Derived categories of quasi-hereditary algebras and their derived composition series

We study composition series of derived module categories in the sense of Angeleri H\"ugel, K\"onig&Liu for quasi-hereditary algebras. More precisely, we show that having a composition series with all factors being derived categories of vector spaces does not characterise derived categories of quasi-hereditay algebras. This gives a negative answer to a question of Liu&Yang and the proof also confirms part of a conjecture of Bobi\'nski&Malicki. In another direction, we show that derived categories of quasi-hereditary algebras can have composition series with lots of different lengths and composition factors. In other words, there is no Jordan-H\"older property for composition series of derived categories of quasi-hereditary algebras.


Introduction
Triangulated categories are used and studied in different areas of mathematics and theoretical physics -algebraic geometry (for example, with applications to classical problems in birational geometry, see e.g. [15,66]), representation theory (with relations to cluster algebras, starting with [16] and perverse sheaves [7] used in the proof of the Kazhdan-Lusztig conjectures), algebraic topology, string theory (via Kontsevich's Homological Mirror Symmetry conjecture [44]), ... . In general, triangulated categories are rather complicated structures and therefore techniques allowing a decomposition into more accessible pieces are important. In this article, we will focus on decompositions of triangulated categories T of the form called recollements, which have properties similar to short exact sequences, see the discussion below. We refer to [7] and Section 2 for the precise definition. Quasi-hereditary algebras form an important class of finite dimensional algebras with relations to Lie theory (in fact this was the original motivation [63]) and also exceptional sequences in algebraic geometry (see e.g. [33] and [17]). The category of finitely generated modules over a quasi-hereditary algebra is an example of a highest weight category and conversely, every highest weight category with finitely many simple objects is of this form [22,Theorem 3.6]. Highest weight categories are also discussed in Krause's article in this volume [45]: in particular, the category of strict polynomial functors admits the structure of a highest weight category. Moreover, work of Dlab & Ringel [26] shows that every finite dimensional algebra admits a 'resolution' by a quasi-hereditary algebra. A generalisation of this result led to Iyama's proof of the finiteness of Auslander's representation dimension [37]. Examples of quasi-hereditary algebras include blocks of category O and Schur algebras (see e.g. the articles by Krause & Külshammer in this volume [45,46]). It is well-known that quasi-hereditary algebras may be defined in terms of sequences of recollements of abelian categories, see [56] and also [45].
Recollements of derived categories induce long exact sequences in K-groups (see e.g. [53]), Hochschild homology and cyclic homology, see [41,Remark 3.2] for the latter two. They also give rise to long exact sequences involving Hochschild cohomology groups of all the algebras in the recollement, see e.g. [30,Corollary 3]. Moreover, recollements of derived module categories allow to reduce the proof of classical homological conjectures to simpler and smaller algebras, see e.g. [32]. Also t-structures on T ′ and T ′′ can be glued to a t-structure on T -in fact, this was one of the main motivations in [7], where recollements arising from stratifications of topological spaces where used to construct so called perverse t-structures giving rise to perverse sheaves. Summing up, recollements behave similar as short exact sequences.
This view was the starting point for a recent series of articles by Angeleri Hügel, König, Liu (and Yang), see [3,4,5], where one can find some historical background and also Yang's ICRA talk [67]. We give a brief account of part of this work here.
In the presence of a notion of short exact sequence one can define simple objects as those which do not appear as middle terms of non-trivial short exact sequences. We call these objects triangulated simple. In analogy with short exact sequences of modules over rings, iteration (i.e. taking recollements of the outer terms T ′ and T ′′ in (1.1) and recollements of their outer terms and so forth until we reach triangulated simple categories) leads to the notion of (triangulated) composition series (sometimes also called stratification of triangulated categories). We call the triangulated simple categories appearing in this process triangulated composition factors. Angeleri Hügel, König & Liu [3, Example 6.1] (see also Remark 2.14 (ii)) show that the derived category D = D(Rep K) of representations of the Kronecker quiver K admits a triangulated composition series of infinite length, where all but one composition factor are not derived module categories. It is well-known that D also has a composition series of length 2 with factors derived categories of vector spaces. Because of this observation and 'a lack of techniques to study the general triangulated categories appearing in triangulated composition series', they decided to restrict to triangulated composition series involving only derived module categories. We call these composition series derived composition series and the corresponding simple factors derived simple. We can now state the main questions of this article.
It is known that the answer is negative in general, for non-uniqueness see [20] (this involves non-artinian rings) and [5] (for counterexamples involving algebras of infinite global dimension). Moreover, it is not too surprising that the length of a derived composition series need not be finite, see e.g. [3, Example 6.2]. Question 1.1 was open for quasi-hereditary algebras and due to the existence of full exceptional sequences with good properties (given for example by the standard modules) it seems that there was some hope for a positive answer in this case. Building on work of Liu & Yang [51], we show that the derived category of the algebra A ′ 2 := kQ 2 /I ′ 2 given by has at least two derived composition series of different length. Since A ′ 2 has global dimension 2 it is quasi-hereditary [25,Theorem 2]. Moreover, since A ′ 2 is gentle (see Definition 4.1) its derived category is of tame representation type -indeed, in this case the repetitive algebra is special biserial [59,61]. Finally, all algebras appearing in our derived composition series have finite global dimension and all involved recollements are induced by idempotents. In particular, compared to the examples in [20] and [5], our examples have a quite different flavour.
