of type A ∞∞ . Let R , S be the two distinct regularcomponents of Γ rep( Q ) with X ∈ R and Y ∈ S . By Theorem 3.3, R and S areorthogonal in D b (rep( Q )). In view of Lemma 4.2(1), we obtainHom C ( Q ) ( X, Y ) ∼ = Hom D b (rep( Q )) ( X, Y ) ⊕ D Hom D b (rep( Q )) ( Y, τ D X ) = 0 . The proof of the proposition is completed.Recall that an object X ∈ C ( Q ) is called a brick if End C ( Q ) ( M ) is one-dimensionalover k ; and rigid if Hom C ( Q ) ( X, X [1]) = 0.4.6.
Corollary.
Let Q be an infinite Dynkin quiver. If Q has no infinite path,then every indecomposable object in C ( Q ) is a rigid brick.Proof. Assume that Q has no infinite path. Let X be an indecomposable objectof C ( Q ). By Theorem 4.1(2), τ C X = τ D X . In order to show that X is a rigidbrick, we may assume X lies in the fundamental domain F ( Q ) of C ( Q ). Since theAuslander-Reiten translation τ C for C ( Q ) coincides with the shift functor, X is arigid brick if and only if so is τ n C X for some integer n . As a consequence, we mayfurther assume that both X and τ D X belong to Γ rep( Q ) . Let Γ be the connected component of Γ D b (rep( Q )) containing X . By Theorem3.3(1), Γ is standard with no oriented cycle. Hence, Hom D b (rep( Q )) ( X, τ D X ) = 0,Hom D b (rep( Q )) ( X, τ D X ) = 0 , and End D b (rep( Q )) ( X ) is one-dimensional over k . In LUSTER CATEGORIES 17 view of Lemma 4.2(1), we deduce that End C ( Q ) ( X ) is one-dimensional over k , andHom C ( Q ) ( X, X [1]) ∼ = Hom C ( Q ) ( X, τ C X ) = Hom C ( Q ) ( X, τ D X ) ∼ = Hom D b (rep( Q )) ( X, τ D X ) ⊕ D Hom D b (rep( Q )) ( τ D X, τ D X )= 0 . The proof of the corollary is completed.More generally, a strictly additive subcategory T of C ( Q ) is called rigid ifHom C ( Q ) ( X, Y [1]) = 0, for all objects
X, Y ∈ T ; and maximal rigid if it is rigid andmaximal with respect to the rigidity property. By definition, a weakly cluster-tiltingsubcategory of C ( Q ) is maximal rigid, and the converse is not true in general.4.7. Lemma.
Let Q be an infinite Dynkin quiver with no infinite path. If T is astrictly additive subcategory of C ( Q ) , then it is weakly cluster-tilting if and only ifit is maximal rigid in C ( Q ) .Proof. We need only to show the sufficiency. Let T be a strictly additive subcat-egory of C ( Q ), which is maximal rigid. Let M ∈ C ( Q ) be indecomposable suchthat Hom C ( Q ) ( T , M [1]) = 0. Since C ( Q ) is 2-Calabi-Yau, Hom C ( Q ) ( M, T [1]) = 0.By Corollary 4.6, M is rigid in C ( Q ). Hence, the strictly additive subcategory of C ( Q ) generated by M and T is rigid. Since T is maximal rigid, M ∈ T . The proofof the lemma is completed.The following result is essential for our investigation.4.8. Proposition.
Let Q be a quiver with no infinite path of type A ∞ or A ∞∞ . If X, Y ∈ C ( Q ) are indecomposable, then Hom C ( Q ) ( X, Y ) is of k -dimension at mostone.Proof. Let
X, Y ∈ C ( Q ) be indecomposable and suppose they lie in the funda-mental domain F ( Q ). Since τ C is an auto-equivalence of C , we may assume that τ i C X, τ i C Y ∈ Γ rep( Q ) for i = 0 , ,
2. Then,
X, Y are preprojective or regular repre-sentations. Since Hom C ( Q ) ( X, Y ) ∼ = D Hom C ( Q ) ( Y, τ D X ), we may assume that X is preprojective in case X or Y is not regular. By Lemma 4.2(1), we obtainHom C ( Q ) ( X, Y ) = Hom D b (rep( Q )) ( X, Y ) ⊕ D Hom D b (rep( Q )) ( Y, τ Q X )= Hom rep( Q ) ( X, Y ) ⊕ D Hom rep( Q ) ( Y, τ Q X ) . Assume first that X and Y belong to a connected component Γ of Γ D b (rep( Q )) . By Propositions 3.2 and 3.3(1), Γ is standard of shape ZA ∞ or ZA ∞∞ . Thus,Hom D b (rep( Q )) ( X, Y ) = 0 or Hom D b (rep( Q )) ( Y, τ Q X ) = 0 . Moreover, in view ofProposition 1.3, we see that Hom C ( Q ) ( X, Y ) is at most one-dimensional over k .Assume now that X, Y belong to two different connected components Γ of Γ D b (rep( Q )) . If both X and Y are regular representations in Γ rep( Q ) then, byProposition 2.4, Hom rep( Q ) ( X, Y ) = 0 and Hom rep( Q ) ( Y, τ Q X ) = 0. Therefore,Hom C ( Q ) ( X, Y ) = 0. Otherwise, by our assumption, X is preprojective and Y isregular. Then Hom rep( Q ) ( Y, τ Q X ) = 0 by Theorem 2.2(2), and Hom rep( Q ) ( X, Y ) isat most one-dimensional over k by Lemma 2.5. As a consequence, Hom C ( Q ) ( X, Y )is at most one-dimensional over k . This completes the proof of the proposition.We are ready to obtain our main result of this section. Theorem.
Let Q be a quiver of type A ∞ or A ∞∞ . If Q contains no infinitepath, then C ( Q ) is a cluster category.Proof. Assume that Q contains no infinite path. By Theorem 4.1(1), C ( Q ) is aHom-finite 2-Calabi-Yau triangulated k -category. Since C ( Q ) has a cluster-tiltingsubcategory by Lemma 4.4, we need only to show that the quiver of every cluster-tilting subcategory T of C ( Q ) has no oriented cycle of length one or two; see [3,(II.1.6)]. For this purpose, it suffices to show that T has no short cycle.Suppose that C ( Q ) has a cluster-tilting subcategory with short cycles. In partic-ular, C ( Q ) has some non-zero radical morphisms f : X → Y and g : Y → X , where X, Y ∈ Γ C ( Q ) form a rigid family contained in the fundamental domain F ( Q ). ByCorollary 4.6, X and Y are not isomorphic. For each integer n , since the Auslander-Reiten translation τ C of C ( Q ) coincides with the shift functor, C ( Q ) has non-zeroradical morphisms f n : τ n C X → τ n C Y and g n : τ n C Y → τ n C X , where τ n C X, τ n C Y forma rigid family contained in the fundamental domain of C ( Q ). Therefore, we mayassume that τ i C X, τ i C Y ∈ Γ rep( Q ) , for 0 ≤ i ≤
3. In particular, τ i C Y = τ i D Y = τ i Q Y ,for 0 ≤ i ≤ . Suppose first that rep( Q ) has a non-zero radical morphism f : X → Y . ByTheorem 3.3(4), rad rep( Q ) ( Y, X ) = 0, and thus, Hom rep( Q ) ( Y, X ) = 0 . In view ofLemma 4.2(1), we obtain a non-zero radical morphism g : X → τ Q Y in rep( Q ).Moreover, since Hom C ( Q ) ( X, τ Q Y ) = Hom C ( Q ) ( X, τ C Y ) = Hom C ( Q ) ( X, Y [1]) = 0 , we have Hom D b (rep( Q )) ( X, τ Q Y ) = 0 . That is, Hom rep( Q ) ( X, τ Q Y ) = 0 . Let Γ be the connected component of Γ D b (rep( Q )) containing X . By Propositions3.2 and 3.3(1), Γ is standard of shape ZA ∞ or ZA ∞∞ . If Y ∈ Γ then, by Proposition1.3, both τ D Y and Y lie in the forward rectangle R X of X . Being convex, R X also contains τ D Y . As a consequence, Hom D b (rep( Q )) ( X, τ D Y ) = 0 , a contradiction.Therefore, Y lies in a connected component Ω of Γ D b (rep( Q )) with Ω = Γ .Observing that X, Y are preprojective or regular representations, we deduce fromTheorem 2.2 and Proposition 2.4 that X is preprojective and Y is regular. Thus, Ω is a regular component of Γ rep( Q ) . By Proposition 2.6, Ω has an infinite wing W ( S ) determined by a quasi-simple representation S such that, for any Z ∈ Ω ,Hom rep( Q ) ( X, Z ) = 0 if and only if Z ∈ W ( S ). In particular, τ Q Y, Y ∈ W ( S ), andconsequently, τ Q Y ∈ W ( S ). That is, Hom rep( Q ) ( X, τ Q Y ) = 0, a contradiction.Suppose now that rep( Q ) has a non-zero radical morphism f : Y → τ Q X .Applying Theorem 3.3(1) and (3), we deduce that Hom D b (rep( Q )) ( X, τ Q Y ) = 0 . ByLemma 4.2(1), we have a non-zero radical morphism g : Y → X . This reduces tothe case we have just treated. The proof of the theorem is completed.5. Triangulations of the infinite strip
From now on, we shall study the cluster category of type A ∞∞ from a geometricpoint of view. Our geometric model will be triangulations of the infinite strip,which was introduced by Igusa and Todorov in [11] and further studied by Holmand Jørgensen in [10].For the rest of this paper, we shall denote by B ∞ the infinite strip in the plane,consisting of the points ( x, y ) with 0 ≤ y ≤ . The line defined by y = 0 is called lower boundary line of B ∞ , while the line defined by y = 1 is called upper boundaryline . The points l i = ( i, i ∈ Z , are called upper marked points ; and the points LUSTER CATEGORIES 19 r i = ( − i, i ∈ Z , are called lower marked points . An upper or lower marked pointin B ∞ will be simply called a marked point . The set of marked points in B ∞ willbe written as M . Moreover, we denote by A the set of all two-element subsets of M except the subsets { r i , r j } and { l i , l j } with | i − j | ≤ curve in B ∞ we mean a curve in B ∞ joining two points, called endpoints .Two curves are said to cross provided that their intersection contains a point whichis not an endpoint of any of the two curves. By a simple curve in B ∞ we mean acurve joining two marked points which does not cross itself and intersects the twoboundary lines only at the endpoints. Moreover, a simple curve in B ∞ is called an edge curve if the set of its endpoints is either { l i , l i +1 } or { r i , r i +1 } for some i ∈ Z ;and an arc curve if the set of its endpoints belongs to A .Let p , q be two distinct marked points in B ∞ . There exists a unique isotopy classof simple curves in B ∞ joining p and q , which we denote by [ p , q ], or equivalently,by [ q , p ]. We shall call p , q the endpoints of the isotopy class [ p , q ]. Now, the isotopyclass of an edge curve is called an edge in B ∞ , while the isotopy class of an arccurve is called an arc in B ∞ . An arc in B ∞ is called an upper arc if its endpointsare upper marked points, and a lower arc if its endpoints are lower marked points,and a connecting arc if its endpoints do not lie on the same boundary line.Let u, v be arcs in B ∞ . We shall say that u crosses v , or equivalently, ( u, v ) isa crossing pair , provided that every curve lying in the isotopy class u crosses eachof the curve lying in the isotopy class v . Observe that an arc does not cross itself.For convenience, we state without a proof the following easy observation.5.1. Lemma.
