Torsion pairs in a triangulated category generated by a spherical object
aa r X i v : . [ m a t h . R T ] N ov TORSION PAIRS IN A TRIANGULATED CATEGORYGENERATED BY A SPHERICAL OBJECT
RAQUEL COELHO SIM ˜OES AND DAVID PAUKSZTELLO
Abstract.
We extend Ng’s characterisation of torsion pairs in the 2-Calabi-Yau tri-angulated category generated by a 2-spherical object to the characterisation of torsionpairs in the w -Calabi-Yau triangulated category, T w , generated by a w -spherical ob-ject for any w ∈ Z . Inspired by the combinatorics of T w for w −
1, we alsocharacterise the torsion pairs in certain negative Calabi-Yau orbit categories of thebounded derived category of the path algebra of Dynkin type A . Contents
1. Torsion pairs, extension closure and functorial finiteness 32. Triangulated categories generated by w -spherical objects 43. Extensions in T w with indecomposable outer terms for w = 1 74. Extensions in T w with decomposable outer terms for w = 1 125. The combinatorial model 186. Contravariant-finiteness 187. Torsion pairs and the Ptolemy condition in T w for w = 0 , T T C w ( A n ) 30References 35 Introduction
Calabi-Yau (CY) triangulated categories are triangulated categories that satisfy animportant duality. They are becoming increasingly important throughout mathemat-ics and physics, for example as 3-CY categories arising from Calabi-Yau threefoldsin algebraic geometry and string theory, to 3-CY categories arising in representationtheory coming from quivers with potential. Of particular importance in representationtheory are (2-)cluster categories, which provide categorifications of important aspectsof the theory of cluster algebras. There are higher analogues, so-called w -cluster cate-gories for w >
2, which are w -CY. These give rise to an important family of categoriesof positive CY dimension which satisfy many interesting and important homologicaland combinatorial properties.Throughout this article k will be an algebraically closed field. Let T be a k -lineartriangulated category and w ∈ Z . An object s ∈ T is w -spherical if it is a w -Calabi-Yauobject and its graded endomorphism algebra is given by Hom • ( s, s ) = k [ x ] / ( x ) , where x sits in cohomological degree − w. Mathematics Subject Classification.
Primary: 05E10, 16G20, 16G70, 18E30; Secondary:05C10.
Key words and phrases.
Auslander-Reiten theory; Calabi-Yau triangulated category; sphericalobject; Ptolemy arc; torsion pair.
In particular, s has the ‘same cohomology’ as the w -sphere. We refer the reader toSection 2 for a more precise definition.Let T w be a k -linear triangulated category that is idempotent complete and gen-erated by a w -spherical object. The T w constitute a family of categories which are w -CY whose structure is sufficiently simple to allow concrete computation. As such,they provide a ‘natural laboratory’ in which to explore the properties of CY trian-gulated categories, as witnessed by the intense recent interest in these categories; see[13, 15, 20, 26, 28]. Indeed, for w > T w occurs naturally as a w -cluster category oftype A ∞ .Owing to their importance and ubiquity, much work has been carried out on un-derstanding triangulated categories of positive CY-dimension. However, very littlework has been carried out on understanding the properties of triangulated categoriesof negative CY-dimension, although there is the beginning of a theory emerging in[10, 12, 13, 24]. In [24], it was shown that for w >
1, the category T w has one familyof bounded t-structures and no bounded co-t-structures, whilst for w T w were obtained by Holm andJørgensen in [19, 20] and employed by Ng in [29] to characterise torsion pairs in T .Building on Ng’s ideas, characterisations of torsion pairs have since been given invarious settings, see [3, 21, 22, 23], and used for detailed studies of their mutationtheories [16, 34].The combinatorial models involve setting up a correspondence between indecompos-able objects of the category and certain ‘admissible’ arcs or diagonals of some geometricobject. For T w with w = 1, the combinatorial model consists of ‘( w − ∞ -gon; see Section 6 for precise details. It is a well-known consequence ofthe 2-Calabi-Yau property of cluster categories that the crossing of arcs correspondsto the existence of a non-trivial extension between the corresponding indecomposableobjects. Given two crossing arcs, the admissible Ptolemy arcs are defined to be theadmissible arcs connecting the endpoints of the two crossing arcs.We extend Ng’s characterisation of torsion pairs for T to the entire family: Theorem A.
Let X be a full additive subcategory of T w for w = 1 and X be thecorresponding set of arcs in the appropriate combinatorial model of T w . Then ( X , X ⊥ ) is a torsion pair in T w if and only if(1) for w > , any so-called ‘right fountain’ in X is a so-called ‘left fountain’ and X is closed under taking admissible Ptolemy arcs.(2) for w , any left fountain in X is a right fountain and X is closed undertaking admissible Ptolemy arcs and ‘modified Ptolemy’ arcs. ORSION PAIRS 3
See Sections 7 and 8 for precise statements. The statement for w > w = 2 in [29] because crossings of arcs instead correspond to the existenceof some higher extension instead of simply extensions, which requires a substantiallydifferent approach from [29]. The case w = 1 is degenerate and does not admit sucha combinatorial model; it is treated in the short Section 9. However, surprisingly, for w T w when w < w -CY category, namely the fol-lowing orbit category: C w ( A n ) := D b ( k A n ) / Σ − w τ for w −
1. When w = − C − ( A n ) proved intractable. However, with the induced combinatorial model,the characterisation is tractable and gives us our second main result. Theorem B.
Let X be a full additive subcategory of C w ( A n ) for w − and X be thecorresponding set of arcs in the combinatorial model for C w ( A n ) . Then ( X , X ⊥ ) is atorsion pair in C w ( A n ) if and only if X is closed under taking admissible Ptolemy arcsand ‘modified’ Ptolemy arcs. It is our viewpoint that for w − C w ( A n ) are naturally w -CY, i.e.natural examples of triangulated categories having negative CY dimension. However,even in the case w = − least positive integer d such that Σ d is (isomorphic to) the Serre functor. According to this definition,the CY dimension of C − ( A n ) is 2 n −
1; [14, Theorem 6.1]. Note, however, that in C − ( A n ), the inverse suspension Σ − is also isomorphic to the Serre functor of C − ( A n ).In contrast, by [24, Proposition 2.8], T w is unambiguously w -CY. It was argued in[13] that for C w ( A n ), with w −
1, the ‘correct’ CY dimension should be w , owing tosimilarities in the combinatorics of so-called w -Hom-configurations in the categories C w ( A n ) and T w . We believe the similarities in the combinatorics of torsion pairs inTheorems A and B provide further support for this viewpoint. Moreover, we believethat this means triangulated categories of negative CY dimension are more widespreadthan previously believed, and warrant further, systematic, study. This article shouldbe considered as a step in this direction. Acknowledgments.
This paper was begun while both authors were at the LeibnizUniversit¨at Hannover. The first author gratefully acknowledges the financial supportof the Riemann Center for Geometry and Physics during her stay in Hannover. RCSwould also like to thank Funda¸c˜ao para a Ciˆencia e Tecnologia, for financial supportthrough the grant SFRH/BPD/90538/2012. DP gratefully acknowledges the supportof the EPSRC through the grant EP/K022490/1. We would like to thank the refereefor a careful reading and useful comments and suggestions.1.
Torsion pairs, extension closure and functorial finiteness
A triangulated category T is called Krull-Schmidt if every object t admits a directsum decomposition t = t ⊕ · · · ⊕ t n into indecomposable objects, which is unique upto reordering and isomorphism. We shall denote the collection of (isomorphism classesof) indecomposable objects by ind ( T ). Throughout this paper all categories will beKrull-Schmidt and all subcategories will be full and additive. RAQUEL COELHO SIM ˜OES AND DAVID PAUKSZTELLO A torsion pair in T consists of a pair of full subcategories ( X , Y ), which are closedunder direct summands, and satisfy Hom T ( X , Y ) = 0 and X ∗ Y = T , where X ∗ Y := { t ∈ T | ∃ x → t → y → Σ x with x ∈ X and y ∈ Y } . A torsion pair is called a t-structure when Σ X ⊆ X ( ⇔ Σ − Y ⊆ Y ); see [4]. It is calleda co-t-structure (or weight structure ) when Σ − X ⊆ X ( ⇔ Σ Y ⊆ Y ); see [5, 30]. If T isKrull-Schmidt, a torsion pair ( X , Y ) is called split if for any t ∈ ind ( T ) we have either t ∈ X or t ∈ Y .A subcategory X of T is closed under extensions or extension-closed if given anydistinguished triangle x ′ → x → x ′′ → Σ x ′ in T with x ′ , x ′′ ∈ X then x ∈ X . Theobject x will be called the middle term of the extension. We denote by h X i thesmallest extension-closed subcategory of T containing X .Let C be any category and A be a subcategory. A morphism f : a → c is called a right A -approximation of c if the induced map Hom C ( a ′ , f ) : Hom C ( a ′ , a ) → Hom C ( a ′ , c )is surjective for each object a ′ of A . In the case that any object of C admits a right A -approximation we say that A is a contravariantly finite subcategory of C . There aredual notions of left A -approximation and covariantly finite . If A is both contra- andcovariantly finite, A is called functorially finite . Right (resp. left) A -approximationsare often called A -precovers (resp. A -preenvelopes).These concepts are linked by the following proposition. Proposition 1.1 ([25, Proposition 2.3]) . Let k be an algebraically closed field and T be a k -linear, Krull-Schmidt and Hom-finite triangulated category. The followingconditions are equivalent:(1) ( X , Y ) is a torsion pair;(2) X is an extension-closed contravariantly finite subcategory of T and Y = X ⊥ ;(3) Y is an extension-closed covariantly finite subcategory of T and X = ⊥ Y . Triangulated categories generated by w -spherical objects Let k be an algebraically closed field. Let T be a k -linear triangulated category and w ∈ Z . An object s of T is called w -spherical [32] if it satisfies the following axioms:(S1) it is a w -spherelike object [18], i.e. Hom T ( s, Σ i s ) = (cid:26) k if i = 0 , w ;0 otherwise, and(S2) it is a w -Calabi-Yau object ( w -CY, for short), i.e. there is a functorial isomor-phism Hom T ( s, t ) ≃ D Hom T ( t, Σ w s ), where t ∈ T and D ( − ) := Hom k ( − , k ) isthe usual vector space duality.An object s generates T if thick T ( s ) = T , i.e. the smallest triangulated subcategorycontaining s that is also closed under direct summands is T .Let T w be a k -linear triangulated category that is idempotent complete and gen-erated by a w -spherical object. By [28, Theorem 2.1], T w is unique up to triangleequivalence. Thus, we shall refer to T w as the triangulated category generated by a w -spherical object . The categories T w satisfy many nice properties: • T w is Hom-finite and Krull-Schmidt; • T w has a Serre functor S , i.e. a functor S : T w → T w satisfying a functorialisomorphism Hom T w ( x, y ) ≃ D Hom T w ( y, S x ) for all x, y ∈ T w . Moreover, S = Σ τ , where τ is the Auslander–Reiten translate in T w . • T w is w -CY, i.e. S ≃ Σ w and all objects s, t ∈ T w satisfy (S2). ORSION PAIRS 5 (cid:31) (cid:31) ❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄ ... (cid:10) (cid:10) Σ d X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ X ? ? ⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ Σ − d X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ X (cid:10) (cid:10) H H · · · ? ? ⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ Σ d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ Σ − d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ · · · X I I (cid:10) (cid:10) Σ d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄ Σ − d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧ (cid:31) (cid:31) ❄❄❄ Σ − d X ? ? ⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄ X I I (cid:10) (cid:10) · · · ? ? ⑧⑧⑧⑧⑧ Σ d X ? ? ⑧⑧⑧ Σ d X ? ? ⑧⑧⑧ X ? ? ⑧⑧⑧⑧ Σ − d X ? ? ⑧⑧⑧ Σ − d X ? ? ⑧⑧⑧ · · · X I I Figure 1.
Left:
A component of type Z A ∞ of the AR quiver in thecase w = 1. Here d = w −
1. The other components are obtained byapplying Σ i to this one, for i = 1 , . . . , d − Right:
A component ofthe AR quiver in the case w = 1. It is a homogeneous tube; all othercomponents are obtained by applying Σ i to it, for i ∈ Z .2.1. The AR quiver of T w . For background on Auslander–Reiten (AR) theory wedirect the reader to [1], [2] and, in the triangulated setting [17].The structure of the AR quiver of T w was described in [24] by using a model of T w as a thick subcategory of the derived category of a certain differential graded algebra.The indecomposable objects of T w and the form of the AR quiver of T w was determinedfor w > w in [15, Section 3.3]. We summarisethis below using the notation from [24]. Proposition 2.1.