We use this example as a starting point to construct quasi-hereditary algebras with an arbitrary number of composition series of different length (see Proposition 3.12 [51] showed that the derived categories of a finite dimensional algebra with at most two simple modules admits a derived composition series with factors D b (k) if and only if it is derived equivalent to a quasi-hereditary algebra. We show that this statement fails for algebras with more than two simples. More precisely, the factor algebra 2)) admits a derived composition series with factors D b (k) (Lemma 5.3), is not quasihereditary (see Lemma 5.5) and is the unique algebra in its derived equivalence class (up to Morita equivalence), see Corollary 4.7. The key step is to show the following proposition which has further consequences and also relates to other work, see the remarks below. Remark 1.5. There seems to be no way of distinguishing the derived categories of A 1 and A 2 by known derived invariants. Indeed, Euler forms are derived invariant by [31,Proposition,p. 101]. The matrices of the Euler forms of A 1 and A 2 in the bases given by the simple modules are The corresponding integral bilinear forms are equivalent 1 since Moreover, Ladkani [49] shows that the dimensions of the Hochschild cohomology groups of gentle algebras are completely determined by the corresponding derived invariants of Avella-Alaminos & Geiß [6]. For both A 1 and A 2 this invariant can be computed to be [2,4].
Remark 1.6. The three algebras A 1 , A 2 and A ′ 2 , which play a key role in this article form a complete set of representatives of the derived equivalence classes for gentle algebras with three arrows and four vertices, see Corollary 4.7.
We point out some consequences and related work.
(a) Proposition 1.4 is part of a conjecture by Bobiński & Malicki [11] (A 1 = Λ ′ 0 (1, 0) and A 2 = Λ 0 (2, 1) in their notation, see also the paragraph after Lemma 4.8). We also show that Proposition 1.4 allows to distinguish the derived categories of a whole family of finite dimensional algebras confirming further parts of this conjecture, see Corollaries 4.29 & 4.31. Recently, Amiot confirmed other cases for gentle algebras arising from a torus with one boundary component, see [2]. Upon receiving a preliminary version of this article Grzegorz Bobiński kindly informed us that using an extension of Amiot's techniques, he is able to establish the conjecture in all cases, see [9]. Our alternative approach might nevertheless be useful to understand derived categories of gentle algebras, see also Remark 4.30. (b) It follows from Avella-Alaminos & Geiß' combinatorially defined derived invariants for gentle algebras [6] and considerations on the Euler form, that the derived equivalence class of A 1 contains at most A 2 (up to Morita-equivalence). In combination with Proposition 1.4 this implies that A 1 and A 2 are 'derived unique' algebras, i.e. algebras for which the notions of Morita and derived equivalence coincide, see Definition 4.5 and Corollary 4.7 for details. (c) The algebra A 1 appears in several different places in the literature. We already mentioned [11]. Moreover, Burban & Drozd show that A 1 is derived equivalent to coherent sheaves over a certain non-commutative irreducible nodal cubic curve [19] -Burban also conjectured Corollary 4.7 for A 1 . The algebra A 1 also appears in work of Seidel [64,Section 3] in relation with a Fukaya category of a certain Lefschetz pencil and in work of Kuznetsov [47] 2 on a geometric counterexample to the JHproperty for triangulated composition series, see Remark 2.14 (ii). Finally, Orlov [55, Section 3.1.] points out that A 1 corresponds to the Ising 3-point function and is related to a Landau-Ginzburg model. Moreover, he shows that its derived category may be realized as a thick subcategory of a strong exceptional collection of vector bundles on a three dimensional smooth projective variety 3 .
Structure. Section 2 contains well-known background material: on recollements (and their relation to admissible subcategories), which can be arranged into triangulated composition series leading to the classical notion of exceptional sequences (in particular, we consider the case of derived categories of quasi-hereditary algebras with the exceptional sequences of standard and costandard modules). We also provide examples showing that complete exceptional sequences need not be full and that full exceptional sequences don't give derived composition series in general.
We explain our constructions of quasi-hereditary algebras with derived composition series of different length in Section 3. This gives a negative answer to Question 1.1. The results of this section are not needed in the rest of the text.
In Section 4, we use Proposition 1.4, which we prove in the final Section 6, to describe the derived equivalence classes of gentle algebras with three vertices and four arrows. This implies that the algebra A 2 is not derived equivalent to a quasi-hereditary algebra, which leads to a negative answer to Liu & Yang's Question 1.2 in Section 5. In Section 4, we also show how to reduce the conjecture of Bobiński & Malicki to 'algebras with full relations' and apply this reduction to obtain further parts of this conjecture from Proposition 1.4. We include some background material on the work of Avella-Alaminos & Geiß [6], which is a key ingredient both in our reduction argument and in the proof of Proposition 1.4, see subsection 'Derived equivalences and the AG-invariant'.
Notation. Throughout, let k be an algebraically closed field. All modules are left modules. For a k-algebra A, we denote the derived category of left A-modules D(A − Mod) by D(A) and the bounded derived category of finitely generated left A-modules D b (A − mod) by D b (A). For a set of objects S in a triangulated category T , we write the thick subcategory generated by S as thick(S). We read elements in the path algebra kQ of a quiver Q from right to left.