The following statement hold true for arcs in B ∞ . (1) An upper arc does not cross any lower arc. (2)
An upper arc [ l i , l j ] with i < j crosses a connecting arc [ l r , r s ] if and only if i < r < j . (3) A lower arc [ r i , r j ] with i > j crosses a connecting arc [ l r , r s ] if and only if i > s > j . (4) Two connecting arcs [ l i , r j ] and [ l p , r q ] cross if and only if i > p and j > q , orelse, i < p and j < q . (5) Two upper arcs [ l i , l j ] and [ l p , l q ] with i < j and p < q cross if and only if i < p < j < q or p < i < q < j. (6) Two lower arcs [ r i , r j ] and [ r p , r q ] with i > j and p > q cross if and only if i > p > j > q or p > i > q > j. We denote by arc( B ∞ ) the set of arcs in B ∞ , which is equipped with a naturaltranslation τ : arc( B ∞ ) → arc( B ∞ ) as defined below.5.2. Definition.
For each arc u in B ∞ , we define its translate τ u as follows.(1) If u = [ l i , l j ] with i < j −
1, then τ u = [ l i +1 , l j +1 ].(2) If u = [ r i , r j ] with i > j + 1, then τ u = [ r i +1 , r j +1 ].(3) If u = [ l i , r j ], then τ u = [ l i +1 , r j +1 ].The following statement follows easily from the definition of τ and Lemma 5.1.5.3. Corollary. If u, v arcs in B ∞ , then the following statement hold. (1) The arc u crosses both τ u and τ − u . (2) The pair ( u, v ) is crossing if and only if ( τ u, τ v ) is crossing. We shall see that the connecting arcs in B ∞ are very special. Indeed, we shallneed a partial order on them.5.4. Lemma.
The set of connecting arcs in B ∞ is partially ordered in such a waythat [ l i , r j ] ≤ [ l r , r s ] if and only if i ≤ r and j ≥ s. The following notion is the central objective of study in this section.5.5.
Definition.
A maximal set T of pairwise non-crossing arcs in B ∞ is called a triangulation of B ∞ ; see Figure 5. In this case, C ( T ) denotes the set of connectingarcs of T . · · · · · · l − l − l − l − l − l − l l l l l l l l r r r r r r r r − r − r − r − r − r − r − • • • • • • • • • • • • • •• • • • • • • • • • • • • • Figure 1.
A triangulation of B ∞ It is easy to see that two connecting arcs are comparable with respect to thepartial order defined in Lemma 5.4 if and only if they do not cross each other. Thisgives us immediately the following useful observation.5.6.
Lemma.
Let T be a triangulation of B ∞ . If T contains some connecting arcs,then C ( T ) is well ordered. We shall also need the following easy result.5.7.
Lemma.
Let T be a triangulation of B ∞ , and let p be an integer. (1) If there exist infinitely many i < p such that [ l i , l j i ] ∈ T for some j i ≥ p orinfinitely many j > − p such that [ l i j , r j ] ∈ T for some i j ≥ p , then no l i with i < p is an endpoint of an arc of C ( T ) . (2) If there exist infinitely many i > p such that [ l j i , l i ] ∈ T for some j i ≤ p orinfinitely many j < − p such that [ l i j , r j ] ∈ T for some i j ≤ p , then no l i with i > p is an endpoint of an arc of C ( T ) .Proof. We shall prove only Statement (1). Consider a connecting arc v = [ l r , r s ]with r < p . If the first situation in Statement (1) occurs, then there exists someinteger i < r such that [ l i , l j i ] ∈ T for some j i ≥ p . In this case, v crosses [ l i , l j i ],and hence, v / ∈ T . If the second situation occurs, then there exists some j > s suchthat [ l i j , r j ] ∈ T for some i j ≥ p . In this case, v crosses [ l i j , l j ], and hence, v / ∈ T .The proof of the lemma is completed. LUSTER CATEGORIES 21
Remark.
A similar statement holds for lower marked points.Let T be a triangulation of B ∞ . For each u ∈ arc( B ∞ ), we shall denote by T u the set of arcs of T which cross u .5.8. Lemma.
Let T be a triangulation of B ∞ containing some connecting arcs, andlet u be an arc in B ∞ . If T u is infinite, then some marked point in B ∞ is anendpoint of infinitely many arcs in T u .Proof. Assume that every marked point in B ∞ is an endpoint of at most finitelymany arcs in T u . We shall prove that T u is finite. Consider first the case where u is an upper arc, say u = [ l r , l s ] with r < s −
1. Then every arc in T u hasas an endpoint some marked point l i with r < i < s . Thus, it follows from theassumption that T u is finite. The case where u is a lower arc can be treated in asimilar manner.Consider finally the case where u is a connecting arc, say u = [ l r , r s ]. Weclaim that T u contains at most finitely connecting arcs. Indeed, assume that T u contains some connecting arc v = [ l p , r q ]. Then p > r and q > s , or p < r and q < s . Suppose that the first situation occurs. If v = [ l i , r j ] ∈ T u , then v does notcross v , and hence, either i > p and q ≥ j > s or p ≥ i > r and j > q . Thus,our claim follows from the assumption on the marked points. Similarly, our claimholds if the second situation occurs.Suppose now that T u contains infinitely many upper arcs u i = [ l r i , l s i ] with r i < r < s i , for i = 1 , · · · . By the hypothesis on the marked points, we mayassume that r i +1 < r i , for all i ≥
1. Since the u i do not cross each other, we obtain s i < s i +1 for all i ≥
1. As a consequence, no upper marked point in B ∞ is anendpoint of some connecting arc of T , a contradiction to the hypothesis stated inthe lemma. Similarly, T u contains at most finitely many lower arcs. That is, T u is finite. The proof of the lemma is completed.We say that an upper marked point l i in B ∞ is covered by an upper arc [ l r , l s ] if r < i < s ; and a lower marked point r j is covered by a lower arc [ r p , r q ] if p > j > q .5.9. Lemma.
Let T be a triangulation of B ∞ . If C ( T ) is empty, then one of thefollowing situations occurs. (1) Every upper marked point in B ∞ is covered by infinitely many upper arcs of T and is an endpoint of at most finitely many upper arcs of T . (2) Every lower marked point in B ∞ is covered by infinitely many lower arcs of T and is an endpoint of at most finitely many lower arcs of T .Proof. Let C ( T ) = ∅ . Assume first that each upper marked point l t with t ∈ Z is covered by an upper arc u t = [ l i t , l j t ] of T . We shall show that Statement (1)holds. Suppose on the contrary that some upper marked point, say l , is coveredby only finitely many upper arcs of T . Let r < l r is anendpoint of some arc v covering l of T . Then v = [ l r , l s ] with r < < s. By ourassumption, l r is covered by an upper arc [ l p , l q ] ∈ T with p < r < q . Since [ l p , l q ]does not cross [ l r , l s ], we obtain s ≤ q . That is, [ l p , l q ] covers l , contrary to theminimality of r . This establishes the first part of Statement (1). Next, given any t ∈ Z , each of the [ l i , l t ] with i < i t and the [ l t , l j ] with j > j t crosses the upperarc u t ∈ T , and hence, it does not belong to T . Thus, l t is an endpoint of at mostfinitely many upper arcs of T . This establishes Statement (1). Assume now that some upper marked point, say l , is not covered by any upperarc of T . For any j ∈ Z , the connecting arc [ l , r j ] does not belong to T , andhence, it crosses some arc v ∈ T . By the hypothesis on l , the arc v is a lowerarc which covers the lower marked point r j . That is, every lower marked point iscovered by a lower arc of T . Using a similar argument as above, we are able toshow that Statement (2) holds. The proof of the lemma is completed.Let T be a triangulation of B ∞ . An upper marked point l p is called left T -bounded if [ l i , l p ] / ∈ T for all but finitely many i < p and [ l p , r j ] / ∈ T for all butfinitely many j > − p ; and left T -unbounded if [ l i , l p ] , [ l p , r j ] ∈ T for infinitelymany i < p and infinitely many j > − p .By symmetry, l p is called right T -bounded if [ l p , l i ] T for all but finitely many i > p and [ l p , r j ] / ∈ T for all but finitely many j < − p ; and right T -unbounded if[ l p , l i ] , [ l p , r j ] ∈ T for infinitely many i > p and infinitely many j < − p .In a similar manner, we shall define a lower marked point to be left T -bounded , left T -unbounded , right T -bounded , and right T -unbounded .5.10. Lemma.
Let T be a triangulation of B ∞ with [ l p , r q ] ∈ C ( T ) . (1) If l p is left ( respectively, right ) T -bounded, then some of the l i with i < p ( respectively, i > p ) is an endpoint of some arc of C ( T ) . (2) If r q is left ( respectively, right ) T -bounded, then some of the r j with j > q ( respectively, j < q ) is an endpoint of some arc of C ( T ) .Proof. We shall prove only the first part of Statement (1). Assume that none ofthe l i with i < p is an endpoint of an arc of C ( T ), moreover, [ l p , r j ] ∈ T for atmost finitely many j > − p . Thus, we may assume with no loss of generality that q is maximal such that [ l p , r q ] ∈ T . We shall need to prove that [ l i , l p ] ∈ T forinfinitely many integers i < p .Indeed, in view of the assumption on l p , we see that [ l p , r q ] is a minimal elementin C ( T ). Since [ l p − , r q ] / ∈ T by the assumption, T [ l p − , r q ] contains some arc v .Crossing [ l p − , r q ] but not [ l p , r q ], the arc v is neither a lower arc nor a connectingarc greater than [ l p , r q ] . Then, v is not a connecting arc by the minimality of [ l p , r q ].As a consequence, v = [ l i , l p ] for some i < p −
1. This proves that T [ l p − , r q ] containsonly upper arcs of the form [ l i , l p ] with i < p − . Thus, it suffices to show that T [ l p − , r q ] is infinite. If this is not the case, thenthere exists a minimal r < p − l r , l p ] ∈ T [ l p − , r q ] . Not belonging to T ,the connecting arc [ l r , r q ] crosses some arc w of T . As argued above, w is neithera lower arc nor a connecting arc. Thus w = [ l s , l t ] with s < r < t . If t ≤ p − w crosses [ l r , l p ], a contradiction. Thus p − < t , and hence, w ∈ T [ l p − , r q ] . As we have shown, t = p , a contradiction to the minimality of r . The proof of thelemma is completed.Let Σ be a set of arcs in B ∞ . We shall denote by τ Σ the set of arcs of the form τ u with u ∈ Σ ; and by τ − Σ the set of arcs of the form τ − v with v ∈ Σ . Definition.
