Discrete derived categories I: homomorphisms, autoequivalences and t-structures
DDISCRETE DERIVED CATEGORIES I
HOMOMORPHISMS, AUTOEQUIVALENCES AND T-STRUCTURES
NATHAN BROOMHEAD, DAVID PAUKSZTELLO, AND DAVID PLOOG
Abstract.
Discrete derived categories were studied initially by Vossieck [42] and laterby Bobi´nski, Geiß, Skowro´nski [9]. In this article, we describe the homomorphism ham-mocks and autoequivalences on these categories. We classify silting objects and boundedt-structures.
Contents
1. Discrete derived categories and their AR-quiver 32. Hom spaces: hammocks 53. Twist functors from exceptional cycles 114. Autoequivalence groups of discrete derived categories 155. Hom spaces: dimension bounds and graded structure 196. Reduction to Dynkin type A and classification results 247. A detailed example: Λ(2 , ,
1) 33Appendix A. Notation, terminology and basic notions 36Appendix B. The repetitive algebra and string modules 40
Introduction
In this article, we study the bounded derived categories of finite-dimensional algebras thatare discrete in the sense of Vossieck [42]. Informally speaking, discrete derived categoriescan be thought of as having structure intermediate in complexity between the derivedcategories of hereditary algebras of finite representation type and those of tame type.Note, however, that the algebras with discrete derived categories are not hereditary. Wedefer the precise definition until the beginning of the next section.Understanding homological properties of algebras means understanding the structureof their derived categories. We investigate several key aspects of the structure of discretederived categories: the structure of homomorphism spaces, the autoequivalence groups ofthe categories, and the t-structures and co-t-structures inside discrete derived categories.The study of the structure of algebras with discrete derived categories was begunby Vossieck, who showed that they are always gentle and classified them up to Moritaequivalence. Bobi´nski, Geiß and Skowro´nski [9] obtained a canonical form for the derivedequivalence class of these algebras; see Figure 1. This canonical form is parametrised byintegers n ≥ r ≥ m >
0, and the corresponding algebra denoted by Λ( r, n, m ). Werestrict to parameters n > r , which is precisely the case of finite global dimension. In[9], the authors also determined the components of the Auslander–Reiten (AR) quiver ofderived-discrete algebras and computed the suspension functor.
Mathematics Subject Classification.
Key words and phrases.
Discrete derived category, Auslander–Reiten quiver, Hom-hammock, twistfunctor, silting object, t-structure, string algebra. a r X i v : . [ m a t h . R T ] J un he structure exhibited in [9] is remarkably simple, which brings us to our princi-pal motivation for studying these categories. Discrete derived categories are sufficientlystraightforward to make explicit computation highly accessible but also non-trivial enoughto manifest interesting behaviour. For example, discrete derived categories contain natu-ral examples of spherelike objects in the sense of [22]. If one takes one of these spherelikeobjects and forms the smallest subcategory generated by it, this category is then equiv-alent to a triangulated category generated by a spherical object. Such categories havepreviously been studied in the context of (higher) cluster categories of type A ∞ in [24, 25].Indeed, we shall see that every discrete derived category contains two such higher clustercategories as proper subcategories when the algebra has finite global dimension.Furthermore, the structure of discrete derived categories is highly reminiscent of thecategories of perfect complexes of cluster-tilted algebras of type ˜ A n studied in [4]. Thissuggests approaches developed here to understand discrete derived categories are likelyto find applications more widely in the study of derived categories of gentle algebras.The basis of our work is giving a combinatorial description via AR quivers of whichindecomposable objects admit non-trivial homomorphism spaces between them, so called‘Hom-hammocks’. As a byproduct, we get the following interesting property of thesecategories: the dimensions of the homomorphism spaces between indecomposable objectshave a common bound. In fact, in Theorem 5.1 we show there are unique homomorphisms,up to scalars, whenever r >
1, and in the exceptional case r = 1, the common dimensionbound is 2. We believe this property holds independent interest and warrants furtherinvestigation. See [20] for a different approach to measuring the ‘smallness’ of discretederived categories. As another for categorical size, the Krull–Gabriel dimension of discretederived categories has been computed in [10]; it is at most 2.In Theorem 4.7 we explicitly describe the group of autoequivalences. For this, weintroduce a generalisation of spherical twist functors arising from cycles of exceptionalobjects. The action of these twists on the AR components of Λ( r, n, m ) is a useful tool,which is frequently employed here.In Section 6, we address the classification of bounded t-structures and co-t-structures in D b (Λ( r, n, m )), which are important in understanding the cohomology theories occurringin triangulated categories, and have recently become a focus of intense research as theprincipal ingredients in the study of Bridgeland stability conditions [13], and their co-t-structure analogues [29]. Further investigation into the properties of (co-)t-structuresand the stability manifolds is conducted in the sequel [14]; see also [44].We study the (co-)t-structures indirectly via certain generating sets: silting subcat-egories, which behave like the projective objects of hearts of bounded t-structures andgeneralise tilting objects. In general, one cannot get all bounded t-structures in thisway, but in Proposition 6.1, we show that the heart of each bounded t-structure in D b (Λ( r, n, m )) is equivalent to mod (Γ), where Γ is a finite-dimensional algebra of finiterepresentation type. The upshot is that using the bijections of K¨onig and Yang [32],classifying silting objects is enough to classify all bounded (co-)t-structures. We showthat D b (Λ( r, n, m )) admits a semi-orthogonal decomposition into D b ( k A n + m − ) and thethick subcategory generated by an exceptional object. Using Aihara and Iyama’s siltingreduction [1], we classify the silting objects in Theorem 6.22. We finish with an explicitexample of Λ(2 , ,
1) in Section 7.
Acknowledgments:
We are grateful to Aslak Bakke Buan, Christof Geiß, Martin Kalck,Henning Krause, and Dong Yang for answering our questions and particularly to ananonymous referee for a careful reading and many valuable comments. Much of thispaper was prepared while all three authors worked at Leibniz Universit¨at Hannover. The igure 1. The quiver Q ( r, n, m ) consisting of an oriented cycle of length n with a tail of length m and r consecutive zero relations inside the cycle.second author acknowledges the financial support of the EPSRC of the United Kingdomthrough the grant EP/K022490/1.1. Discrete derived categories and their AR-quiver
We always work over a fixed algebraically closed field k . All modules will be finite-dimensional right modules. Throughout, all subcategories will be additive and closedunder isomorphisms.1.1. Discrete derived categories.
We are interested in k -linear, Hom-finite triangu-lated categories which are small in a certain sense. One precise definition of such smallnessis given by Vossieck [42]; here we present a slight generalisation of his notion. Definition 1.1.
A derived category (or, more generally and intrinsically, a Hom-finitetriangulated category with a bounded t-structure) D is discrete (with respect to this t -structure) , if for every map v : Z → K ( D ) there are only finitely many isomorphismclasses of objects D ∈ D with [ H i ( D )] = v ( i ) ∈ K ( D ) for all i ∈ Z .Let us elaborate on the connection to [42]: Vossieck speaks of finitely supported, positive dimension vectors v ∈ K ( D ) ( Z ) which he can do since he has D = D b (Λ) for a finite-dimensional algebra Λ, so K (Λ) ∼ = Z r . In our slight generalisation of his notion, wecannot do so, but for finite-dimensional algebras the new notion gives back the old one:if v is negative somewhere, there will be no objects of that dimension vector whatsoever.For the same reason, we don’t have to assume that v has finite support: if it doesn’t, theset of objects of that class is empty.Note that our definition of discreteness appears to depend on the choice of bounded t-structure. Throughout this article, we shall be interested in the bounded derived category D b (Λ) of a finite-dimensional algebra Λ. We shall always use discreteness with respect tothe standard t-structure, whose heart is mod (Λ), the category of finite-dimensional rightΛ-modules. However, in [15], the results of this article will be used to show that thecategories studied here are discrete with respect to any bounded t-structure.Obviously, derived categories of path algebras of type ADE Dynkin quivers are exam-ples of discrete categories. Moreover, [42] shows that the bounded derived category of afinite-dimensional algebra Λ, which is not of finite representation type, is discrete if andonly if Λ is Morita equivalent to the bound quiver algebra of a gentle quiver with exactlyone cycle having different numbers of clockwise and anticlockwise orientations.Furthermore, in [9], Bobi´nski, Geiß and Skowro´nski give a derived Morita classificationof such algebras. More precisely, for Λ connected and not of Dynkin type, the derivedcategory D b (Λ) is discrete if and only if Λ is derived equivalent to the path algebraΛ( r, n, m ) for the quiver with relations given in Figure 1, and some values of r, n, m . .2. The AR quiver of D b (Λ( r, n, m )). The algebra Λ( r, n, m ) has finite global di-mension if and only if n > r . In the following, we always make this assumption.
Thereforethe derived category D b (Λ( r, n, m )) enjoys duality in the formHom( A, B ) = Hom( B, S A ) ∗ functorially in A, B ∈ D b (Λ( r, n, m )), where the Serre functor S is given by the Nakayamafunctor. In other words, D b (Λ( r, n, m )) has Auslander–Reiten triangles and translation τ := Σ − S . We will use both notations, depending on the context. Some general prop-erties of D b (Λ( r, n, m )) are: this triangulated category is algebraic, Hom-finite, Krull–Schmidt and indecomposable; see Appendix A.1 for details.We collect together some more special properties of D b (Λ( r, n, m )) which will be cru-cial throughout the paper; the reference is [9]. By [9, Theorem B], the AR quiver of D b (Λ( r, n, m )) has precisely 3 r components; these are denoted by X , . . . , X r − , Y , . . . , Y r − , Z , . . . , Z r − . The X and Y components are of type Z A ∞ , whereas the Z components are of type Z A ∞∞ .It will be convenient to have notation for the subcategories generated by indecomposableobjects of the same type: X := add (cid:91) i X i , Y := add (cid:91) i Y i , Z := add (cid:91) i Z i . For each k = 0 , . . . , r −
1, we label the indecomposable objects in X k , Y k , Z k as follows: X kij ∈ X k with i, j ∈ Z , j ≥ i ; Y kij ∈ Y k with i, j ∈ Z , i ≥ j ; Z kij ∈ Z k with i, j ∈ Z . Properties 1.2.
This labelling is chosen in such a way that the following properties hold:(1) Irreducible morphisms go from an object with coordinate ( i, j ) to objects ( i + 1 , j )and ( i, j + 1) in the same component (when they exist). X coordinates: Y coordinates: Z coordinates: ... (cid:31) (cid:31) ... ( − , (cid:31) (cid:31) ... (0 , (cid:31) (cid:31) ... . . . ( − , (cid:31) (cid:31) (cid:63) (cid:63) (0 , (cid:31) (cid:31) (cid:63) (cid:63) (1 , (cid:31) (cid:31) (cid:63) (cid:63) · · · (cid:63) (cid:63) (0 , (cid:63) (cid:63) (1 , (cid:63) (cid:63) · · · · · · (cid:31) (cid:31) (0 , (cid:31) (cid:31) (1 , (cid:31) (cid:31) · · · (0 , − (cid:31) (cid:31) (cid:63) (cid:63) (1 , (cid:31) (cid:31) (cid:63) (cid:63) (2 , (cid:31) (cid:31) (cid:63) (cid:63) . . . (cid:63) (cid:63) ... (1 , − (cid:63) (cid:63) ... (2 , (cid:63) (cid:63) ... ... ... (cid:31) (cid:31) ... ( − , (cid:31) (cid:31) ... (0 , (cid:31) (cid:31) ... . . . ( − , (cid:31) (cid:31) (cid:63) (cid:63) (0 , (cid:31) (cid:31) (cid:63) (cid:63) (1 , (cid:31) (cid:31) (cid:63) (cid:63) · · · (cid:31) (cid:31) (cid:63) (cid:63) (0 , (cid:31) (cid:31) (cid:63) (cid:63) (1 , (cid:31) (cid:31) (cid:63) (cid:63) · · · (0 , − (cid:31) (cid:31) (cid:63) (cid:63) (1 , (cid:31) (cid:31) (cid:63) (cid:63) (2 , (cid:31) (cid:31) (cid:63) (cid:63) . . . (cid:63) (cid:63) ... (1 , − (cid:63) (cid:63) ... (2 , (cid:63) (cid:63) ... ... (2) The AR translate of an object with coordinate ( i, j ) is the object with coordinate( i − , j −
1) in the same component, i.e. τ X ki,j = X ki − ,j − etc.(3) The suspension of indecomposable objects is given below, with k = 0 , . . . , r − X kij = X k +1 ij , Σ X r − ij = X i + r + m,j + r + m , Σ Y kij = Y k +1 ij , Σ Y r − ij = Y i + r − n,j + r − n , Σ Z kij = Z k +1 ij , Σ Z r − ij = Z i + r + m,j + r − n In particular, Σ r | X = τ − m − r and Σ r | Y = τ n − r on objects.
4) There are distinguished triangles, for any i, j, d ∈ Z with d ≥ X ki,i + d (cid:47) (cid:47) Z kij (cid:47) (cid:47) Z ki + d +1 ,j (cid:47) (cid:47) Σ X ki,i + d ,Y kj + d,j (cid:47) (cid:47) Z kij (cid:47) (cid:47) Z ki,j + d +1 (cid:47) (cid:47) Σ Y kj + d,j . (5) There are chains of non-zero morphisms for any i ∈ Z and k = 0 , . . . , r − X kii (cid:47) (cid:47) X ki,i +1 (cid:47) (cid:47) ··· (cid:47) (cid:47) Z ki,i − (cid:47) (cid:47) Z kii (cid:47) (cid:47) Z ki,i +1 (cid:47) (cid:47) ··· (cid:47) (cid:47) Σ X ki +1 ,i − (cid:47) (cid:47) Σ X ki,i − (cid:47) (cid:47) Σ X ki − ,i − ,Y kii (cid:47) (cid:47) Y ki +1 ,i (cid:47) (cid:47) ··· (cid:47) (cid:47) Z ki − ,i (cid:47) (cid:47) Z kii (cid:47) (cid:47) Z ki +1 ,i (cid:47) (cid:47) ··· (cid:47) (cid:47) Σ Y ki − ,i − (cid:47) (cid:47) Σ Y ki − ,i − (cid:47) (cid:47) Σ Y ki − ,i − . Later, we will often use the ‘height’ of indecomposable objects in X or Y components.For X kij ∈ ind ( X k ), we set h ( X kij ) = j − i and call it the height of X kij in the component X k . Similarly, for Y kij ∈ ind ( Y k ), we set h ( Y kij ) = i − j and call it the height of Y kij in thecomponent Y k . The mouth of an X or Y component consists of all objects of height 0.2. Hom spaces: hammocks
For brevity, we will write Λ := Λ( r, n, m ). In this section, for a fixed indecomposableobject A ∈ D b (Λ) we compute the so-called ‘Hom-hammock’ of A , i.e. the set of indecom-posables B ∈ D b (Λ) with Hom( A, B ) (cid:54) = 0. By duality, this also gives the contravariantHom-hammocks: Hom( − , A ) = Hom( S − A, − ) ∗ . Therefore we generally refrain fromlisting the Hom( − , A ) hammocks explicitly.The precise description of the hammocks is slightly technical. However, the result isquite simple, and the following schematic indicates the hammocks Hom( X, − ) (cid:54) = 0 andHom( Z, − ) (cid:54) = 0 for indecomposables X ∈ X and Z ∈ Z : X S X Z S Z Y X Y X Y X Z Z Z Hammocks from the mouth.
We start with a description of the Hom-hammocksof objects at the mouths of all Z A ∞ components. The proof relies on Happel’s triangleequivalence of D b (Λ( r, n, m )) with the stable module category of the repetitive algebra ofΛ( r, n, m ). As the repetitive algebras are special biserial algebras, the well-known theoryof string (and band) modules provides a useful tool to understand the indecomposableobjects and homomorphisms between them; we summarise this theory in Appendix B.To make our statements of Hom-hammocks more readable, we employ the languageof rays and corays. Let V = V i,j be an indecomposable object of D b (Λ( r, n, m )) withcoordinates ( i, j ). Recall the conventions that j ≥ i if V ∈ X whereas i ≥ j if V ∈ Y .Denoting the AR component of V by C and its objects by V a,b , the following six definitionsgive the rays/corays from/to/through V , respectively ray + ( V i,j ) := { V i,j + l ∈ C | l ∈ N } , coray + ( V i,j ) := { V i + l,j ∈ C | l ∈ N } , ray − ( V i,j ) := { V i,j − l ∈ C | l ∈ N } , coray − ( V i,j ) := { V i − l,j ∈ C | l ∈ N } , ray ± ( V i,j ) := { V i,j + l ∈ C | l ∈ Z } , coray ± ( V i,j ) := { V i + l,j ∈ C | l ∈ Z } . Note that, because of the orientation of the components, the (positive) ray of an inde-composable X kii ∈ X k at the mouth consists of indecomposables in X k reached by arrows oing out of X kii , while in the Y components the (negative) ray of Y kii contains objectswhich have arrows going in to it.For the next statement, whose proof is deferred to Lemma B.7, recall that the Serrefunctor is given by suspension and AR translation: S = Σ τ . Also, rays and corayscommute with these three functors. Lemma 2.1.
Let A ∈ ind ( D b (Λ( r, n, m ))) with r > and let i, k ∈ Z , ≤ k < r . Then Hom( X kii , A ) = k if A ∈ ray + ( X kii ) ∪ coray − ( S X kii ) ∪ ray ± ( Z kii ) , Hom( Y kii , A ) = k if A ∈ coray + ( Y kii ) ∪ ray − ( S Y kii ) ∪ coray ± ( Z kii ) , Hom(
A, X kii ) = k if A ∈ coray − ( X kii ) ∪ ray + ( S − X kii ) ∪ ray ± ( S − Z kii ) , Hom(
A, Y kii ) = k if A ∈ ray − ( Y kii ) ∪ coray + ( S − Y kii ) ∪ coray ± ( S − Z kii ) and in all other cases the Hom spaces are zero. For r = 1 the Hom-spaces are as above,except Hom( X ii , X i,i + m ) = k . Hom-hammocks for objects in X components. Assume A = X kij ∈ ind ( X k ).In order to describe the various Hom-hammocks conveniently, we set A := X kjj to be the intersection of the coray through A with the mouth of X k , and A := X kii to be the intersection of the ray through A with the mouth of X k .By definition, A and A have height 0. If A sits at the mouth, then A = A = A .We now write down some standard triangles involving the objects A , A and A . Thefollowing lemma is completely general and holds in any Z A ∞ component of the AR quiverof a Krull–Schmidt triangulated category — we use the notation introduced above forthe X components of D b (Λ( r, n, m )). Lemma 2.2.
Let A be an indecomposable object of height h ( A ) ≥ in a Z A ∞ componentof a Krull–Schmidt triangulated category. Let A (cid:48) u −→ A ⊕ C v −→ A (cid:48)(cid:48) → Σ A (cid:48) be the ARtriangle with A at its apex; assuming C = 0 if h ( A ) = 1 . Then there are triangles A −→ A v (cid:48)(cid:48) −→ A (cid:48)(cid:48) −→ Σ A and A (cid:48) u (cid:48) −→ A −→ A −→ Σ A (cid:48) where u (cid:48) and v (cid:48)(cid:48) are induced by u and v , respectively.Proof. By Lemma 2.1 the composition, A → A , of irreducible maps along a ray is non-zero. Likewise the composition, A → A , of irreducible maps along a coray is non-zero.We proceed by induction on h ( A ). If h ( A ) = 1, then both triangles coincide with theAR triangle A → A → A → Σ A ; in particular, A (cid:48) = A and A (cid:48)(cid:48) = A .Assume h ( A ) >
1. We shall show the existence of one triangle, the other one is dual.Consider the AR triangle together with the split triangle A → A ⊕ C −→ C → Σ A . Thesetriangles fit into the following commutative diagram arising from the octahedral axiom. A (cid:0) (cid:1) (cid:15) (cid:15) A v (cid:48) (cid:15) (cid:15) A (cid:48) u = (cid:0) u (cid:48) u (cid:48)(cid:48) (cid:1) (cid:47) (cid:47) A ⊕ C v =( v (cid:48) v (cid:48)(cid:48) ) (cid:47) (cid:47) (0 1) (cid:15) (cid:15) A (cid:48)(cid:48) (cid:15) (cid:15) A (cid:48) u (cid:48)(cid:48) (cid:47) (cid:47) C (cid:47) (cid:47) D Since h ( A (cid:48) ) = h ( A ) −
1, by induction there is a triangle A (cid:48) → A (cid:48) u (cid:48)(cid:48) −→ C → Σ( A ). Thus D = Σ( A (cid:48) ). From A (cid:48) = A we get the desired triangle. (cid:3) e introduce notation for line segments in the AR quiver: given two indecomposableobjects A, B ∈ D b (Λ( r, n, m )) which lie on a ray or coray (so in particular sit in the samecomponent), then the finite set consisting of these two objects and all indecomposableslying between them on the (co)ray is denoted by AB . Finally, we recall our conventionthat X r = X and note that ( S A ) = Σ τ ( A ). Lemma 2.3.