Preliminaries: recollements, composition series and exceptional sequences
The following notion is classical, see e.g. [12]. For a subset S of a triangulated category T , we define the right orthogonal subcategory S ⊥ := {T ∈ T | Hom T (S, T [i]) = 0 for all S ∈ S and all i ∈ Z}, which is a triangulated subcategory of T . The left orthogonal subcategory ⊥ S is defined dually. For an admissible subcategory A, the corresponding quotient T → T /A has good properties (see e.g. [7,Section 1.4.4] and also [54,Section 9]) leading to the notion of a recollement. Proposition 2.3. Let A ⊆ T be admissible. Then the following statements hold.
(a) The canonical triangulated quotient functor j * : T → T /A admits both a left adjoint j ! and a right adjoint j * . This gives rise to a recollement, i.e a sequence where i * is the canonical inclusion with left adjoint i * and right adjoint i ! .
(b) Conversely, let j * : T → Q be a triangulated quotient functor (i.e. j * induces and equivalence T / ker j * → Q) with left adjoint j ! and right adjoint j * . Then ker j * ⊆ T is admissible.
(c) The right adjoint j * induces a triangle equivalence j * : T /A → A ⊥ . Dually, the left ajoint j ! yields a triangle equivalence j ! : T /A → ⊥ A. In particular, j * and j ! are fully faithful.
Proof. Part (a) & (b) are [54, Proposition 9.1.18] and its dual. To see part (c), we note that the composition γ of the inclusion A ⊥ ⊆ T followed by the natural projection j * : T → T /A is an equivalence, see e.g. [54,Proposition 9.1.16]. Using the adjunction (j * , j * ) one can check that j * : T /A → A ⊥ is well-defined and right adjoint to the equivalence γ : A ⊥ → T /A. Since right adjoints are unique j * has to be a quasi-inverse to γ completing the proof.
where the morphisms starting from and ending at X are the units and counits of the adjunctions.
By Proposition 2.3, the following well-known example yields a recollement starting from the left hand side. This is used in Lemma 2.15 to construct recollements from exceptional sequences.
Let E in T be an exceptional object. Then the thick subcategory thick(E) ⊆ T generated by E is admissible. Indeed the right adjoint is given by RHom A (E, −) L ⊗ k E and the left adjoint is where (−) * = Hom k (−, k) denotes the k-duality. More generally, one can replace T by a k-linear algebraic triangulated category, such that dim k i∈Z Hom T (X, In combination with Proposition 2.3, the following well-known proposition gives examples for recollements starting from a fixed right hand side. Proposition 2.8. Let A be a finite dimensional k-algebra and e ∈ A be an idempotent such that eAe has finite global dimension. Then there is a triple of adjoint triangle functors i.e. (j ! , j ! ) and (j * , j * ) are adjoint pairs. Moreover, j * = j ! is a triangulated quotient functor.
Proof. Already on the abelian level, we have a triple of adjoint functors Viewing recollements as analogues of short exact sequences for triangulated categories, leads to the notions of triangulated simple categories and triangulated composition series, which we introduce below. The main results of this article deal with the special case of derived simple categories and derived composition series, see Definition 2.22. However, for examples from algebraic geometry and some general statements (e.g. Lemma 2.15) it is convenient to introduce this terminology. (b) Indecomposable Calabi-Yau categories C (e.g. derived categories of connected Calabi-Yau varieties, cluster categories, singularity categories of isolated Gorenstein singularities) are triangulated simple. Indeed assume that there exists a non-trivial recollement, i.e. an admissible subcategory A ⊂ N . It follows from the triangles in Remark 2.5 that C is generated by A and A ⊥ . Using the Calabi-Yau property, we see that A ⊥ = ⊥ A and therefore C ∼ = A ⊕ A ⊥ .
Definition 2.12. Let T be a triangulated category. A triangulated composition series of T is a binary tree constructed by iteratively taking recollements. Starting with a recollement (T 0 , T , T 1 ) of T and continuing with recollements of T 0 and T 1 and so forth until triangulated simple categories are reached.
We refer to Example 2.23, Lemma 2.15 and Section 3 for examples.
Exceptional sequences and derived categories of quasi-hereditary algebras.
Remark 2.14. (i) It is well-known that full exceptional sequences are complete. We proceed by induction. Let T = thick(E) for an exceptional object E. Then T ∼ = D b (k) where E is identified with k -in particular, any object in T is a direct sum of shifts of E. Therefore, this exceptional sequence cannot be extended. Assume that the statement is already shown for a full exceptional sequence of length at most n − 1. Let (E 1 , . . . , E n ) be a full exceptional sequence and assume that there is an exceptional sequence (E 1 , . . . , This yields the following equalities of subcategories is a full exceptional sequence in this subcategory. By induction it is complete which contradicts the existence of the full exceptional sequence (E, E i , . . . , E n ). This finishes the proof.
(ii) The converse fails already for the derived category of the algebra A 1 from Proposition 1.4, which has global dimension 2 and hence is quasi-hereditary [25, Theorem 2] (we will see later (Corollary 2.19) that this implies that D b (A) admits a full exceptional sequence). Bondal (see e.g. [47]) observed that the exceptional collection (E) of length 1 is complete, where To see this one can check that the Euler forms on ⊥ E and E ⊥ are anti-symmetric and therefore these categories don't contain exceptional objects. This is sometimes referred to as a 'failure of Jordan-Hölder Theorem' for semiorthogonal decompositions (or triangulated composition series in our language) and was used by Kuznetsov [47] to construct new geometric counter-examples to the Jordan-Hölder property.
