Shod String Algebras

Jennifer Bélanger  Cecilia Tosar
Département de Mathématiques  Departamento de Matemática IME
Université de Sherbrooke  Universidade de São Paulo
Sherbrooke, Québec, Canada J1K 2R1  CP 66281 São Paulo, Brazil
jennifer.belanger@usherbrooke.ca  cecilia@ime.usp.br

Abstract. In this paper, we give a simple combinatorial criterion allowing to verify whether a string algebra is shod or not. As a consequence, we classify all algebras with a discrete derived category which are derived equivalent to shod algebras.

1. Introduction

We consider the artin algebras A which satisfy the property that for each indecomposable A-module M, either the projective dimension of M is at most one or the injective dimension of M is at most one. The algebras satisfying this condition are called shod algebras [8]. The class of shod algebras, introduced by Coelho and Lanzilotta, contains all tilted, and more generally all quasi-tilted algebras, and has been recently the object of extensive investigations (see, for instance, [7], [8], [19], [21]).

The class of quasi-tilted algebras was introduced by Happel, Reiten and Smalø in order to give a general tilting theory for abelian categories [15]. Any artin k-algebra A = End_{H}(T)^{op} where H is a locally finite hereditary abelian k-category with tilting object T, is called a quasi-tilted algebra. The following characterization, also due to Happel, Reiten and Smalø, has been very useful when dealing with this class of algebras: An algebra A is quasi-tilted if and only if it is shod and of global dimension at most two [15].

We are interested in the problem of characterizing the shod algebras in terms of bound quivers. In this paper, we consider shod string algebras. String algebras play an important rôle in the representation theory of algebras (see, for instance, [6], [24]). Huard and Liu have shown that a string algebra A = kQ/I over an algebraically closed field k is quasi-tilted if and only if (Q, I) contains no double-zero [18]. In our case, we give a combinatorial criterion allowing us to verify whether a string algebra is shod or not. Our characterization will give the complete classification of the shod string algebras.

We now state our main theorem. For the definitions of consecutive-zeros, of overlapping zero-relations and of start-tight and end-tight, we refer to the subsection 2.3 and the section 3.

Theorem 1.1. Let k be an algebraically closed field and let A = kQ/I be a string algebra. Then A is shod if and only if (Q, I) satisfies the following conditions:

(1) Every consecutive-zero in (Q, I) contains at most two zero-relations. In case there are two, they are overlapping and the path containing these two zerorelations is either start-tight or end-tight.

(2) (Q, I) contains no full subcategory of the following forms or the duals (i’) and (ii’) of (i) and (ii) respectively.

[figure: (i); ∘; (ii); ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; (iii); ∘; ∘; ∘; ∘; ∘; ∘]

Observe that the bound quiver (iii) is self-dual.

Since a strictly shod algebra A = kQ/I has global dimension 3, we obtain from [11] (see also [10]) that (Q, I) admits a path containing two overlapping zero-relations. So, as an important consequence of theorem 1.1, we obtain that if A = kQ/I is a shod string algebra, then A is strictly shod if and only if (Q, I) admits a consecutive-zero.

In particular, the class of string algebras contains the gentle ones. Gentle algebras were introduced by Assem and Skowronski in order to characterize the algebras derived equivalent to a hereditary algebra of type 𝔸̃_{n} [4]. Another example of gentle algebras is the class of algebras derived equivalent to a hereditary algebra of type 𝔸_{n}, described by Assem and Happel [3]. In [23], Vossieck classified the algebras with discrete derived category and this classification involves also the gentle algebras.

As a corollary to theorem 1.1, we present in this paper a characterization of the shod gentle algebras. This characterization allows us to classify the shod algebras derived equivalent to a hereditary algebra of type 𝔸_{n} or 𝔸̃_{n} and the shod algebras with discrete derived category not of Dynkin type. The class of algebras with discrete derived category not of Dynkin type is an example of a class of algebras which are not derived equivalent to a quasi-tilted algebra. But many algebras in that class are derived equivalent to shod algebras. We end this paper by showing which algebras with discrete derived category are derived equivalent to shod algebras.

Theorem 1.2. Let A be an algebra with discrete derived category. The following conditions are equivalent:

(i) A is derived equivalent to a shod algebra.

(ii) A is derived equivalent to a hereditary algebra of Dynkin type or to Λ(r, n, m) with r > 0, n ⩾ 3r and m ⩾ 2r.