A set ∆ of arcs in B ∞ is called compact if it admits a finitesubset Σ such that every arc in ∆ crosses some arc of τ Σ and some arc of τ − Σ .By definition, the empty subset of arc( B ∞ ) is compact. Moreover, as shownbelow, any finite set of arcs in B ∞ is compact. LUSTER CATEGORIES 23
Lemma.
Let ∆ be a set of arcs in B ∞ , and let Ω be a co-finite subset of ∆.If Ω is compact, then ∆ is compact.Proof. Assume that Ω is compact. Let Σ be a finite subset of Ω satisfying thecondition stated in Definition 5.11. Let Θ be the union of Σ and the complementof Ω in ∆ , which is finite by the hypothesis. It is evident that τ Σ ⊆ τ Θ and τ − Σ ⊆ τ − Θ . Let u be an arc in B ∞ . If u ∈ Ω , then it crosses some arc of τ Σ andsome arc of τ − Σ . Otherwise, u ∈ Θ , and by Corollary 5.3, u crosses τ u and τ − u .The proof of the lemma is completed.The following notion is important for us to determine the cluster-tilting subcat-egories in the next section.5.13. Definition.
A triangulation T of B ∞ is called compact provided that T u iscompact, for every arc u in B ∞ .5.14. Lemma.
Let T be a compact triangulation of B ∞ , and let p be an integer. (1) If [ l i , l p ] ∈ T for infinitely many integers i < p , then l p is left T -unbounded. (2) If [ l p , l i ] ∈ T for infinitely many integers i > p , then l p is right T -unbounded. (3) If [ r j , r p ] ∈ T for infinitely many integers j > p , then r p is left T -unbounded. (4) If [ r p , r j ] ∈ T for infinitely many integers j < p , then r p is right T -unbounded.Proof. We shall prove only Statement (1). Assume that [ l i , l p ] ∈ T for infinitelymany i < p . We shall need to show that [ l p , r j ] ∈ T for infinitely many integers j > − p . Suppose on the contrary that this is not the case. Then, there existsan integer q such that [ l p , r j ] / ∈ T for every j > q . Consider the connecting arc u = [ l p − , r q ]. By the assumption, [ l i , l p ] ∈ T u for infinitely many i < p −
1. Since T u is compact, there exists a finite subset Σ of T u satisfying the condition statedin Definition 5.11. Let t < p − u = [ l t , l p ] ∈ Σ . Observethat T u contains some arc w = [ l r , l p ] with r < t .We claim that w does not cross τ − v for any arc v ∈ Σ . Indeed, this is trivially thecase if v is a lower arc. Suppose that v is a connecting arc in Σ . Then v = [ l m , r n ]with m > p − n > q , or else, m < p − n < q . Since v does not crossany of the infinitely many arcs [ l i , l p ] in T u with i < p −
1, we obtain m > p − n > q . By the assumption on q , we obtain m > p , and hence, w = [ l r , l p ] doesnot cross τ − v = [ l m − , r n − ]. Suppose now that v is an upper arc in Σ . Then v = [ l m , l n ] with m < p − < n . Since v does not cross any of the infinitely manyarcs [ l i , l p ] ∈ T u with i < p , we have n = p , that is, v = [ l m , l p ] with m < p − t , we obtain t ≤ m , and hence, w = [ l r , l p ] does not cross τ − v = [ l m − , l p − ]. This establishes our claim, which is contrary to the propertystated in Definition 5.11. The proof of the lemma is completed.The following result exhibits a characteristic property of a compact triangulation.5.15. Proposition.
Let T be a triangulation of B ∞ . If T is compact, then C ( T ) is a double-infinite chain as follows : · · · < u − < u − < u < u < u < · · · Proof.
Let T be compact. First, we claim that C ( T ) is non-empty. Assume thatthis is not the case. By Lemma 5.9, we may assume that every upper marked pointis covered by infinitely many upper arcs of T and is an endpoint of at most finitely many upper arcs of T . Consider the connecting arc u = [ l , r ] . Since every upperarc covering l crosses u , the set U ( T u ) of upper arcs of T u is infinite. Beingcompact, T u has a finite subset Σ satisfying the condition stated in Definition5.11. Let r , s be the least and the greatest, respectively, such that l r , l s areendpoints of some arcs of Σ . Since each upper marked point is an endpoint of atmost finitely many arcs of U ( T u ), the set T u contains an upper arc u = [ l r , l s ]with r < r − s > s + 1. Given any arc v ∈ Σ , by the assumption, either v = [ l r , l s ] with r ≤ r < s ≤ s or v is a lower arc. In either case, u does notcross τ − v or τ v , a contradiction. This establishes our claim.Next, suppose on the contrary that C ( T ) contains a minimal element [ l p , r q ].Since the arcs of T do not cross each other, we deduce from the minimality of [ l p , r q ]that [ l i , r j ] T for all i, j with i < p . By Lemmas 5.10 and 5.14, [ l p , r j ] ∈ T forsome j > q , contrary to the minimality of [ l p , r q ]. Similarly, one can show that C ( T ) contains no maximal element. Being well ordered and interval-finite, C ( T )is a double infinite chain. The proof of the lemma is completed.Let T be a triangulation of B ∞ . A marked point p in B ∞ is called a left T -fountain base if p is left T -unbounded but right T -bounded. In this case, if p = l p ,then the set of arcs of the form [ l i , l p ] with i < p and [ l p , r j ] with j > − p is calleda left fountain of T at p ; and if p = r q , then the set of arcs of the form [ r i , r q ] with i > q and [ l j , r q ] with j < − q is called a left fountain of T at p . Similarly, p is said to be a right T -fountain base if p is right T -unbounded butleft T -bounded. In this case, if p = l p , then the set of arcs of the form [ l p , l i ] with p < i and [ l p , r j ] with j < − p is called a right fountain of T at p ; and if p = r q ,then the set of arcs of the form [ r q , r i ] with q > i and [ l j , r q ] with j > − q is calleda right fountain of T at p . Furthermore, p is called a full T -fountain base if p is left and right T -unbounded;and in this case, the set of arcs of T which have p as an endpoint is called a fullfountain of T at p . For brevity, a marked point is called a T - fountain base if it is a left, right orfull T -fountain base; and a left, right or full fountain of T will be simply called a fountain . If p is a T -fountain base, then the fountain at p will be denoted by F T ( p ).5.16. Lemma.
Let T be a triangulation of B ∞ . (1) If p is a full T -fountain base in B ∞ , then it is the unique T -fountain base andis an endpoint of all connecting arcs of T . (2) If p , q are two distinct T -fountain bases in B ∞ , then they are the only T -fountain bases with one being a left T -fountain base and the other one being aright T -fountain base.Proof. Assume that some l p is left T -unbounded. We claim that l p is the only left T -unbounded marked point, and none of the l i with i < p is an endpoint of anyconnecting arc in B ∞ . Indeed, the second part of this claim follows from Lemma5.7(1). As a consequence, none of the l i with i < p and the r j with j ∈ Z is left T -unbounded. Now, choose arbitrarily a connecting arc [ l p , r q ] of T . Since the arcsof T do not cross each other, [ l i , l j ] T for all i, j with i < p < j. In particular,no l j with j > p is left T -unbounded. This establishes the claim.Suppose now that p is a full T -fountain base. We shall consider only the casewhere p is an upper marked point, say p = l p . It follows from the above claim and LUSTER CATEGORIES 25 its dual version that p is the only T -fountain base. Moreover, none of the l i with i < p or i > p is an end-point of some connecting arc of T . As a consequence, p isan endpoint of all connecting arcs of T . This establishes Statement (1). Similarly,Statement (2) follows immediately from the above claim and its dual version. Theproof of the lemma is completed.Next we shall find some sufficient conditions for a triangulation to be compact.5.17. Lemma.
Let T be a triangulation of B ∞ , and let v be an arc in B ∞ . If v crosses infinitely many arcs of a full fountain of T , then T v is compact.Proof. Assume that v crosses infinitely many arcs of a full fountain F T ( p ) of T . Weshall consider only the case where p is an upper marked point, say p = l p for some p ∈ Z . In particular, v is not be a lower arc.Assume first that v is an upper arc. It is easy to see that v = [ l r , l s ] with r < p < s . Let i < r be maximal such that v = [ l i , l p ] ∈ T , and let j > s beminimal such that v = [ l p , l j ] ∈ T . Suppose that u is an arc lying in T v but notin F T ( l p ). Then u is not a lower arc, and by Lemma 5.16(1), it is an upper arc.Since u does not cross v or v , we see that u = [ l i , l j ] with i ≤ i < r < j < p or p < i < s < j ≤ j . Therefore, T v ∩ F T ( p ) is co-finite in T v .We claim that T v ∩ F T ( p ) is compact. Indeed, v , v ∈ T v ∩ F T ( l p ) with τ v =[ l i +1 , l p +1 ] and τ − v = [ l p − , l j − ]. Let u ∈ T v ∩ F T ( l p ). If u is an upper arc,then u = [ l m , l p ] with m < r or u = [ l p , l n ] with n > s . By the maximality of i and the minimality of j , we obtain u = [ l m , l p ] with m ≤ i or u = [ l p , l n ]with j ≤ n . In the first situation, since m < i + 1 < r + 1 ≤ p < p + 1 and m < r ≤ p − < p < s ≤ j −
1, we see that u crosses both τ v and τ − v . In thesecond situation, since i + 1 ≤ r < p < p + 1 ≤ s < n and p < s ≤ j − < n , wesee that u crosses both τ v and τ − v . This establishes our claim, and hence, T v iscompact by Lemma 5.12.Consider next the case where v is a connecting arc, say v = [ l r , r s ]. Then r = p .We shall consider only the case where r < p . Let i < r be maximal such that v = [ l i , l p ] ∈ T and let j > s be minimal such that v = [ l p , r j ] ∈ T . Moreover,since l p is right T -unbounded, there exists a maximal integer t < s such that w = [ l p , r t ] ∈ T . Let u be an arc lying in T v but not in F T ( p ). By Lemma5.16(1), u is not a connecting arc. Since u does not cross any of the arcs v , v , w ,in case u is a lower arc, we have u = [ r i , r j ] ∈ T with j ≥ i > s > j ≥ t ; andotherwise, u = [ l i , l j ] ∈ T with i ≤ i < r < j < p . Therefore, T v ∩ F T ( p ) isco-finite in T v .By Lemma 5.12, it remains to show that T v ∩ F T ( p ) is compact. Indeed, v , v ∈ T v ∩ F T ( p ) with τ v = [ l i +1 , l p +1 ] and τ − v = [ l p − , r j − ]. Let u be an arbitraryarc in T v ∩ F T ( l p ) . Then u is not a lower arc. If u is a connecting arc, then u = [ l p , r j ]with j > s . By the minimality of j , we have j ≥ j . In this case, u crosses both τ v = [ l i +1 , l p +1 ] and τ − v = [ l p − , r j − ]. If u is an upper arc, then it followsfrom the maximality of i that u = [ l i , l p ] with i ≤ i . It is then easy to see that u crosses τ v and τ − v . The proof of the lemma is completed.Given a marked point p in B ∞ , we shall denote by E T ( p ) the set of arcs of T which have p as an endpoint. Lemma.
