Discrete derived categories I: homomorphisms, autoequivalences and t-structures

Discrete derived categories were studied initially by Vossieck and later by Bobi\'nski, Gei\ss, Skowro\'nski. In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures.


Introduction
In this article, we study the bounded derived categories of finite-dimensional algebras that are discrete in the sense of Vossieck [35]. Informally speaking, discrete derived categories can be thought of as having structure intermediate in complexity between the derived categories of hereditary algebras of finite representation type and those of tame type. Note, however, that the algebras with discrete derived categories are not hereditary. We defer the precise definition until the beginning of the next section.
The study of the structure of discrete derived categories was begun by Bobiński, Geiß and Skowroński in [8], who obtained a canonical form for the derived equivalence class of algebras with a given discrete derived category; see Figure 1. This canonical form is parametrised by integers n ≥ r ≥ 1 and m > 0, and the corresponding algebra denoted by Λ(r, n, m). We restrict our attention to the case n > r, which ensures finite global dimension. In [8], the authors also determined the components of the Auslander-Reiten (AR) quiver of derived-discrete algebras and computed the suspension functor.
The structure exhibited in [8] is remarkably simple, which brings us to our principal motivation for studying these categories. Discrete derived categories are sufficiently simple to make explicit computation highly accessible but also non-trivial enough to manifest interesting behaviour. In particular, sitting naturally inside these categories are the (higher) cluster categories of type A ∞ studied in [18] and [19], and an abundance of spherelike objects in the sense of [17]. Moreover, their structure is highly reminiscent of that of the bounded derived categories of cluster-tilted algebras of typeÃ n studied in [4], suggesting approaches developed here to understand discrete derived categories are likely to find applications more widely in the study of derived categories of gentle algebras.
Understanding homological properties of algebras means understanding the structure of their derived categories. We investigate several key aspects of the structure of discrete derived categories: the structure of homomorphism spaces, the autoequivalence groups of the categories, and the t-structures and co-t-structures inside discrete derived categories.
The basis of our work is giving a combinatorial description via AR quivers of which indecomposable objects admit non-trivial homomorphism spaces between them, so called 'Hom-hammocks'. As a byproduct, we get the following interesting property of these categories: the dimensions of the homomorphism spaces between indecomposable objects have 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 dimension bound is 2. We believe this property holds independent interest and warrants further investigation. See [15] for a different approach to measuring the 'smallness' of discrete derived categories.
In Theorem 4.6 we explicitly describe the group of autoequivalences. For this, we introduce a generalisation of spherical twist functors arising from cycles of exceptional objects. The action of these twists on the AR components of Λ(r, n, m) is a useful tool, which is employed here and will also be used in the sequel.
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 occurring in triangulated categories, and have recently become a focus of intense research as the principal ingredients in the study of Bridgeland stability conditions [11], and their cot-structure analogues [23]. Further investigation into the properties of (co-)t-structures and the stability manifolds is the subject of the sequel.
We study the (co-)t-structures indirectly via certain generating sets: silting subcategories, which behave like the projective objects of hearts and generalise tilting objects. In general, one cannot get all bounded t-structures in this way, but in Proposition 6.1, we show that the heart of each bounded t-structure in D b (Λ) is equivalent to mod(Γ), where Γ is a finite-dimensional algebra of finite representation type. The upshot is that using some correspondences of König and Yang [26], classifying silting objects is enough to classify all bounded (co-)t-structures. We show that D b (Λ(r, n, m)) admits a semiorthogonal decomposition into D b (kA n+m−1 ) and the thick subcategory generated by an exceptional object. Using Aihara and Iyama's silting reduction [1], we classify the silting objects in Theorem 6.18. We finish with an explicit example of Λ(2, 3, 1) in Section 7.
Later, we will often use the 'height' of indecomposable objects in X or Y components. For X k ij ∈ ind(X k ), we set h(X k ij ) = j − i and call it the height of X k ij in the component X k . Similarly, for Y k ij ∈ ind(Y k ), we set h(Y k ij ) = i − j and call it the height of Y k ij in the component Y k . The mouth of an X or Y component consists of all objects of height 0.

Hom spaces: hammocks
In this section, for a fixed indecomposable object A ∈ D b (Λ) we compute the so-called 'Hom-hammock' of A, i.e. the set of indecomposables B ∈ D b (Λ) with Hom(A, B) = 0.
The precise description of the hammocks is slightly technical. However, the result is quite simple, and the following schematic indicates the hammocks Hom(X, −) = 0 and Hom(Z, −) = 0 for indecomposables X ∈ X and Z ∈ Z: 00 00 11 11 Hammocks from the mouth. We start with a description of the Hom-hammocks of objects at the mouths of all ZA ∞ components. The proof relies on Happel's equivalence of D b (Λ(r, n, m)) with the stable module category of the repetitive algebra of Λ(r, n, m).
The well-known theory of string (and band) modules, which we summarise in Appendix B, provides a useful tool to understand the indecomposable objects and homomorphisms between them.
To make our statements of Hom-hammocks more readable, we employ the language of rays and corays. Let V = V i,j be an indecomposable object of D b (Λ(r, n, m)) with coordinates (i, j). Recall the conventions that Denoting the AR component of V by C and its objects by V a,b , we define the rays and corays from, to, and through V by 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 ray of an indecomposable X k ii ∈ X k at the mouth consists of indecomposables in X k reached by arrows going out of X k ii , while in the Y components the ray of Y k ii contains objects which have arrows going in to it.
For the next statement, whose proof is deferred to Lemma B.4, recall that the Serre functor is given by suspension and AR translation: S = Στ . Also, rays and corays commute with these three functors.
Lemma 2.1. Let A ∈ ind(D b (Λ(r, n, m))) with r > 1 and let i, k ∈ Z, 0 ≤ k < r. Then and in all other cases the Hom spaces are zero. For r = 1 the Hom-spaces are as above, except Hom(X 0 ii , X 0 i,i+m ) = k 2 . 2.2. Hom-hammocks for objects in X components. Assume A = X k ij ∈ ind(X k ). In order to describe the various Hom-hammocks conveniently, we set A 0 := X k jj to be the intersection of the coray through A with the mouth of X k , and 0 A := X k ii to be the intersection of the ray through A with the mouth of X k . By definition, A 0 and 0 A have height 0. If A sits at the mouth, then A = A 0 = 0 A. We introduce notation for line segements in the AR quiver: given two indecomposable objects A, B ∈ D b (Λ(r, n, m)) which lie on a ray or coray (so in particular sit in the same component), then the finite set consisting of these two objects and all indecomposables lying between them on the (co-)ray is denoted by AB. Finally, before we state the proposition, recall our convention that X r = X 0 and note that 0 (SA) = Στ ( 0 A). Proposition 2.2 (Hammocks Hom(X k , −)). Let A = X k ij ∈ ind(X k ) and assume r > 1. For any indecomposable object B ∈ ind(D b (Λ)) the following cases apply: B ∈ X k : then Hom(A, B) = 0 ⇐⇒ B ∈ ray + (AA 0 ); B ∈ X k+1 : then Hom(A, B) = 0 ⇐⇒ B ∈ coray − ( 0 (SA), SA); B ∈ Z k : then Hom(A, B) = 0 ⇐⇒ B ∈ ray ± (Z k ii Z k ji ) 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 0 : then Hom(A, B) = 0 ⇐⇒ B ∈ ray + (AA 0 ) ∪ coray − ( 0 (τ −m A), τ −m A).
Proof. The main tool in the proof of this, and the following propositions, will be induction on the height of A -the induction base step is proved in Lemma 2.1 which gives the hammocks for indecomposables of height 0. We give a careful exposition for the first claim, 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 0 , and the line segment AA 0 . The existence of non-zero homomorphisms A → B for objects B ∈ R(A) follows directly from the properties of the AR quiver. For the vanishing statement, we proceed by induction on the height of A. If A sits on the mouth of X k , then Lemma 2.1 states indeed that the Hom(A, B) = 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 diamond in the AR mesh which has A as the top vertex, and the corresponding AR triangle we are in the degenerate case with C = 0.) It is clear from the definitions that A 0 = A 0 , A 0 = C 0 and there are inclusions R(A ) ⊂ R(A) ⊂ R(A ) ∪ R(A ). We start with an 6 object B ∈ X k such that B / ∈ R(A ) ∪ R(A ). By the induction hypothesis, we know that R(A ), R(C) and R(A ) are the Hom-hammocks in X k for A , C, A , respectively. Since B is contained in none of them, we see that Hom(A , B) = Hom(C, B) = Hom(A , 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 ) ∪ R(A ))\R(A) which can be seen to be the line segment A A 0 . Again we work up from the mouth: Hom(A, A 0 ) = 0 and Hom(A, τ A 0 ) = 0 by Lemma 2.1, as before. The extension D 1 given by τ A 0 → D 1 → A 0 → Στ A 0 is the indecomposable object of height 1 on A A 0 . Applying Hom(A, −) to this triangle, we find Hom(A, D 1 ) = 0, as required. The same reasoning works for the objects of heights 2, . . . , h − 1 on the segment.
Case B ∈ X k+1 : We start by showing the existence of non-zero homomorphisms to indecomposable objects in the desired region. For any B in this region, it follows directly from the properties of the AR quiver that there is a non-zero homomorphism from B to SA. However by Serre duality we see that Hom(A, B) = Hom(B, SA) * = 0 as required. The statement that Hom(A, B) = 0 for all other B ∈ X k+1 can be proved by an induction argument which is analogous to the one given in the first case above.
Case B ∈ Z k : For any indecomposable object A = X k ij ∈ X k , write V (A) for the region in Z k specified in the statement, i.e. the region bounded by the rays through Z k ii and Z k ji . We start by proving that Hom(A, B) = 0 for B ∈ V (A). The first chain of morphisms in Properties 1.1 (5), implies that Hom(A, B) = 0 for any B ∈ ray ± (Z k ii ). For . Applying Hom(A, −) leaves us with the exact sequence Hom(A, X k i,i+s−1 ) → Hom(A, B) → Hom(A, B ) → Hom(A, ΣX k i,i+s−1 ). By looking at the Hom-hammocks in the X -components that we already know, we see that the left-hand term vanishes as X k i,i+s−1 is on the same ray as A but has strictly lower height. Similarly, we observe that the right-hand term of the sequence vanishes: 0 = Hom(X k i,i+s−1 , τ A) = Hom(A, ΣX k i,i+s−1 ). Hence Hom(A, B ) = Hom(A, B) = 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 k ii ). For h > 0, as before we consider the AR mesh which has A as its top vertex: Remaining cases: These comprise vanishing statements for entire AR components, namely Hom(X k , X j ) = 0 for j = k, k + 1, and Hom(X k , Y j ) = 0 for any j, and Hom(X k , Z j ) = 0 for j = 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 seen using induction on height and considering a square in the AR mesh. The vanishing to the Z k components with k = j follows similarly.

