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

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## Abstract

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.

### Keywords

Discrete derived category Auslander–Reiten quiver Hom-hammock Twist functor Silting object t-Structure String algebra### Mathematics Subject Classification

16G10 16G70 18E30## 1 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 \(n\ge r \ge 1\) and \(m>0\), and the corresponding algebra denoted by \(\Lambda (r,n,m)\). 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 \(A_\infty \) 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 type \(\tilde{A}_n\) 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 \(\Lambda (r,n,m)\) is a useful tool, which is frequently employed here.

In Sect. 7, we address the classification of bounded t-structures and co-t-structures in \(\mathsf {D}^b(\Lambda (r,n,m))\), 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 [13], and their co-t-structure analogues [29]. Further investigation into the properties of (co-)t-structures and the stability manifolds is conducted in the sequel [14]; see also [38].

We study (co-)t-structures indirectly via silting subcategories, which generalise tilting objects and behave like the projective objects of hearts of bounded t-structures. In general, one cannot get all bounded t-structures in this way, but in Proposition 7.1, we show that the heart of each bounded t-structure in \(\mathsf {D}^b(\Lambda (r,n,m))\) is equivalent to \(\mathsf {mod}(\Gamma )\), where \(\Gamma \) is a finite-dimensional algebra of finite representation type. The upshot is that using the bijections of König and Yang [33], classifying silting objects is enough to classify all bounded (co-)t-structures. We show that \(\mathsf {D}^b(\Lambda (r,n,m))\) admits a semi-orthogonal decomposition into \(\mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) and the thick subcategory generated by an exceptional object. Using Aihara and Iyama’s silting reduction [1], we classify the silting objects in Theorem 7.22. We finish with an explicit example of \(\Lambda (2,3,1)\) in Sect. 8.

## 2 Discrete derived categories and their AR-quiver

We always work over a fixed algebraically closed field \({\mathbf {k}}\). All modules will be finite-dimensional right modules. Throughout, all subcategories will be additive and closed under isomorphisms.

### 2.1 Discrete derived categories

We are interested in \({\mathbf {k}}\)-linear, Hom-finite triangulated categories which are small in a certain sense. One precise definition of such smallness is given by Vossieck [44]; here we present a slight generalisation of his notion.

### Definition 2.1

A derived category (or, more generally and intrinsically, a Hom-finite triangulated category with a bounded t-structure) \(\mathsf D\) is *discrete (with respect to this**t*-*structure)*, if for every map \(v:{\mathbb {Z}}\rightarrow K_0(\mathsf D)\) there are only finitely many isomorphism classes of objects \(D\in \mathsf D\) with \([H^i(D)]=v(i)\in K_0(\mathsf D)\) for all \(i\in {\mathbb {Z}}\).

Let us elaborate on the connection to [44]: Vossieck speaks of *finitely supported, positive* dimension vectors \(v\in K_0(\mathsf D)^{({\mathbb {Z}})}\) which he can do since he has \(\mathsf D=\mathsf {D}^b(\Lambda )\) for a finite-dimensional algebra \(\Lambda \), so \(K_0(\Lambda )\cong {\mathbb {Z}}^r\). In our slight generalisation of his notion, we cannot do so, but for finite-dimensional algebras the new notion gives back the old one: if *v* is negative somewhere, there will be no objects of that dimension vector whatsoever. For the same reason, we don’t have to assume that *v* has finite support: if it doesn’t, the set of objects of that class is empty.

Note that our definition of discreteness appears to depend on the choice of bounded t-structure. Throughout this article, we shall be interested in the bounded derived category \(\mathsf {D}^b(\Lambda )\) of a finite-dimensional algebra \(\Lambda \). We shall always use discreteness with respect to the standard t-structure, whose heart is \(\mathsf {mod}(\Lambda )\), the category of finite-dimensional right \(\Lambda \)-modules. However, in [15], the results of this article will be used to show that the categories studied here are discrete with respect to any bounded t-structure.

Obviously, derived categories of path algebras of type ADE Dynkin quivers are examples of discrete categories. Moreover, [44] shows that the bounded derived category of a finite-dimensional algebra \(\Lambda \), which is not of derived-finite representation type, is discrete if and only if \(\Lambda \) is Morita equivalent to the bound quiver algebra of a gentle quiver with exactly one cycle having different numbers of clockwise and anticlockwise orientations.

*r*,

*n*,

*m*.

### 2.2 The AR quiver of \(\varvec{\mathsf {D}^b(\Lambda (r,n,m))}\)

*In the following, we always make this assumption.*Therefore the derived category \(\mathsf {D}^b(\Lambda (r,n,m))\) enjoys duality in the form

*r*components; these are denoted by

### Properties 2.2

- (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. \(\tau X^k_{i,j} = X^k_{i-1,j-1}\) etc. - (3)The suspension of indecomposable objects is given below, with \(k=0,\ldots ,r-2\): In particular, \(\Sigma ^r|_\mathcal {X}= \tau ^{-m-r}\) and \(\Sigma ^r|_\mathcal {Y}= \tau ^{n-r}\) on objects.
- (4)There are distinguished triangles, for any \(i,j,d\in {\mathbb {Z}}\) with \(d\ge 0\):
- (5)There are chains of non-zero morphisms for any \(i\in {\mathbb {Z}}\) and \(k=0,\ldots ,r-1\):

Later, we will often use the ‘height’ of indecomposable objects in \(\mathcal {X}\) or \(\mathcal {Y}\) components. For \(X^k_{ij} \in \mathsf {ind}(\mathcal {X}^k)\), we set \(h(X^k_{ij}) = j-i\) and call it the *height* of \(X^k_{ij}\) in the component \(\mathcal {X}^k\). Similarly, for \(Y^k_{ij} \in \mathsf {ind}(\mathcal {Y}^k)\), we set \(h(Y^k_{ij}) = i-j\) and call it the *height* of \(Y^k_{ij}\) in the component \(\mathcal {Y}^k\). The *mouth* of an \(\mathcal {X}\) or \(\mathcal {Y}\) component consists of all objects of height 0.

## 3 Hom spaces: hammocks

For brevity, we will write \(\Lambda \, {:}{=}\, \Lambda (r,n,m)\). In this section, for a fixed indecomposable object \(A\in \mathsf {D}^b(\Lambda )\) we compute the so-called ‘Hom-hammock’ of *A*, i.e. the set of indecomposables \(B\in \mathsf {D}^b(\Lambda )\) with \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0\). By duality, this also gives the contravariant Hom-hammocks: \({{\mathrm{\mathrm {Hom}}}}(-,A) = {{\mathrm{\mathrm {Hom}}}}(\mathsf {S}^{-1}A,-)^*\). Therefore we generally refrain from listing the \({{\mathrm{\mathrm {Hom}}}}(-,A)\) hammocks explicitly.

### 3.1 Hammocks from the mouth

We start with a description of the Hom-hammocks of objects at the mouths of all \({\mathbb {Z}}A_\infty \) components. The proof relies on Happel’s triangle equivalence of \(\mathsf {D}^b(\Lambda (r,n,m))\) with the stable module category of the repetitive algebra of \(\Lambda (r,n,m)\). 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.

*i*,

*j*). Recall the conventions that \(j\ge i\) if \(V\in \mathcal {X}\) whereas \(i\ge j\) if \(V\in \mathcal {Y}\). Denoting the AR component of

*V*by \({\mathcal C}\) and its objects by \(V_{a,b}\), the following six definitions give the

*rays/corays from/to/through*

*V*, respectivelyNote that, because of the orientation of the components, the (positive) ray of an indecomposable \(X^k_{ii}\in \mathcal {X}^k\) at the mouth consists of indecomposables in \(\mathcal {X}^k\) reached by arrows going out of \(X^k_{ii}\), while in the \(\mathcal {Y}\) 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: \(\mathsf {S}= \Sigma \tau \). Also, rays and corays commute with these three functors.

### Lemma 3.1

### 3.2 Hom-hammocks for objects in \(\mathcal {X}\) components

\(A_0 \,{:}{=}\, X^k_{jj}\) to be the intersection of the coray through

*A*with the mouth of \(\mathcal {X}^k\), and\(_0 A \,{:}{=}\, X^k_{ii}\) to be the intersection of the ray through

*A*with the mouth of \(\mathcal {X}^k\).

*A*sits at the mouth, then \(A = A_0 = {}_0A\).

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 \({\mathbb {Z}}A_\infty \) component of the AR quiver of a Krull–Schmidt triangulated category—we use the notation introduced above for the \(\mathcal {X}\) components of \(\mathsf {D}^b(\Lambda (r,n,m))\).

### Lemma 3.2

*A*be an indecomposable object of height \(h(A) \ge 1\) in a \({\mathbb {Z}}A_\infty \) component of a Krull–Schmidt triangulated category. Let Open image in new window be the AR triangle with

*A*at its apex; assuming \(C = 0\) if \(h(A)=1\). Then there are triangles

*u*and

*v*, respectively.

### Proof

By Lemma 3.1 the composition, \({}_0A \rightarrow A\), of irreducible maps along a ray is non-zero. Likewise the composition, \(A \rightarrow A_0\), of irreducible maps along a coray is non-zero.

We proceed by induction on \(h(A)\). If \(h(A) = 1\), then both triangles coincide with the AR triangle \({}_0A \rightarrow A \rightarrow A_0 \rightarrow \Sigma {}_0A\); in particular, \(A' = {}_0 A\) and \(A'' = A_0\).

We introduce notation for line segments in the AR quiver: given two indecomposable objects \(A,B\in \mathsf {D}^b(\Lambda (r,n,m))\) 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 \(\overline{AB}\). Finally, we recall our convention that \(\mathcal {X}^r=\mathcal {X}^0\) and note that \(_0(\mathsf {S}A) = \Sigma \tau ({}_0A)\).

### Lemma 3.3

Consider \(\mathsf {D}^b(\Lambda (r,n,m))\) with \(r > 1\). If \(A \in \mathsf {ind}(\mathcal {X}) \cup \mathsf {ind}(\mathcal {Y})\) then for each indecomposable object \(B \in \mathsf {ray}_{\! +}(\overline{AA_0})\) we have \({{\mathrm{\mathrm {Hom}}}}(A,B) \ne 0\).

Note that we shall treat the case \(r=1\) in Proposition 6.2 below; we continue to use the notation for the \(\mathcal {X}\) components, however, the argument applies also to the \(\mathcal {Y}\) components.

### Proof

Let *A* be an indecomposable object in an \(\mathcal {X}\) component. Let \(B \in \mathsf {ray}_{\! +}(\overline{AA_0})\). If \(B \in \mathsf {ray}_{\! +}(A) \cup \overline{AA_0} \cup \mathsf {ray}_{\! +}(A_0)\) (see Fig. 2), then \({{\mathrm{\mathrm {Hom}}}}(A,B) \ne 0\), using Serre duality and Lemma 3.1 if \(B \in \mathsf {ray}_{\! +}(A_0)\) (Fig. 2).

*B*lies in the interior of the region \(\mathsf {ray}_{\! +}(\overline{AA_0})\). Consider the following part of the AR quiver of \(\mathsf {D}\):where \(B''\) is one irreducible morphism closer to \(\mathsf {ray}_{\! +}(A_0)\) and \({}_0 B \rightarrow B \rightarrow B'' \rightarrow \Sigma ({}_0 B)\) is the triangle from Lemma 3.2. Moreover, since \(\Sigma \) 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 \(\mathsf {ray}_{\! +}(A_0)\). By induction, \({{\mathrm{\mathrm {Hom}}}}(A,B'') \ne 0\). Since \({}_0 B \ne A_0\) because \(B \notin \mathsf {ray}_{\! +}(A_0)\), we have \({{\mathrm{\mathrm {Hom}}}}(A,{}_0 B) = 0\) by Lemma 3.1. Applying \({{\mathrm{\mathrm {Hom}}}}(A,-)\) to the triangle involving \(B''\) above produces a long exact sequence in which the vanishing of \({{\mathrm{\mathrm {Hom}}}}(A,\Sigma ({}_0 B))\) gives \({{\mathrm{\mathrm {Hom}}}}(A,B) \ne 0\). By Lemma 3.1, *A* admits nontrivial morphisms to precisely \(A_0\) and \(\mathsf {S}({}_0 A)\) on the mouth of an \(\mathcal {X}\) component. Since \(r \ge 2\), \(\Sigma ({}_0 B)\) and \({}_0 A\) lie in different components of the AR quiver so \(\Sigma ({}_0 B) \ne {}_0 A\). If \(\Sigma ({}_0 B) = \mathsf {S}({}_0 A)\) then \({}_0 B = \tau ({}_0 A)\), which contradicts \(B \in \mathsf {ray}_{\! +}(\overline{AA_0})\). Hence, \({{\mathrm{\mathrm {Hom}}}}(A,\Sigma ({}_0 B)) = 0\).\(\square \)

### Proposition 3.4

(Hammocks \({{\mathrm{\mathrm {Hom}}}}(\mathcal {X}^k,-)\)) Let \(A=X^k_{ij}\in \mathsf {ind}(\mathcal {X}^k)\) and assume \(r>1\).

- \(B\in \mathcal {X}^k\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! +}(\overline{AA_0})\);

- \(B\in \mathcal {X}^{k+1}\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {coray}_{\! -}(\overline{{}_0(\mathsf {S}A),\mathsf {S}A})\);

- \(B\in \mathcal {Z}^k\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! \pm }(\overline{Z^k_{ii}Z^k_{ji}})\)

and \({{\mathrm{\mathrm {Hom}}}}(A,B)=0\) for all other \(B\in \mathsf {ind}(\mathsf {D}^b(\Lambda ))\).

- \(B\in \mathcal {X}^0\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! +}(\overline{AA_0}) \cup \mathsf {coray}_{\! -}(\overline{_0(\tau ^{-m}A),\tau ^{-m}A})\).

