Discrete derived categories I: homomorphisms, autoequivalences and tstructures
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 tstructures.
Keywords
Discrete derived category Auslander–Reiten quiver Homhammock Twist functor Silting object tStructure String algebraMathematics Subject Classification
16G10 16G70 18E301 Introduction
In this article, we study the bounded derived categories of finitedimensional 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 tstructures and cotstructures 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 deriveddiscrete 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 nontrivial 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 clustertilted 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 nontrivial homomorphism spaces between them, so called ‘Homhammocks’. 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 tstructures and cotstructures 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 cotstructure analogues [29]. Further investigation into the properties of (co)tstructures and the stability manifolds is conducted in the sequel [14]; see also [38].
We study (co)tstructures indirectly via silting subcategories, which generalise tilting objects and behave like the projective objects of hearts of bounded tstructures. In general, one cannot get all bounded tstructures in this way, but in Proposition 7.1, we show that the heart of each bounded tstructure in \(\mathsf {D}^b(\Lambda (r,n,m))\) is equivalent to \(\mathsf {mod}(\Gamma )\), where \(\Gamma \) is a finitedimensional 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)tstructures. We show that \(\mathsf {D}^b(\Lambda (r,n,m))\) admits a semiorthogonal decomposition into \(\mathsf {D}^b({\mathbf {k}}A_{n+m1})\) 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 ARquiver
We always work over a fixed algebraically closed field \({\mathbf {k}}\). All modules will be finitedimensional right modules. Throughout, all subcategories will be additive and closed under isomorphisms.
2.1 Discrete derived categories
We are interested in \({\mathbf {k}}\)linear, Homfinite 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 Homfinite triangulated category with a bounded tstructure) \(\mathsf D\) is discrete (with respect to this tstructure), 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 finitedimensional algebra \(\Lambda \), so \(K_0(\Lambda )\cong {\mathbb {Z}}^r\). In our slight generalisation of his notion, we cannot do so, but for finitedimensional 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 tstructure. Throughout this article, we shall be interested in the bounded derived category \(\mathsf {D}^b(\Lambda )\) of a finitedimensional algebra \(\Lambda \). We shall always use discreteness with respect to the standard tstructure, whose heart is \(\mathsf {mod}(\Lambda )\), the category of finitedimensional 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 tstructure.
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 finitedimensional algebra \(\Lambda \), which is not of derivedfinite 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.
2.2 The AR quiver of \(\varvec{\mathsf {D}^b(\Lambda (r,n,m))}\)
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 \((i1,j1)\) in the same component, i.e. \(\tau X^k_{i,j} = X^k_{i1,j1}\) etc.
 (3)The suspension of indecomposable objects is given below, with \(k=0,\ldots ,r2\): In particular, \(\Sigma ^r_\mathcal {X}= \tau ^{mr}\) and \(\Sigma ^r_\mathcal {Y}= \tau ^{nr}\) on objects.
 (4)There are distinguished triangles, for any \(i,j,d\in {\mathbb {Z}}\) with \(d\ge 0\):
 (5)There are chains of nonzero morphisms for any \(i\in {\mathbb {Z}}\) and \(k=0,\ldots ,r1\):
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 \(h(X^k_{ij}) = ji\) 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 \(h(Y^k_{ij}) = ij\) 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 socalled ‘Homhammock’ 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 Homhammocks: \({{\mathrm{\mathrm {Hom}}}}(,A) = {{\mathrm{\mathrm {Hom}}}}(\mathsf {S}^{1}A,)^*\). 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 Homhammocks 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 wellknown 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.
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 Homhammocks 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 \(\mathcal {X}^k\).
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
Proof
By Lemma 3.1 the composition, \({}_0A \rightarrow A\), of irreducible maps along a ray is nonzero. Likewise the composition, \(A \rightarrow A_0\), of irreducible maps along a coray is nonzero.
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).
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 nonzero 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 \(h(A') = h(A'') = h1\) and \(h(C) = h2\). (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 Homhammocks 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 \(2,\ldots ,h1\) on the segment.
Case \(B\in \mathcal {X}^{k+1}\): We start by showing the existence of nonzero 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 nonzero 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.
For the Homvanishing 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 nonzero 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 Homhammocks 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 Homhammocks 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.
 \({\mathcal U}:\)