Hom-hammocks for objects in
. This case is similar to the one above. Put ii to be the intersection of the coray through A with the mouth of Y k , and A 0 := Y k jj to be the intersection of the ray through A with the mouth of Y k . The remaining hammock ray + (coray + (A 0 )) ⊂ Y 1 is not shown.
and assume r > 1. For any indecomposable object B ∈ ind(D b (Λ)) the following cases apply: For r = 1, these results still hold, except that the Y-clauses are replaced by Proof. These statements are analogous to those of Proposition 2.2. In this case, the vanishing statement for full components consists of Hom(Y k , X j ) = 0 for any j, and Hom(Y k , Y j ) = 0 for j = k, k + 1, and Hom(Y k , Z j ) = 0 for j = k.

2.4.
Hom-hammocks for objects in Z components. Let A = Z k ij ∈ ind(Z k ). By Lemma 2.1 we know that the following objects are well defined: A 0 := the unique object at the mouth of an X component for which Hom(A, A 0 ) = 0, A 0 := the unique object at the mouth of a Y component for which Hom(A, A 0 ) = 0.
Proof. The cases B ∈ X k+1 and B ∈ Y k+1 follow by Serre duality from Proposition 2.2 and Proposition 2.3, respectively. 8 Thus let B = Z l ab ∈ Z l be an indecomposable object in a Z component. There are two special distinguished triangles associated with B; see Section 1: / / ΣY l bb where 0 B = X l aa is the unique object at the mouth of a X component with Hom( 0 B, B) = 0 and similarly 0 B = Y l bb is unique at a Y mouth with Hom( 0 B, B) = 0. We get two exact sequences by applying Hom(A, −): Case l = k: Again, we first show that the dimension function hom(A, −) is constant on certain regions of Z k . In particular, we have Half of the first equality follows through the chain of equivalences . Likewise one obtains Hom(A, Σ 0 B) = 0 ⇐⇒ B ∈ ray ± (τ A), giving the first equality. Using the other triangle, the second equality is analogous.
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 admit nontrivial morphisms from A. Using (2) and analogous reasoning shows that no objects in 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 1 (A, A) = 0 by using (2) above. The existence of the stalk complex of a projective module in the Z component, Lemma B.6, 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 that Hom 1 (A, A) = 0 for all A ∈ Z. Finally, R = coray + (ray + (A)) is the non-vanishing hammock simply by Hom(A, A) = 0 and using either (1) or (2).

