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

Discrete derived categories were studied initially by Vossieck (J Algebra 243:168–176, 2001) and later by Bobiński et al. (Cent Eur J Math 2:19–49, 2004). 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 [44]. 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.
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 study of the structure of algebras with discrete derived categories was begun by Vossieck, who showed that they are always gentle and classified them up to Morita equivalence. Bobiński et al. [9] obtained a canonical form for the derived equivalence class of these algebras; see Fig. 1. This canonical form is parametrised by integers n ≥ r ≥ 1 and m > 0, and the corresponding algebra denoted by (r, n, m). We restrict 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 of derived-discrete algebras and computed the suspension functor.
The structure exhibited in [9] is remarkably simple, which brings us to our principal motivation for studying these categories: they are sufficiently straightforward to make explicit computation highly accessible but also non-trivial enough to manifest interesting behaviour. For example, discrete derived categories contain natural examples of spherelike objects in the sense of [22]. The smallest subcategory generated by such a spherelike object has been studied in [25,32] and also in the context of (higher) cluster categories of type A ∞ in [24]. Indeed, in Proposition 6.4 we show that every discrete derived category contains two such higher cluster categories, up to triangle equivalence, as proper subcategories when the algebra has finite global dimension.
Furthermore, the structure of discrete derived categories is highly reminiscent of the categories of perfect complexes of cluster-tilted algebras of typeÃ n studied in [4]. This suggests approaches developed here to understand discrete derived categories are likely to 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 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 6.1 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 in [15], we investigate it further. See [20] for a different approach to capturing the 'smallness' of discrete derived categories. As another measure for categorical size, the Krull-Gabriel dimension of discrete derived categories has been computed in [10]; it is at most 2.
In Theorem 5.7 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 (r, n, m) is a useful tool, which is frequently employed here.
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 indecomposable objects of the same type: For each k = 0, . . . , r − 1, we label the indecomposable objects in X k , Y k , Z k as follows:

Properties 2.2 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).
(2) The AR translate of an object with coordinate (i, j) is the object with coordinate (i − 1, j − 1) in the same component, i.e. τ X k i, j = X k i−1, j−1 etc. (3) The suspension of indecomposable objects is given below, with k = 0, . . . , r − 2: In particular, r | X = τ −m−r and r | Y = τ n−r on objects. (4) There are distinguished triangles, for any i, j, d ∈ Z with d ≥ 0: (5) There are chains of non-zero morphisms for any i ∈ Z and k = 0, . . . , r − 1: Later, we will often use the 'height' of indecomposable objects in X or Y components. For X k i j ∈ ind(X k ), we set h(X k i j ) = j − i and call it the height of X k i j in the component X k . Similarly, for Y k i j ∈ ind(Y k ), we set h(Y k i j ) = i − j and call it the height of Y k i j in the component Y k . The mouth of an X or Y component consists of all objects of height 0.

Hom spaces: hammocks
For brevity, we will write := (r, n, m). In this section, for a fixed indecomposable object A ∈ D b ( ) we compute the so-called 'Hom-hammock' of A, i.e. the set of indecomposables B ∈ D b ( ) with Hom(A, B) = 0. By duality, this also gives the contravariant Homhammocks: Hom(−, A) = Hom(S −1 A, −) * . Therefore we generally refrain from listing the Hom(−, A) hammocks explicitly.
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:

Hammocks from the mouth
We start with a description of the Hom-hammocks of objects at the mouths of all Z A ∞ components. The proof relies on Happel's triangle equivalence of D b ( (r, n, m)) with the stable module category of the repetitive algebra of (r, n, m). As the repetitive algebras are special biserial algebras, the well-known theory of string (and band) modules provides a useful tool to understand the indecomposable objects and homomorphisms between them; we summarise this theory Appendix B.
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 ( (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 , the following six definitions give the rays/corays from/to/through V , respectively Note that, because of the orientation of the components, the (positive) 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 (negative) 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.7, recall that the Serre functor is given by suspension and AR translation: S = τ . Also, rays and corays commute with these three functors. Lemma 3. 1 Let A ∈ ind(D b ( (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 .

Hom-hammocks for objects in X components
Assume A = X k i j ∈ ind(X k ). In order to describe the various Hom-hammocks conveniently, we set A 0 := X k j j 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 now write down some standard triangles involving the objects 0 A, A 0 and A. The following lemma is completely general and holds in any Z A ∞ component of the AR quiver of a Krull-Schmidt triangulated category-we use the notation introduced above for the X components of D b ( (r, n, m)). Proof By Lemma 3.1 the composition, 0 A → A, of irreducible maps along a ray is non-zero. Likewise the composition, A → A 0 , of irreducible maps along a coray is non-zero.

Lemma 3.2 Let A be an indecomposable object of height h(A) ≥ 1 in a Z
We proceed by induction on h(A). If h(A) = 1, then both triangles coincide with the AR 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 0 − → A. These triangles fit into the following commutative diagram arising from the octahedral axiom.
we get the desired triangle as the rightmost vertical triangle in the diagram above.
We introduce notation for line segments in the AR quiver: given two indecomposable objects A, B ∈ D b ( (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, we recall our convention that X r = X 0 and note that 0 (SA) = τ ( 0 A).
Note that we shall treat the case r = 1 in Proposition 6.2 below; we continue to use the notation for the X components, however, the argument applies also to the Y components.
By the considerations above, we may assume that B lies in the interior of the region ray + (A A 0 ). Consider the following part of the AR quiver of D: where B is one irreducible morphism closer to ray + (A 0 ) and 0 B → B → B → ( 0 B) is the triangle from Lemma 3.2. Moreover, since is an autoequivalence, any (co)suspension of 0 B and B 0 must also lie on the mouth. We proceed by induction up each ray in the interior of the hammock starting with the ray closest to ray + (A 0 ). By induction, Hom(A, B ) = 0. Since 0 B = A 0 because B / ∈ ray + (A 0 ), we have Hom(A, 0 B) = 0 by Lemma 3.1. Applying Hom(A, −) to the triangle involving B above produces a long exact sequence in which the vanishing of Hom(A, ( 0 B)) gives Hom(A, B) = 0. By Lemma 3.1, A admits nontrivial morphisms to precisely A 0 and S( 0 A) on the mouth of an X component. Since r ≥ 2, ( 0 B) and 0 A lie in different components of the AR quiver so Proposition 3.4 (Hammocks Hom(X k , −)) Let A = X k i j ∈ ind(X k ) and assume r > 1. For any indecomposable object B ∈ ind(D b ( )) the following cases apply: and Hom(A, B) = 0 for all other B ∈ ind(D b ( )). For r = 1, these results still hold, except that the X -clauses are replaced by 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 3.1 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 6.2.
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 A A 0 . The existence of non-zero homomorphisms A → B for objects B ∈ R(A) follows directly from Lemma 3.3.
For the vanishing statement, we proceed by induction on the height of A. If A sits on the mouth of X k , then Lemma 3.1 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 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 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 dual of Lemma 3.3 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 i j ∈ 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 1.2(5), implies that Hom(A, B) = 0 for any B ∈ ray ± (Z k ii ). For any other . Applying Hom(A, −) leaves us with the exact sequence 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: For h > 0, as before we consider the AR mesh which has A as its top vertex: A → A ⊕ C → A → A . For any Z ∈ ind(Z k ), we apply Hom(−, Z ) to this triangle and find that Hom(A, Z ) = 0 implies Hom(A , Z ) = 0 or Hom(A , Z ) = 0. Therefore Hom(A, B) = 0 for all where the final equality is clear from the definitions.
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 3.1: 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 Y components
. This case is similar to the one above. Put 0 A := Y k ii to be the intersection of the coray through A with the mouth of Y k , and A 0 := Y k j j to be the intersection of the ray through A with the mouth of Y k .