The upwardsopen region including \(\mathsf {ray}_{\! +}(\tau A) \backslash \{\tau A\}\) but excluding \(\mathsf {coray}_{\! }(\tau A)\);
 \({\mathcal L}:\)

The leftopen region including \(\mathsf {ray}_{\! }(\tau A) \cup \mathsf {coray}_{\! }(\tau A)\);
 \({\mathcal D}:\)

The downwardsopen region including \(\mathsf {coray}_{\! +}(\tau A)\backslash \{\tau A\}\) but excluding \(\mathsf {ray}_{\! }(\tau A)\);
 \({\mathcal R}:\)

The rightopen 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 nontrivial morphisms from A. Using (2) and analogous reasoning shows that no objects in \({\mathcal D}\) admit nontrivial morphisms from A. Nonexistence of nontrivial 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 nonvanishing 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.
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
 (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 1cycle’ 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 nontrivial 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 \((E_1,\ldots ,E_{n1})\) 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 ncycle 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 ncycles 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 ncycle 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 ncycle \(E_*\) consists of sheaves, then this forces \(k_i=d\) to be maximal for all i, as nonzero 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 2cycle 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 ncycle 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.
This also works if \(n=2\) and \(k_1=k_2\) (with unchanged lefthand 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 \(\mathsf {E}^\perp \).
Remark 4.6
An object \(X\in \mathsf D\) is called dspherelike 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 \(E_n\notin \mathsf {thick}_{}(E_1,\ldots ,E_{n1})\). Recall that an exceptional ncycle \((E_1,\ldots ,E_n)\) 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 ncycle in \(\mathsf D\). Then there exists a \((k+1n)\)spherelike object \(X\in \mathsf D\) with nonzero maps \(X\rightarrow E_1\) and \(\Sigma ^{n1k+k_n}E_n\rightarrow X\).
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 \(l_i {:}{=}k_1+\cdots +k_i + 1i\).
Since \(i+1<n\), \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_i,E_{i+2})=0\) by (ii) and the definition of exceptional cycles, hence \({{\mathrm{\mathrm {Hom}}}}^\bullet (X_{i+1},E_{i+2}) = {{\mathrm{\mathrm {Hom}}}}^\bullet (\Sigma ^{l_i1}E_{i+1},E_{i+2})\). 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 1dimensional, and situated in degree \(l_i+k_{i+1}  1 = l_{i+1}\).
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 injectiveprojective 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 Homfinite, 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 \(k_*=(1,\ldots ,1,1r)\), and \(F_*\) forms an exceptional \((nr)\)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 \(X^0_{ii}=\tau ^{1i}(X^0_{11})\) is. This point also gives the second condition of exceptional cycles: for \(i=1,\ldots ,m+r1\), we have \(\mathsf {S}E_i = \Sigma \tau X^0_{m+r+1i,m+r+1i} = \Sigma X^0_{m+ri,m+ri} = \Sigma E_{i+1}\) and at the boundary step we have \(\mathsf {S}E_{m+r} = \Sigma \tau X^0_{11} = \Sigma X^0_{00} = \Sigma ^{1r} X^0_{m+r,m+r} = \Sigma ^{1r} E_1\), where we freely make use of the results stated in Sect. 2. Hence the degree shifts of the sequence \(E_*\) are \(k_1=\ldots =k_{m+r1}=1\) and \(k_{m+r}=1r\). 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 \(\mathsf {S}F_{nr} = \Sigma \tau Y^0_{11} = \Sigma Y^0_{00} = \Sigma ^{1+r} Y^0_{nr,nr} = \Sigma ^{1+r} F_1\). \(\square \)
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. \(\Sigma ^r_\mathcal {X}= \tau ^{mr}\) and \(\Sigma ^r_\mathcal {Y}= \tau ^{nr}\) on objects.
Note that the righthand 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 \(\mathsf {T}\!_\mathcal {X}(E_i)=\Sigma ^{1k_{i1}}(E_{i1})\), and furthermore \(k_1=\cdots =k_{m+r1}=1\) and \(k_{m+r}=1r\) 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.
Remaining cases: Analogous reasoning shows \(\mathsf {T}\!_{F_*}(F_i)=\tau ^{1}(F_i)\) for all \(i=1,\ldots ,nr\), 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 wellknown 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 \((n+m1)\)subtorus of the subgroup \(({\mathbf {k}}^*)^{m+n}\) of arrowscaling automorphisms consists of inner automorphisms. Furthermore, the automorphism scaling all arrows simultaneously by the same number is easily seen not to be inner, hence, \({{\mathrm{\mathsf {Out}}}}(\Lambda )={\mathbf {k}}^*\). 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 1dimensional, 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 \(f_0 \Sigma ^{r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,rn}={{\mathrm{{\mathsf {id}}}}}\) 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 \(\Sigma ^{r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,rn}\) fixes all objects of \(\mathsf {D}\), hence \(f_0\Sigma ^{r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,rn} = {{\mathrm{{\mathsf {id}}}}}\) 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 \(c=k(nr)\) for some \(k\in {\mathbb {Z}}\). Hence \(k=l\) and \(\psi = \Sigma ^{lr}\mathsf {T}\!_\mathcal {X}^{\,l(m+r)}\mathsf {T}\!_\mathcal {Y}^{\,l(nr)}=f_0^{l}\). So \(g=f_0^{l}\) and altogether, \( g\Sigma ^a\mathsf {T}\!_\mathcal {X}^{\,b}\mathsf {T}\!_\mathcal {Y}^{\,c} = (f_0 \Sigma ^{r} \mathsf {T}\!_\mathcal {X}^{\,m+r} \mathsf {T}\!_\mathcal {Y}^{\,nr})^{l}\) is a power of the stated relation.
Step 6: \({{\mathrm{\mathsf {Aut}}}}(\mathsf {D}) \cong {\mathbb {Z}}^2 \times {\mathbb {Z}}/(r,n,m) \times {\mathbf {k}}^*\).
Question
 (1)
\(\Sigma ^r_\mathcal {X}= \tau ^{mr}\) and \(\Sigma ^r_\mathcal {Y}= \tau ^{nr}\)
 (2)
\(\mathsf {T}\!_\mathcal {X}_\mathcal {X}= \tau ^{1}\) and \(\mathsf {T}\!_\mathcal {Y}_\mathcal {Y}= \tau ^{1}\)
 (3)
\(\Sigma ^{r} = \mathsf {T}\!_\mathcal {X}^{\,m+r} \, \mathsf {T}\!_\mathcal {Y}^{\,rn}\)
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 Homhammocks. 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.
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 nonzero 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 nonvanishing 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
Proof
The argument is similar to the computation of the Homhammocks in the \(\mathcal {Z}\) components from Sect. 3. We proceed in several steps.
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 1dimensional 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 nonzero 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.
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 nonzero. 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 Homhammocks 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 selfextensions 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.
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 Homfinite 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 semiorthogonal 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 dspherelike; see [22] for details. Assuming \(\mathsf {D}\) has a Serre functor \(\mathsf {S}\), a dspherelike object D is called dspherical if \(\mathsf {S}(D)=\Sigma ^dD\); see [27, §8].
Proposition 6.4