2. Preliminaries

We first fix some terminology and notations. Let Q be a finite quiver. For an arrow α of Q, we denote by s(α) its source and by t(α) its target. We associate to α its formal inverse α^{−1} with source s(α^{−1}) = t(α) and target t(α^{−1}) = s(α). We agree that (α^{−1})^{−1} = α. A path p of length n, with n > 0, from a source a to a target c is a sequence (a|α_{1}, α_{2}, …, α_{n}|c) with α_{i} an arrow of Q for all i, where 1 ≤ i ≤ n, such that s(α_{1}) = a, t(α_{n}) = c and t(α_{i}) = s(α_{i+1}) for all i such that 1 ≤ i < n. Such a path is briefly denoted by α_{1}α_{2} ⋯ α_{n}. We denote by ϵ_{a} the stationary path at a vertex a. A walk in Q of length n, with n > 0, is a sequence w = c_{1} ⋯ c_{n} with c_{i} an arrow or the inverse of an arrow such that s(c_{i+1}) = t(c_{i}) for i where 1 ≤ i < n and we say that w_{1} = c_{r} ⋯ c_{t} with 1 ≤ r ≤ t ≤ n is a subwalk of w. We define s(w) = s(c_{1}) and t(w) = t(c_{n}) and we say that w is a walk from s(w) to t(w). Finally, we define w^{−1} = c_{n}^{−1} ⋯ c_{1}^{−1}. A trivial walk at a vertex a of Q is the stationary path ϵ_{a}.

A walk w in Q is called reduced if w is trivial or if w = c_{1} ⋯ c_{n} is such that c_{i+1} ≠ c_{i}^{−1} for all i where 1 ≤ i < n. A non-trivial reduced walk w = c_{1} ⋯ c_{n} is called a reduced cycle if s(w) = t(w) and c_{n} ≠ c_{1}^{−1}.

Let w be a non-trivial reduced walk in Q and let w_{1}, w_{2}, ..., w_{l} be subwalks of w. We say that w_{1} and w_{2} point to the same direction in w, if there are paths p and q of Q such that either w_{1} = p and w_{2} = q or w_{1} = p^{−1} and w_{2} = q^{−1}. Otherwise, we say that w_{1} and w_{2} point to opposite directions in w. More generally, we say that w_{1}, w_{2}, ..., w_{l} point to the same direction in w, if there are paths p_{1}, p_{2}, ..., p_{l} of Q such that either w_{1} = p_{1}, w_{2} = p_{2}, ..., w_{l} = p_{l} or w_{1} = p_{1}^{−1}, w_{2} = p_{2}^{−1}, ..., w_{l} = p_{l}^{−1}.

Let (Q, I) be a bound quiver. A path p in Q is called a zero-path if p belongs to the ideal I. A zero-relation on (Q, I) is a zero-path such that none of its proper subpaths is a zero-path. Let w = c_{1} ⋯ c_{n} be a non-trivial reduced walk in Q. We say that a subwalk u = c_{i} ⋯ c_{i+r} is a zero-relation contained in w if u = p or u = p^{−1} with p a zero-relation on (Q, I). Note that a zero-relation on (Q, I) may appear many times in a reduced walk. By saying that a reduced cycle w = c_{1} ⋯ c_{n} contains no zero-relation, we mean that, for all i such that 1 ≤ i ≤ n, there is no zero-relation contained in the reduced walk c_{i} ⋯ c_{n}c_{1} ⋯ c_{i−1}. A path in Q is a non-zero path if it contains no zero-relation. We say that a path p in Q contains two overlapping zerorelations if p = p_{1}p_{2}p_{3} where p_{i} are non-trivial non-zero paths such that p_{1}p_{2} and p_{2}p_{3} are zero-relations.

Throughout this paper, by algebra we mean a basic, connected, finite dimensional algebra over an algebraically closed field k. Thus, an algebra A is isomorphic to kQ/I with (Q, I) a finite bound quiver [9]. We identify finite dimensional right A-modules with finite dimensional representations of (Q, I). The global dimension of an algebra A is denoted by gl.dim A and for an A-module M, we denote by pd_{A}M and id_{A}M the projective dimension and the injective dimension of M, respectively.

Finally, we say that an algebra A ≅ kQ/I is triangular, if its ordinary quiver Q contains no oriented cycle.