### 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 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*\(B\in \mathcal {X}^k\): For any indecomposable object \(A\in \mathcal {X}^k\), write *R*(*A*) for the subset of \(\mathcal {X}^k\) specified in the statement, i.e. bounded by the rays out of *A* and \(A_0\), and the line segment \(\overline{AA_0}\). The existence of non-zero homomorphisms \(A\rightarrow B\) for objects \(B\in R(A)\) 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 \(\mathcal {X}^k\), then Lemma 3.1 states indeed that the \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0\) if and only if *B* is in the ray of *A*. Note that *R*(*A*) is precisely \(\mathsf {ray}_{\! +}(A)\) in this case.

Now let \(A\in \mathcal {X}^k\) 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 \(A'\rightarrow A\oplus C\rightarrow A''\rightarrow \Sigma A'\), where \(h(A') = h(A'') = h-1\) and \(h(C) = h-2\). (If \(h=1\), 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 \(R(A'')\subset R(A)\subset R(A')\cup R(A'')\). We start with an object \(B\in \mathcal {X}^k\) such that \(B\notin R(A')\cup R(A'')\). By the induction hypothesis, we know that \(R(A')\), *R*(*C*) and \(R(A'')\) are the Hom-hammocks in \(\mathcal {X}^k\) for \(A'\), *C*, \(A''\), respectively. Since *B* is contained in none of them, we see that \({{\mathrm{\mathrm {Hom}}}}(A',B)={{\mathrm{\mathrm {Hom}}}}(C,B)={{\mathrm{\mathrm {Hom}}}}(A'',B)=0\). Applying \({{\mathrm{\mathrm {Hom}}}}(-,B)\) to the given AR triangle shows \({{\mathrm{\mathrm {Hom}}}}(A,B)=0\).

It remains to show that \({{\mathrm{\mathrm {Hom}}}}(A,D)=0\) for objects \(D \in (R(A')\cup R(A'')) \backslash R(A)\) which can be seen to be the line segment \(\overline{A'A'_0}\). Again we work up from the mouth: \({{\mathrm{\mathrm {Hom}}}}(A,A'_0)= 0\) and \({{\mathrm{\mathrm {Hom}}}}(A,\tau A'_0)=0\) by Lemma 3.1, as before. The extension \(D_1\) given by \(\tau A'_0\rightarrow D_1\rightarrow A'_0\rightarrow \Sigma \tau A'_0{}{}{}\) is the indecomposable object of height 1 on \(\overline{A'A'_0}\). Applying \({{\mathrm{\mathrm {Hom}}}}(A,-)\) to this triangle, we find \({{\mathrm{\mathrm {Hom}}}}(A,D_1)=0\), as required. The same reasoning works for the objects of heights \(2,\ldots ,h-1\) on the segment.

*Case*\(B\in \mathcal {X}^{k+1}\): 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 \(\mathsf {S}A\). However, by Serre duality we see that \({{\mathrm{\mathrm {Hom}}}}(A,B) = {{\mathrm{\mathrm {Hom}}}}(B, \mathsf {S}A)^* \ne 0\), as required. The statement that \({{\mathrm{\mathrm {Hom}}}}(A,B) = 0\) for all other \(B\in \mathcal {X}^{k+1}\) can be proved by an induction argument which is analogous to the one given in the first case above.

*Case*\(B\in \mathcal {Z}^k\): For any indecomposable object \(A=X^k_{ij}\in \mathcal {X}^k\), write

*V*(

*A*) for the region in \(\mathcal {Z}^k\) 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 \({{\mathrm{\mathrm {Hom}}}}(A,B) \ne 0\) for \(B \in V(A)\). The first chain of morphisms in Properties 1.2(5), implies that \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0\) for any \(B\in \mathsf {ray}_{\! \pm }(Z^k_{ii})\). For any other \(B'=Z^{k}_{i+s,t} \in V(A)\), so \(t \in {\mathbb {Z}}\) and \(s \in \{1,\ldots ,h(A) = j-i\}\), we consider the special triangle \(X^k_{i,i+s-1}\rightarrow B\rightarrow B'\rightarrow \Sigma X^k_{i,i+s-1}{}{}{}\) from Properties 1.2(4), where \(B=Z^k_{it}\in \mathsf {ray}_{\! \pm }(Z^k_{ii})\). Applying \({{\mathrm{\mathrm {Hom}}}}(A,-)\) leaves us with the exact sequence

*A*but has strictly lower height. Similarly, we observe that the right-hand term of the sequence vanishes: \(0 = {{\mathrm{\mathrm {Hom}}}}(X^k_{i,i+s-1}, \tau A) = {{\mathrm{\mathrm {Hom}}}}(A,\Sigma X^k_{i,i+s-1})\). Hence \({{\mathrm{\mathrm {Hom}}}}(A,B') = {{\mathrm{\mathrm {Hom}}}}(A,B) \ne 0\).

For the Hom-vanishing part of the statement, we again use induction on the height \(h\,{:}{=}\,h(A)\ge 0\). For \(h=0\), Lemma 3.1 gives \(V(A)=\mathsf {ray}_{\! \pm }(Z^k_{ii})\). For \(h>0\), as before we consider the AR mesh which has *A* as its top vertex: \(A'\rightarrow A\oplus C\rightarrow A''\rightarrow \Sigma A'{}{}{}\). For any \(Z\in \mathsf {ind}(\mathcal {Z}^k)\), we apply \({{\mathrm{\mathrm {Hom}}}}(-,Z)\) to this triangle and find that \({{\mathrm{\mathrm {Hom}}}}(A,Z)\ne 0\) implies \({{\mathrm{\mathrm {Hom}}}}(A',Z)\ne 0\) or \({{\mathrm{\mathrm {Hom}}}}(A'',Z)\ne 0\). Therefore \({{\mathrm{\mathrm {Hom}}}}(A,B) = 0\) for all \(B \notin V(A')\cup V(A'') = V(A)\), where the final equality is clear from the definitions.

*Remaining cases: * These comprise vanishing statements for entire AR components, namely \({{\mathrm{\mathrm {Hom}}}}(\mathcal {X}^k,\mathcal {X}^j)=0\) for \(j\ne k,k+1\), and \({{\mathrm{\mathrm {Hom}}}}(\mathcal {X}^k,\mathcal {Y}^j)=0\) for any *j*, and \({{\mathrm{\mathrm {Hom}}}}(\mathcal {X}^k,\mathcal {Z}^j)=0\) for \(j\ne k\). 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 \(\mathcal {X}\) and \(\mathcal {Y}\), Hom vanishing can be seen using induction on height and considering a square in the AR mesh. The vanishing to the \(\mathcal {Z}^k\) components with \(k\ne j\) follows similarly. \(\square \)

### 3.3 Hom-hammocks for objects in \(\mathcal {Y}\) components

\(^0\! A \,{:}{=}\, Y^k_{ii}\) to be the intersection of the coray through

*A*with the mouth of \(\mathcal {Y}^k\), and\(A^0 \,{:}{=}\, Y^k_{jj}\) to be the intersection of the ray through

*A*with the mouth of \(\mathcal {Y}^k\).

### Proposition 3.5

(Hammocks \({{\mathrm{\mathrm {Hom}}}}(\mathcal {Y}^k,-))\) Let \(A=Y^k_{ij}\in \mathsf {ind}(\mathcal {Y}^k)\) and assume \(r>1\).

- \(B\in \mathcal {Y}^k\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {coray}_{\! +}(\overline{AA^0})\);

- \(B\in \mathcal {Y}^{k+1}\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! -}(\overline{{}^0(\mathsf {S}A),\mathsf {S}A})\);

- \(B\in \mathcal {Z}^k\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {coray}_{\! \pm }(\overline{Z^k_{ii}Z^k_{ij}})\)

and \({{\mathrm{\mathrm {Hom}}}}(A,B)=0\) for all other \(B\in \mathsf {ind}(\mathsf {D}^b(\Lambda ))\).

- \(B\in \mathcal {Y}^0\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {coray}_{\! +}(\overline{AA^0}) \cup \mathsf {ray}_{\! -}(\overline{^0(\tau ^n A),\tau ^n A})\).

### Proof

These statements are analogous to those of Proposition 3.4. \(\square \)

### 3.4 Hom-hammocks for objects in \(\mathcal {Z}\) components

\(A_0 \,{:}{=}\) the unique object at the mouth of an \(\mathcal {X}\) component for which \({{\mathrm{\mathrm {Hom}}}}(A, A_0) \ne 0\),

\(A^0 \,{:}{=}\) the unique object at the mouth of a \(\mathcal {Y}\) component for which \({{\mathrm{\mathrm {Hom}}}}(A, A^0) \ne 0\).

### Proposition 3.6

(Hammocks \({{\mathrm{\mathrm {Hom}}}}(\mathcal {Z}^k,-))\) Let \(A=Z^k_{ij}\in \mathsf {ind}(\mathcal {Z}^k)\) and assume \(r>1\).

- \(B\in \mathcal {X}^{k+1}\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! +}(\mathsf {coray}_{\! -}(A_0))\);

- \(B\in \mathcal {Y}^{k+1}\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! -}(\mathsf {coray}_{\! +}(A^0))\);

- \(B\in \mathcal {Z}^k\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! +}(\mathsf {coray}_{\! +}(A))\);

- \(B\in \mathcal {Z}^{k+1}\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! -}(\mathsf {coray}_{\! -}(\mathsf {S}A))\)

and \({{\mathrm{\mathrm {Hom}}}}(A,B)=0\) for all other \(B\in \mathsf {ind}(\mathsf {D}^b(\Lambda ))\).

- \(B\in \mathcal {Z}^0\):
then \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0 \Longleftrightarrow B \in \mathsf {ray}_{\! +}(\mathsf {coray}_{\! +}(A)) \cup \mathsf {ray}_{\! -}(\mathsf {coray}_{\! -}(\mathsf {S}A))\).

The hammocks described in Proposition 3.6 are illustrated in Fig. 3.

### Proof

The cases \(B\in \mathcal {X}^{k+1}\) and \(B\in \mathcal {Y}^{k+1}\) follow by Serre duality from Proposition 3.4 and Proposition 3.5, respectively.

*B*; see Properties 1.2(4):where \({}_0B=X^l_{aa}\) is the unique object at the mouth of a \(\mathcal {X}\) component with \({{\mathrm{\mathrm {Hom}}}}({}_0B,B)\ne 0\) and similarly \({}^0B=Y^l_{bb}\) is unique at a \(\mathcal {Y}\) mouth with \({{\mathrm{\mathrm {Hom}}}}({}^0B,B)\ne 0\). We get two exact sequences by applying \({{\mathrm{\mathrm {Hom}}}}(A,-)\):

*Case*\(l\ne k, k+1\): In this case \({{\mathrm{\mathrm {Hom}}}}(A,B)={{\mathrm{\mathrm {Hom}}}}(A,B')={{\mathrm{\mathrm {Hom}}}}(A,B'')\) follows from the above triangles via these exact sequences and Lemma 3.1. But this implies \({{\mathrm{\mathrm {Hom}}}}(A,B)={{\mathrm{\mathrm {Hom}}}}(A,Z)\) for all \(Z \in \mathcal {Z}^l\) and in particular \({{\mathrm{\mathrm {Hom}}}}(A,B)={{\mathrm{\mathrm {Hom}}}}(A,\Sigma ^{cr}B)\) for all \(c\in {\mathbb {Z}}\). It follows that \({{\mathrm{\mathrm {Hom}}}}(A,B)=0\) as \(\Lambda (r,n,m)\) has finite global dimension.

*Case*\(l=k\): Again, we first show that the dimension function \(\hom (A,-)\) is constant on certain regions of \(\mathcal {Z}^k\). In particular, we have

- \({\mathcal U}:\)
The upwards-open region including \(\mathsf {ray}_{\! +}(\tau A) \backslash \{\tau A\}\) but excluding \(\mathsf {coray}_{\! -}(\tau A)\);

- \({\mathcal L}:\)
The left-open region including \(\mathsf {ray}_{\! -}(\tau A) \cup \mathsf {coray}_{\! -}(\tau A)\);

- \({\mathcal D}:\)
The downwards-open region including \(\mathsf {coray}_{\! +}(\tau A)\backslash \{\tau A\}\) but excluding \(\mathsf {ray}_{\! -}(\tau A)\);

- \({\mathcal R}:\)
The right-open region excluding \(\mathsf {ray}_{\! \pm }(\tau A) \cup \mathsf {coray}_{\! \pm }(\tau A)\).

Using (1) above coupled with the fact that \({\mathcal U}\) contains infinitely many objects \(\Sigma ^{-rc} A\) with \(c \in {\mathbb {N}}\), shows by the finite global dimension of \(\Lambda (r,n,m)\) that no objects in \({\mathcal U}\) admit non-trivial morphisms from *A*. Using (2) and analogous reasoning shows that no objects in \({\mathcal D}\) admit non-trivial morphisms from *A*. Non-existence of non-trivial morphisms from *A* to objects in \({\mathcal L}\) follows as soon as \({{\mathrm{\mathrm {Hom}}}}(A,\tau A) = {{\mathrm{\mathrm {Hom}}}}^1(A,A)=0\) by using (2) above. The existence of the stalk complex of a projective module in the \(\mathcal {Z}\) component, Lemma B.9, coupled with the transitivity of the action of the automorphism group of \(\mathsf {D}^b(\Lambda (r,n,m))\) on the \(\mathcal {Z}\) component, which is proved in Sect. 5 using only Lemma 3.1, shows that \({{\mathrm{\mathrm {Hom}}}}^1(A,A)=0\) for all \(A\in \mathcal {Z}\).