Twist functors from exceptional cycles
In this purely categorical section, we consider an abstract source of autoequivalences coming from exceptional cycles. These generalise the tubular mutations from [28] as well as spherical twists. In fact, a quite general and categorical construction has been given in [32]. However, for our purposes this is still a little bit too special, as the Serre functor will act with different degree shifts on the objects in our exceptional cycles. We also give a quick proof using spanning classes.
Let D be a k-linear, Hom-finite algebraic triangulated category. Assume that D has a Serre functor S and is indecomposable; see Appendix A.1 for these notions. Recall that an object E ∈ D is called exceptional if Hom • (E, E) = k · id E . For any object A ∈ D we define the functor F A : D → D, X → Hom • (A, X) ⊗ A and note that there is a canonical evaluation morphism F A → id of functors. Also note that for two objects A 1 , A 2 ∈ D there is a common evaluation morphism In fact, for any sequence of objects A * = (A 1 , . . . , A n ), we define the associated twist functor T A * as the cone of the evaluation morphism -this gives a well-defined, exact functor by our assumption that D is algebraic: These functors behave well under equivalences: Proof. This follows the standard argument for spherical twists: For F A * we have Conjugating the evaluation functor morphism F A * → id with ϕ, we find that ϕT A * ϕ −1 is the cone of the conjugated evaluation functor morphism F ϕ(A * ) → id which is the evaluation morphism for ϕ(A * ). Hence that cone is T ϕ(A * ) .
Definition. A sequence (E 1 , . . . , E n ) of objects of D is called an exceptional n-cycle if (1) every E i is an exceptional object, This definition assumes n ≥ 2 but a single object E should be considered an 'exceptional 1-cycle' (a linguistic oxymoron) if E is a spherical object, i.e. there is an integer k with S(E) ∼ = Σ k (E) and Hom • (E, E) = k ⊕ Σ −k k. In this light, the above definition, and statement and proof of Theorem 3.4 are generalisations of the treatment of spherical objects and their twist functors as in [21, §8].
In an exceptional cycle, the only non-trivial morphisms among the E i apart from the identities are given by α i : E i → Σ k i E i+1 . This explains the terminology: the subsequence (E 1 , . . . , E n−1 ) is an honest exceptional sequence, but the full set (E 1 , . . . , E n ) is notthe morphism α n : E n → Σ kn E 1 prevents it from being one, and instead creates a cycle.
Remark 3.2. 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 1 + · · · + k n . Thus the Calabi-Yau dimension of each object in the cycle is k/n. Example 3.3. We mention that this severely restricts the existence of exceptional ncycles of geometric origin: Let X be a smooth, projective variety over k of dimension d and let D := D b (cohX) be its bounded derived category. The Serre functor of D is given by S(−) = Σ d (−) ⊗ ω X and in particular, is given by an autoequivalence of the standard heart followed by an iterated suspension. If E * is any exceptional n-cycle in D, we find If furthermore the exceptional n-cycle E * consists of sheaves, then this forces k i = d to be maximal for all i, as non-zero extensions among sheaves can only exist in degrees between 0 and d. However, 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 an exceptional 2-cycle and, by the next theorem, gives rise to an autoequivalence of D b (X).
Theorem 3.4. Let E * = (E 1 , · · · , E n ) be an exceptional n-cycle in D. Then the twist functor 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} and Ω := 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 spanning classes 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 notational simplicity, we will treat E 1 and α 1 : The cone of the evaluation morphism is easily seen to sit in the following triangle so that T E * (E 1 ) = Σ 1−kn E n . The zero morphism in the middle follows trivially from Hom(E 1 , Σ 1−kn E n ) = 0 which always holds by assumption unless n = 2 and k 1 = 1 − k 2 . The third map is indeed the one specified above; this can be formally checked with the octahedral axiom, or one can use the vanishing of the composition of two adjacent maps in a triangle. It is straightforward to check the special case n = 2 and k 1 + k 2 = 1. Likewise, we find F E * (E 2 ) = Σ −k 1 E 1 ⊕ E 2 and T E * (E 2 ) = Σ 1−k 1 E 1 . Now consider the following diagram of distinguished triangles, where the vertical maps are induced by α 1 : Hence, the commutativity of the right-hand square forces T E * (α 1 ) = −Σ 1−kn α n .
Step 2: The above computation shows that the functor T E * is fully faithful when restricted to E. It is also obvious from the construction of the twist that T E * is the identity when restricted 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 1−k i−1 E i−1 , X) = 0. Finally, we use Serre duality and the defining property of E * to see that Combining all these statements, we deduce that T E * is fully faithful when restricted to the spanning class Ω, hence bona fide fully faithful by general theory; see e.g. [21,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 canonical map of functors S −1 T E * S → id which can be spelled out in the following diagram: It is easy to check that the left-hand vertical arrow is an isomorphism whenever we plug in objects from Ω: both vector spaces are zero for objects from E ⊥ , and the claim follows Hence T E * commutes with the Serre functor on Ω, and so by more general theory is essentially surjective; see [21,Corollary 1.56], this is the place where we need the assumption that D is indecomposable.
Remark 3.5. We point out that the twist T E * defined above is an instance of a spherical functor [3], given by the following data: It is easy to see that R is right adjoint to S and that T E * coincides with the cone of the adjunction morphism SR → id.
One condition for S to be a spherical functor is that the cone of id → RS should be an autoequivalence of D b (k n ). This amounts to precisely the computation of T E * on the E i and α i carried out above.

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 denote these by T X and T Y and prove some crucial properties: they commute with each other, act transitively on the indecomposables of each Z k component and provide a factorisation of the Auslander-Reiten translation: T X T Y = τ −1 . Moreover, T X acts trivially on Y and T Y acts trivially on X ; see Proposition 4.3 and Corollary 4.4 for the precise assertions. We then give an explicit description of the group of autoequivalences of D b (Λ(r, n, m)) in Theorem 4.6.
The category D = D b (Λ(r, n, m)) with n > r is Hom-finite, indecomposable, algebraic and has Serre duality (see Appendix A.1). Therefore we can apply the results of the previous section to D.
Our first observation is that every sequence of m + r consecutive objects at the mouth of X 0 is an exceptional (m + r)-cycle; likewise, every sequence of n − r consecutive objects at the mouth of Y 0 is an exceptional (n − r)-cycle, by which we mean a (r + 1)-spherical object in case n − r = 1. For the moment, we specify two concrete sequences: Proof. The object X 0 11 is exceptional by Lemma 2.1, hence any object at the mouth is. This point also gives the second condition of exceptional cycles: for and at the boundary step we have SE m+r = Στ X 0 11 = ΣX 0 00 = Σ 1−r X 0 m+r,m+r = Σ 1−r E 1 , where we freely make use of the results stated in Section 1. Hence the degree shifts of the sequence E * are k 1 = . . . = k m+r−1 = 1 and k m+r = 1 − r. Finally, the required vanishing Hom(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 SF n−r = Στ Y 0 11 = ΣY 0 00 = Σ 1+r Y 0 n−r,n−r = Σ 1+r F 1 . However, the actual choice of exceptional cycle is not relevant as the following easy lemma shows. It also allows us to write T X instead of T E * and T Y instead of T F * .
Lemma 4.2. Any two exceptional cycles E * , E * at the mouths of X components differ by suspensions and AR translations, and the associated twist functors coincide: Proof. A suitable iterated suspension will move E * into the X component that E * inhabits, and two exceptional cycles at the mouth of the same AR component obviously differ by some power of the AR translation. Thus we can write E * = Σ a τ b E * for some a, b ∈ Z. We point out that the suspension and the AR translation commute with all autoequivalences (it is a general and easy fact that the Serre functor does). Finally, we Proof of the proposition. By Lemma 2.1, we have Hom For the action of T X on the X components, we recall that the proof of Theorem 3.4 showed T X (E i ) = Σ 1−k i−1 (E i−1 ), and furthermore k 1 = . . . = k m+r−1 = 1 and k m+r = 1−r from Lemma 4.1. Hence T X (E i ) = τ −1 (E i ) for all i -as explained in Lemma 4.2, this holds for any exceptional cycle at an X mouth. Since T X is an equivalence and X k are AR components of type ZA ∞ , this forces ii and the triangle defining T E * (Z 0 ij ) is one of the special triangles of Properties 1.1(4): Application of AR translations extends this computation to arbitary Z ∈ Z 0 , and suspending extends it to all Z components, thus T X (Z 0 i,j ) = Z 0 i+1,j . Analogous reasoning shows T F * (F i ) = τ −1 (F i ) for all i = 1, . . . , n − r, and the rest of the above proof works as well: , now using the other special triangle.
The following technical lemma about the additive closures of the X and Y components will be used later on, but is also interesting in its own right. Using the twist functors, the proof is easy. Lemma 4.5. Each of X and Y is a thick triangulated subcategory of D.
Proof. The proof of Proposition 4.3 contains the fact thick(E * ) ⊥ = Y. Perpendicular subcategories are always closed under extensions and direct summands; since thick(E * ) is by construction a triangulated subcategory, the orthogonal complement Y is triangulated as well.
Proof. The suspension commutes with all triangle functors. The remaining relations stated in the theorem follow from Proposition 4.3.
Let G := Σ, T X , T Y be the subgroup of Aut(D b (Λ)) generated by the suspension and twist functors. This is an abelian group with three generators and one relation Σ r = T m+r X T r−n Y . By elementary algebra, there is an isomorphism of groups G ∼ = Z 2 × Z/ with = gcd(r, n, m) = gcd(r, m + r, r − n). Furthermore, G is a normal subgroup: the suspension Σ commutes with all exact endofunctors, and for the twists this is the statement of Lemma 3.1.
The inclusion G ⊂ Aut(D b (Λ)) has a section: choosing an object Z ∈ ind(Z), we map σ : This is possible since G acts transitivitely on ind(Z) and autoequivalences preserve the types of AR components.
We are left to classify the outer automorphisms of Λ(r, n, m). Scaling of arrows leads to a subgroup (k * ) m+n of Aut(Λ(r, n, m)). However, choosing an indecomposable idempotent e (i.e. a vertex) together with a scalar λ ∈ k * produces a unit u = 1 Λ + (λ − 1)e, and hence an inner automorphism c u ∈ Aut(Λ). It is easy to check that c u (α) = 1 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(r, n, m) shows that an (n + m − 1)-subtorus of the subgroup (k * ) m+n of arrowscaling automorphisms consists of inner automorphisms. Furthermore, the automorphism scaling all arrows simultaneously by the same number is easily seen not to be inner, hence, ker(σ) = k * .