The indecomposable objects of T w are precisely the (co)suspensionsof a family of objects X r for r > . If w = 1 then the AR quiver of T w consists of | w − | copies of Z A ∞ . If w = 1 then the AR quiver consists of a homogeneous tubeand all its (co)suspensions. See Figure 1. Hom-hammocks in T w for w = 1 . The notion of a Hom-hammock, introducedin [6], is well known in the setting of Auslander–Reiten theory. To describe the Hom-hammocks in T w conveniently, we need to introduce some notation regarding rays andcorays, which is borrowed from [7].Consider the object Σ i X j and make the following definitions (see Figure 2 for anillustration): ray + (Σ i X j ) := { Σ i − nd X j + n | n > } , coray + (Σ i X j ) := { Σ i X k | k j } ; ray − (Σ i X j ) := { Σ i + nd X j − n | n j } , coray − (Σ i X j ) := { Σ i X k | k > j } . Given a set S ⊆ ind ( T w ) we make the obvious definitions of rays and corays deter-mined by S , for example, ray + ( S ) := S s ∈ S ray + ( s ) . For an object a ∈ ind ( T w ) define L ( a ) ∈ ray − ( a ) to be the unique object lying onthe mouth of the component. Analogously, define R ( a ) ∈ coray + ( a ) to be the uniqueobject lying on the mouth. Thus, if a itself lies on the mouth, then a = L ( a ) = R ( a ).Given two indecomposable objects a, b ∈ ind ( T w ) that lie on the same ray or corayin the AR quiver of T w , then the finite set consisting of these two objects and allindecomposables lying between them on the (co)ray is denoted by ab . In an abuse ofnotation, we identify ray + ( a ) ∩ coray − ( b ) with its indecomposable additive generator.Following the usage prevalent in algebraic geometry, for objects a, b ∈ T w we sethom T w ( a, b ) := dim k Hom T w ( a, b ). For a ∈ ind ( T w ), define the forward Hom-hammock RAQUEL COELHO SIM ˜OES AND DAVID PAUKSZTELLO coray − ( L ( a )) coray − ( a ) ray + ( a ) ray + ( R ( a )) coray + ( a ) ray − ( a ) aL ( a ) R ( a ) H − ( a ) H + ( a ) Figure 2.
The forward and backward Hom-hammocks of a .and the backward Hom-hammock of a as, respectively, H + ( a ) := ray + ( aR ( a )) and H − ( a ) := coray − ( L ( a ) a ) . Proposition 2.2 ([24, Propositions 3.2 and 3.3]) . Let a, b ∈ ind ( T w ) for w = 1 .(i) If w = 0 , then hom T w ( a, b ) = (cid:26) if b ∈ H + ( a ) ∪ H − ( S a ) , otherwise.(ii) If w = 0 , then hom T w ( a, b ) = if b ∈ H + ( a ) ∪ H − ( S a ) \ { a } , if b = a, otherwise. Factorisation properties.
Later, it will be important to know how morphismsbetween indecomposable objects of T w factor. This is dealt with in the followingproposition, which generalises the statements of [20, Propositions 2.1 and 2.2]. Proposition 2.3.
Suppose a, b and c are in ind ( T w ) for w ∈ Z \ { , } .(i) Suppose that c ∈ H + ( a ) ∩ H + ( b ) and b ∈ H + ( a ) . If g : a → b is a nonzeromorphism, then each morphism f : a → c factors through g as a g −→ b −→ c .(ii) Suppose that c ∈ H − ( S a ) ∩ H − ( S b ) and b ∈ H + ( a ) . If g : a → b is a nonzeromorphism, then each morphism f : a → c factors through g as a g −→ b −→ c .Dually,( i ′ ) Suppose that c ∈ H − ( a ) ∩ H − ( b ) and b ∈ H − ( a ) . If g : b → a is a nonzeromorphism, then each morphism f : c → a factors through g as c −→ b g −→ a .( ii ′ ) Suppose that c ∈ H + ( S − a ) ∩ H + ( S − b ) and b ∈ H − ( a ) . If g : b → a is a nonzeromorphism, then each morphism f : c → a factors through g as c −→ b g −→ a .Proof. For w > T w when w − (cid:3) When w = 0, each a ∈ ind ( T ) has a two-dimensional endomorphism space and wetweak the result for this case. Proposition 2.4.
Suppose a , b and c are indecomposable objects in T .(i) If a , b and c are pairwise non-isomorphic, then the statements in Proposition 2.3hold without modification. ORSION PAIRS 7 (ii) Suppose a = b and b ∈ H + ( a ) ∪ H − ( a ) . Then for any non-isomorphism f ∈ Hom T ( a, a ) there exist nonzero maps g : a → b and h : b → a such that f = hg .Proof. For statement (i) use one-dimensionality of the Hom-spaces and [20] as above.Let f : a → a be a nonzero non-isomorphism and consider the AR triangle τ a → e ⊕ e → a → Σ τ a . Let b ∈ H + ( a ) ∪ H − ( a ) \ { a, e , e } . By the almost split propertyof the second map in the AR triangle, there is a commutative diagram: a f (cid:15) (cid:15) ∃ (cid:2) k k (cid:3) { { ✇✇✇✇✇✇✇✇✇ τ a / / e ⊕ e (cid:2) ǫ ǫ (cid:3) / / a / / Σ τ a. By part (i) the maps k i : a → e i factor through b as a g −→ b h i −→ e i . Note that thesame map g : a → b can be chosen for each factorisation by the one-dimensionality ofthe Hom-spaces. Thus, f = (cid:2) ǫ ǫ (cid:3)(cid:2) k k (cid:3) = (cid:2) ǫ ǫ (cid:3)(cid:2) h gh g (cid:3) = ǫ h g + ǫ h g = ( ǫ h + ǫ h ) g =: hg, giving the required factorisation. If b ∈ { e , e } , then we argue dually using the otherAR triangle a → e ′ ⊕ e ′ → τ − a → Σ a . (cid:3) Remark 2.5.
Let a, b ∈ ind ( T ), a = b and b ∈ H + ( a ). Let g ∈ Hom T ( a, b ) and h ∈ Hom T ( b, a ) be nonzero maps. Then { id a , hg } forms a basis of Hom T ( a, a ).3. Extensions in T w with indecomposable outer terms for w = 1In this section, we describe how to compute the middle terms of extensions in T w for which the outer terms are indecomposable.3.1. A necessary condition.
In this subsection we assume only that T is a Krull-Schmidt triangulated category. Given a triangle a → e → b → Σ a we give necessaryconditions that the object e must satisfy with respect to a and b . The material iswell-known to experts, but we give brief proofs for the convenience of the reader.In a distinguished triangle x f −→ y g −→ z −→ Σ x , the object z is called the cone of f and written cone ( f ), and the object x is called the cocone of g and written cocone ( g ).The following lemma is straightforward. Lemma 3.1.
Let a f −→ e g −→ b h −→ Σ a be a distinguished triangle in a Krull-Schmidttriangulated category T . We have the following isomorphism of triangles: a (cid:2) f (cid:3) / / ≃ e ⊕ e / / ≃ cone ( (cid:2) f (cid:3) ) / / ≃ Σ a , ≃ a (cid:2) f (cid:3) / / e ⊕ e (cid:2) g (cid:3) / / b ⊕ e (cid:2) h (cid:3) / / Σ a Dually when taking the cocone of a map of the form (cid:2) g (cid:3) . An analogue of the following lemma is contained in the proof of [8, Proposition 8.3].
Lemma 3.2.
Let T be a Krull-Schmidt triangulated category and suppose that a " f ... f n −→ n M i =1 e i (cid:2) g ··· g n (cid:3) −→ b −→ Σ a RAQUEL COELHO SIM ˜OES AND DAVID PAUKSZTELLO is a non-split distinguished triangle for some a, b ∈ ind ( T ) . Then:(1) The f i and g i are each nonzero.If additionally, hom( x, y ) for each x, y ∈ ind ( T ) and Ext ( x, x ) = 0 for all x ∈ ind ( T ) , then we also have(2) Ext ( e i , a ) = 0 and Ext ( b, e i ) = 0 .(3) The multiplicity of each indecomposable summand of the middle term is at mostone.Proof. Statement (1) follows immediately from Lemma 3.1. For (2), apply the functors
Hom ( − , a ) and Hom ( b, − ) to the distinguished triangle and use one-dimensionality ofthe Hom-spaces and the vanishing self-extension property.Without loss of generality, to show (3) it is enough to show that a non-split triangleof the form a f −→ e m −→ b −→ Σ a with b ∈ ind ( T ), is forced to satisfy m
1. Supposefor a contradiction that m >
1. Write 0 = f := (cid:2) f · · · f m (cid:3) t . Since hom( a, e ) = 1,we have f i = − λ i f with λ i ∈ k for 2 i m . Let D be the matrix with 1s alongthe leading diagonal and λ i for 2 i m down the first column. Then D defines anisomorphism, making the first square in the following diagram of triangles commute: a f / / e m / / D (cid:15) (cid:15) b / / ≃ (cid:15) (cid:15) Σ aa f ′ / / e m / / / / b ′ ⊕ e m − / / Σ a, where f ′ := (cid:2) f · · · (cid:3) t . Lemma 3.1 tells us that cone ( f ′ ) = b ′ ⊕ e m − , where b ′ = cone ( f ), and the Five Lemma for triangles gives b ≃ b ′ ⊕ e m − . If b ′ = 0, thenwe get the desired contradiction to the indecomposability of b . If b ′ = 0, then f is anisomorphism, meaning that we started with a split triangle, also a contradiction. (cid:3) Ext-hammocks.
As with Hom-spaces, we write ext T w ( b, a ) := dim k Ext T w ( b, a )for a, b ∈ ind ( T w ). The Ext-hammocks for a can be obtained by combining Propo-sition 2.2 with Serre duality. The forward and backward Ext-hammocks of a are,respectively, E + ( a ) := H + ( τ − a ) and E − ( a ) := H − (Σ a ) . Proposition 3.3.
Suppose that w ∈ Z \ { } and a, b ∈ ind ( T w ) .(i) If w = 0 , then ext T w ( b, a ) = (cid:26) if b ∈ E + ( a ) ∪ E − ( a ) , otherwise.(ii) If w = 0 , then ext T ( b, a ) = if b ∈ E + ( a ) ∪ E − ( a ) \ { Σ a } , if b = Σ a, otherwise. Consider the object X r ∈ ind ( T w ) for r >
0. The Ext-hammocks of X r are given by E + ( X r ) = r [ i =0 ray + (Σ − d X r − i ) and E − ( X r ) = r [ i =0 coray − (Σ id +1 X r − i ) . These are indicated graphically in Figure 3.
ORSION PAIRS 9
Cohomology of the middle terms.
In this section we compute the cohomologyof the middle terms of extensions in T w for w = 1 whose outer terms are indecompos-able. Since the action of Σ and τ is transitive on the AR quiver of T w , without loss ofgenerality we may restrict our attention to the objects X r for r > E + ( X r ). Note that the non-trivial extensionsoccurring in this Ext-hammock have the form(1) X r −→ E −→ Σ − sd X r + s − i f −→ Σ X r for s > , and 1 i r + 1 . Lemma 3.4.
Let w ∈ Z \ { , } . Consider a triangle of the form (1) above. Then: H n ( E ) = k for n = d, d, . . . , sd, k for n = 0 , − d, . . . , − ( r − i ) d, k for n = − ( r − i + 1) d, − ( r − i + 2) d, . . . , − rd, otherwise,where when i = r + 1 , we take the second condition to be empty.Proof. Suppose | d | >
1. Applying the functor H n ( − ) to the distinguished triangle (1)and using the fact that, by the proofs of [24, Propositions 3.2–3.4], H n ( X t ) = (cid:26) k if n = 0 , − d, − d, . . . , − td, i r : H ( s − j ) d ( E ) ∼ −→ H − jd ( X r + s − i ) for 0 j < s ; H − jd ( X r ) ֒ −→ H − jd ( E ) − ։ H − ( j + s ) d ( X r + s − i ) for 0 j r − i ;(2) H − jd ( X r ) ∼ −→ H − jd ( E ) for r − i + 1 j r. When i = r + 1, the sequence (2) degenerates into the isomorphism on the third line.Now suppose d = 1. Then the short exact sequences (2) above are connected intothe long exact sequence H − ( r − i ) ( X r ) ֒ −→ H − ( r − i ) ( E ) −→ H − ( r + s − i ) ( X r + s − i ) H − ( r − i ) ( f ) −→ H − ( r − i − ( X r ) −→ · · ·· · · −→ H − s − ( X r + s − i ) H − ( f ) −→ H ( X r ) −→ H ( E ) − ։ H − s ( X r + s − i ) . For i = r and i = r + 1 there is nothing to prove: in the first case, there is alreadyonly one short exact sequence, and in the second, (2) degenerates into an isomor-phism. Thus, we need only consider the cases 1 i < r . There are nonzero maps g : Σ − s X r + s − i → Σ r +2 − i X i − and h : Σ r +2 − i X i − → Σ X r by Proposition 2.2. Thusthe map f in triangle (1) factors as f = hg by Proposition 2.3(ii). It follows that H ( f ) = H ( h ) H ( g ). Now for 0 j r − i we have H − j (Σ r +2 − i X i − ) = H r +2 − i − j ( X i − ) = 0because r + 2 − i − j > H n ( X t ) = 0 for any n > t > d > H − j ( f ) for 0 j r − i in the long exactsequence above are zero, which thus decomposes back into the short exact sequences(2) allowing us to again read off the cohomology of E . (cid:3) We now deal with the Ext-hammock E − ( X r ). Note that the non-trivial extensionsoccurring in this Ext-hammock have the form(3) X r −→ E −→ Σ id X r + s − i f −→ Σ X r for s > i r. Lemma 3.5.