Finally, \({\mathcal R}= \mathsf {coray}_{\! +}(\mathsf {ray}_{\! +}(A))\) is the non-vanishing hammock simply by \({{\mathrm{\mathrm {Hom}}}}(A,A)\ne 0\) and using either (1) or (2).

*Case*\(l=k+1\): This is analogous to the previous case.\(\square \)

## 4 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.

*exceptional*if \({{\mathrm{\mathrm {Hom}}}}^\bullet (E,E)={\mathbf {k}}\cdot {{\mathrm{{\mathsf {id}}}}}_E\). For any object \(A\in \mathsf D\) we define the functor

*twist functor*\(\mathsf {T}\!_{A_*}\) as the cone of the evaluation morphism—this gives a well-defined, exact functor by our assumption that \(\mathsf {D}\) is algebraic; see [22, §2.1] for details:

### Lemma 4.1

Let Open image in new window be a triangle equivalence of algebraic \({\mathbf {k}}\)-linear triangulated categories induced from a dg functor, and let \(A_{*}=(A_1,\ldots ,A_n)\) be any sequence of objects. Then there are functor isomorphisms \(\mathsf {F}\!_{\varphi (A_*)} = \varphi \mathsf {F}\!_{A_{*}}\varphi ^{-1}\) and \(\mathsf {T}\!_{\varphi (A_*)} = \varphi \mathsf {T}\!_{A_{*}}\varphi ^{-1}\).

### Proof

### Definition 4.2

*exceptional*

*n*-

*cycle*if

- (1)
every \(E_i\) is an exceptional object,

- (2)
there are integers \(k_i\) such that \(\mathsf {S}(E_i)\cong \Sigma ^{k_i}(E_{i+1})\) for all

*i*(where \(E_{n+1}{:}{=}E_1\)), - (3)
\({{\mathrm{\mathrm {Hom}}}}^\bullet (E_i,E_j)=0\) unless \(j=i\) or \(j=i+1\).

This definition assumes \(n\ge 2\) 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 \(\mathsf {S}(E)\cong \Sigma ^k(E)\) and \({{\mathrm{\mathrm {Hom}}}}^\bullet (E,E)={\mathbf {k}}\oplus \Sigma ^{-k}{\mathbf {k}}\). 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 \(\alpha _i: E_i\rightarrow \Sigma ^{k_i}E_{i+1}\). This explains the terminology: the subsequence \((E_1,\ldots ,E_{n-1})\) is an honest exceptional sequence, but the full set \((E_1,\ldots ,E_n)\) is not—the morphism \(\alpha _n: E_n\rightarrow \Sigma ^{k_n}E_1\) 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 \(\mathsf {S}(E_i)\cong \Sigma ^{k_i}E_{i+1}\) for all *i*, applying the Serre functor *n* times yields \(\mathsf {S}^n(E_i)\cong \Sigma ^k E_i\), where \(k{:}{=}k_1+\cdots +k_n\). 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 \({\mathbf {k}}\) of dimension *d* and let \(\mathsf D{:}{=}\mathsf {D}^b(\mathsf {coh} X)\) be its bounded derived category. The Serre functor of \(\mathsf D\) is given by \(\mathsf {S}(-) = \Sigma ^d(-)\otimes \omega _X\) 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 \(\mathsf D\), we find \(\mathsf {S}^n(E_i)=\Sigma ^{dn} E_i\otimes \omega ^n_X \cong \Sigma ^k E_i\), hence \(k=k_1+\cdots +k_n=dn\) and \(E_i\otimes \omega _X^n\cong E_i\). 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, \(\mathsf {S}E_i=\Sigma ^d E_i\otimes \omega _X\cong \Sigma ^d E_{i+1}\) implies \(E_{i+1}\cong E_i\otimes \omega _X\) for all *i*.

As an example, let *X* be an Enriques surface. Its structure sheaf \({\mathcal O}_X\) is exceptional, and the canonical bundle \(\omega _X\) has minimal order 2. In particular, \(({\mathcal O}_X,\omega _X)\) forms an exceptional 2-cycle and, by the next theorem, gives rise to an autoequivalence of \(\mathsf {D}^b(X)\).

### Theorem 4.5

Let \(E_* = (E_1,\ldots ,E_n)\) be an exceptional *n*-cycle in \(\mathsf D\). Then the twist functor \(\mathsf {T}\!_{E_*}\) is an autoequivalence of \(\mathsf D\).

### Proof

We define two classes of objects of \(\mathsf D\) by \(\mathsf {E}{:}{=}\{ \Sigma ^l E_i \mid l\in {\mathbb {Z}}, i=1,\ldots ,n \}\) and \(\Omega \,{:}{=}\, \mathsf {E}\cup \mathsf {E}^\perp \). Note that \(\mathsf {E}\) and hence \(\Omega \) are closed under suspensions and cosuspensions. It is a simple and standard fact that \(\Omega \) is a spanning class for \(\mathsf D\), i.e. \(\Omega ^\perp =0\) and \({}^\perp \Omega =0\); the latter equality depends on the Serre condition \(\mathsf {S}(E_i)\cong \Sigma ^{k_i}(E_{i+1})\). Note that spanning classes are often called ‘(weak) generating sets’ in the literature.

*Step 1:*We start by computing \(\mathsf {T}\!_{E_*}\) on the objects \(E_i\) and the maps \(\alpha _i\). For notational simplicity, we will treat \(E_1\) and \(\alpha _1: E_1\rightarrow \Sigma ^{k_1}E_2\). It follows immediately from the definition of exceptional cycle that \(\mathsf {F}\!\!\,_{E_*}(E_1)=E_1\oplus \Sigma ^{-k_n}E_n\). The cone of the evaluation morphism is easily seen to sit in the following triangleso that \(\mathsf {T}\!_{E_*}(E_1) = \Sigma ^{1-k_n}E_n\). 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.

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 \(\mathsf {T}\!_{E_*}\) is fully faithful when restricted to \(\mathsf {E}\). It is also obvious from the construction of the twist that \(\mathsf {T}\!_{E_*}\) is the identity when restricted to \(\mathsf {E}^\perp \).

*Step 3:*With \(\mathsf {T}\!_{E_*}\) fully faithful, the defining property of Serre functors gives a canonical map of functors \(\mathsf {S}^{-1}\mathsf {T}\!_{E_*}\mathsf {S}\rightarrow \mathsf {T}\!_{E_*}\) 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 \(\Omega \): both vector spaces are zero for objects from \(\mathsf {E}^\perp \); for the top row, use \({{\mathrm{\mathrm {Hom}}}}^\bullet (E_i,\mathsf {S}(-)) = {{\mathrm{\mathrm {Hom}}}}^\bullet (\mathsf {S}^{-1}(E_i),-) = {{\mathrm{\mathrm {Hom}}}}^\bullet (\Sigma ^{-k_{i-1}}E_{i-1},-)\). For objects \(E_i\), again use \(\mathsf {S}(E_i)\cong \Sigma ^{k_i}E_{i+1}\). Hence \(\mathsf {T}\!_{E_*}\) commutes with the Serre functor on \(\Omega \), and so by more general theory is essentially surjective; see [27, Corollary 1.56], this is the place where we need the assumption that \(\mathsf D\) is indecomposable.\(\square \)

### Remark 4.6

*R*is right adjoint to

*S*and that \(\mathsf {T}\!_{E_*}\) coincides with the cone of the adjunction morphism \(SR\rightarrow {{\mathrm{{\mathsf {id}}}}}\).

An object \(X\in \mathsf D\) is called *d*-*spherelike* if \({{\mathrm{\mathrm {Hom}}}}^\bullet (X,X) = {\mathbf {k}}\oplus \Sigma ^{-d}{\mathbf {k}}\); see [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 \(E_* = (E_1,\ldots ,E_n)\)*irredundant* if \(E_n\notin \mathsf {thick}_{}(E_1,\ldots ,E_{n-1})\). Recall that an exceptional *n*-cycle \((E_1,\ldots ,E_n)\) comes with a tuple of integers \((k_1,\ldots ,k_n)\) and that we have set \(k = k_1+\cdots +k_n\).

### Proposition 4.7

Let \(E_* = (E_1,\ldots ,E_n)\) be an irredundant exceptional *n*-cycle in \(\mathsf D\). Then there exists a \((k+1-n)\)-spherelike object \(X\in \mathsf D\) with non-zero maps \(X\rightarrow E_1\) and \(\Sigma ^{n-1-k+k_n}E_n\rightarrow X\).

### Proof

- (i)
\(X_i\) is exceptional,

- (ii)
\(X_i \in \mathsf {thick}_{}(E_1,\ldots ,E_i)\),

- (iii)
\({{\mathrm{\mathrm {Hom}}}}^\bullet (X_i,E_{i+1}) = \Sigma ^{-l_i}{\mathbf {k}}\) with \(l_i {:}{=}k_1+\cdots +k_i + 1-i\).

Since \(i+1<n\), \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_i,E_{i+2})=0\) by (ii) and the definition of exceptional cycles, hence \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_{i+1},E_{i+2}) = {{\mathrm{\mathrm {Hom}}}}^\bullet (\Sigma ^{l_i-1}E_{i+1},E_{i+2})\). As \(\alpha _{i+1}: E_{i+1}\rightarrow \Sigma ^{k_{i+1}}E_{i+2}\) generates \({{\mathrm{\mathrm {Hom}}}}^\bullet (E_{i+1},E_{i+2})\), we find that \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_{i+1},E_{i+2})\) is 1-dimensional, and situated in degree \(l_i+k_{i+1} - 1 = l_{i+1}\).

*f*sends the identity to the morphism \(X_{n-1} \rightarrow \Sigma ^{l_{n-1}}E_n\) defining \(X_n\). Hence

*f*is an isomorphism, thus \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_n, \Sigma ^{l_{n-1}}E_n) = 0\) and we arrive at the isomorphism Open image in new window.

*g*is a map of two 1-dimensional complexes. This map cannot be an isomorphism, because this would force \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_n,X_n)=0\), hence \(X_n=0\), which implies \(X_{n-1} \cong \Sigma ^{l_{n-1}} E_n\). But we have \(X_{n-1}\in \mathsf {thick}_{}(E_1,\ldots ,E_{n-1})\) by (ii) and also \(E_n\notin \mathsf {thick}_{}(E_1,\ldots ,E_{n-1})\) as \(E_*\) is irredundant; which gives a contradiction. Therefore we find that \(g=0\) and thus,

### Example 4.8

The additional hypothesis on \(E_*\) is necessary: consider \(\mathsf {D}= \mathsf {D}^b({\mathbf {k}}A_3)\) for the \(A_3\)-quiver \(1\rightarrow 2\rightarrow 3\). Denoting the injective-projective module by \(M=P(1)=I(3)\), the sequence \(E_*=(S(1),S(2),S(3),M)\) is an exceptional cycle with \(k_*=(1,1,0,0)\). The cycle is redundant because \(M\in \mathsf {thick}_{}(S(1),S(2),S(3))\); note that (*S*(1), *S*(2), *S*(3)) is a full exceptional collection for \(\mathsf {D}\).

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 \(\mathsf {T}\!_{E_*}=\tau ^{-1}\) on objects. In light of this example, it would also be interesting to investigate twists coming from redundant exceptional cycles further.

## 5 Autoequivalence groups of discrete derived categories

We now use the general machinery of the previous section to show that categories \(\mathsf {D}^b(\Lambda (r,n,m))\) possess two very interesting and useful autoequivalences. We will denote these by \(\mathsf {T}\!_\mathcal {X}\) and \(\mathsf {T}\!_\mathcal {Y}\) and prove some crucial properties: they commute with each other, act transitively on the indecomposables of each \(\mathcal {Z}^k\) component and provide a weak factorisation of the Auslander–Reiten translation: \(\mathsf {T}\!_\mathcal {X}\mathsf {T}\!_\mathcal {Y}= \tau ^{-1}\) on objects. Moreover, \(\mathsf {T}\!_\mathcal {X}\) acts trivially on \(\mathcal {Y}\) and \(\mathsf {T}\!_\mathcal {Y}\) acts trivially on \(\mathcal {X}\); see Proposition 5.4 and Corollary 5.5 for the precise assertions. We then give an explicit description of the group of autoequivalences of \(\mathsf {D}^b(\Lambda (r,n,m))\) in Theorem 5.7.

The category \(\mathsf D= \mathsf {D}^b(\Lambda (r,n,m))\) 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 \(\mathsf D\).

### Lemma 5.1

\(E_*\) forms an exceptional \((m+r)\)-cycle in \(\mathsf D\) with \(k_*=(1,\ldots ,1,1-r)\), and \(F_*\) forms an exceptional \((n-r)\)-cycle in \(\mathsf D\) with \(k_*=(1,\ldots ,1,1+r)\).

### Proof

The object \(X^0_{11}\) is exceptional by Lemma 3.1, hence any object at the mouth \(X^0_{ii}=\tau ^{1-i}(X^0_{11})\) is. This point also gives the second condition of exceptional cycles: for \(i=1,\ldots ,m+r-1\), we have \(\mathsf {S}E_i = \Sigma \tau X^0_{m+r+1-i,m+r+1-i} = \Sigma X^0_{m+r-i,m+r-i} = \Sigma E_{i+1}\) and at the boundary step we have \(\mathsf {S}E_{m+r} = \Sigma \tau X^0_{11} = \Sigma X^0_{00} = \Sigma ^{1-r} X^0_{m+r,m+r} = \Sigma ^{1-r} E_1\), where we freely make use of the results stated in Sect. 2. Hence the degree shifts of the sequence \(E_*\) are \(k_1=\ldots =k_{m+r-1}=1\) and \(k_{m+r}=1-r\). Finally, the required vanishing \({{\mathrm{\mathrm {Hom}}}}(E_i,E_j)=0\) unless \(j=i+1\) or \(i=j\) again follows from Lemma 3.1.