Consider D b (Λ( r, n, m )) with r > . If A ∈ ind ( X ) ∪ ind ( Y ) then for eachindecomposable object B ∈ ray + ( AA ) we have Hom(
A, B ) (cid:54) = 0 . Note that we shall treat the case r = 1 in Proposition 5.2 below; we continue to use thenotation for the X components, however, the argument applies also to the Y components. Proof.
First observe that any indecomposable object B lying in an X or Y componentadmits morphisms to precisely two objects on the mouth, and precisely one object onthe mouth of the same component, since B lies in precisely one ray and one coray.Let A be an indecomposable object in an X or Y component. If B ∈ ray + ( A ) or B ∈ AA or B ∈ ray + ( A ), then Hom( A, B ) (cid:54) = 0, using Serre duality for the thirdstatement. Let B ∈ ray + ( AA ); the rays and corays of ray + ( A ) ∪ AA ∪ ray + ( A ) areindicated in the left-hand sketch in Figure 2. Consider the following part of the ARquiver of D : C (cid:31) (cid:31) B (cid:31) (cid:31) B (cid:48) (cid:63) (cid:63) B (cid:48)(cid:48) (cid:31) (cid:31) C (cid:63) (cid:63) B (cid:63) (cid:63) B where C is one irreducible morphism closer to ray + ( A ), B (cid:48) one closer to AA and B (cid:48)(cid:48) onecloser to ray + ( A ), if B is in the interior of the region ray + ( AA ). Note that the triangles C → C → C (cid:48)(cid:48) → Σ C, B → B → B (cid:48)(cid:48) → Σ B and B (cid:48) → B → B → Σ B (cid:48) are those from Lemma 2.2 and C (cid:48)(cid:48) = B . Furthermore, since Σ is an autoequivalence, any(co)suspension of C , B and B must also lie on the mouth.The idea is to proceed by induction up each ray of the hammock starting with the lowestray, ray + ( A ). By the observation above if B ∈ ray + ( A ) ∪ AA then Hom( A, B ) (cid:54) = 0 andwe are done. By induction, we may assume B / ∈ ray + ( A ) ∪ AA and that Hom( A, B (cid:48) ) (cid:54) = 0and Hom( A, B (cid:48)(cid:48) ) (cid:54) = 0.Since B (cid:54) = A (cid:54) = B , we have Hom( A, B ) = Hom( A, B ) = 0. Applying Hom( A, − )to the triangles involving B (cid:48) and B (cid:48)(cid:48) above produces long exact sequences in which thevanishing of one of Hom( A, Σ( B )) and Hom( A, Σ − B ) is enough to give Hom( A, B ) (cid:54) = 0.However, in the case r = 2 it may happen that Σ( B ) = Σ − B = S ( A ) and by Serreduality Hom( A, Σ( B )) = Hom( A, Σ − B ) (cid:54) = 0. In this case, starting with the inductionfrom the topmost ray, ray + ( A ), instead will give us that Hom( A, C ) (cid:54) = 0. Now we onlyrequire the vanishing of Hom( A, Σ( C )) to give us Hom( A, B ) (cid:54) = 0. However, we haveΣ( C ) (cid:54) = Σ( B ) = S ( A ). Since A admits morphisms only to the objects A and S ( A )on the mouth of a Z A ∞ component, we get Hom( A, Σ( C )) = 0. We can now resume thestandard induction. (cid:3) Proposition 2.4 (Hammocks Hom( X k , − )) . Let A = X kij ∈ ind ( X k ) and assume r > .For any indecomposable object B ∈ ind ( D b (Λ)) the following cases apply: A ray + ( AA ) ⊂ X S A S A Σ A coray − ( S A, S A ) ⊂ X Z , Z , Z , ray ± ( Z , Z , ) ⊂ Z Figure 2.
Hom hammocks Hom( A, − ) = Hom( − , S A ) ∗ (cid:54) = 0 for A = X , . A = X , , Σ A = X , (if r ≥ , S A = Σ τ A = X , , S A = X , B ∈ X k : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray + ( AA ) ; B ∈ X k +1 : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ coray − ( ( S A ) , S A ) ; B ∈ Z k : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray ± ( Z kii Z kji ) and Hom(
A, B ) = 0 for all other B ∈ ind ( D b (Λ)) .For r = 1 , these results still hold, except that the X -clauses are replaced by B ∈ X : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray + ( AA ) ∪ coray − ( ( τ − m A ) , τ − m A ) .Proof. The main tool in the proof of this, and the following propositions, will be inductionon the height of A — the induction base step is proved in Lemma 2.1 which gives thehammocks for indecomposables of height 0. We give a careful exposition for the firstclaim, and for r >
1. The r = 1 case will be treated in Proposition 5.2. Case B ∈ X k : For any indecomposable object A ∈ X k , write R ( A ) for the subset of X k specified in the statement, i.e. bounded by the rays out of A and A , and the line segment AA . The existence of non-zero homomorphisms A → B for objects B ∈ R ( A ) followsdirectly from Lemma 2.3.For the vanishing statement, we proceed by induction on the height of A . If A sits onthe mouth of X k , then Lemma 2.1 states indeed that the Hom( A, B ) (cid:54) = 0 if and only if B is in the ray of A . Note that R ( A ) is precisely ray + ( A ) in this case.Now let A ∈ X k be any object of height h := h ( A ) >
0. We consider the diamondin the AR mesh which has A as the top vertex, and the corresponding AR triangle A (cid:48) → A ⊕ C → A (cid:48)(cid:48) → Σ A (cid:48) , where h ( A (cid:48) ) = h ( A (cid:48)(cid:48) ) = h − h ( C ) = h −
2. (If h = 1,we are in the degenerate case with C = 0.) It is clear from the definitions that A = A (cid:48)(cid:48) , A (cid:48) = C and there are inclusions R ( A (cid:48)(cid:48) ) ⊂ R ( A ) ⊂ R ( A (cid:48) ) ∪ R ( A (cid:48)(cid:48) ). We start with anobject B ∈ X k such that B / ∈ R ( A (cid:48) ) ∪ R ( A (cid:48)(cid:48) ). By the induction hypothesis, we know that R ( A (cid:48) ), R ( C ) and R ( A (cid:48)(cid:48) ) are the Hom-hammocks in X k for A (cid:48) , C , A (cid:48)(cid:48) , respectively. Since B is contained in none of them, we see that Hom( A (cid:48) , B ) = Hom( C, B ) = Hom( A (cid:48)(cid:48) , B ) = 0.Applying Hom( − , B ) to the given AR triangle shows Hom( A, B ) = 0.It remains to show that Hom(
A, D ) = 0 for objects D ∈ ( R ( A (cid:48) ) ∪ R ( A (cid:48)(cid:48) )) \ R ( A )which can be seen to be the line segment A (cid:48) A (cid:48) . Again we work up from the mouth:Hom( A, A (cid:48) ) = 0 and Hom( A, τ A (cid:48) ) = 0 by Lemma 2.1, as before. The extension D givenby τ A (cid:48) → D → A (cid:48) → Σ τ A (cid:48) is the indecomposable object of height 1 on A (cid:48) A (cid:48) . ApplyingHom( A, − ) to this triangle, we find Hom( A, D ) = 0, as required. The same reasoningworks for the objects of heights 2 , . . . , h − ase B ∈ X k +1 : We start by showing the existence of non-zero homomorphisms toindecomposable objects in the desired region. For any B in this region, it follows directlyfrom the dual of Lemma 2.3 that there is a non-zero homomorphism from B to S A .However, by Serre duality we see that Hom( A, B ) = Hom( B, S A ) ∗ (cid:54) = 0, as required. Thestatement that Hom( A, B ) = 0 for all other B ∈ X k +1 can be proved by an inductionargument which is analogous to the one given in the first case above. Case B ∈ Z k : For any indecomposable object A = X kij ∈ X k , write V ( A ) for theregion in Z k specified in the statement, i.e. the region bounded by the rays through Z kii and Z kji . We start by proving that Hom( A, B ) (cid:54) = 0 for B ∈ V ( A ). The first chain ofmorphisms in Properties 1.2(5), implies that Hom( A, B ) (cid:54) = 0 for any B ∈ ray ± ( Z kii ). Forany other B (cid:48) = Z ki + s,t ∈ V ( A ), so t ∈ Z and s ∈ { , . . . , h ( A ) = j − i } , we considerthe special triangle X ki,i + s − → B → B (cid:48) → Σ X ki,i + s − from Properties 1.2(4), where B = Z kit ∈ ray ± ( Z kii ). Applying Hom( A, − ) leaves us with the exact sequenceHom( A, X ki,i + s − ) → Hom(
A, B ) → Hom(
A, B (cid:48) ) → Hom( A, Σ X ki,i + s − ) . By looking at the Hom-hammocks in the X -components that we already know, we seethat the left-hand term vanishes as X ki,i + s − is on the same ray as A but has strictlylower height. Similarly, we observe that the right-hand term of the sequence vanishes:0 = Hom( X ki,i + s − , τ A ) = Hom( A, Σ X ki,i + s − ). Hence Hom( A, B (cid:48) ) = Hom(
A, B ) (cid:54) = 0.For the Hom-vanishing part of the statement, we again use induction on the height h := h ( A ) ≥
0. For h = 0, Lemma 2.1 gives V ( A ) = ray ± ( Z kii ). For h >
0, as beforewe consider the AR mesh which has A as its top vertex: A (cid:48) → A ⊕ C → A (cid:48)(cid:48) → Σ A (cid:48) .For any Z ∈ ind ( Z k ), we apply Hom( − , Z ) to this triangle and find that Hom( A, Z ) (cid:54) = 0implies Hom( A (cid:48) , Z ) (cid:54) = 0 or Hom( A (cid:48)(cid:48) , Z ) (cid:54) = 0. Therefore Hom( A, B ) = 0 for all
B / ∈ V ( A (cid:48) ) ∪ V ( A (cid:48)(cid:48) ) = V ( A ), where the final equality is clear from the definitions. Remaining cases:
These comprise vanishing statements for entire AR components, namelyHom( X k , X j ) = 0 for j (cid:54) = k, k + 1, and Hom( X k , Y j ) = 0 for any j , and Hom( X k , Z j ) = 0for j (cid:54) = k . All of those follow at once from Lemma 2.1: with no non-zero maps from A to the mouths of the specified components of type X and Y , Hom vanishing can be seenusing induction on height and considering a square in the AR mesh. The vanishing tothe Z k components with k (cid:54) = j follows similarly. (cid:3) Hom-hammocks for objects in Y components. Assume A = Y kij ∈ ind ( Y k ).This case is similar to the one above. Put A := Y kii to be the intersection of the coray through A with the mouth of Y k , and A := Y kjj to be the intersection of the ray through A with the mouth of Y k . Proposition 2.5 (Hammocks Hom( Y k , − )) . Let A = Y kij ∈ ind ( Y k ) and assume r > .For any indecomposable object B ∈ ind ( D b (Λ)) the following cases apply: B ∈ Y k : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ coray + ( AA ) ; B ∈ Y k +1 : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray − ( ( S A ) , S A ) ; B ∈ Z k : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ coray ± ( Z kii Z kij ) and Hom(
A, B ) = 0 for all other B ∈ ind ( D b (Λ)) .For r = 1 , these results still hold, except that the Y -clauses are replaced by B ∈ Y : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ coray + ( AA ) ∪ ray − ( ( τ n A ) , τ n A ) .Proof. These statements are analogous to those of Proposition 2.4. (cid:3) ray + ( coray − ( A )) ⊂ X AτA ray + ( coray + ( A )) ⊂ Z S A Σ A ray − ( coray − ( S A )) ⊂ Z Figure 3.
Hammocks Hom( A, − ) = Hom( − , S A ) ∗ (cid:54) = 0 for A ∈ ind ( Z ).The remaining hammock ray + ( coray + ( A )) ⊂ Y is not shown.2.4. Hom-hammocks for objects in Z components. Let A = Z kij ∈ ind ( Z k ). ByLemma 2.1 we know that the following objects are well defined: A := the unique object at the mouth of an X component for which Hom( A, A ) (cid:54) = 0, A := the unique object at the mouth of a Y component for which Hom( A, A ) (cid:54) = 0.In fact, A ∈ X k +1 and A ∈ Y k +1 . Proposition 2.6 (Hammocks Hom( Z k , − )) . Let A = Z kij ∈ ind ( Z k ) and assume r > .For any indecomposable object B ∈ ind ( D b (Λ)) the following cases apply: B ∈ X k +1 : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray + ( coray − ( A )) ; B ∈ Y k +1 : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray − ( coray + ( A )) ; B ∈ Z k : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray + ( coray + ( A )) ; B ∈ Z k +1 : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray − ( coray − ( S A )) and Hom(
A, B ) = 0 for all other B ∈ ind ( D b (Λ)) .For r = 1 , these results still hold, with the Z -clauses replaced by B ∈ Z : then Hom(
A, B ) (cid:54) = 0 ⇐⇒ B ∈ ray + ( coray + ( A )) ∪ ray − ( coray − ( S A )) .Proof. The cases B ∈ X k +1 and B ∈ Y k +1 follow by Serre duality from Proposition 2.4and Proposition 2.5, respectively.Thus let B = Z lab ∈ Z l be an indecomposable object in a Z component. There are twospecial distinguished triangles associated with B ; see Properties 1.2(4): B (cid:47) (cid:47) B (cid:47) (cid:47) B (cid:48) (cid:47) (cid:47) Σ BX laa (cid:47) (cid:47) Z lab (cid:47) (cid:47) Z la +1 ,b (cid:47) (cid:47) Σ X laa and B (cid:47) (cid:47) B (cid:47) (cid:47) B (cid:48)(cid:48) (cid:47) (cid:47) Σ BY lbb (cid:47) (cid:47) Z lab (cid:47) (cid:47) Z la,b +1 (cid:47) (cid:47) Σ Y lbb where B = X laa is the unique object at the mouth of a X component with Hom( B, B ) (cid:54) =0 and similarly B = Y lbb is unique at a Y mouth with Hom( B, B ) (cid:54) = 0. We get twoexact sequences by applying Hom( A, − ):Hom( A, B ) −→ Hom(
A, B ) −→ Hom(
A, B (cid:48) ) −→ Hom( A, Σ B ) , Hom( A, B ) −→ Hom(
A, B ) −→ Hom(
A, B (cid:48)(cid:48) ) −→ Hom( A, Σ B ) . Case l (cid:54) = k, k + 1 : In this case Hom(
A, B ) = Hom(
A, B (cid:48) ) = Hom(
A, B (cid:48)(cid:48) ) follows from theabove triangles via these exact sequences and Lemma 2.1. But this implies Hom(
A, B ) = om( A, Z ) for all Z ∈ Z l and in particular Hom( A, B ) = Hom( A, Σ cr B ) for all c ∈ Z . Itfollows that Hom( A, B ) = 0 as Λ( r, n, m ) has finite global dimension.
Case l = k : Again, we first show that the dimension function hom( A, − ) is constant oncertain regions of Z k . In particular, we haveHom( A, B ) = Hom(
A, B (cid:48) ) for
B / ∈ ray ± ( S A ) ∪ ray ± ( τ A );(1) Hom( A, B ) = Hom(
A, B (cid:48)(cid:48) ) for
B / ∈ coray ± ( S A ) ∪ coray ± ( τ A ) . (2)Half of the first equality follows through the chain of equivalencesHom( A, B ) (cid:54) = 0 ⇐⇒ A ∈ ray ± ( S − Z kaa ) ⇐⇒ A ∈ ray ± ( S − B ) ⇐⇒ B ∈ ray ± ( S A ) . Likewise one obtains Hom( A, Σ B ) (cid:54) = 0 ⇐⇒ B ∈ ray ± ( τ A ), giving the first equality.Using the other triangle, the second equality is analogous.The component Z k is divided by ray ± ( τ A ) and coray ± ( τ A ) into four regions: U : The upwards-open region including ray + ( τ A ) \{ τ A } butexcluding coray − ( τ A ); L : The left-open region including ray − ( τ A ) ∪ coray − ( τ A ); D : The downwards-open region including coray + ( τ A ) \{ τ A } but excluding ray − ( τ A ); R : The right-open region excluding ray ± ( τ A ) ∪ coray ± ( τ A ). • • τA A UD RL
Using (1) above coupled with the fact that U contains infinitely many objects Σ − rc A with c ∈ N , shows by the finite global dimension of Λ( r, n, m ) that no objects in U admitnon-trivial morphisms from A . Using (2) and analogous reasoning shows that no objectsin D admit non-trivial morphisms from A . Non-existence of non-trivial morphisms from A to objects in L follows as soon as Hom( A, τ A ) = Hom ( A, A ) = 0 by using (2) above. Theexistence of the stalk complex of a projective module in the Z component, Lemma B.9,coupled with the transitivity of the action of the automorphism group of D b (Λ( r, n, m ))on the Z component, which is proved in Section 4 using only Lemma 2.1, shows thatHom ( A, A ) = 0 for all A ∈ Z .Finally, R = coray + ( ray + ( A )) is the non-vanishing hammock simply by Hom( A, A ) (cid:54) = 0and using either (1) or (2). Case l = k + 1 : This is analogous to the previous case. (cid:3)
Remark 2.7.
In the case that r >
1, Propositions 2.4, 2.5 and 2.6 say that each compo-nent of the AR quiver of D b (Λ( r, n, m )) is standard, i.e. that there are no morphisms inthe infinite radical. Note that the components are not standard when r = 1.3. Twist functors from exceptional cycles
In this purely categorical section, we consider an abstract source of autoequivalencescoming from exceptional cycles. These generalise the tubular mutations from [34] as wellas spherical twists. In fact, a quite general and categorical construction has been givenin [39]. However, for our purposes this is still a little bit too special, as the Serre functorwill act with different degree shifts on the objects in our exceptional cycles. We also givea quick proof using spanning classes.Let D be a k -linear, Hom-finite algebraic triangulated category. Assume that D has aSerre functor S and is indecomposable; see Appendix A.1 for these notions. Recall thatan object E ∈ D is called exceptional if Hom • ( E, E ) = k · id E . For any object A ∈ D wedefine the functor F A : D → D , X (cid:55)→ Hom • ( A, X ) ⊗ A nd note that there is a canonical evaluation morphism F A → id of functors. Also notethat for two objects A , A ∈ D there is a common evaluation morphism F A ⊕ F A → id .In fact, for any sequence of objects A ∗ = ( A , . . . , A n ), we define the associated twistfunctor T A ∗ as the cone of the evaluation morphism — this gives a well-defined, exactfunctor by our assumption that D is algebraic; see [22, § F A ∗ → id D → T A ∗ → Σ F A ∗ with F A ∗ := F A ⊕ · · · ⊕ F A n These functors behave well under equivalences:
Lemma 3.1.
Let ϕ : D ∼ → D (cid:48) be a triangle equivalence of algebraic k -linear triangulatedcategories induced from a dg functor, and let A ∗ = ( A , . . . , A n ) be any sequence of objects.Then there are functor isomorphisms F ϕ ( A ∗ ) = ϕ F A ∗ ϕ − and T ϕ ( A ∗ ) = ϕ T A ∗ ϕ − .Proof. This follows the standard argument for spherical twists: For F A ∗ we have ϕ F A ∗ ϕ − = (cid:77) i Hom • ( A i , ϕ − ( − )) ⊗ ϕ ( A i ) = (cid:77) i Hom • ( ϕ ( A i ) , − ) ⊗ ϕ ( A i ) = F ϕ ( A ∗ ) . Conjugating the evaluation functor morphism F A ∗ → id with ϕ , we find that ϕ T A ∗ ϕ − is the cone of the conjugated evaluation functor morphism F ϕ ( A ∗ ) → id which is theevaluation morphism for ϕ ( A ∗ ). Hence that cone is T ϕ ( A ∗ ) . (cid:3) Definition 3.2.