Let w ∈ Z \ { , } . Consider a triangle of the form (3) above. Then: H n ( E ) = k for n = 0 , d, . . . , ( i − d, k for n = ( r + i ) d − , ( r + i + 1) d − , . . . , ( r + s + i ) d − , otherwise,where when i = 0 we assume the first condition to be empty.Proof. Apply the functor H n ( − ) to the triangle (3) and note that the long exactcohomology sequence decomposes into exact sequences H − jd − ( E ) ֒ −→ H ( i − j ) d ( X r + s − i ) H − jd − ( f ) −→ H − jd ( X r ) − ։ H − jd ( E ) . Since H − jd − ( f ) = 0 for 0 j < i and i + r j i + r + s , we have H − jd ( X r ) ≃ H − jd ( E ) for 0 j < i and H − jd − ( E ) ≃ H ( i − j ) d ( X r + s − i ) for i + r j i + r + s .The map f : Σ id X r + s − i → Σ X r factors as Σ id X r + s − i h −→ Σ id X r − i g −→ Σ X r byProposition 2.3. The map g is induced from an inclusion map of the underlying DGmodules; see [24, Section 2] for precise details. As such H − jd − ( g ) : H ( i − j ) d ( X r − i ) −→ H − jd ( X r ) is nonzero and thus an isomorphism (by one-dimensionality) for i j i + r , and zero otherwise. Similarly the induced map H ( i − j ) d ( h ) : H jd − ( X r + s − i ) → H − jd − ( X r − i ) is an isomorphism for i j i + r , and zero otherwise. Since H ( f ) = H ( g ) H ( h ), it follows that H − jd − ( f ) is an isomorphism for i j i + r . Now onecan read off the cohomology of E from the sequences above. (cid:3) Lemma 3.6.
In the case w = 0 , the statements of Lemmas 3.4 and 3.5 also hold, withthe modification that there are, up to equivalence, two extensions, X r → E → Σ X r → Σ X r , one whose middle term has cohomology as in Lemma 3.4, and one whose middle termhas trivial cohomology.Proof. In the case that w = 0, d = − T consists ofonly one Z A ∞ component. However, the extensions with indecomposable outer termsare formulated exactly as in Lemmas 3.4 and 3.5. The only difference occurs be-cause the Ext-hammocks E + ( X r ) and E − ( X r ) have non-trivial intersection E + ( X r ) ∩ E − ( X r ) = { Σ X r } . The two-dimensional Ext-space, Ext (Σ X r , X r ) = Hom (Σ X r , Σ X r )(see Proposition 2.2), has a basis { id a , f } where f can be chosen to be a non-isomorphismfactoring through any indecomposable object in H + (Σ X r ); see Remark 2.5. The cor-responding extensions are: X r −→ E −→ Σ X r f −→ Σ X r and X r −→ −→ Σ X r id −→ Σ X r . The first triangle is (equivalent to) the AR triangle, thus its cohomology is known.However, one can also argue exactly as in the case d = 1 in the proof of Lemma 3.4.It is clear that the middle term of the second triangle has trivial cohomology. (cid:3) Graphical calculus.
The main technical result of this section is the followingcomputation of the middle terms of extensions whose outer terms are indecomposable.It is analogous to the graphical calculus in [8, Corollary 8.5]. The strategy of our proofis inspired by [26, Section 8].
Theorem 3.7.
Let a, b ∈ ind ( T w ) for w = 0 , . Suppose Ext T ( b, a ) = 0 . Let a → e → b → Σ a be the unique non-split extension of b by a . Then e decomposes as e = e ⊕ e ,where e i is either indecomposable or zero for i = 1 , . Moreover, the e i can be computedby the following graphical calculus:(i) If b ∈ E + ( a ) then e = ray + ( a ) ∩ coray − ( b ) and e = coray + ( a ) ∩ ray − ( b ) . ORSION PAIRS 11 a Σ abe e a Σ a S aL ( S a ) b S − b R ( S − b ) e e Figure 3.
Top: Middle terms of extensions whose outer terms lie inthe same component. Bottom: Middle terms of extensions whose outerterms lie in different components. Top and bottom: Shaded regions arethe Ext-hammocks E + ( a ) and E − ( a ). (ii) If b ∈ E − ( a ) then e = coray − ( L ( S a )) ∩ ray − ( b ) and e = coray + ( a ) ∩ ray + ( R ( S − b )) .If any of the intersections in parts (i) and (ii) are empty, then we interpret the corre-sponding object as being the zero object. See Figure 3 for an illustration.Proof. Without loss of generality, we may assume that a = X r for some r >
0. Firstlyconsider triangle (1): X r → E → Σ − sd X r + s − i → Σ X r Suppose E = L ni =1 E m ( i ) i with E i ∈ ind ( T w ) and m ( i ) >
0. Note that when w = 0 , m ( i )
1. Moreover, the only indecomposable objectssatisfying the necessary conditions of Lemma 3.2 are { X r − i , Σ − sd X r + s } . Call theseobjects candidates . Note that when i = r + 1, the candidate is simply Σ − sd X r + s . Wenow use Lemma 3.4 to identify whether these two indecomposable summands appearin E with multiplicity 0 or 1. Write E = E ⊕ E with E indecomposable.Suppose that d < −
1. In this case the cohomology of the indecomposable objects X t is concentrated in non-negative degrees. By Lemma 3.4 the lowest degree in which E has non-trivial cohomology is sd . Thus, Σ − sd X r + s must be a direct summand of E . Set E = Σ − sd X r + s , and note that E has one-dimensional cohomology in degrees sd, ( s − d, . . . , − rd . This leaves E with one-dimensional cohomology in degrees0 , − d, . . . , − ( r − i ) d . The only candidate object with cohomology in these degrees is X r − i , giving the unique non-split triangle as X r → Σ − sd X r + s ⊕ X r − i → Σ − sd X r + s − i → Σ X r . Inspecting the AR quiver gives: ray + ( X r ) ∩ coray − (Σ − sd X r + s − i ) = { Σ − sd X r + s } , coray + ( X r ) ∩ ray − (Σ − sd X r + s − i ) = { X r − i } . The case d > X t nowhave non-trivial cohomology only in non-positive degrees, giving statement (i). Ananalogous argument applied to the triangle (3) using Lemma 3.5 gives (ii). (cid:3) Proposition 3.8. If w = 0 and a, b ∈ ind ( T ) with b = Σ a , then the statement ofTheorem 3.7 also holds. If b = Σ a , then there are two non-split triangles a −→ e ⊕ e −→ Σ a f −→ Σ a and a −→ −→ Σ a id −→ Σ a, where f is a non-isomorphism. The first is computed as in Theorem 3.7(i), the secondcorresponds to Theorem 3.7(ii).Proof. Here we have d = − E + ( X r ). Again write E = E ⊕ E with E indecomposable. First observe that Lemma 3.4 implies thatone summand of E is Σ s X t for some t >
0. Lemma 3.2(1) and Proposition 2.2 meanthat Σ s X r + s is the only possibility, which we take to be E . This means that E hascohomology in degrees 0 , , . . . , r − i , and so there is precisely one summand of E that consists of an unsuspended X t for some 0 t r − i . Note that if t > r − i then X t has cohomology in too many degrees. Inspecting the AR quiver and usingProposition 2.2 again, we see that if t < r − i , there is no map X t → Σ − s X r + s − i , andthus by Lemma 3.2(1) such X t cannot be a summand of E . This leaves only X r − i itself, giving E = Σ − s X r + s ⊕ X r − i again, as claimed.The argument for the triangle (3) is analogous, however, one must deal with the case X r → E → Σ X r → Σ X r separately. As remarked in Lemma 3.6, the two trianglesare the standard triangle X r −→ −→ Σ X r id −→ Σ X r and the AR triangle. The ARtriangle puts us in case (i) of the theorem, and the second triangle puts us in case (ii)of the theorem with empty intersections, and therefore zero middle term. (cid:3) Remark 3.9.
In the case that w = 0 and a is an indecomposable object not lyingon the boundary of the AR quiver we have Ext T ( a, a ) = k . The middle term of theself-extension a → e ⊕ e → a → Σ a is computed using Theorem 3.7(ii): one shouldregard the second occurrence of a as being S a and lying in a ‘different AR component’.4. Extensions in T w with decomposable outer terms for w = 1When computing the extension closure of a set of objects, it is useful to be ableto reduce to computing only the middle terms of extensions whose outer terms areindecomposable. The aim of this section is to establish this for the categories T w byproving the following theorem. Theorem 4.1.
Let w ∈ Z \ { } . Let { a i } ni =1 and { b j } mj =1 be sets of (not necessarilypairwise non-isomorphic) indecomposable objects of T w . Any extension of the form n M i =1 a i → e → m M j =1 b j → Σ n M i =1 a i ORSION PAIRS 13 can be computed iteratively from extensions whose outer terms are indecomposable andbuilt from { a i } ni =1 and { b j } mj =1 . Factorisation-free extensions.
We start with a short general subsection. Here T will be a k -linear, Hom-finite, Krull-Schmidt triangulated category. Definition 4.2.
A collection of morphisms { h i : b i → Σ a } mi =1 in T will be called factorisation-free if for each i = j there is no map β : b i → b j such that h i = h j β . Lemma 4.3.
Let a, b , b ∈ ind ( T ) . Suppose h : b → Σ a and h : b → Σ a arenon-zero maps such that there exists β : b → b with h = h β . Then, there is anisomorphism of triangles: a / / ≃ e / / ≃ b ⊕ b (cid:2) h h (cid:3) / / ≃ Σ a ≃ a / / b ⊕ f / / b ⊕ b (cid:2) h (cid:3) / / Σ a, where f is the cocone of h , i.e. a −→ f −→ b h −→ Σ a .Proof. This follows from the Five Lemma for triangulated categories by consideringthe isomorphism F : b ⊕ b → b ⊕ b given by F = (cid:2) β (cid:3) and using Lemma 3.1. (cid:3) Corollary 4.4.
Any extension a → e → L mi =1 b i → Σ a in T is isomorphic to a directsum of the triangles a → E → B → Σ a and → B ′ → B ′ → Σ0 , in which B is thesum of objects b i such that { h i : b i → Σ a } is factorisation-free and B ′ is the sum ofthe remaining summands. Remark 4.5.
In the case of T w , Corollary 4.4 implies that we need only considerextensions of the form a → e → L mi =1 b i → Σ a in which the b i are pairwise non-isomorphic: any morphism b i → Σ a factors through an isomorphism b i → b i .We now show that factorisation-freeness in T w is essentially the same as Hom-orthogonality. Lemma 4.6.
Suppose b , b ∈ E + ( a ) ∪ E − ( a ) are indecomposable objects such that b i = Σ a for i = 1 , . The pair of morphisms { h i : b i → a } i =1 is factorisation-free ifand only if b and b are Hom-orthogonal.Proof. One direction is clear. For the other, suppose
Hom ( b , b ) = 0. We claim that { h , h } is not factorisation-free. Case b ∈ H + ( τ − a ) : In this case, either b ∈ H + ( b ) or b , Σ a ∈ H − ( S b ). Eitherway, we have that h factors through h , by Propositions 2.3 and 2.4 and by one-dimensionality of Hom ( b , Σ a ). Case b ∈ H − (Σ a ) : Since
Hom ( b , b ) = 0, we have two subcases. Subcase b ∈ H + ( b ) : If w = 0 ,
2, then b must lie in H − (Σ a ) since Hom ( b , Σ a ) = 0.Hence, by Proposition 2.3, h factors through h . If w = 0 or 2, then b can lie eitherin H − (Σ a ), in which case h factors through h , or H + ( τ − a ). In the latter case, notethat b , Σ a ∈ H − ( S b ), and so, by Propositions 2.3 and 2.4, h factors through h . Subcase b ∈ H − ( S b ) : If w = 0 ,
2, then b H + ( τ − a ) ∪ H − (Σ a ), and so Hom ( b , Σ a ) =0, a contradiction. So, w must be 0 or 2. But then we have b , Σ a ∈ H + ( b ), and soagain, by Propositions 2.3 and 2.4, h factors through h . (cid:3) Extensions for which the first term is indecomposable.
For this subsectionwe invoke the following setup.
Setup 4.7.
Let w ∈ Z \ { } and fix a ∈ ind ( T w ). Suppose b i ∈ ind ( T w ) are such that b i ∈ E + ( a ) ∪ E − ( a ) for i = 1 , . . . , m . Consider the triangles a −→ e ′ i ⊕ e ′′ i −→ b i h i −→ Σ a from Theorem 3.7 and Proposition 3.8. We shall always assume that e ′′ i ∈ coray + ( a ).We now compute the middle terms of triangles of the form a −→ e −→ m M i =1 b i (cid:2) h ··· h m (cid:3) −→ Σ a, where { h i : b i → Σ a } mi =1 is factorisation-free, and each map h i : b i → Σ a is nonzero. If w = 0, we further assume that b i = Σ a . In particular, we always have ext( b i , a ) = 1.In light of Lemma 4.6, the assumption that { h i : b i → Σ a } mi =1 is factorisation-freeboils down to assuming that the b i are pairwise Hom-orthogonal.To simplify our arguments we need the following technical definition. Definition 4.8.