Let T be a triangulation of B ∞ with p a left or right T -fountainbase, and let v be an arc in B ∞ . If v crosses infinitely many arcs of F T ( p ) , then T v ∩ F T ( p ) is compact and co-finite in E T ( p ) .Proof. We shall consider only the case where p = l p for some p ∈ Z , which is a left T -fountain base. Assume that v crosses infinitely many arcs of F T ( p ). Since everylower arc crosses at most finitely many arcs of E T ( l p ), the arc v is not a lower arc.Moreover, since T is right T -bounded, v has as an endpoint some marked point l r with r < p . That is, v = [ l r , l s ] with p < s or v = [ l r , r s ].Let w be an arc of F T ( p ), which does not cross v . If v = [ l r , l s ] with r < p < s ,then w = [ l j , l p ] with r ≤ j < p −
1. If v = [ l r , r s ], then w = [ l j , l p ] with r ≤ j < p − w = [ l i , r t ] with r ≤ i < p − − p < t ≤ s . Therefore, all butfinitely many arcs of F T ( p ) cross v .Now, let m < r be maximal such that v = [ l m , l p ] ∈ T . It is clear that v ∈ T v ∩ F T ( l p ). Let u ∈ T v ∩ F T ( l p ) . If u is a connecting arc, then u = [ l p , r t ]for some t > − p , which clearly crosses τ v = [ l m +1 , l p +1 ] . Otherwise, u = [ l t , l p ]for some t < p −
1. By the maximality of m , we obtain t ≤ m , and thus, u crosses τ v = [ l m +1 , l p +1 ] . Next, in case v = [ l r , l s ], let n > − p be minimal such that [ l p , r n ] ∈ T ; and incase v = [ l r , r s ], let n > max {− p, s } be minimal such that [ l p , r n ] ∈ T . In eithercase, set v = [ l p , r n ] , which clearly belongs to T v ∩ F T ( l p ). Let u ∈ T v ∩ F T ( l p ) . If u is an upper arc, then u = [ l t , l p ] with t < r , which crosses τ − v = [ l p − , r n − ] . Otherwise, u = [ l p , r t ], where t > − p , and t > s in case v = [ l r , r t ]. By theminimality of n, we obtain t ≥ n , and hence, u crosses τ − v = [ l p − , r n − ] . Setting Σ = { v , v } , we see that T v ∩ F T ( p ) is compact. The proof of the lemma iscompleted.Let T be a triangulation of B ∞ . We shall say that a marked point in B ∞ is T -bounded if it is both left and right T -bounded, or equivalently, p is an endpointof at most finitely many arcs of T .5.19. Lemma.
Let T be a triangulation of B ∞ . If every marked point in B ∞ is either T -bounded or an endpoint of infinitely many arcs of C ( T ) , then every marked pointis either T -bounded or a T -fountain base.Proof. Assume that every marked point is either T -bounded or an endpoint ofinfinitely many arcs of C ( T ). Let p be a marked point, which is an endpoint ofinfinitely many arcs of C ( T ). We need to show that p is a T -fountain base. Firstof all, we claim that C ( T ) is a double infinite chain. Indeed, suppose that C ( T ) hasa least element [ l m , r n ]. Since the arcs of T do not cross each other, [ l i , r j ] / ∈ T for all i, j with i < m or j > n . By Lemma 5.10, [ l i , l m ] ∈ T for infinitely many i < m ; and [ r j , r n ] ∈ T for infinitely many j > n . By the hypothesis stated in thelemma, C ( T ) has connecting arcs [ l m , r n ] with n = n and [ l m , r n ] with m = m .By the minimality of [ l m , r n ], we obtain m > m and n < n . Then, [ l m , r n ]crosses [ l m , r n ], which is absurd. Similarly, we can show that C ( T ) has no greatestelement. This establishes our claim.For the rest of the proof, we shall consider only the case where p = l p such that[ l p , r j ] ∈ T for infinitely many j > − p . We claim that l p is left T -unbounded, thatis, [ l i , l p ] ∈ T for infinitely many i < p . Indeed, assume that this is not the case.Define s = p − l j , l p ] T for every j < p −
1; and otherwise, let s < p − l s , l p ] ∈ T . By Lemma 5.7(1), l s is not an endpoint of any LUSTER CATEGORIES 27 arc of C ( T ). By the hypothesis stated in the lemma, l s is an endpoint of at mostfinitely many arcs of T . Set t = s − l i , l s ] / ∈ T for every i < s −
1; andotherwise, let t < s − l t , l s ] ∈ T . Consider the upper arc v = [ l t , l p ] which, by Lemma 5.7(1), does not cross any arc of C ( T ). Suppose that v crosses some upper arc u in T . Since u does not cross any connecting arc of T having l p as an endpoint, u = [ l t , l s ] with t < t < s < p . If s < s < p , then s < p −
1, and by the definition of s , the arc [ l s , l p ] belongs to T and crosses u , acontradiction. If t < s < s , then t < s −
1, and by the definition of t , the arc [ l t , l s ]belongs to T and crosses u , a contradiction. Thus, s = s , which is a contradictionto the definition of t . Therefore, v does not cross any arc of T , and hence, v ∈ T ,a contradiction to the minimality of s . This establishes our claim. In particular, l p is a left T -fountain whenever it is right T -bounded.Assume now that l p is not right T -bounded. We shall consider only the casewhere [ l p , r i ] / ∈ T for all but finitely many i < − p . In particular, there exists aminimal integer q such that [ l p , r q ] ∈ T . Since l p is not right T -bounded, [ l p , l j ] ∈ T for infinitely many j > p . By Lemma 5.7(2), no l j with j > p is an endpoint ofan arc of C ( T ). Thus, [ l p , r q ] is a maximal element in C ( T ), a contradiction. Thisshows that [ l p , r i ] ∈ T for infinitely many i < − p . Using a similar argument asabove, we see that l p is right T -unbounded. That is, l p is a full T -fountain. Theproof of the lemma is completed.We are ready to obtain our main result of this section, which gives an easycriterion for a triangulation of B ∞ to be compact.5.20. Theorem. If T is a triangulation of B ∞ , then it is compact if and onlyif it contains some connecting arcs such that every marked point in B ∞ is either T -bounded or an endpoint of infinitely many connecting arcs of T .Proof. By Lemmas 5.14 and 5.15, we shall need only to prove the sufficiency. Forthis purpose, assume that T is a triangulation of B ∞ such that C ( T ) is non-emptyand every marked point in B ∞ is either T -bounded or an endpoint of infinitelymany arcs of C ( T ).Fix an arc v in B ∞ . We shall need to prove that T v is compact. By Lemma 5.12,we may assume that T v is infinite. Then, by Lemma 5.8, some marked point is anendpoint of infinitely many arcs of T v ; and by Lemma 5.19, such a marked pointis a T -fountain base. Denoting by t the number of such T -fountain bases, we have t ≤ p i , with i ∈ { , t } , be the T -fountain bases such that T v ∩ F T ( p i ) is infinite. By Lemma 5.17, we may assume that each p i with i ∈ { , t } is a left or right T -fountain base. By Lemma 5.18, each T v ∩ F T ( p i ) with i ∈ { , t } is compact and co-finite in E T ( p i ). It is then easy to see that ∪ ≤ i ≤ t T v ∩ F T ( p i ) iscompact. We claim that ∪ ≤ i ≤ t T v ∩ F T ( p i ) is co-finite in T v . Indeed, given anymarked point q in B ∞ , we set Ω ( q ) = (cid:26) E T ( q ) \ ( T v ∩ F T ( q )) , if q ∈ { p , p t } , T v ∩ E T ( q ) , if q
6∈ { p , p t } , which is always finite by Lemma 5.18 and the assumption.Suppose first that v is an upper arc, say v = [ l r , l s ] with r < s −
1. Let u bean arc lying in T v but not in F T ( p ) ∪ F T ( p t ). Since u crosses [ l r , l s ], there existssome i with r < i < s such that u ∈ E T ( l i ) , and by definition, u ∈ Ω ( l i ). That is, u ∈ ∪ r r . Then, F T ( p ) contains some connecting arc w = [ l p , r q ] with q > s . Let u be an arc lying in T v but not in ∪ ≤ i ≤ t F T ( p i ) . If u is an upper arc then, since it does not cross w , we obtain u = [ l i , l j ] with i < r < j ≤ p ; and by definition, u ∈ Ω ( l j ) for some r < j ≤ p . If u isa connecting arc, since it does not cross w , we deduce from Lemma 5.7(1) that u = [ l i , r j ] with i ≥ p and q ≥ j > s ; and by definition, u ∈ Ω ( r j ) for some q ≥ j > s . If u is a lower arc, since u does not cross w , we obtain u = [ r j , r i ] with q ≥ j > s ; and by definition, u ∈ Ω ( r j ) for some q ≥ j > s . In any case, u belongsto the finite union of the Ω ( l i ) with r < i ≤ p and the Ω ( r j ) with q ≥ j > s . Thatis, our claim holds in this case. By Lemma 5.12, T v is compact. The proof of thetheorem is completed.Here is an example of a compact triangulation of B ∞ having two fountains. · · · · · ·· · · · · · l − l − l − l l l l l l l l l l l r r r r r r r r r r r r − r − r − • • • • • • • • • • • • • •• • • • • • • • • • • • • • Geometric Realization of cluster categories of type A ∞∞ The objective of this section is to study the cluster structure of a cluster categoryof type A ∞∞ in terms of triangulations of the infinite strip B ∞ , as studied in Section5. We start this section with some algebraic considerations. Throughout, Q standsfor a canonical quiver with no infinite path of type A ∞∞ , that is, its vertices are theintegers and its arrows are of form n → ( n + 1) or ( n + 1) → n . Let a i , b i , i ∈ Z , bethe sources and the sinks in Q respectively such that b i − < a i < b i . We denote by p i : a i b i and q i : a i b i − , i ∈ Z , the maximal paths in Q . It will be convenientto picture Q as follows: a − q − { { {; {; {; {; {; p − " " "b"b"b"b a p ` ` ` ` q } } }= }= }= }= a q ~ ~ ~> ~> ~> ~> p " " "b"b"b"b"b · · · b − b − b b · · · Let S be a set of paths having pairwise distinct starting points. For p, q ∈ S ,define an order (cid:22) on S so that p (cid:22) q if and only if s ( p ) ≤ s ( q ). In case p ≺ q andthere exists no u in S such that p ≺ u ≺ q, we write p = σ S ( q ) and q = σ − S ( p ) . This yields an injective map σ S : S → S , called source translation for S . LUSTER CATEGORIES 29
Let Q R stand for the union of the maximal paths p i with i ∈ Z and the trivialpaths ε a with a being a middle point of some q j with j ∈ Z . Dually, Q L standsfor union of the maximal paths q i with i ∈ Z and the trivial paths ε b with b being a middle point of some p j with j ∈ Z . Recall from Lemma 2.3 that theAuslander-Reiten quiver Γ rep( Q ) of rep( Q ) has two regular components R R and R L such that the quasi-simple objects in R R are the string representations M ( p )with p ∈ Q R , while those in R L are the string representations M ( q ) with q ∈ Q L .For convenience, we quote the following result stated in [2, (5.13),(5.14)].6.1. Lemma.