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 that the dimensions of homomorphism spaces between indecomposable objects have a common bound. We also present the endomorphism complexes in Lemma 5.3.

5.1.
Hom space dimension bounds. The bounds are given in the the following theorem; for more precise information in case r = 1 see Proposition 5.2.
Proof. Our strategy for establishing the dimension bound follows that of the proofs of the Hom-hammocks. Let A, B ∈ ind(D b (Λ(r, n, m))) and assume r > 1. In this proof, we use the abbreviated notation hom = dim Hom; see Appendix A. We want to show hom(A, B) ≤ 1 by considering the various components separately.
Case A ∈ X k or Y k : Consider first A, B ∈ X k and perform induction on the height of A. If A = A 0 sits at the mouth, then hom(A, B) ≤ 1 by Lemma 2.1. For A higher up, and assuming Hom(A, B) = 0, which means B ∈ ray + (AA 0 ), we consider the triangle and A := cocone(g) otherwise, where by abuse of notation the intersection means the indecomposable additive generator of the specified subcategory.
Using hammocks from Proposition 2.2, we see that Hom(A , B) = 0 if B ∈ ray + (A) and Hom(A , B) = 0 otherwise. Thus from the exact sequence The induction hypothesis then gives hom(A, B) ≤ 1.
The subcase 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 morphisms from A (these form a ray in Z k ) and similarly ray − (B) ⊂ X k , and application of Proposition 2.4. An obvious modification, which we leave to the reader, extends the argument 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 Proposition 2.4 that the dimensions in the two non-vanishing regions ray + (coray + (A)) and ray − (coray − (SA)) are constant. Since the Z components contain the simple S(0) and the twist functors together with the suspension act transitively on Z, it is clear that hom(A, A) = hom(A, SA) * = 1. This completes the proof.
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. Otherwise we consider the AR mesh which has A at the top, and let A and A be the two indecomposibles of height h(A) − 1. There are two triangles: where, as before, 0 A and A 0 are the unique indecomposable objects on the mouth which are contained in respectively ray − (A) and coray + (A). Applying the functor Hom(X, −) to both triangles we obtain two exact sequences: Since 0 A and A 0 lie on the mouth of the component, Lemma 2.1 implies that the outer terms have dimension at most 2. Using the fact that X 0 and 0 SX are the only objects of the Hom-hammock from X lying on the mouth, Lemma 2.1 actually yields: The spaces are 2-dimensional precisely when A belongs to the intersections of the (co)rays on the right-hand side, which can only happen when 0 SX = X 0 . The set of rays and corays listed above divide the component into regions. In this proof, each region is considered to be closed below and open above.
Step 2: The function hom(X, −) is constant on each region, and changes by at most 1 when crossing a (co)ray if 0 SX = X 0 , and by at most 2 otherwise.
The first claim is clear from exact sequences (3) and (4). We show the second claim for rays; for corays the argument is similar. We get hom(X, A) ≤ hom(X, A ) + hom(X, 0 A) from sequence (3). This yields the stated upper bound for hom(X, A), as hom(X, 0 A) ≤ 1 when 0 SX = X 0 and hom(X, 0 A) ≤ 2 otherwise. For the lower bound, instead observe that hom(X, A ) ≤ hom(X, Σ 0 A) + hom(X, A), again from sequence (3).
Step 3: ψ = 0 unless A ∈ ray + (Σ −1 0 SX) and µ = 0 unless A ∈ coray − (ΣX 0 ) If A / ∈ ray + (Σ −1 X 0 ) ∪ ray + (Σ −1 0 SX) then hom(X, Σ 0 A) = 0 and so ψ = 0 trivially. Therefore, we just need to consider A ∈ ray + (Σ −1 X 0 ) but A / ∈ ray + (Σ −1 0 SX), and in this case hom(X, Σ 0 A) = 1. It is clear that the maps going down the coray from X to X 0 span a 1-dimensional subspace of Hom(X, Σ 0 A), which therefore is the whole space. Using properties of the ZA ∞ mesh, the composition of such maps with a map along ray + (X 0 ) from X 0 to ΣA defines a non-zero element in Hom(X, Σ A). Thus the map Hom(X, Σ 0 A) → Hom(X, ΣA) 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 morphisms in Properties 1.1 (5) to show that the map Hom(X, Step 4: If ray + (Σ −1 X 0 ) (or coray − (Σ 0 SX), respectively) does not coincide with one of the other three (co)rays, then crossing it does not affect the value of hom(X, −).
Step 5: There are three possible configurations of rays and corays determining the regions where hom(X, −) is constant.
It follows from Step 4 that it suffices to consider the remaining rays and corays, ray + (Σ −1 0 SX), ray + (Σ 0 SX), ray + (X 0 ) and coray − (ΣX 0 ), coray − (Σ 0 SX), coray − (X 0 ), 16 for determining the regional constants hom(X, −). Note that these are precisely the rays and corays required to bound the regions ray + (XX 0 ) and coray − ( 0 (SX), SX) of the statement of the proposition. Considering their relative positions on the mouth, Σ −1 0 SX is always furthest to the left and ΣX 0 is furthest to the right, while 0 SX can lie to the left, or to the right, or coincide with X 0 , depending on the height of X. We consider now the case where 0 SX is to the left of X 0 . We label the regions in the following diagram by letters A-M (this is the order in which we treat them, and the subscripts indicate the claimed hom(X, −) for the region): First 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, −) ≥ 1 on regions H, I, K and L; and on F, G, J and K using Serre duality. However regions F-I are reached by crossing a single ray or coray from one of the regions A-E. By Step 2 we thus get hom(X, −) = 1 on regions F-I. Now look at the element A ∈ ray + (S 0 X) ∩ coray − (X 0 ); this is the object of minimal height in region K. We can see that A ∈ coray + (X) and the map down the coray from X to A 0 , factors through the map from A to A 0 . Therefore the map δ in the second exact sequence (4) is non-zero. It is clear that A / ∈ coray − (ΣX 0 ) so µ = 0 by Step 3 above. We deduce from sequence (4) that hom(X, A) > hom(X, A ), so hom(X, A) > 1 since A is in region G. Since A is an object in region K, which can be reached from region D by crossing just two rays, Step 2 now gives hom(X, −) = 2 on region K.
Finally we now take up A ∈ ray + (Σ −1 S 0 X) ∩ coray − (ΣX 0 ), the object of minimal height in region M. It is clear that A / ∈ ray + (S 0 X) ∪ ray + (X 0 ), so hom(X, 0 A) = 0. A short calculation shows A ∈ ray + (X), and again using the chain of morphisms in Properties 1.1(5), we see that there is a map X → Σ 0 A = S 0 X factoring through A . Looking at the sequence (3) it follows that hom(X, A) < hom(X, A ) = 1 since A is in region L. Therefore, hom(X, −) = 0 on region M. For region J, we see that since it is sandwiched between regions K and M, hom(X, −) = 1 here.
This deals with the case that 0 SX lies to the left of X 0 . If instead it lies to the right, analogous reasoning applies. Finally, if 0 SX = X 0 , matters are simpler: in that case, the regions C and F-I all vanish.