Let b ∈ E + ( a ) ∪ E − ( a ). We define the extended ray and extendedcoray of b with respect to a as follows: exray a ( b ) := ( ray + ( L ( b )) ∪ coray − ( S L ( b )) if b ∈ E + ( a ) , coray − ( R ( b )) ∪ ray + ( S − R ( b )) if b ∈ E − ( b ) , and excoray a ( b ) := ( coray − ( R ( b )) ∪ ray + ( S − R ( b )) if b ∈ E + ( a ) , ray + ( L ( b )) ∪ coray − ( S L ( b )) if b ∈ E − ( a ) . Note that every element in E + ( a ) ∪ E + ( a ) lies in exray a ( b ) for some b ∈ ( τ − a ) R ( τ − a ).We can define a total order on extended rays of elements in E + ( a ) ∪ E − ( a ) as follows.Given x, y ∈ E + ( a ) ∪ E − ( a ), we say that exray a ( x ) exray a ( y ) if and only if there is k > x, y ∈ E + ( a ) and L ( x ) = τ k L ( y );(2) x, y ∈ E − ( a ) and R ( x ) = τ k R ( y );(3) x ∈ E + ( a ) , y ∈ E − ( a ) and S L ( x ) = τ k R ( y ), or(4) x ∈ E − ( a ) , y ∈ E + ( a ) and R ( x ) = τ k S L ( y ). Lemma 4.9.
Suppose b , b ∈ ind ( T w ) are Hom-orthogonal objects in E + ( a ) ∪ E − ( a ) such that exray a ( b ) < exray a ( b ) . Then(1) Ext ( b , e ′ ) = 0 but Ext ( b , e ′′ ) = 0 ;(2) Ext ( b , e ′′ ) = 0 but Ext ( b , e ′ ) = 0 .Proof. This follows by a case analysis in examining the various positions b and b canhave inside E + ( a ) ∪ E − ( a ) and then comparing the Ext-hammocks of e ′ i and e ′′ i withthe positions of b i , for i = 1 , (cid:3) Remark 4.10.
By Theorem 3.7 and Proposition 3.8, the non-split triangles corre-sponding to the non-vanishing extensions
Ext ( b , e ′′ ) and Ext ( b , e ′ ) above are e ′′ → x ⊕ e ′′ → b → Σ e ′′ and e ′ → x ⊕ e ′ → b → Σ e ′ , respectively, where x = excoray a ( b ) ∩ exray a ( b ). Lemma 4.11.
Suppose b , b ∈ ind ( T w ) are Hom-orthogonal objects in E + ( a ) ∪ E − ( a ) such that exray a ( b ) < exray a ( b ) . We have: ORSION PAIRS 15
XY Z Σ a Σ b b Σ a Σ b b b Figure 4.
The regions used in the proof of Lemma 4.11. The Hom-hammocks for Σ a , Σ b and b are shown in solid, broken and dottedlines, respectively. (1) If w = 0 then Ext ( b , b ) = 0 = Ext ( b , b ) .(2) If w = 0 then Ext ( b , b ) = 0 = Ext ( b , b ) provided there is b ∈ E + ( a ) ∪ E − ( a ) for which b , b and b are pairwise Hom-orthogonal and exray a ( b ) < exray a ( b ) < exray a ( b ) .Proof. Statement (1) can be checked by a case analysis as in Lemma 4.9.Now let w = 0. We shall only show that if Ext ( b , b ) = 0 then there is no b satisfyingthe conditions of statement (2), since the argument is dual for Ext ( b , b ) = 0. Suppose Ext ( b , b ) = 0. We only consider the case when b ∈ E − ( a ); the case when b ∈ E + ( a )is similar.Since b and b are Hom-orthogonal and b ∈ ( E + ( b ) ∪ E − ( b )) ∩ ( E + ( a ) ∪ E − ( a )), wehave that b must lie in the region X ∪ Y ∪ Z indicated in Figure 4. If b ∈ Y ∪ Z , then exray a ( b ) < exray a ( b ), contradicting the hypothesis. Hence, b ∈ X . If there is anindecomposable object b in E + ( a ) ∪ E − ( a ) such that exray a ( b ) < exray a ( b ) < exray a ( b ),then b must lie in the shaded area of lower sketch in Figure 4. But then we have either Hom ( b , b ) = 0 or Hom ( b , b ) = 0, meaning that b , b and b are not pairwise Hom-orthogonal. (cid:3) We are now ready to compute the middle term of the extension in Setup 4.7.
Proposition 4.12.
Suppose we are in Setup 4.7 and we have ordered the objects b i such that exray a ( b i ) < exray a ( b i +1 ) , for i < m . In the non-split triangle a → e → m M i =1 b i → Σ a, we have e ≃ e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m , where x i = exray a ( b i ) ∩ excoray a ( b i +1 ) for i = 1 , . . . , m − .Proof. We proceed by induction on m . For m = 1 this is Theorem 3.7 and Proposi-tion 3.8. Let m > a → e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m − → L m − i =1 b i → Σ a and a → e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m → L mi =2 b i → Σ a . Consider a diagram coming from the octahedral axiom in which the first column is thesplit triangle: Σ − ( L m − i =1 b i ) (cid:15) (cid:15) Σ − ( L m − i =1 b i ) (cid:15) (cid:15) Σ − e / / Σ − ( L mi =1 b i ) / / (cid:15) (cid:15) a / / (cid:15) (cid:15) e Σ − e / / Σ − b m α / / (cid:15) (cid:15) f / / (cid:15) (cid:15) e L m − i =1 b i L m − i =1 b i where f ≃ e ′ ⊕ x ⊕· · ·⊕ x m − ⊕ e ′′ m − . If α = 0, then e ≃ b m ⊕ e ′ ⊕ x ⊕· · ·⊕ x m − ⊕ e ′′ m − .Suppose now that α = 0. By applying Hom (Σ − b m , − ) to the family of triangles { e ′ i +1 → e ′ i ⊕ x i → b i → Σ e ′ i +1 } i m − , and using Lemmas 4.9 and 4.11, we have Ext ( b m , x i ) = 0, for i = 1 , . . . , m −
2. On the other hand, ext ( b m , e ′′ m − ) = 1 andthe corresponding non-split triangle is e ′′ m − → e ′′ m ⊕ x m − → b m → Σ e ′′ m − . Hence, byLemma 3.1(i), we have e ≃ e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m .Now consider the following application of the octahedral axiom:Σ − ( L mi =2 b i ) (cid:15) (cid:15) Σ − ( L mi =2 b i ) (cid:15) (cid:15) Σ − e / / Σ − ( L mi =1 b i ) / / (cid:15) (cid:15) a / / (cid:15) (cid:15) e Σ − e / / Σ − b α ′ / / (cid:15) (cid:15) f ′ / / (cid:15) (cid:15) e L m − i =2 b i L m − i =2 b i where f ′ = e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m . Using Lemmas 4.9 and 4.11, we have that e ≃ b ⊕ e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m if α ′ = 0, and e ≃ e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m otherwise.Since the cone, e , must coincide in both diagrams, we must have α = 0 , α ′ = 0 and e ≃ e ′ ⊕ x ⊕ · · · ⊕ x m − ⊕ e ′′ m . (cid:3) Extensions involving Σ a . In the previous subsection we excluded b i = Σ a foreach i from our analysis. The following proposition deals with this case. Proposition 4.13.
Let w = 0 and b , . . . , b k ∈ E + ( a ) ∪ E − ( a ) with b i = Σ a , for i = 1 , . . . , k . Consider the triangle a −→ e −→ ( k M i =1 b i ) ⊕ Σ a (cid:2) H h (cid:3) −→ Σ a, where H = (cid:2) h · · · h k (cid:3) . Then either e ≃ L ki =1 b i or e ≃ cocone ( h ) ⊕ Σ a . ORSION PAIRS 17
Proof.
Suppose h is an isomorphism. Then clearly every map h i factors through h . ByLemma 4.3, and because the cocone of h is zero, we have an isomorphism of triangles: a / / e / / ≃ ( L ki =1 b i ) ⊕ Σ a (cid:2) H h (cid:3) / / ≃ F Σ aa / / L ki =1 b i / / ( L ki =1 b i ) ⊕ Σ a (cid:2) h (cid:3) / / Σ a, where F is the matrix with 1s along the leading diagonal and h − h i as the entry inposition ( k + 1 , i ), for i = 1 , . . . , k .Now suppose h is a non-isomorphism. By Proposition 2.4(ii), h factors through anyof the h i ’s. Write, for instance, h = h β , with β : Σ a → b nonzero. Again, by Lemma4.3, we have an isomorphism of triangles: a / / e / / ≃ ( L ki =1 b i ) ⊕ Σ a (cid:2) H h (cid:3) / / ≃ F Σ aa / / f ⊕ Σ a / / ( L ki =1 b i ) ⊕ Σ a (cid:2) H (cid:3) / / Σ a, where F is the matrix with 1s along the leading diagonal and β as the entry in position(1 , k + 1) and f = cocone ( h ). (cid:3) Proof of Theorem 4.1.
We are now ready to prove the main result of thissection.
Lemma 4.14.
Any extension of the form a → e → L mi =1 b i → Σ a in T w , w ∈ Z \ { } ,can be computed iteratively from extensions whose outer terms are indecomposable andinvolving only a, b , b , . . . , b m and objects constructed from these objects.Proof. Immediate from Corollary 4.4 and Propositions 4.12 and 4.13. (cid:3)
Proof of Theorem 4.1.
We proceed by induction on n . For n = 1, this is Lemma 4.14.Suppose n > a (cid:15) (cid:15) a (cid:15) (cid:15) Σ − ( L mi =1 b i ) / / L ni =1 a i / / (cid:15) (cid:15) e / / (cid:15) (cid:15) L mi =1 b i Σ − ( L mi =1 b i ) / / L ni =2 a i / / (cid:15) (cid:15) L ki =1 x i / / (cid:15) (cid:15) L mi =1 b i Σ a Σ a By induction, the triangle L ni =2 a i → L ki =1 x i → L mi =1 b i → Σ( L ni =2 a i ) is constructedfrom extensions whose outer terms are indecomposable, which are built from the a i ’sand the b j ’s. Therefore, by Lemma 4.14, it follows that so is a → e → L ki =1 x i → Σ a ,as required. (cid:3) (cid:31) (cid:31) ❄❄❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄❄❄❄ ... (cid:31) (cid:31) ❄❄❄❄❄❄❄ ( − d − , − d ) ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , d ) ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , d ) ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , d ) ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − , d ) ? ? ⑧⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ · · · ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , − d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − , d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ · · · ( − d − , − d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , − d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − d − , d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( − , d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ ( d − , d ) ? ? ⑧⑧⑧⑧⑧⑧ (cid:31) (cid:31) ❄❄❄❄❄❄ · · · ? ? ⑧⑧⑧⑧⑧⑧ ( − d − , − d ) ? ? ⑧⑧⑧⑧⑧⑧ ( − d − , − d ) ? ? ⑧⑧⑧⑧⑧⑧ ( − d − , ? ? ⑧⑧⑧⑧⑧⑧ ( − , d ) ? ? ⑧⑧⑧⑧⑧⑧ ( d − , d ) ? ? ⑧⑧⑧⑧⑧⑧ · · · Figure 5.
A component of type Z A ∞ with the endpoints of the d -admissible arcs described.5. The combinatorial model
Here we recall the combinatorial model for T w from [20] in the case w > w ∞ -gon’. Namely, weregard each pair of integers ( t, u ) as an arc connecting the integers t and u .For w ∈ Z \ { } set d = w −
1. A pair of integers ( t, u ) is called a d -admissible arc if(i) for w >
2, one has u − t > w and u − t ≡ d ;(ii) for w = 0, a ( − t, u ) is one with u − t
0; and(iii) for w −
1, one has u − t w and u − t ≡ d ; Notation.
Whenever the value of w is not specified, the arc incident with the distinctintegers t and u is denoted by { t, u } . In other words, assuming t > u , { t, u } = ( t, u )when w − { t, u } = ( u, t ) when w > length of the arc ( t, u ) is | u − t | .When d is clear from context we refer to d -admissible arcs simply as admissiblearcs . Figure 5 shows how admissible arcs correspond to the indecomposable objectsof T w when w = 1. We note that the action of the suspension, AR translate and Serrefunctor in this model are given byΣ( t, u ) = ( t − , u − τ ( t, u ) = ( t − d, u − d ) S ( t, u ) = ( t − w, u − w ) . Let A be a collection of d -admissible arcs. Then an integer t is called a left fountain of A if A contains infinitely many arcs of the form { s, t } with s t . Dually, one definesa right fountain of A ; a fountain of A is both a left fountain and a right fountain.By Proposition 1.1, to obtain a characterisation of torsion pairs in T w we need tocharacterise extension-closed contravariantly finite subcategories X of T w . The com-putations of Sections 3 and 4 will be used to characterise the extension-closed subcat-egories. Below, we state a characterisation of contravariantly finite subcategories in T w . 6. Contravariant-finiteness
The characterisation of contravariantly finite subcategories in T w is as follows: Proposition 6.1.
Suppose w ∈ Z \ { } . A full subcategory X of T w is contravariantlyfinite if and only if, for each i ∈ Z , an infinitude of objects in X ∩ ray + (Σ i X ) forcesan infinitude of objects in X ∩ coray − ( S Σ i X ) . This can be neatly described in terms of fountains.