For piecewise hereditary algebras (i.e. finite dimensional algebras which are derived equivalent to abelian categories of global dimension 1) the notions of full and complete exceptional sequences coincide, see e.g. [4] together with Lemma 2.15 and also [24] for the special case of exceptional sequences of quiver representations.
It follows from Example 2.7 & Proposition 2.3 that full exceptional sequences give rise to triangulated composition series with composition factors D b (k) and vice versa. This is summarized in the following well-known lemma.
Then every full exceptional sequences in T gives rise to a triangulated composition series with factors D b (k). Conversely, every such composition series yields a full exceptional sequence.
Conversely, assume that T has a composition series with factors D b (k). In particular, we obtain a recollement of the form shows that T ′ respectively T ′′ identify with E ⊥ and ⊥ E. By assumption these categories again admit recollements involving D b (k) as one of the factors. Iterating this process yields an exceptional sequence. The standard triangles associated with recollements (Remark 2.5) imply that this sequence is full.
We turn to an example which will be important in the sequel. Derived categories of quasihereditary algebras admit full exceptional sequences, for example given by standard modules. We start with the definition of a quasi-hereditary algebra due to Scott [63] (cf. [51]).
Definition 2.16. Let A be a finite dimensional k-algebra and let e ∈ A be an idempotent. The two-sided ideal AeA is called heredity if eAe is a semi-simple algebra and AeA is projective as a left A-module. The algebra A is called quasi-hereditary if there exists a chain of two-sided ideals In particular, semi-simple algebras and all quotient algebras A/J i are quasi-hereditary.
Remark 2.17. (a) Let A be a quasi-hereditary algebra. One can refine the chain (2.4) in such a way that all heredity ideals J i /J i−1 are given by primitive idempotents.
(b) Let A be a finite dimensional k-algebra and let e ∈ A be an idempotent. The canonical functor ι :  In the situation of Lemma 2.18, all triangulated categories appearing in the triangulated composition series are derived modules categories. In a series of papers Angeleri Hügel, König, Liu (and Yang) [3,4,5] studied triangulated composition series of this form. We introduce some terminology for later use.
Remark 2.21. Every triangulated simple algebra is derived simple.
Following [3,Section 5] we can now introduce the notion of composition series of derived module categories -these can be thought of as analogues of composition series for modules over rings.
Definition 2.22. A composition series of the derived module category D b (A) of a finite dimensional k-algebra A is a triangulated composition series (see Definition 2.12) such that all triangulated categories appearing in the binary tree are equivalent to derived categories of finite dimensional algebras. It is also called derived composition series of A.
The following example shows that full exceptional sequences need not give rise to derived composition series. In fact the exceptional sequence studied here also leads to our counterexample for the question of Liu & Yang, see Section 5.
Example 2.23. Let A = A 2 be the algebra from Proposition 1.4 and consider the full exceptional sequence of A-modules (S 2 , P 1 , P 3 ), which by Proposition 2.3 gives rise to a recollement (thick( where one arrow is in degree 0 and the other arrow is in degree 2. By definition, the graded algebra G is isomorphic to the cohomology of the dg endomorphism algebra End(S 2 ⊕ P 1 ), which can be equipped with a minimal A ∞ -structure such that there is an A ∞ -quasi-isomorphism (see [39] and also [42,Section 3.3

.] and references in there)
Since the quiver of H * (End(S 2 ⊕ P 1 )) ∼ = G is directed and has only two vertices this A ∞ -algebra has no higher multiplications, see for example [42,Section 3.5]. This shows that there is a quasiisomorphism of dg algebras End(S 2 ⊕ P 1 ) ∼ = G, where G is considered as a dg algebra with trivial differential. In combination with Keller's Morita theorem for triangulated categories (see e.g. [43, Theorem 3.8 b)]) this yields triangle equivalences One can show that per G is not triangle equivalent to the derived category of a k-algebra, see e.g.

Quasi-hereditary algebras with non-unique derived composition series
The aim of this section is to construct a family of examples having different derived composition series. More precisely, given any natural number n we construct a finite dimensional quasi-hereditary algebra A such that D * (A) has at least 2 n derived composition series of pairwise different length and at least n different derived simple factors occur. In particular, this gives a negative answer to the uniqueness part of Question 1.1 for quasi-hereditary algebras. The results of this section are not needed in the rest of the text. In this section, we write D * (A) for the derived categories Examples arising from generalized Fibonacci algebras. For l ∈ Z ≥1 , we consider a family of algebras B l := kQ l /I l given by quivers Q l 1 a b 3 a [ [ 4 a P P P P P P P g g P P P P P P P with relations I l := (ba). Note that B 1 = A ′ 2 is the algebra from the introduction (1.2). We show below that the algebras B l are quasi-hereditary of global dimension 2 and that they give a negative answer to Question 1.1. In other words, they do not satisfy the derived Jordan-Hölder property as studied by Angeleri-Hügel, König & Liu [3]. (a) B is a finite dimensional k-algebra.
Proof. (a) One can check that all paths of length greater than 4 are contained in I l .
(b, c) The projective resolutions of the simple B l modules S i are given as follows: This yields (b) and (c).  with relations bc i and c j a. Moreover, these algebras are isomorphic to the generalized Fibonacci algebra G l := A 3 ((1, 1), (l)) as studied by Liu & Yang [51] and it is shown in loc. cit. that the G l have global dimension 3.
Proposition 3.4. Let l ∈ Z ≥1 then B = B l has at least two non-equivalent derived composition series.