The same reasoning works for \(\mathcal {Y}\), now with the boundary step degree computation \(\mathsf {S}F_{n-r} = \Sigma \tau Y^0_{11} = \Sigma Y^0_{00} = \Sigma ^{1+r} Y^0_{n-r,n-r} = \Sigma ^{1+r} F_1\). \(\square \)

The next lemma shows that the functors \(\mathsf {F}\!\!\,_{E_*}\) and \(\mathsf {F}\!\!\,_{F_*}\) of the last section take on a particularly simple form, where we use the notation \({}_0X,X_0,{}^0Y,Y^0\) from Sects. 3.2, 3.3:

### Lemma 5.2

### Proof

This follows immediately from the definition of these functors in Sect. 4, Proposition 3.4 and Properties 1.2(3), i.e. \(\Sigma ^r|_\mathcal {X}= \tau ^{-m-r}\) and \(\Sigma ^r|_\mathcal {Y}= \tau ^{n-r}\) on objects.

Note that the right-hand sides extend to direct sums. Another description of \(\mathsf {F}\!\!\,_{E_*}(X)\) is as the minimal approximation of *X* with respect to the mouth of \(\mathcal {X}^0\), and analogously for \(\mathsf {F}\!\!\,_{F_*}\).\(\square \)

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 \(\mathsf {T}\!_\mathcal {X}\) instead of \(\mathsf {T}\!_{E_*}\) and \(\mathsf {T}\!_\mathcal {Y}\) instead of \(\mathsf {T}\!_{F_*}\).

### Lemma 5.3

Any two exceptional cycles \(E_*, E'_*\) at the mouths of \(\mathcal {X}\) components differ by suspensions and AR translations, and the associated twist functors coincide: \(\mathsf {T}\!_{E_*} = \mathsf {T}\!_{E'_*}\).

### Proof

A suitable iterated suspension will move \(E'_*\) into the \(\mathcal {X}\) 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 \(E'_* = \Sigma ^a\tau ^b E_*\) for some \(a,b\in {\mathbb {Z}}\). 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 \(\mathsf {T}\!_{E'_*} = \mathsf {T}\!_{\Sigma ^a\tau ^bE_*} = \Sigma ^a\tau ^b\mathsf {T}\!_{E_*}\Sigma ^{-a}\tau ^{-b} = \mathsf {T}\!_{E_*}\), using Lemma 4.1.\(\square \)

### Proposition 5.4

### Corollary 5.5

The twist functors \(\mathsf {T}\!_\mathcal {X}\) and \(\mathsf {T}\!_\mathcal {Y}\) act simply transitively on each component \(\mathcal {Z}^k\) and factorise the inverse AR translation: \(\mathsf {T}\!_\mathcal {X}\mathsf {T}\!_\mathcal {Y}= \mathsf {T}\!_\mathcal {Y}\mathsf {T}\!_\mathcal {X}= \tau ^{-1}\) on the objects of \(\mathsf {D}^b(\Lambda )\). Moreover, \(\mathsf {T}\!_\mathcal {X}\), \(\mathsf {T}\!_\mathcal {Y}\) and \(\Sigma \) act transitively on \(\mathsf {ind}(\mathcal {Z})\).

### Proof of the proposition

By Lemma 3.1, we have \({{\mathrm{\mathrm {Hom}}}}^\bullet (X^k_{ii},Y)=0\) for all \(Y\in \mathcal {Y}\). This immediately implies \(\mathsf {T}\!_\mathcal {X}|_\mathcal {Y}= {{\mathrm{{\mathsf {id}}}}}\).

*Action of*\(\mathsf {T}\!_\mathcal {X}\)*on objects of*\(\mathcal {X}\): we recall that the proof of Theorem 4.5 showed \(\mathsf {T}\!_\mathcal {X}(E_i)=\Sigma ^{1-k_{i-1}}(E_{i-1})\), and furthermore \(k_1=\cdots =k_{m+r-1}=1\) and \(k_{m+r}=1-r\) from Lemma 5.1. Hence \(\mathsf {T}\!_\mathcal {X}(E_i)=\tau ^{-1}(E_i)\) for all *i*—as explained in Lemma 5.3, this holds for any exceptional cycle at an \(\mathcal {X}\) mouth. Since \(\mathsf {T}\!_\mathcal {X}\) is an equivalence and each \(\mathcal {X}\) component is of type \({\mathbb {Z}}A_\infty \), this forces \(\mathsf {T}\!_{E_*}|_{\mathcal {X}}=\tau ^{-1}\) on objects.

*Action of*\(\mathsf {T}\!_\mathcal {X}\)

*on objects of*\(\mathcal {Z}\): Pick \(Z^0_{ij}\in \mathcal {Z}^0\) with \(1\le i\le m+r\). Using \(\mathsf {T}\!_\mathcal {X}= \mathsf {T}\!_{E_*}\) with the cycle originally specified, i.e. \(E_{m+r}=X^0_{11}\), we invoke Lemma 3.1 once more to get \({\mathbf {k}}= {{\mathrm{\mathrm {Hom}}}}^\bullet (X^0_{ii},Z^0_{ij}) = {{\mathrm{\mathrm {Hom}}}}^\bullet (E_{m+r+1-i},Z^0_{ij})\), and \(0 = {{\mathrm{\mathrm {Hom}}}}^\bullet (E_l,Z^0_{ij})\) for all \(l\ne m+r+1-i\). So \(\mathsf {F}\!\!\,_{E_*}(Z^0_{ij}) = X^0_{ii}\) and the triangle defining \(\mathsf {T}\!_{E_*}(Z^0_{ij})\) is one of the special triangles of Properties 1.2(4):Application of AR translations extends this computation to arbitary \(Z\in \mathcal {Z}^0\), and suspending extends it to all \(\mathcal {Z}\) components, thus \(\mathsf {T}\!_\mathcal {X}(Z^0_{i,j}) = Z^0_{i+1,j}\).

*Remaining cases:* Analogous reasoning shows \(\mathsf {T}\!_{F_*}(F_i)=\tau ^{-1}(F_i)\) for all \(i=1,\ldots ,n-r\), and the rest of the above proof works as well: \(\mathsf {T}\!_\mathcal {Y}(Z^0_{i,j})=Z^0_{i,j+1}\), now using the other special triangle. \(\square \)

The following technical lemma about the additive closures of the \(\mathcal {X}\) and \(\mathcal {Y}\) 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 \(\mathcal {X}\) and \(\mathcal {Y}\) is a thick triangulated subcategory of \(\mathsf D\).

### Proof

The proof of Proposition 5.4 contains the fact \(\mathsf {thick}_{}(E_*)^\perp = \mathcal {Y}\). Perpendicular subcategories are always closed under extensions and direct summands; since \(\mathsf {thick}_{}(E_*)\) is by construction a triangulated subcategory, the orthogonal complement \(\mathcal {Y}\) is triangulated as well.\(\square \)

Our results enable us to compute the group of autoequivalences of \(\mathsf {D}^b(\Lambda (r,n,m))\). For \(\Lambda (1,2,0)\), König and Yang [33, Lemma 9.3] showed \({{\mathrm{\mathsf {Aut}}}}(\mathsf {D}^b(\Lambda (1,2,0))) \cong {\mathbb {Z}}^2 \times {\mathbf {k}}^*\).

### Theorem 5.7

### Proof

In this proof, we will write \(\mathsf {D}= \mathsf {D}^b(\Lambda (r,n,m))\) and \(\Lambda = \Lambda (r,n,m)\).

*Step 1:*\({{\mathrm{\mathsf {Out}}}}(\Lambda ) = {\mathbf {k}}^*\)*from common scaling of arrows.*

Recall that units \(u\in \Lambda \) induce inner automorphisms \(c_u(\alpha )=u\alpha u^{-1}\), and thus a normal subgroup \({{\mathrm{\mathsf {Inn}}}}(\Lambda )\subseteq {{\mathrm{\mathsf {Aut}}}}(\Lambda )\). It is a well-known fact that inner automorphisms induce autoequivalences of \(\mathsf {mod}(\Lambda )\) and \(\mathsf {D}^b(\Lambda )\) which are isomorphic to the identity; see [46, §3]. The quotient group \({{\mathrm{\mathsf {Out}}}}(\Lambda ) = {{\mathrm{\mathsf {Aut}}}}(\Lambda )/{{\mathrm{\mathsf {Inn}}}}(\Lambda )\) acts faithfully on modules. The form of the quiver and the relations for \(\Lambda (r,n,m)\) imply that algebra automorphisms can only act by scaling arrows.

Scaling of arrows leads to a subgroup \(({\mathbf {k}}^*)^{m+n}\) of \({{\mathrm{\mathsf {Aut}}}}(\Lambda )\). However, choosing an indecomposable idempotent *e* (i.e. a vertex) together with a scalar \(\lambda \in {\mathbf {k}}^*\) produces a unit \(u = 1_\Lambda + (\lambda -1)e\), and hence an inner automorphism \(c_u\in {{\mathrm{\mathsf {Aut}}}}(\Lambda )\). It is easy to check that \(c_u(\alpha )=\frac{1}{\lambda }\alpha \) if \(\alpha \) ends at *e*, and \(c_u(\alpha )=\lambda \alpha \) if \(\alpha \) starts at *e*, and \(c_u(\alpha )=\alpha \) otherwise. Since the quiver of \(\Lambda \) has one cycle, we see that an \((n+m-1)\)-subtorus of the subgroup \(({\mathbf {k}}^*)^{m+n}\) 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, \({{\mathrm{\mathsf {Out}}}}(\Lambda )={\mathbf {k}}^*\). Note that the class of an automorphism scaling precisely one arrow also generates \({{\mathrm{\mathsf {Out}}}}(\Lambda )\).

*Step 2:*\(\varphi \in {{\mathrm{\mathsf {Aut}}}}(\mathsf {D})\)*is isomorphic to the identity on objects*\(\Longleftrightarrow \)\(\varphi \in {{\mathrm{\mathsf {Out}}}}(\Lambda )\).

By Step 1, it is clear that algebra automorphisms act trivially on objects. Let now \(\varphi \in {{\mathrm{\mathsf {Aut}}}}(\mathsf {D})\) fixing all objects. In particular, \(\varphi \) fixes the abelian category \(\mathsf {mod}(\Lambda )\) and the object \(\Lambda \), thus giving rise to \(\varphi : \Lambda \rightarrow \Lambda \), i.e. an automorphism which by Step 1 can be taken to be outer.

*Step 3:**The subgroup*\({\langle \Sigma ,\mathsf {T}\!_\mathcal {X},\mathsf {T}\!_\mathcal {Y},{{\mathrm{\mathsf {Out}}}}(\Lambda )\rangle }\)*is abelian.*

The suspension commutes with all exact functors. Next, to see \([\mathsf {T}\!_\mathcal {X},\mathsf {T}\!_\mathcal {Y}]={{\mathrm{{\mathsf {id}}}}}\), we fix exceptional cycles \(E_*\) for \(\mathcal {X}\) and \(F_*\) for \(\mathcal {Y}\); then \(\mathsf {T}\!_{E*}\mathsf {T}\!_{F_*}(\mathsf {T}\!_{E_*})^{-1}= \mathsf {T}\!_{\mathsf {T}\!_{E_*}(F_*)} = \mathsf {T}\!_{F_*}\) by Lemma 4.1 and Proposition 5.4. Let \(f\in {{\mathrm{\mathsf {Out}}}}(\Lambda )\). Then we have \([f,\mathsf {T}\!_\mathcal {X}]=[f,\mathsf {T}\!_\mathcal {Y}]={{\mathrm{{\mathsf {id}}}}}\) by the same lemma, now using \(f(E_*)=E_*\) and \(f(F_*)=F_*\) from Step 2.

*Step 4:*\({{\mathrm{\mathsf {Aut}}}}(\mathsf {D})\)*is generated by*\(\Sigma ,\mathsf {T}\!_\mathcal {X},\mathsf {T}\!_\mathcal {Y},{{\mathrm{\mathsf {Out}}}}(\Lambda )\).

Fix a \(Z\in \mathsf {ind}(\mathcal {Z})\). For any \(\varphi \in {{\mathrm{\mathsf {Aut}}}}(\mathsf {D})\), there are \(a,b,c\in {\mathbb {Z}}\) with \(\Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}\mathsf {T}\!_\mathcal {Y}^{\,c}(Z) = \varphi (Z)\), since the suspension and the twist functors act transitively on \(\mathsf {ind}(\mathcal {Z})\) by Corollary 5.5. Therefore, \(\psi {:}{=}\Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}\mathsf {T}\!_\mathcal {Y}^{\,c}\varphi ^{-1}\) fixes *Z*. Moreover, since all autoequivalences commute with \(\tau \) (because they commute with the Serre functor \(\mathsf {S}= \Sigma \tau \) and with \(\Sigma \)) and \(\mathcal {Z}\) is a \({\mathbb {Z}}A_\infty ^\infty \)-component, either \(\psi \) is the identity on \(\mathsf {ind}(\mathcal {Z})\) or else \(\psi \) flips \(\mathsf {ind}(\mathcal {Z})\) along the \(Z\tau (Z)\) axis. However, the latter possibility is excluded by the action of \(\Sigma ^r|_\mathcal {Z}\); see Properties 1.2(3).

By Properties 1.2(4), every indecomposable object of \(\mathcal {X}\) or \(\mathcal {Y}\) is a cone of a morphism \(Z_1\rightarrow Z_2\) for some \(Z_1,Z_2\in \mathsf {ind}(\mathcal {Z})\). Moreover, the morphism \(Z_1\rightarrow Z_2\) 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 \(\mathcal {Z}\) are 1-dimensional, even for \(r=1\).) Hence \(\varphi \) actually fixes all indecomposable objects and thus all objects of \(\mathsf {D}^b(\Lambda )\).