A sequence ( E , . . . , E n ) of objects of D is an exceptional n -cycle if(1) every E i is an exceptional object,(2) there are integers k i such that S ( E i ) ∼ = Σ k i ( E i +1 ) for all i (where E n +1 := E ),(3) Hom • ( E i , E j ) = 0 unless j = i or j = i + 1.This definition assumes n ≥ E should be considered an ‘exceptional1-cycle’ if E is a spherical object, i.e. there is an integer k with S ( E ) ∼ = Σ k ( E ) andHom • ( E, E ) = k ⊕ Σ − k k . In this light, the above definition, and statement and proofof Theorem 3.5 are generalisations of the treatment of spherical objects and their twistfunctors as in [27, § E i apart from theidentities are given by α i : E i → Σ k i E i +1 . This explains the terminology: the subsequence( E , . . . , E n − ) is an honest exceptional sequence, but the full set ( E , . . . , E n ) is not —the morphism α n : E n → Σ k n E prevents it from being one, and instead creates a cycle. Remark 3.3.
All objects in an exceptional n -cycle are fractional Calabi–Yau: since S ( E i ) ∼ = Σ k i E i +1 for all i , applying the Serre functor n times yields S n ( E i ) ∼ = Σ k E i , where k := k + · · · + k n . Thus the Calabi–Yau dimension of each object in the cycle is k/n . Example 3.4.
We mention that this severely restricts the existence of exceptional n -cycles of geometric origin: Let X be a smooth, projective variety over k of dimension d and let D := D b ( coh X ) be its bounded derived category. The Serre functor of D is givenby S ( − ) = Σ d ( − ) ⊗ ω X and in particular, is given by an autoequivalence of the standardheart followed by an iterated suspension. If E ∗ is any exceptional n -cycle in D , we find S n ( E i ) = Σ dn E i ⊗ ω nX ∼ = Σ k E i , hence k = k + · · · + k n = dn and E i ⊗ ω nX ∼ = E i . Iffurthermore the exceptional n -cycle E ∗ consists of sheaves, then this forces k i = d to bemaximal for all i , as non-zero extensions among sheaves can only exist in degrees between0 and d . However, S E i = Σ d E i ⊗ ω X ∼ = Σ d E i +1 implies E i +1 ∼ = E i ⊗ ω X for all i .As an example, let X be an Enriques surface. Its structure sheaf O X is exceptional,and the canonical bundle ω X has minimal order 2. In particular, ( O X , ω X ) forms anexceptional 2-cycle and, by the next theorem, gives rise to an autoequivalence of D b ( X ). heorem 3.5. Let E ∗ = ( E , . . . , E n ) be an exceptional n -cycle in D . Then the twistfunctor T E ∗ is an autoequivalence of D .Proof. We define two classes of objects of D by E := { Σ l E i | l ∈ Z , i = 1 , . . . , n } andΩ := E ∪ E ⊥ . Note that E and hence Ω are closed under suspensions and cosuspensions.It is a simple and standard fact that Ω is a spanning class for D , i.e. Ω ⊥ = 0 and ⊥ Ω = 0;the latter equality depends on the existence of a Serre functor for D . Note that spanningclasses are often called ‘(weak) generating sets’ in the literature. Step 1:
We start by computing T E ∗ on the objects E i and the maps α i . For notationalsimplicity, we will treat E and α : E → Σ k E . It follows immediately from the def-inition of exceptional cycle that F E ∗ ( E ) = E ⊕ Σ − k n E n . The cone of the evaluationmorphism is easily seen to sit in the following triangle E ⊕ Σ − k n E n id ⊕ Σ − kn α n (cid:47) (cid:47) E (cid:47) (cid:47) Σ − k n E n ( − Σ − kn α n , id ) t (cid:47) (cid:47) Σ E ⊕ Σ − k n E n , so that T E ∗ ( E ) = Σ − k n E n . The left-hand morphism has an obvious splitting, this impliesthe zero morphism in the middle. The third map is indeed the one specified above; thiscan be formally checked with the octahedral axiom, or one can use the vanishing of thecomposition of two adjacent maps in a triangle.Likewise, we find F E ∗ ( E ) = Σ − k E ⊕ E and T E ∗ ( E ) = Σ − k E . Now consider thefollowing diagram of distinguished triangles, where the vertical maps are induced by α : E ⊕ Σ − k n E n id ⊕ Σ − kn α n (cid:47) (cid:47) (cid:0) id (cid:1) (cid:15) (cid:15) E (cid:47) (cid:47) α (cid:15) (cid:15) Σ − k n E n ( − Σ − kn α n , id ) t (cid:47) (cid:47) T ( α ) (cid:15) (cid:15) Σ E ⊕ Σ − k n E n (cid:0) id (cid:1) (cid:15) (cid:15) E ⊕ Σ k E α ⊕ id (cid:47) (cid:47) Σ k E (cid:47) (cid:47) Σ E id , − Σ α ) t (cid:47) (cid:47) Σ E ⊕ Σ k E Hence, the commutativity of the right-hand square forces T E ∗ ( α ) = − Σ − k n α n .This also works if n = 2 and k = − k (with unchanged left-hand vertical arrow). Step 2:
The above computation shows that the functor T E ∗ is fully faithful when restrictedto E . It is also obvious from the construction of the twist that T E ∗ is the identity whenrestricted to E ⊥ .Let E i ∈ E and X ∈ E ⊥ . Then Hom • ( E i , X ) = 0 and also Hom • ( T E ∗ ( E i ) , T E ∗ ( X )) =Hom • (Σ − k i − E i − , X ) = 0. Finally, we use Serre duality and the defining property of E ∗ to see thatHom • ( X, E i ) = Hom • ( X, Σ − k i − S ( E i − )) ∼ = Hom • ( E i − , Σ k i − X ) ∗ = 0 . Combining all these statements, we deduce that T E ∗ is fully faithful when restrictedto the spanning class Ω, hence bona fide fully faithful by general theory; see e.g. [27,Proposition 1.49]. Note that T E ∗ has left and rights adjoints as the identity and F E ∗ do. Step 3:
With T E ∗ fully faithful, the defining property of Serre functors gives a canonicalmap of functors S − T E ∗ S → T E ∗ which can be spelled out in the following diagram: (cid:76) i Hom • ( E i , S ( − )) ⊗ S − ( E i ) (cid:47) (cid:47) (cid:15) (cid:15) id (cid:47) (cid:47) (cid:15) (cid:15) S − T E ∗ S (cid:15) (cid:15) (cid:76) i Hom • ( E i , − ) ⊗ E i (cid:47) (cid:47) id (cid:47) (cid:47) T E ∗ It is easy to check that the left-hand vertical arrow is an isomorphism whenever we plugin objects from Ω: both vector spaces are zero for objects from E ⊥ ; for the top row, use om • ( E i , S ( − )) = Hom • ( S − ( E i ) , − ) = Hom • (Σ − k i − E i − , − ). For objects E i , again use S ( E i ) ∼ = Σ k i E i +1 . Hence T E ∗ commutes with the Serre functor on Ω, and so by moregeneral theory is essentially surjective; see [27, Corollary 1.56], this is the place where weneed the assumption that D is indecomposable. (cid:3) Remark 3.6.
We point out that the twist T E ∗ defined above is an instance of a sphericalfunctor [3], given by the following data: S : D b ( k n ) → D , ( V • , . . . , V • n ) (cid:55)→ V • ⊗ k E ⊕ · · · ⊕ V • n ⊗ k E n ,R : D → D b ( k n ) , X (cid:55)→ (Hom • ( E , X ) , . . . , Hom • ( E n , X ))where D b ( k n ) = (cid:76) n D b ( k ) is a decomposable category. It is easy to see that R is rightadjoint to S and that T E ∗ coincides with the cone of the adjunction morphism SR → id .An object X ∈ D is called d -spherelike if Hom • ( X, X ) = k ⊕ Σ − d k ; see [22] and alsoSection 5.3. We will now show that reasonable exceptionable cycles come with a spherelikeobject. For this purpose, we call an exceptional cycle E ∗ = ( E , . . . , E n ) irredundant if E n / ∈ thick ( E , . . . , E n − ). Recall that an exceptional n -cycle ( E , . . . , E n ) comes with atuple of integers ( k , . . . , k n ) and that we have set k = k + · · · + k n . Proposition 3.7.
Let E ∗ = ( E , . . . , E n ) be an irredundant exceptional n -cycle in D .Then there exists a ( k + 1 − n ) -spherelike object X ∈ D with non-zero maps X → E and Σ n − − k + k n E n → X .Proof. Inductively, we construct a series of objects X , . . . , X n with the following prop-erties for i < n :(i) X i is exceptional,(ii) X i ∈ thick ( E , . . . , E i ),(iii) Hom • ( X i , E i +1 ) = Σ − l i k with l i := k + · · · + k i + 1 − i .These conditions are satisfied for X := E , because Hom • ( E , E ) is generated by α : E → Σ k E . With X i already constructed, by (iii) there is a unique object X i +1 with a non-split distinguished triangle X i +1 → X i → Σ l i E i +1 → Σ X i +1 . Moreover, ( X i , E i +1 ) is an exceptional pair with just one (graded) morphism by (ii) and(iii). Hence in the above triangle, the object X i +1 is, up to suspension, the left mutationof that pair. In particular, X i +1 is exceptional. By construction, X i +1 satisfies (ii).If i + 1 < n , then Hom • ( X i , E i +2 ) = 0 by (ii) and the definition of exceptional cycles,hence Hom • ( X i +1 , E i +2 ) = Hom • (Σ l i − E i +1 , E i +2 ). As α i +1 : E i +1 → Σ k i +1 E i +2 generatesHom • ( E i +1 , E i +2 ), we find that Hom • ( X i +1 , E i +2 ) is 1-dimensional, and situated in degree l i + k i +1 − l i +1 .Having constructed X n − in this fashion, we can use (iii) to define X n → X n − → Σ l n − E n → Σ X n . This triangle induces a commutative diagram of complexes of k -vector spacesHom • ( X n , X n ) (cid:47) (cid:47) Hom • ( X n , X n − ) (cid:47) (cid:47) Hom • ( X n , Σ l n − E n )Hom • ( X n − , X n ) (cid:47) (cid:47) (cid:79) (cid:79) Hom • ( X n − , X n − ) (cid:47) (cid:47) (cid:79) (cid:79) Hom • ( X n − , Σ l n − E n ) (cid:79) (cid:79) Hom • (Σ l n − E n , X n ) (cid:47) (cid:47) (cid:79) (cid:79) Hom • (Σ l n − E n , X n − ) (cid:47) (cid:47) g (cid:79) (cid:79) Hom • (Σ l n − E n , Σ l n − E n ) f (cid:79) (cid:79) nd we know that Hom • ( X n − , X n − ) = Hom • (Σ l n − E n , Σ l n − E n ) = k , since X n − and E n are exceptional. Moreover, we get Hom • ( X n − , Σ l n − E n ) = k from applying Hom • ( − , E n )to the triangle defining X n − and using X n − ∈ (cid:104) E , , . . . , E n − (cid:105) , none of which map to E n . In particular, the map f sends the identity to the morphism X n − → Σ l n − E n defining X n . Hence f is an isomorphism, thus Hom • ( X n , Σ l n − E n ) = 0 and we arrive atthe isomorphism Hom • ( X n , X n ) ∼ → Hom • ( X n , X n − ).We turn to Hom • (Σ l n − E n , X n − ). By (ii) and Hom • ( E n , E i ) = 0 for 1 < i < n ,Hom • (Σ l n − E n , X n − ) = Hom • (Σ l n − E n , X n − ) = · · · = Hom • (Σ l n − E n , X ) = Σ n − − k k , where k = k + · · · + k n as before. Now g is a map of two 1-dimensional complexes.This map cannot be an isomorphism, because this would force Hom • ( X n , X n ) = 0, hence X n = 0 but we have X n − ∈ thick ( E , . . . , E n − ) by (ii) and also E n / ∈ thick ( E , . . . , E n − )as E ∗ is irredundant. Therefore we findHom • ( X n , X n ) ∼ = Hom • ( X n , X n − ) ∼ = k ⊕ Σ n − − k k . Hence X := X n is indeed ( k + 1 − n )-spherelike. The degrees of non-zero maps inHom • ( E n , X ) and Hom • ( X, E ) are computed with the same methods as above. (cid:3) Example 3.8.
The additional hypothesis on E ∗ is necessary: consider D = D b ( k A ) forthe A -quiver 1 → →
3. Denoting the injective-projective module by M = P (1) = I (3),the sequence E ∗ = ( S (1) , S (2) , S (3) , M ) is an exceptional cycle with k ∗ = (1 , , , M ∈ thick ( S (1) , S (2) , S (3)); note that ( S (1) , S (2) , S (3)) isa full exceptional collection for D .Following the iterative construction of the above proof, we get X = S (1), X = I (2)and X = M . This forces X = X = 0, and we do not get a spherelike object in this case.Note that E ∗ still gives a twist autoequivalence, which for this example is just T E ∗ = τ − .4. Autoequivalence groups of discrete derived categories
We now use the general machinery of the previous section to show that categories D b (Λ( r, n, m )) possess two very interesting and useful autoequivalences. We will denotethese by T X and T Y and prove some crucial properties: they commute with each other, acttransitively on the indecomposables of each Z k component and provide a weak factori-sation of the Auslander–Reiten translation: T X T Y = τ − on objects. Moreover, T X actstrivially on Y and T Y acts trivially on X ; see Proposition 4.4 and Corollary 4.5 for theprecise assertions. We then give an explicit description of the group of autoequivalencesof D b (Λ( r, n, m )) in Theorem 4.7.The category D = D b (Λ( r, n, m )) with n > r is Hom-finite, indecomposable, algebraicand has Serre duality (see Appendix A.1). Therefore we can apply the results of theprevious section to D .Our first observation is that every sequence of m + r consecutive objects at the mouthof X is an exceptional ( m + r )-cycle; likewise, every sequence of n − r consecutive objectsat the mouth of Y is an exceptional ( n − r )-cycle, by which we mean a ( r + 1)-sphericalobject in case n − r = 1. For the moment, we specify two concrete sequences: E ∗ = ( E , . . . , E m + r ) := ( X m + r,m + r , . . . , X ) , i.e. E i = X m + r +1 − i,m + r +1 − i ,F ∗ = ( F , . . . , F n − r ) := ( Y n − r,n − r , . . . , Y ) , i.e. F i = Y n − r +1 − i,n − r +1 − i . Lemma 4.1. E ∗ forms an exceptional ( m + r ) -cycle in D with k ∗ = (1 , . . . , , − r ) , and F ∗ forms an exceptional ( n − r ) -cycle in D with k ∗ = (1 , . . . , , r ) . roof. The object X is exceptional by Lemma 2.1, hence any object at the mouth X ii = τ − i ( X ) is. This point also gives the second condition of exceptional cycles: for i = 1 , . . . , m + r −
1, we have S E i = Σ τ X m + r +1 − i,m + r +1 − i = Σ X m + r − i,m + r − i = Σ E i +1 and at the boundary step we have S E m + r = Σ τ X = Σ X = Σ − r X m + r,m + r = Σ − r E ,where we freely make use of the results stated in Section 1. Hence the degree shifts of thesequence E ∗ are k = . . . = k m + r − = 1 and k m + r = 1 − r . Finally, the required vanishingHom( E i , E j ) = 0 unless j = i + 1 or i = j again follows from Lemma 2.1.The same reasoning works for Y , now with the boundary step degree computation S F n − r = Σ τ Y = Σ Y = Σ r Y n − r,n − r = Σ r F . (cid:3) The next lemma shows that the functors F E ∗ and F F ∗ of the last section take on aparticularly simple form, where we use the notation X , X , Y, Y from Sections 2.2,2.3: Lemma 4.2.
For X ∈ ind ( X ) and Y ∈ ind ( Y ) , F E ∗ ( X ) = X ⊕ S − X , F F ∗ ( Y ) = Y ⊕ S − X . Proof.
This follows immediately from the definition of these functors in Section 3, Propo-sition 2.4 and Properties 1.2(3), i.e. Σ r | X = τ − m − r and Σ r | Y = τ n − r on objects.Note that the right-hand sides extend to direct sums. Another description of F E ∗ ( X )is as the minimal approximation of X with respect to the mouth of X , and analogouslyfor F F ∗ . (cid:3) The actual choice of exceptional cycle is not relevant as the following easy lemmashows. We only state it for E ∗ but the similar statement holds for F ∗ , with the sameproof. This allows us to write T X instead of T E ∗ and T Y instead of T F ∗ . Lemma 4.3.
Any two exceptional cycles E ∗ , E (cid:48)∗ at the mouths of X components differ bysuspensions and AR translations, and the associated twist functors coincide: T E ∗ = T E (cid:48)∗ .Proof. A suitable iterated suspension will move E (cid:48)∗ into the X component that E ∗ inhab-its, and two exceptional cycles at the mouth of the same AR component obviously differby some power of the AR translation. Thus we can write E (cid:48)∗ = Σ a τ b E ∗ for some a, b ∈ Z .We point out that the suspension and the AR translation commute with all autoequiva-lences (it is a general and easy fact that the Serre functor does, see [27, Lemma 1.30]).Finally, we have T E (cid:48)∗ = T Σ a τ b E ∗ = Σ a τ b T E ∗ Σ − a τ − b = T E ∗ , using Lemma 3.1. (cid:3) Proposition 4.4.
The twist functors T X and T Y act as follows on objects of D b (Λ) , where X ∈ X , Y ∈ Y and k = 0 , . . . , r − and i, j ∈ Z : T X ( X ) = τ − ( X ) , T X ( Y ) = Y, T X ( Z ki,j ) = Z ki +1 ,j , T Y ( Y ) = τ − ( Y ) , T Y ( X ) = X, T Y ( Z ki,j ) = Z ki,j +1 . Corollary 4.5.
The twist functors T X and T Y act simply transitively on each component Z k and factorise the inverse AR translation: T X T Y = T Y T X = τ − on the objects of D b (Λ) .Moreover, T X , T Y and Σ act transitively on ind ( Z ) .Proof of the proposition. By Lemma 2.1, we have Hom • ( X kii , Y ) = 0 for all Y ∈ Y . Thisimmediately implies T X | Y = id . Action of T X on objects of X : we recall that the proof of Theorem 3.5 showed T X ( E i ) =Σ − k i − ( E i − ), and furthermore k = . . . = k m + r − = 1 and k m + r = 1 − r from Lemma 4.1.Hence T X ( E i ) = τ − ( E i ) for all i — as explained in Lemma 4.3, this holds for anyexceptional cycle at an X mouth. Since T X is an equivalence and each X component isof type Z A ∞ , this forces T E ∗ | X = τ − on objects. ction of T X on objects of Z : Pick Z ij ∈ Z with 1 ≤ i ≤ m + r . Using T X = T E ∗ withthe cycle originally specified, i.e. E m + r = X , we invoke Lemma 2.1 once more to get k = Hom • ( X ii , Z ij ) = Hom • ( E m + r +1 − i , Z ij ), and 0 = Hom • ( E l , Z ij ) for all l (cid:54) = m + r +1 − i .So F E ∗ ( Z ij ) = X ii and the triangle defining T E ∗ ( Z ij ) is one of the special triangles ofProperties 1.2(4): F E ∗ ( Z ij ) (cid:47) (cid:47) Z ij (cid:47) (cid:47) T E ∗ ( Z i,j ) (cid:47) (cid:47) Σ F E ∗ ( Z i,j ) X ii (cid:47) (cid:47) Z ij (cid:47) (cid:47) Z i +1 ,j (cid:47) (cid:47) Σ X i,i Application of AR translations extends this computation to arbitary Z ∈ Z , and sus-pending extends it to all Z components, thus T X ( Z i,j ) = Z i +1 ,j . Remaining cases:
Analogous reasoning shows T F ∗ ( F i ) = τ − ( F i ) for all i = 1 , . . . , n − r ,and the rest of the above proof works as well: T Y ( Z i,j ) = Z i,j +1 , now using the otherspecial triangle. (cid:3) The following technical lemma about the additive closures of the X and Y componentswill be used later on, but is also interesting in its own right. Using the twist functors,the proof is easy. Lemma 4.6.