2.1. Shod Algebras. We recall the definition of shod algebras.

Definition 2.1. [8] An artin algebra A which satisfies the property that, for each indecomposable A-module M, either pd_{A}M ≤ 1 or id_{A}M ≤ 1, is called a shod algebra (for small homological dimension). We say that an algebra is strictly shod if it is shod but not quasi-tilted.

The following reformulation of a result of Kleiner, Skowroński and Zacharia will be very useful [19].

Theorem 2.2. ([19], see also [2]) Let A = kQ/I be a shod algebra, then every full subcategory of A is shod.

2.2. String Algebras. We recall the definition of string algebras.

Definition 2.3. [6] An algebra A is called a string algebra if A ≅ kQ/I with (Q, I) a bound quiver satisfying the following conditions:

(1) The ideal I is generated by a set of paths.

(2) Each vertex of Q is the source of at most two arrows and the target of at most two arrows.

(3) For an arrow α, there exist at most one arrow β and at most one arrow γ such that αβ and γα do not belong to I.

In the sequel, whenever we write that A = kQ/I is a string algebra, we mean that the bound quiver (Q, I) satisfies the above conditions.

We now give the constructions of string modules and band modules. Let A = kQ/I be a string algebra.

A reduced walk in Q is a string if it contains no zero-relation. Let w be a string. We define the string module M(w) as follows. Firstly, if w = ϵ_{a} is the trivial path at a, then M(w) is defined to be the simple A-module S(a) at a. Secondly, assume that w = c_{1} ⋯ c_{n} is a non-trivial string (so that c_{i} is an arrow or the inverse of an arrow for all i where 1 ≤ i ≤ n). For i such that 1 ≤ i ≤ n + 1, we define U_{i} = k and, for i such that 1 ≤ i ≤ n, let U_{c_{i}} be the identity map from U_{i} to U_{i+1} if c_{i} is an arrow and otherwise, let U_{c_{i}} be the identity map from U_{i+1} to U_{i}. The string module M(w) is defined as follows: for a vertex a in Q, M(w)_{a} is the direct sum of all the vector spaces U_{i} such that s(c_{i}) = a together, in case t(c_{n}) = a, with the vector space U_{n+1}; if a does not appear in w, M(w)_{a} = 0. For an arrow α in Q, if α appears in w, then M(w)_{α} is the direct sum of the maps U_{c_{i}} such that c_{i} = α or c_{i} = α^{−1}; otherwise M(w)_{α} is the zero map.

A reduced cycle w = c_{1} ⋯ c_{n} in Q is a band if w it is not a power of a reduced cycle of smaller length and none of its powers contains a zero-relation. Let ϕ be an indecomposable automorphism of a k-vector space V. For i such that 1 ≤ i ≤ n, we define V (i) = V. For i such that 1 ≤ i ≤ n − 1, let f_{c_{i}} be the identity map from V (i) to V (i + 1), if c_{i} is an arrow and otherwise, the identity map from V (i + 1) to V (i). We finally let f_{c_{n}} = ϕ : V (n) → V (1), if c_{n} is an arrow and otherwise f_{c_{n}} = ϕ^{−1} : V (1) → V (n). The band module N(w, ϕ) is defined as follows: for a vertex a in Q, N(w, ϕ)_{a} is the direct sum of the vector spaces V (i) such that s(c_{i}) = a; if a does not appear in w, N(w, ϕ)_{a} = 0. For an arrow α in Q, if α appears in w, then N(w, ϕ)_{α} is the direct sum of the maps f_{c_{i}} such that c_{i} = α or c_{i} = α^{−1}; otherwise N(w, ϕ)_{α} is the zero map.

It is shown in [6] that, if A is a string algebra, then every indecomposable A-module is either a string module or a band module.

2.3. Consecutive-zeros. The following definition is particulary useful for our classification of the shod string algebras.

Definition 2.4. Let (Q, I) be a bound quiver. Let w be a reduced walk in (Q, I) containing at least two zero-relations such that all zero-relations point to the same direction in w. Then any subwalk of w containing at least two zero-relations is called a consecutive-zero.

Note that, for instance, we can get a consecutive-zero from a reduced cycle containing exactly one zero-relation. Also, a path containing two overlapping zero-relations is a consecutive-zero.

Notice that the definition of consecutive-zero is a generalization of that of doublezero ([16], see also [1]). Thus, we may reformulate the characterization of quasi-tilted string algebras due to Huard and Liu as follows [18].