Thus, by Step 2, \(\psi \in {{\mathrm{\mathsf {Out}}}}(\Lambda )\) and \(\varphi \in {\langle \Sigma ,\mathsf {T}\!_\mathcal {X},\mathsf {T}\!_\mathcal {Y},{{\mathrm{\mathsf {Out}}}}(\Lambda )\rangle }\).

*Step 5:*\({{\mathrm{\mathsf {Aut}}}}(\mathsf {D})\)*is abelian with one relation*\(f_0 \Sigma ^{-r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,r-n}={{\mathrm{{\mathsf {id}}}}}\)*for some*\(f_0\in {{\mathrm{\mathsf {Out}}}}(\Lambda )\).

By Steps 3 and 4, \({{\mathrm{\mathsf {Aut}}}}(\mathsf {D})={\langle \Sigma ,\mathsf {T}\!_\mathcal {X},\mathsf {T}\!_\mathcal {Y},{{\mathrm{\mathsf {Out}}}}(\Lambda )\rangle }\) is abelian. Properties 1.2(3) and Proposition 5.4 imply that the autoequivalence \(\Sigma ^{-r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,r-n}\) fixes all objects of \(\mathsf {D}\), hence \(f_0\Sigma ^{-r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,r-n} = {{\mathrm{{\mathsf {id}}}}}\) for a unique automorphism \(f_0\in {{\mathrm{\mathsf {Out}}}}(\Lambda )\).

Let now \(a,b,c\in {\mathbb {Z}}\) and \(g\in {{\mathrm{\mathsf {Out}}}}(\Lambda )\) such that \(g \Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}\mathsf {T}\!_\mathcal {Y}^{\,c} = {{\mathrm{{\mathsf {id}}}}}\). In particular, \(\psi {:}{=}\, \Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}\mathsf {T}\!_\mathcal {Y}^{\,c}\) fixes all objects. From \(X=\psi (X)=\Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}(X)=\Sigma ^a\tau ^{-b}(X)\) we deduce first \(a=lr\) for some \(l\in {\mathbb {Z}}\) and then \(b=-l(m+r)\); whereas \(Y=\psi (Y)\) similarly implies \(a=kr\) and \(c=k(n-r)\) for some \(k\in {\mathbb {Z}}\). Hence \(k=l\) and \(\psi = \Sigma ^{lr}\mathsf {T}\!_\mathcal {X}^{\,-l(m+r)}\mathsf {T}\!_\mathcal {Y}^{\,l(n-r)}=f_0^{l}\). So \(g=f_0^{-l}\) and altogether, \( g\Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}\mathsf {T}\!_\mathcal {Y}^{\,c} = (f_0 \Sigma ^{-r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,n-r})^{-l}\) is a power of the stated relation.

*Step 6:*\({{\mathrm{\mathsf {Aut}}}}(\mathsf {D}) \cong {\mathbb {Z}}^2 \times {\mathbb {Z}}/(r,n,m) \times {\mathbf {k}}^*\).

*A*be a free abelian group of finite rank and \(a_0\in A\), \(f_0\in {\mathbf {k}}^*\). Write \(a_0=da_1\) with \(d\in {\mathbb {Z}}\) and \(a_1\) indivisible. Choose \(f_1\in {\mathbf {k}}^*\) with \(f_1^d=f_0\)—this is possible because \({\mathbf {k}}\) is algebraicaly closed. Now fix a group homomorphism \(\nu : A\rightarrow {\mathbb {Z}}\) with \(\nu (a_1)=1\)—this is possible because \(a_1\) is indivisible. Consider the diagram with exact rowswhere \(\alpha (na_0,1) = (na_0,f_0^n)\) and \(\beta (a,f) = (a,ff_1^{\nu (a)})\). Both maps are easily checked to be group homomorphisms and bijective. Moreover, the left-hand square commutes:

### Question

- (1)
\(\Sigma ^r|_\mathcal {X}= \tau ^{-m-r}\) and \(\Sigma ^r|_\mathcal {Y}= \tau ^{n-r}\)

- (2)
\(\mathsf {T}\!_\mathcal {X}|_\mathcal {X}= \tau ^{-1}\) and \(\mathsf {T}\!_\mathcal {Y}|_\mathcal {Y}= \tau ^{-1}\)

- (3)
\(\Sigma ^{r} = \mathsf {T}\!_\mathcal {X}^{\,m+r} \, \mathsf {T}\!_\mathcal {Y}^{\,r-n}\)

## 6 Hom spaces: dimension bounds and graded structure

In this section, we prove a strong result about \(\mathsf {D}^b(\Lambda ){:}{=}\mathsf {D}^b(\Lambda (r,n,m))\) 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.

### 6.1 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.

### Theorem 6.1

Let *A*, *B* be indecomposable objects of \(\mathsf {D}^b(\Lambda (r,n,m))\) where \(n>r\). If \(r\ge 2\), then \(\dim {{\mathrm{\mathrm {Hom}}}}(A,B) \le 1\) and if \(r=1\), then \(\dim {{\mathrm{\mathrm {Hom}}}}(A,B) \le 2\).

### Proof

Our strategy for establishing the dimension bound follows that of the proofs of the Hom-hammocks. Let \(A,B\in \mathsf {ind}(\mathsf {D}^b(\Lambda (r,n,m)))\) and assume \(r>1\). In this proof, we use the abbreviation \(\hom = \dim {{\mathrm{\mathrm {Hom}}}}\). We want to show \(\hom (A,B)\le 1\) by considering the various components separately.

*Case*\(A\in \mathcal {X}^k\) or \(\mathcal {Y}^k\): Consider first \(A,B\in \mathcal {X}^k\) and perform induction on the height of

*A*. If \(A=A_0\) sits at the mouth, then \(\hom (A,B)\le 1\) by Lemma 3.1. For

*A*higher up, and assuming \({{\mathrm{\mathrm {Hom}}}}(A,B)\ne 0\), which means \(B\in \mathsf {ray}_{\! +}(\overline{AA_0})\), we consider one of the triangles from Lemma 3.2

The subcase \(B\in \mathcal {X}^{k+1}\) follows from the above by Serre duality.

Furthermore, the above argument applies without change to \(B\in \mathcal {Z}^k\)—with \(\mathsf {ray}_{\! +}(A)\subset \mathcal {Z}^k\) understood to mean the subset of indecomposables of \(\mathcal {Z}^k\) admitting non-zero morphisms from *A* (these form a ray in \(\mathcal {Z}^k\)) and similarly \(\mathsf {ray}_{\! -}(B)\subset \mathcal {X}^k\), and application of Proposition 3.6. An obvious modification, which we leave to the reader, extends the argument to \(B\in \mathcal {Z}^{k+1}\). The statements for \(A\in \mathcal {Y}\) are completely analogous.

*Case*\(A\in \mathcal {Z}^k\): In light of Serre duality, we don’t need to deal with \(B \in \mathcal {X}\) or \(B \in \mathcal {Y}\). Therefore we turn to \(B\in \mathcal {Z}\). However, we already know from the proof of Proposition 3.6 that the dimensions in the two non-vanishing regions \(\mathsf {ray}_{\! +}(\mathsf {coray}_{\! +}(A))\) and \(\mathsf {ray}_{\! -}(\mathsf {coray}_{\! -}(\mathsf {S}A))\) are constant. Since the \(\mathcal {Z}\) components contain the simple *S*(0) and the twist functors together with the suspension act transitively on \(\mathcal {Z}\), it is clear that \(\hom (A,A)=\hom (A,\mathsf {S}A)^*=1\). This completes the proof.\(\square \)

### Proposition 6.2

*A*in the heavily shaded square have \(\dim {{\mathrm{\mathrm {Hom}}}}(X,A)=2\):

### Proof

The argument is similar to the computation of the Hom-hammocks in the \(\mathcal {Z}\) components from Sect. 3. We proceed in several steps.

*Step 1:*For any \(A\in \mathsf {ind}(\mathcal {X})\) 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):

*X*lying on the mouth, Lemma 3.1 actually yields:

*A*belongs to the intersections of the (co)rays on the right-hand side, which can only happen when \({}_0 \mathsf {S}X = X_0\). 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*\({}_0 \mathsf {S}X \ne X_0\), *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 \(\hom (X,A) \le \hom (X,A'')+ \hom (X, {}_0 A)\) from sequence (3). This yields the stated upper bound for \(\hom (X,A) \), as \(\hom (X, {}_0 A) \le 1\) when \({}_0 \mathsf {S}X \ne X_0\) and \(\hom (X, {}_0 A) \le 2\) otherwise. For the lower bound, instead observe that \(\hom (X,A'')\le \hom (X,\Sigma {}_0 A) +\hom (X,A)\), again from sequence (3).

*Step 3:*\(\psi =0\)*unless*\(A \in \mathsf {ray}_{\! +}(\Sigma ^{-1} \,{}_0 \mathsf {S}X)\)*and*\(\mu =0\)*unless*\(A \in \mathsf {coray}_{\! -}(\Sigma X_0)\).

If \(A \notin \mathsf {ray}_{\! +}(\Sigma ^{-1} \, X_0) \cup \mathsf {ray}_{\! +}(\Sigma ^{-1} \,{}_0 \mathsf {S}X)\) then \(\hom (X,\Sigma {}_0 A)=0\) and so \(\psi =0\) trivially. Therefore, we just need to consider \(A \in \mathsf {ray}_{\! +}(\Sigma ^{-1} \, X_0)\) but \(A\notin \mathsf {ray}_{\! +}(\Sigma ^{-1} \,{}_0 \mathsf {S}X)\), and in this case \(\hom (X,\Sigma \, {}_0A)=1\). It is clear that the maps going down the coray from *X* to \(X_0\) span a 1-dimensional subspace of \({{\mathrm{\mathrm {Hom}}}}(X,\Sigma \, {}_0A)\), which therefore is the whole space. Using properties of the \({\mathbb {Z}}A_\infty \) mesh, the composition of such maps with a map along \(\mathsf {ray}_{\! +}(X_0)\) from \(X_0\) to \(\Sigma A\) defines a non-zero element in \({{\mathrm{\mathrm {Hom}}}}(X,\Sigma \, A)\). Thus the map \({{\mathrm{\mathrm {Hom}}}}(X,\Sigma {}_0 A) \rightarrow {{\mathrm{\mathrm {Hom}}}}(X, \Sigma A)\) in the sequence (3) is injective and it follows that \(\psi =0\). 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 \({{\mathrm{\mathrm {Hom}}}}(X,\Sigma ^{-1} A) \rightarrow {{\mathrm{\mathrm {Hom}}}}(X, \Sigma ^{-1} A_0)\) in the sequence (4) is surjective.

*Step 4:**If*\(\mathsf {ray}_{\! +}(\Sigma ^{-1} X_0)\) (*or*\(\mathsf {coray}_{\! -}(\Sigma {}_0 \mathsf {S}X)\), *respectively*) *does not coincide with one of the other three (co)rays, then crossing it does not affect the value of*\(\hom (X,-)\).

Suppose \(\mathsf {ray}_{\! +}(\Sigma ^{-1}\, X_0) \ni A\) doesn’t coincide with \(\mathsf {ray}_{\! +}(X_0)\), \(\mathsf {ray}_{\! +}({}_0\mathsf {S}X)\) or \(\mathsf {ray}_{\! +}(\Sigma ^{-1}{}_0\mathsf {S}X)\). Thus \(\hom (X,{}_0A)=0\), and from Step 3 the map \(\psi =0\), hence \({{\mathrm{\mathrm {Hom}}}}(X,A) = {{\mathrm{\mathrm {Hom}}}}(X,A'')\). Similarly, suppose \(A \in \mathsf {coray}_{\! -}(\Sigma \,{}_0 \mathsf {S}X)\) and this doesn’t coincide with any of the other corays. Then \(\hom (X,A_0)=0\) and \(\mu =0\) and again the claim follows.

*Step 5:** There are three possible configurations of rays and corays determining the regions where*\(\hom (X, -)\)*is constant.*

*X*. We consider now the case where \({}_0 \mathsf {S}X\) 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 \(\hom (X,-) \ge 1\) 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 \(A \in \mathsf {ray}_{\! +}(\mathsf {S}{}_0X) \cap \mathsf {coray}_{\! -}(X_0)\); this is the object of minimal height in region K. We can see that \(A \in \mathsf {coray}_{\! +}(X)\) and the map down the coray from *X* to \(A_0\), factors through the map from *A* to \(A_0\). Therefore the map \(\delta \) in the second exact sequence (4) is non-zero. It is clear that \( A \notin \mathsf {coray}_{\! -}(\Sigma X_0)\) so \(\mu =0\) 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 \(A \in \mathsf {ray}_{\! +}(\mathsf {S}{}_0X) \cap \mathsf {coray}_{\! -}(\Sigma X_0)\), the object of minimal height in region L. Observe that \(A'' \in \mathsf {ray}_{\! +}(\tau ^{-1}\mathsf {S}{}_0X) \cap \mathsf {coray}_{\! -}(\Sigma X_0) = {{\mathrm{\mathsf {add}}}}\Sigma X\) from which we can see that the map to \({{\mathrm{\mathrm {Hom}}}}(X,{}_0 A)\) in (3) is surjective. Now \(A \notin \mathsf {ray}_{\! +}(\Sigma ^{-1}{}_0 \mathsf {S}X)\), so \(\psi =0\) 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 \(A \in \mathsf {ray}_{\! +}(\Sigma ^{-1}\mathsf {S}{}_0X) \cap \mathsf {coray}_{\! -}(\Sigma X_0)\), the object of minimal height in region M. It is clear that \(A \notin \mathsf {ray}_{\! +}(\mathsf {S}{}_0X) \cup \mathsf {ray}_{\! +}(X_0)\), so \(\hom (X,{}_0 A) = 0\). A short calculation shows \(A'' \in \mathsf {ray}_{\! +}(X)\), and again using the chain of morphisms in Properties 1.2(5), we see that there is a map \(X \rightarrow \Sigma {}_0 A = \mathsf {S}{}_0 X\) 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 \({}_0 \mathsf {S}X\) lies to the left of \(X_0\). If instead it lies to the right, analogous reasoning applies. Finally, if \({}_0 \mathsf {S}X = X_0\), matters are simpler: in that case, the regions C and F–I all vanish. \(\square \)

### 6.2 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 [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.

### Lemma 6.3

In words, the functions \(\delta ^+\) and \(\delta ^-\) determine the ranges of self-extensions of positive and negative degree, respectively. We point out that the result holds for all \(r\ge 1\).