(a) A derived composition series of length l + 2 with all composition factors given by D * (k).
(b) A derived composition series of length l + 1 with l composition factors given by D * (k) and one composition factor given by D * (G l ).
In particular, the derived JH property fails for these algebras.
Proof. The existence of (a) follows from the fact that A l is quasi-hereditary (Corollary 3. Remark 3.5. Dong Yang pointed out that Vossieck's derived discrete algebras (of finite global dimension) satisfy the derived JH property. Recently, this was also shown in a work of Yongyun Qin for all derived discrete algebras, see [57]. The algebra B 1 = A ′ 2 is gentle and therefore its derived category has tame representation type. In this sense our example is as small as possible.
The idea which we used to modify the derived simple algebras G k to obtain quasi-hereditary algebras A k seems to work for all derived simple two-vertex algebras of finite global dimension, see Liu & Yang [51] for a list of these algebras. This leads to the following question. In the next paragraph, we 'glue' copies of the algebras B l together using certain triangular matrix algebras. The algebras constructed in this way are again quasi-hereditary. Extending and building on Proposition 3.4, we show that they can have an arbitrary number of derived composition series of different length (Proposition 3.12).
Glueing. Let B ′ = kQ ′ /I ′ and B ′′ = kQ ′′ /I ′′ be finite dimensional algebras and let a ∈ Q ′ 0 and b ∈ Q ′′ 0 be vertices. We write B = B ′ a → b B ′′ for the triangular matrix algebra which may also be written as B = kQ/I, where Q is obtained from the disjoint union of Q ′ and Q ′′ by adding an arrow a → b and I := I ′ + I ′′ . This construction is associative: given quiver algebras A = kQ/I, B = kR/J, C = kS/K and vertices a ∈ Q 0 , b, b ′ ∈ R 0 and c ∈ S 0 , we have . Then we have the following fact.  ), e.g. by [5].

Derived equivalence classification of certain gentle algebras
Definition 4.1. Let Q be a finite quiver with set of arrows Q 1 . A gentle algebra is a finite dimensional k-algebra kQ/I such that: (G1) At any vertex, there are at most two incoming and at most two outgoing arrows.
(G2) I is a two-sided admissible ideal, which is generated by paths of length two.
(G3) For each arrow β ∈ Q 1 , there is at most one arrow α ∈ Q 1 such that 0 = αβ ∈ I and at most one arrow γ ∈ Q 1 such that 0 = βγ ∈ I.
(G4) For each arrow β ∈ Q 1 , there is at most one arrow α ∈ Q 1 such that αβ / ∈ I and at most one arrow γ ∈ Q 1 such that βγ / ∈ I.
Remark 4.2. It is well-known that gentle algebras can also be characterised as those finite dimensional algebras with special biserial repetitive algebras, see for example Schröer [61, Section 4] and also Ringel [59]. Proof. We check this case by case. For this we give a list of all gentle algebras kQ/I of finite global dimension with three vertices and four arrows -in particular, the quivers Q do not contain loops, see e.g. [36]. Therefore, there have to be two vertices which are connected by at least two arrows. There are the following four possibilities to extend this to a connected gentle quiver Q with three vertices, four arrows and no loops. Namely, Up to algebra isomorphism the algebra A 1 from Proposition 1.4 above is the unique gentle algebra with underlying quiver Q 1 . Moreover, A 3 = kQ 3 /(ca, ab, dc) is (up to algebra isomorphism) the unique gentle algebra with underlying quiver Q 3 and dually Q op 3 gives rise to A op 3 (again unique up to isomorphism). There are two gentle algebra structures (of finite global dimension) on Q 2 up to isomorphism (A 2 and A ′ 2 := kQ 2 /I ′ 2 from (1.2), which is the starting point for our examples in Section 3). The rank of the symmetrized Euler form for all algebras (except for A 1 and A 2 ) arising in this way is 2.
Case 2: Two-cycle. Assume that Q contains a subquiver of the form 1 x * * 2. y j j In addition to Q 3 , Q op 3 there is the following family of quivers 3 1 2 x y z 2 z 1 where the edges z 1 and z 2 can have an arbitrary orientation. One can check that the rank of the symmetrized Euler-form is 2 for all of these algebras.
There is no way to define a finite dimensional gentle algebra of finite global dimension on the following quiver: Summing up, A 1 and A 2 are the only gentle algebras of finite global dimension with three vertices and four arrows such that the rank of the symmetrized Euler form is 1.
Remark 4.4. One can compute that all gentle algebras of finite global dimension with three vertices & four arrows have AG-invariant [2,4]. This can also be deduced from [11, Lemma 3.1] as all these gentle algebras are degenerate in the sense of [11], see e.g. the proof of Corollary 4.7.
Moreover, the AG-invariant is a complete derived invariant for gentle algebras with at most two vertices. Indeed this follows from the classification of Bessenrodt & Holm [8,Example 3.7] in combination with Ladkani [49] and the definition of the AG-invariant, which detects oriented cycles with full zero relations. In particular, Proposition 1.4 provides a minimal 4 example showing that the AG-invariant is not sufficient to distinguish derived categories of gentle algebras, see [11] & [2] for further examples.
Definition 4.5. We call a noetherian ring A derived unique 5 if every ring B which is derived equivalent to A is already Morita equivalent to A. In other words, the derived equivalence class and the Morita equivalence class of A coincide. (2) In algebraic geometry, Bondal & Orlov [14] showed that the derived category D b (coh X) of a smooth projective variety X with ample canonical or anticanonical bundle determines X. It would be interesting to look for analogous results for derived unique algebras.