5.2.
Graded endomorphism algebras. In this section we use the Hom-hammocks and universal hom space dimension bounds to recover some results of Bobiński on the graded endomorphism algebras of algebras with discrete derived categories; see [7,Section 4]. Our approach is somewhat different, so we provide proofs for the convenience of the reader. Using these descriptions we give a coarse classification of the homological properties of the indecomposable objects of discrete derived categories.
In order to conveniently write down the endomorphism complexes, we define four func- In words, the functions δ + and δ − determine the ranges of self-extensions of positive and negative degree, respectively. We point out that the result holds for all r ≥ 1.
Proof. Let A ∈ ind(X ), assuming r > 1. Suspending if necessary, we may suppose that We start with the first possibility: d = lr for some l ∈ Z. By Properties 1.1 (3) and (2), which is an indecomposable object in X 0 sharing its height h = j − i with A. Again using Proposition 2.2, we can reformulate the claim as follows: where the set of h + 1 objects in ( ) are precisely the objects in ray + (AA 0 ) of height h.
We now turn to the other possibility, d = 1 + lr for some l ∈ Z. Here we get . As we know from Theorem 5.1, all Hom spaces have dimension 1 when r > 1, these two computations give For r = 1 and A = X 0 ij ∈ ind(X ), by Proposition 2.2 the hammock Hom(A, −) = 0 is ray + (AA 0 ) ∪ coray − ( 0 (SA), SA). We treat each part separately: and, noting SA = X 0 i+m,j+m , 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 the endomorphism complex of B ∈ ind(Y) is proved in the same way, now using h = i − j, Σ r = τ n−r and the hammocks specified by Proposition 2.3.

Coarse classification of objects.
Our previous results allow us to give a crude grouping 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 Proposition 5.4. Each object A ∈ ind(D b (Λ(r, n, m))) is of exactly one type below: Remark 5.5. In fact, the direct sum E 1 ⊕ E 2 of two exceptional objects E 1 and E 2 with Hom • (E 1 , E 2 ) = Hom • (E 2 , E 1 ) = 0 is a 0-spherelike object. Examples for r > 1 are given by taking E 1 ∈ X and E 2 ∈ Y at the mouths. The theory of spherelike objects also applies in this degenerate case, but is less interesting [17,Appendix].
Proof. We know from Lemma B.6 that the projective module P (n − r) ∈ Z. This is an exceptional object by Proposition 2.4. As the autoequivalence group acts transitively on ind(Z) by Corollary 4.4, every indecomposable object of Z is exceptional. The remaining parts 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.  Next, B ∈ Y with h(B) = n − r − 1 is spherical if and only if Στ B = SB = Σ 1+r B = Στ n−r B, so that here we get τ n−r−1 B = B which is possibly only for n = r + 1.

Reduction to Dynkin type A and classification results
Two keys for understanding the homological properties of algebras are t-structures and co-t-structures, especially bounded ones. The main theorem of [26], cited in the appendix as Theorem A.6, states that for finite-dimensional algebras, bounded co-t-structures are in bijection with silting objects, which are in turn in bijection with bounded t-structures whose 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 to classify silting objects in D b (Λ(r, n, m)). This is the main goal of this section. In the first part, we prove that any bounded t-structure in D b (Λ(r, n, m)) is length, then we obtain a semi-orthogonal decompositon D b (Λ(r, n, m)) = D b (kA n+m−1 ), Z , for some trivial thick subcategory Z, and use this to bootstrap Keller-Vossieck's classification of silting objects in the bounded derived categories of path algebras of Dynkin type A to get 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 a finite number of indecomposable objects up to isomorphism, and is a length category.
We prove each statement in the following two lemmas. The first is a generalisation of the corresponding statement for the algebra Λ(1, 2, 0) proved in [26]; the second is a general statement about Hom-finite abelian categories. Lemma 6.2. Any heart of a t-structure of a discrete derived category has a finite number of indecomposable objects up to isomorphism.
Proof. We use the fact that there can be no negative extensions between objects in the heart H of a t-structure (X, Y). Suppose H contains an indecomposable Z ∈ ind(Z). Then any other indecomposable object in H must lie outside the hammocks Hom <0 (Z, −) = 0 and Hom <0 (−, Z) = 0. Looking at the complement of these Hom-hammocks, it is clear that all objects of ind(H) ∩ Z must be (co)suspensions of a finite set of objects. At most one suspension can sit in the heart H; hence ind(H) ∩ Z is finite.
In the X and Y components we use a similar argument: by Proposition 5.4, any object X l i,j which is sufficiently high up in an X component -here j − i ≥ r + m − 1 will do -has a negative self extension. Such objects can't lie in the heart and so again, up to (co)suspension, ind(H) ∩ X is finite. Similarly, for any object Y = Y l i,j sufficiently far down in an Y component -here i − j ≥ n − r − 1 suffices -the hammocks Hom <0 (Y, −) = 0 or Hom <0 (−, Y ) = 0 contain every indecomposable object of height less than r − n. The same argument then shows that ind(H) ∩ Y is finite as well. Proof. We show the stronger statement that any object A ∈ H has only finitely many subobjects. This certainly implies that ascending and descending chains of inclusions have to become stationary, and hence that H is a finite length category.
Suppose that there is an infinite set of pairwise non-isomorphic subobjects of A. Writing each of these subobjects as a direct sum of (the finitely many) indecomposables, we find that there is a U ∈ ind(H) such that U, U ⊕2 , . . . occur as summands of these subobjects, hence U, U ⊕2 , . . . ⊂ A. However, the number hom(U, A) < ∞ is a bound on the number of copies of U which can occur in A -a contradiction.
Remark 6.4. Proposition 6.1 means 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 finite representation type.
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-orthogonal decomposition, one of whose orthogonal subcategories is the bounded derived category of a path algebra of Dynkin type A.