### Proof

Let \(A\in \mathsf {ind}(\mathcal {X})\), assuming \(r>1\). Suspending if necessary, we may suppose that \(A = X^0_{ij}\). We are looking for all \(d\in {\mathbb {Z}}\) with \({{\mathrm{\mathrm {Hom}}}}^d(A,A)={{\mathrm{\mathrm {Hom}}}}(A,\Sigma ^dA)\ne 0\). By Proposition 3.4, this is only possible for either \(d \equiv 0\) or \(d \equiv 1\) modulo *r*.

*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 \(\mathsf {ray}_{\! +}(\overline{AA_0})\) of height

*h*. We now turn to the other possibility, \(d=1+lr\) for some \(l\in {\mathbb {Z}}\). Here we getAs we know from Theorem 6.1, all Hom spaces have dimension 1 when \(r>1\), these two computations give

### 6.3 Coarse classification of objects

Our previous results allow us to give a crude grouping of the indecomposable objects of \(\mathsf {D}^b(\Lambda (r,n,m))\). In the \(\mathcal {X}\) and \(\mathcal {Y}\) components, the distinction depends on the height of an object, i.e. the distance from the mouth; see page 5. Recall that an object *D* of a \({\mathbf {k}}\)-linear Hom-finite triangulated category \(\mathsf {D}\) is *exceptional* if \(\hom ^*(D,D)=1\), then \({{\mathrm{\mathrm {Hom}}}}^\bullet (D,D)={\mathbf {k}}\); see section “Exceptional sequences and semi-orthogonal decompositions” in “Appendix 1”, and *D* is called *spherelike* if \(\hom ^*(D,D)=2\), then \({{\mathrm{\mathrm {Hom}}}}^\bullet (D,D)={\mathbf {k}}\oplus \Sigma ^{-d}{\mathbf {k}}\) as graded vector spaces for some \(d\in {\mathbb {Z}}\) and *D* is called *d*-spherelike; see [22] for details. Assuming \(\mathsf {D}\) has a Serre functor \(\mathsf {S}\), a *d*-spherelike object *D* is called *d*-*spherical* if \(\mathsf {S}(D)=\Sigma ^dD\); see [27, §8].

### Proposition 6.4

Exceptional if \(A\in \mathcal {Z}\), or \(A\in \mathcal {X}\) with \(h(A)<m+r-1\), or \(A\in \mathcal {Y}\) with \(h(A)<n-r-1\).

\((1-r)\)-spherelike if \(A\in \mathcal {X}\) with \(h(A)=m+r-1\).

\((1+r)\)-spherelike if \(A\in \mathcal {Y}\) with \(h(A)=n-r-1\).

\(\dim {{\mathrm{\mathrm {Hom}}}}^*(A,A)\ge 3\) with \({{\mathrm{\mathrm {Hom}}}}^{<0}(A,A)\ne 0\) otherwise.

### Remark 6.5

In fact, the direct sum \(E_1\oplus E_2\) of two exceptional objects \(E_1\) and \(E_2\) with \({{\mathrm{\mathrm {Hom}}}}^\bullet (E_1,E_2) = {{\mathrm{\mathrm {Hom}}}}^\bullet (E_2,E_1) = 0\) is a 0-spherelike object. Examples for \(r>1\) are given by taking \(E_1\in \mathcal {X}\) and \(E_2\in \mathcal {Y}\) at the mouths. The theory of spherelike objects also applies in this degenerate case, but is less interesting [22, Appendix].

### Remark 6.6

We can infer the existence of \((1-r)\)-spherelike indecomposable objects in \(\mathcal {X}\) and \((1+r)\)-spherelike objects in \(\mathcal {Y}\) also from Proposition 4.7 and Lemma 5.1. To any reasonable \({\mathbf {k}}\)-linear triangulated category, [23] associates a poset derived from indecomposable spherelike objects. In [23, §6], these posets are computed for derived-discrete algebras.

### Proof

We know from Lemma B.9 that the projective module \(P(n-r)\in \mathcal {Z}\). This is an exceptional object by Proposition 3.6. As the autoequivalence group acts transitively on \(\mathsf {ind}(\mathcal {Z})\) by Corollary 5.5, every indecomposable object of \(\mathcal {Z}\) is exceptional. The remaining parts of the proposition all follow from Lemma 6.3. We only give the argument for \(A\in \mathsf {ind}(\mathcal {X})\), as the one for indecomposable objects of \(\mathcal {Y}\) runs entirely parallel.

Observing the trivial inequalities \(0\le \delta ^+(A)\le \delta ^-(A)\), we see that *A* is exceptional if and only if \(1=\dim {{\mathrm{\mathrm {Hom}}}}^*(A,A)=1+\delta ^+(A)+\delta ^-(A)\). In turn, this happens precisely if \(\delta ^-(A)=0\), which means \(h<m+r-1\).

Similarly, *A* is spherelike if and only if \(2=\dim {{\mathrm{\mathrm {Hom}}}}^*(A,A)=1+\delta ^+(A)+\delta ^-(A)\) which is equivalent to \(\delta ^+(A)=0\) and \(\delta ^-(A)=1\). The only solution of these equations is \(h=m+r-1\). Furthermore, in this case the endomorphism complex is \({{\mathrm{\mathrm {Hom}}}}^\bullet (A,A) = {\mathbf {k}}\oplus \Sigma ^{r-1}{\mathbf {k}}\), so that *A* is indeed \((1-r)\)-spherelike.\(\square \)

### Corollary 6.7

0-spherical if and only if \(m=0\), \(r=1\) and

*A*sits at an \(\mathcal {X}\)-mouth;*n*-spherical if and only if \(n=r+1\) and*A*sits at a \(\mathcal {Y}\)-mouth.

### Proof

The only candidates for spherical objects are the spherelike objects listed in Proposition 6.4. Start with \(A\in \mathcal {X}\) with \(h(A)=m+r-1\). Then *A* is spherical if and only if \(\mathsf {S}A = \Sigma ^{1-r} A\). By \(\mathsf {S}= \Sigma \tau \) and \(\Sigma ^{1-r} = \Sigma \tau ^{m+r}\) (Properties 1.2(3)), this is equivalent to \(\tau ^{m+r-1}A=A\) which happens precisely if \(m+r=1\). The only solution for this equation is \(m=0\), \(r=1\).

Next, \(B\in \mathcal {Y}\) with \(h(B)=n-r-1\) is spherical if and only if \(\Sigma \tau B = \mathsf {S}B = \Sigma ^{1+r}B = \Sigma \tau ^{n-r} B\), so that here we get \(\tau ^{n-r-1}B=B\) which is possibly only for \(n=r+1\). \(\square \)

## 7 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 [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 \(\mathsf {D}^b(\Lambda (r,n,m))\) has length heart, and hence to classify both bounded t-structures and bounded co-t-structures it is sufficient to classify silting objects in \(\mathsf {D}^b(\Lambda (r,n,m))\). This is the main goal of this section. In the first part, we prove that any bounded t-structure in \(\mathsf {D}^b(\Lambda (r,n,m))\) is length, then we obtain a semi-orthogonal decompositon \(\mathsf {D}^b(\Lambda (r,n,m)) = {\langle \mathsf {D}^b({\mathbf {k}}A_{n+m-1}),\mathsf {Z}\rangle }\), for some trivial thick subcategory \(\mathsf {Z}\), 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.

### 7.1 All hearts in \(\varvec{\mathsf {D}^b(\Lambda (r,n,m))}\) are length

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 \(\Lambda (1,2,0)\) proved in [33]; the third is a general statement about Hom-finite abelian categories.

### Lemma 7.2

(cf. [25, Lemma 4.1]) Let \(\mathsf {D}\) be a triangulated category equipped with a t-structure \((\mathsf {X},\mathsf {Y})\) with heart \(\mathsf {H}= \mathsf {X}\cap \Sigma \mathsf {Y}\). Then at most one suspension of any object of \(\mathsf {D}\) may lie in the heart \(\mathsf {H}\).

### Proof

Let \(0 \ne H \in \mathsf {H}\). We show that \(\Sigma ^n H \notin \mathsf {H}\) for any \(n \ne 0\). First suppose that \(\Sigma ^n H \in \mathsf {H}\) for some \(n > 0\). Then \(H \in \Sigma ^{-n} \mathsf {H}\). We have \(\Sigma ^{-n} \mathsf {H}\subseteq \Sigma ^{-n} \Sigma \mathsf {Y}\subseteq \mathsf {Y}\). The condition \({{\mathrm{\mathrm {Hom}}}}(\mathsf {X},\mathsf {Y})=0\) then implies that \({{\mathrm{\mathrm {Hom}}}}(H,H)=0\), a contradiction.

Now suppose \(\Sigma ^{-n} H \in \mathsf {H}\) for some \(n>0\). In this case we have \(\Sigma ^{n} \mathsf {H}\subseteq \Sigma ^{n} \mathsf {X}\subseteq \Sigma \mathsf {X}\), whence the condition \({{\mathrm{\mathrm {Hom}}}}(\Sigma \mathsf {X},\Sigma \mathsf {Y})=0\) gives the required contradiction. \(\square \)

### 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 \(\mathsf {H}\) of a t-structure \((\mathsf {X},\mathsf {Y})\). Suppose \(\mathsf {H}\) contains an indecomposable \(Z\in \mathsf {ind}(\mathcal {Z})\). Then any other indecomposable object in \(\mathsf {H}\) must lie outside the hammocks \({{\mathrm{\mathrm {Hom}}}}^{<0}(Z,-) \ne 0\) and \({{\mathrm{\mathrm {Hom}}}}^{<0}(-,Z) \ne 0\). By Properties 1.2(3), the action of \(\Sigma ^r\) partitions \(\mathcal {Z}^k\) into strips \(r+m\) rays thick. These are further partitioned into rectangles of size \((r+m) \times (n-r)\) such that every rectangle in each strip is sent to precisely one rectangle in each other strip by some power of \(\Sigma ^r\); see Fig. 4 for an illustration. The complement of the Hom-hammocks \({{\mathrm{\mathrm {Hom}}}}^{<0}(Z,-)\) and \({{\mathrm{\mathrm {Hom}}}}^{<0}(-,Z)\) intersects finitely many such rectangles up to the action of \(\Sigma ^r\). Since each \(\mathcal {Z}^k\) component is just a suspension of each of the other \(\mathcal {Z}\) components, it follows that the objects of \(\mathsf {ind}(\mathsf {H})\cap \mathcal {Z}\) 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 \(\mathsf {H}\); hence \(\mathsf {ind}(\mathsf {H})\cap \mathcal {Z}\) is finite.

Now consider the \(\mathcal {X}\) component. By Proposition 6.4, any object \(X_{i,j}^l\) which is sufficiently high up in an \(\mathcal {X}\) component—here \(j-i \ge r+m-1\) 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, \(\mathsf {ind}(\mathsf {H})\cap \mathcal {X}\) is finite. The argument for the \(\mathcal {Y}\) component is similar. \(\square \)

### Lemma 7.4

Let \(\mathsf {H}\) be a Hom-finite abelian category with finitely many indecomposable objects. Then \(\mathsf {H}\) is a finite length category.

### Proof

Since \(\mathsf {H}\) is a Hom-finite, \({\mathbf {k}}\)-linear abelian category, it is Krull–Schmidt; see [6]. Now let *L* be the direct sum of all indecomposable objects (up to isomorphism) of \(\mathsf {H}\). By assumption, this sum is finite and hence \(L\in \mathsf {H}\). We define the function \(d:\text {Ob}(\mathsf {H})\rightarrow {\mathbb {N}}\), \(A\mapsto \dim {{\mathrm{\mathrm {Hom}}}}(L,A)\).

If \(A\subset B\) is a subobject, we obtain exact sequences \(0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0\) and \(0 \rightarrow {{\mathrm{\mathrm {Hom}}}}(L,A) \rightarrow {{\mathrm{\mathrm {Hom}}}}(L,B) \rightarrow {{\mathrm{\mathrm {Hom}}}}(L,C)\). This shows \(d(A)\le d(B)\). Moreover, if \(d(A)=d(B)\), then the induced map \({{\mathrm{\mathrm {Hom}}}}(L,B) \rightarrow {{\mathrm{\mathrm {Hom}}}}(L,C)\) is zero. For some \(s\in {\mathbb {N}}\), there is a surjection \(p: L^{\oplus s}\twoheadrightarrow B\), inducing a further surjection \(q: L^{\oplus s}\twoheadrightarrow C\). However, we also get \(0 \rightarrow {{\mathrm{\mathrm {Hom}}}}(L^{\oplus s},A) \rightarrow {{\mathrm{\mathrm {Hom}}}}(L^{\oplus s},B) \xrightarrow {v} {{\mathrm{\mathrm {Hom}}}}(L^{\oplus s},C)\). The dimensions of the first two Hom spaces are \(sd(A)=sd(B)\), so that \(v=0\). Since \(v(p)=q\) by construction, this forces \(C=0\).