Proposition 3.5 (Hammocks Hom
and assume r > 1. For any indecomposable object B ∈ ind(D b ( )) the following cases apply: and Hom(A, B) = 0 for all other B ∈ ind(D b ( )). For r = 1, these results still hold, except that the Y-clauses are replaced by Proof These statements are analogous to those of Proposition 3.4.

Hom-hammocks for objects in Z components
Let A = Z k i j ∈ ind(Z k ). By Lemma 3.1 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.
In fact, A 0 ∈ X k+1 and A 0 ∈ Y k+1 . Proposition 3.6 (Hammocks Hom(Z k , −)) Let A = Z k i j ∈ ind(Z k ) and assume r > 1. For any indecomposable object B ∈ ind(D b ( )) the following cases apply: Half of the first equality follows through the chain of equivalences . Likewise one obtains Hom(A, 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 −rc A with c ∈ N, shows by the finite global dimension of (r, n, m) that no objects in U admit non-trivial 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.9, coupled with the transitivity of the action of the automorphism group of D b ( (r, n, m)) on the Z component, which is proved in Sect. 5 using only Lemma 3.1, shows that Hom 1 (A, A) = 0 for all A ∈ Z.
Case l = k + 1: This is analogous to the previous case.
Remark 3.7 In the case that r > 1, Propositions 3.4, 3.5 and 3.6 say that each component of the AR quiver of D b ( (r, n, m)) is standard, i.e. that there are no morphisms in the infinite radical. Note that the components are not standard when r = 1.

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 [35] as well as spherical twists. In fact, a quite general and categorical construction has been given in [43]. 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 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 F A 1 ⊕ F A 2 → id. 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; see [22, §2.1] for details: 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 * ) .
This definition assumes n ≥ 2 but a single object E should be considered an 'exceptional 1-cycle' if E is a spherical object, i.e. there is an integer k with S(E) ∼ = k (E) and Hom • (E, E) = k ⊕ −k k. In this light, the above definition, and statement and proof of Theorem 4.5 are generalisations of the treatment of spherical objects and their twist functors as in [27, §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 → 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 not-the morphism α n : E n → k n E 1 prevents it from being one, and instead creates a cycle.

Remark 4.3
All objects in an exceptional n-cycle are fractional Calabi-Yau: since S(E i ) ∼ = k i E i+1 for all i, applying the Serre functor n times yields S n (E i ) ∼ = k E i , where k:=k 1 + · · · + k n . Thus the Calabi-Yau dimension of each object in the cycle is k/n. Example 4.4 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 (cohX ) be its bounded derived category. The Serre functor of D is given by S(−) = 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 S n (E i ) = dn E i ⊗ ω n X ∼ = k E i , hence k = k 1 + · · · + k n = dn and E i ⊗ ω n X ∼ = E i . 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 ). Proof We define two classes of objects of D by E:={ l E i | l ∈ Z, i = 1, . . . , n} and := E ∪ E ⊥ . Note that E and hence are closed under suspensions and cosuspensions. It is a simple and standard fact that is a spanning class for D, i.e. ⊥ = 0 and ⊥ = 0; the latter equality depends on the Serre condition S(E i ) ∼ = k i (E i+1 ). 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 : E 1 → k 1 E 2 . It follows immediately from the definition of exceptional cycle that F E * (E 1 ) = E 1 ⊕ −k n E n . The cone of the evaluation morphism is easily seen to sit in the following triangle The left-hand morphism has an obvious splitting, this implies the zero morphism in the middle. 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.
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 This also works if n = 2 and k 1 = −k 2 (with unchanged left-hand vertical arrow).
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 ⊥ .
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 , 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 canonical map of functors S −1 T E * S → T E * 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 : both vector spaces are zero for objects from E ⊥ ; for the top row, use Hence T E * commutes with the Serre functor on , and so by more general theory is essentially surjective; see [27,Corollary 1.56], this is the place where we need the assumption that D is indecomposable.
Remark 4. 6 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 S R → id. [22] and also Sect. 6.3. We will now show that reasonable exceptional cycles come with a spherelike object. For this purpose, we call an exceptional cycle Recall that an exceptional n-cycle (E 1 , . . . , E n ) comes with a tuple of integers (k 1 , . . . , k n ) and that we have set k = k 1 + · · · + k n .
Proof Inductively, we construct a series of objects X 1 , . . . , X n with the following properties for i < n: These conditions are satisfied for X 1 :=E 1 , because Hom • (E 1 , E 2 ) is generated by α 1 : Assume X i with i < n − 1 has already been constructed. By (iii) there is a unique object X i+1 with a non-split distinguished triangle is an exceptional sequence with just one (graded) morphism by (ii) and (iii). So in the above triangle, the object X i+1 is, up to suspension, the left mutation of that pair. In particular, X i+1 is exceptional. By construction, X i+1 satisfies (ii).