We finish with a complete description of derived equivalence classes of gentle algebras of finite global dimension with three vertices and four arrows, this also follows from Bobiński [9]. In particular, the algebras A 1 and A 2 are derived unique.
Proof. We first show that A 1 & A 2 are derived unique. Only this part is used in Section 5.
be a triangle equivalence, with i = 1 or 2. The main result of Schröer & Zimmermann [62] shows that B is Morita equivalent to a gentle algebra C = kQ/I. Since the rank of the Grothendieck group is a derived invariant Q has three vertices. The number of arrows is a derived invariant by work of Avella-Alaminos & Geiß [6]. So Q has four arrows. The rank of the symmetrized Euler form (cf. In order to complete the description of the derived equivalence classes, it remains to show that all gentle algebras of finite global dimension and with three vertices and four arrows are derived equivalent to A ′ 2 . This follows from Bobiński & Malicki [11, Theorem 2] once we show that all these algebras are degenerate in their sense. This can either be checked by direct computation using the list in Proposition 4.3 or one can proceed as follows. By definition a gentle two-cycle algebra (all our algebras are of this form) is either degenerate or non-degenerate. Moreover, this property is invariant under derived equivalences. Bobiński & Malicki [11,Theorem 1] give a list of representatives of derived equivalence classes of non-degenerate gentle two-cycle algebras. One can check that there is no representative which has three vertices and finite global dimension. Finiteness of global dimension is invariant under derived equivalences. Since, by assumption, our algebras are of finite global dimension, they cannot be derived equivalent to a non-degenerate algebra and therefore are indeed degenerate. This completes the proof.
Derived equivalences and the AG-invariant. This subsection contains background material on work of Avella-Alaminos & Geiß [6], who describe the structure of certain characteristic components of the Auslander-Reiten quiver of the derived category of a gentle algebra (of finite global dimension), leading to the definition of a derived invariant (called AG-invariant).
Building on this, we modify given derived equivalences such that they identify certain prescribed objects (Lemma 4.22). In special cases, this yields derived equivalences between corner algebras, by passing to triangulated quotient categories (Corollary 4.23). This is used to simplify a conjecture of Bobiński & Malicki [11] in the next subsection (Corollary 4.29) and also in the proof of Proposition 1.4 in Section 6.
We start with a general lemma, which we apply to gentle algebras in Corollary 4.23.
where e i ∈ A and e j ∈ B are the idempotents corresponding to S i and S j , respectively.
Proof. Our assumptions imply that the triangle equivalence  (i) C has a boundary, i.e. C contains an Auslander-Reiten triangle τ X → Y → X → νX such that Y is indecomposable.
Conversely, every AR- In particular, derived autoequivalences act transitively on the boundary of a CC. Proof. If C is of type ZA ∞ or ZA ∞ /τ r , then there is a unique boundary component. By definition the AR-translation τ acts transitively on it. Therefore, we can take ψ = τ t for some t ∈ Z.
If C is of type ZA n , then there are two boundary components. The AR-translation acts transitively on each of these components and the shift functor [1] identifies the two components. So either ψ = τ t [1] or ψ = τ t for some t ∈ Z will identify X and Y in this case.
We call a CC C homogeneous if C = ZA ∞ /τ and non-homogeneous otherwise. Proof. Since ψ is a triangle equivalence it commutes with Serre-functors ν and shift functors [1]. In particular, it commutes with the AR-translations τ = ν • [−1]. Therefore, ψ maps AR-components to AR-components and boundary objects to boundary objects. Finally, by assumption C = ZA ∞ /τ appears in (i) -(iii) of Proposition 4.10, thus ψ(C) = ZA ∞ /τ occurs in this list as well. So ψ(C) is a CC by Proposition 4.10.
Remark 4.13. In general, derived autoequivalences can identify homogeneous CCs with noncharacteristic components. For example, this happens for the Kronecker quiver.
The AG-invariant. One can check 6 that the shift functor (on the stable category of the repetitive algebra this is given by the inverse syzygy functor) acts on characteristic components. Avella-Alaminos & Geiß [6]   where ν denotes the Nakayama=Serre functor of D b (Λ). In other words, X is mi ni -fractionally CY.  Therefore, we call φ Λ the AG-invariant of Λ.
Boundaries of CCs, permitted threads and modifications of derived equivalences. The objects in the boundary of CCs are classified, see e.g. [6,Section 2.3]. For our purposes it is enough to understand which Λ-modules are in the boundary. We need the following definition.
Definition 4.17. Let A = kQ/I be a gentle algebra. A non-trivial permitted thread of A is a maximal path p in (Q, I), i.e. p is not contained in I but any path in Q with subpath p is contained in I. A trivial permitted thread is a trivial path 1 v where v is a vertex in Q such that (a) there is at most one arrow α ending in v.  A permitted thread is a trivial or non-trivial permitted thread.
Building on [18,65], Avella-Alaminos & Geiß [6] show the following result, cf. also [29].    Proof. Assume that φ B = [n, n]. It follows from [10, Lemma 3.2] that the underlying quiver Q of B has n vertices and n arrows. Since Q is connected (indeed otherwise φ B has at least two summands), the unoriented graph underlying Q is a cycle of length n. It follows from [6, Section 7] that φ B has exactly two summands. Contradiction. So we see that indeed n = m.