Theorem 2.5. [18] Let A = kQ/I be a string algebra. Then, A is quasi-tilted if and only if (Q, I) contains no consecutive-zero.

2.4. Derived categories. For an algebra A, we denote by D^{b}(modA), or briefly by D^{b}(A), the derived category of bounded complexes of finite dimensional A-modules. This category has the structure of a triangulated category. We say that two algebras A and B are derived equivalent if D^{b}(A) and D^{b}(B) are equivalent as triangulated categories, and we denote it by D^{b}(A) ≅ D^{b}(B). We also say that A and B are tilting-cotilting equivalent if there exist a sequence of algebras A = A_{0}, ⋯, A_{m} = B and a sequence of modules T_{A_{i}}^{(i)}, with 0 ⩽ i ⩽ m − 1, such that A_{i+1} = End(T_{A_{i}}^{(i)}) and T_{A_{i}}^{(i)} is either a tilting or a cotilting A_{i}-module. It is well-known that if A and B are tilting-cotilting equivalent algebras then they are derived equivalent. In general, the converse is not true. For details about these notions we refer to [12].

Now, if A is derived equivalent to B, with B a quasi-tilted algebra, then A and B are tilting-cotilting equivalent [14]. Following [14], we say that A is a piecewise hereditary algebra if A is derived equivalent to a quasi-tilted algebra. Also, in [14], the authors prove that if A is a piecewise hereditary algebra then A is triangular. In [13], Happel shows that a piecewise hereditary algebra is derived equivalent either to a hereditary algebra or to a canonical algebra. An algebra A derived equivalent to a (hereditary) path algebra kΔ is called derived hereditary of type Δ.

3. Shod String Algebras

In this section, we classify the shod string algebras.

Lemma 3.1. Let A = kQ/I be a string algebra. If (Q, I) admits a consecutive-zero u containing no overlapping zero-relations, then A is not shod.

Proof: Since u is a consecutive-zero in (Q, I), u contains at least two zero-relations. Moreover, u contains no overlapping zero-relations. Then, there exists w = z_{1}vz_{2} a subwalk of u such that w contains two zero-relations z_{1} and z_{2} with v a string and where z_{1} and z_{2} point to the same direction in w. Without loss of generality, we may assume that w = α_{1} ⋯ α_{s}vβ_{1} ⋯ β_{t} where α_{i} and β_{j} are arrows for 1 ≤ i ≤ s and for 1 ≤ j ≤ t and where z_{1} = α_{1} ⋯ α_{s} and z_{2} = β_{1} ⋯ β_{t} are the two zero-relations contained in w. Notice that α_{s}v and vβ_{1} are strings, since u contains no overlapping zero-relations. We consider the string module M(v). There is a path z_{2} such that vβ_{1} is a string and vz_{2} admits exactly one zero-relation which contains z_{2}. Then, from [17, (2.3)], pd_{A}M(v) > 1. Moreover, there is a path z_{1} such that α_{s}v is a string and z_{1}v admits exactly one zero-relation which contains z_{1}. Hence, from [17, (2.3)], id_{A}M(v) > 1. Therefore, A is not shod. □

The following definition will be useful to express the next lemma.

Definition 3.2. Let (Q, I) be a bound quiver. Let p = p_{1}p_{2}p_{3} be a path in Q containing two overlapping zero-relations p_{1}p_{2} and p_{2}p_{3}. If p_{1} (or p_{3}) is an arrow, we say that p is start-tight (or end-tight, respectively).

Example 3.3. Let (Q, I) be the following bound quiver:

[figure: ∘; α_{1}; ∘; α_{2}; ∘; α_{3}; ∘; α_{4}; ∘; α_{5}; ∘]

where a dotted line on a path means that the composition of the arrows on this path is a generator of the ideal I. First, we notice that the path α_{1}α_{2}α_{3}α_{4} contains two overlapping zero-relations, namely α_{1}α_{2}α_{3} and α_{2}α_{3}α_{4}, and that α_{1}α_{2}α_{3}α_{4} is start-tight and end-tight. Also, we observe that α_{2}α_{3}α_{4}α_{5} is start-tight and end-tight. However, the path α_{1}α_{2}α_{3}α_{4}α_{5} contains the overlapping zero-relations α_{1}α_{2}α_{3} and α_{3}α_{4}α_{5}, so α_{1}α_{2}α_{3}α_{4}α_{5} is neither start-tight nor end-tight.