Each of X and Y is a thick triangulated subcategory of D .Proof. The proof of Proposition 4.4 contains the fact thick ( E ∗ ) ⊥ = Y . Perpendicularsubcategories are always closed under extensions and direct summands; since thick ( E ∗ ) isby construction a triangulated subcategory, the orthogonal complement Y is triangulatedas well. (cid:3) Our results enable us to compute the group of autoequivalences of D b (Λ( r, n, m )). ForΛ(1 , , Aut ( D b (Λ(1 , , ∼ = Z × k ∗ . Theorem 4.7.
The group of autoequivalences of D b (Λ( r, n, m )) is an abelian group gen-erated by T X , T Y , Σ and Out (Λ( r, n, m )) = k ∗ , subject to one relation Σ r = f T m + r X T r − n Y for some f ∈ Out (Λ( r, n, m )) . As an abstract group,
Aut ( D b (Λ( r, n, m ))) ∼ = Z × Z /(cid:96) × k ∗ , where (cid:96) := gcd( r, n, m ) .Proof. In this proof, we will write D = D b (Λ( r, n, m )) and Λ = Λ( r, n, m ). Step 1:
Out (Λ) = k ∗ from common scaling of arrows. It is a well-known fact that inner automorphisms induce autoequivalences of mod (Λ)and D b (Λ) which are isomorphic to the identity; see [45, § Out (Λ) =
Aut (Λ) / Inn (Λ) acts faithfully on modules. The form of the quiver and the relations forΛ( r, n, m ) imply that algebra automorphisms can only act by scaling arrows.Scaling of arrows leads to a subgroup ( k ∗ ) m + n of Aut (Λ). However, choosing an inde-composable idempotent e (i.e. a vertex) together with a scalar λ ∈ k ∗ produces a unit u = 1 Λ + ( λ − e , and hence an inner automorphism c u ∈ Aut (Λ). It is easy to checkthat c u ( α ) = λ α if α ends at e , and c u ( α ) = λα if α starts at e , and c u ( α ) = α otherwise.The form of the quiver of Λ shows that an ( n + m − k ∗ ) m + n of arrow-scaling automorphisms consists of inner automorphisms. Furthermore, the au-tomorphism scaling all arrows simultaneously by the same number is easily seen not tobe inner, hence, Out (Λ) = k ∗ . Step 2: ϕ ∈ Aut ( D ) is the identity on objects ⇐⇒ ϕ ∈ Out (Λ) . y Step 1, it is clear that algebra automorphisms act trivially on objects. Let now ϕ ∈ Aut ( D ) fixing all objects. In particular, ϕ fixes the abelian category mod (Λ) and theobject Λ, thus giving rise to ϕ : Λ → Λ, i.e. an automorphism which by Step 1 can betaken to be outer.
Step 3: The subgroup (cid:104) Σ , T X , T Y , Out (Λ) (cid:105) is abelian.
The suspension commutes with all exact functors. Next, to see [ T X , T Y ] = id , wefix exceptional cycles E ∗ for X and F ∗ for Y ; then T E ∗ T F ∗ ( T E ∗ ) − = T T E ∗ ( F ∗ ) = T F ∗ byLemma 3.1 and Proposition 4.4. Let f ∈ Out (Λ). Then we have [ f, T X ] = [ f, T Y ] = id bythe same lemma, now using f ( E ∗ ) = E ∗ and f ( F ∗ ) = F ∗ from Step 2. Step 4:
Aut ( D ) is generated by Σ , T X , T Y , Out (Λ) . Fix a Z ∈ ind ( Z ). For any ϕ ∈ Aut ( D ), there are a, b, c ∈ Z with Σ a T b X T c Y ( Z ) = ϕ ( Z ),since the suspension and the twist functors act transitively on ind ( Z ) by Corollary 4.5.Therefore, ψ := Σ a T b X T c Y ϕ − fixes Z . Moreover, since all autoequivalences commute with τ (because they commute with the Serre functor S = Σ τ and with Σ) and Z is a Z A ∞∞ -component, either ψ is the identity on ind ( Z ) or else ψ flips ind ( Z ) along the Zτ ( Z ) axis.However, the latter possibility is excluded by the action of Σ r | Z ; see Properties 1.2(3).By Properties 1.2(4), every indecomposable object of X or Y is a cone of a morphism Z → Z for some Z , Z ∈ ind ( Z ). Moreover, the morphism Z → Z is unique up toscalars by Theorem 5.1. (The proofs in that section make no use of the autoequivalencegroup. Note that by the proof of Theorem 5.1, morphism spaces between indecomposableobjects in Z are 1-dimensional, even for r = 1.) Hence ϕ actually fixes all indecomposableobjects and thus all objects of D b (Λ).Thus, by Step 2, ψ ∈ Out (Λ) and ϕ ∈ (cid:104) Σ , T X , T Y , Out (Λ) (cid:105) . Step 5:
Aut ( D ) is abelian with one relation f Σ − r T m + r X T r − n Y = id for some f ∈ Out (Λ) . By Steps 3 and 4,
Aut ( D ) = (cid:104) Σ , T X , T Y , Out (Λ) (cid:105) is abelian. Properties 1.2(3) andProposition 4.4 imply that the autoequivalence Σ − r T m + r X T r − n Y fixes all objects of D , hence f Σ − r T m + r X T r − n Y = id for a unique automorphism f ∈ Out (Λ).Let now be a, b, c ∈ Z and g ∈ Out (Λ) such that g Σ a T b X T c Y = id . In particular, ψ := Σ a T b X T c Y fixes all objects. From X = ψ ( X ) = Σ a T b X ( X ) = Σ a τ − b ( X ) we deduce first a = lr for some l ∈ Z and then b = − l ( m + r ); whereas Y = ψ ( Y ) similarly implies a = kr and c = k ( n − r ) for some k ∈ Z . Hence k = l and ψ = Σ lr T − l ( m + r ) X T l ( n − r ) Y = f l . So g = f − l and altogether, g Σ a T b X T c Y = ( f Σ − r T m + r X T n − r Y ) − l is a power of the stated relation. Step 6:
Aut ( D ) ∼ = Z × Z / ( r, n, m ) × k ∗ . This is elementary algebra: let A be a free abelian group of finite rank and a ∈ A , f ∈ k ∗ . Write a = da with d ∈ Z and a indivisible. Choose f ∈ k ∗ with f d = f — this is possible because k is algebraicaly closed. Now fix a group homomorphism ν : A → Z with ν ( a ) = 1 — this is possible because a is indivisible. Consider thediagram with exact rows0 (cid:47) (cid:47) { ( na , f n ) | n ∈ Z } (cid:47) (cid:47) A × k ∗ (cid:47) (cid:47) A × k ∗ / (cid:104) ( a , f ) (cid:105) (cid:47) (cid:47) (cid:47) (cid:47) { ( na , | n ∈ Z } (cid:47) (cid:47) α (cid:79) (cid:79) A × k ∗ (cid:47) (cid:47) β (cid:79) (cid:79) A × k ∗ / (cid:104) ( a , (cid:105) (cid:47) (cid:47) α ( na ,
1) = ( na , f n ) and β ( a, f ) = ( a, f f ν ( a )1 ). Both maps are easily checked tobe group homomorphisms and bijective. Moreover, the left-hand square commutes: β ( na ,
1) = ( na , f ν ( na )1 ) = ( na , f ndν ( a )1 ) = ( na , f nν ( a )0 ) = ( na , f n ) = α ( na , . herefore we obtain an induced isomorphism between the right-hand quotients: A × k ∗ / (cid:104) ( a , f ) (cid:105) ∼ = A × k ∗ / (cid:104) ( a , (cid:105) = A/ (cid:104) a (cid:105) × k ∗ . For the case at hand, A = Z and a = ( r, n, m ) ∈ Z and hence A/a ∼ = Z × Z /(cid:96) withthe greatest common divisor (cid:96) = ( r, n, m ), by the theory of elementary divisors. (cid:3) Question.
It is natural to speculate about the action of the various functors on maps.More precisely, we ask whether(1) Σ r | X = τ − m − r and Σ r | Y = τ n − r (2) T X | X = τ − and T Y | Y = τ − (3) Σ r = T m + r X T r − n Y hold as functors. In all cases, we know these relations hold on objects. Note that (1) and(2) together imply (3), and that (3) means f = id in Theorem 4.7.5. Hom spaces: dimension bounds and graded structure
In this section, we prove a strong result about D b (Λ) := D b (Λ( r, n, m )) which says thatthe dimensions of homomorphism spaces between indecomposable objects have a commonbound. We also present the endomorphism complexes in Lemma 5.3.5.1. Hom space dimension bounds.
The bounds are given in the the following theo-rem; for more precise information in case r = 1 see Proposition 5.2. Theorem 5.1.
Let
A, B be indecomposable objects of D b (Λ( r, n, m )) where n > r . If r ≥ , then dim Hom( A, B ) ≤ and if r = 1 , then dim Hom( A, B ) ≤ .Proof. Our strategy for establishing the dimension bound follows that of the proofs ofthe Hom-hammocks. Let
A, B ∈ ind ( D b (Λ( r, n, m ))) and assume r >
1. In this proof, weuse the abbreviation hom = dim Hom. We want to show hom(
A, B ) ≤ Case A ∈ X k or Y k : Consider first
A, B ∈ X k and perform induction on the height of A .If A = A sits at the mouth, then hom( A, B ) ≤ A higher up, andassuming Hom( A, B ) (cid:54) = 0, which means B ∈ ray + ( AA ), we consider one of the trianglesfrom Lemma 2.2 A −→ A g −→ A (cid:48)(cid:48) −→ Σ A. Using the Hom-hammock Proposition 2.4, we see that Hom • ( A (cid:48)(cid:48) , B ) = 0 if B ∈ ray + ( A )and Hom • ( A, B ) = 0 otherwise. Thus the exact sequenceHom(Σ( A ) , B ) −→ Hom( A (cid:48)(cid:48) , B ) −→ Hom(
A, B ) −→ Hom( A, B ) −→ Hom(Σ − A (cid:48)(cid:48) , B )yields hom( A, B ) ≤ hom( A (cid:48)(cid:48) , B ) if B ∈ ray + ( A ) and hom( A, B ) ≤ hom( A, B ) otherwise.The induction hypothesis then gives hom(
A, B ) ≤ B ∈ X k +1 follows from the above by Serre duality.Furthermore, the above argument applies without change to B ∈ Z k — with ray + ( A ) ⊂Z k understood to mean the subset of indecomposables of Z k admitting non-zero mor-phisms from A (these form a ray in Z k ) and similarly ray − ( B ) ⊂ X k , and applicationof Proposition 2.6. An obvious modification, which we leave to the reader, extends theargument to B ∈ Z k +1 . The statements for A ∈ Y are completely analogous. Case A ∈ Z k : In light of Serre duality, we don’t need to deal with B ∈ X or B ∈ Y .Therefore we turn to B ∈ Z . However, we already know from the proof of Propo-sition 2.6 that the dimensions in the two non-vanishing regions ray + ( coray + ( A )) and ray − ( coray − ( S A )) are constant. Since the Z components contain the simple S (0) and he twist functors together with the suspension act transitively on Z , it is clear thathom( A, A ) = hom( A, S A ) ∗ = 1. This completes the proof. (cid:3) Proposition 5.2.
Let r = 1 and X, A ∈ ind ( X ) . Then hom( X, A ) = 2 ⇐⇒ A ∈ ray + ( XX ) ∩ coray − ( ( S X ) , S X ) . The following diagram illustrates the proposition: all indecomposables A in the heavilyshaded square have dim Hom( X, A ) = 2: X S XX ( S X ) coray − ( ( S X ) , S X ) ray + ( XX ) Proof.
The argument is similar to the computation of the Hom-hammocks in the Z components from Section 2. We proceed in several steps. Step 1:
For any A ∈ ind ( X ) of height 0 the claim follows from Lemma 2.1. Other-wise we consider the AR mesh which has A at the top, and let A (cid:48) and A (cid:48)(cid:48) be the twoindecomposibles of height h ( A ) −
1. There are two triangles (see Lemma 2.2): A −→ A −→ A (cid:48)(cid:48) −→ Σ( A ) = Σ A, (ray) A (cid:48) −→ A −→ A −→ Σ A (cid:48) , (coray)where, as before, A and A are the unique indecomposable objects on the mouth whichare contained in respectively ray − ( A ) and coray + ( A ). Applying the functor Hom( X, − ) toboth triangles we obtain two exact sequences:Hom( X, A ) −→ Hom(
X, A ) ϕ −→ Hom(
X, A (cid:48)(cid:48) ) ψ −→ Hom( X, Σ A ) , (3) Hom( X, Σ − A ) µ −→ Hom(
X, A (cid:48) ) −→ Hom(
X, A ) δ −→ Hom(
X, A ) . (4)Since A and A lie on the mouth of the component, Lemma 2.1 implies that the outerterms have dimension at most 2. Using the fact that X and S X are the only objects ofthe Hom-hammock from X lying on the mouth, Lemma 2.1 actually yields:hom( X, A ) > ⇐⇒ A ∈ ray + ( X ) ∪ ray + ( S X ) , hom( X, Σ A ) > ⇐⇒ A ∈ ray + (Σ − X ) ∪ ray + (Σ − S X ) , hom( X, Σ − A ) > ⇐⇒ A ∈ coray − (Σ X ) ∪ coray − (Σ S X ) , hom( X, A ) > ⇐⇒ A ∈ coray − ( X ) ∪ coray − ( S X ) . The spaces are 2-dimensional precisely when A belongs to the intersections of the (co)rayson the right-hand side, which can only happen when S X = X . The set of rays andcorays listed above divide the component into regions. In this proof, each region isconsidered to be closed below and open above. tep 2: The function hom( X, − ) is constant on each region, and changes by at most 1when crossing a (co)ray if S X (cid:54) = X , and by at most 2 otherwise. The first claim is clear from exact sequences (3) and (4). We show the second claim forrays; for corays the argument is similar. We get hom(
X, A ) ≤ hom( X, A (cid:48)(cid:48) ) + hom( X, A )from sequence (3). This yields the stated upper bound for hom( X, A ), as hom( X, A ) ≤ S X (cid:54) = X and hom( X, A ) ≤ X, A (cid:48)(cid:48) ) ≤ hom( X, Σ A ) + hom( X, A ), again from sequence (3).
Step 3: ψ = 0 unless A ∈ ray + (Σ − S X ) and µ = 0 unless A ∈ coray − (Σ X )If A / ∈ ray + (Σ − X ) ∪ ray + (Σ − S X ) then hom( X, Σ A ) = 0 and so ψ = 0 trivially.Therefore, we just need to consider A ∈ ray + (Σ − X ) but A / ∈ ray + (Σ − S X ), and inthis case hom( X, Σ A ) = 1. It is clear that the maps going down the coray from X to X span a 1-dimensional subspace of Hom( X, Σ A ), which therefore is the wholespace. Using properties of the Z A ∞ mesh, the composition of such maps with a mapalong ray + ( X ) from X to Σ A defines a non-zero element in Hom( X, Σ A ). Thus themap Hom( X, Σ A ) → Hom( X, Σ A ) in the sequence (3) is injective and it follows that ψ = 0. The proof of the second statement is similar: here we use the chain of morphismsin Properties 1.2(5) to show that the map Hom( X, Σ − A ) → Hom( X, Σ − A ) in thesequence (4) is surjective. Step 4: If ray + (Σ − X ) (or coray − (Σ S X ) , respectively) does not coincide with one of theother three (co)rays, then crossing it does not affect the value of hom( X, − ) . Suppose ray + (Σ − X ) (cid:51) A doesn’t coincide with ray + ( X ), ray + ( S X ) or ray + (Σ − S X ).Thus hom( X, A ) = 0, and from Step 3 the map ψ = 0, hence Hom( X, A ) = Hom(
X, A (cid:48)(cid:48) ).Similarly, suppose A ∈ coray − (Σ S X ) and this doesn’t coincide with any of the othercorays. Then hom( X, A ) = 0 and µ = 0 and again the claim follows. Step 5: There are three possible configurations of rays and corays determining the regionswhere hom( X, − ) is constant. It follows from Step 4 that it suffices to consider the remaining rays and corays, ray + (Σ − S X ) , ray + (Σ S X ) , ray + ( X ) and coray − (Σ X ) , coray − (Σ S X ) , coray − ( X ) , for determining the regional constants hom( X, − ). Note that these are precisely therays and corays required to bound the regions ray + ( XX ) and coray − ( ( S X ) , S X ) of thestatement of the proposition. Considering their relative positions on the mouth, Σ − S X is always furthest to the left and Σ X is furthest to the right, while S X can lie to theleft, or to the right, or coincide with X , depending on the height of X . We consider nowthe case where S X is to the left of X . We label the regions in the following diagramby letters A–M (this is the order in which we treat them, and the subscripts indicate theclaimed hom( X, − ) for the region): A B C D E F G H I J K L M Σ − S X S X X Σ X irst we note that regions A–E all contain part of the mouth and so hom( X, − ) = 0 here.Looking at the maps from X that exist in the AR component we see that hom( X, − ) ≥ X, − ) = 1 on regions F–I.Now look at the element A ∈ ray + ( S X ) ∩ coray − ( X ); this is the object of minimalheight in region K. We can see that A ∈ coray + ( X ) and the map down the coray from X to A , factors through the map from A to A . Therefore the map δ in the second exactsequence (4) is non-zero. It is clear that A / ∈ coray − (Σ X ) so µ = 0 by Step 3 above. Wededuce from sequence (4) that hom( X, A ) > hom( X, A (cid:48) ), so hom(
X, A ) > A (cid:48) isin region G. Since A is an object in region K, which can be reached from region D bycrossing just two rays, Step 2 now gives hom( X, − ) = 2 on region K.In the same vein, consider A ∈ ray + ( S X ) ∩ coray − (Σ X ), the object of minimal heightin region L. Observe that A (cid:48)(cid:48) ∈ ray + ( τ − S X ) ∩ coray − (Σ X ) = add Σ X from which wecan see that the map to Hom( X, A ) in (3) is surjective. Now A / ∈ ray + (Σ − S X ), so ψ = 0 by Step 3 and hence hom( X, A ) = hom(
X, A (cid:48)(cid:48) ). With A (cid:48)(cid:48) in region I where wealready know hom( X, A (cid:48)(cid:48) ) = 1, we get hom( X, − ) = 1 on region L.Finally we now take up A ∈ ray + (Σ − S X ) ∩ coray − (Σ X ), the object of minimalheight in region M. It is clear that A / ∈ ray + ( S X ) ∪ ray + ( X ), so hom( X, A ) = 0.A short calculation shows A (cid:48)(cid:48) ∈ ray + ( X ), and again using the chain of morphisms inProperties 1.2(5), we see that there is a map X → Σ A = S X factoring through A (cid:48)(cid:48) .Looking at the sequence (3) it follows that hom( X, A ) < hom( X, A (cid:48)(cid:48) ) = 1 since A (cid:48)(cid:48) is inregion L. Therefore, hom( X, − ) = 0 on region M. For region J, we see that since it issandwiched between regions K and M, hom( X, − ) = 1 here.This deals with the case that S X lies to the left of X . If instead it lies to the right,analogous reasoning applies. Finally, if S X = X , matters are simpler: in that case, theregions C and F–I all vanish. (cid:3) Graded endomorphism algebras.
In this section we use the Hom-hammocks anduniversal hom space dimension bounds to recover some results of Bobi´nski on the gradedendomorphism algebras of algebras with discrete derived categories; see [8, Section 4].Our approach is somewhat different, so we provide proofs for the convenience of thereader. Using these descriptions, we give a coarse classification of indecomposable objectsof discrete derived categories in terms of their homological properties.In order to conveniently write down the endomorphism complexes, we define four func-tions δ + X , δ −X , δ + Y , δ −Y : N → N by δ + X ( h ) := (cid:22) hm + r (cid:23) , δ −X ( h ) := (cid:22) h + 1 m + r (cid:23) , δ + Y ( h ) := (cid:22) h + 1 n − r (cid:23) , δ −Y ( h ) := (cid:22) hn − r (cid:23) . We write δ ± ( A ) to mean δ ±X ( h ( A )) or δ ±Y ( h ( A )) for A ∈ ind ( X ) or A ∈ ind ( Y ), respectively. Lemma 5.3.