ORSION PAIRS 19
Corollary 6.2.
Suppose w ∈ Z \ { } . Let X be a full subcategory of T w and X be theset of admissible arcs corresponding to ind ( X ) . Then(1) If w > , X is contravariantly finite in T w if and only if every right fountainin X is also a left fountain.(2) If w , X is contravariantly finite in T w if and only if every left fountain in X is also a right fountain.Proof. For w >
2, a right fountain at a given integer corresponds to having infinitelymany objects from a ray, ray + (Σ i X ), in the subcategory X . The corresponding leftfountain consists of infinitely many objects from coray − ( S Σ i X ). For w ray + (Σ i X ) in X , and the corresponding right fountain consists ofinfinitely many objects from coray − ( S Σ i X ). The result then follows from Proposition6.1. (cid:3) When w = 2 Propositon 6.1 is [29, Theorem 2.2]; for w > w
0, sowe refrain from giving full details.A crucial part of the arguments in [20] and [29] is showing that the composition oftwo so-called ‘backward maps’ is zero. The term ‘backward map’ was first used in theproof of [29, Theorem 2.2] but not explicitly defined.
Definition 6.3.
Assume w ∈ Z \ { } . A backward map in T w is a non-zero morphismbetween indecomposable objects f : a → b with b ∈ H − ( S a ). In T we additionallyrequire that b is not isomorphic to a .We require a new argument for the case w = 0. While thinking about the case w = 0, we noticed an unexplained step in the ‘backward maps’ part of the proofin [29, Theorem 2.2]. For the convenience of the reader, we prove the required zerocomposition of ‘backward maps’ in T w for any w = 1. Lemma 6.4.
Let w ∈ Z \ { } . The composition of two backward maps in T w is zero.Proof. We deal with three cases: w = 0, w = 2 and w = 0 ,
2. Suppose f : a → b and g : b → c are backward maps. We start with the easier general case. Case w = 0 , : Denote the components of the AR quiver of T w by C , . . . , C | d |− , where,as usual, d = w −
1. When w = 0 , | d | >
1, i.e. the AR quiver has more thanone component. By applying a power of the suspension if necessary, we may assumethat a ∈ C . Since f and g are backward maps, it follows that b ∈ C and c ∈ C ,where the subscript is interpreted modulo | d | . When | d | > Hom T w ( a, c ) = 0since H + ( a ) ∪ H − ( S a ) ⊆ C ∪ C and c ∈ C . If | d | = 2 a direct computation showsthat H − ( S b ) ∩ H + ( a ) = ∅ , whence Hom T w ( a, c ) = 0. Thus backward maps compose tozero whenever w = 0 , Case w = 0 : By Proposition 2.4 the map gf factors as a f −→ b g −→ a g −→ c , where g f is a non-isomorphism and g is unique up to scalars. Applying Proposition 2.4 againgives that g factors through R ( a ) as g : a h −→ R ( a ) h ′ −→ c . Since a is not isomorphicto b , we have R ( a ) / ∈ H + ( b ) ∪ H − ( b ), whence the composite g g = 0, giving gf = 0,as claimed. Case w = 2 : Assume for a contradiction that gf = 0. By Proposition 2.3, the map g factors through a as g : b h −→ a h ′ −→ c . By [20, Proposition 2.1 (i)], the map h is a composition of irreducible maps h = h n · · · h say. Thus, the map gf factors as a f −→ b h −→ a h ′ −→ c and we have a nonzero map hf : a → a . By Proposition 2.2,hom T ( a, a ) = 1, whence hf = λ id a with λ = 0. It follows that λ ( h n − · · · h f ) is aright inverse for h n , i.e. h n is a split epimorphism. This contradicts the irreducibilityof h n , whence the original composition gf must have been zero. (cid:3) To show how the composition of two backward maps being zero is used in the proofof Proposition 6.1, we sketch this part of the argument for w = 0. Proof of Proposition 6.1.
Firstly, we show the forward implication for w = 0. Suppose,without loss of generality, that there are infinitely many objects s i ∈ ray + ( X ) occurringas objects in X . We want to show that this implies there are infinitely many objects c ∈ coray − ( X ) occurring in X , recalling that S X = X . The diagrams in [29, Proofof Theorem 2.2] may be useful to help understand our arguments.Since X is contravariantly finite, there is a right X -approximation x → c , where x = x ⊕ · · · ⊕ x n . We may assume that the map x → c is nonzero from each summand.Since c ∈ H − ( S s i ) = H − ( s i ) for each i ∈ N , the map s i → c factors as s i → x → c .In particular, there is a summand, x i say, of x such that the map s i f −→ x i g −→ c isnonzero. Now inspecting the Hom-hammocks shows that x i either lies on coray − ( X )or lies in the region of the AR quiver of T bounded by the following: coray − ( X ) , coray − ( R ( s i )) , and, ray + ( L ( c )) , ray + ( X ) . If x i ∈ coray − ( X ) there is nothing to show, so suppose that x i / ∈ coray − ( X ).Now by Lemma 6.4, unless x i ≃ s i , we have gf = 0; a contradiction. Now, since x contains only finitely many indecomposable summands, only finitely many of the s i may occur as summands of x . Taking an s j that is not a summand of x thus yields therequired contradiction. This shows that each of the x i must lie on coray − ( X ) above c .Repeating this argument indefinitely for c ∈ coray − ( X ) further and further from themouth then gives infinitely many objects of coray ( X ) in X , as claimed.The reverse implication works in exactly the same way as in [29, Theorem 2.2]. (cid:3) Torsion pairs and the Ptolemy condition in T w for w = 0 , w ∈ Z \ { , } . We will give a combinatorial description ofthe extension closure of a subcategory of T w using the combinatorial model presentedin Section 5. This description is in terms of Ptolemy diagrams of different classes,which include those defined in [29].For a ∈ ind ( T w ) denote the corresponding admissible arc by a . If a = ( t, u ) then the starting point is s ( a ) = t and the ending point is t ( a ) = u .Recall that when w −
1, the first coordinate of an admissible arc is strictly biggerthan the second coordinate, and when w > Definitions 7.1.
Let a = { t, u } , b = { v, w } , with t > u and v > w , be two admissiblearcs of T w .(1) The arcs a and b are said to cross if either u < w < t < v or w < u < v < t .(2) If a and b are crossing arcs, then the Ptolemy arcs of class I associated to a and b are the remaining four arcs connecting the vertices incident with a or b ,i.e. the set of Ptolemy arcs is {{ x, y } | x, y ∈ { t, u, v, w } , x = y, { x, y } 6 = a , b } .(3) The distance between a and b is defined as d ( a , b ) := min {| t − v | , | u − w | , | t − w | , | u − v |} . (4) The arcs a and b are neighbouring if they do not cross and d ( a , b ) = 1. ORSION PAIRS 21 (5) If a and b are neighbouring arcs and d ( a , b ) is given by the distance betweenvertices x and x −
1, then the corresponding
Ptolemy arc of class II is the arcconnecting the vertices incident with a or b which are not x and x − Figure 6.
The Ptolemy arcs of class I.
11 1
Figure 7.
The Ptolemy arcs of class II.Recall that for a full subcategory X of T w , h X i denotes the smallest extension-closedsubcategory of T w containing X . In this section, we prove the following main result. Theorem 7.2.
Let X be a full additive subcategory of T w and let X be the arcs corre-sponding to the objects of ind ( X ) . Then the objects of ind ( h X i ) correspond to the arcsof (1) for w > , the closure of X under admissible Ptolemy arcs of class I,(2) for w − , the closure of X under admissible Ptolemy arcs of classes I and II. Putting this together with Propositions 1.1 and 6.2 gives us part of Theorem A:
Corollary 7.3.
Let w ∈ Z \ { , } , X be a full additive subcategory of T w and X be thecorresponding set of arcs. Then ( X , X ⊥ ) is a torsion pair in T w if and only if(1) for w > , any right fountain in X is also a left fountain and X is closed undertaking admissible Ptolemy arcs of class I.(2) for w − , any left fountain in X is also a right fountain and X is closed undertaking admissible Ptolemy arcs of classes I and II. Combinatorial description of the Ext-hammocks.
We first need to intro-duce some notation regarding partial fountains , which is borrowed from [13].
Notation.
Let v > t > u be integers such that { t, u } and { t, v } are d -admissible arcs.Define the partial right fountain at t starting at v and the partial left fountain at t starting at u by RF ( t ; v ) := {{ t, x } d -admissible | x > v } ; LF ( t ; u ) := {{ t, y } d -admissible | y u } . Let V ⊆ Z such that { t, v } is a d -admissible arc for each v ∈ V . Write RF ( V ; t ) := [ v ∈ V RF ( v ; t ) and LF ( V ; t ) := [ v ∈ V LF ( v ; t ) . Below is a description of the Ext-hammocks in terms of partial fountains.
Lemma 7.4.
Let a, b ∈ ind ( T w ) , and V a = { s ( a ) + id | i = 1 , . . . , k } , where k > issuch that t ( a ) − s ( a ) = kd + 1 .(1) If w > , then Ext T w ( b, a ) = 0 if and only if b ∈ LF ( V a ; s ( a ) − ∪ RF ( V a ; t ( a ) + d ) . In particular, b ∈ E + ( a ) ⇐⇒ b ∈ RF ( V a ; t ( a ) + d ) and b ∈ E − ( a ) ⇐⇒ b ∈ LF ( V a ; s ( a ) − . (2) If w , then Ext T w ( b, a ) = 0 if and only if b ∈ RF ( V a ; s ( a ) − ∪ LF ( V a ; t ( a ) + d ) . In particular, b ∈ E + ( a ) ⇐⇒ b ∈ LF ( V a ; t ( a ) + d ) and b ∈ E − ( a ) ⇐⇒ b ∈ RF ( V a ; s ( a ) − . Proof.
The case when w − w = 0, the firstcoordinate of an admissible arc is greater than or equal to the second coordinate, andso the result is the same as in w −
1. When w >
2, the admissible arcs are orderedfrom left to right, and so we swap RF ( − ; − ) and LF ( − ; − ). (cid:3) The following two propositions are direct consequences of the previous lemma.
Proposition 7.5.
Let V a be as in Lemma 7.4. If w > , and a, b ∈ ind ( T w ) , then Ext T w ( b, a ) = 0 if and only if a and b cross and b is incident with a vertex in V a . Proposition 7.6.
Let V a be as in Lemma 7.4. If w − , and a, b ∈ ind ( T w ) , then Ext T w ( b, a ) = 0 if and only if we are in one of the following situations:(1) a and b cross and b is incident with a vertex in V a \ { t ( a ) − } ;(2) a and b are neighbouring arcs such that b is incident with s ( a ) − or t ( a ) − ;(3) b = ( s ( a ) − , t ( a ) − , i.e. b = Σ a . The middle terms of extensions correspond to admissible Ptolemy arcs.
We now define some arcs associated with a, b ∈ ind ( T w ) for which Ext T w ( b, a ) = 0. Apriori these arcs need not be admissible, but when they are, they will correspond tothe indecomposable summands of the middle term of the extension. Definition 7.7.
Let a, b ∈ ind ( T w ) and suppose Ext T w ( b, a ) = 0.(1) If b ∈ E + ( a ) then e := ( s ( a ) , t ( b )) and e := ( s ( b ) , t ( a )).(2) If b ∈ E − ( a ) then e := ( s ( b ) , s ( a )) and e := ( t ( b ) , t ( a )). Proposition 7.8.
Let a, b ∈ ind ( T w ) be such that Ext T w ( b, a ) = 0 , and e and e be asin Theorem 3.7.(1) Suppose b ∈ E + ( a ) . The arc e is always admissible. The arc e is admissibleif and only if s ( b ) = t ( a ) − .(2) Suppose b ∈ E − ( a ) . For w > , the arc e is admissible if and only if s ( b ) s ( a ) . For w ∈ Z \ { , } , the arc e is admissible if and only if t ( b ) = t ( a ) − .Moreover, e i is nonzero if and only if e i , which is the corresponding arc, is admissible. ORSION PAIRS 23
Proof.
We will first see when e and e are admissible. Let k and V a be as in Lemma7.4, and k ′ > t ( b ) − s ( b ) = k ′ d + 1.Suppose b ∈ E + ( a ). Then s ( b ) ∈ V a , and so we have: • t ( e ) − s ( e ) = ( k ′ + i ) d + 1, for some i ∈ { , . . . , k } ; and • t ( e ) − s ( e ) = ( k − i ) d + 1.Hence, e is always admissible, and e is admissible if and only if k − i >
1, i.e. s ( b ) = t ( a ) − b ∈ E − ( a ). Then t ( b ) ∈ V a , and so we have: • t ( e ) − s ( e ) = ( k ′ − i ) d + 1, for some i ∈ { , . . . , k } ; and • t ( e ) − s ( e ) = ( k − i ) d + 1.Hence, e is admissible if and only if k ′ − i >
1, i.e. s ( b ) < s ( a ) − w > s ( b ) > s ( a ) if w −
1. On the other hand, e is admissible if and only if k − i > t ( b ) = t ( a ) − • For x ∈ ind ( T w ), the arc corresponding to L ( S x ) is ( s ( x ) − w, s ( x )) and the arccorresponding to R ( S − x ) is ( t ( x ) , t ( x ) + w ). • Two indecomposable objects lie in the same ray (resp. coray) if and only if thefirst (resp. second) coordinate of the corresponding arcs coincides. (cid:3)
Corollary 7.9. If w − and a and b are neighbouring admissible arcs, then thecorresponding Ptolemy arcs of class II are always admissible. Definition 7.10.