Lemma 3.4. Let A = kQ/I be a string algebra. If A is shod, then any path in (Q, I) containing two overlapping zero-relations is either start-tight or end-tight.

Proof: Assume that there is a path α_{1} ⋯ α_{t} in (Q, I) containing two overlapping zerorelations, α_{1} ⋯ α_{s} and α_{r} ⋯ α_{t} such that 1 < r ≤ s < t, which is neither start-tight nor end-tight. Then, r > 2 and t > s + 1. Notice that α_{r} ⋯ α_{s}, α_{r} ⋯ α_{s}α_{s+1} and α_{r−1}α_{r} ⋯ α_{s} are strings. Consider the string module M(α_{r} ⋯ α_{s}). There is a path α_{s+1} ⋯ α_{t} such that α_{r} ⋯ α_{s}α_{s+1} is a string and such that α_{r} ⋯ α_{s}α_{s+1} ⋯ α_{t} admits exactly one zero-relation which contains α_{s+1} ⋯ α_{t}. Then, from [17, (2.3)], we have pd_{A}M(α_{r} ⋯ α_{s}) > 1. Moreover, there is a path α_{1} ⋯ α_{r−1} such that α_{r−1}α_{r} ⋯ α_{s} is a string and such that α_{1} ⋯ α_{r−1}α_{r} ⋯ α_{s} admits exactly one zero-relation which contains α_{1} ⋯ α_{r−1}. Hence, from [17, (2.3)], id_{A}M(α_{r} ⋯ α_{s}) > 1. Therefore, A is not shod. □

Lemma 3.5. Let A = kQ/I be a string algebra. Then A is not shod if A contains a full subcategory of the following form (or its dual):

[figure: ∘; β_{1}; ∘; α_{1}; ∘; α_{2}; β_{2}; ∘; α_{3}; ∘; ∘]

Proof: Assume there exists a full subcategory B = kQ′/I′ of A where (Q′, I′) is the bound quiver above. Observe that α_{2}^{−1}β_{2} and α_{1}β_{2} are strings in B. Consider the string B-module M(β_{2}). First, there is a path α_{2}α_{3} such that α_{2}^{−1}β_{2} is a string and such that α_{3}^{−1}α_{2}^{−1}β_{2} admits exactly one zero-relation which contains α_{2}α_{3}. Then, from [17, (2.3)], pd_{B}M(β_{2}) > 1. Moreover, there is a path β_{1}α_{1} such that α_{1}β_{2} is a string and such that β_{1}α_{1}β_{2} admits exactly one zero-relation which contains β_{1}α_{1}. Hence, from [17, (2.3)], id_{B}M(β_{2}) > 1. Therefore, B is not shod and it follows from 2.2 that A is not shod. □

Lemma 3.6. Let A = kQ/I be a string algebra. Then A is not shod if A contains a full subcategory of the following form (or its dual):

[figure: ∘; α_{1}; ∘; α_{2}; β_{1}; ∘; α_{3}; ∘; α_{4}; ∘; ∘]

Proof: Assume there exists a full subcategory B = kQ′/I′ of A where (Q′, I′) is the bound quiver above. Observe that β_{1}^{−1}α_{2} and β_{1}^{−1}α_{2}α_{3} are strings in B. Consider the string B-module M(β_{1}^{−1}α_{2}). First, there is a path α_{3}α_{4} such that β_{1}^{−1}α_{2}α_{3} is a string and such that α_{2}α_{3}α_{4} admits exactly one zero-relation which contains α_{3}α_{4}. Then, from [17, (2.3)], pd_{B}M(β_{1}^{−1}α_{2}) > 1. Moreover, there is a non-zero path α_{1} such that α_{1}β_{1} and α_{1}α_{2} are zero paths. Hence, from [17, (2.4)], id_{B}M(β_{1}^{−1}α_{2}) > 1. Therefore, B is not shod and it follows from 2.2 that A is not shod. □

Lemma 3.7. Let A = kQ/I be a string algebra. Then A is not shod if A contains a full subcategory of the following form:

[figure: ∘; β_{2}; ∘; α_{1}; ∘; α_{2}; β_{1}; ∘; α_{3}; ∘; ∘]