Exceptional if \(A\in \mathcal {Z}\), or \(A\in \mathcal {X}\) with \(h(A)<m+r1\), or \(A\in \mathcal {Y}\) with \(h(A)<nr1\).

\((1r)\)spherelike if \(A\in \mathcal {X}\) with \(h(A)=m+r1\).

\((1+r)\)spherelike if \(A\in \mathcal {Y}\) with \(h(A)=nr1\).

\(\dim {{\mathrm{\mathrm {Hom}}}}^*(A,A)\ge 3\) 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 0spherelike 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 \((1r)\)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 deriveddiscrete algebras.
Proof
We know from Lemma B.9 that the projective module \(P(nr)\in \mathcal {Z}\). 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 \(h<m+r1\).
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 \(h=m+r1\). Furthermore, in this case the endomorphism complex is \({{\mathrm{\mathrm {Hom}}}}^\bullet (A,A) = {\mathbf {k}}\oplus \Sigma ^{r1}{\mathbf {k}}\), so that A is indeed \((1r)\)spherelike.\(\square \)
Corollary 6.7

0spherical if and only if \(m=0\), \(r=1\) and A sits at an \(\mathcal {X}\)mouth;

nspherical 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 \(h(A)=m+r1\). Then A is spherical if and only if \(\mathsf {S}A = \Sigma ^{1r} A\). By \(\mathsf {S}= \Sigma \tau \) and \(\Sigma ^{1r} = \Sigma \tau ^{m+r}\) (Properties 1.2(3)), this is equivalent to \(\tau ^{m+r1}A=A\) which happens precisely if \(m+r=1\). The only solution for this equation is \(m=0\), \(r=1\).
Next, \(B\in \mathcal {Y}\) with \(h(B)=nr1\) is spherical if and only if \(\Sigma \tau B = \mathsf {S}B = \Sigma ^{1+r}B = \Sigma \tau ^{nr} B\), so that here we get \(\tau ^{nr1}B=B\) 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 tstructures and cotstructures, especially bounded ones. The main theorem of [33], cited in the appendix as Theorem A.8, states that for finitedimensional algebras, bounded cotstructures are in bijection with silting objects, which are in turn in bijection with bounded tstructures whose heart is a length category; see sections “Torsion pairs, tstructures and cotstructures” and “König–Yang bijections” in “Appendix 1” for a more detailed overview.
It turns out, however, that any bounded tstructure in \(\mathsf {D}^b(\Lambda (r,n,m))\) has length heart, and hence to classify both bounded tstructures and bounded cotstructures 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 tstructure in \(\mathsf {D}^b(\Lambda (r,n,m))\) is length, then we obtain a semiorthogonal decompositon \(\mathsf {D}^b(\Lambda (r,n,m)) = {\langle \mathsf {D}^b({\mathbf {k}}A_{n+m1}),\mathsf {Z}\rangle }\), 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 tstructure 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 tstructures, 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 Homfinite abelian categories.
Lemma 7.2
(cf. [25, Lemma 4.1]) Let \(\mathsf {D}\) be a triangulated category equipped with a tstructure \((\mathsf {X},\mathsf {Y})\) 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 tstructure 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 tstructure \((\mathsf {X},\mathsf {Y})\). 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 \((r+m) \times (nr)\) 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 Homhammocks \({{\mathrm{\mathrm {Hom}}}}^{<0}(Z,)\) 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 \(ji \ge r+m1\) will do—has a negative selfextension. 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 Homfinite abelian category with finitely many indecomposable objects. Then \(\mathsf {H}\) is a finite length category.
Proof