Let a, b ∈ ind ( T w ). Write a ∗ b := add a ∗ add b , where the starproduct is defined on page 4. We define E ( a, b ) := ind ( a ∗ b ∪ b ∗ a ) \ { a, b } . Denotethe set of arcs corresponding to the objects in E ( a, b ) by E ( a , b ). Remark 7.11.
By Remark 4.5, E ( a, b ) consists of the indecomposable objects whichoccur as summands in the middle terms of non-split extensions a → e → b → Σ a and b → f → a → Σ b . Remark 7.12.
Note that the only case in Proposition 7.6 where a and b are neithercrossing nor neighbouring arcs is when w = − a lies on the mouth of the AR quiverand b = Σ a . In this case, a / ∈ E + (Σ a ) ∪ E − (Σ a ), and so Ext T − ( a, b ) = 0. On the otherhand, ext T − ( b, a ) = 1 and the extension of b (= Σ a ) by a is a −→ −→ Σ a id −→ Σ a .Therefore E ( a, b ) = ∅ .We have the following corollary to Proposition 7.8. Corollary 7.13.
Let a, b ∈ ind ( T w ) and a , b the corresponding admissible arcs. Then E ( a , b ) ⊆ { admissible Ptolemy arcs incident with the endpoints of a and b } . Proof.
By Remark 7.11, we have E ( a, b ) = { e , e , f , f } , where the e i ’s and f i ’s aresuch that the extensions are a → e ⊕ e → b → Σ a and b → f ⊕ f → a → Σ b . Notethat some of these objects may be zero. We will only check that e and e correspondto admissible Ptolemy arcs when nonzero, as the proof is analogous for f and f .Suppose e i = 0. By Remark 7.12, a and b are either crossing or neighbouring arcs.By Proposition 7.8, e i is admissible. It remains to check that e i is a Ptolemy arc. Bydefinition, the Ptolemy arcs and the arc e i connect endpoints of a and b . Ptolemyarcs of class I cover all the possibilities for this connection, so the only non-trivial casethat we need to consider is when a and b are neighbours. The problem here lies in thefact that the Ptolemy arc of class II might not be the only admissible arc connectingthe endpoints of a and b . Namely, when w = −
1, the arc ( x, x −
1) connecting the two closest endpoints of a and b is also admissible, and it is not in general a Ptolemyarc of class II.Since Ext ( b, a ) = 0, we have b incident with either s ( a ) − t ( a ) − • ( s ( a ) , s ( b )), if b is incident with s ( a ) − • ( t ( a ) , s ( b )) or ( t ( a ) , t ( b )), if b is incident with t ( a ) − e i is not any of these arcs, and therefore e i is the Ptolemyarc of class II. (cid:3) Let X be a full subcategory of T w . By the above and Theorem 4.1, the arcs cor-responding to objects of ind ( h X i ) are a subset of the closure of X under admissiblePtolemy arcs. We now need to show that the inclusion in Corollary 7.13 is in fact anequality.7.3. The extension closure.
Let a, b ∈ ind ( T w ). Combinatorially, Ptolemy arcs ofclass I and II arise out of the following situations, respectively: • w ∈ Z \ { , } and a and b are crossing arcs, • w − a and b are neighbouring arcs.We shall show that in each of the two cases above E ( a , b ) is precisely the set of allthe admissible Ptolemy arcs of the appropriate class associated to a and b . First, letus consider the case when a and b are neighbours. Proposition 7.14.
Let w − and a, b ∈ ind ( T w ) be such that a and b are neighbour-ing arcs. Then Ext T w ( a, b ) = 0 or Ext T w ( b, a ) = 0 , and E ( a , b ) is the set of admissiblePtolemy arcs of class II associated to a and b .Proof. If b is incident with s ( a ) − t ( a ) −
1, then
Ext T w ( b, a ) = 0, by Proposition 7.8.Dually, if b is incident with s ( a ) + 1 or t ( a ) + 1, then Ext T w ( a, b ) = 0. In both cases,the middle term of the extension must be nonzero, since b = Σ a, Σ − a . Therefore, E ( a , b ) = ∅ , and so in particular, its cardinality is greater than or equal to one.Note that the set of (admissible) Ptolemy arcs of class II associated to a and b hascardinality one or two. If the cardinality is one, then by Corollary 7.13, E ( a , b ) mustbe equal to the set of admissible Ptolemy arcs associated to a and b . Now, supposethere are two admissible Ptolemy arcs associated to a and b . Then these must be ofthe form ( x, y ) and ( x − , y + 1), with x > y + 3, for w = −
1. In this case we haveextensions in both directions, giving rise to the Ptolemy arcs ( x, x −
1) and ( y, y − (cid:3) We now turn our attention to the case when a and b cross. Proposition 7.15.
Let a, b ∈ ind ( T w ) . If Ext T w ( b, a ) = 0 and a , b are crossing arcs,then the set of admissible Ptolemy arcs of class I associated to a and b is contained in E ( a , b ) .Proof. Firstly, let us consider the case when w > Case 1: b ∈ E + ( a ). We need to check whether ( s ( a ) , s ( b )) and ( t ( a ) , t ( b )) are admis-sible. Note that these two arcs are Ptolemy arcs associated to a and b , but they donot correspond to the middle term of the extension a → e → b → Σ a .We have s ( b ) = s ( a ) + id , for some i = 1 , . . . , k because s ( b ) ∈ V a . Hence, s ( b ) − s ( a ) = id , and t ( b ) − t ( a ) = ( i + k ′ + k ) d . Therefore, the arcs ( s ( a ) , s ( b )) and ( t ( a ) , t ( b ))are admissible if and only if w = 2. Indeed, Ext T ( a, b ) ≃ D Ext T ( b, a ) since T is 2-CY. Since a ∈ E − ( b ), ( s ( a ) , s ( b )) and ( t ( a ) , t ( b )) are the arcs corresponding to the ORSION PAIRS 25 indecomposable summands of the middle term of b → e ′ → a → Σ b , by Proposition7.8. Case 2: b ∈ E − ( a ). We need to check whether ( s ( b ) , t ( a )) and ( s ( a ) , t ( b )) are ad-missible. We have t ( b ) = s ( a ) + id , for some i = 1 , . . . , k , and so, t ( b ) − s ( a ) = id and t ( a ) − s ( b ) = ( k + k ′ − i ) d + 2. Therefore, ( s ( b ) , t ( a )) and ( s ( a ) , t ( b )) are ad-missible if and only if w = 2, as in case 1. As we have seen above, when w = 2, Ext T ( a, b ) ≃ Ext T ( b, a ) = 0, and here we have a ∈ E + ( b ). Hence, by Proposition 7.8,( s ( b ) , t ( a )) and ( s ( a ) , t ( b )) are the arcs corresponding to the indecomposable sum-mands of the middle term of the triangle b → e ′ → a → Σ b .Now, let w −
1. We need to consider the following three cases.
Case 1: b crosses a , it is to the left of a , and s ( b ) ∈ V a . As we have seen in case 1 for w >
2, we have s ( b ) − s ( a ) = id and t ( b ) − t ( a ) = ( i + k + k ′ ) d . However, d − w −
1, and so the arcs ( s ( a ) , s ( b )) and ( t ( a ) , t ( b )) are never admissible. Case 2: b crosses a , it is to the right of a , and t ( b ) ∈ V a . As in case 2 for w >
2, wehave t ( b ) − s ( a ) = id and t ( a ) − s ( b ) = ( k + k ′ + i ) d + 2. But again, since d − l > t ( b ) − s ( a ) = ld + 1 or t ( a ) − s ( b ) = ld + 1. Therefore,( s ( b ) , t ( a )) and ( s ( a ) , t ( b )) are never admissible. Case 3: b = Σ a , i.e. b = ( s ( a ) − , t ( a ) − a does not lie on the mouth if w = −
1. It is easy to check that ( s ( a ) , t ( b )) and ( s ( b ) , t ( a )) are never admissible. Onthe other hand, ( s ( a ) , s ( b )) and ( t ( a ) , t ( b )) are admissible if and only if w = −
1. Fromnow on let w = −
1. We have
Ext T − ( a, Σ a ) ≃ D Hom T − ( a, τ S a ) = 0 if and only if τ S a ∈ H − ( S a ), which always holds unless a lies on the mouth of the AR quiver.Since, by hypothesis, a does not lie on the mouth, we have Ext T − ( a, Σ a ) = 0 andΣ a ∈ E − ( a ). So ( s ( a ) , s ( a ) −
1) and ( t ( a ) , t ( a ) − a and Σ a , are the arcs corresponding to theindecomposable summands of the middle term of the triangle Σ a → e → a → Σ a . (cid:3) In contrast to the case w = 2, for w ∈ Z \ { , , } , there are crossing arcs whosecorresponding indecomposable objects do not have extensions between them. We mustcheck that these yield no admissible Ptolemy arcs. Lemma 7.16.
Suppose, in addition to w ∈ Z \{ , } , that w = 2 and let a, b ∈ ind ( T w ) be such that Ext T w ( a, b ) = 0 = Ext T w ( b, a ) and a and b cross. Then none of theassociated Ptolemy arcs of class I is admissible.Proof. Let w > a and b cross each other in such a way that s ( a )
1, nor can we have t ( a ) = s ( b ) + jd , for some j > a and b are admissible arcs, we have t ( a ) − s ( a ) = kd + 1 and t ( b ) − s ( b ) = k ′ d + 1, for some k, k ′ >
1. We also have s ( b ) = s ( a ) + x for some x >
1, and t ( a ) = s ( b ) + y , for some y > t ( b ) − s ( a ) = t ( b ) − s ( b ) + s ( b ) − s ( a ) = k ′ d + x + 1, we conclude that thePtolemy arc ( s ( a ) , t ( b )) is admissible if and only if k ′ d + x + 1 = ld + 1, for some l > x = ( l − k ′ ) d , and l − k ′ > s ( a ) < s ( b ). However, thiscontradicts the hypothesis, and therefore this Ptolemy arc is not admissible.The arc ( s ( b ) , t ( a )) is admissible if and only if t ( a ) − s ( b ) = kd + x + 1 = ld + 1, forsome l >
1, i.e. x = ( k − l ) d . Since x, d >
1, we must also have k − l >
1, but thiscontradicts the hypothesis.Using the fact that t ( a ) = s ( b ) + y , for some y >
1, we can similarly show that( t ( a ) , t ( b )) is not admissible. Finally, ( s ( a ) , s ( b )) is admissible if and only if s ( b ) − s ( a ) = x = ld + 1, for some l >
1. Suppose, for a contradiction, that this holds, i.e. s ( b ) = s ( a ) + ld + 1, for some l >
1. Then we have t ( a ) = s ( b ) + ( k − l ) d , with k − l >
1, a contradiction.The proof for the case when w − (cid:3) Corollary 7.17.
Let a, b ∈ ind ( T w ) be such that a and b are crossing arcs. Then E ( a , b ) is the set of admissible Ptolemy arcs of class I associated to a and b .Proof. This follows from Corollary 7.13, Proposition 7.15 and Lemma 7.16. (cid:3)
Putting these together with Corollary 7.13 and Proposition 7.14 yields the followingcorollary, which together with Theorem 4.1 gives Theorem 7.2.
Corollary 7.18.
Let a, b ∈ ind ( T w ) and a , b be the corresponding admissible arcs.Then E ( a , b ) = { admissible Ptolemy arcs incident with the endpoints of a and b } . Torsion pairs and the Ptolemy condition in T In this section we complete the proof of Theorem A. Throughout this section w = 0.The AR quiver of T has only one component since | d | = 1. Recall that T is 0-CY,i.e. for a ∈ T we have Σ a = τ − a , and S a = a . In this case, the admissible arcs arepairs ( t, u ) with t > u . In particular, for a ∈ ind ( T ) lying on the mouth of the ARquiver, the corresponding arc a is a loop , i.e. a pair of integers ( x, x ).To classify extension closed subcategories of T we need to introduce a new class ofPtolemy arcs. Definition 8.1.
Let a and b be ( − a = b . We say a and b are adjacent if they are incident with a commonvertex x . In this case, the Ptolemy arcs of class III associated to a and b are the loopat x and the arc connecting the other two vertices. See Figure 8 for an illustration. Figure 8.
Ptolemy arcs of class III.Note that the notion of crossing arcs only makes sense for non-loops. However, weadmit loops in the notion of neighbouring arcs. Note also that Ptolemy arcs in T arealways admissible.The aim of this section is to prove the following theorem. Theorem 8.2.
Let X be a full additive subcategory of T and X be the arcs correspond-ing to the objects of ind ( X ) . Then the objects of ind ( h X i ) correspond to the arcs of theclosure of X under Ptolemy arcs of classes I, II, and III. Putting this together with Propositions 1.1 and 6.2 completes Theorem A.
ORSION PAIRS 27
Corollary 8.3.