6.2.
A semi-orthogonal decomposition: reduction to Dynkin type A. We start by showing that the derived categories of derived-discrete algebras always arise as extensions of derived categories of path algebras of type A by a single exceptional object.
and there is a semi-orthogonal decomposition D b (Λ(r, n, m)) = D b (kA n+m−1 ), Z . In particular, Z is functorially finite in D b (Λ(r, n, m)). Moreover, D b (Λ(r, n, m)) has a full exceptional 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 claim see [9,Theorem 3.2]. Furthermore D b (Λ) = Z ⊥ , Z is the standard semi-orthogonal decomposition for an exceptional object; see Appendix A.7 for details. Another way to see this: inspect the Hom-hammocks of Section 2 and apply Lemma A.1 to find that the subcategory Z is functorially finite in D b (Λ). Applying [25,Proposition 1.3] gives the admissibility of Z.
Lemma B.6 places the indecomposable projective P (n − r) in the Z component of the AR quiver of D b (Λ). Using the transitive action of the autoequivalence group on ind(Z), see Corollary 4.4, 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/Λe n−r Λ) → D b (Λ) with essential image thick D b (Λ) (e n−r Λ) ⊥ = Z ⊥ ; see, for example, [2, Lemma 3.4]. Inspecting the Gabriel quiver of Λ/Λe n−r Λ, we see that this quiver satisfies the criteria of [5,Theorem]. 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: Therefore Λ/Λe n−r Λ is an iterated tilted algebra of type A n+m−1 . It is well known that this implies D b (Λ/Λe n−r Λ) D b (kA n+m−1 ); see [16]. Combining these pieces, we get Z ⊥ D b (kA n+m−1 ). The final claim about D b (Λ) having a full exceptional sequence follows at once from the fact that D b (kA n+m−1 ) has one.  where f V : V → B V is a minimal left B-approximation of V . Note that here, in contrast to elsewhere in this paper, we require that the approximation is minimal to ensure welldefinedness of the map G N .
In light of Proposition 6.5, the natural choice for a functorially finite thick subcategory to which we can apply Theorem 6.7 is Z for some Z in the Z components. For silting reduction 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 a small generalisation of the statement we need, which we specialise in the subsequent corollary. Note that the corollary makes a statement regarding simple-minded collections, which are not used, and thus not defined, elsewhere in this paper, but will be used in the sequel. For the precise definition of this notion see [26].  Theorem 6.7 coupled with Proposition 6.5 tells us that all silting objects in D b (Λ) containing Z can be obtained by lifting silting objects in Z ⊥ D b (kA n+m−1 ) back up to D b (Λ). In other words, any silting object in D b (Λ) can be described by a pair (Z, M ) consisting of an indecomposable object Z ∈ Z and a silting object M ∈ Z ⊥ D b (kA n+m−1 ).

· · ·
Following [25], a quiver Q = (Q 0 , Q 1 ) is called an A t -quiver if |Q 0 | = t, its underlying graph is a tree, and Q 1 decomposes into a disjoint union Q 1 = Q α ∪ Q β such that at any vertex at most one arrow from Q α ends, at most one arrow from Q α starts, at most one arrow from Q β ends and at most one arrow from Q β starts.
For a vertex x, define maps s α , e α , s β , e β : Q 0 → N by s α (x) := #{y ∈ Q 0 | the shortest walk from x to y starts with an arrow in Q α }; e α (x) := #{y ∈ Q 0 | 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 one map ϕ Q := (g Q (x), h Q (x)) : Q 0 → (ZA t ) 0 , where g Q and h Q correspond to the coordinates in the AR quiver of D b (kA t ), such that h Q (x) = 1 + e α (x) + s β (x) and g Q (y) = g Q (x) for 23 each 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 0 {g Q (x)} = 0. By abuse of notation we identify the object T Q := ϕ Q (Q 0 ) with the direct sum of the indecomposables lying at the corresponding coordinates. This map gives rise to the following classification result. Theorem 6.10 ( [25], Section 4). The assignment Q → T Q induces a bijection from isomorphism classes of A t -quivers and tilting objects T in D b (kA t ) which satisfy 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 = t i=1 T i can be re-ordered to give a strong, full exceptional collection {T 1 , · · · , T t }, see [25,Section 5.2]. We now have the following classification of silting objects in D b (Λ).
Moreover, all silting objects of D b (kA t ) occur in this way.
The machinery above is rather technical, so we give a quick example of the classification of tilting (and hence silting) objects in D b (kA 3 ).
We indicate the corresponding tilting objects in the following sketch: In each sketch the triangle depicts the standard heart for the quiver 1 −→ 2 −→ 3 whose indecomposable projectives have coordinates (0, i) for i = 1, 2, 3.
From this one sees there are twelve tilting objects in D b (mod(kA 3 )) up to suspension: 6.5. Classification of silting objects for derived-discrete algebras. Silting objects in D b (Λ) correspond to pairs (Z, M ), where Z ∈ ind(Z) and M is a silting object of Z ⊥ D b (kA n+m−1 ). However, a silting object in D b (Λ) may have more than one indecomposable summand in the Z components. Thus, using silting reduction, we will obtain multiple descriptions of the same object. To rectify this problem, we classify 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 ∈ ind(Z j ) and define if i = j and ray(Z ) = ray(Z ); ray(Z ) < ray(Z ) otherwise, where ray(Z a ij ) ≤ ray(Z a kl ) if and only if i ≤ k and coray(Z a ij ) ≤ coray(Z a kl ) if and only if j ≤ l. Equivalently, for Z ∈ ind(Z i ), the total order is defined by successor sets, Lemma 6.13. The relation defined above defines a total order on indecomposables in the Z components.
Proof. Anti-symmetry: Suppose Z Z and Z Z with Z ∈ ind(Z i ) and Z ∈ 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 ) and ray(τ −1 Σ i−j Z ) ≤ ray(Z). In particular, it follows that ray(τ −1 Z ) ≤ ray(Σ j−i Z) ≤ ray(Z ), which is a contradition, since ray(τ −1 Z ) > ray(Z ). The same argument works when i > j.