The endomorphism complexes of A ∈ ind ( X ) and B ∈ ind ( Y ) are Hom • ( A, A ) = δ + ( A ) (cid:77) l =0 Σ − lr k ⊕ δ − ( A ) (cid:77) l =1 Σ lr − k and Hom • ( B, B ) = δ − ( B ) (cid:77) l =0 Σ lr k ⊕ δ + ( B ) (cid:77) l =1 Σ − lr − k . In words, the functions δ + and δ − determine the ranges of self-extensions of positive andnegative degree, respectively. We point out that the result holds for all r ≥ roof. Let A ∈ ind ( X ), assuming r >
1. Suspending if necessary, we may suppose that A = X ij . We are looking for all d ∈ Z with Hom d ( A, A ) = Hom( A, Σ d A ) (cid:54) = 0. ByProposition 2.4, this is only possible for either d ≡ d ≡ r .We start with the first possibility: d = lr for some l ∈ Z . By Properties 1.2(3) and (2),Σ lr A = τ − l ( m + r ) A = X i + l ( m + r ) ,j + l ( m + r ) which is an indecomposable object in X sharing its height h = j − i with A . Again usingProposition 2.4, we can reformulate the claim as follows:Hom lr ( A, A ) (cid:54) = 0 ⇐⇒ Σ lr A ∈ ray + ( AA ) ⇐⇒ Σ lr A = X i + l ( m + r ) ,j + l ( m + r ) ∈ { A, τ − A, . . . , τ − h A } = { X ij , X i +1 ,j +1 , . . . , X i + h,j + h }⇐⇒ i ≤ i + l ( m + r ) ≤ i + h ⇐⇒ ≤ l ( m + r ) ≤ h ⇐⇒ ≤ l ≤ δ + X ( h ) = δ + ( A ) , where the set of h + 1 objects in the second line are precisely the objects in ray + ( AA ) ofheight h . We now turn to the other possibility, d = 1 + lr for some l ∈ Z . Here we getHom lr ( A, A ) (cid:54) = 0 ⇐⇒ Σ lr A ∈ ray + ( S A, S A ) ⇐⇒ Σ lr A = X i + l ( m + r ) ,j + l ( m + r ) ∈ { τ h S A, . . . , S A } = { X i − h − ,j − h − , . . . , X i − ,j − }⇐⇒ i − h − ≤ i + l ( m + r ) ≤ i − ⇐⇒ − h − ≤ l ( m + r ) ≤ − ⇐⇒ ≤ − l ≤ δ −X ( h ) = δ − ( A ) . As we know from Theorem 5.1, all Hom spaces have dimension 1 when r >
1, these twocomputations giveHom • ( A, A ) = (cid:77) l ∈ Z Σ − l Hom( A, Σ l A ) = δ + ( A ) (cid:77) l =0 Σ − lr k ⊕ δ − ( A ) (cid:77) l =1 Σ lr − k . For r = 1 and A = X ij ∈ ind ( X ), by Proposition 2.4 the hammock Hom( A, − ) (cid:54) = 0 is ray + ( AA ) ∪ coray − ( ( S A ) , S A ). We treat each part separately:Σ l A = τ − l ( m +1) X ij = X i + l ( m +1) ,j + l ( m +1) ∈ ray + ( AA ) ⇐⇒ ≤ l ( m + 1) ≤ h ⇐⇒ ≤ l ≤ δ + ( h )and, noting S A = X i + m,j + m ,Σ l A ∈ coray − ( ( S A ) , S A ) ⇐⇒ m − h ≤ l ( m + 1) ≤ m ⇐⇒ ≤ − l ≤ (cid:106) h − mm + 1 (cid:107) = 1 + δ − ( h ) . The last inequality translates to the same degree range as in the statement of the lemma —note the index shift by 1. The claim for Hom • ( B, B ) for B ∈ ind ( Y ) is proved in the sameway, now using h = i − j , Σ r = τ n − r and the hammocks specified by Proposition 2.5. (cid:3) Coarse classification of objects.
Our previous results allow us to give a crudegrouping of the indecomposable objects of D b (Λ( r, n, m )). In the X and Y components,the distinction depends on the height of an object, i.e. the distance from the mouth;see page 5. Recall that an object D of a k -linear Hom-finite triangulated category D is exceptional if hom ∗ ( D, D ) = 1, then Hom • ( D, D ) = k ; see Appendix A.7, and D is called spherelike if hom ∗ ( D, D ) = 2, then Hom • ( D, D ) = k ⊕ Σ − d k as graded vector spaces forsome d ∈ Z and D is called d -spherelike; see [22] for details. Assuming D has a Serrefunctor S , a d -spherelike object D is called d -spherical if S ( D ) = Σ d D ; see [27, § roposition 5.4. Each object A ∈ ind ( D b (Λ( r, n, m ))) is of exactly one type below: • Exceptional if A ∈ Z , or A ∈ X with h ( A ) < m + r − , or A ∈ Y with h ( A ) < n − r − . • (1 − r ) -spherelike if A ∈ X with h ( A ) = m + r − . • (1 + r ) -spherelike if A ∈ Y with h ( A ) = n − r − . • dim Hom ∗ ( A, A ) ≥ with Hom < ( A, A ) (cid:54) = 0 otherwise. Remark 5.5.
In fact, the direct sum E ⊕ E of two exceptional objects E and E withHom • ( E , E ) = Hom • ( E , E ) = 0 is a 0-spherelike object. Examples for r > E ∈ X and E ∈ Y at the mouths. The theory of spherelike objects alsoapplies in this degenerate case, but is less interesting [22, Appendix]. Remark 5.6.
We can infer the existence of (1 − r )-spherelike indecomposable objectsin X and (1 + r )-spherelike objects in Y also from Proposition 3.7 and Lemma 4.1.To any reasonable k -linear triangulated category, [23] associates a poset derived fromindecomposable spherelike objects. In [23, § Proof.
We know from Lemma B.9 that the projective module P ( n − r ) ∈ Z . This is anexceptional object by Proposition 2.6. As the autoequivalence group acts transitively on ind ( Z ) by Corollary 4.5, every indecomposable object of Z is exceptional. The remainingparts of the proposition all follow from Lemma 5.3. We only give the argument for A ∈ ind ( X ), as the one for indecomposable objects of Y runs entirely parallel.Observing the trivial inequalities 0 ≤ δ + ( A ) ≤ δ − ( A ), we see that A is exceptional ifand only if 1 = dim Hom ∗ ( A, A ) = 1 + δ + ( A ) + δ − ( A ). In turn, this happens precisely if δ − ( A ) = 0, which means h < m + r − A is spherelike if and only if 2 = dim Hom ∗ ( A, A ) = 1 + δ + ( A ) + δ − ( A )which is equivalent to δ + ( A ) = 0 and δ − ( A ) = 1. The only solution of these equations is h = m + r −
1. Furthermore, in this case the endomorphism complex is Hom • ( A, A ) = k ⊕ Σ r − k , so that A is indeed (1 − r )-spherelike. (cid:3) Corollary 5.7.
Spherical objects exist in D b (Λ( r, n, m )) only if m = 0 , r = 1 or n − r = 1 .More precisely, A ∈ ind ( D b (Λ( r, n, m ))) is • m = 0 , r = 1 and A sits at an X -mouth; • n -spherical if and only if n = r + 1 and A sits at a Y -mouth.Proof. The only candidates for spherical objects are the spherelike objects listed in Propo-sition 5.4. Start with A ∈ X with h ( A ) = m + r −
1. Then A is spherical if and onlyif S A = Σ − r A . By S = Σ τ and Σ − r = Σ τ m + r (Properties 1.2(3)), this is equivalent to τ m + r − A = A which happens precisely if m + r = 1. The only solution for this equationis m = 0, r = 1.Next, B ∈ Y with h ( B ) = n − r − τ B = S B = Σ r B =Σ τ n − r B , so that here we get τ n − r − B = B which is possibly only for n = r + 1. (cid:3) Reduction to Dynkin type A and classification results Two keys for understanding the homological properties of algebras are t-structures andco-t-structures, especially bounded ones. The main theorem of [32], cited in the appendixas Theorem A.8, states that for finite-dimensional algebras, bounded co-t-structures arein bijection with silting objects, which are in turn in bijection with bounded t-structureswhose heart is a length category; see Appendices A.5 and A.6 for a more detailed overview.It turns out, however, that any bounded t-structure in D b (Λ( r, n, m )) has length heart,and hence to classify both bounded t-structures and bounded co-t-structures it is sufficient o classify silting objects in D b (Λ( r, n, m )). This is the main goal of this section. In thefirst part, we prove that any bounded t-structure in D b (Λ( r, n, m )) is length, then weobtain a semi-orthogonal decompositon D b (Λ( r, n, m )) = (cid:104) D b ( k A n + m − ) , Z (cid:105) , for sometrivial thick subcategory Z , and use this to bootstrap Keller–Vossieck’s classification ofsilting objects in the bounded derived categories of path algebras of Dynkin type A toget a classification of silting objects in discrete derived categories.6.1. All hearts in D b (Λ( r, n, m )) are length. The main result of this section is:
Proposition 6.1.
Any heart of a t-structure of a discrete derived category has only afinite number of indecomposable objects up to isomorphism, and is a length category.
We prove these statements separately in the following lemmas. The first lemma is ageneral statement regarding t-structures, which is well known to experts, and includedfor the convenience of the reader. The second is a generalisation of the correspondingstatement for the algebra Λ(1 , ,
0) proved in [32]; the third is a general statement aboutHom-finite abelian categories.
Lemma 6.2 (cf. [25, Lemma 4.1]) . Let D be a triangulated category equipped with a t-structure ( X , Y ) with heart H = X ∩ Σ Y . Then at most one suspension of any object of D may lie in the heart H .Proof. Let 0 (cid:54) = H ∈ H . We show that Σ n H / ∈ H for any n (cid:54) = 0. First suppose thatΣ n H ∈ H for some n >
0. Then H ∈ Σ − n H . We have Σ − n H ⊆ Σ − n Σ Y ⊆ Y . Thecondition Hom( X , Y ) = 0 then implies that Hom( H, H ) = 0, a contradiction.Now suppose Σ − n H ∈ H for some n >
0. In this case we have Σ n H ⊆ Σ n X ⊆ Σ X ,whence the condition Hom(Σ X , Σ Y ) = 0 gives the required contradiction. (cid:3) Lemma 6.3.
Any heart of a t-structure of a discrete derived category has a finite numberof indecomposable objects up to isomorphism.Proof.
We use the fact that there can be no negative extensions between objects in theheart H of a t-structure ( X , Y ). Suppose H contains an indecomposable Z ∈ ind ( Z ). Thenany other indecomposable object in H must lie outside the hammocks Hom < ( Z, − ) (cid:54) = 0and Hom < ( − , Z ) (cid:54) = 0. Looking at the complement of these Hom-hammocks, it is clearthat all objects of ind ( H ) ∩Z must be (co)suspensions of a finite set of objects, see Figure 4for an illustration. Lemma 6.2 implies that at most one suspension can sit in the heart H ; hence ind ( H ) ∩ Z is finite.Now consider the X component. By Proposition 5.4, any object X li,j which is suffi-ciently high up in an X component — here j − i ≥ r + m − ind ( H ) ∩ X is finite. The argument for the Y componentis similar. (cid:3) Lemma 6.4.
Let H be a Hom-finite abelian category with finitely many indecomposableobjects. Then H is a finite length category.Proof. Since H is a Hom-finite, k -linear abelian category, it is Krull–Schmidt; see [6].Now let L be the direct sum of all indecomposable objects (up to isomorphism) of H . Byassumption, this sum is finite and hence L ∈ H . We define the function d : Ob( H ) → N , A (cid:55)→ dim Hom( L, A ).If A ⊂ B is a subobject, we obtain exact sequences 0 → A → B → C → → Hom(
L, A ) → Hom(
L, B ) → Hom(
L, C ). This shows d ( A ) ≤ d ( B ). Moreover, if d ( A ) = d ( B ), then the induced map Hom( L, B ) → Hom(
L, C ) is zero. For some s ∈ N , here is a surjection p : L ⊕ s (cid:16) B , inducing a further surjection q : L ⊕ s (cid:16) C . However,we also get 0 → Hom( L ⊕ s , A ) → Hom( L ⊕ s , B ) v −→ Hom( L ⊕ s , C ). The dimensions of thefirst two Hom spaces are sd ( A ) = sd ( B ), so that v = 0. Since v ( p ) = q by construction,this forces C = 0.Hence for B ∈ H , the function d can only take the values 1 , . . . , d ( B ) − B must stabilise. (cid:3) Remark 6.5.
Proposition 6.1 means that the heart of each bounded t-structure in D b (Λ( r, n, m )) is equivalent to mod (Γ), for a finite-dimensional algebra Γ of finite repre-sentation type. Note that, by work of Schr¨oer and Zimmermann [41], Γ is again gentle.Knowing this, we can now turn our attention solely to classifying the silting objects.The first step in our approach is to decompose D b (Λ( r, n, m )) into a semi-orthogonaldecomposition, one of whose orthogonal subcategories is the bounded derived categoryof a path algebra of Dynkin type A .6.2. A semi-orthogonal decomposition: reduction to Dynkin type A . We startby showing that the derived categories of derived-discrete algebras always arise as exten-sions of derived categories of path algebras of type A by a single exceptional object. Proposition 6.6.
Let Z ∈ ind ( Z ) and Z = thick D b (Λ) ( Z ) . Then Z ⊥ (cid:39) D b ( k A n + m − ) and there is a semi-orthogonal decomposition D b (Λ( r, n, m )) = (cid:104) D b ( k A n + m − ) , Z (cid:105) . Inparticular, Z is functorially finite in D b (Λ( r, n, m )) . Moreover, D b (Λ( r, n, m )) has a fullexceptional sequence.Proof. By Proposition 5.4, the object Z is exceptional. This implies, on general grounds,that the thick hull of Z just consists of sums, summands and (co)suspensions: Z = add (Σ i Z | i ∈ Z ) and that Z is an admissible subcategory of D b (Λ); for this last claimsee [11, Theorem 3.2]. Furthermore D b (Λ) = (cid:104) Z ⊥ , Z (cid:105) is the standard semi-orthogonaldecomposition for an exceptional object; see Appendix A.7 for details.Lemma B.9 places the indecomposable projective P ( n − r ) in the Z component ofthe AR quiver of D b (Λ). Using the transitive action of the autoequivalence group on ind ( Z ), see Corollary 4.5, we thus can assume, without loss of generality, that Z = P ( n − r ) = e n − r Λ. There is a full embedding ι : D b (Λ / Λ e n − r Λ) → D b (Λ) with essentialimage thick D b (Λ) ( e n − r Λ) ⊥ = Z ⊥ ; see, for example, [2, Lemma 3.4]. Inspecting the Gabrielquiver of Λ / Λ e n − r Λ, we see that this quiver satisfies the criteria of [5, Theorem, p. 2122].For the convenience of the reader, we list those criteria which are relevant for our case,where we have specialised the conditions of [5] to bound quivers: ( α ) The underlying graph is a tree.( α ) All relations are zero-relations of length two.( α ) Each vertex has at most four neighbours.( α ) A vertex with three neighbourssits in a full subgraph of the form: Therefore Λ / Λ e n − r Λ is an iterated tilted algebra of type A n + m − . It is well knownthat this implies D b (Λ / Λ e n − r Λ) (cid:39) D b ( k A n + m − ); see [21]. Combining these pieces, weget Z ⊥ (cid:39) D b ( k A n + m − ). The final claim about D b (Λ) having a full exceptional sequencefollows at once from the fact that D b ( k A n + m − ) has one. (cid:3) Remark 6.7.
The subcategory of type D b ( k A n + m − ) can be explicitly identified in theAR quiver of D b (Λ( r, n, m )); see Figure 4. The choice of right orthogonal to Z wasarbitrary, since Serre duality provides an equivalence ⊥ Z → Z ⊥ , X (cid:55)→ S ( X ). We mentionin passing that the thick subcategory Z is equivalent to D b ( k A ). Y Z Z Z Y X Z Y X Z Y X Z Y X Z Y X Z Z Y X Figure 4.
Above: D b ( k A ) ∼ = thick ( Z ) ⊥ (cid:44) → D b (Λ(2 , , X = Σ X , Y = Σ Y , Z = Σ Z not shown.Below: AR quiver of D b ( k A ) with its D b (Λ(2 , , D b ( k A n + m − ) are well understood from work of Keller andVossieck in [31]. We shall now bootstrap their classification to discrete derived cate-gories using the technique of silting reduction Aihara and Iyama in [1].6.3. Silting reduction.
The main technical tool in the classification is the followingresult of Aihara and Iyama in [1]:
Theorem 6.8 (Silting reduction [1, Theorem 2.37]) . Let D be a Krull–Schmidt triangu-lated category, U ⊂ D a thick, contravariantly finite subcategory and F : D → D / U thecanonical functor. Then for any silting subcategory N of U , there is an injective map { silting subcategories M of D | N ⊆ M } (cid:44) → { silting subcategories of D / U } , M (cid:55)→ F ( M ) . If U is functorially finite in D , then the map is bijective. We are working towards an explicit description of the inverse map G in Proposition 6.15.The subcategory B := susp Σ N is the ‘co-aisle’ of the co-t-structure associated to N (seeTheorem A.8) and thus covariantly finite in U . Putting this together with U beingfunctorially finite in D , it gives rise to a co-t-structure ( A , B ) in D , where A := ⊥ B . Nowlet K be a silting subcategory of U ⊥ and consider the approximation triangle of K ∈ K with respect to the co-t-structure ( A , B ), A K → K → B K → Σ A K with A K ∈ A and B K ∈ B . In their proof of Theorem 6.8 in [1], Aihara and Iyama showthat G ( K ) := add ( N ∪ { A K | K ∈ K } ) is a silting subcategory of D . Definition 6.9.
Assume the notation and hypotheses of Theorem 6.8 above. Given asilting subcategory N of U , by abuse of notation we write G N for the map G N : U ⊥ → D ,which for V ∈ U ⊥ , is defined by G N ( V ) −→ V f V −→ B V −→ Σ G N ( V ) , where f V : V → B V is a minimal left B -approximation of V . Note that here, in contrastto elsewhere in this paper, we require that the approximation is minimal to ensure well-definedness of the map G N . Furthermore, we stress here that G N is a map not a functor. n light of Proposition 6.6, the natural choice for a functorially finite thick subcategoryto which we can apply Theorem 6.8 is Z for some Z in the Z components. For siltingreduction to work, we first need to establish that any silting subcategory of D b (Λ( r, n, m ))contains an indecomposable object from the Z components. The following lemma is asmall generalisation of the statement we need, which we specialise in the subsequentcorollary. Simple-minded collections (see [32] for the definition) are also an importantfocus of current research. Therefore, while we do not use them in this paper, it is usefulto highlight in the corollary below that the following lemma also applies to them. Lemma 6.10. If M is a subcategory of D b (Λ) such that thick ( M ) = D b (Λ) , then M contains an indecomposable object from the Z components. Corollary 6.11.