Let σ R and σ L be the source translations for Q R and Q L , respectively. (1) If p ∈ Q R , then τ Q M ( p ) = M ( σ R ( p )) . (2) If q ∈ Q L , then τ Q M ( q ) = M ( σ − L ( q )) . We shall choose the vertex set of the Auslander-Reiten quiver Γ C ( Q ) of C ( Q ) tobe the fundamental domain F ( Q ) of C ( Q ), which consists of the regular represen-tations in Γ rep( Q ) and the objects in the connecting component of Γ D b (rep( Q )) . ByProposition 4.5, Γ C ( Q ) has exactly three connected components, namely, the con-necting component C of shape ZA ∞∞ , and the two orthogonal regular components R R and R L of shape ZA ∞ . As stated in Proposition 4.5(3), the two regular com-ponents R R and R L are orthogonal. We shall describe morphisms from an objectin C to an object in R R or R L . For this purpose, we shall need some notationfor C . First, observe that C has a section; see [17, 2.1], which is generated by theprojective representations in Γ rep( Q ) as follows: ...P a P b ; ; ;{;{;{;{ P a P b − " " "b"b"b ; ; ;{;{;{;{ P a − ... We shall denote by R the unique double infinite sectional path in C containingthe path P b P a , which corresponds to the path p in Q ; and by L the uniquedouble infinite sectional path containing the path P b − P a , which correspondsto the path q in Q . For each i ∈ Z , put R i = τ i C R and L i = τ i C L . Observe thateach object in C lies in a unique R i with i ∈ Z and in a unique L j with j ∈ Z .6.2. Proposition.
Let M be an object in C . For each integer i , we have (1) M ∈ R i if and only if Hom C ( Q ) ( M, τ i C M ( p )) = 0; and in this case, for any Y ∈ R R , Hom C ( Q ) ( M, Y ) = 0 if and only if Y ∈ W ( τ i C M ( p )); (2) M ∈ L i if and only if Hom C ( Q ) ( M, τ i C M ( q )) = 0; and in this case, for any Y ∈ R L , Hom C ( Q ) ( M, Y ) = 0 if and only if Y ∈ W ( τ i C M ( q )) .Proof. We shall only prove (1) since the proof of (2) is similar. For each i ∈ Z ,write the path q i as follows: b i − = a i,ℓ i a i,ℓ i − o o · · · o o a i, o o a i, = a i , o o where ℓ i is the length of q i . Write σ = σ R , the source translation for Q R . Bydefinition, σ j ( p i ) = ε a i,j for 0 < j < ℓ i , and σ ℓ i ( p i ) = p i − . Setting t = 0 and t i = t i +1 + ℓ i +1 = ℓ + · · · + ℓ i +1 for i ≤ −
1, we get a sequence of non-negativeintegers: 0 = t < t − < · · · < t i < t i − < · · · For each j ≥ , there exists a unique i ≤ t i ≤ j < t i − . In this case, itis easy to see that σ j ( p ) = (cid:26) p i , if j = t i ; ε a i,j − ti , if t i < j < t i − .Consider the section in C generated by the representations P a with a ∈ Q .Recall that R is the double infinite sectional path in C containing P b P a . Foreach x ∈ Q with x ≤
0, there exists in C a unique sectional path u x : P x M x with M x ∈ R , whose length is denoted by d x . On the other hand, there exists aunique path ρ x ∈ Q R with ρ x (cid:22) p such that x ∈ ρ x . We claim that ρ x = σ d x ( p ).Indeed, considering the rectangle in C with vertices P b i , M b i , P a i and M a i as follows: M b i P b i u bi = = =}=}=} ! ! !a!a!a M a i ,P a i u ai < < <|<|<| we see that d b i = d a i , for any i ≥
0. Now, for each x ∈ Q with x ≤
0, there existsa unique integer r ≤ p r − ≺ ρ x (cid:22) p r . If ρ x = p r , then x ∈ p r andhence, P x lies on the sectional path P b r P a r . Considering the rectangle in C withvertices P b r , M b r , P x and M x , we see that d x = d b r . Now, the path u b r : P b r M b r is the composite of the path P b r P a r +1 corresponding to q r +1 and the path u a r +1 : P a r +1 M a r +1 . Hence, d x = d b r = d a r +1 + ℓ r +1 = d b r +1 + ℓ r +1 = ℓ + · · · + ℓ r +1 = t r . This yields ρ x = p r = σ t r ( p ) = σ d x ( p ). Suppose now that p r − ≺ ρ x ≺ p r . Then, x lies in the path q r : a r b r − , that is, x = a r,l for some 0 < l < ℓ r . In this case, ρ x = ε a r,l = σ t r + l ( p ). On the other hand, the path u x : P x M x is the compositeof the path P x = P a r,l / / · · · / / P a r, = P a r , corresponding to a r,l ← · · · ← a r, = a r , and the path u a r : P a r M a r . Thisyields d x = l + d a r = l + d b r = l + t r . Since t r < t r + l < t r − , we obtain σ d x ( p ) = σ l + t r ( p ) = ε a r,l = ρ x . This establishes our claim.Let M ∈ R , which we assume to be a successor of P b in R . Then, M = τ − s C P x for a unique pair ( s, x ), where s is a non-negative integer and x ∈ Q with x ≤ . LUSTER CATEGORIES 31
As shown above, x ∈ ρ x = σ d x ( p ), where d x is the length of the sectional path u x : P x M x in C , where M x ∈ R . Since R has a subpath M x M of length d x , we see that M = τ − d x C P x . Since τ C is an auto-equivalence of C ( Q ), we obtainHom C ( Q ) ( M, M ( p )) ∼ = Hom C ( Q ) ( P x , τ d x C M ( p )) ∼ = Hom C ( Q ) ( P x , τ d xQ M ( p ))= Hom C ( Q ) ( P x , M ( σ d x ( p )))= Hom C ( Q ) ( P x , M ( ρ x )) ∼ = Hom rep( Q ) ( P x , M ( ρ x )) = 0 , where the first equality follows from Lemma 6.1 and the last isomorphism followsfrom Proposition 4.3. In case M is a predecessor of P b in R , we may show by adual argument that Hom C ( Q ) ( M, M ( p )) = 0 . That is, Hom C ( Q ) ( M, M ( p )) = 0for all M ∈ R . As a consequence, if M ∈ R i with i ∈ Z , say M = τ i C U with U ∈ R , then Hom C ( Q ) ( M, τ i C M ( p )) ∼ = Hom C ( Q ) ( U, M ( p )) = 0 . Conversely, let M ∈ R i with i ∈ Z , say M = τ i C U for some U ∈ R , be such thatHom C ( Q ) ( M, M ( p )) = 0. Since τ − i Q M ( p ) = τ − i C M ( p ) , by Lemma 6.1, we haveHom C ( Q ) ( U, M ( σ − i ( p ))) ∼ = Hom C ( Q ) ( U, τ − i Q M ( p )) ∼ = Hom C ( Q ) ( M, M ( p )) = 0 . On the other hand, as we have shown, Hom C ( Q ) ( U, M ( p )) = 0. If U is a preprojec-tive representation then, by Proposition 2.6, M ( σ − i ( p )) ∈ W ( M ( p )), and hence i = 0. Otherwise, U = τ j C P x for some j > x ∈ Q . In this case, we obtainHom C ( Q ) ( P x , M ( σ − j − i ( p ))) ∼ = Hom C ( Q ) ( P x , τ − j Q M ( σ − i ( p ))) ∼ = Hom C ( Q ) ( τ j C P x , M ( σ − i ( p )))= Hom C ( Q ) ( U, M ( σ − i ( p ))) = 0 . Similarly, Hom C ( Q ) ( P x , M ( σ − j ( p ))) ∼ = Hom C ( Q ) ( U, M ( p )) = 0 . By Proposition2.6, M ( σ − j − i ( p )) ∈ W ( M ( σ − j ( p )), and hence i = 0.More generally, let M ∈ R j with j ∈ Z , say M = τ j C U for some U ∈ R , be suchthat Hom C ( Q ) ( M, τ i C M ( p )) = 0 . This yieldsHom C ( Q ) ( U, τ i − j C M ( p )) ∼ = Hom C ( Q ) ( M, τ i C M ( p )) = 0 . As we have just shown, τ i − j C M ( p ) ∈ W ( M ( p )), and hence j = i .Finally, let M ∈ R i with i ∈ Z and Y ∈ R R . Then, Hom C ( Q ) ( M, τ i C M ( p )) = 0 . If M is a representation, then Hom C ( Q ) ( M, Y ) ∼ = Hom rep( Q ) ( M, Y ) by Proposition4.3. It follows from Proposition 2.6 that Hom C ( Q ) ( M, Y ) = 0 , if and only if, Y ∈ W ( τ i C M ( p )). Otherwise, M = τ j C P y for some j > y ∈ Q . In particular, P y ∈ R i − j , and hence Hom C ( Q ) ( P y , τ i − j C M ( p )) = 0 . On the other hand,Hom C ( Q ) ( M, Y ) ∼ = Hom C ( Q ) ( P y , τ − j C Y ) ∼ = Hom rep( Q ) ( P y , τ − j C Y ) . By Proposition 2.6, Hom C ( Q ) ( M, Y ) = 0 if and only if τ − j C Y ∈ W ( τ i − j C M ( p )). Thelatter is equivalent to Y ∈ W ( τ i C M ( p )). The proof of the proposition is completed.We shall parameterize the indecomposable objects of C ( Q ) by the arcs in B ∞ ,that is, to define a bijection ϕ : F ( Q ) → arc( B ∞ ) , where F ( Q ) is the fundamental domain of C ( Q ). Recall that F ( Q ) consists of theobjects in C , R R and R L . First, for each object M in C , there exists a unique pairof integers ( i, j ) such that M = L i ∩ R j . Sending M to the connecting arc [ l i , r j ]in B ∞ yields a bijection ϕ C from the objects in C onto the connecting arcs in B ∞ .Put ϕ ( M ) = ϕ C ( M ) , for each object in C . Next, consider S L = τ − C M ( q ) , a quasi-simple object in R L . For each i ∈ Z ,denote by L + i the ray in R L starting with τ i C S L , and by L − i the coray ending with τ i C S L . For each object X in R L , there exists a unique pair of integers ( i, j ) with i ≤ j such that X = L − i ∩ L + j , and we set ϕ L ( X ) = [ l i − , l j +1 ] ∈ arc( B ∞ ). Thisdefines a bijection ϕ L from the objects in R L onto the upper arcs in B ∞ . Put ϕ ( X ) = ϕ L ( X ) , for all X ∈ R L . In this way, the quasi-simple objects in R L arethose mapped by ϕ to [ l i , l j ] with | i − j | = 2.Finally, consider S R = τ − C M ( p ) , a quasi-simple object in R R . For i ∈ Z , denoteby R + i the ray in R R starting with τ i C S R ; and by R − i the coray ending with τ i C S R .For each object Y ∈ R R , there exists a unique pair of integers ( i, j ) with i ≥ j such that Y = R + i ∩ R − j , and we set ϕ R ( Y ) = [ r i +1 , r j − ] ∈ arc( B ∞ ). This yieldsa bijection ϕ R from the objects in R R onto the lower arcs [ r i , r j ] in B ∞ . Set ϕ ( Y ) = ϕ R ( Y ) , for all Y ∈ R R . Observe that the quasi-simple objects in R R arethose mapped by ϕ to [ r i , r j ] with | i − j | = 2.This yields the desired bijection ϕ : F ( Q ) → arc( B ∞ ) . To simplify the notation,for each object X in F ( Q ) and each arc u in B ∞ , we write a X = ϕ ( X ) and M u = ϕ − ( u ) . Example.