by (ii) and the definition of exceptional cycles, hence Hom
Having constructed X n−1 in this fashion, we can use (iii) to define This triangle induces a commutative diagram of complexes of k-vector spaces and we know that Hom • (X n−1 , X n−1 ) = Hom • ( l n−1 E n , l n−1 E n ) = k, since X n−1 and E n are exceptional. Moreover, we get Hom • (X n−1 , l n−1 E n ) = k from applying Hom • (−, E n ) to the triangle defining X n−1 and using X n−2 ∈ E 1 , , . . . , E n−2 , none of which map to E n . In particular, the map f sends the identity to the morphism X n−1 → l n−1 E n defining X n . Hence f is an isomorphism, thus Hom • (X n , l n−1 E n ) = 0 and we arrive at the isomorphism where k = k 1 + · · · + 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, which implies X n−1 ∼ = l n−1 E n . But we have X n−1 ∈ thick(E 1 , . . . , E n−1 ) by (ii) and also E n / ∈ thick(E 1 , . . . , E n−1 ) as E * is irredundant; which gives a contradiction. Therefore we find that g = 0 and thus, The degrees of non-zero maps in Hom • (E n , X ) and Hom • (X, E 1 ) are computed with the same methods as above.

Example 4.8 The additional hypothesis on
Following the iterative construction of the above proof, we get X 1 = S(1), X 2 = I (2) and X 3 = M. This forces X = X 4 = 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 −1 on objects. In light of this example, it would also be interesting to investigate twists coming from redundant exceptional cycles further.

Autoequivalence groups of discrete derived categories
We now use the general machinery of the previous section to show that categories D b ( (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 weak factorisation of the Auslander-Reiten translation: T X T Y = τ −1 on objects. Moreover, T X acts trivially on Y and T Y acts trivially on X ; see Proposition 5.4 and Corollary 5.5 for the precise assertions. We then give an explicit description of the group of autoequivalences of D b ( (r, n, m)) in Theorem 5.7.
The category D = D b ( (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 3.1, hence any object at the mouth is. This point also gives the second condition of exceptional cycles: and at the boundary step we have SE m+r = τ X 0 11 = X 0 00 = 1−r X 0 m+r,m+r = 1−r E 1 , where we freely make use of the results stated in Sect. 2. 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 3.1.
The same reasoning works for Y, now with the boundary step degree computation The next lemma shows that the functors F E * and F F * of the last section take on a particularly simple form, where we use the notation 0 X, X 0 , 0 Y, Y 0 from Sects. 3.2, 3.3: Proof This follows immediately from the definition of these functors in Sect. 4, Proposition 3.4 and Properties 1.2(3), i.e. r | X = τ −m−r and 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 0 , and analogously for F F * .
The actual choice of exceptional cycle is not relevant as the following easy lemma shows. We only state it for E * but the analogous statement holds for F * , with the same proof. This allows us to write T X instead of T E * and T Y instead of T F * .

Lemma 5.3
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 * = 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, see [27,Lemma 1.30]). Finally, we have

Proposition 5.4
The twist functors T X and T Y act as follows on objects of D b ( ), where X ∈ X , Y ∈ Y and k = 0, . . . , r − 1 and i, j ∈ Z:

Corollary 5.5 The twist functors T X and T Y act simply transitively on each component Z k and factorise the inverse AR translation:
Moreover, T X , T Y and act transitively on ind(Z).

Proof of the proposition By Lemma 3.1, we have Hom
Action of T X on objects of X : we recall that the proof of Theorem 4.5 showed , and furthermore k 1 = · · · = k m+r −1 = 1 and k m+r = 1−r from Lemma 5.1.
for all i-as explained in Lemma 5.3, this holds for any exceptional cycle at an X mouth. Since T X is an equivalence and each X component is of type Action of T X on objects of Z: is one of the special triangles of Properties 1.2 (4): Application of AR translations extends this computation to arbitary Z ∈ Z 0 , and suspending extends it to all Z components, thus . . , 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 5.6
Each of X and Y is a thick triangulated subcategory of D.
Proof The proof of Proposition 5.4 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 In this proof, we will write D = D b ( (r, n, m)) and = (r, n, m).
Recall that units u ∈ induce inner automorphisms c u (α) = uαu −1 , and thus a normal subgroup Inn( ) ⊆ Aut( ). It is a well-known fact that inner automorphisms induce autoequivalences of mod( ) and D b ( ) which are isomorphic to the identity; see [46, §3]. The quotient group Out( ) = Aut( )/ Inn( ) acts faithfully on modules. The form of the quiver and the relations for (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( ). However, choosing an indecomposable idempotent e (i.e. a vertex) together with a scalar λ ∈ k * produces a unit u = 1 + (λ − 1)e, and hence an inner automorphism c u ∈ Aut( ). 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. Since the quiver of has one cycle, we see that an (n + m − 1)-subtorus of the subgroup (k * ) m+n of arrow-scaling automorphisms consists of inner automorphisms. Furthermore, the automorphism scaling all arrows simultaneously by the same number is easily seen not to be inner, hence, Out( ) = k * . Note that the class of an automorphism scaling precisely one arrow also generates Out( ).
Step 2: ϕ ∈ Aut(D) is isomorphic to the identity on objects ⇐⇒ ϕ ∈ Out( ). By 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( ) and the object , thus giving rise to ϕ : → , i.e. an automorphism which by Step 1 can be taken to be outer. Step The suspension commutes with all exact functors. Next, to see Step 4: , since the suspension and the twist functors act transitively on ind(Z) by Corollary 5.5. Therefore, ψ: Moreover, since all autoequivalences commute with τ (because they commute with the Serre functor S = τ and with ) and Z is a Z A ∞ ∞component, either ψ is the identity on ind(Z) or else ψ flips ind(Z) along the Z τ (Z ) axis. However, the latter possibility is excluded by the action of 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 1 → Z 2 for some Z 1 , Z 2 ∈ ind(Z). Moreover, the morphism Z 1 → Z 2 is unique up to scalars by Theorem 6.1. (The proofs in that section make no use of the autoequivalence group. Note that by the proof of Theorem 6.1, morphism spaces between indecomposable objects in Z are 1-dimensional, even for r = 1.) Hence ϕ actually fixes all indecomposable objects and thus all objects of D b ( ).