Any silting subcategory of D b (Λ) and any simple-minded collection in D b (Λ) contain objects from some Z component.Proof of lemma. By Lemma 4.6, the additive closure of the X components of D b (Λ) is athick subcategory of D b (Λ), and likewise for the additive closure of the Y components.Furthermore, these two subcategories are fully orthogonal by Propositions 2.4 and 2.5,so that their sum is a thick subcategory of D b (Λ) as well. Therefore we cannot have M ⊂ X ⊕ Y as that would force D b (Λ) = thick ( M ) = X ⊕ Y , a contradiction. (cid:3)
Theorem 6.8 coupled with Proposition 6.6 tells us that all silting objects in D b (Λ)containing Z can be obtained by lifting silting objects in Z ⊥ (cid:39) D b ( k A n + m − ) back up to D b (Λ). In other words, any silting object in D b (Λ) can be described by a pair ( Z, M (cid:48) ) con-sisting of an indecomposable object Z ∈ Z and a silting object M (cid:48) ∈ Z ⊥ (cid:39) D b ( k A n + m − ).We now make a brief expository digression explaining Keller and Vossieck’s clas-sification of silting subcategories of D b ( k A t ), from which the silting subcategories of D b (Λ( r, n, m )) can be ‘glued’.6.4. Classification of silting objects in Dynkin type A . Consider the following di-agram of the AR quiver of D b ( k A t ) with coordinates ( g, h ) with g ∈ Z and h ∈ { , . . . , t } . · · · (cid:31) (cid:31) ( − , (cid:31) (cid:31) (0 , (cid:31) (cid:31) (1 , (cid:31) (cid:31) (2 , (cid:31) (cid:31) · · · ( − , (cid:31) (cid:31) (cid:63) (cid:63) (0 , (cid:31) (cid:31) (cid:63) (cid:63) (1 , (cid:31) (cid:31) (cid:63) (cid:63) (2 , (cid:31) (cid:31) (cid:63) (cid:63) (3 , (cid:31) (cid:31) (cid:63) (cid:63) · · · (cid:63) (cid:63) (0 , (cid:63) (cid:63) (1 , (cid:63) (cid:63) (2 , (cid:63) (cid:63) (3 , (cid:63) (cid:63) · · · Given an indecomposable object U ∈ D b ( k A t ) we write its coordinates as ( g ( U ) , h ( U )).Following [31], a quiver Q = ( Q , Q ) is called an A t -quiver if | Q | = t , its underlyinggraph is a tree, and Q decomposes into a disjoint union Q = Q α ∪ Q β such that atany vertex at most one arrow from Q α ends, at most one arrow from Q α starts, at mostone arrow from Q β ends and at most one arrow from Q β starts. One should think of an A t -quiver as a ‘gentle tree quiver’, where gentle is used in the sense of gentle algebras.We define maps s α , e α , s β , e β : Q → N by s α ( x ) := { y ∈ Q | the shortest walk from x to y starts with an arrow in Q α } ; e α ( x ) := { y ∈ Q | the shortest walk from y to x ends with an arrow in Q α } . The functions s β and e β are defined analogously. With these maps, there is precisely onemap ϕ Q := ( g Q ( x ) , h Q ( x )) : Q → ( Z A t ) , where g Q and h Q correspond to the coordinatesin the AR quiver of D b ( k A t ), such that h Q ( x ) = 1 + e α ( x ) + s β ( x ) and g Q ( y ) = g Q ( x ) foreach arrow x −→ y in Q α , and g Q ( y ) = g Q ( x ) + e α ( x ) + s α ( x ) + 1 for each arrow x −→ y in Q β , and finally normalised by min x ∈ Q { g Q ( x ) } = 0. y abuse of notation we identify the object T Q := ϕ Q ( Q ) with the direct sum ofthe indecomposables lying at the corresponding coordinates. This map gives rise to thefollowing classification result. Theorem 6.12 ([31], Section 4) . The assignment Q (cid:55)→ T Q induces a bijection betweenisomorphism classes of A t -quivers and tilting objects T in D b ( k A t ) satisfying the condition min { g ( U ) | U is an indecomposable summand of T } = 0 . Note that in Dynkin type A t , the summands of any tilting object T = (cid:76) ti =1 T i can bere-ordered to give a strong, full exceptional sequence { T , · · · , T t } , see [31, Section 5.2].We now have the following classification of silting objects in D b ( k A t ). Theorem 6.13 ([31], Theorem 5.3) . Let T = T ⊕ · · · ⊕ T t be a tilting object in D b ( k A t ) whose summands form an exceptional collection. Let p : { , . . . , t } → N be a weaklyincreasing function. Then Σ p (1) T ⊕ · · · ⊕ Σ p ( t ) T t is a silting object in D b ( k A t ) . Moreover,all silting objects of D b ( k A t ) occur in this way. The machinery above is slightly technical, so we give a quick example of the classifica-tion of tilting (and hence silting) objects in D b ( k A ). Example 6.14 (Classification of tilting objects in D b ( k A )) . When t = 3, up to isomor-phism there are the following possible A -quivers:1 α −→ α −→ , α −→ β −→ , α −→ β ←− , α ←− β −→ , β −→ β −→ , β −→ α −→ . Computing the ϕ Q for each of the above quivers gives the following, where each 3-tupledenotes ( ϕ Q (1) , ϕ Q (2) , ϕ Q (3)):((0 , , (0 , , (0 , , ((0 , , (0 , , (2 , , ((1 , , (1 , , (0 , , ((0 , , (0 , , (1 , , ((0 , , (1 , , (2 , , ((0 , , (2 , , (2 , . We indicate the corresponding tilting objects in the following sketch:
In each sketch the triangle depicts the standard heart for the quiver 1 ←− ←− , , (0 , , (0 , U with minimal g ( U ) = 0. In partic-ular, these are precisely the exceptional sequences in D b ( k A ) containing one of P ( i ) for1 ≤ i ≤ U with minimal g ( U ) = 1. These correspondprecisely to τ − applied to each of the diagrams to . Observe that τ − = Σ and τ − = Σ . Therefore, up to suspension, we pick up only four more tilting objects. Nextwe consider those for which there exists an indecomposable summand U with minimal g ( U ) = 2, which correspond precisely to τ − applied to each of the diagrams to . Wehave τ − = Σ , τ − = Σ , τ − = Σ τ − , and τ − = Σ τ − , which leaves, upto suspension, only τ − and τ − as new tilting objects. Continuing in this way, one ees that, up to suspension, these are all tilting objects. Hence, there are twelve tiltingobjects in D b ( k A ) up to suspension: = P (1) ⊕ P (2) ⊕ P (3) , = P (1) ⊕ P (3) ⊕ S (3) , = P (3) ⊕ I (2) ⊕ S (2) , = P (2) ⊕ P (3) ⊕ S (2) , = P (3) ⊕ I (2) ⊕ S (3) , = P (3) ⊕ S (3) ⊕ Σ S (2) ,τ − = S (2) ⊕ I (2) ⊕ Σ P (1) , τ − = S (2) ⊕ Σ P (1) ⊕ Σ P (3) , τ − = Σ P (1) ⊕ Σ P (2) ⊕ S (3) ,τ − = I (2) ⊕ Σ P (1) ⊕ S (3) , τ − = S (3) ⊕ Σ P (2) ⊕ Σ S (2) , τ − = S (3) ⊕ Σ S (2) ⊕ Σ P (1) . Classification of silting objects for derived-discrete algebras.
As this sectionis rather technical, the reader may find it helpful to refer to the detailed example, Λ(2 , , G Z : Z ⊥ → D b (Λ( r, n, m )) from Defini-tion 6.9, where Z = thick ( Z ) for some fixed, arbitrary, indecomposable object Z ∈ ind ( Z ).We first explicitly compute the map G Z : ind ( Z ⊥ ) → D b (Λ( r, n, m )) on objects in thecase Z = Z , . Proposition 6.15. If r > and Z = Z , , and G := G Z , , then G ( U ) = U for all butfinitely many (up to positive suspension) U ∈ ind ( Z ⊥ ) . The exceptions are:(1) G (Σ i X ,j ) = Σ i Z j +1 , for ≤ j < r + m − and i ≥ .(2) G (Σ i Y j, ) = Σ i Z ,j +1 for ≤ j < n − r − and i ≥ .(3) G (Σ i Z j, ) = Σ i X j, − for ≤ j ≤ r + m − and i ≥ .(4) G (Σ i Z ,j ) = Σ i Y − ,j for r − n + 1 ≤ j ≤ − and i ≥ .(5) G (Σ i Z − r − m, ) = (cid:26) Σ i X − r − m, − for ≤ i ≤ r, Σ i Z ,n − r for i > r. Proposition 6.16. If r = 1 and Z = Z , , and G := G Z , , then G ( U ) = U for all butfinitely many (up to positive suspension) U ∈ ind ( Z ⊥ ) . The exceptions are:(1) G (Σ i X m, m + j ) = Σ i Z j +1 , for ≤ j < m and i ≥ .(2) G (Σ i Y − n + j, − n ) = Σ i Z ,j +1 for ≤ j < n − and i ≥ .(3) G (Σ i Z j, − n ) = Σ i X j,m for < j < m + 1 and i ≥ .(4) G (Σ i Z m +1 ,j ) = Σ i Y − n,j for − n < j < − n and i ≥ .(5) G (Σ i Z , − n ) = (cid:26) X ,m for i = 0 , Σ i Z ,n − for i > . Proof of Propositions 6.15 and 6.16.
We do the calculations for the generic case with r > G is definedvia the ‘co-aisle’ of the co-t-structure ( A , B ) with B = susp Σ Z = add { Σ i Z | i ≥ } .Using Proposition 2.6, one can easily compute A = ⊥ B . If U ∈ A , then G ( U ) = U , soexamining A ∩ Z ⊥ gives the list of exceptions above.We now compute the cocones G ( U ) directly using the triangles from Properties 1.2(4):(1) The relevant triangles here are Z j +1 , → X ,j → Z , → Σ Z j +1 , for 0 ≤ j 1, where we note that Σ Z j +1 , = Z j +1 , .(2) Here we have Y j, → Z , → Z ,j +1 → Σ Y j, for 0 ≤ j < n − r − 1, again notingthat Z ,j +1 = Σ − Z ,j +1 .(3) The triangles are X j, − → Z j, → Z , → Σ X j, − for 1 ≤ j ≤ r + m − Y − ,j → Z ,j → Z , → Σ Y − ,j for r − n + 1 ≤ j ≤ − ≤ i ≤ r , the relevant triangle belongs with the family in (3) above, andcan be computed analogously. However, when i > r , we need to take the coconeof the morphism Σ i Z − r − m, → Σ i (cid:0) Z − r − m,r − n ⊕ Z , (cid:1) . We claim that the cone of − r − m, → Z − r − m,r − n ⊕ Z , is Z ,r − n . To show this, we compute the cocone of Z − r − m,r − n ⊕ Z , → Z ,r − n via the following octahedron: Z − r − m,r − n (cid:15) (cid:15) Z − r − m,r − n (cid:15) (cid:15) C (cid:47) (cid:47) Z − r − m,r − n ⊕ Z , (cid:47) (cid:47) (cid:15) (cid:15) Z ,r − n (cid:15) (cid:15) C (cid:47) (cid:47) Z , (cid:47) (cid:47) Σ X − r − m, − , where the second column is the split triangle, and the third column is a standardtriangle from Properties 1.2(4). The triangle forming the bottom row is noneother than X − r − m, − → Z − r − m, → Z , → Σ X − r − m, − , which computes thecocone C = Z − r − m, as claimed. (cid:3) Corollary 6.17. Let Z ∈ ind ( Z ) be arbitrary. If U ∈ Z ⊥ is indecomposable then G Z ( U ) is also indecomposable.Proof. Since the autoequivalences T X , T Y and Σ act transitively on the Z components, itis sufficient to see this for Z = Z , . This is clear from the computations in (the proofof) Proposition 6.15 above. (cid:3) Silting objects in D b (Λ) correspond to pairs ( Z, M (cid:48) ), where Z ∈ ind ( Z ) and M (cid:48) is asilting object of Z ⊥ (cid:39) D b ( k A n + m − ). However, a silting object in D b (Λ) may have morethan one indecomposable summand in the Z components. Thus, using silting reduction,we will obtain multiple descriptions of the same object. To rectify this problem, weclassify silting objects for which Z ∈ ind ( Z ) is minimal with respect to a total order on ind ( Z ) defined as follows. Let Z ∈ ind ( Z i ) and Z (cid:48) ∈ ind ( Z j ) and define Z (cid:22) Z (cid:48) ⇐⇒ ray (Σ j − i Z ) ≤ ray ( Z (cid:48) ) if i < j ; ray ( τ − Σ j − i Z ) ≤ ray ( Z (cid:48) ) if i > j ; coray ( Z ) ≤ coray ( Z (cid:48) ) if i = j and ray ( Z ) = ray ( Z (cid:48) ); ray ( Z ) < ray ( Z (cid:48) ) if i = j and ray ( Z ) (cid:54) = ray ( Z (cid:48) ) , where ray ( Z aij ) ≤ ray ( Z akl ) if and only if i ≤ k and coray ( Z aij ) ≤ coray ( Z akl ) if and only if j ≤ l . Equivalently, for Z ∈ ind ( Z i ), the total order is defined by successor sets, { ˜ Z ∈ ind ( Z ) | Z (cid:22) Z (cid:48) } = ray + ( Z ) ∪ ray ± ( coray + ( τ − Z )) ∪ ray ± ( coray + (Σ { i +1 ,...,r − } Z )) ∪ ray ± ( coray + ( τ − Σ { ,...,i − } Z )) . The following diagrams indicate the indecomposables Z ∈ Z with Z (cid:22) Z (cid:48) :Σ i − j Z (cid:48) Z i , i < j Z (cid:48) Z j τ Σ i − j Z (cid:48) Z i , i > j Lemma 6.18. The relation (cid:22) defines a total order on ind ( Z ) . roof. Anti-symmetry: Suppose Z (cid:22) Z (cid:48) and Z (cid:48) (cid:22) Z with Z ∈ ind ( Z i ) and Z (cid:48) ∈ ind ( Z j ). If i = j , then anti-symmetry is clear. For a contradiction, suppose i < j .Then ray (Σ j − i Z ) ≤ ray ( Z (cid:48) ) and ray ( τ − Σ i − j Z (cid:48) ) ≤ ray ( Z ). In particular, it follows that ray ( τ − Z (cid:48) ) ≤ ray (Σ j − i Z ) ≤ ray ( Z (cid:48) ), which is a contradition, since ray ( τ − Z (cid:48) ) > ray ( Z (cid:48) ).The same argument works when i > j . Transitivity: Suppose Z (cid:22) Z (cid:48) and Z (cid:48) (cid:22) Z (cid:48)(cid:48) with Z ∈ ind ( Z i ), Z (cid:48) ∈ ind ( Z j ) and Z (cid:48)(cid:48) ∈ ind ( Z k ). One simply analyses the different possibilities for i , j and k . We do the case i > j and j < k ; the rest are similar. The first inequality means that ray ( τ − Σ j − i Z ) ≤ ray ( Z (cid:48) ) and the second inequality means that ray (Σ k − j Z (cid:48) ) ≤ ray ( Z (cid:48)(cid:48) ). There are twosubcases: first assume i ≤ k . In this case, apply τ Σ k − j to the condition arising fromthe first inequality and combine this with the second inequality to get ray (Σ k − i Z ) ≤ ray ( τ Σ k − j Z (cid:48) ) < ray (Σ k − j Z (cid:48) ) ≤ ray ( Z (cid:48)(cid:48) ). Now assume that i > k and apply Σ k − j to thecondition arising from the first inequality and combine with the second inequality to get ray ( τ − Σ k − i Z ) ≤ ray (Σ k − j Z (cid:48) ) ≤ ray ( Z (cid:48) ). Totality: Suppose Z ∈ ind ( Z i ) and Z (cid:48) ∈ ind ( Z j ). If i = j then it is clear that either Z (cid:22) Z (cid:48) or Z (cid:48) (cid:22) Z . Now suppose i < j . If ray (Σ j − i Z ) ≤ ray ( Z (cid:48) ) then Z (cid:22) Z (cid:48) and we aredone, so suppose that ray (Σ j − i Z ) > ray ( Z (cid:48) ). Then it follows that ray (Σ i − j Z (cid:48) ) < ray ( Z ),in which case, because τ − increases the index of the ray by 1, one gets ray ( τ − Σ i − j Z (cid:48) ) ≤ ray ( Z ) and hence Z (cid:48) (cid:22) Z . A similar argument holds in the case i > j . Thus, (cid:22) is indeeda total order. (cid:3) Using Corollary 6.17, we now ensure we identify each silting subcategory of M of D b (Λ)as precisely one pair ( Z, M (cid:48) ), with M (cid:48) a silting object of Z ⊥ (cid:39) D b ( k A n − m +1 ) by insistingthat Z (cid:22) Z (cid:48) for each Z (cid:48) ∈ ind ( Z ) ∩ add M (cid:48) . Definition 6.19. We define the following additive subcategory of D : Z ⊥≺ := add { U ∈ ind ( Z ⊥ ) | G Z ( U ) ∈ Z and G Z ( U ) ≺ Z } . With the identification of D b ( k A n + m − ) in D b (Λ( r, n, m )) of Remark 6.7, using Propo-sition 6.15, we now give an explicit description of the additive subcategory Z ⊥≺ .Recall from Proposition 6.6 that Z ⊥ (cid:39) D b ( k A n + m − ). Let Γ := k A n + m − be the pathalgebra of the A n + m − quiver with the linear orientation:1 2 (cid:111) (cid:111) (cid:111) (cid:111) n + m − n + m − . (cid:111) (cid:111) Consider the unique Σ i mod (Γ) ⊂ D b (Γ) that contains the indecomposable objects in Z ⊥ ∩ Z admitting non-zero morphisms to Z . In Lemma 6.20 below, when we specify mod (Γ), we shall mean precisely this copy sitting inside D b ( k A n + m − ). Lemma 6.20. With the conventions described above, the additive subcategory Z ⊥≺ is Z ⊥≺ = add { Σ i A | i ≤ − r } ∪ add { Σ i B | − r ≤ i < } ∪ add ( C ) , where the sets of indecomposables A , B and C are defined as follows: A := { X ∈ mod (Γ) | Hom Γ ( P ( r + m ) , X ) (cid:54) = 0 } ; B := A ∩ { X ∈ mod (Γ) | Hom Γ ( P ( r + m + 1) , X ) (cid:54) = 0 } (empty when n − r = 1 ); C := { P ( r + m − , . . . , P ( n + m − } (empty when n − r = 1 ),where P ( i ) is the indecomposable projective at vertex i for the path algebra Γ = k A n + m − .Proof. This is a direct computation using Proposition 6.15, the total order on the inde-composable objects of the Z components of Lemma 6.18, and the identification of thesubcategory from Remark 6.7. (cid:3) o illustrate Lemma 6.20, we sketch the additive subcategory Z ⊥≺ in the case of Λ(2 , , Z = Z , below. Σ ≤− r A (cid:122) (cid:125)(cid:124) (cid:123) Σ − B (cid:122) (cid:125)(cid:124) (cid:123) C (cid:122)(cid:125)(cid:124)(cid:123) We summarise this discussion in the following proposition, and obtain the main theoremof the section as a corollary. Proposition 6.21. Suppose Z ∈ ind ( Z ) and write Z = thick D b (Λ) ( Z ) . Then there is abijection between(1) Silting subcategories M of D b (Λ) with Z ∈ M and Z (cid:22) ind ( Z ) ∩ M .(2) Silting subcategories N of Z ⊥ with N ∩ Z ⊥≺ = ∅ . Theorem 6.22. In D b (Λ( r, n, m )) there are bijections between(1) Pairs ( Z, N ) where Z ∈ ind ( Z ) and N is a silting subcategory of D b ( k A m + n − ) containing no objects in the additive subcategory Z ⊥≺ .(2) Silting subcategories of D b (Λ( r, n, m )) .(3) Bounded t-structures in D b (Λ( r, n, m )) .(4) Bounded co-t-structures in D b (Λ( r, n, m )) . A detailed example: Λ(2 , , , , 1) in detail. Let Z = Z , and write Z = thick ( Z ). Take the convention for homological degree as in Lemma 6.20. With thisconvention, we identify the indecomposable objects in Z ⊥ and of D b ( k A ) as follows: Z , − P (3) X , Z , − (cid:55)→ P (2) I (2) X , X , Z , − P (1) S (2) S (3)Using Lemma 6.20, Theorem 6.13 and the explicit calulation of the tilting objects, up tosuspension, in Example 6.14, we compute the twelve families of silting objects in D b ( k A )that lift to silting objects in D b (Λ(2 , , Z , as the minimal indecomposablesummand in the Z components. The results of this computation are presented in Table 1.We make the following observation regarding tilting objects in D b (Λ(2 , , Proposition 7.1. Let Λ = Λ(2 , , . Fir any Z ∈ ind ( Z ) , put Z = thick ( Z ) and F Z : D b (Λ) → Z ⊥ (cid:39) D b ( k A ) . Then:(1) There are precisely six tilting objects in D b (Λ) containing Z as a summand.