Let X be a full additive subcategory of T and X be the correspondingset of arcs. Then ( X , X ⊥ ) is a torsion pair in T w if and only if any left fountain in X is also a right fountain and X is closed under taking admissible Ptolemy arcs of classesI, II and III. Combinatorial description of the Ext-hammocks.
The following proposi-tions are direct consequences of Lemma 7.4(2) and give us a combinatorial descriptionof the arcs b for which there is an extension of b by a . Proposition 8.4.
Let a, b ∈ ind ( T ) , and assume a has length greater than or equal toone. We have Ext T ( b, a ) = 0 if and only if b satisfies one of the following conditions:(1) b crosses a ,(2) b is a neighbour of a incident with t ( a ) − or s ( a ) − ,(3) s ( b ) = t ( a ) and t ( b ) t ( a ) − ,(4) t ( b ) = t ( a ) and s ( b ) > s ( a ) − ,(5) s ( b ) = s ( a ) and t ( a ) − t ( b ) s ( a ) − . Proposition 8.5. If a is a loop, and b ∈ ind ( T ) , then Ext T ( b, a ) = 0 if and only if b satisfies condition (2) or (5) of Proposition 8.4. The middle terms of extensions correspond to admissible Ptolemy arcs.
Recall the definitions of e and e from Definition 7.7. The next proposition showsthat, like the case w ∈ Z \ { , } , these arcs, when admissible, correspond to theindecomposable summands of the middle term of the extension. Proposition 8.6.
Let a, b ∈ ind ( T ) be such that Ext T ( b, a ) = 0 , and let e and e beas in Theorem 3.7.(1) Suppose b ∈ E + ( a ) . The arc e is always admissible. The arc e is admissibleif and only if s ( b ) > t ( a ) .(2) Suppose b ∈ E − ( a ) . The arc e is admissible if and only if s ( b ) > s ( a ) . Thearc e is admissible if and only if t ( b ) t ( a ) .Moreover, e i is nonzero if and only if e i , which is the corresponding arc, is admissible.Proof. Since w = 0, the arcs e and e are admissible if and only if their first componentis greater than or equal to the second component. The proof of the last statement isthe same as in Proposition 7.8. (cid:3) Recall the definitions of E ( a, b ) and E ( a , b ) from Definition 7.10. We end this sub-section by checking that E ( a , b ) is contained in the set of Ptolemy arcs associated to a and b . Remark 8.7.
The only cases in Propositions 8.4 and 8.5 where a and b are neithercrossing, neighbouring nor adjacent arcs are when:(i) a is a loop and b = ( t ( a ) , t ( a ) − a = ( s ( a ) , s ( a ) −
1) and b = ( t ( a ) , t ( a )).The extensions of b by a are a f −→ a −→ b −→ Σ a in (i) and a −→ b g −→ b −→ Σ a in (ii), where f and g are non-isomorphisms. On the other hand, Ext T ( a, b ) = 0.Therefore E ( a , b ) = ∅ in these cases. Corollary 8.8.
Let a, b ∈ ind ( T ) and a , b be the corresponding admissible arcs. Then E ( a , b ) ⊆ { admissible Ptolemy arcs incident with the endpoints of a and b } . Proof.
The proof is similar to that of Corollary 7.13, when a and b are crossing orneighbouring arcs. When a and b are adjacent, any arc connecting endpoints of a and b are Ptolemy arcs of class III, so in particular, e i is a Ptolemy arc. The result thenfollows by Remark 4.5; cf. Remark 7.11. (cid:3) The extension closure.
Let a, b ∈ ind ( T ). Combinatorially, Ptolemy arcs ofclass I, II and III arise out of the following situations, respectively: • a and b are crossing arcs, • a and b are neighbouring arcs, • a and b are adjacent arcs.We shall show that in each of the three cases above E ( a , b ) is precisely the set ofall the admissible Ptolemy arcs of the appropriate class associated to a and b . Weconsider each situation in turn. Proposition 8.9. If a and b cross each other then Ext T ( b, a ) = 0 , Ext T ( a, b ) = 0 and E ( a , b ) is the set of the Ptolemy arcs of class I associated to a and b .Proof. The fact that there are extensions in both directions is an immediate conse-quence of Proposition 8.4(1).We can assume that b crosses a to the right, as the other case is dual. We mustshow that the four Ptolemy arcs associated to a and b lie in E ( a , b ).On one hand, b ∈ E − ( a ) and so the admissible Ptolemy arcs ( s ( b ) , s ( a )) and( t ( b ) , t ( a )) lie in E ( a , b ), by Proposition 8.6(2). On the other hand, a ∈ E + ( b ), and sothe other two admissible Ptolemy arcs, namely ( s ( b ) , t ( a )) and ( s ( a ) , t ( b )), also lie in E ( a , b ), by Proposition 8.6(1). (cid:3) Proposition 8.10. If a and b are neighbouring arcs, then Ext T ( b, a ) = 0 or Ext T ( a, b ) =0 , and E ( a , b ) is the set of Ptolemy arcs of class II associated to a and b .Proof. The proof is similar to that of Proposition 7.14. The only difference is that b can be Σ a or Σ − a , namely when a is a loop. In these cases, the extension hasdimension two, and the middle term of the extension whose middle term is nonzerocorresponds to the Ptolemy arc of associated to a and b . (cid:3) Proposition 8.11. If a and b are adjacent arcs, then Ext T ( b, a ) = 0 or Ext T ( a, b ) =0 , and E ( a , b ) contains the set of Ptolemy arcs of class III associated to a and b .Proof. Let us fix an arc a and check the possibilities for b . Case 1: b and a are adjacent at t ( a ) and t ( b ) < t ( a ). We have Ext T ( b, a ) = 0 and b ∈ E + ( a ). So, by Proposition 8.6(1), the two Ptolemy arcs associated to a and b liein E ( a , b ). Case 2: b and a are adjacent at t ( a ) and s ( b ) > s ( a ). Then Ext T ( b, a ) = 0 and b ∈ E − ( a ). So, by Proposition 8.6(2), the middle term of the extension has twoindecomposable summands, which correspond to the Ptolemy arcs associated to a and b . Case 3: b and a are adjacent at s ( a ) and t ( a ) t ( b ) < s ( a ). In this case we also have Ext T ( b, a ) = 0 and b ∈ E − ( a ). By Proposition 8.6(2), the two Ptolemy arcs of classIII associated to a and b lie in E ( a , b ).The remaining three cases are dual. (cid:3) Putting these together with Corollary 8.8 yields the following corollary, which inturn, together with Theorem 4.1, gives Theorem 8.2.
Corollary 8.12.
Let a, b ∈ ind ( T ) and a , b be the corresponding admissible arcs.Then E ( a , b ) = { admissible Ptolemy arcs incident with the endpoints of a and b } . ORSION PAIRS 29 Torsion pairs and extensions in T There is no combinatorial model of T in terms of admissible arcs of the infinity-gon,and thus no characterisation of torsion pairs in terms of Ptolemy diagrams. However,the classification of torsion pairs in T is quite simple, see Theorem 9.1 below.Recall that the AR quiver of T consists of Z copies of the homogeneous tube below: X + + X + + k k X + + k k X , , k k · · · . k k Let T be the additive (even abelian) category generated by this tube and T n := Σ n T .There is a Z -indexed family of split t-structures in T , ( X n , Y n ) given by X n := add [ i > n T i and Y n := add [ i Theorem 9.1. The only torsion pairs in T are the (de)suspensions of the standardt-structure, i.e. ( X n , Y n ) for n ∈ Z , and the trivial torsion pairs ( T , and (0 , T ) . Before proving Theorem 9.1 we need two preliminary results. The first one describesthe Hom- and Ext-hammocks of T . Proposition 9.2 ([24, Proposition 3.4]) . Consider the indecomposable object X r in T and let b ∈ ind ( T ) be any other indecomposable. Then ext T ( b, X r ) = hom T ( X r , b ) = (cid:26) min { r, s } + 1 if b = X s or b = Σ X s , otherwise. Note that the first equality above follows from the 1-Calabi-Yau property. Lemma 9.3. All torsion pairs in T are split.Proof. Suppose ( X , Y ) is a non-split torsion pair in T and that t ∈ ind ( T ) doesnot belong to either X or Y . Therefore, there is a non-trivial approximation triangle x → t → y → Σ x with x ∈ X and y ∈ Y . The object t lies in some homogeneous tube, T k say, and thus t = Σ k X u for some u > 0. By Proposition 9.2 there are maps to t onlyfrom the tubes T k − and T k . Suppose x contains a summand Σ k − X r ∈ T k − . By theproperties of homogeneous tubes (see for example [33, Chapter X] and combine it withthe properties of derived categories of hereditary categories [17]), the maps Σ k − X r → t factor through Σ k − X s for all s > r . Hence, no finite sum of indecomposable objectsfrom T k − can be a summand of x . The only possibility remaining is that x containsa summand from T k . But since X is extension-closed, we get T k ⊆ X , whence t ∈ X ;a contradiction. Thus, any torsion pair ( X , Y ) is split. (cid:3) Proof of Theorem 9.1. Let ( X , Y ) be a torsion pair in T and suppose T k ⊆ X . ByProposition 9.2, Hom T ( T k , T k +1 ) = 0, whence T k +1 ⊆ X since ( X , Y ) is split byLemma 9.3. Thus, if T k ⊆ X then T j ⊆ X for each j > k . There is either a minimalsuch k , in which case ( X , Y ) = ( X k , Y k ), or there is not, in which case ( X , Y ) = ( T , Y gives the other trivial torsion pair (0 , T ). (cid:3) Remark 9.4. In [24] it was shown that T has only one family of non-trivial (bounded)t-structures, namely the ( X n , Y n ). Here we have shown that this family of boundedt-structures are the only non-trivial torsion pairs in T . Extensions with indecomposable outer terms in T . Whilst this is notneeded in the classification of torsion pairs in T , for the sake of completeness, weinclude a brief description.Without loss of generality we may consider extensions starting at X r for some r > X r have the following form for s > X r → E → X s → Σ X r and X r → F → Σ X s → Σ X r . The first extension X r → E → X s → Σ X r has all three objects lying in the heart T . Thus, it is enough to compute the extensions in the abelian category T , which isa special case of [3, Lemma 5.1].The middle term of the second extension is isomorphic to cone ( f ) = (Σ ker f ) ⊕ coker f , where f : X s → X r , where we use the fact that T is hereditary. We can nowcompute the cones of these morphisms in T using, for example, [33, Chapter X].We summarise these considerations below. Proposition 9.5. Consider a non-trivial extension X r → E → B → Σ X r in T ,where B is either X s or Σ X s for some s > . We interpret X − as the zero object.(i) If B = X s , write n = min { r, s } and m = max { r, s } . Then the n + 1 extensionsare X r → X m + i ⊕ X n − i → X s → Σ X r for i n + 1 .(ii) If B = Σ X s for s > r then the r + 1 extensions are X r → Σ X s − r − i ⊕ X i − → Σ X s → Σ X r for i r + 1 .(iii) If B = Σ X s for s < r then the s + 1 extensions are X r → X r − s − i ⊕ Σ X i − → Σ X s → Σ X r for i s + 1 . Torsion pairs in C w ( A n )Throughout this section w − m = − w + 1 and we shall consider the orbitcategory C m := C m ( A n ) = D b ( k A n ) / Σ m τ, where τ denotes the AR translate of D b ( k A n ). For more detailed background onthese categories we refer the reader to the papers [10, 12, 13]. These categories aretriangulated by Keller’s Theorem [27], and satisfy Σ w ≃ S , where S = Σ τ is the Serrefunctor. In particular, they can be considered to be w -CY.10.1. The combinatorial model for C m . Given two indecomposable objects a and b in T w , we say that a is an innerarc of b if one has t ( b ) < t ( a ) < s ( a ) < s ( b ).It was shown in [13, Theorem 5.1, Corollary 6.3] that the combinatorial model in T w induces a combinatorial model in C m as follows: C m is equivalent to the full subcategory C w of T w whose set of indecomposable objects correspond to the admissible innerarcsof an admissible arc a of length | ( n + 1)( − w − 1) + 1 | . Note that this equivalence isnot a triangle equivalence.We briefly recall the explicit description of the induced combinatorial model. Let P n,m be the regular N -gon, where N = m ( n + 1) − 2, with vertices numbered clock-wise from 1 to N . All operations on vertices of P n,m will be done modulo N , withrepresentatives 1 , . . . , N . An m -diagonal of P n,m is a diagonal that divides P n,m intotwo polygons each of whose number of vertices is divisible by m .The AR quiver of C m is equivalent to the stable translation quiver Γ( n, m ) whosevertices are the m -diagonals P n,m . We denote a vertex of Γ( n, m ) by { i, j } , where i and j are vertices of P n,m . The arrows of Γ( n, m ) are obtained in the following way:given