By Steps 3 and 4, Aut(D) =
, T X , T Y , Out( ) is abelian. Properties 1.2(3) and Proposition 5.4 imply that the autoequivalence −r T m+r This is elementary algebra: let A be a free abelian group of finite rank and a 0 ∈ A, ). Both maps are easily checked to be group homomorphisms and bijective. Moreover, the left-hand square commutes: Therefore we obtain an induced isomorphism between the right-hand quotients: For the case at hand, A = Z 3 and a 0 = (r, n, m) ∈ Z 3 and hence A/a 0 ∼ = Z 2 × Z/ with the greatest common divisor = (r, n, m), by the theory of elementary divisors.
Question It is natural to speculate about the action of the various functors on maps. More precisely, we ask whether 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 0 = id in Theorem 5.7.

Hom spaces: dimension bounds and graded structure
In this section, we prove a strong result about D b ( ):=D b ( (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 6.3.

Hom space dimension bounds
The bounds are given in the the following theorem; for more precise information in case r = 1 see Proposition 6.2. Proof Our strategy for establishing the dimension bound follows that of the proofs of the Hom-hammocks. Let A, B ∈ ind(D b ( (r, n, m))) and assume r > 1. In this proof, we use the abbreviation hom = dim Hom. We want to show hom(A, B) ≤ 1 by considering the various components separately. 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 3.6. 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 3.6 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. Proposition 6.2 Let r = 1 and X, A ∈ ind(X ). Then The following diagram illustrates the proposition: all indecomposables A in the heavily shaded square have dim Hom(X, A) = 2: X SX Proof The argument is similar to the computation of the Hom-hammocks in the Z components from Sect. 3. We proceed in several steps.
Step 1: For any A ∈ ind(X ) of height 0 the claim follows from Lemma 3.1. Otherwise we consider the AR mesh which has A at the top, and let A and A be the two indecomposables of height h(A) − 1. There are two triangles (see Lemma 3.2): 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 3.1 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 3.1 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, 0 A) + hom(X, A), again from sequence (3).
Step 3: ψ = 0 unless A ∈ ray + ( −1 0 SX ) and μ = 0 unless A ∈ coray − ( X 0 ). If A / ∈ ray + ( −1 X 0 ) ∪ ray + ( −1 0 SX ) then hom(X, 0 A) = 0 and so ψ = 0 trivially. Therefore, we just need to consider A ∈ ray + ( −1 X 0 ) but A / ∈ ray + ( −1 0 SX ), and in this case hom(X, 0 A) = 1. It is clear that the maps going down the coray from X to X 0 span a 1dimensional subspace of Hom(X, 0 A), which therefore is the whole space. Using properties of the Z A ∞ mesh, the composition of such maps with a map along ray + (X 0 ) from X 0 to A defines a non-zero element in Hom(X, A). Thus the map Hom(X, 0 A) → Hom(X, 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 morphisms in Properties 1.2 (5) to show that the map Hom(X, −1 A) → Hom(X, −1 A 0 ) in the sequence (4) is surjective.
Step 4: If ray + ( −1 X 0 ) (or coray − ( 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, for determining the regional constants hom(X, −). Note that these are precisely the rays and corays required to bound the regions ray + (X X 0 ) and coray − ( 0 (SX ), SX ) of the statement of the proposition. Considering their relative positions on the mouth, −1 0 SX is always furthest to the left and X 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 − ( X 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.
In the same vein, consider A ∈ ray + (S 0 X ) ∩ coray − ( X 0 ), the object of minimal height in region L. Observe that A ∈ ray + (τ −1 S 0 X ) ∩ coray − ( X 0 ) = add X from which we can see that the map to Hom(X, 0 A) in (3) is surjective. Now A / ∈ ray + ( −1 0 SX ), so ψ = 0 by Step 3 and hence hom(X, A) = hom(X, A ). With A in region I where we already know hom(X, A ) = 1, we get hom(X, −) = 1 on region L.
Finally we now take up A ∈ ray + ( −1 S 0 X ) ∩ coray − ( X 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 1.2(5), we see that there is a map X → 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.

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 [8,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 indecomposable objects of discrete derived categories in terms of their homological properties.
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 A = X 0 i j . We are looking for all d ∈ Z with Hom d (A, A) = Hom(A, d A) = 0. By Proposition 3.4, this is only possible for either d ≡ 0 or d ≡ 1 modulo r .
We start with the first possibility: d = lr for some l ∈ Z. By Properties 1.2(3) and (2), which is an indecomposable object in X 0 sharing its height h = j − i with A. Again using Proposition 3.4, we can reformulate the claim as follows: where the set of h + 1 objects in the second line are precisely the objects in ray + (A A 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 6.1, all Hom spaces have dimension 1 when r > 1, these two computations give For r = 1 and A = X 0 i j ∈ ind(X ), by Proposition 3.4 the hammock Hom(A, −) = 0 is ray + (A A 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 Hom • (B, B) for B ∈ ind(Y) is proved in the same way, now using h = i − j, r = τ n−r and the hammocks specified by Proposition 3.5.

Coarse classification of objects
Our previous results allow us to give a crude grouping of the indecomposable objects of D b ( (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  Hom <0 (A, A) = 0 otherwise. Remark 6.5 In fact, the direct sum E 1 ⊕ E 2 of two exceptional objects E 1 and E 2 with

Proposition 6.4 Each object A ∈ ind(D b ( (r, n, m))) is of exactly one type below:
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 [22,Appendix].
Proof We know from Lemma B.9 that the projective module P(n − r ) ∈ Z. This is an exceptional object by Proposition 3.6. As the autoequivalence group acts transitively on ind(Z) by Corollary 5.5, every indecomposable object of Z is exceptional. The remaining parts of the proposition all follow from Lemma 6.3. We only give the argument for A ∈ ind(X ), as the one for indecomposable objects of Y runs entirely parallel.