(2) If T ∈ D b (Λ) is a tilting object containing Z as a summand then F Z ( T ) is a tiltingobject in Z ⊥ .Proof. The proof is a direct computation. Without loss of generality, we may set Z = Z , .Consider the additive subcategory T := (cid:0) (cid:84) n (cid:54) =0 ⊥ (Σ n Z ) (cid:1) ∩ (cid:0) (cid:84) n (cid:54) =0 (Σ n Z ) ⊥ (cid:1) ∩ Z ⊥ . Thesubcategory T consists of the thick subcategory Z ⊥ ∩ ⊥ Z (cid:39) D b ( k ), which has just oneindecomposable object in each homological degree, together with finitely many indecom-posables in homological degrees 0,1 and 2. ilting object in k A silting family in Λ(2 , , T ⊕ T ⊕ T Σ i T ⊕ Σ j T ⊕ Σ k T P (1) ⊕ P (2) ⊕ P (3) j ≥ i and k ≥ max { j, − } P (1) ⊕ P (3) ⊕ S (3) k ≥ j ≥ max { i, − } P (2) ⊕ S (2) ⊕ P (3) j ≥ i and k ≥ max { i, − } S (2) ⊕ P (3) ⊕ I (2) j ≥ − k ≥ max { i, j, − } P (3) ⊕ I (2) ⊕ S (3) k ≥ j ≥ i ≥ − P (3) ⊕ S (3) ⊕ Σ S (2) k ≥ j ≥ i ≥ − S (2) ⊕ I (2) ⊕ Σ P (1) k ≥ j ≥ max { i, − } I (2) ⊕ S (3) ⊕ Σ P (1) k ≥ i and j ≥ i ≥ − S (2) ⊕ Σ P (1) ⊕ Σ P (3) j ≥ i and k ≥ max { j, − } Σ P (1) ⊕ S (3) ⊕ Σ P (2) j ≥ − k ≥ max { i, j } S (3) ⊕ Σ P (2) ⊕ Σ S (2) k ≥ j ≥ i ≥ − S (3) ⊕ Σ S (2) ⊕ Σ P (1) k ≥ j ≥ i ≥ − Table 1. The twelve tilting objects in k A giving rise to the silting objectscontaining Z , as the (cid:22) -minimal summand in Z for Λ(2 , , Z ⊥ ∩ ⊥ Z shows thatunless the object lies in homological degree 0, 1 or 2, there is not sufficient intersectionwith T to give rise to a tilting object. Thus we must form tilting objects from only finitelymany indecomposables. A detailed analysis of the Hom-hammocks of these finitely manyindecomposables gives rise to the six tilting objects obtained from Z , and the followingobjects: Z − , ⊕ X − , − ⊕ X , , X − , − ⊕ X − , − ⊕ X , , X − , − ⊕ X − , − ⊕ X , X − , − ⊕ X , ⊕ X , , X − , − ⊕ X , ⊕ X , , X − , − ⊕ X , ⊕ Z , . The second claim can be directly computed. (cid:3) Our computations lead us to state the following conjecture: Conjecture 7.2. For an arbitrary Z ∈ ind ( Z ) , writing Λ = Λ( r, n, m ) and Z = thick ( Z ) and F Z : D b (Λ) → Z ⊥ (cid:39) D b ( k A n + m − ) , we have:(1) There are finitely many tilting objects in D b (Λ) containing Z as a summand.(2) If T ∈ D b (Λ) is a tilting object containing Z as a summand then F Z ( T ) is a tiltingobject in Z ⊥ . An explicit example for a bounded t-structure in D b (Λ(2 , , We finish bychoosing a silting object N ∈ D b ( k A ), assembling this with Z = Z , to the associatedsilting object M ∈ D b (Λ(2 , , D b (Λ(2 , , M .Let us start with the silting object N = Σ − S (2) ⊕ P (1) ⊕ Σ P (3) ∈ D b ( k A )and set Z = Z and Z = thick ( Z ). As explained above, N corresponds to the object M (cid:48) = Σ − X , ⊕ X , ⊕ Σ Z , − = X − , − ⊕ X , ⊕ Z , − ∈ Z ⊥ . By Proposition 6.15, M (cid:48) lifts under G Z to the silting object M = Z , ⊕ X − , − ⊕ X , ⊕ Z , − ∈ D b (Λ(2 , , . X (0,0) X (0,0) Y (0,0) Y Z Z the four summands of M , with Z the top one in Z ;and positive (Σ > M ) and negative suspensions (Σ < M ), respectively;and coaisle Y M = (Σ ≥ M ) ⊥ and aisle X M = (Σ < M ) ⊥ , respectively.The corresponding co-t-structure ( A M , B M ) is right adjacent in the sense of [12] to thet-structure ( X M , Y M ), i.e. B M = X M and A M := ⊥ B M = ⊥ susp M = ⊥ (Σ ≥ M ).Recall how to obtain from the silting object M a bounded t-structure ( X M , Y M ) andbounded co-t-structure ( A M , B M ), using the bijections of K¨onig and Yang [32]: X M := (Σ < M ) ⊥ = susp M and Y M := (Σ ≥ M ) ⊥ , A M := ⊥ (Σ ≥ M ) = cosusp Σ − M and B M := (Σ < M ) ⊥ = susp M. ppendix A. Notation, terminology and basic notions In this section we collect some notation and basic terminology, which is mostly standard.We always work over an algebraically closed field k and denote the dual of a vectorspace V by V ∗ . Throughout, D will be a k -linear triangulated category with suspension(otherwise know as shift or translation) functor Σ : D → D .For two objects A, B ∈ D , we use the shorthand Hom i ( A, B ) = Hom( A, Σ i B ) resem-bling Ext spaces in abelian categories, and hom( A, B ) = dim Hom( A, B ) for dimensionsof homomorphism spaces. We writeHom > ( A, B ) = (cid:77) i> Hom( A, Σ i B ) and Hom • ( A, B ) = (cid:77) i ∈ Z Σ − i Hom( A, Σ i B )for aggregated homomorphism spaces (and similarly for obvious variants) and for thehomomorphism complex, a complex of vector spaces with zero differential.A.1. Properties of triangulated categories and their subcategories. A k -lineartriangulated category D is said to be algebraic : if D arises as the homotopy category of a k -linear differential graded category;see [30]. Examples are bounded derived categories of k -linear abelian categories. Hom-finite : if dim Hom( D , D ) < ∞ for all objects D , D ∈ D . The bounded derivedcategory D b (Λ) of any finite-dimensional k -algebra Λ is Hom-finite. Krull–Schmidt : if every object of D is isomorphic to a finite direct sum of objects allof whose endomorphism rings are local. In this case, the direct sum decomposition isunique up to isomorphism. Bounded derived categories of k -linear Hom-finite abeliancategories are Krull–Schmidt; see [6]. indecomposable : if for every decomposition D ∼ = D ⊕ D with triangulated categories D and D either D ∼ = 0 or D ∼ = 0. The derived category of a finite-dimensional algebrais indecomposable if (and only if) the associated Gabriel quiver is connected.in possession of Serre duality : if there is an equivalence S : D ∼ → D with Hom( D , D ) ∼ =Hom( D , S D ) ∗ , bifunctorially in D , D ∈ D . Such an autoequivalence is canonicaland unique, if it exists, and called the Serre functor of D .The existence of a Serre functor is equivalent to the existence of Auslander–Reiten trian-gles; see [37, § I.2]. If Λ is a finite-dimensional k -algebra, then D b (Λ) has Serre duality ifand only if Λ has finite global dimension; in this case, the Auslander–Reiten translationis given by the cosuspended Serre functor: τ = Σ − S .We conclude that D b (Λ( r, n, m )) is algebraic, Hom-finite, Krull–Schmidt and indecom-posable for all choices of r, n, m . It has Serre duality if and only if n > r , which we alwaysassume in this article.A.2. Subcategories of triangulated categories. Let C be a collection of objects of D , regarded as a full subcategory. We recall the following terminology: C ⊥ , the right orthogonal to C , the full subcategory of D ∈ D with Hom( C , D ) = 0, ⊥ C , the left orthogonal to C , the full subcategory of D ∈ D with Hom( D, C ) = 0.If C is closed under suspensions and cosuspensions, then C ⊥ and ⊥ C are tri-angulated subcategories of D . thick ( C ), the thick subcategory generated by C , the smallest triangulated subcategory of D containing C which is also closed under taking direct summands. usp ( C )and cosusp ( C ), the (co-)suspended subcategory generated by C , the smallest full subcategory of D containing C which is closed under (co-)suspension, extensions and takingdirect summands. add ( C ), the additive subcategory of D containing C , the smallest full subcategory of D containing C which is closed under finite coproducts and direct summands. ind ( C ), the set of indecomposable objects of C , up to isomorphism. (cid:104) C (cid:105) , the smallest full subcategory of D containing C that is closed under extensions,i.e. if C (cid:48) → C → C (cid:48)(cid:48) → Σ C (cid:48) is a triangle with C (cid:48) , C (cid:48)(cid:48) ∈ C then C ∈ C .The ordered extension closure of a pair of subcategories ( C , C ) of D is defined as C ∗ C := add { D ∈ D | C → D → C → Σ C for C ∈ C and C ∈ C } . This operation is associative and C is extension closed in D if and only if C ∗ C ⊆ C .A.3. Approximations and adjoints. For this section only, suppose D is an additivecategory and C a full subcategory of D .Recall that C is called right admissible in D if the inclusion functor C (cid:44) → D admits aright adjoint. Analogously for left admissible . A subcategory C is called admissible if itis both left and right admissible.Often, one does not need admissibility but only approximate admissibility. A right C -approximation of an object D ∈ D is a morphism C → D with C ∈ C such that theinduced maps Hom( C (cid:48) , C ) → Hom( C (cid:48) , D ) are surjective for all C (cid:48) ∈ C . A morphism f : C → D is called a minimal right C -approximation if f g = f is only possible forisomorphisms g : C → C . Dually for (minimal) left C -approximations . We say C is • contravariantly finite in D if all objects of D have right C -approximations; • covariantly finite in D if all objects of D have left C -approximations; • functorially finite in D if it is contravariantly finite and covariantly finite in D .Note that in the case that D is a Hom-finite, k -linear, Krull–Schmidt category, the exis-tence of a C -approximation guarantees the existence (and uniqueness, up to isomorphism)of a minimal C -approximation.Sometimes, right C -approximations are called C -precovers and left C -approximationsare called C -preenvelopes . If for all D ∈ D the induced map Hom( C (cid:48) , C ) → Hom( C (cid:48) , D )above were bijective instead of surjective, then C would be even right admissible. In thissense, the morphism C → D ‘approximates’ the (possibly nonexistent) right adjoint tothe inclusion functor.For Krull–Schmidt triangulated categories D , these concepts coincide: Proposition A.1 ([31, Proposition 1.3]) . Let D be a Krull–Schmidt triangulated categoryand let C ⊂ D a suspended subcategory. Then C is contravariantly finite in D if and onlyif C is right admissible. Dually for covariantly finite cosuspended subcategories. Thus, a thick subcategory C of D is functorially finite if and only if it is admissible.Functorial finiteness can often be deduced from Hom-finiteness. More precisely, let H D := { C ∈ ind ( C ) | Hom( D, C ) (cid:54) = 0 } , H D := { C ∈ ind ( C ) | Hom( C, D ) (cid:54) = 0 } . Lemma A.2. Let D be a Hom-finite, Krull–Schmidt category with a subcategory C . Ifthe set H D is finite for all D ∈ ind ( D ) , then C is covariantly finite in D . Dually, if H D is finite for all D ∈ ind ( D ) , then C is contravariantly finite in D . roof. For D ∈ ind ( D ), the direct sum (cid:76) C ∈ H D C ⊗ Hom( D, C ) ∗ is a well-defined object of D by the assumption on H D . Hence the natural morphism D → (cid:76) C ∈ H D C ⊗ Hom( D, C ) ∗ is a (not necessarily minimal) left C -approximation of D . Therefore, indecomposableobjects of D have left C -approximations; as D is Krull–Schmidt, all objects of D do and C is covariantly finite in D . Dually for contravariant finiteness. (cid:3) Corollary A.3. Let D be a Hom-finite, Krull–Schmidt category with a subcategory C containing only finitely many indecomposable objects. Then C is functorially finite in D . A.4. Silting subcategories. Silting objects are a generalisation of tilting objects, whichwere introduced in [31]. However, we follow the terminology of [1]. Note that all subcat-egories are assumed to be additive and closed under isomorphisms.Let M be a subcategory of a triangulated category D . • M is called a partial silting subcategory if Hom > ( M , M ) = 0. • M is called a silting subcategory if it is partial silting and thick D ( M ) = D . • An object D ∈ D is called a silting object if add ( D ) is a silting subcategory. • Two silting objects D, D (cid:48) ∈ D are equivalent if and only if add ( D ) = add ( D (cid:48) ).For reasonable categories, there is a strong connection between silting objects andsilting subcategories; see [1, Theorem. 2.27]: Lemma A.4. Let D be a Hom-finite, Krull–Schmidt triangulated category. Then D hasa silting object if and only if D has a silting subcategory and K ( D ) is free of finite rank. In particular, if a category D as in the lemma has a silting object D , thenrk K ( D ) = { isomorphism classes of indecomposable summands of D } and in particular, the right-hand side is independent of the silting object. We record twofurther easy observations: Lemma A.5. A partial silting subcategory is extension-closed.Proof. If M ⊂ D is partial silting, then any extension M (cid:48) −→ D −→ M (cid:48)(cid:48) e −→ Σ M (cid:48) with M (cid:48) , M (cid:48)(cid:48) ∈ M has e ∈ Hom( M (cid:48)(cid:48) , Σ M (cid:48) ) = 0, so that the extension is trivial. In other words,the extension closure M ∗ M is built from direct sums only. (cid:3) Lemma A.6. If D is a Hom-finite, Krull–Schmidt triangulated category with a siltingobject, then any (additive) subcategory N of a silting subcategory M is functorially finitein M .Proof. The existence of a silting object implies that M and N are each additively generatedby finitely many objects. Now apply Corollary A.3. (cid:3) A.5. Torsion pairs, t-structures and co-t-structures. We assume again that D isa k -linear triangulated category. A pair ( X , Y ) of full subcategories closed under directsummands is called a torsion pair if Hom( X , Y ) = 0 and D = X ∗ Y ; see [28].Both X and Y are then extension closed. By definition, for every D ∈ D there is atriangle X → D → Y → Σ X with X ∈ X and Y ∈ Y . The map X → D is a right X -approximation and D → Y is a left Y -approximation, i.e. X is contravariantly finiteand Y is covariantly finite in D . The triangle is called the approximation triangle of D .By abuse of terminology, we shall call X the aisle and Y the co-aisle of the torsion pair.The abuse arises as this terminology is normally reserved for the case that ( X , Y ) is at-structure (see below).The torsion pair ( X , Y ) will be called bounded if (cid:83) i ∈ Z Σ i X = (cid:83) i ∈ Z Σ i Y = D . Torsionpairs appear in three important guises, namely ( X , Y ) is called a t-structure [7] if Σ X ⊆ X ( ⇐⇒ Σ − Y ⊆ Y ); • co-t-structure [36] (also weight structure [12]) if Σ − X ⊆ X ( ⇐⇒ Σ Y ⊆ Y ); • stable t-structure (also semi-orthogonal decomposition ) if Σ X = X ( ⇐⇒ Σ Y = Y ).For historical reasons, when the terminology ‘semi-orthogonal decomposition’ is used thetorsion pair is often written as (cid:104) Y , X (cid:105) . Furthermore, a t-structure is stable if and only ifit is also a co-t-structure.If ( X , Y ) is a t-structure then its heart H = X ∩ Σ Y is an abelian subcategory of D ; see[7, Theorem 1.3.6]. A bounded t-structure is determined by its heart via X = susp H and Y = cosusp Σ − H ; see, for example, [13, Section 3].If ( X , Y ) is a co-t-structure then its co-heart M = X ∩ Σ − Y is a partial silting sub-category of D ; see, for instance, [35, Corollary 5.9]. Note that, if M is abelian thenit is semisimple. A co-t-structure is bounded if and only if M is a silting subcategory.Moreover, a bounded co-t-structure is determined by its co-heart ([1, Proposition 2.23]): X = cosusp M = (cid:91) l ≥ Σ − l M ∗ Σ − l +1 M ∗ · · · ∗ M , and, Y = susp M = (cid:91) l ≥ M ∗ Σ M ∗ · · · ∗ Σ l M . Remark A.7. If ( X , Y ) is a t-structure then the approximation triangle is functorial andcalled the truncation triangle , with X → D being a right minimal X -approximation calledthe right truncation and D → Y a left minimal Y -approximation called the left truncation of D . Another way to express this functoriality is: the inclusion X (cid:44) → D has a rightadjoint (given by D (cid:55)→ X ) and Y (cid:44) → D has a left adjoint. In particular, truncations areminimal approximations. We mention that ‘t-structure’ is an abbreviation for ‘truncationstructure’.A.6. K¨onig-Yang bijections. The notions of silting subcategories, t-structures and co-t-structures for finite dimensional k -algebras are related by the following bijections ofK¨onig and Yang. Before we state them, recall an abelian category A is called a lengthcategory if it is both artinian and noetherian. Theorem A.8 ([32, Theorem 6.1]) . Let Λ be a finite dimensional k -algebra. There arebijections between(i) equivalence classes of silting objects in K b ( proj (Λ)) ,(ii) bounded t-structures in D b ( mod (Λ)) whose heart is a length category,(iii) bounded co-t-structures in K b ( proj (Λ)) . Under these bijections, a silting subcategory M ⊂ D is mapped to thet-structure ( X M , Y M ) := (Σ < M ) ⊥ , Σ ≥ M ) ⊥ ) = ( susp M , Σ < M ) ⊥ );co-t-structure ( A M , B M ) := ⊥ (Σ ≥ M ) , Σ < M ) ⊥ ) = ( cosusp Σ − M , susp M ) . A.7. Exceptional sequences and semi-orthogonal decompositions. The notionof semi-orthogonal decomposition D = (cid:104) C , C (cid:105) is synonymous with that of a stablet-structure ( C , C ), see A.5, and leads to equivalences C ∼ = D / C and C ∼ = D / C .An admissible subcategory C ⊂ D produces two semi-orthogonal decompositions D = (cid:104) C , ⊥ C (cid:105) = (cid:104) C ⊥ , C (cid:105) .An object E of a k -linear triangulated category D is exceptional if Hom( E, E ) = k and Hom (cid:54) =0 ( E, E ) = 0, i.e. E has the smallest possible graded endormorphism ring.Exceptional objects are characterised by the following property (which is used in thetext): thick D ( E ) = add (Σ i E | i ∈ Z ). Morever, the subcategory thick D ( E ) is thenadmissible by [11, Theorem 3.2]. Hence an exceptional object E leads to semi-orthogonaldecompositions D = (cid:104) thick D ( E ) ⊥ , thick D ( E ) (cid:105) . n exceptional sequence in D is a tuple ( E , . . . , E t ) of exceptional objects such thatHom • ( E i , E j ) = 0 for all i > j . The sequence is full if thick D ( E , . . . , E t ) = D and strong ifHom • ( E i , E j ) = Hom( E i , E j ), i.e. all homomorphisms occur in degree zero. A full, strongexceptional sequence ( E , . . . , E t ) gives rise to a tilting object E ⊕ · · · ⊕ E t . Similarly, afull exceptional sequence ( E , . . . , E t ) with Hom > ( E i , E j ) = 0 for all i, j gives rise to asilting object. Appendix B. The repetitive algebra and string modules For a finite-dimensional algebra Λ, Happel showed in [21] that there is a full embedding F : D b (Λ) → mod ( ˆΛ ), where mod ( ˆΛ ) denotes the stable module category of the repetitivealgebra ˆΛ , and F is called the Happel functor . A finite-dimensional algebra Λ is gentleif and only if its repetitive algebra is special biserial (see [40, Proposition]). For such analgebra, there is a convenient description of all the indecomposable objects of mod ( ˆΛ )using string and band modules; see [40]. Since the algebras Λ( r, n, m ) are gentle, thismachinery applies. Moreover, only string modules occur; indeed it is this absence of