In Figure 2 below, the two black dots are objects in R L , mapped by ϕ to [ l , l ] and [ l , l ], respectively. We see that the quasi-socle of ϕ − [ l , l ] is ϕ − [ l , l ] and its quasi-top is ϕ − [ l , l ]. L +7 L +4 L − L − S L • • Figure 2.
The regular component R L .The following easy observation describes the Auslander-Reiten translation andthe arrows in Γ C ( Q ) in terms of the arcs in B ∞ . Recall that arc( B ∞ ) is equippedwith a translation τ defined in Definition 5.2.6.3. Lemma.
Let u, v be two distinct arcs in B ∞ . (1) In any case, τ C M u = M τu , and τ − C M u = M τ − u . LUSTER CATEGORIES 33 (2) If u = [ l i , r j ] , then there exists an arrow M u → M v in Γ C ( Q ) if and only if v = [ l i , r j − ] or v = [ l i − , r j ] . (3) If u = [ l i , l j ] with i ≤ j − , then there exists an arrow M u → M v in Γ C ( Q ) ifand only if v = [ l i , l j − ] with i < j − or v = [ l i − , l j ] . (4) If u = [ r i , r j ] with i ≥ j + 2 , then there exists an arrow M u → M v in Γ C ( Q ) ifand only if v = [ r i − , r j ] with i > j + 2 or v = [ r i , r j − ] . The following result is essential in our investigation, and characterizes the rigidityof a pair of indecomposable objects of C ( Q ) by the non-crossing property of thecorresponding arcs.6.4. Theorem.
Let u, v be arcs in B ∞ . If M u , M v are the corresponding objects inΓ C ( Q ) , then ( u, v ) is a crossing pair if and only if Hom C ( Q ) ( M u , M v [1]) = 0 , orequivalently, Hom C ( Q ) ( M v , M u [1]) = 0 . Proof.
Let u, v be distinct arcs in B ∞ . Since C ( Q ) is 2-CY, the last two statementsstated in the proposition are equivalent. If one of u, v is an upper arc and the otherone is a lower arc, then one of M u , M v lies in R L and the other one lies in R R .In this case, the arcs u, v do not cross. Since R L , R R are orthogonal in C ( Q ) byProposition 4.5(3), the results holds true.Consider now the case where u and v are connecting arcs. Then M u , M v ∈ C . For any integer i , by Corollary 5.3, the arcs u, v cross if and only if τ i u, τ i v cross.On the other hand, since τ C is an automorphism of C ( Q ), we haveHom C ( Q ) ( M τ i u , M τ i v [1]) = Hom C ( Q ) ( τ i C M u , τ i C M v [1]) ∼ = Hom C ( Q ) ( M u , M v [1]) . Therefore, there is no loss of generality in assuming that M u and τ C M v = M τv arepreprojective representations in Γ rep( Q ) .Suppose first that u, v cross. By Lemma 5.1(4), there is no loss of generality inassuming that u = [ l p , r q ] and v = [ l i , r j ] with i < p and j < q . In view of Lemma6.3(1), we obtain a path M u = M [ l p , r q ] −→ M [ l p , r q − ] −→ · · · −→ M [ l p , r j +1 ] −→ M [ l p − , r j +1 ] −→ M [ l p − , r j +1 ] −→ · · · −→ M [ l i +1 , r j +1 ] = M τv in C . Since C is a standard component of Γ D b (rep( Q )) of shape ZA ∞∞ ; see (3.3), wededuce from Corollary 1.4 that Hom D b (rep( Q )) ( M u , M τv ) = 0 , and consequently,Hom C ( Q ) ( M u , M v [1]) = Hom C ( Q ) ( M u , τ C M v ) = Hom C ( Q ) ( M u , M τv ) = 0 . Suppose conversely that Hom C ( Q ) ( M u , M τv ) = 0 . Since M u , M τv are assumed tobe representations, by Lemma 4.2(1), we have Hom D b (rep( Q )) ( M u , M τv ) = 0 orHom D b (rep( Q )) ( M v , M τu ) ∼ = Hom D b (rep( Q )) ( τ D M v , τ D M τu )= Hom D b (rep( Q )) ( M τv , τ D M u ) = 0 . Since C is standard in D b (rep( Q )), we obtain a path M u M τv or M v M τu ,that is, a path M [ l p , r q ] M [ l i +1 , r j +1 ] or M [ l i , r j ] M [ l p +1 , r q +1 ] in C . By Lemma6.3(1), p ≤ i + 1 and q ≤ j + 1 in the first case, and i ≤ p + 1 and j ≤ q + 1 in thesecond case. By Lemma 5.1(4), the arcs u, v cross.Consider next the case where v, u are upper arcs, say u = [ l p , l q ] and v = [ l i , l j ]with p ≤ q − i ≤ j −
2. Then M u , M v ∈ R L . Assume first that u and v cross. By Lemma 5.1(5), we may assume that i < p < j < q . In view of Lemma 6.3(3),we see that R L contains a path M u = M [ l p , l q ] −→ M [ l p , l q − ] −→ · · · −→ M [ l p , l j +2 ] −→ M [ l p , l j +1 ] −→ M [ l p − , l j +1 ] −→ M [ l p − , l j +1 ] −→ · · · −→ M [ l i +1 , l j +1 ] = M τv , lying in the forward rectangle of M u . By Proposition 1.3, Hom rep( Q ) ( M u , M τv ) = 0 , and consequently, we obtainHom C ( Q ) ( M u , M v [1]) = Hom C ( Q ) ( M u , M τv ) = 0 . Conversely, assume that Hom C ( Q ) ( M u , M v [1]) = Hom C ( Q ) ( M u , M τv ) is non-zero. By Lemma 4.2(1), we have Hom D b (rep( Q )) ( M u , M τv ) = 0 orHom D b (rep( Q )) ( M v , M τu ) ∼ = Hom D b (rep( Q )) ( τ D M v , τ D M τu )= Hom D b (rep( Q )) ( M τv , τ D M u ) = 0 . Suppose that the first case occurs. We claim that i < p < j < q . Since R L isstandard in D b (rep( Q )) of shape ZA ∞ , by Proposition 1.3, M τv lies in the forwardrectangle of M u . In particular, R L contains a path ρ : M u = M [ l p , l q ] M [ l i +1 , l j +1 ] . By Lemma 6.3(3), we obtain p ≥ i + 1 > i and q ≥ j + 1 > j . If ρ is trivial, then p = i + 1 and q = j + 1, and hence, p = i + 1 ≤ j − < j by our assumption.If ρ is a sectional path, then we deduce from Lemma 6.3(3) that p = i + 1 and q > j + 1, or else, p > i + 1 and q = j + 1. In both cases, since i + 1 ≤ j − p ≤ q − p < j . If ρ is non-sectional, then M [ l p , l q ] is not quasi-simple, that is, p < q −
2. Recalling that M [ l p , l q ] = L − p +1 ∩ L + q − and M [ l i +1 , l j +1 ] = L − i +2 ∩ L + i , we may choose ρ to be the composite of a sectional path M [ l p , l q ] M contained in the coray L − p +1 and a sectional path M M [ l i +1 , l i +1 ] contained in the ray L + i . Then, M = L − p +1 ∩ L + i , and consequently, p + 1 ≤ j , thatis, p < j . This proves our claim. Similarly, if the second case occurs, we can showthat p < i < q < j . By Lemma 5.1, the arcs u, v cross. The case where both u and v are lower arcs can be treated in a similar way.Consider further the case where one of u, v is a connecting arc and the otherone is an upper arc, say u = [ l p , l q ] with p ≤ q − v = [ l i , r j ]. We have M u = L − p +1 ∩ L + q − ∈ R L and M v = L i ∩ L j ∈ C . Using Corollary 5.3 and thefact that τ C is an automorphism of C ( Q ), we may assume that M v is a prepro-jective representation. By Corollary 4.3, Hom C ( Q ) ( M v , M u [1]) = 0 if and only ifHom D b (rep( Q )) ( M v , M u [1]) = 0 . Since M u [1] = τ C M u = M τu = M [ l p +1 , l q +1 ] , thelatter condition by Proposition 6.2(2) is equivalent to M [ l p +1 , l q +1 ] ∈ W ( τ i C M ( q )).Since τ i C M ( q ) = τ i +1 C S L = L + i +1 ∩ L − i +1 and M [ l p +1 , l q +1 ] = L − p +2 ∩ L + q , we see that M [ l p +1 , l q +1 ] ∈ W ( τ i C M ( q )) if and only if i + 1 ≥ p + 2 and q ≥ i + 1, that is, p < i < q . This last condition by Lemma 5.1 is equivalent to u, v crossing. Thecase where one of u, v is a connecting arc and the other one is a lower arc can betreated in a similar manner. The proof of the Theorem is completed.Given a strictly additive subcategory T of C ( Q ), we shall write arc( T ) for theset of arcs a T with T ∈ T ∩ F ( Q ) . As an immediate consequence of Theorem 6.4and Lemma 4.7, we obtain the following result.6.5.