Reduction to Dynkin type A and classification results
Two keys for understanding the homological properties of algebras are t-structures and cot-structures, especially bounded ones. The main theorem of [33], cited in the appendix as Theorem A.8, 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 sections "Torsion pairs, t-structures and co-t-structures" and "König-Yang bijections" in "Appendix 1" for a more detailed overview. It turns out, however, that any bounded t-structure in D b ( (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 ( (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 ( (r, n, m)) is length, then we obtain a semi-orthogonal decompositon D b ( (r, n, m)) = D b (k A 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. D b ( (r, n, m)) are length

All hearts in
The main result of this section is:

Proposition 7.1 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 these statements separately in the following lemmas. The first lemma is a general statement regarding t-structures, which is well known to experts, and included for the convenience of the reader. The second is a generalisation of the corresponding statement for the algebra (1, 2, 0) proved in [33]; the third is a general statement about Hom-finite abelian categories. Proof Let 0 = H ∈ H. We show that n H / ∈ H for any n = 0. First suppose that n H ∈ H for some n > 0. Then H ∈ −n H. We have −n H ⊆ −n Y ⊆ Y. The condition Hom(X, Y) = 0 then implies that Hom(H, H ) = 0, a contradiction. Now suppose −n H ∈ H for some n > 0. In this case we have n H ⊆ n X ⊆ X, whence the condition Hom( X, Y) = 0 gives the required contradiction.

Lemma 7.3 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. By Properties 1.2(3), the action of r partitions Z k into strips r +m rays thick. These are further partitioned into rectangles of size (r + m) × (n − r ) such that every rectangle in each strip is sent to precisely one rectangle in each other strip by some power of r ; see Fig. 4 for an illustration. The complement of the Hom-hammocks Hom <0 (Z , −) and Hom <0 (−, Z ) intersects finitely many such rectangles up to the action of r . Since each Z k component is just a suspension of each of the other Z components, it follows that the objects of ind(H) ∩ Z must be (co)suspensions of a finite set of objects. Now, Lemma 7.2 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 6.4, 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 cannot lie in the heart ([25, Lemma 4.1(a)], for instance) and so again, up to (co)suspension, ind(H) ∩ X is finite. The argument for the Y component is similar.

Lemma 7.4 Let H be a Hom-finite abelian category with finitely many indecomposable objects. 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. By assumption, this sum is finite and hence L ∈ H. We define the function d :  ( (r, n, m)) is equivalent to mod( ), for a finite-dimensional algebra of finite representation type. Note that, by work of Schröer and Zimmermann [42], 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 ( (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.

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. ( (r, n, m) ( (r, n, m)). Moreover, D b ( (r, n, m)) has a full exceptional sequence.
Proof By Proposition 6.4, 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( i Z | i ∈ Z) and that Z is an admissible subcategory of D b ( ); for this last claim see [11,Theorem 3.2]. Furthermore D b ( ) = Z ⊥ , Z is the standard semi-orthogonal decomposition for an exceptional object; see Appendix A.7 for details. Lemma B.9 places the indecomposable projective P(n − r ) in the Z component of the AR quiver of D b ( ). Using the transitive action of the autoequivalence group on ind(Z), see Corollary 5.5, we thus can assume, without loss of generality, that Z = P(n − r ) = e n−r . There is a full embedding ι : D b ( / e n−r ) → D b ( ) with essential image thick D b ( ) (e n−r ) ⊥ = Z ⊥ ; see, for example, [2,Lemma 3.4]. Inspecting the Gabriel quiver of / e n−r , 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: Therefore / e n−r is an iterated tilted algebra of type A n+m−1 . It is well known that this implies D b ( / e n−r ) D b (k A n+m−1 ); see [21]. Combining these pieces, we . The final claim about D b ( ) having a full exceptional sequence follows at once from the fact that D b (k A n+m−1 ) has one.

Remark 7.7
The subcategory of type D b (k A n+m−1 ) can be explicitly identified in the AR quiver of D b ( (r, n, m)); see Fig. 4. The choice of right orthogonal to Z was arbitrary, since Serre duality provides an equivalence ⊥ Z → Z ⊥ , X → S(X ). We mention in passing that the thick subcategory Z is equivalent to D b (k A 1 ).
The silting objects of D b (k A n+m−1 ) are well understood from work of Keller and Vossieck in [31]. We shall now bootstrap their classification to discrete derived categories using the technique of silting reduction introduced by Aihara and Iyama in [1].

Silting reduction
The main technical tool in the classification is the following result of Aihara and Iyama in  ( (2, 5, 2)). Remaining components 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 7.15. The subcategory B:= susp N is the 'co-aisle' of the co-t-structure associated to N (see Theorem A.8) and thus covariantly finite in U. Putting this together with U being functorially finite in D, it gives rise to a co-t-structure (A, B) in D, where A:= ⊥ B. Now let K be a silting subcategory of U ⊥ and consider the approximation triangle of K ∈ K with respect to the co-t-structure (A, B), In their proof of Theorem 7.8 in [1], Aihara and Iyama show that is a silting subcategory of D. Definition 7.9 Assume the notation and hypotheses of Theorem 7.8 above. Given a silting 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 Note that here, in contrast to elsewhere in this paper, we require that the approximation is minimal to ensure well-definition of the map G N . Furthermore, we stress here that G N is a map not a functor.
In light of Proposition 7.6, the natural choice for a functorially finite thick subcategory to which we can apply Theorem 7.8 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 ( (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. Simple-minded collections (see [33] for the definition) are also an important focus of current research. Therefore, while we do not use them in this paper, it is useful to highlight in the corollary below that the following lemma also applies to them. In other words, any silting object in D b ( ) can be described by a pair (Z , M ) consisting of an indecomposable object Z ∈ Z and a silting object M ∈ Z ⊥ D b (k A n+m−1 ).
We now make a brief expository digression explaining Keller and Vossieck's classification of silting subcategories of D b (k A t ), from which the silting subcategories of D b ( (r, n, m)) can be 'glued'.

Classification of silting objects in Dynkin type A
Consider the following diagram of the AR quiver of D b (k A t ) with coordinates (g, h) with g ∈ Z and h ∈ {1, . . . , t}.
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 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. 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 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 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 7.12 ([31], Section 4) The assignment Q → T Q induces a bijection between isomorphism 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 = t i=1 T i can be re-ordered to give a strong, full exceptional sequence {T 1 , . . . , T t }, see [31,Section 5.2]. We now have the following classification of silting objects in D b (k A t ).