Proof: Assume there exists a full subcategory B = kQ′/I′ of A where (Q′, I′) is the bound quiver above. Observe that β_{1}^{−1}α_{2}β_{2}^{−1} is a string in B. Consider the string B-module M(β_{1}^{−1}α_{2}β_{2}^{−1}). First, there is a non-zero path α_{3} such that α_{2}α_{3} and β_{2}α_{3} are zero paths. Then, from [17, (2.4)], pd_{B}M(β_{1}^{−1}α_{2}β_{2}^{−1}) > 1. Moreover, there is a non-zero path α_{1} such that α_{1}β_{1} and α_{1}α_{2} are zero paths. Hence, from [17, (2.4)], id_{B}M(β_{1}^{−1}α_{2}β_{2}^{−1}) > 1. Therefore, B is not shod and it follows from 2.2 that A is not shod. □

We now give the proof of the theorem 1.1 stated in the introduction. Proof of the theorem 1.1: First, suppose that (Q, I) does not satisfy the condition (1), then there exists a consecutive-zero u in (Q, I) containing more than two zero-relations or there exists a consecutive-zero u in (Q, I) containing exactly two zero-relations which are not overlapping or where the path containing these two zerorelations is neither start-tight nor end-tight. Therefore, in both cases, either u contains two zero-relations which are not overlapping or u contains a path having two overlapping zero-relations which is neither start-tight nor end-tight. Hence, it follows from 3.1 and 3.4 that A is not shod. Finally, suppose that (Q, I) does not satisfy the condition (2), then it follows from 3.5, 3.6 and 3.7 that A is not shod.

Conversely, assume that (Q, I) satisfies the conditions (1) and (2) and, suppose to the contrary that A is not shod. Then, there exists an indecomposable A-module M such that pd_{A}M > 1 and id_{A}M > 1. Moreover, M is either a string module or a band module [6].

First, suppose that M = M(w) is a string module. Let w = p_{1}^{−1}q_{1} ⋯ p_{n}^{−1}q_{n} where p_{i} and q_{j} are paths which are non-trivial for 1 < i ≤ n and 1 ≤ j < n.

(a) First, we consider the case where p_{1} is non-trivial and q_{n} is trivial. It follows from [18, (2.1)] that one of the following holds:

(A) There is a non-trivial path z_{1} = αz1^{′}, with α an arrow, such that p_{1}z_{1} is a zero path, while p_{1}α is a non-zero path;

(B_{j}) There is a non-trivial path z_{j}, with 1 < j ≤ n, such that p_{j}z_{j} and q_{j−1}z_{j} are both zero paths;

(C) There is a zero path z_{n+1} = βzn+1^{′}, with β an arrow, such that wz_{n+1} is a reduced walk.

Observe that we can write w = q_{0}p_{1}^{−1} ⋯ p_{n}^{−1} where q_{0} is a trivial path. It follows from [18, (2.1)] that one of the following holds:

(A′) There is a zero path y_{0} = y0^{′}γ, with γ an arrow, such that y_{0}w is a reduced walk;

(Bi^{′}) There is a non-trivial path y_{i} with 1 ≤ i < n, such that y_{i}q_{i} and y_{i}p_{i} are zero paths;

(C′) There is a non-trivial path y_{n} = yn^{′}δ, with δ an arrow, such that y_{n}p_{n} is a zero path, while δp_{n} is a non-zero path.

- If the cases (A) and (A′) hold, then γα is a zero-relation, because p_{1}α is a non-zero path. If y0^{′} is a zero path, then the reduced walk y_{0}α contains a consecutive-zero having two zero-relations which are not overlapping. Now, suppose that y0^{′} is a non-zero path, then the reduced walk y_{0}α contains two overlapping zero-relations. If z_{1} is a zero path, then the reduced walk y_{0}z_{1} contains a consecutive-zero having two zero-relations which are not overlapping. If z_{1} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′).