Theorem. If T is a strictly additive subcategory of C ( Q ) , then T is weaklycluster-tilting if and only if arc( T ) is a triangulation of B ∞ . LUSTER CATEGORIES 35
Next, we shall describe the triangulations of B ∞ that correspond to the cluster-tilting subcategories of C ( Q ). Consider again the bijection ϕ : F ( Q ) → arc( B ∞ ) . For each object M ∈ F ( Q ), we have M = ϕ − ( a M ) = M a M , and hence, τ C M = τ C M a M = M τa M = ϕ − ( τ a M ) , that is, τ a M = ϕ ( τ C M ) = a τ C M . The following easy result will be useful.6.6.
Corollary. If M, N are in F ( Q ) , then ( a N , τ a M ) is crossing if and only if Hom C ( Q ) ( M, N ) = 0 .Proof. Let
M, N ∈ F ( Q ). Since C ( Q ) is 2-Calabi-Yau, we obtainHom C ( Q ) ( N, τ C M [1]) = Hom C ( Q ) ( N, M [2]) ∼ = D Hom C ( Q ) ( M, N ) . By Theorem 6.4, a N and τ a M cross if and only if Hom C ( Q ) ( N, τ C M [1]) = 0, orequivalently, Hom C ( Q ) ( M, N ) = 0. The proof of the corollary is completed.6.7. Lemma.
Let T be a weakly cluster-tilting subcategory of C ( Q ) , containing anon-zero morphism f : X → Y which embeds in an exact triangle X f / / Y g / / Z h / / X [1] in C ( Q ) . If X, Y are indecomposable, then so is Z .Proof. Suppose that
X, Y are indecomposable and Z = Z ⊕ Z with Z , Z non-zero. Write g = ( g , g ) T and h = ( h , h ). Assume that Z [ − ∈ T . ThenHom C ( Q ) ( Y, Z ) = 0, and hence, g = 0. Since g is a pseudo-kernel of h , we deducethat h is a monomorphism. Since C ( Q ) is triangulated, h is a section, and since X is indecomposable, h is an isomorphism. Observing that f [1] ◦ h = 0, we obtain f = 0, a contradiction. Thus Z [ −
6∈ T , and similarly, Z [ −
6∈ T . Since T is weakly cluster-tilting, there exists T i ∈ T such that Hom C ( Q ) ( T i , Z i ) = 0 , for i = 1 , . Set T = X ⊕ Y ⊕ T ⊕ T . Since Hom C ( Q ) ( T, X [1]) = 0 , applyingHom C ( Q ) ( T, − ) to the triangle stated in the lemma yields a projective presentationof the right End( T )-module Hom C ( Q ) ( T, Z ) as follows:Hom C ( Q ) ( T, X ) / / Hom C ( Q ) ( T, Y ) / / Hom C ( Q ) ( T, Z ) / / . Since Hom C ( Q ) ( T, Z ) = Hom C ( Q ) ( T, Z ) ⊕ Hom C ( Q ) ( T, Z ) is decomposable, so isHom C ( Q ) ( T, Y ), a contradiction to Y being indecomposable in C ( Q ). The proof ofthe lemma is completed.If p , q are two marked points of B ∞ , then we define p < q as follows. If p is anupper marked point and q is a lower marked point, then p < q and q < q . If p = l i and q = l j , then p < q if and only if i < j . If p = r i and q = r j , then p < q if andonly if i < j .6.8. Lemma.
Let T be a weakly cluster-tilting subcategory of C ( Q ) , and let M, N, L be indecomposable objects in C ( Q ) with M, N ∈ T . If f : M → N and g : N → L are non-zero morphisms in C ( Q ) , then Hom C ( Q ) ( M, L ) is generated by gf over k .Proof. We may assume that
M, N, L all lie in the fundamental domain F ( Q ) andHom C ( Q ) ( M, L ) = 0. Let f : M → N and g : N → L be non-zero morphisms in C ( Q ). Since Hom C ( Q ) ( M, L ) is of k -dimension at most one by Proposition 4.8, itsuffices to show that gf = 0. Assume on the contrary that gf = 0. In particular, f is not an isomorphism. Since N is indecomposable, f is not a section. Thus, C ( Q )contains a non-split exact triangle( ∗ ) M f / / N v / / C w / / M [1] . By Lemma 6.7, C is indecomposable, which we may assume to be in F ( Q ).Since each of f, v, w is non-zero, by Proposition 4.8, each of Hom C ( Q ) ( M, N ),Hom C ( Q ) ( N, C ) and Hom C ( Q ) ( C, M [1]) is one-dimensional. Since f and v are notisomorphisms, M, N and C are pairwise non-isomorphic. We shall need two crucialfacts as follows.(1) Each pair of arcs ( a N , τ a M ) , ( a C , τ a N ) , ( a C , a M ) is crossing. (2) Each pair of arcs ( a N , a M ) , ( a C , a N ) , ( a C , τ a M ) is non-crossing. Indeed, Statement (1) follows immediately from Corollary 6.6 and Theorem6.4. Since T is rigid, Hom C ( Q ) ( M, N [1]) = 0, and by Theorem 6.4, ( a M , a N ) isnon-crossing. Moreover, since Hom C ( Q ) ( C, C [1]) = 0 by Corollary 4.6, applyingHom C ( Q ) ( C, − ) to the triangle ( ∗ ) yields an exact sequenceHom C ( Q ) ( C, C ) / / Hom C ( Q ) ( C, M [1]) / / Hom C ( Q ) ( C, N [1]) / / . Since Hom C ( Q ) ( C, M [1]) is one-dimensional, we have Hom C ( Q ) ( C, N [1]) = 0 , andby Theorem 6.4, ( a C , a N ) is non-crossing. Further, since Hom C ( Q ) ( M, M [1]) = 0 , applying Hom C ( Q ) ( M, − ) to the triangle ( ∗ ) yields an exact sequenceHom C ( Q ) ( M, M ) / / Hom C ( Q ) ( M, N ) / / Hom C ( Q ) ( M, C ) / / . Since Hom C ( Q ) ( M, N ) is one-dimensional, we have Hom C ( Q ) ( M, C ) = 0, and thus,( a C , τ a M ) is non-crossing by Corollary 6.6. This establishes Statement (2).Denote by p , q with p < q the endpoints of a M , and by τ p , τ q the endpoints of τ a M . Observe that p , q , τ p , τ q are pairwise distinct. Let Ω be a simply connectedregion in B ∞ enclosed by a closed simple curve which is the composite of twoedge arcs u ′ ∈ [ p , τ p ] and v ′ ∈ [ q , τ q ], an arc curve u ∈ [ p , τ q ] and a simple curve v ∈ [ τ p , q ].We shall say that an arc w in B ∞ crosses Ω if every simple curve in the isotopyclass of w intersects the interior region of Ω . Since ( a N , τ a M ) is crossing, a N crosses Ω . Since u ′ and v ′ are edge curves, either a N crosses both u and v , or else, a N hasas endpoint one of the marked points p , q , τ p , τ q . In the first situation, a N crosses a M , a contradiction to Statement (2).If a N shares an endpoint with τ a M , then this contradicts the fact that ( a N , τ a M )is crossing. Therefore, a M , a N share an endpoint, which we may assume to be p .Write a N = [ p , r ], where r is a marked point different from any of p , q , τ p , τ q .Since ( a C , τ a N ) is crossing and ( a C , a N ) is non-crossing, a similar argument showsthat a C , a N share an endpoint. Since ( a C , a M ) is crossing, this common endpoint isdifferent from p , and hence, it is r .Finally, since ( a C , a M ) is crossing and ( a C , τ a M ) is non-crossing, a C , τ a M sharean endpoint. Since ( a C , τ a N ) is crossing, this endpoint is different from τ p , andhence, it is τ q . This yields a C = [ r , τ q ].Since Hom C ( Q ) ( M, L ) is nonzero, by Corollary 6.6, ( a L , τ a M ) is crossing. Sim-ilarly, since g : N → L is nonzero, ( a L , τ a N ) is crossing. Let Ω ′ be a simplyconnected region in B ∞ enclosed by a closed simple curve which is the compositeof three arcs u ∈ [ τ p , τ r ], u ∈ [ τ r , τ q ], and u ∈ [ τ q , τ p ]. Observe that u is LUSTER CATEGORIES 37 isotopic to τ a N and u is isotopic to τ a M . Since we know that a L crosses both τ a M , τ a N , the arc a L cannot cross the other arc, which is u . Consider now Ω ′′ a simply connected region in B ∞ enclosed by a closed simple curve which is thecomposite of three arcs v = u ∈ [ τ q , τ r ], v ∈ [ τ r , r ], and v ∈ [ r , τ q ]. Observethat v is an edge arc. We claim that a L does not cross Ω ′′ . Suppose the contrary.Since v is an edge arc, either a L crosses both v , v , or else a L only crosses one of v , v but then has endpoint τ r or r . Since we already know that a L does not cross u = v , the first case does not occur. Hence, a L crosses v and has endpoint τ r ,which contradicts the fact that ( a L , τ a N ) is crossing. This proves our claim that a L does not cross Ω ′′ . Since a C is isotopic to v , this yields that a L does not cross a C .By Theorem 6.4, Hom C ( Q ) ( C [ − , L ) = 0. Applying Hom C ( Q )( − , L ) to theexact triangle ( ∗ ), we obtain the following exact sequenceHom C ( Q ) ( C, L ) / / Hom C ( Q ) ( N, L ) / / Hom C ( Q ) ( M, L ) / / . Since both Hom C ( Q ) ( N, L ) and Hom C ( Q ) ( M, L ) are one-dimensional by Proposition4.8, Hom C ( Q ) ( f, L ) : Hom C ( Q ) ( N, L ) → Hom C ( Q ) ( M, L ) is an isomorphism. Inparticular, gf = 0. This completes the proof of the lemma.Now, we are ready to determine the cluster-tilting subcategories of C ( Q ) in termsof the triangulations of B ∞ .6.9. Theorem.