bandmodules that is responsible for discreteness. Thus, we shall omit any further reference toband modules.In this section we shall recall the construction of the repetitive algebra, the descriptionof string modules and the maps between them. We then apply these results to thederived-discrete algebras Λ( r, n, m ).B.1. The repetitive algebra. The notion of a repetitive algebra was introduced byHughes and Waschb¨usch in [26]. The standard references are [26, 38, 40]. The relationsfor Λ( r, n, m ) are also recalled in [9]. The following summary is based on [40].Let Q = ( Q , Q ) be a finite, connected quiver with vertices Q and arrows Q . A path p in Q is a sequence of arrows p = a a · · · a t with s ( a i +1 ) = e ( a i ) for 1 ≤ i < t .The start of p , s ( p ) = s ( a ) and the end of p , e ( p ) = e ( a t ). The path p is said to have length t . Note there is a trivial path of length 0, e v , corresponding to each vertex v ∈ Q .The concatenation p p of paths p and p is defined if and only if e ( p ) = s ( p ). Apath q is called a subpath of a path p if p = p qp for some (not necessarily non-trivial)paths p and p . Write Pa for the set of paths of Q . A relation for Q is a non-zerolinear combination of paths of length at least 2 which have the same starting points andend points. A zero-relation is a relation of the form p (sometimes written p = 0). A commutativity relation is a relation of the form p − q .Now let ρ be a set of zero- and commutativity relations for Q and consider the pathalgebra arising from the bound quiver Λ := k Q/ (cid:104) ρ (cid:105) . Two paths p and p in Q are equivalent if p = p (cid:48) vp (cid:48)(cid:48) and p = p (cid:48) wp (cid:48)(cid:48) , where v − w or w − v is a commutativity relationin ρ . Note that this generates an equivalence relation on Pa ; we denote the equivalenceclass of a path p by p . A path p in Q is called a path in ( Q, ρ ) if for each p (cid:48) ∈ p , p (cid:48) doesnot have a subpath belonging to ρ . A path a · · · a n is called maximal if ba · · · a n and a · · · a n c are not paths in ( Q, ρ ) for each b and c such that e ( b ) = s ( a ) and e ( a n ) = s ( c ).The repetitive algebra ˆΛ := k ˆ Q/ (cid:104) ˆ ρ (cid:105) , where ˆ Q = ( ˆ Q , ˆ Q ) is specified by: • the vertex set is given by ˆ Q := Z × Q ; • for each arrow a : x → y in Q there is an arrow ( i, a ) : ( i, x ) → ( i, y ) in ˆ Q ; • for each maximal path p in ( Q, ρ ), there is a connecting arrow ˆ p : ( i, y ) → ( i +1 , x )in ˆ Q , where s ( p ) = x and e ( p ) = y . f p is a path in Q , the corresponding path in ( i, Q ) is denoted ( i, p ). Let p = p p be amaximal path in ( Q, ρ ). Then the path ( i, p )( i, ˆ p )( i + 1 , p ) is called a full path in ˆ Q .We now define the relations: • ˆ ρ inherits the relations from ρ , i.e. for paths p , p and p in Q , if p ∈ ρ (resp. p − p ∈ ρ ) then ( i, p ) ∈ ˆ ρ (resp. ( i, p ) − ( i, p ) ∈ ˆ ρ ) for all i ∈ Z . • Let p be a path that contains a connecting arrow. If p is not a subpath of a fullpath then p ∈ ˆ ρ . • Let p = p p p and q = q q q be maximal paths in ( Q, ρ ) with p = q . Then( i, p )( i, ˆ p )( i + 1 , p ) − ( i, q )( i, ˆ q )( i + 1 , q ) ∈ ˆ ρ for all i ∈ Z .Denote the set of paths in ( ˆ Q, ˆ ρ ) by (cid:99) Pa .In ˆ Q ( r, n, m ) there are Z copies of each vertex in Q ( r, n, m ), labelled ( i, x ) for x ∈ Q ( r, n, m ) and i ∈ Z . Likewise, there are Z copies of each arrow in Q ( r, n, m ), for each i ∈ Z we have: • the arrows ( i, a j ) : ( i, j ) → ( i, j + 1) for − m ≤ j ≤ − • the arrows ( i, b j ) : ( i, j ) → ( i, j + 1) for 0 ≤ j ≤ n − r ; • the arrows ( i, c j ) : ( i, j ) → ( i, j + 1) for n − r + 1 ≤ j ≤ n − 1, where n ≡ • the arrows ( i, x j ) : ( i, j + 1) → ( i + 1 , j ) for n − r + 1 ≤ j ≤ n − 1, where n ≡ • the arrows ( i, y ) : ( i, n − r + 1) → ( i + 1 , − m ).In an abuse of notation, we write down only one copy of each vertex and arrow in thefollowing shorthand version of the quiver ˆ Q ( r, n, m ). b (cid:47) (cid:47) · · · b n − r − (cid:47) (cid:47) n − r b n − r (cid:36) (cid:36) − m a − m (cid:47) (cid:47) − m a − m (cid:47) (cid:47) · · · a − (cid:47) (cid:47) − a − (cid:47) (cid:47) b (cid:49) (cid:49) x n − (cid:29) (cid:29) n − r + 1 c n − r +1 (cid:114) (cid:114) y (cid:79) (cid:79) n − c n − (cid:93) (cid:93) x n − (cid:43) (cid:43) · · · c n − (cid:108) (cid:108) x n − r +2 (cid:45) (cid:45) n − r + 2 c n − r +2 (cid:107) (cid:107) x n − r +1 (cid:50) (cid:50) Following the rules above, we can read off the following relations for ˆΛ ( r, n, m ); see [9,Section 3]. The degrees of the arrows should be inferred by the presence of the connectingarrows labelled x and y of degree 1. We have the following relations: • c k c k +1 = 0 for k = n − r, . . . , n − 1, where c n − r := b n − r and c n := b ; • x k x k − = 0 for k = n − r + 2 , . . . , n − • yx n − = 0 if m = 0, and a − x n − = 0 if m > • c n − r +1 x n − r +1 − ya − m · · · b n − r = 0 if r > • c k x k − x k − c k − = 0 for k = n − r + 2 , . . . , n − r > • x n − c n − − b · · · b n − r ya − m · · · a − = 0 if r > 1, and in the case r = 1 we have ya − m · · · b n − − b · · · b n − ya − m · · · a − = 0; • Any path starting at ( i, k ) and ending at ( i + 1 , k + 1), with k (cid:54) = 0 and − m ≤ k ≤ n − r , that contains y as a subpath is zero.B.2. String modules. Let Λ = k Q/ (cid:104) ρ (cid:105) be a special biserial algebra. We describe stringsfor the bound quiver ( Q, ρ ), which give rise to string modules. The references are [16]and [43]. We remind the reader that all modules are right modules.For each arrow a ∈ Q , introduce a formal inverse ¯ a = a − with s (¯ a ) = e ( a ) and e (¯ a ) = s ( a ). For a path p = a · · · a n the inverse path ¯ p = ¯ a n · · · ¯ a .A walk w of length l > Q, ρ ) is a sequence w = w · · · w l , satisfying the usualconcatenation requirements, where each w i is either an arrow or an inverse arrow. Formal nverses of walks are defined in the obvious way. Starting and ending vertices of walksand their inverses are defined analogously to those for paths.A walk is called a string if it contains neither subwalks of the form a ¯ a or ¯ aa for some a ∈ Q , nor a subwalk v such that v ∈ ρ or ¯ v ∈ ρ . We also define two strings oflength zero , namely, for each x ∈ Q there are trivial strings + x and 1 − x . We write s (1 ± x ) = e (1 ± x ) = x and set (1 ± x ) − = 1 ∓ x .For technical reasons, in order to define composition of strings with trivial strings, weneed to introduce string functions σ, ε : Q → {− , } satisfying the following properties: • If a (cid:54) = a ∈ Q with s ( a ) = s ( a ) then σ ( a ) = − σ ( a ). • If b (cid:54) = b ∈ Q with e ( b ) = e ( b ) then ε ( b ) = − ε ( b ). • If a, b ∈ Q are such that ab / ∈ ρ then σ ( b ) = − ε ( a ).The choice of such string functions is completely arbitrary. An explicit algorithm forchoosing such functions is given in [16, p. 158]. The functions σ and ε can be extendedto strings as follows. If a ∈ Q , define σ (¯ a ) = ε ( a ) and ε (¯ a ) = σ ( a ). If w = w · · · w n isa string, define σ ( w ) = σ ( w ) and ε ( w ) = ε ( w n ). Finally, for x ∈ Q define σ (1 ± x ) = ∓ ε (1 ± x ) = ± v = v · · · v m and w = w · · · w n of length at least 1 this is done in the obvious way: the composition vw is defined if vw = v · · · v m w · · · w n is a string. However, if w = 1 ± x then vw is definedif e ( v ) = x and ε ( v ) = ± 1. Analogously, if v = 1 ± x then vw is defined if s ( w ) = x and σ ( w ) = ∓ 1. Note that given arbitary strings v and w whose composition vw is defined,we necessarily have σ ( w ) = − ε ( v ). However, in the case of a special biserial algebra, thiscondition is not sufficient for a string to be defined.Modulo the equivalence relation w ∼ ¯ w , the strings form an indexing set for the so-called string modules of Λ. We shall write M ( w ) for the corresponding string module. Wedirect the reader to [16, Section 3] for precise details on how to pass to a representation-theoretic description of the modules. Example B.1. Consider ˆΛ (2 , , a = a − , b = b , c = b , d = c , x = x and y = y to avoid cumbersome subscripts.Consider the string ( − , x )( − , ¯ c )( − , ¯ b )( − , x )(0 , ¯ c )(0 , ¯ b ), which we write as x ¯ c ¯ bx ¯ c ¯ b forshort, with ¯ b = (0 , ¯ b ) to determine the ‘degrees’ of each of the arrows. This can berepresented pictorially by the diagram below. • (0 ,b ) (cid:127) (cid:127) • ( − ,b ) (cid:127) (cid:127) ( − ,x ) (cid:31) (cid:31) • (0 ,c ) (cid:127) (cid:127) • ( − ,x ) (cid:31) (cid:31) • ( − ,c ) (cid:127) (cid:127) •• In this picture, we read from left to right, direct arrows point downwards and to the rightand inverse arrows point downwards and to the left.B.3. Irreducible maps between string modules and a linear order. A completedescription of the irreducible maps between string modules was obtained in [16]. Given astring w , the irreducible maps whose source is the string module M ( w ) can be determinedby modifying w in a minimal way either on the left, or on the right.We describe the algorithm that modifies w on the left, i.e. that keeps the endpoint of w fixed, to produce a new string w [1]. This yields an irreducible morphism M ( w ) → M ( w [1]). Adding a hook on the left: If there exists a ∈ Q such that a w is defined, then let a · · · a n be the maximal direct string starting at s ( a ). Then w [1] := ¯ a n · · · ¯ a a w ;the irreducible map is the natural inclusion.(2) Removing a cohook on the left: If there is no a ∈ Q such that aw is defined, then w = v · · · v n − ¯ v n w (cid:48) with v i ∈ Q and a string w (cid:48) , where v · · · v n − is a maximaldirect substring at the beginning of w . Then w [1] := w (cid:48) ; the irreducible map isthe natural projection map.There is a dual algorithm, which adds a hook or removes a cohook on the right to outputthe string [1] w . The inverse operations are written w [ − 1] and [ − w , respectively. For n ∈ Z we define w [ n ] = w [1] · · · [1]; similarly for [ n ] w .We illustrate these concepts in the diagrams below; in the left-hand diagram, we adda hook, in the right-hand diagram, we remove a cohook. • v (cid:31) (cid:31) • a (cid:127) (cid:127) a (cid:31) (cid:31) •• w • (cid:55)→ • • w • • v n (cid:31) (cid:31) • v n +1 (cid:127) (cid:127) w (cid:48) • (cid:55)→ • w (cid:48) •• a n (cid:127) (cid:127) •• These operations give rise to AR sequences/triangles: w [1] (cid:31) (cid:31) w (cid:63) (cid:63) (cid:31) (cid:31) [1] w [1] = [1]( w [1]) = ([1] w )[1] . [1] w (cid:63) (cid:63) The process of adding a hook or removing a cohook determines a total order on stringsending at a given vertex whose modules lie in the same component of the AR quiver.This process can be generalised to produce a total order on all strings ending at a givenvertex. This is the Geiß total order [19], which we describe next.Let x ∈ Q . There is a linear order on strings w and v in ( Q, ρ ) such that e ( w ) = e ( v ) = x and ε ( w ) = ε ( v ) = t with t ∈ {− , } . Namely, v < w ⇐⇒ either w = w (cid:48) v, where w (cid:48) = w (cid:48) · · · w (cid:48) n with w (cid:48) n ∈ Q ;or v = v (cid:48) w, where v (cid:48) = v (cid:48) · · · v (cid:48) m with ¯ v (cid:48) m ∈ Q ;or v = v (cid:48) c, w = w (cid:48) c with w (cid:48) n ∈ Q and ¯ v (cid:48) m ∈ Q , where w (cid:48) , v (cid:48) and c are strings. It may be useful to illustrate this definition with a picture.Below we indicate arrows of either direction by short wiggly lines, (sub)strings by longwiggly lines, a direct arrow points downwards and to the right, an inverse arrow points ownwards and to the left.Case 1: w = • w (cid:48) • • w (cid:48) n − • w (cid:48) n (cid:31) (cid:31) • v • Case 2: • v (cid:48) m (cid:127) (cid:127) w • v = • v (cid:48) • • v (cid:48) m − • w = • w (cid:48) • • w (cid:48) n − • w (cid:48) n (cid:31) (cid:31) Case 3: • v (cid:48) m (cid:127) (cid:127) c • v = • v (cid:48) • • v (cid:48) m − • Example B.2. Consider ˆΛ (2 , , a = a − , b = b , c = b , d = c , x = x and y = y to avoid cumbersome subscripts. Wehave the following linear order on the strings ending at vertex (0 , − (0 , − < ¯ dy − < a ¯ dy − < · · · < cya ¯ dy − < y − < cy − < ¯ xbcy − < · · · < abc ¯ xbcy − < bcy − , where we write y − = ( − , y ) for short, 1 (0 , − denotes the trivial string at vertex (0 , − cya ¯ dy − < y − and abc ¯ xbcy − < bcy − , which correspond to removing a cohook.The AR triangle starting at cy − is ¯ xbcy − (cid:31) (cid:31) cy − (cid:63) (cid:63) (cid:31) (cid:31) ¯ xbc − .c − (cid:63) (cid:63) Remark B.3. Note that ( i, a − ) and ( i, c n − ) are two arrows in ˆ Q ( r, n, m ) ending atthe vertex ( i, ε : ˆ Q ( r, n, m ) → {− , } we have ε (cid:0) ( i, a − ) (cid:1) = − ε (cid:0) ( i, c n − ) (cid:1) . Thus, ( i, a − ) and ( i, c n − ) are not comparable in the totalorder defined above (because their ε values differ).B.4. Maps between string modules. It is straightforward to compute the maps be-tween string modules. This was first observed in [17] and later generalised in [18] and[33]. We follow the neat exposition given in [41, Section 2].For a string w , define the set of factor strings , Fac ( w ), to be the set of decompositions w = def with d, e, f ∈ St , where d = d · · · d n and f = f · · · f m , in which we require d to be trivial or d n ∈ Q − and f to be trivial or f ∈ Q . Similarly, the set of substrings , Sub ( w ), is the set of decompositions in which we require d to be trivial or d n ∈ Q and f to be trivial or f ∈ Q − . A picture may be useful: on the left we illustrate a factorstring decomposition and on the right, a substring decomposition. • d n (cid:127) (cid:127) e • f (cid:31) (cid:31) • d ··· d n − • • f ··· f m • • d ··· d n − • d n (cid:31) (cid:31) • f (cid:127) (cid:127) f ··· f m •• e • pair (( d , e , f ) , ( d , e , f )) ∈ Fac ( v ) × Sub ( w ) is called admissible if e = e or e = ¯ e . Then the main results of [17, 18] and [33] assert: Theorem B.4. Let v, w ∈ St and suppose M v and M w are their corresponding stringmodules. Then hom( M v , M w ) = { admissible pairs in Fac ( v ) × Sub ( w ) } . The following corollary is immediate. Corollary B.5. Suppose v and w are strings such that e ( v ) = e ( w ) and ε ( v ) = ε ( w ) ,making v and w comparable in the Geiß total order. If v ≤ w then there is a non-zeromorphism M ( v ) → M ( w ) . B.5. Strings and maps for derived-discrete algebras. Here we list some pertinentfacts about strings and string modules for discrete derived categories from [9], and estab-lish some additional routine but useful properties. Lemma B.6 ([9]) . Denote the simple modules of Λ( r, n, m ) by S ( i ) for − m ≤ i < n . Inthe coordinate system introduced in Properties 1.2, Z , = S (0) . Then:(i) If m > then S ( − 1) = X , ; in particular there is a simple module on the mouthof the X component.(ii) If r < n then S ( n − r ) lies on the mouth of the Y component. The embedding mod (Λ( r, n, m )) (cid:44) → mod ( ˆΛ ( r, n, m )) maps simple modules S ( i ) (cid:55)→ S (0 , i );the latter corresponds to the trivial string 1 (0 ,i ) . Since morphisms to and from a simplemodule cannot factor through projective-injective modules (recall that ˆΛ is a self-injectivealgebra), we obtain both Hom ( S (0 , i ) , X ) = Hom( S (0 , i ) , X ) and Hom ( X, S (0 , i )) =Hom( X, S (0 , i )) for any X ∈ mod ( ˆΛ ( r, n, m )). Lemma B.7. Let A ∈ ind ( D b (Λ( r, n, m ))) with r > and let i, k ∈ Z , ≤ k < r . Then Hom( X kii , A ) = k if A ∈ ray + ( X kii ) ∪ coray − ( S X kii ) ∪ ray ± ( Z kii ) , Hom( Y kii , A ) = k if A ∈ coray + ( Y kii ) ∪ ray − ( S Y kii ) ∪ coray ± ( Z kii ) , and in all other cases the Hom spaces are zero. For r = 1 the Hom-spaces are as above,except Hom( X ii , X i,i + m ) = k . The other two statements of Lemma 2.1 follow from these by Serre duality. Proof. Case m > : Since, by [9, Theorem B], the action of τ and Σ together is transitiveon the set of objects at the mouths of the X components, by Lemma B.6, we mayassume that X kii = S (0 , − (0 , − in the Geiß total order.By Corollary B.5, it follows that each object in this totally ordered set admits a morphismfrom S (0 , − ˆΛ (2 , , S (0 , − w be a string that admits a substring decomposition def ∈ Sub ( w ) with e = 1 (0 , − or ¯ e = 1 (0 , − . We claim that w = de (= d ) or w = ef (= f ). This is clear sincethere is only one arrow ending at (0 , − , a − ) when m > − , y ) when m = 1. These strings (or their inverses) are precisely the strings listed in Geiß total orderin Properties 1.2(5). Therefore, by Theorem B.4, these are precisely the indecomposableobjects admitting a morphism from S (0 , − ray + ( S (0 , coray − ( S S (0 , X components, and ray ± ( Z , ) in the Z component.The Hom-hammock of objects admitting morphisms from S (0 , n − r ), which is in the Y component for any m , can be obtained in an analogous fashion. ase m = 0 : We use an embedding D b (Λ( r, n, (cid:44) → D b (Λ( r, n, P ( − 1) lies on the mouth of the X component. ApplyingLemma A.2 to P = thick D b (Λ( r,n, ( P ( − P ⊥ (cid:39) D b (Λ( r, n, P ⊥ by the case m > (cid:3) Remark B.8. In the ‘extended ray’ of strings ending at 1 (0 , − , to obtain the part of thislinearly ordered set corresponding to coray − ( S S (0 , − (0 , − with the direct arrow a (0 , − (for m > 1) or y − (for m = 1). Wethus obtain strings starting with the corresponding inverse arrow, which gives the coray.The next fact is used in particular for Proposition 6.6, the classification of silting objects. Lemma B.9. 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