- If the cases (A) and (Bi^{′}) hold, then the reduced walk y_{i}p_{i}q_{i−1}^{−1} ⋯ q_{1}^{−1}p_{1}z_{1} contains a consecutive-zero having two zero-relations which, from (1), must be overlapping. Hence, i = 1. Since (Q, I) satisfies (1), the path containing these two overlapping zero-relations contained in y_{1}p_{1}z_{1} is either start-tight or end-tight. Let u_{1}u_{2}u_{3} be this path. If z_{1} is a zero path, then the reduced walk y_{1}p_{1}z_{1} contains a consecutive-zero having two zero-relations which are not overlapping. Suppose z_{1} is a non-zero path. First, the path u_{1}u_{2}u_{3} cannot be end-tight, because p_{1}α is a nonzero path. Hence, u_{1}u_{2}u_{3} is start-tight and u_{1} is an arrow. Let α_{1} be this arrow. Since α_{1}u_{2}u_{3} is contained in y_{1}p_{1}z_{1}, we infer that y_{1} is a path with terminal arrow α_{1}. Recall that y_{1}q_{1} is a zero path. Suppose that α_{1}q_{1} is a non-zero path. If y_{1} is a zero path, then the reduced walk y_{1}p_{1}z_{1} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{1} is a non-zero path, then (Q, I) contains a full subcategory of the form (i). Now, suppose that α_{1}q_{1} is a zero path. Since the path α_{1}u_{2}u_{3} cannot also be end-tight, then (Q, I) contains a full subcategory of the form (ii).

- If the cases (A) and (C′) hold, then the reduced walk y_{n}p_{n}q_{n−1}^{−1} ⋯ q_{1}^{−1}p_{1}z_{1} contains a consecutive-zero having two zerorelations which, from (1), must be overlapping. Hence, n = 1 and y_{1}p_{1}z_{1} contains two overlapping zero-relations, but the path containing these two overlapping zero-relations is neither start-tight nor end-tight, because δp_{1} and p_{1}α are non-zero paths.

- Suppose that the cases (B_{j}) and (A′) hold, then the reduced walk y_{0}p_{1}^{−1}q_{1} ⋯ p_{j−1}^{−1}q_{j−1}z_{j} contains a consecutive-zero having two zerorelations which are not overlapping, because j > 1.

- If the cases (B_{j}) and (Bi^{′}) hold. If j ≤ i, then the reduced walk y_{i}p_{i}q_{i−1}^{−1} ⋯ q_{j}^{−1}p_{j}z_{j} contains a consecutive-zero having two zero-relations which, from (1), must be overlapping. Hence, i = j. From (1), the path containing these two overlapping zero-relations contained in y_{i}p_{i}z_{i} is either start-tight or end-tight. Let u_{1}u_{2}u_{3} be this path. Suppose first that u_{1}u_{2}u_{3} is start-tight, then u_{1} is an arrow. Let α_{1} be this arrow. Recall that y_{i}q_{i} is a zero path. Suppose α_{1}q_{i} is a nonzero path. If y_{i} is a zero path, then the reduced walk y_{i}p_{i}z_{i} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{i} is a non-zero path, then (Q, I) contains a full subcategory of the form (i). Now, suppose that α_{1}q_{i} is a zero path. If α_{1}u_{2}u_{3} is not also endtight, then (Q, I) contains a full subcategory of the form (ii). Suppose that α_{1}u_{2}u_{3} is also end-tight, then u_{3} is an arrow. Let α_{t} be this arrow. Recall that q_{i−1}z_{i} is a zero path. If q_{i−1}α_{t} is a zero path, then (Q, I) contains a full subcategory of the form (iii). Suppose that q_{i−1}α_{t} is a non-zero path. If z_{i} is a zero path, then the reduced walk y_{i}p_{i}z_{i} contains a consecutive-zero having two zero-relations which are not overlapping. If z_{i} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′). Now, suppose that u_{1}u_{2}u_{3} is end-tight, then u_{3} is an arrow. Let α_{t} be this arrow. Recall that q_{i−1}z_{i} is a zero path. Suppose that q_{i−1}α_{t} is a non-zero path. If z_{i} is a zero path, then y_{i}p_{i}z_{i} contains a consecutive-zero having two zero-relations which are not overlapping. If z_{i} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′). Now, suppose that q_{i−1}α_{t} is a zero path. If u_{1}u_{2}α_{t} is not also start-tight, then (Q, I) contains a full subcategory of the form (ii′). Suppose that u_{1}u_{2}α_{t} is also start-tight. Recall that y_{i}q_{i} is a zero path. If α_{1}q_{i} is a zero path, then (Q, I) contains a full subcategory of the form (iii). Suppose that α_{1}q_{i} is a non-zero path. If y_{i} is a zero path, then y_{i}p_{i}z_{i} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{i} is a non-zero path, then (Q, I) contains a full subcategory of the form (i). Now, consider the case where j > i. Then the reduced walk y_{i}q_{i}p_{i+1}^{−1} ⋯ p_{j−1}^{−1}q_{j−1}z_{j} contains a consecutive-zero having two zerorelations which, from (1), must be overlapping. Hence, i = j − 1. From (1), the path containing these two overlapping zero-relations contained in y_{i}q_{i}z_{i+1} is either start-tight or end-tight. Let u_{1}u_{2}u_{3} be this path. Suppose first that u_{1}u_{2}u_{3} is start-tight, then u_{1} is an arrow. Let α_{1} be this arrow. Recall that y_{i}p_{i} is a zero path. Suppose α_{1}p_{i} is a non-zero path. If y_{i} is a zero path, then the reduced walk y_{i}q_{i}z_{i+1} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{i} is a non-zero path, then (Q, I) contains a full subcategory of the form (i). Now, suppose that α_{1}p_{i} is a zero path. If α_{1}u_{2}u_{3} is not also endtight, then (Q, I) contains a full subcategory of the form (ii). Suppose that α_{1}u_{2}u_{3} is also end-tight, then u_{3} is an arrow. Let α_{t} be this arrow. Recall that p_{i+1}z_{i+1} is a zero path. If p_{i+1}α_{t} is a zero path, then (Q, I) contains a full subcategory of the form (iii). Suppose that p_{i+1}α_{t} is a nonzero path. If z_{i+1} is a zero path, then the reduced walk y_{i}q_{i}z_{i+1} contains a consecutive-zero having two zero-relations which are not overlapping.