In the situation of Setup 4.19, the derived equivalence can be adapted such that it identifies given objects X ∈ D b (A), Y ∈ D b (B) contained in boundaries of CCs (cf. also [9, Corollary 1.4]).  Then there exists a derived equivalence , where e v and e w are the idempotents corresponding to the vertices v and w.
Remark 4.27. Since the rank of the Grothendieck group is a derived invariant, it follows that D b (B(p, r)) ∼ = D b (B(p ′ , r ′ )) implies p = p ′ in part (b) of the conjecture. By the same argument (a) holds if p = p ′ .
Bobiński & Malicki [11, paragraph after Conjecture 1] check that part (a) holds for r ≡ 0 (mod 2) and also D b (B(p, r)) ∼ = D b (B(p ′ , r ′ )) implies r ≡ r ′ (mod 2). Indeed the symmetrized Euler form of A(p) has rank p, whereas the rank of the symmetrized Euler form of B(p, r) is p + 1 if r is even and p if r is odd.
The next result will be used in Corollary 4.29 to show that one of the algebras in the conjecture can be assumed to be B(p, p). Bobiński [9,Corollary 2.2] shows the converse statements of this proposition and combines them with Amiot's results [2] to prove Conjecture 4.26 in full generality.  (ii) If B(p, r) is not derived equivalent to A(p), then B(p + 1, r) and A(p + 1) are not derived equivalent.
Using Proposition 4.28 iteratively, we can now reduce the conjecture as follows.

Exceptional sequences and quasi-hereditary algebras -a negative answer to a question of Liu & Yang
The aim of this section is to give a negative answer to Question 1.2 of Liu & Yang (cf. [51, Question 1.1]), which we restate below for the convenience of the reader. We refer to Section 2 for unexplained terminology. Proposition 5.6. The derived category D b (A 2 ) of the gentle algebra A 2 admits a composition series by derived categories D b (k) but A 2 is not derived equivalent to a quasi-hereditary algebra.
Proof. The first statement is Lemma 5.3. The second statement is a combination of Lemma 5.5 and Corollary 4.7.
Remark 5.7. For finite dimensional quiver algebras A with two vertices Liu & Yang show that Question 5.1 has a positive answer. In other words if D b (A) admits a composition series by derived categories D b (k), then A is derived equivalent to a quasi-hereditary algebra. In that sense A 2 is a minimal counterexample to the question. We refer to [56,Section 5] for a first answer to this question -unfortunately, we are not able to use this characterisation to obtain an alternative proof of Proposition 1.4.

Proof of Proposition 1.4
Outline of the proof. We give a proof by contradiction consisting of the following two steps: (i) Assume that there exists a triangle equivalence Φ : Rickard's derived Morita theory [58] shows that there exists a tilting object T ∈ D b (A 1 ) such that there are isomorphisms of graded algebras i∈Z where A op 2 is concentrated in degree 0. Moreover, Φ(T ) ∼ = A 2 in D b (A 2 ) and we can assume We show that T 1 can be chosen to be the string module 1 a − → 2 b − → 3 and Φ(T 1 ) ∼ = P 2 .
(ii) Let T 1 be as in (i). Using the repetitive algebra of A 1 , we show that there is no indecompos- as graded algebras with C concentrated in degree 0.
It would be interesting to show an analogue of step (ii). Namely, that there is no indecomposable exceptional object T ′ such that there are isomorphisms of graded algebras where C p+1 denotes a quiver consisting of a single oriented (p+1)-cycle, J ⊆ kC p+1 is the two-sided ideal generated by the arrows and P ′ is isomorphic to P 1 or P p+2 .
Step (ii). We use Happel's triangle equivalence H : D b (A) ∼ = A − mod for finite dimensional algebras A of finite global dimension (see [31]) to translate the claim to a question about (string) modules over the repetitive algebra A 1 . We refer to [18,23,62,65] for more details on string module combinatorics. We begin by describing the repetitive algebra A 1 following Ringel [59] and Schröer [61]. Let where by abuse of notation we write T 1 for H(T 1 ) and T ′ for H(T ′ ). In particular, T ′ is given by a string module as band modules always have self-extensions.
(e) Crawley-Boevey showed that homomorphism spaces Hom(M, N ) between two (indecomposable) string modules M & N have bases given by graph maps [23]. In conjunction with Proposition 3.7 of Schröer & Zimmermann [62] and (6.8), it follows that End A1 (T 1 ⊕ T ′ ) is generated by (weakly) one-sided graph maps, i.e. maps between string modules S 1 = EαF and S 2 = Eβ − F ′ defined as identity map from the factorstring E of S 1 to the substring E of S 2 and as zero everywhere else. Here α and β are arrows in the quiver Q 1 , F is a substring of S 1 and F ′ is a factorstring of S 2 .
(f) Using Happel's equivalence H, (6.2) translates to the following statement. The stable endo- We show that this leads to a contradiction, see the last paragraph 'Final step' below. By part (e), every stable morphism from T 1 to T ′ is a linear combination of equivalence classes of (weakly) one-sided graph maps, which are given by a factorstring E f in T 1 and a corresponding substring E f in T ′ . In turn, stable morphisms from T ′ to T 1 are generated by equivalence classes of maps given by substrings E s in T 1 and corresponding factorstrings E s in T ′ . Since Hom A1 (T 1 , T ′ ) ∼ = kx and Hom A1 (T ′ , T 1 ) ∼ = ky are one dimensional, there are (weakly) one sided graph maps x ∈ Hom A1 (T 1 , T ′ ) and y ∈ Hom A1 (T ′ , T 1 ) representing x ∈ Hom A1 (T 1 , T ′ ) and y ∈ Hom A1 (T ′ , T 1 ).