Since \(\mathsf {H}\) is a Homfinite, \({\mathbf {k}}\)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 nontrivial 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 tstructure in \(\mathsf {D}^b(\Lambda (r,n,m))\) is equivalent to \(\mathsf {mod}(\Gamma )\), for a finitedimensional 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 semiorthogonal decomposition, one of whose orthogonal subcategories is the bounded derived category of a path algebra of Dynkin type A.
7.2 A semiorthogonal decomposition: reduction to Dynkin type A
We start by showing that the derived categories of deriveddiscrete 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 \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m1})\) and there is a semiorthogonal decomposition \(\mathsf {D}^b(\Lambda (r,n,m)) = {\langle \mathsf {D}^b({\mathbf {k}}A_{n+m1}), \mathsf {Z}\rangle }\). 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 semiorthogonal decomposition for an exceptional object; see Appendix A.7 for details.
Lemma B.9 places the indecomposable projective \(P(nr)\) 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 \(Z=P(nr)=e_{nr}\Lambda \). There is a full embedding \(\iota : \mathsf {D}^b(\Lambda / \Lambda e_{nr} \Lambda ) \rightarrow \mathsf {D}^b(\Lambda )\) with essential image \(\mathsf {thick}_{\mathsf {D}^b(\Lambda )}(e_{nr} \Lambda )^\perp = \mathsf {Z}^\perp \); see, for example, [2, Lemma 3.4]. Inspecting the Gabriel quiver of \(\Lambda / \Lambda e_{nr} \Lambda \), 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 \(\Lambda / \Lambda e_{nr} \Lambda \) is an iterated tilted algebra of type \(A_{n+m1}\). It is well known that this implies \(\mathsf {D}^b(\Lambda / \Lambda e_{nr} \Lambda ) \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m1})\); see [21]. Combining these pieces, we get \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m1})\). The final claim about \(\mathsf {D}^b(\Lambda )\) having a full exceptional sequence follows at once from the fact that \(\mathsf {D}^b({\mathbf {k}}A_{n+m1})\) has one. \(\square \)
Remark 7.7
The subcategory of type \(\mathsf {D}^b({\mathbf {k}}A_{n+m1})\) 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 \(\mathsf {D}^b({\mathbf {k}}A_{n+m1})\) 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
Definition 7.9
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. Simpleminded 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 simpleminded 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 \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m1})\) 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 \(M'\in \mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{n+m1})\).
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
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 reordered 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.
7.5 Classification of silting objects for deriveddiscrete 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 \(0 \le j < r +m1\) and \(i \ge 0\).
 (2)
\(G(\Sigma ^i Y^1_{j,0}) = \Sigma ^i Z^0_{0,j+1}\) for \(0 \le j < nr1\) and \(i \ge 0\).
 (3)
\(G(\Sigma ^i Z^1_{j,0}) = \Sigma ^i X^1_{j,1}\) for \(1 \le j \le r+m1\) and \(i \ge 0\).
 (4)
\(G(\Sigma ^i Z^1_{0,j}) = \Sigma ^i Y^1_{1,j}\) for \(rn+1 \le j \le 1\) and \(i \ge 0\).
 (5)
\(G(\Sigma ^i Z^1_{rm,0}) = \left\{ \begin{array}{ll} \Sigma ^i X^1_{rm,1} &{} \text {for } 0 \le i \le r, \\ \Sigma ^i Z^1_{0,nr} &{} \text {for } i > r. \end{array} \right. \)
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)
\(G(\Sigma ^i Y_{1n+j,1n}) = \Sigma ^i Z_{0,j+1}\) for \(0 \le j < n2\) and \(i \ge 0\).
 (3)
\(G(\Sigma ^i Z_{j,1n}) = \Sigma ^i X_{j,m}\) for \(0< j < m+1\) and \(i \ge 0\).