Let Q be a quiver with no infinite path of type A ∞∞ . The followingthree statements are equivalent for a strictly additive subcategory T of C ( Q ) . (1) The subcategory T is cluster-tilting. (2) arc( T ) is a compact triangulation of B ∞ . (3) arc( T ) contains some connecting arcs and every marked point in B ∞ is either arc( T ) -bounded or a arc( T ) -fountain base.In this case, moreover, arc( T ) has at most two fountains, and if it has two fountains,then one is a left fountain and the other one is a right fountain.Proof. In view of Theorem 5.20 and Lemmas 5.16 and 5.19, it suffices to showthe equivalence of the first two statements. Assume first that arc( T ) is a compacttriangulation of B ∞ . In particular, T is weakly cluster-tilting. We shall need toprove that T is functorially finite. We first show that every indecomposable object M ∈ C ( Q ) admits a right T -approximation. Since C ( Q ) is 2-Calabi-Yau, for eachindecomposable object N ∈ C ( Q ), we haveHom C ( Q ) ( N, M ) ∼ = D Hom C ( Q ) ( τ − C M, N [1]) , which is of k -dimension at most one by Proposition 4.8.By the assumption, the set arc( T ) τ − a M of arcs of arc( T ) which cross τ − a M iscompact. Let Σ be a finite subset of arc( T ) τ − a M satisfying the condition stated inDefinition 5.11. Observe that τ − C M = τ − C M a M = M τ − a M . If v ∈ Σ , since v crosses τ − a M , we deduce from Theorem 6.4 thatHom C ( Q ) ( M v , M ) ∼ = D Hom C ( Q ) ( τ − C M, M v [1]) = D Hom C ( Q ) ( M τ − a M , M v [1]) = 0 . In particular, we may choose a nonzero morphism f v : M v → M for each v ∈ Σ .Set f = ⊕ v ∈ Σ f v : ⊕ v ∈ Σ M v → M, which we claim is a right T -approximation of M . Indeed, assume that Hom C ( Q ) ( N, M ) = 0, for some indecomposable object N ∈ T . Then Hom C ( Q ) ( τ − C M, N [1]) = 0, and by Theorem 6.4, a N crosses τ − a M , that is, a N ∈ arc( T ) τ − a M . By assumption, there exists an arc w ∈ Σ such that a N crosses τ − w. This implies thatHom C ( Q ) ( N, M w ) ∼ = D Hom C ( Q ) ( τ − C M w , N [1]) = 0 . Let g w : N → M w be a non-zero morphism. By Lemma 6.8, every morphism g : N → M is a multiple of f w g w . In particular, g factors through f . This establishesour claim. Using the dual of Lemma 6.8 and the compactness of arc( T ) τa M , wemay show that M admits a left T -approximation. This establishes the sufficiency.Conversely, assume that T is functorially finite in C ( Q ). Fix u ∈ arc( B ∞ ) . Byassumption, M τu admits a minimal right T -approximation f : ⊕ w ∈ Σ − M w → M τu ,where Σ − is a finite subset of arc( T ). For any w ∈ Σ − , since f is right minimal,restricting f to M w yields a non-zero morphism f w : M w → M τu . Observing that M u = τ − C M τu , we obtainHom C ( Q ) ( M u , M w [1]) = Hom C ( Q ) ( τ − C M τu , M w [1]) ∼ = D Hom C ( Q ) ( M w , M τu ) = 0 . By Theorem 6.4, w crosses u . This shows that Σ − ⊆ arc( T ) u . On the other hand,for v ∈ arc( T ) u , since M τu = τ C M u = M u [1], we deduce from Theorem 6.4 thatHom C ( Q ) ( M v , M τu ) = Hom C ( Q ) ( M v , M u [1]) ∼ = D Hom C ( Q ) ( M u , M v [1]) = 0 . Thus, there exists a nonzero morphism g : M v → M τu , which factors through f : ⊕ w ∈ Σ − M w → M τu . In particular, there exists v ∈ Σ − such thatHom C ( Q ) ( M τ − v , M v [1]) = Hom C ( Q ) ( τ − C M v , M v [1]) ∼ = D Hom C ( Q ) ( M v , M v ) = 0 . That is, v crosses τ − v . Similarly, considering a minimal left T -approximation for M τ − u , we may show that there exists a finite subset Σ + of arc( T ) u such that every v ∈ arc( T ) u crosses τ v for some arc v ∈ Σ + . In particular, arc( T ) u is compact.Hence, arc( T ) is compact. The proof of the theorem is completed.Here is an example of a triangulation of B ∞ , having two fountains, correspondingto a cluster-tilting subcategory of C ( Q ). · · · · · ·· · · · · · l − l − l − l l l l l l l l l l l r r r r r r r r r r r r − r − r − • • • • • • • • • • • • • •• • • • • • • • • • • • • • Let T be a cluster-tilting subcategory of C ( Q ). We would like to have a com-binatorial description of the irreducible morphisms of T in terms of the arcs ofarc( T ). This would give a combinatorial description of the quiver Q T of T . Giventwo arcs u, v ∈ arc( T ), we shall write u ⊢ v if u, v share an endpoint p and v isobtained by rotating u in the counter-clockwise direction around p . LUSTER CATEGORIES 39
Proposition.
Let T be a cluster-tilting subcategory of C ( Q ) . If M, N arenon-isomorphic objects in F ( Q ) ∩ T , then Hom C ( Q ) ( M, N ) = 0 if and only if a M ⊢ a N .Proof. Let
M, N ∈ F ( Q ) ∩ T be non-isomorphic. Suppose that Hom C ( Q ) ( M, N ) =0. Then Hom C ( Q ) ( M, τ − C N [1]) = 0. Since T is cluster-tilting, Hom C ( Q ) ( M, N [1]) =0. Hence, by Theorem 6.4, a M , τ − a N cross but a M , a N do not cross. This clearlymeans that a M , a N share an endpoint p . Suppose first that a N = [ l p , l q ] with p < q −
1. The conditions that a M , τ − a N cross but a M , a N do not cross clearlyimply that p = l p . If a M is not a connecting arc, then a M = [ l p , l r ] with r > q + 1or a M = [ l r , l p ] with r < p −
1. In both cases, we see that a M ⊢ a N . Clearly, if a M is a connecting arc, then a M ⊢ a N as well. A similar argument handles thecase where a N lies on the lower boundary component. So assume a N = [ l p , r q ].Then the conditions that a M , τ − a N cross but a M , a N do not cross give the followingpossibilities for a M . If p = l p , then a M = [ l p , r i ] with i > q or a M = [ l t , l p ] with t < p −
1. If p = r q , then a M = [ l j , r q ] with j > p or a M = [ r q , r s ] with s < q − a M ⊢ a N . This proves the necessity. Conversely, assume that a M ⊢ a N . Then a M , a N share some endpoint. With no loss of generality, we mayassume that l p is a common endpoint of a M , a N . Then a small neighborhood of l p looks as follows: l p l p − •• a M a N In view of this figure, we see that a M crosses τ − a N . By Theorem 6.4, we obtainHom C ( Q ) ( M, N ) ∼ = Hom C ( T ) ( M, τ − C N [1]) = 0 . The proof of the proposition is completed.Let T be a cluster-tilting subcategory of C ( Q ). Given two arcs u, v ∈ arc( T ),we shall say that v covers u with respect to ⊢ provided that u ⊢ v and there existsno w ∈ arc( T ) such that u ⊢ w and w ⊢ v . As an easy consequence of Proposition6.10, Lemmas 6.8 and 4.8, we obtain the following result.6.11. Proposition.
Let T be a cluster-tilting subcategory of C ( Q ) . If M, N areobjects in F ( Q ) ∩ T , then Q T has an arrow M → N if and only if a N covers a M with respect to ⊢ . If p , q are two marked points on the same boundary line of B ∞ , then the linesegment between p and q is called a boundary segment . The following is an easyconsequence of Proposition 6.11.6.12. Corollary.
Let T be a cluster-tilting subcategory of C ( Q ) . If M, N ∈ F ( Q ) ∩ T , then M, N lie in the same connected component of Q T if and onlyif there exists a simple closed curve S in B ∞ which is the composite of some arcsof arc( T ) and possibly some boundary segments such that the region enclosed by S contains a M , a N and at most finitely many arcs. We conclude this section with the following statement.
Proposition.
Let T be a cluster-tilting subcategory of C ( Q ) . If arc( T ) has m full fountains and n non-full fountains, then Q T has m + n + 1 connectedcomponents. In particular, Q T is connected if and only if arc( T ) has no fountain,or equivalently, every marked point in B ∞ is an endpoint of at most finitely manyarcs of arc( T ) .Proof. If arc( T ) has no fountain then, by Corollary 6.12, any pair of objects M, N of F ( Q ) ∩T lie in the same connected component of Q T . Therefore, we may assumethat arc( T ) has at least one fountain.Suppose first that arc( T ) has either a full fountain (which is then the uniquefountain) or two non-full fountains (that is, one left fountain and one right fountain).We shall construct non-empty subsets Σ , Σ , Σ of arc( T ) such that arc( T ) is adisjoint union of Σ , Σ , Σ . Moreover, for M, N ∈ F ( Q ) ∩ T , the arcs a M , a N lie in the same connected component of Q T if and only if a M , a N ∈ Σ i for some1 ≤ i ≤ l p for some integer p . Let Σ be the set of non-upper arcs ofarc( T ), and Σ be the set of arcs of arc( T ) of form [ l i , l j ] with i, j ≤ p , and Σ bethe set of arcs of arc( T ) of form [ l i , l j ] with i, j ≥ p . By Corollary 6.12, Σ , Σ , Σ satisfy the desired properties.For the second case, assume that there exists a left fountain and a right fountainwhose fountain bases lie on the same boundary line of B ∞ . We may assume thatthese fountain bases are l p , l q with p < q . In this case, let Σ be the set of arcsof arc( T ) which are connecting arcs, lower arcs, or upper arcs of form [ l i , l j ] with p ≤ i, j ≤ q , and Σ be the set of arcs of arc( T ) of form [ l i , l j ] with i, j ≤ p , and Σ be the set of arcs of arc( T ) of form [ l i , l j ] with i, j ≥ q . By Corollary 6.12, Σ , Σ , Σ satisfy the desired properties.For the third case, assume that there exists a left fountain and a right fountainwhose fountain bases do not lie on the same boundary line of B ∞ . We may assumethat these fountain bases are l p , r q with l p a left fountain base and r q a rightfountain base. In this case, let Σ be the arcs of arc( T ) which are connecting arcs,lower arcs of form [ r i , r j ] with i, j ≥ q or upper arcs of form [ l i , l j ] with i, j ≥ p .Let Σ be the arcs of arc( T ) of form [ l i , l j ] with i, j ≤ p , and Σ be the arcsof arc( T ) of form [ r i , r j ] with i, j ≤ q . By Corollary 6.12, Σ , Σ , Σ satisfy thedesired properties.It remains to consider the case where arc( T ) has a unique fountain, which isnot a full fountain. We may assume that l p is the fountain base, which is a leftarc( T )-fountain base. In this case, let Σ be the set of arcs of arc( T ) of form [ l i , l j ]with i, j ≤ p , and Σ be the set of other arcs of arc( T ). By Corollary 6.12, Σ , Σ satisfy the desired properties. The proof of the proposition is completed. References [1]
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Shiping Liu, D´epartement de math´ematiques, Universit´e de Sherbrooke, Sherbrooke,Qu´ebec, Canada
E-mail address : [email protected] Charles Paquette, Department of Mathematics, University of Connecticut, Storrs,CT, 06269-3009, USA
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