⊕ · · · ⊕ 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 classification of tilting (and hence silting) objects in D b (k A 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, 1), (0, 2), (0, 3). These are precisely the tilting objects having an indecomposable summand U with minimal g(U ) = 0. In particular, these are precisely the exceptional sequences in D b (k A 3 ) containing one of P(i) for 1 ≤ i ≤ 3 as a least element.
To obtain all tilting objects (up to suspension), we next consider those for which there exists an indecomposable summand U with minimal g(U ) = 1. These correspond precisely to τ −1 applied to each of the diagrams 1 to 6 . Observe that τ −1 5 = 1 and τ −1 6 = 2 . Therefore, up to suspension, we pick up only four more tilting objects. Next we consider those for which there exists an indecomposable summand U with minimal g(U ) = 2, which correspond precisely to τ −2 applied to each of the diagrams 1 to 6 . We have τ −2 3 = 3 , τ −2 4 = 4 , τ −2 5 = τ −1 1 , and τ −2 6 = τ −1 2 , which leaves, up to suspension, only τ −2 1 and τ −2 2 as new tilting objects. Continuing in this way, one sees that, up to suspension, these are all tilting objects. Hence, there are twelve tilting objects in D b (k A 3 ) up to suspension:

Classification of silting objects for derived-discrete algebras
As this section is rather technical, the reader may find it helpful to refer to the detailed example, (2, 3, 1) studied in Sect. 8 whilst reading this section.
We first start with some preliminary results regarding the indecomposability of the images of indecomposable objects under the map G Z : Z ⊥ → D b ( (r, n, m)) from Definition 7.9, 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 ( (r, n, m)) on objects in the case Z = Z 0 0,0 .

Proposition 7.15
If r > 1 and Z = Z 0 0,0 , and G:=G Z 0,0 , then G(U ) = U for all but finitely many (up to positive suspension) U ∈ ind(Z ⊥ ). The exceptions are: Proposition 7.16 If r = 1 and Z = Z 0,0 , and G:=G Z 0,0 , then G(U ) = U for all but finitely many (up to positive suspension) U ∈ ind(Z ⊥ ). The exceptions are: Proof of Propositions 7.15 and 7. 16 We do the calculations for the generic case with r > 1 in Proposition 7.15; those for Proposition 7.16 are similar. The function G is defined via the 'co-aisle' of the co-t-structure (A, B) with B = susp Z 0 = add{ i Z 0 | i ≥ 1}. Using Proposition 3.6, one can easily compute A = ⊥ B. If U ∈ A, then G(U ) = U , so examining 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 0 When 0 ≤ i ≤ r , the relevant triangle belongs with the family in (3) above, and can be computed analogously. However, when i > r , we need to take the cocone of the morphism i Z −r −m,0 → i Z 1 −r −m,r −n ⊕ Z 1 0,0 . We claim that the cone of Z −r −m,0 → Z 1 −r −m,r −n ⊕ Z 1 0,0 is Z 1 0,r −n . To show this, we compute the cocone of Z 1 −r −m,r −n ⊕ Z 1 0,0 → Z 1 0,r −n via the following octahedron: where the second column is the split triangle, and the third column is a standard triangle from Properties 1.2(4). The triangle forming the bottom row is none other than X 1 −r −m,−1 → Z 1 −r −m,0 → Z 1 0,0 → X 1 −r −m,−1 , which computes the cocone C = Z 1 −r −m,0 as claimed.
Corollary 7. 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 act transitively on the Z components, it is sufficient to see this for Z = Z 0 0,0 . This is clear from the computations in (the proof of) Proposition 7.15 above.
Silting objects in D b ( ) correspond to pairs (Z , M ), where Z ∈ ind(Z) and M is a silting object of Z ⊥ D b (k A n+m−1 ). However, a silting object in D b ( ) 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 where ray(Z a i j ) ≤ ray(Z a kl ) if and only if i ≤ k and coray(Z a i j ) ≤ coray(Z a kl ) if and only if j ≤ l. Equivalently, for Z ∈ ind(Z i ), the total order is defined by successor sets, The following diagrams indicate the indecomposables Z ∈ Z with Z Z :

Lemma 7.18 The relation defines a total order on ind(Z).
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( j−i Z ) ≤ ray(Z ) and ray(τ −1 i− j Z ) ≤ ray(Z ). In particular, it follows that ray(τ −1 Z ) ≤ ray( 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 j−i Z ) ≤ ray(Z ) and the second inequality means that ray( k− j Z ) ≤ ray(Z ). There are two subcases: first assume i ≤ k. In this case, apply τ k− j to the condition arising from the first inequality and combine this with the second inequality to get ray( k−i Z ) ≤ ray(τ k− j Z ) < ray( k− j Z ) ≤ ray(Z ). Now assume that i > k and apply k− j to the condition arising from the first inequality and combine with the second inequality to get ray(τ −1 k−i Z ) ≤ ray( 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( j−i Z ) ≤ ray(Z ) then Z Z and we are done, so suppose that ray( j−i Z ) > ray(Z ). Then it follows that ray( i− j Z ) < ray(Z ), in which case, because τ −1 increases the index of the ray by 1, one gets ray(τ −1 i− j Z ) ≤ ray(Z ) and hence Z Z . A similar argument holds in the case i > j. Thus, is indeed a total order. Using Corollary 7.17, we now ensure we identify each silting subcategory of M of D b ( ) as precisely one pair (Z , M ), with M a silting object of Definition 7. 19 We define the following additive subcategory of D: ( (r, n, m)) of Remark 7.7, using Proposition 7.15, we now give an explicit description of the additive subcategory Z ⊥ ≺ . Recall from Proposition 7.6 that Z ⊥ D b (k A n+m−1 ). Let := k A n+m−1 be the path algebra of the A n+m−1 quiver with the linear orientation: Consider the unique i mod( ) ⊂ D b ( ) that contains the indecomposable objects in Z ⊥ ∩Z admitting non-zero morphisms to Z . In Lemma 7.20 below, when we specify mod( ), we shall mean precisely this copy sitting inside D b (k A n+m−1 ).

Lemma 7.20 With the conventions described above, the additive subcategory
where the sets of indecomposables A, B and C are defined as follows: Proof This is a direct computation using Proposition 7.15, the total order on the indecomposable objects of the Z components of Lemma 7.18, and the identification of the subcategory from Remark 7.7.
We summarise this discussion in the following proposition, and obtain the main theorem of the section as a corollary.  ( (r, n, m)).