 (4)
\(G(\Sigma ^i Z_{m+1,j}) = \Sigma ^i Y_{n,j}\) for \(22n< j < 1n \) and \(i \ge 0\).
 (5)
\(G(\Sigma ^i Z_{0,1n}) = \left\{ \begin{array}{ll} X_{0,m} &{} \text {for } i=0, \\ \Sigma ^i Z_{0,n1} &{} \text {for } i > 0. \end{array} \right. \)
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 ‘coaisle’ of the cotstructure \((\mathsf {A},\mathsf {B})\) 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.
 (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 \(0 \le j < r+m1\), 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 \(0 \le j < nr1\), 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 \(1 \le j \le r+m1\).
 (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 \(rn+1 \le j \le 1\).
 (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 \(\Sigma ^i Z_{rm,0} \rightarrow \Sigma ^i \big ( Z^1_{rm,rn} \oplus Z^1_{0,0} \big )\). We claim that the cone of \(Z_{rm,0} \rightarrow Z^1_{rm,rn} \oplus Z^1_{0,0}\) is \(Z^1_{0,rn}\). To show this, we compute the cocone of \( Z^1_{rm,rn} \oplus Z^1_{0,0} \rightarrow Z^1_{0,rn}\) 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 \(X^1_{rm,1}\rightarrow Z^1_{rm,0}\rightarrow Z^1_{0,0}\rightarrow \Sigma X^1_{rm,1}\), which computes the cocone \(C = Z^1_{rm,0}\) 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
Antisymmetry: 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 antisymmetry is clear. For a contradiction, suppose \(i<j\). Then \(\mathsf {ray}(\Sigma ^{ji}Z) \le \mathsf {ray}(Z')\) and \(\mathsf {ray}(\tau ^{1}\Sigma ^{ij}Z')\le \mathsf {ray}(Z)\). In particular, it follows that \(\mathsf {ray}(\tau ^{1} Z') \le \mathsf {ray}(\Sigma ^{ji} Z) \le \mathsf {ray}(Z')\), 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 \(\mathsf {ray}(\tau ^{1}\Sigma ^{ji}Z) \le \mathsf {ray}(Z')\) and the second inequality means that \(\mathsf {ray}(\Sigma ^{kj}Z') \le \mathsf {ray}(Z'')\). There are two subcases: first assume \(i\le k\). In this case, apply \(\tau \Sigma ^{kj}\) to the condition arising from the first inequality and combine this with the second inequality to get \(\mathsf {ray}(\Sigma ^{ki}Z)\le \mathsf {ray}(\tau \Sigma ^{kj} Z') < \mathsf {ray}(\Sigma ^{kj} Z') \le \mathsf {ray}(Z'')\). Now assume that \(i > k\) and apply \(\Sigma ^{kj}\) to the condition arising from the first inequality and combine with the second inequality to get \(\mathsf {ray}(\tau ^{1}\Sigma ^{ki}Z) \le \mathsf {ray}(\Sigma ^{kj}Z') \le \mathsf {ray}(Z')\).
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 \(\mathsf {ray}(\Sigma ^{ji}Z) \le \mathsf {ray}(Z')\) then \(Z \preceq Z'\) and we are done, so suppose that \(\mathsf {ray}(\Sigma ^{ji}Z) > \mathsf {ray}(Z')\). Then it follows that \(\mathsf {ray}(\Sigma ^{ij}Z') < \mathsf {ray}(Z)\), in which case, because \(\tau ^{1}\) increases the index of the ray by 1, one gets \(\mathsf {ray}(\tau ^{1}\Sigma ^{ij}Z')\le \mathsf {ray}(Z)\) 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 \(\mathsf {Z}^\perp \simeq \mathsf {D}^b({\mathbf {k}}A_{nm+1})\) 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 \(\mathsf {D}^b({\mathbf {k}}A_{n+m1})\) 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 \(\mathsf {Z}^\perp _{\prec }\).
Lemma 7.20
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 \(\mathsf {D}^b({\mathbf {k}}A_{m+n1})\) containing no objects in the additive subcategory \(\mathsf {Z}^\perp _{\prec }\).
 (2)
Silting subcategories of \(\mathsf {D}^b(\Lambda (r,n,m))\).
 (3)
Bounded tstructures in \(\mathsf {D}^b(\Lambda (r,n,m))\).
 (4)
Bounded cotstructures 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 \(\Lambda (2,3,1)\) 