Transitivity: Suppose Z Z and Z Z with Z ∈ ind(Z i ), Z ∈ ind(Z j ) and Z ∈ 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 −1 Σ j−i Z) ≤ ray(Z ) and the second inequality means that ray(Σ k−j Z ) ≤ ray(Z ). There are two subcases: first assume i ≤ k. In this case, apply τ Σ k−j to the condition arising from the first inequality and combine this with the second inequality to get ray(Σ k−i Z) ≤ ray(τ Σ k−j Z ) < ray(Σ k−j Z ) ≤ ray(Z ). Now assume that i > k and apply Σ k−j to the condition arising from the first inequality and combine with the second inequality to get ray(τ −1 Σ k−i Z) ≤ ray(Σ k−j Z ) ≤ ray(Z ).
Totality: Suppose Z ∈ ind(Z i ) and Z ∈ ind(Z j ). If i = j then it is clear that either Z Z or Z Z. Now suppose i < j. If ray(Σ j−i Z) ≤ ray(Z ) then Z Z and we are done, so suppose that ray(Σ j−i Z) > ray(Z ). Then it follows that ray(Σ i−j Z ) < ray(Z), in which case, because τ −1 increases the index of the ray by 1, one gets ray(τ −1 Σ i−j Z ) ≤ ray(Z) and hence Z Z. A similar argument holds in the case i > j. Thus, is indeed a total order.
In order to make the next definition, we need the following lemma. Lemma 6.14. If U ∈ ind(Z ⊥ ) then G Z 0 (U ) is also indecomposable.
Proof. Let U ∈ ind(Z ⊥ ). The Hom-hammocks for the Z components, Proposition 2.4, imply that U admits non-trivial morphisms to at most two suspensions of Z 0 .
Suppose U admits a non-trivial morphism to precisely one suspension of Z 0 . Completing this morphism to a distinguished triangle yields one of the triangles listed in Properties 1.1 (4). In particular, the cocone of the morphism, which defines G Z 0 (U ), is indecomposable. Now suppose U admits non-trivial morphisms to precisely two suspensions of Z 0 . Again, examining the Hom-hammocks incident with Z shows that this can only occur when U and the two suspensions of Z 0 sit in a mesh in one of the Z components. In this case, the cocone can be read off directly from the AR quiver; in particular, it is indecomposable.
We now ensure we identify each silting subcategory of M of D b (Λ) as precisely one pair (Z 0 , M ), with M a silting object of Z ⊥ D b (kA n−m+1 ) by insisting that Z 0 Z for each Z ∈ ind(Z) ∩ add M . Recall the map G from page 22 and define Z ⊥ ≺ := add{U ∈ ind(Z ⊥ ) | G Z 0 (U ) ∈ Z and G Z 0 (U ) ≺ Z 0 }, considered as an additive subcategory.
With the identification of D b (kA n+m−1 ) in D b (Λ(r, n, m)) of Remark 6.6, we can give an explicit description of the additive subcategory Z ⊥ ≺ . We first explicitly compute the map G Z 0 : ind(Z ⊥ ) → D b (Λ(r, n, m)) on objects, in the case Z 0 = Z 0 0,0 .
for the following pairs of U and i: Proof. The 'co-aisle' of the co-t-structure (A, B) with respect to which the function G is defined is given by B = susp ΣZ 0 = add{Σ i Z 0 | i ≥ 1}. Using Proposition 2.4, one can easily compute A := ⊥ B. If U ∈ A, then G Z 0 (U ) = U , so examining A ∩ Z ⊥ gives the list of exceptions above. One now computes the cocones G Z 0 (U ) directly using the triangles described in Lemma 6.14 above.
We can now use Proposition 6.15 to describe the additive subcategory Z ⊥ ≺ explicitly. By the standard heart of D b (kA n+m−1 ), we mean the one corresponding to the module category of the path algebra Γ = kA n+m−1 with the linear orientation 1 ← 2 ← · · · ← n + m − 1. We take as cohomological degree zero in D b (Γ) the standard heart containing the unique indecomposable objects in Z ⊥ ∩ Z admitting non-zero morphisms to Z 0 . Corollary 6.16. With the conventions described above, the additive subcategory Z ⊥ ≺ is where the sets of indecomposables A, B and C are defined as follows: A := {X ∈ mod(Γ) | Hom Γ (P r+m , X) = 0}; B := A ∩ {X ∈ mod(Γ) | Hom Γ (P r+m+1 , X) = 0} (empty when n − r = 1); C := {P (r + m − 1), . . . , P (n + m − 1)} (empty when n − r = 1).
We summarise this discussion in the following proposition, and obtain the main theorem of the section as a corollary. In this section we examine the algebra Λ(2, 3, 1) in detail. Let Z 0 = Z 0 0,0 and, as usual, write Z = thick(Z 0 ). Take the convention for homological degree as in Corollary 6.16. With this convention, we have the following identification of indecomposable objects in Z ⊥ and D b (kA 3 ): Using Corollary 6.16, Theorem 6.11 and the explicit calulation of the tilting objects, up to suspension, in Example 6.12, we compute the twelve families of silting objects in D b (kA 3 ) that lift to silting objects in D b (Λ(2, 3, 1)) containing Z 0 0,0 as the minimal indecomposable summand in the Z components. The results of this computation are presented in Table 1.
(2) If T ∈ D b (Λ) is a tilting object containing Z as a summand then F Z (T ) is a tilting object in Z ⊥ .
Proof. The proof is a direct computation. Without loss of generality, we may set Z = Z 0 0,0 . Consider the additive subcategory T := n =0 The subcategory T consists of the thick subcategory Z ⊥ ∩ ⊥ Z D b (k), which has just one indecomposable object in each homological degree, together with finitely many indecomposables in homological degrees 0,1 and 2.
Examining the Hom-hammocks from each of the indecomposables in Z ⊥ ∩ ⊥ Z shows that unless the object lies in homological degree 0, 1 or 2, there is not sufficient intersection tilting object silting family with T to give rise to a tilting object. Thus we must form tilting objects from only finitely many indecomposables. A detailed analysis of the Hom-hammocks of these finitely many indecomposables gives rise to the six tilting objects obtained from Z 0 0,0 and the following objects: The second claim can be directly computed.
Our computations lead us to state the following conjecture: Conjecture 7.2. Let Λ = Λ(r, n, m), Z ∈ ind(Z), write Z = thick(Z) and F Z : D b (Λ) → Z ⊥ D b (kA n+m−1 ), as usual. Then: (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 tilting object in Z ⊥ .