A detailed example: (2, 3, 1)
In this section we examine the algebra (2, 3, 1) in detail. Let Z = Z 0 0,0 and write Z = thick(Z ). Take the convention for homological degree as in Lemma 7.20. With this convention, we identify the indecomposable objects in Z ⊥ and of D b (k A 3 ) as follows: Using Lemma 7.20, Theorem 7.13 and the explicit calulation of the tilting objects, up to suspension, in Example 7.14, we compute the twelve families of silting objects in D b (k A 3 ) that lift to silting objects in D b ( (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.
We make the following observation regarding tilting objects in D b ( (2, 3, 1).
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 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 Table 1 The twelve tilting objects in k A 3 giving rise to the silting objects containing Z 0 0,0 as the -minimal summand in Z for (2,3,1) Tilting object in k A 3 Silting family in (2, 3, 1) 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 8.2
For an arbitrary Z ∈ ind(Z), writing = (r, n, m) and Z = thick(Z ) and F Z : (1) There are finitely many tilting objects in D b ( ) containing Z as a summand.
(2) If T ∈ D b ( ) is a tilting object containing Z as a summand then F Z (T ) is a tilting object in Z ⊥ . D b ( (2, 3, 1
Let us start with the silting object and set Z = Z 0 00 and Z = thick(Z ). As explained above, N corresponds to the object By Proposition 7.15, M lifts under G Z to the silting object ( (2, 3, 1)).
The corresponding co-t-structure t h ethick 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), t h eadditive subcategory of D containing C, the smallest full subcategory of D containing C which is closed under finite coproducts and direct summands. ind(C), t h es e to findecomposable 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 → C 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 Capproximation 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.
Note that in the case that D is a Hom-finite, k-linear, Krull-Schmidt category, the existence 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-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 (possibly nonexistent) right adjoint to the inclusion functor.
For Krull-Schmidt triangulated categories D, these concepts coincide: 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, which were introduced in [31]. However, we follow the terminology of [1]. Note that all subcategories are assumed to be additive and closed under isomorphisms. Let M be a subcategory of a triangulated category D. In particular, if a category D as in the lemma has a silting object D, then rk K 0 (D) = #{isomorphism classes of indecomposable summands of D} and in particular, the right-hand side is independent of the silting object. We record two further easy observations: Proof The existence of a silting object implies that M and N are each additively generated by finitely many objects. Now apply Corollary A.3.

A.5 Torsion pairs, t-structures and co-t-structures
We assume again that D is a k-linear triangulated category. A pair (X, Y) of full subcategories closed under direct summands 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 a triangle X → D → Y → 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 finite and 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 a t-structure (see below).
The torsion pair (X, Y) will be called bounded if i∈Z i X = i∈Z i Y = D. Torsion pairs appear in three important guises, namely (X, Y) is called a [37] (also weight structure [12] For historical reasons, when the terminology 'semi-orthogonal decomposition' is used the torsion pair is often written as Y, X . Furthermore, a t-structure is stable if and only if it is also a co-t-structure. If (X, Y) is a t-structure then its heart H = X ∩ 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 −1 H; see, for example, [13,Section 3].
If (X, Y) is a co-t-structure then its co-heart M = X∩ −1 Y is a partial silting subcategory of D; see, for instance, [36,Corollary 5.9]. Note that, if M is abelian then it 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]): Remark A.7 If (X, Y) is a t-structure then the approximation triangle is functorial and called the truncation triangle, with X → D being a right minimal X-approximation called the 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 → D has a right adjoint (given by D → X ) and Y → D has a left adjoint. In particular, truncations are minimal approximations. We mention that 't-structure' is an abbreviation for 'truncation structure'.

A.6 König-Yang bijections
The notions of silting subcategories, t-structures and co-t-structures for finite dimensional k-algebras are related by the following bijections of König and Yang. Before we state them, recall an abelian category A is called a length category if it is both artinian and noetherian.

(ii) bounded t-structures in D b (mod( )) whose heart is a length category, (iii) bounded co-t-structures in K b (proj( )).
Under these bijections, a silting subcategory M ⊂ D b (mod( )) for of finite global dimension is mapped to the

A.7 Exceptional sequences and semi-orthogonal decompositions
The notion of semi-orthogonal decomposition D = C 1 , C 2 is synonymous with that of a stable t-structure (C 2 , C 1 ), see Appendix A.5, and leads to equivalences C 1 ∼ = D/C 2 and C 2 ∼ = D/C 1 . An admissible subcategory C ⊂ D produces two semi-orthogonal decomposi- An object E of a k-linear triangulated category D is exceptional if Hom(E, E) = k and Hom =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 the text): thick D (E) = add( i E | i ∈ Z). Morever, the subcategory thick D (E) is then admissible by [11,Theorem 3.2]. Hence an exceptional object E leads to semi-orthogonal decompositions An exceptional sequence in D is a tuple (E 1 , . . . , E t ) of exceptional objects such that i.e. all homomorphisms occur in degree zero. A full, strong exceptional sequence (E 1 , . . . , E t ) gives rise to a tilting object E 1 ⊕ · · · ⊕ E t . Similarly, a full exceptional sequence (E 1 , . . . , E t ) with Hom >0 (E i , E j ) = 0 for all i, j gives rise to a silting object.

Appendix B: The repetitive algebra and string modules
For a finite-dimensional algebra , Happel showed in [21] that there is a full embedding F : D b ( ) → mod(ˆ ), where mod(ˆ ) denotes the stable module category of the repetitive algebraˆ , and F is called the Happel functor. A finite-dimensional algebra is gentle if and only if its repetitive algebra is special biserial (see [41,Proposition]). For such an algebra, there is a convenient description of all the indecomposable objects of mod(ˆ ) using string and band modules; see [41]. Since the algebras (r, n, m) are gentle, this machinery applies. Moreover, only string modules occur; indeed it is this absence of band modules that is responsible for discreteness. Thus, we shall omit any further reference to band modules.
In this section we shall recall the construction of the repetitive algebra, the description of string modules and the maps between them. We then apply these results to the derived-discrete algebras (r, n, m).