Hence for \(B\in \mathsf {H}\), the function *d* can only take the values \(1,\ldots ,d(B)-1\) on non-trivial subobjects. Thus ascending or descending chains of subobjects of *B* must stabilise. \(\square \)

### Remark 7.5

Proposition 7.1 means that the heart of each bounded t-structure in \(\mathsf {D}^b(\Lambda (r,n,m))\) is equivalent to \(\mathsf {mod}(\Gamma )\), for a finite-dimensional algebra \(\Gamma \) of finite representation type. Note that, by work of Schröer and Zimmermann [42], \(\Gamma \) 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 \(\mathsf {D}^b(\Lambda (r,n,m))\) into a semi-orthogonal decomposition, one of whose orthogonal subcategories is the bounded derived category of a path algebra of Dynkin type *A*.

### 7.2 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.

### Proposition 7.6

Let \(Z\in \mathsf {ind}(\mathcal {Z})\) and \(\mathsf {Z}=\mathsf {thick}_{\mathsf {D}^b(\Lambda )}(Z)\). Then \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) and there is a semi-orthogonal decomposition \(\mathsf {D}^b(\Lambda (r,n,m)) = {\langle \mathsf {D}^b({\mathbf {k}}A_{n+m-1}), \mathsf {Z}\rangle }\). In particular, \(\mathsf {Z}\) is functorially finite in \(\mathsf {D}^b(\Lambda (r,n,m))\). Moreover, \(\mathsf {D}^b(\Lambda (r,n,m))\) 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: \(\mathsf {Z}= {{\mathrm{\mathsf {add}}}}(\Sigma ^i Z \mid i \in {\mathbb {Z}})\) and that \(\mathsf {Z}\) is an admissible subcategory of \(\mathsf {D}^b(\Lambda )\); for this last claim see [11, Theorem 3.2]. Furthermore \(\mathsf {D}^b(\Lambda ) = {\langle \mathsf {Z}^\perp ,\mathsf {Z}\rangle }\) 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 \(\mathcal {Z}\) component of the AR quiver of \(\mathsf {D}^b(\Lambda )\). Using the transitive action of the autoequivalence group on \(\mathsf {ind}(\mathcal {Z})\), see Corollary 5.5, we thus can assume, without loss of generality, that \(Z=P(n-r)=e_{n-r}\Lambda \). There is a full embedding \(\iota : \mathsf {D}^b(\Lambda / \Lambda e_{n-r} \Lambda ) \rightarrow \mathsf {D}^b(\Lambda )\) with essential image \(\mathsf {thick}_{\mathsf {D}^b(\Lambda )}(e_{n-r} \Lambda )^\perp = \mathsf {Z}^\perp \); see, for example, [2, Lemma 3.4]. Inspecting the Gabriel quiver of \(\Lambda / \Lambda e_{n-r} \Lambda \), 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 \(\Lambda / \Lambda e_{n-r} \Lambda \) is an iterated tilted algebra of type \(A_{n+m-1}\). It is well known that this implies \(\mathsf {D}^b(\Lambda / \Lambda e_{n-r} \Lambda ) \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m-1})\); see [21]. Combining these pieces, we get \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m-1})\). The final claim about \(\mathsf {D}^b(\Lambda )\) having a full exceptional sequence follows at once from the fact that \(\mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) has one. \(\square \)

### Remark 7.7

The subcategory of type \(\mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) can be explicitly identified in the AR quiver of \(\mathsf {D}^b(\Lambda (r,n,m))\); see Fig. 4. The choice of right orthogonal to *Z* was arbitrary, since Serre duality provides an equivalence \(^\perp \mathsf {Z}\rightarrow \mathsf {Z}^\perp \), \(X\mapsto \mathsf {S}(X)\). We mention in passing that the thick subcategory \(\mathsf {Z}\) is equivalent to \(\mathsf {D}^b({\mathbf {k}}A_1)\).

The silting objects of \(\mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) 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].

### 7.3 Silting reduction

The main technical tool in the classification is the following result of Aihara and Iyama in [1]:

### Theorem 7.8

*G*in Proposition 7.15. The subcategory \(\mathsf {B}{:}{=}{{\mathrm{\mathsf {susp}}}}{\Sigma \mathsf {N}}\) is the ‘co-aisle’ of the co-t-structure associated to \(\mathsf {N}\) (see Theorem A.8) and thus covariantly finite in \(\mathsf {U}\). Putting this together with \(\mathsf {U}\) being functorially finite in \(\mathsf {D}\), it gives rise to a co-t-structure \((\mathsf {A},\mathsf {B})\) in \(\mathsf {D}\), where \(\mathsf {A}{:}{=}{}^\perp \mathsf {B}\). Now let \(\mathsf {K}\) be a silting subcategory of \(\mathsf {U}^\perp \) and consider the approximation triangle of \(K \in \mathsf {K}\) with respect to the co-t-structure \((\mathsf {A},\mathsf {B})\),

### Definition 7.9

*minimal*left \(\mathsf {B}\)-approximation of

*V*. Note that here, in contrast to elsewhere in this paper, we require that the approximation is minimal to ensure well-definition of the map \(G_{\mathsf {N}}\). Furthermore, we stress here that \(G_{\mathsf {N}}\) 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 \(\mathsf {Z}\) for some *Z* in the \(\mathcal {Z}\) components. For silting reduction to work, we first need to establish that any silting subcategory of \(\mathsf {D}^b(\Lambda (r,n,m))\) contains an indecomposable object from the \(\mathcal {Z}\) 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.

### Lemma 7.10

If \(\mathsf {M}\) is a subcategory of \(\mathsf {D}^b(\Lambda )\) such that \(\mathsf {thick}_{}(\mathsf {M})=\mathsf {D}^b(\Lambda )\), then \(\mathsf {M}\) contains an indecomposable object from the \(\mathcal {Z}\) components.

### Corollary 7.11

Any silting subcategory of \(\mathsf {D}^b(\Lambda )\) and any simple-minded collection in \(\mathsf {D}^b(\Lambda )\) contain objects from some \(\mathcal {Z}\) component.

### Proof of lemma

By Lemma 5.6, the additive closure of the \(\mathcal {X}\) components of \(\mathsf {D}^b(\Lambda )\) is a thick subcategory of \(\mathsf {D}^b(\Lambda )\), and likewise for the additive closure of the \(\mathcal {Y}\) components. Furthermore, these two subcategories are fully orthogonal by Propositions 3.4 and 3.5, so that their sum is a thick subcategory of \(\mathsf {D}^b(\Lambda )\) as well. Therefore we cannot have \(\mathsf {M}\subset \mathcal {X}\oplus \mathcal {Y}\) as that would force \(\mathsf {D}^b(\Lambda ) = \mathsf {thick}_{}(\mathsf {M}) = \mathcal {X}\oplus \mathcal {Y}\), a contradiction. \(\square \)

Theorem 7.8 coupled with Proposition 7.6 tells us that all silting objects in \(\mathsf {D}^b(\Lambda )\) containing *Z* can be obtained by lifting silting objects in \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) back up to \(\mathsf {D}^b(\Lambda )\). In other words, any silting object in \(\mathsf {D}^b(\Lambda )\) can be described by a pair \((Z,M')\) consisting of an indecomposable object \(Z\in \mathcal {Z}\) and a silting object \(M'\in \mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m-1})\).

We now make a brief expository digression explaining Keller and Vossieck’s classification of silting subcategories of \(\mathsf {D}^b({\mathbf {k}}A_t)\), from which the silting subcategories of \(\mathsf {D}^b(\Lambda (r,n,m))\) can be ‘glued’.

### 7.4 Classification of silting objects in Dynkin type *A*

*g*,

*h*) with \(g \in {\mathbb {Z}}\) and \(h\in \{1,\ldots ,t\}\).Given an indecomposable object \(U \in \mathsf {D}^b({\mathbf {k}}A_t)\) we write its coordinates as (

*g*(

*U*),

*h*(

*U*)).

Following [31], a quiver \(Q=(Q_0,Q_1)\) is called an \({\mathbb {A}}_t\)-*quiver* if \(|Q_0|=t\), its underlying graph is a tree, and \(Q_1\) decomposes into a disjoint union \(Q_1 = Q_{\alpha } \cup Q_{\beta }\) such that at any vertex at most one arrow from \(Q_{\alpha }\) ends, at most one arrow from \(Q_{\alpha }\) starts, at most one arrow from \(Q_{\beta }\) ends and at most one arrow from \(Q_{\beta }\) starts. One should think of an \({\mathbb {A}}_t\)-quiver as a ‘gentle tree quiver’, where gentle is used in the sense of gentle algebras.

By abuse of notation we identify the object \(T_Q {:}{=}\varphi _Q(Q_0)\) 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 \(Q \mapsto T_Q\) induces a bijection between isomorphism classes of \({\mathbb {A}}_t\)-quivers and tilting objects *T* in \(\mathsf {D}^b({\mathbf {k}}A_t)\) satisfying the condition \(\min \{g(U) \mid U \text { is an indecomposable summand of } T\} =0\).

Note that in Dynkin type \(A_t\), the summands of any tilting object \(T=\bigoplus _{i=1}^t T_i\) can be re-ordered to give a strong, full exceptional sequence \(\{T_1,\ldots ,T_t\}\), see [31, Section 5.2]. We now have the following classification of silting objects in \(\mathsf {D}^b({\mathbf {k}}A_t)\).

### Theorem 7.13

([31], Theorem 5.3) Let \(T=T_1\oplus \cdots \oplus T_t\) be a tilting object in \(\mathsf {D}^b({\mathbf {k}}A_t)\) whose summands form an exceptional collection. Let \(p: \{1,\ldots ,t\} \rightarrow {\mathbb {N}}\) be a weakly increasing function. Then \(\Sigma ^{p(1)} T_1\oplus \cdots \oplus \Sigma ^{p(t)} T_t\) is a silting object in \(\mathsf {D}^b({\mathbf {k}}A_t)\). Moreover, all silting objects of \(\mathsf {D}^b({\mathbf {k}}A_t)\) 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 \(\mathsf {D}^b({\mathbf {k}}A_3)\).

### Example 7.14

In each sketch the triangle depicts the standard heart for the quiver \(1 \longleftarrow 2 \longleftarrow 3\) 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 \(\mathsf {D}^b({\mathbf {k}}A_3)\) containing one of *P*(*i*) for \(1 \le i \le 3\) as a least element.

*U*with minimal \(g(U) = 1\). These correspond precisely to \(\tau ^{-1}\) applied to each of the diagrams Open image in new window to Open image in new window. Observe that Open image in new window and Open image in new window. 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 \(\tau ^{-2}\) applied to each of the diagrams Open image in new window to Open image in new window. We have Open image in new window, Open image in new window, Open image in new window, and Open image in new window, which leaves, up to suspension, only Open image in new window and Open image in new window 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 \(\mathsf {D}^b({\mathbf {k}}A_3)\) up to suspension:

### 7.5 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, \(\Lambda (2,3,1)\) 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 \(G_Z : \mathsf {Z}^\perp \rightarrow \mathsf {D}^b(\Lambda (r,n,m))\) from Definition 7.9, where \(\mathsf {Z}= \mathsf {thick}_{}(Z)\) for some fixed, arbitrary, indecomposable object \(Z \in \mathsf {ind}(\mathcal {Z})\).

We first explicitly compute the map \(G_Z: \mathsf {ind}(\mathsf {Z}^\perp ) \rightarrow \mathsf {D}^b(\Lambda (r,n,m))\) on objects in the case \(Z = Z^0_{0,0}\).

### Proposition 7.15

- (1)
\(G(\Sigma ^i X^1_{0,j}) = \Sigma ^i Z^0_{j+1,0}\) for \(0 \le j < r +m-1\) and \(i \ge 0\).

- (2)
\(G(\Sigma ^i Y^1_{j,0}) = \Sigma ^i Z^0_{0,j+1}\) for \(0 \le j < n-r-1\) and \(i \ge 0\).

- (3)
\(G(\Sigma ^i Z^1_{j,0}) = \Sigma ^i X^1_{j,-1}\) for \(1 \le j \le r+m-1\) and \(i \ge 0\).

- (4)
\(G(\Sigma ^i Z^1_{0,j}) = \Sigma ^i Y^1_{-1,j}\) for \(r-n+1 \le j \le -1\) and \(i \ge 0\).

- (5)
\(G(\Sigma ^i Z^1_{-r-m,0}) = \left\{ \begin{array}{ll} \Sigma ^i X^1_{-r-m,-1} &{} \text {for } 0 \le i \le r, \\ \Sigma ^i Z^1_{0,n-r} &{} \text {for } i > r. \end{array} \right. \)

### Proposition 7.16

- (1)
\(G(\Sigma ^i X_{1+m,1+m+j}) = \Sigma ^i Z_{j+1,0}\) for \(0 \le j < m\) and \(i \ge 0\).

- (2)
\(G(\Sigma ^i Y_{1-n+j,1-n}) = \Sigma ^i Z_{0,j+1}\) for \(0 \le j < n-2\) and \(i \ge 0\).

- (3)
\(G(\Sigma ^i Z_{j,1-n}) = \Sigma ^i X_{j,m}\) for \(0< j < m+1\) and \(i \ge 0\).

- (4)
\(G(\Sigma ^i Z_{m+1,j}) = \Sigma ^i Y_{-n,j}\) for \(2-2n< j < 1-n \) and \(i \ge 0\).