Appendix 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 vector space 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 aggregated homomorphism spaces (and similarly for obvious variants) and for the homomorphism complex, a complex of vector spaces with zero differential.
A.1. Properties of triangulated categories and their subcategories. A k-linear triangulated category D is said to be algebraic: if D arises as the homotopy category of a k-linear differential graded category; see [24]. Examples are bounded derived categories of k-linear abelian categories. Hom-finite: if dim Hom(D 1 , D 2 ) < ∞ for all objects D 1 , D 2 ∈ D. The bounded derived category 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 all of whose endomorphism rings are local. In this case, the direct sum decomposition is unique up to isomorphism. Bounded derived categories of k-linear Hom-finite abelian categories are Krull-Schmidt. The existence of a Serre functor is equivalent to the existence of Auslander-Reiten triangles; see [30, §I.2]. If Λ is a finite-dimensional k-algebra, then D b (Λ) has Serre duality if and only if Λ has finite global dimension; in this case, the Serre functor is given by the suspended Auslander-Reiten translation: S = Στ .
We conclude that D b (Λ(r, n, m)) is algebraic, Hom-finite, Krull-Schmidt and indecomposable for all choices of r, n, m. It has Serre duality if and only if n > r, which we always assume 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: 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 triangulated 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. susp(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 taking direct 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. C , the smallest full subcategory of D containing C that is closed under extensions, i.e. if C → C → C → ΣC is a triangle with C , C ∈ C then C ∈ C.
The ordered extension closure of a pair of subcategories (C 1 , C 2 ) of D is defined as 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 additive category and C a full subcategory of D.
Recall that C is called right admissible in D if the inclusion functor C → D admits a right adjoint. Analogously for left admissible. A subcategory C is called admissible if it is 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 the induced maps Hom(C , C) → Hom(C , D) are surjective for all C ∈ C. A morphism f : C → D is called a minimal right C-approximation if f g = f is only possible for isomorphisms 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. Sometimes, right C-approximations are called C-precovers and left C-approximations are called C-preenvelopes. If for all D ∈ D the induced map Hom(C , C) → Hom(C , D) above were bijective instead of surjective, then C would be even right admissible. In this sense, the morphism C → D 'approximates' the right adjoint to the inclusion functor.
This relationship is even stronger for Krull-Schmidt triangulated categories D: by [25, Proposition 1.3], a suspended subcategory C of D is contravariantly finite if and only if C is right admissible in D; dually for covariantly finite cosuspended subcategories. Thus, a thick subcategory C of D is functorially finite if and only if it is admissible.
If p is a path in Q, the corresponding path in (i, Q) is denoted (i, p). Let p = p 1 p 2 be a maximal path in (Q, ρ). Then the path (i, p 2 )(i,p)(i + 1, p 1 ) is called a full path inQ. We now define the relations: •ρ inherits the relations from ρ, i.e. for paths p, p 1 and p 2 in Q, if p ∈ ρ (resp. p 1 − p 2 ∈ ρ) then (i, p) ∈ρ (resp. (i, p 1 ) − (i, p 2 ) ∈ρ) for all i ∈ Z. • Let p be a path that contains a connecting arrow. If p is not a subpath of a full path then p ∈ρ. • Let p = p 1 p 2 p 3 and q = q 1 q 2 q 3 be maximal paths in (Q, ρ) with p 2 = q 2 . Then (i, p 3 )(i,p)(i + 1, p 1 ) − (i, q 3 )(i,q)(i + 1, q 1 ) ∈ρ for all i ∈ Z.
Denote the set of paths in (Q,ρ) by Pa.
It is sometimes easier to viewQ as a type of Z-graded quiver, where the inherited arrows have degree zero and the connecting arrows have degree one. For the algebrâ Λ(r, n, m), the quiverQ(r, n, m) is shown in the figure below, where the inherited arrows a i , b i , c i have degree zero and x i , y have degree one.
B.2. String modules. If Λ = kQ/ ρ is gentle, then its repetitive algebraΛ = kQ/ ρ is special-biserial and the indecomposableΛ-modules can be described explicitly using the formalism of string and band modules. In the case of the algebras Λ(r, n, m) of interest in this paper, only string modules occur as non-trivial indecomposable modules in the stable module category mod(Λ(r, n, m)). Thus we describe only the strings, omitting any further reference to bands. For each arrow a ∈Q 1 , introduce a formal inverseā with s(ā) = e(a) and e(ā) = s(a). For a path p = a 1 · · · a n the inverse pathp =ā n · · ·ā 1 .
A walk w of length l > 0 in (Q,ρ) is a sequence w = w 1 · · · w l , satisfying the usual concatenation requirements, where each w i is either an arrow or an inverse arrow. Formal inverses of walks are defined in the obvious way. Starting and ending vertices of walks and their inverses are defined analogously to those for paths.
A walk is called a string if it contains neither subwalks of the form aā orāa for some a ∈Q 1 , nor a subwalk v such that v ∈ρ orv ∈ρ. Denote the set of strings of (Q,ρ) by St. Modulo the equivalence relation w ∼w, the strings form an indexing set for the so-called string modules ofΛ; see [12] for precise details on how to pass to a representation-theoretic description of the modules.
Let (i, x) be a vertex inQ. From [14], there is a linear order on strings w and v in (Q,ρ) such that e(w) = e(v) = (i, x): namely, v < w ⇐⇒    either w = w v, where w = w 1 · · · w n with w n ∈Q 1 ; or v = v w, where v = v 1 · · · v m with v m ∈Q −1 1 ; or v = v c, w = w c with w n ∈Q 1 and v m ∈Q −1 1 . 34 Let w be a string. We define the successor w [1] to be the minimal string such that e(w [1]) = e(w) and w < w [1] in the linear order on strings ending at e(w). Similarly, define [1]w to be the minimal string such that s([1]w) = s(w) and w < [1]w in the linear order on strings starting at s(w). The predecessor operations are defined analogously and denoted by w[−1] and [−1]w, respectively. For n ∈ Z, we denote by w[n] = w[1] · · · [1] the n-times repeated sucessor operation; similarly for [n]w. Note that, applying the machinery of Butler and Ringel in [12], couched in the language of 'starting/ending at peaks/deeps', the strings w [1] and [1]w sit in the following AR mesh: B.3. Maps between string modules. It is straightforward to compute the maps between string modules. This was first observed in [13] and later generalised in [27]. We follow the neat exposition given in [34,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 1 · · · d n and f = f 1 · · · f m , in which we require d to be trivial or d n ∈ Q −1 1 and f to be trivial or f 1 ∈ Q 1 . Similarly, the set of substrings, Sub(w), is the set of decompositions in which we require d to be trivial or d n ∈ Q 1 and f to be trivial or f 1 ∈ Q −1 1 . A pair ((d 1 , e 1 , f 1 ), (d 2 , e 2 , f 2 )) ∈ Fac(v) × Sub(w) is called admissible if e 1 = e 2 or e 1 =ē 2 . Then the main results of [13] and [27] assert: B.4. Strings and maps for derived-discrete algebras. Here we list some pertinent facts about strings and string modules for discrete derived categories from [8], and establish some additional routine but useful properties. First we start with a lemma, which is collection of facts from [8]. Lemma B.3 ([8]). Denote the simple modules of Λ(r, n, m) by S(i) for −m ≤ i < n. In the coordinate system introduced in Properties 1.1, Z 0 0,0 = S(0). Then: (i) If m > 0 then S(−1) = X 1 0,0 ; in particular there is a simple module on the mouth of the X component. (ii) If r < n then S(n − r) lies on the mouth of the Y component.
Lemma B.4. Let A ∈ ind(D b (Λ(r, n, m))) with r > 1 and let i, k ∈ Z, 0 ≤ k < r. Then , and in all other cases the Hom spaces are zero. For r = 1 the Hom-spaces are as above, except Hom(X 0 ii , X 0 i,i+m ) = k 2 . Proof. First, note that the other two statements of Lemma 2.1 follow from these by Serre duality. We prove the statement in two cases, m > 0 and m = 0.
Case m > 0: Since the action of τ and Σ together is transitive on the set of objects at the mouths of the X components, by Lemma B.3, we may assume that X k ii = S(0, −1). By Proposition B.2, there is a morphism S(0, −1) to an indecomposable object X if and only if the string w corresponding to X admits a substring string decomposition w = def ∈ Sub(w) such that e = 1 (0,−1) orē = 1 (0,−1) . The unique direct arrow ending at (0, −1) is a (0,−2) when m > 1 and y −1 when m = 1. Likewise, the unique inverse arrow starting at (0, −1) isā (0,−2) when m > 1 andȳ −1 when m = 1. Hence, strings w with substring decompositions w = def , with e = 1 ± (0,−1) either end with the direct arrow a (0,−2) if m > 1 or y −1 if m = 1, or start with the inverse arrowā (0,−2) if m > 1 orȳ −1 if m = 1, or is precisely the trivial string 1 ± (0,−1) . Moving along a ray or coray corresponds to carrying out the successor operation [1]. Thus, the strings corresponding to objects on the same ray all end at the same vertex and those on the same coray all start at the same vertex. The chain of morphisms in Properties 1.1(5) corresponds to the linearly ordered set of strings ending at 1 (0,−1) . Thus, the 'extended ray' of Properties 1.1(5) consists of precisely the strings with substring decompositions def with e = 1 ± (0,−1) . Reading this off gives ray + (S(0, 1)) and coray − (SS(0, 1)) in the X components, and ray ± (Z 1 0,0 ) 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.
Case m = 0: We shall use an embedding D b (Λ(r, n, 0)) → D b (Λ(r, n, 1)). By [8, Lemma 3.1], the (stalk complex of the) indecomposable projective P (−1) lies on the mouth of the X component. Applying Lemma A.1 to P = thick D b (Λ(r,n,1)) (P (−1)) means that P ⊥ D b (Λ(r, n, 0)) as in the proof of Proposition 6.5. We can now use the case m > 0 to compute the Hom-hammocks in P ⊥ to give the result.
Remark B.5. In the 'extended ray' of strings ending at 1 (0,−1) , to obtain the part of this linearly ordered set corresponding to coray − (SS(0, −1)), we consider the inverse strings of those ending at 1 (0,−1) with the direct arrow a (0,−2) (for m > 1) or y −1 (for m = 1). We thus obtain strings starting with the corresponding inverse arrow, which gives the coray.
In the classification of silting objects for derived-discrete algebras, we need to explicitly locate the indecomposable projective Λ-module P (n − r) in the AR quiver of D b (Λ).
Lemma B.6. The projective P (n − r) ∈ Z in the AR quiver of D b (Λ(r, n, m)).