B.1 The repetitive algebra
The notion of a repetitive algebra was introduced by Hughes and Waschbüsch in [26]. The standard references are [26,40,41]. The relations for (r, n, m) are also recalled in [9]. The following summary is based on [41]. Following the rules above, we can read off the following relations forˆ (r, n, m); see [9,Section 3]. The degrees of the arrows should be inferred by the presence of the connecting arrows 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 0 ; • x k x k−1 = 0 for k = n − r + 2, . . . , n − 1; n−r ya −m · · · a −1 = 0 if r > 1, and in the case r = 1 we have ya −m · · · b n−1 − b 0 · · · b n−1 ya −m · · · a −1 = 0; • Any path starting at (i, k) and ending at (i + 1, k + 1), with k = 0 and −m ≤ k ≤ n − r , that contains y as a subpath is zero.

B.2 String modules
Let = kQ/ ρ be a special biserial algebra. We describe strings for the bound quiver (Q, ρ), which give rise to string modules. The references are [16] and [45]. We remind the reader that all modules are right modules.
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. We also define two strings of length zero, namely, for each x ∈ Q 0 there are trivial strings 1 + x and 1 − x . We write s(1 ± x ) = e(1 ± x ) = x and set (1 ± x ) −1 = 1 ∓ x . For technical reasons, in order to define composition of strings with trivial strings, we need to introduce string functions σ, ε : Q 1 → {−1, 1} satisfying the following properties: • If a 1 = a 2 ∈ Q 1 with s(a 1 ) = s(a 2 ) then σ (a 1 ) = −σ (a 2 ).
We are now able to define compositions of strings. For strings v = v 1 · · · v m and w = w 1 · · · w n of length at least 1 this is done in the obvious way: the composition vw is defined if vw = v 1 · · · v m w 1 · · · w n is a string. However, if w = 1 ± x then vw is defined if 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, this condition 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 . We shall write M(w) for the corresponding string module. We direct the reader to [16,Section 3] for precise details on how to pass to a representation-theoretic description of the modules. (2, 3, 1), where we relabel the arrows in the figure above as a = a −1 , b = b 0 , c = b 1 , d = c 2 , x = x 2 and y = y to avoid cumbersome subscripts. Consider the string (−2, x)(−1,c)(−1,b)(−1, x)(0,c)(0,b), which we write as xcbxcb 0 for short, withb 0 = (0,b) to determine the 'degrees' of each of the arrows. This can be represented pictorially by the diagram below.

Example B.1 Considerˆ
• In this picture, we read from left to right, direct arrows point downwards and to the right and inverse arrows point downwards and to the left.

B.3 Irreducible maps between string modules and a linear order
A complete description of the irreducible maps between string modules was obtained in [16]. Given a string w, the irreducible maps whose source is the string module M(w) can be determined by 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]).
(1) Adding a hook on the left: If there exists a 0 ∈ Q 1 such that a 0 w is defined, then let a 1 · · · a n be the maximal direct string starting at s(a 0 ). Then w [1]:=ā n · · ·ā 1 a 0 w; the irreducible map is the natural inclusion. (2) Removing a cohook on the left: If there is no a ∈ Q 1 such that aw is defined, then w = v 1 · · · v n−1vn w with v i ∈ Q 1 and a string w , where v 1 · · · v n−1 is a maximal direct substring at the beginning of w. Then w [1]:=w ; the irreducible map is the natural projection map.
There is a dual algorithm, which adds a hook or removes a cohook on the right to output the string [1] We illustrate these concepts in the diagrams below; in the left-hand diagram, we add a hook, in the right-hand diagram, we remove a cohook.
These operations give rise to AR sequences/triangles:  [1]. [1]w The process of adding a hook or removing a cohook determines a total order on strings ending 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 given vertex. This is the Geiß total order [19], which we describe next. This is motivated by the fact that v ≤ w implies that Hom(M(v), M(w)) = 0, see Corollary B.5 below.
Let x ∈ Q 0 . 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 ∈ {−1, 1}. Namely, this. All the inequalities above correspond to adding a hook, except for cyad y −1 < y −1 and abcxbcy −1 < bcy −1 , which correspond to removing a cohook.

B.4 Maps between string modules
It is straightforward to compute the maps between string modules. This was first observed in [17] and later generalised in [18] and [34]. We follow the neat exposition given in [42, §2]. For a string w, define the set of factor strings, Fac(w), to be the set of decompositions w = de f 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 picture may be useful: on the left we illustrate a factor string decomposition and on the right, a substring decomposition.

B.5 Strings and maps for derived-discrete algebras
Here we list some pertinent facts about strings and string modules for discrete derived categories from [9], and establish some additional routine but useful properties.
Lemma B.6 ([9]) Denote the simple modules of (r, n, m) by S(i) for −m ≤ i < n. In the coordinate system introduced in Properties 1.2(1), Z 0 0,0 = S(0). Then: (1) If m > 0 then S(−1) = X 1 0,0 ; in particular there is a simple module on the mouth of the X component. (2) If r < n then S(n − r ) lies on the mouth of the Y component.
Lemma B.7 Let A ∈ ind(D b ( (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 .
The other two statements of Lemma 3.1 follow from these by Serre duality.
Proof Case m > 0: Since, by [9,Theorem B], the action of τ and together is transitive on the set of objects at the mouths of the X components, by Lemma B.6, we may assume that X k ii = S(0, −1). Note that the chain of morphisms in Properties 1.2(5) corresponds to the totally ordered set of strings ending at 1 (0,−1) in the Geiß total order. By Corollary B.5, it follows that each object in this totally ordered set admits a morphism from S(0, −1). This totally ordered set is shown in Example B.2 for the algebraˆ (2, 3, 1). Now it remains to show that these are the only objects admitting morphisms from S(0, −1). Let w be a string that admits a substring decomposition de f ∈ Sub(w) with e = 1 (0,−1) or e = 1 (0,−1) . We claim that w = de(= d) or w = e f (= f ). This is clear since there is only one arrow ending at (0, −1), namely (0, a −2 ) when m > 1 and (−1, y) when m = 1. These strings (or their inverses) are precisely the strings listed in Geiß total order in Properties 1.2(5). Therefore, by Theorem B.4, these are precisely the indecomposable objects admitting a morphism from S(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.
Remark B.8 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.
The next fact is used in particular for Proposition 7.6, the classification of silting objects.

Lemma B.9
The projective P(n − r ) ∈ Z in the AR quiver of D b ( (r, n, m)).