- (5)
\(G(\Sigma ^i Z_{0,1-n}) = \left\{ \begin{array}{ll} X_{0,m} &{} \text {for } i=0, \\ \Sigma ^i Z_{0,n-1} &{} \text {for } i > 0. \end{array} \right. \)

### 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 \((\mathsf {A},\mathsf {B})\) with \(\mathsf {B}= {{\mathrm{\mathsf {susp}}}}{\Sigma Z_0}={{\mathrm{\mathsf {add}}}}\{\Sigma ^i Z_0 \mid i\ge 1\}\). Using Proposition 3.6, one can easily compute \(\mathsf {A}= {}^\perp \mathsf {B}\). If \(U \in \mathsf {A}\), then \(G(U) = U\), so examining \(\mathsf {A}\cap \mathsf {Z}^\perp \) gives the list of exceptions above.

*G*(

*U*) directly using the triangles from Properties 1.2(4):

- (1)
The relevant triangles here are \(Z^0_{j+1,0}\rightarrow X^1_{0,j}\rightarrow Z^1_{0,0}\rightarrow \Sigma Z^0_{j+1,0}\) for \(0 \le j < r+m-1\), where we note that \(\Sigma Z^0_{j+1,0} = Z^1_{j+1,0}\).

- (2)
Here we have \(Y^1_{j,0}\rightarrow Z^1_{0,0}\rightarrow Z^1_{0,j+1}\rightarrow \Sigma Y^1_{j,0}\) for \(0 \le j < n-r-1\), again noting that \(Z^0_{0,j+1} = \Sigma ^{-1} Z^1_{0,j+1}\).

- (3)
The triangles are \(X^1_{j,-1}\rightarrow Z^1_{j,0}\rightarrow Z^1_{0,0}\rightarrow \Sigma X^1_{j,-1}\) for \(1 \le j \le r+m-1\).

- (4)
The triangles are \(Y^1_{-1,j}\rightarrow Z^1_{0,j}\rightarrow Z^1_{0,0}\rightarrow \Sigma Y^1_{-1,j}\) for \(r-n+1 \le j \le -1\).

- (5)When \(0 \le i \le r\), 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 \(\Sigma ^i Z_{-r-m,0} \rightarrow \Sigma ^i \big ( Z^1_{-r-m,r-n} \oplus Z^1_{0,0} \big )\). We claim that the cone of \(Z_{-r-m,0} \rightarrow Z^1_{-r-m,r-n} \oplus Z^1_{0,0}\) is \(Z^1_{0,r-n}\). To show this, we compute the cocone of \( Z^1_{-r-m,r-n} \oplus Z^1_{0,0} \rightarrow Z^1_{0,r-n}\) 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 \(X^1_{-r-m,-1}\rightarrow Z^1_{-r-m,0}\rightarrow Z^1_{0,0}\rightarrow \Sigma X^1_{-r-m,-1}\), which computes the cocone \(C = Z^1_{-r-m,0}\) as claimed. \(\square \)

### Corollary 7.17

Let \(Z \in \mathsf {ind}(\mathcal {Z})\) be arbitrary. If \(U \in \mathsf {Z}^\perp \) is indecomposable then \(G_Z(U)\) is also indecomposable.

### Proof

Since the autoequivalences \(\mathsf {T}\!_{\mathcal {X}}\), \(\mathsf {T}\!_{\mathcal {Y}}\) and \(\Sigma \) act transitively on the \(\mathcal {Z}\) 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.\(\square \)

### Lemma 7.18

The relation \(\preceq \) defines a total order on \(\mathsf {ind}(\mathcal {Z})\).

### Proof

*Anti-symmetry:* Suppose \(Z \preceq Z'\) and \(Z' \preceq Z\) with \(Z\in \mathsf {ind}(\mathcal {Z}^i)\) and \(Z'\in \mathsf {ind}(\mathcal {Z}^j)\). If \(i=j\), then anti-symmetry is clear. For a contradiction, suppose \(i<j\). Then \(\mathsf {ray}(\Sigma ^{j-i}Z) \le \mathsf {ray}(Z')\) and \(\mathsf {ray}(\tau ^{-1}\Sigma ^{i-j}Z')\le \mathsf {ray}(Z)\). In particular, it follows that \(\mathsf {ray}(\tau ^{-1} Z') \le \mathsf {ray}(\Sigma ^{j-i} Z) \le \mathsf {ray}(Z')\), which is a contradition, since \(\mathsf {ray}(\tau ^{-1}Z') > \mathsf {ray}(Z')\). The same argument works when \(i>j\).

*Transitivity:* Suppose \(Z \preceq Z'\) and \(Z' \preceq Z''\) with \(Z\in \mathsf {ind}(\mathcal {Z}^i)\), \(Z'\in \mathsf {ind}(\mathcal {Z}^j)\) and \(Z''\in \mathsf {ind}(\mathcal {Z}^k)\). 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 \(\mathsf {ray}(\tau ^{-1}\Sigma ^{j-i}Z) \le \mathsf {ray}(Z')\) and the second inequality means that \(\mathsf {ray}(\Sigma ^{k-j}Z') \le \mathsf {ray}(Z'')\). There are two subcases: first assume \(i\le k\). In this case, apply \(\tau \Sigma ^{k-j}\) to the condition arising from the first inequality and combine this with the second inequality to get \(\mathsf {ray}(\Sigma ^{k-i}Z)\le \mathsf {ray}(\tau \Sigma ^{k-j} Z') < \mathsf {ray}(\Sigma ^{k-j} Z') \le \mathsf {ray}(Z'')\). Now assume that \(i > k\) and apply \(\Sigma ^{k-j}\) to the condition arising from the first inequality and combine with the second inequality to get \(\mathsf {ray}(\tau ^{-1}\Sigma ^{k-i}Z) \le \mathsf {ray}(\Sigma ^{k-j}Z') \le \mathsf {ray}(Z')\).

*Totality:* Suppose \(Z\in \mathsf {ind}(\mathcal {Z}^i)\) and \(Z'\in \mathsf {ind}(\mathcal {Z}^j)\). If \(i=j\) then it is clear that either \(Z \preceq Z'\) or \(Z' \preceq Z\). Now suppose \(i<j\). If \(\mathsf {ray}(\Sigma ^{j-i}Z) \le \mathsf {ray}(Z')\) then \(Z \preceq Z'\) and we are done, so suppose that \(\mathsf {ray}(\Sigma ^{j-i}Z) > \mathsf {ray}(Z')\). Then it follows that \(\mathsf {ray}(\Sigma ^{i-j}Z') < \mathsf {ray}(Z)\), in which case, because \(\tau ^{-1}\) increases the index of the ray by 1, one gets \(\mathsf {ray}(\tau ^{-1}\Sigma ^{i-j}Z')\le \mathsf {ray}(Z)\) and hence \(Z' \preceq Z\). A similar argument holds in the case \(i>j\). Thus, \(\preceq \) is indeed a total order.\(\square \)

Using Corollary 7.17, we now ensure we identify each silting subcategory of \(\mathsf {M}\) of \(\mathsf {D}^b(\Lambda )\) as precisely one pair \((Z,M')\), with \(\mathsf {M}'\) a silting object of \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n-m+1})\) by insisting that \(Z \preceq Z'\) for each \(Z' \in \mathsf {ind}(\mathcal {Z}) \cap {{\mathrm{\mathsf {add}}}}{M'}\).

### Definition 7.19

With the identification of \(\mathsf {D}^b({\mathbf {k}}A_{n+m-1})\) in \(\mathsf {D}^b(\Lambda (r,n,m))\) of Remark 7.7, using Proposition 7.15, we now give an explicit description of the additive subcategory \(\mathsf {Z}^\perp _{\prec }\).

*Z*. In Lemma 7.20 below, when we specify \(\mathsf {mod}(\Gamma )\), we shall mean precisely this copy sitting inside \(\mathsf {D}^b({\mathbf {k}}A_{n+m-1})\).

### Lemma 7.20

*P*(

*i*) is the indecomposable projective at vertex

*i*for the path algebra \(\Gamma = {\mathbf {k}}A_{n+m-1}\).

### Proof

This is a direct computation using Proposition 7.15, the total order on the indecomposable objects of the \(\mathcal {Z}\) components of Lemma 7.18, and the identification of the subcategory from Remark 7.7. \(\square \)

We summarise this discussion in the following proposition, and obtain the main theorem of the section as a corollary.

### Proposition 7.21

- (1)
Silting subcategories \(\mathsf {M}\) of \(\mathsf {D}^b(\Lambda )\) with \(Z \in \mathsf {M}\) and \(Z \preceq \mathsf {ind}(\mathcal {Z})\cap \mathsf {M}\).

- (2)
Silting subcategories \(\mathsf {N}\) of \(\mathsf {Z}^\perp \) with \(\mathsf {N}\cap \mathsf {Z}^\perp _{\prec } = \varnothing \).

### Theorem 7.22

- (1)
Pairs \((Z,\mathsf {N})\) where \(Z \in \mathsf {ind}(\mathcal {Z})\) and \(\mathsf {N}\) is a silting subcategory of \(\mathsf {D}^b({\mathbf {k}}A_{m+n-1})\) containing no objects in the additive subcategory \(\mathsf {Z}^\perp _{\prec }\).

- (2)
Silting subcategories of \(\mathsf {D}^b(\Lambda (r,n,m))\).

- (3)
Bounded t-structures in \(\mathsf {D}^b(\Lambda (r,n,m))\).

- (4)
Bounded co-t-structures in \(\mathsf {D}^b(\Lambda (r,n,m))\).

## 8 A detailed example: \(\Lambda (2,3,1)\)

The twelve tilting objects in \({\mathbf {k}}A_3\) giving rise to the silting objects containing \(Z^0_{0,0}\) as the \(\preceq \)-minimal summand in \(\mathcal {Z}\) for \(\Lambda (2,3,1)\)

Tilting object in \({\mathbf {k}}A_3\) | Silting family in \(\Lambda (2,3,1)\) |
---|---|

\(T_1\)\(\oplus \)\(T_2\)\(\oplus \)\(T_3\) | \(\Sigma ^iT_1 \oplus \Sigma ^jT_2 \oplus \Sigma ^kT_3\) |

| \(j\ge i\) and \(k\ge \max \{j,-1\}\) |

| \(k\ge j \ge \max \{i,-1\}\) |

| \(j\ge i\) and \(k\ge \max \{i,-1\}\) |

| \(j\ge -1\) and \(k\ge \max \{i,j,-1\}\) |

| \(k\ge j\ge i\ge -1\) |

| \(k\ge j\ge i\ge -1\) |

| \(k\ge j\ge \max \{i,-1\}\) |

| \(k\ge i\) and \(j\ge i\ge -1\) |

| \(j\ge i\) and \(k\ge \max \{j,-2\}\) |

\(\Sigma P(1)\)\(\oplus \) | \(j\ge -1\) and \(k\ge \max \{i,j\}\) |

| \(k\ge j\ge i\ge -1\) |

| \(k\ge j\ge i\ge -1\) |

We make the following observation regarding tilting objects in \(\mathsf {D}^b(\Lambda (2,3,1)\).

### Proposition 8.1

- (1)
There are precisely six tilting objects in \(\mathsf {D}^b(\Lambda )\) containing

*Z*as a summand. - (2)
If \(T \in \mathsf {D}^b(\Lambda )\) is a tilting object containing

*Z*as a summand then \(F_Z(T)\) is a tilting object in \(\mathsf {Z}^\perp \).

### Proof

The proof is a direct computation. Without loss of generality, we may set \(Z=Z^0_{0,0}\). Consider the additive subcategory \(\mathsf {T}{:}{=}\big (\bigcap _{n\ne 0} {}^\perp (\Sigma ^n Z)\big ) \cap \big (\bigcap _{n\ne 0} (\Sigma ^n Z)^\perp \big ) \cap \mathsf {Z}^\perp \). The subcategory \(\mathsf {T}\) consists of the thick subcategory \(\mathsf {Z}^\perp \cap {}^\perp \mathsf {Z}\simeq \mathsf {D}^b({\mathbf {k}})\), which has just one indecomposable object in each homological degree, together with finitely many indecomposables in homological degrees 0, 1 and 2.

Our computations lead us to state the following conjecture:

### Conjecture 8.2

- (1)
There are finitely many tilting objects in \(\mathsf {D}^b(\Lambda )\) containing

*Z*as a summand. - (2)
If \(T \in \mathsf {D}^b(\Lambda )\) is a tilting object containing

*Z*as a summand then \(F_Z(T)\) is a tilting object in \(\mathsf {Z}^\perp \).

### 8.1 An explicit example for a bounded t-structure in \(\varvec{\mathsf {D}^b(\Lambda (2,3,1))}\)

We finish by choosing a silting object \(N\in \mathsf {D}^b({\mathbf {k}}A_3)\), assembling this with \(Z=Z^0_{0,0}\) to the associated silting object \(M\in \mathsf {D}^b(\Lambda (2,3,1))\) and computing the bounded t-structure in \(\mathsf {D}^b(\Lambda (2,3,1))\) induced by *M*.

*N*corresponds to the object

The corresponding co-t-structure \((\mathsf {A}_M,\mathsf {B}_M)\) is right adjacent in the sense of [12] to the t-structure \((\mathsf {X}_M,\mathsf {Y}_M)\), i.e. \(\mathsf {B}_M = \mathsf {X}_M\) and \(\mathsf {A}_M {:}{=}{}^\perp \mathsf {B}_M = {}^\perp {{\mathrm{\mathsf {susp}}}}{M} = {}^\perp (\Sigma ^{\ge 0} M)\).

*M*using the bijections of König and Yang [33]:

## Notes

### Acknowledgments

We are grateful to Aslak Bakke Buan, Christof Geiß, Martin Kalck, Henning Krause, and Dong Yang for answering our questions and particularly to an anonymous referee for a careful reading and many valuable comments. Much of this paper was prepared while all three authors worked at Leibniz Universität Hannover. The second author acknowledges the financial support of the EPSRC of the United Kingdom through the grant EP/K022490/1.

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