\(T_1\) \(\oplus \) \(T_2\) \(\oplus \) \(T_3\)  \(\Sigma ^iT_1 \oplus \Sigma ^jT_2 \oplus \Sigma ^kT_3\) 
P(1) \(\oplus \) P(2) \(\oplus \) P(3)  \(j\ge i\) and \(k\ge \max \{j,1\}\) 
P(1) \(\oplus \) P(3) \(\oplus \) S(3)  \(k\ge j \ge \max \{i,1\}\) 
P(2) \(\oplus \) S(2) \(\oplus \) P(3)  \(j\ge i\) and \(k\ge \max \{i,1\}\) 
S(2) \(\oplus \) P(3) \(\oplus \) I(2)  \(j\ge 1\) and \(k\ge \max \{i,j,1\}\) 
P(3) \(\oplus \) I(2) \(\oplus \) S(3)  \(k\ge j\ge i\ge 1\) 
P(3) \(\oplus \) S(3) \(\oplus \) \(\Sigma S(2)\)  \(k\ge j\ge i\ge 1\) 
S(2) \(\oplus \) I(2) \(\oplus \) \(\Sigma P(1)\)  \(k\ge j\ge \max \{i,1\}\) 
I(2) \(\oplus \) S(3) \(\oplus \) \(\Sigma P(1)\)  \(k\ge i\) and \(j\ge i\ge 1\) 
S(2) \(\oplus \) \(\Sigma P(1)\) \(\oplus \) \(\Sigma P(3)\)  \(j\ge i\) and \(k\ge \max \{j,2\}\) 
\(\Sigma P(1)\) \(\oplus \) S(3) \(\oplus \) \(\Sigma P(2)\)  \(j\ge 1\) and \(k\ge \max \{i,j\}\) 
S(3) \(\oplus \) \(\Sigma P(2)\) \(\oplus \) \(\Sigma S(2)\)  \(k\ge j\ge i\ge 1\) 
S(3) \(\oplus \) \(\Sigma S(2)\) \(\oplus \) \(\Sigma ^2 P(1)\)  \(k\ge j\ge i\ge 1\) 
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 tstructure 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 tstructure in \(\mathsf {D}^b(\Lambda (2,3,1))\) induced by M.
The corresponding cotstructure \((\mathsf {A}_M,\mathsf {B}_M)\) is right adjacent in the sense of [12] to the tstructure \((\mathsf {X}_M,\mathsf {Y}_M)\), i.e. \(\mathsf {B}_M = \mathsf {X}_M\) and \(\mathsf {A}_M {:}{=}{}^\perp \mathsf {B}_M = {}^\perp {{\mathrm{\mathsf {susp}}}}{M} = {}^\perp (\Sigma ^{\ge 0} M)\).
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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