two m -diagonals D and D ′ with a vertex i in common, there is an arrow from D to D ′ in Γ( n, m ) if and only if D ′ can be obtained from D by rotating clockwise ORSION PAIRS 31 m steps around i . The translation automorphism τ : Γ( n, m ) → Γ( n, m ) sends an m -diagonal { i, j } to τ ( { i, j } ) := { i − m, j − m } . Note that Σ { i, j } = { i + 1 , j + 1 } .Figure 9 shows an example of this stable translation quiver. { , } { , } { , }{ , } { , } { , }{ , } { , } { , }{ , } { , } { , }{ , } { , } { , }{ , } { , } { , } Figure 9. The AR quiver of C ( A ) is equivalent to Γ(3 , m -diagonals and Ptolemy diago-nals of class II in the same manner as in T w . The main result of this section is: Theorem 10.1. Let X be a full additive subcategory of C m and X the set of m -diagonalscorresponding to the objects of ind ( X ) . Then the objects of ind ( h X i ) correspond to the m -diagonals of the closure of X under Ptolemy m -diagonals of classes I and II. Since C m has finitely many indecomposable objects up to isomorphism, any subcat-egory of C m is contravariantly finite. Thus, Theorem B is an immediate corollary ofTheorem 10.1 and Proposition 1.1. Corollary 10.2. Let X be a full additive subcategory of C m and X the correspondingset of m -diagonals. Then ( X , X ⊥ ) is a torsion pair in C m if and only if X is closedunder Ptolemy m -diagonals of classes I and II. Remark 10.3. The characterisation of torsion pairs in C m does not follow immediatelyfrom that in T w : C w is not a triangulated subcategory of T w ; see [13, Theorem 5.1]. Infact, if there is an extension in T w between two objects of C w then there is an extensionin C m between their images, but the converse is not true. As an example, consider m = 2 , n = 3 and a = (7 , 0) to be the arc that defines the equivalence. The admissiblearcs (2 , 1) and (6 , 5) are objects in C w and the corresponding images { , } and { , } are 2-diagonals of a hexagon. It is easy to check that Ext T − ((1 , , (5 , Ext C ( { , } , { , } ) = 0.10.2. Combinatorial description of the Ext-hammocks. We require the follow-ing notation to describe the Hom- and Ext-hammocks in C m . Notation. Given the vertices i , i , . . . , i k of P n,m , we write C ( i , i , . . . , i k ) to meanthat i , i , . . . , i k , i follow each other under the clockwise circular order on the bound-ary of P n,m . Lemma 10.4. Let a, b ∈ ind ( C m ) and a = { a , a } , with a < a , b = { b , b } be thecorresponding m -diagonals of P n,m . We have Hom C m ( a, b ) = 0 if and only if b satisfiesthe following condition: b = a + im, b = a + jm, for some i, j > , and C ( a , b , a , b ) . Proof. Explicit computation using the combinatorial model of C m . (cid:3) Using the Auslander–Reiten formula, we obtain the Ext-hammocks: Corollary 10.5. We have Ext C m ( b, a ) = 0 if and only if the following condition holds: b = a + im, b = a + jm, for some i, j > , and C ( a + m, b , a + m, b ) . Corollary 10.5 can be unpacked into the following more readable statement. Corollary 10.6. Let a, b ∈ ind ( C m ) and a = { a , a } , with a < a , be the m -diagonalcorresponding to a . Then Ext C m ( b, a ) = 0 if and only if b satisfies one of the following:(1) b is a neighbour of a incident with a + 1 ,(2) b is a neighbour of a incident with a + 1 ,(3) b crosses a in such a way that b is obtained by adding multiples of m to theendpoints of a ,(4) b = { a + 1 , a + 1 } , i.e. b = Σ a . Figure 10 shows where the indecomposable objects corresponding to the arcs thatsatisfy the conditions in Corollary 10.6 lie in the AR quiver. a τ − a (2)(3) (4)(1) Figure 10. Ext C m ( − , a ) = 0.10.3. Graphical calculus. To prove Theorem 10.1 we need a graphical calculus anal-ogous to Theorem 3.7. Before doing this, we briefly recall some useful notation andfacts about C m .Write inj ( k A n ) for the subcategory of mod ( k A n ) containing the injective k A n -modules. We have the following fundamental domain for C m in D b ( k A n ): F = ind ( m [ i =1 Σ i − mod ( k A n ) ∪ Σ m ( mod ( k A n ) \ inj ( k A n ))) , From now on, we identify objects in ind ( C m ) with their representatives in F . Given X ∈ ind ( D b ( k A n )), we denote by d ( X ) the degree of X , i.e. the integer such that X = Σ d ( X ) X for X ∈ ind ( mod ( k A n )). Lemma 10.7. Let A, B ∈ F , P be a projective k A n -module and C ∈ ind ( D b ( k A n )) .(1) Hom D b ( k A n ) ( A, τ k Σ km B ) = 0 , for every k = 0 , .(2) Hom D b ( k A n ) ( A, τ k Σ km B ) = 0 for at most one value of k .(3) If Hom D b ( k A n ) ( P, C ) = 0 , then C ∈ ind ( mod ( k A n )) .Proof. Statements (1) and (2) are proved for m = 1 in [10, Proposition 2.1]; the prooffor m > (cid:3) For a ∈ ind ( C m ) the starting and ending frames of a are: F s ( a ) := { b ∈ ind ( C m ) | Hom C m ( a, b ) = 0 , Ext C m ( b, a ) = 0 } ; F e ( a ) := { b ∈ ind ( C m ) | Hom C m ( b, a ) = 0 , Ext C m ( a, b ) = 0 } , cf. Lemma 3.2(ii), ray + ( a ) ∪ coray + ( a ) and ray − ( a ) ∪ coray − ( a ) in Theorem 3.7.We are now ready to state the graphical calculus result. For w = − ORSION PAIRS 33 Proposition 10.8. Let a, b ∈ ind ( C m ) be such that Ext C m ( b, a ) = 0 . Then ext C m ( b, a ) =1 and the unique (up to equivalence) non-split extension a → e → b → Σ a has middleterm e whose summands are given by F s ( a ) ∩ F e ( b ) . If this intersection is empty, thenwe interpret e to be the zero object.Proof. We can assume, without loss of generality, that the quiver of type A n has thelinear orientation n −→ n − −→ · · · −→ 1. Recall that we see a and b as sittinginside the fundamental domain F . Suppose Ext C m ( b, a ) = 0. By taking a suitable ARtranslate of a and b , we may assume that a is a projective k A n -module. Case 1: Assume b is a non-projective k A n -module. Then τ b ∈ mod ( k A n ) ⊆ F and so Ext C m ( b, a ) ≃ D Hom C m ( a, τ b ) = D Hom D b ( k A n ) ( a, τ b ) ⊕ D Hom D b ( k A n ) ( a, τ Σ m b ) . Since τ b is a module, d ( τ Σ m b ) = m − m , so d ( τ Σ m b ) > 1. Hence, thesecond summand is zero, since a is a projective module. Therefore Ext C m ( b, a ) ≃ Hom k A n ( a, τ b ). Hence, ext C m ( b, a ) = 1, since Hom spaces in type A are either zero orone dimensional. The result then follows from [8, Corollary 8.5]. Case 2: Assume b is a projective indecomposable module. Then τ b = Σ − I for someindecomposable injective I . In C m we have τ b ≃ Σ m − τ I , which lies in F . Hence, Ext C m ( b, a ) ≃ D Hom C m ( a, τ b ) = D Hom D b ( k A n ) ( a, Σ m τ I ) ⊕ D Hom D b ( k A n ) ( a, τ Σ m − I ) . The second summand is always zero since d ( τ Σ m − I ) > a is projective. Wehave d ( τ Σ m − I ) = m − m − 1. For any m > 2, we have m − > 1. Thus, if d ( τ Σ m − I ) = m − 1, the first summand is also zero giving a contradiction. Analogouslyif d ( τ Σ m − I ) = m − m > 2. To avoid a contradiction, we must have m = 2 and d ( τ Σ I ) = 0 and I ∼ = P ( n ) ∼ = I (1) the indecomposable projective-injective, whence a ∼ = P (1). Now,0 = Ext C ( b, a ) ≃ D Hom D b ( A n ) ( a, Σ τ P ( n )) = D Hom k A n ( a, I ( n )) , where one-dimensionality follows from being in type A . Since I ( n ) = S ( n ), i.e. thesimple at vertex n , and a is projective, we get a = P ( n ). In C we have Σ P ( n ) ∼ = P (1),whence the non-split triangle is P ( n ) → → P (1) → Σ P ( n ). However, F s ( a ) ∩ F e ( b ) = ∅ , giving the claim in this case . Case 3: b = Σ i B , for some indecomposable non-injective module B and 0 < i < m .If i = 1, assume also that B is non-projective. Then τ b lies in F and d ( τ b ) > Ext C m ( b, a ) ≃ D Hom C m ( a, τ b ) = D Hom D b ( k A n ) ( a, τ b ) ⊕ D Hom D b ( k A n ) ( a, Σ m τ b ) . Since d (Σ m τ b ) > d ( τ b ) > 1, both summands are zero; a contradiction, so thiscase provides no extensions. Case 4: b = Σ B , where B is projective. Then τ b is an injective module, and thus liesin F . Hence Ext C m ( b, a ) ≃ D Hom C m ( a, τ b ) = Hom D b ( k A n ) ( a, τ b ) ⊕ D Hom D b ( k A n ) ( a, Σ m τ b ) . The second summand is zero because d (Σ m τ b ) > 1. Therefore ext C m ( b, a ) = 1, since Ext C m ( b, a ) = 0 by assumption. We construct a non-split triangle with first term a and last term b . Let a = P ( i ) and b = Σ P ( j ).For i < j we have Hom D b ( k A n ) ( a, τ b ) = Hom D b ( k A n ) ( P ( i ) , I ( j )) = 0, giving a con-tradiction. Thus i > j . For i > j , there is a short exact sequence 0 → P ( j ) → P ( i ) → E → 0, which induces a triangle P ( j ) → P ( i ) → E → Σ P ( j ). Shiftingthis triangle gives the desired triangle. It is easy to check that the cokernel E is a be ab ea be a bea be e ab e e a b ab Figure 11. The middle term of the extension of b by a . The arrowsin the third case mean that the distance between the correspondingendpoints is a multiple of m .the unique object in F s ( P ( i )) ∩ F e (Σ P ( j )). For i = j , we get the standard triangle P ( i ) → → Σ P ( i ) → Σ P ( i ) in which the last map is an isomorphism. Moreover, F s ( P ( i )) ∩ F e (Σ P ( i )) = ∅ , corresponding to the zero middle term. (cid:3) Using Proposition 10.8, one can check that the m -diagonals corresponding to themiddle term are as in Figure 11.The arguments of Section 4 can be applied to C m to give the following proposition,which makes it sufficient to consider only extensions between indecomposable objects. Proposition 10.9. Let { a i } ni =1 and { b j } mj =1 be sets of (not necessarily pairwise non-isomorphic) indecomposable objects of C m . Any extension of the form n M i =1 a i → e → m M j =1 b j → Σ n M i =1 a i can be computed iteratively from extensions whose outer terms are indecomposable andbuilt from { a i } ni =1 and { b j } mj =1 . Extension closure. Let a and b be m -diagonals and recall the notion of E ( a , b )from Section 7. ORSION PAIRS 35 Remark 10.10. Let m = 2, a = { i, i + 1 } and b = Σ a . Note that a and b areincident with the vertex i + 1, and this is the only case when extensions occur betweennoncrossing and non-neighbouring m -diagonals. In this case, the middle term of theextension of b by a is zero, and Ext C ( a, b ) ≃ D Hom C ( a, τ − a ) = 0. Therefore, E ( a , b ) = ∅ .Figure 11 (together with Proposition 10.9) shows us that E ( a , b ) is contained in theset of Ptolemy m -diagonals associated to a and b . Our aim is to prove that thesetwo sets are in fact equal, giving Theorem 10.1. This follows from the following threepropositions and Proposition 10.9. Proposition 10.11. Let a, b ∈ ind ( C m ) be such that a and b are neighbouring m -diagonals. Then E ( a , b ) contains all the Ptolemy m -diagonals of class II associated to a and b .Proof. Similar to the proof of Proposition 7.14. (cid:3) Proposition 10.12. Let a, b ∈ ind ( C m ) be such that a and b are crossing m -diagonalsand Ext C m ( b, a ) = 0 . Then E ( a , b ) contains all the Ptolemy m -diagonals of class Iassociated to a and b .Proof. It follows immediately from the fact that the Ptolemy diagonals of class I as-sociated to a and b other than e and e (see Figure 11) are not m -diagonals. (cid:3) Proposition 10.13. Let a, b ∈ ind ( C m ) be such that a and b are crossing m -diagonalsand Ext C m ( a, b ) = 0 = Ext C m ( b, a ) . Then none of the corresponding Ptolemy diagonalsis an m -diagonal.Proof. By Remark 10.3, there is no extension between the corresponding d = ( w − T w . Hence, by Lemma 7.16 the corresponding Ptolemy diagonals ofclass I are not m -diagonals. (cid:3) References [1] I. Assem, D. Simson, A. Skowro´nski, Elements of the Representation Theory of AssociativeAlgebras. 1: Techniques of Representation Theory , London Math. Soc. Stud. Texts, vol ,Cambridge University Press (2006).[2] M. 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Centro de An´alise Funcional, Estruturas Lineares e Aplicac¸˜oes, Faculdade deCiˆencias da Universidade de Lisboa, Campo Grande, Edif´ıcio C6, Piso 2, 1749-016,Lisboa, Portugal E-mail address : [email protected] School of Mathematics, The University of Manchester, Oxford Road, Manchester,M13 9PL, United Kingdom. E-mail address ::