If the substring E f and the factorstring E s overlap inside T ′ , then 0 = yx ∈ End A1 (T 1 ).
Since this space is one dimensional, we have yx = λ · 1 T1 for some λ ∈ k * . By our assumption yx = yx = 0, so the identity endomorphism of T 1 factors over a projective injective A 1module. This would imply that T 1 ∼ = 0 in the stable module category. Contradiction. So E f and E s must not overlap in T ′ . This is indicated in the picture above.
(g) We claim that choosing E f = T 1 = E s implies xy = 0, which contradicts our assumption.
We assume without loss of generality that x and y are given as in the picture above (in particular, x is left-sided and y is right-sided in the terminology of [62]. Since T ′ is a string module we may write T ′ = M (S) for a string S. Let S − be the inverse string of S, see e.g. [62, p.4]. Then there is a canonical isomorphism γ : M (S − ) → M (S). Now yγ and x are both left-sided graph maps in the terminology of [62]. It follows from [62, Lemma 3.3] that xy · γ = xyγ = 0. (the lemma is applicable since T 1 is not simple, see Zimmermann's correction [71] based on Zhou's thesis [68, preliminary chapter]). In particular, xy = 0 as claimed.
(h) Without loss of generality, we may assume E f = 1 or E f = 1 a − → 2.
To see this, we observe that the only other choice for E f is T 1 . If E f = T 1 , then E s = T 1 by (g). Next, k-duality D = Hom k (−, k) defines an exact anti-autoequivalence of A 1 − mod.
Since C ∼ = C op , the pair of modules T 1 , T ′ satisfies (6.2) if and only if the pair D(T 1 ), D(T ′ ) does. Under this duality D(E s ) = D(T 1 ) is a factor string in D(T 1 ). This proves the claim.
The following construction is well-known, see e.g. [62,Lemma 3.5]. It is used in the final step below to show that every candidate for the string module T ′ has non-trivial self-extensions, contradicting (6.8).
Lemma 6.2. Let S 1 be the following string Here U l , U r , F l , F r are allowed to be empty -for example, (6.9) shows S 2 with empty F l . There exists a short exact sequence provided M 1 and M 2 are well-defined strings. This sequence does not split provided at least one of U l and F l is non-empty and at least one of U r and F l is non-empty.
Final step. Assume that there exists T ′ satisfying (6.2). Then the preliminary step (f) above implies that there is a pair (E s , E f ) consisting of a substring E s and a factorstring E f in T 1 . Moreover, E s appears as a factorstring and E f as a substring in T ′ which do not overlap. By step (h), we may assume E f = 1 or 1 a − → 2. We treat the case E f = 1 first and show that the other case follows from this.
We prove below that we can assume that T ′ has the following form (with E f = 1 appearing as a substring on the left): We note that we have to start with the arrow a. Otherwise, the morphism x : T 1 → T ′ would factor over the indecomposable projective A 1 -module P 0 (see (6.6)), so x = 0 ∈ Hom A1 (T 1 , T ′ ). Moreover, the string defining T ′ has to reach vertex 3 at some point (otherwise we don't get a non-zero morphism to T 1 ). This forces S to have the following shape (indeed otherwise we can never reach a vertex smaller than 2 since the longest paths without relations in ( Q 1 , I) have length 2).
We apply Lemma 6.2 to obtain a non-trivial self-extension of the string T ′ contradicting (6.8). In order to do this, we write T ′ in two different ways: exist, Lemma 6.2 shows Ext 1 A1 (T ′ , T ′ ) = 0 as desired. It remains to show that we can assume that T ′ starts as indicated in the picture (S0) above. Since we want 1 to be a substring there are the following two other possibilities -where S 1 and S 2 have to be non-trivial (indeed, otherwise there is no non-zero morphism T ′ → T 1 ). If T ′ starts as in (S1), we apply the syzygy functor Ω to T 1 ⊕ T ′ . Using (6.6), we compute This is a shifted version of (S0) and we have already seen that this leads to a contradiction. Since Ω is an autoequivalence, we deduce that T ′ cannot be of the form (S1).
In case T ′ starts as in (S2), we can repeatedly apply the Auslander-Reiten translation τ (viewed as an autoequivalence of A 1 − mod) to both T ′ and T 1 . Since A 1 is special biserial [59,61], the action of τ on strings is well understood (see e.g. [65,Thm 4.1]).
if n is even; if n is odd.
If we apply τ to T ′ , we 'remove a hook' from the left of the string T ′ : where S ′ 2 is non-zero (indeed otherwise there is no non-zero morphism τ (T ′ ) → τ (T 1 )). After repeatedly applying τ , we have removed all these hooks from T ′ and reach a (shifted) version of (S0) or (S1) (indeed otherwise T ′ would have the following form  This is a shifted version of (S0) and we have already seen that this leads to a contradiction. Since Ω −1 is an autoequivalence, we deduce that T ′ cannot be of the form (S4). This completes the proof. I thank the anonymous referee for reading the article carefully and for many helpful comments and suggestions, which helped to improve the paper. I also thank Pieter Belmans, Sefi Ladkani and Yuya Mizuno, for helpful remarks on an earlier version.