If z_{i+1} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′). Now, suppose that u_{1}u_{2}u_{3} is end-tight, then u_{3} is an arrow. Let α_{t} be this arrow. Recall that p_{i+1}z_{i+1} is a zero path. Suppose that p_{i+1}α_{t} is a non-zero path. If z_{i+1} is a zero path, then y_{i}q_{i}z_{i+1} contains a consecutivezero having two zero-relations which are not overlapping. If z_{i+1} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′). Now, suppose that p_{i+1}α_{t} is a zero path. If u_{1}u_{2}α_{t} is not also start-tight, then (Q, I) contains a full subcategory of the form (ii′). Suppose that u_{1}u_{2}α_{t} is also start-tight. Recall that y_{i}p_{i} is a zero path. If α_{1}p_{i} is a zero path, then (Q, I) contains a full subcategory of the form (iii). Suppose that α_{1}p_{i} is a non-zero path. If y_{i} is a zero path, then y_{i}q_{i}z_{i+1} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{i} is a non-zero path, then (Q, I) contains a full subcategory of the form (i).

- If the cases (B_{j}) and (C′) hold, then the reduced walk y_{n}p_{n}q_{n−1}^{−1} ⋯ q_{j}^{−1}p_{j}z_{j} contains a consecutive-zero having two zerorelations which, from (1), must be overlapping. Hence, n = j. From (1), the path containing these two overlapping zero-relations contained in y_{n}p_{n}z_{n} is either start-tight or end-tight. Let u_{1}u_{2}u_{3} be this path. If y_{n} is a zero path, then y_{n}p_{n}z_{n} contains a consecutive-zero having two zero-relations which are not overlapping. Suppose that y_{n} is a non-zero path. Since δp_{n} is a non-zero path, the path u_{1}u_{2}u_{3} cannot be start-tight. Hence, it is end-tight and u_{3} is an arrow. Let α_{t} be this arrow. Recall that q_{n−1}z_{n} is a zero path. If q_{n−1}α_{t} is a zero path, then (Q, I) contains a full subcategory of the form (ii′), because δp_{n} is a non-zero path. Suppose that q_{n−1}α_{t} is a non-zero path. If z_{n} is a zero path, then y_{n}p_{n}z_{n} contains a consecutive-zero having two zero-relations which are not overlapping. If z_{n} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′).

- If the cases (C) and (A′) hold, then the reduced walk y_{0}p_{1}^{−1}q_{1} ⋯ p_{n}^{−1}z_{n+1} contains a consecutive-zero having two zero-relations which are not overlapping.

- If the cases (C) and (Bi^{′}) hold, then the reduced walk y_{i}q_{i}p_{i+1}^{−1} ⋯ p_{n}^{−1}z_{n+1} contains a consecutive-zero having two zero-relations which are not overlapping, because i < n.

- If the cases (C) and (C′) hold, then δβ is a zero-relation, because δp_{n} is a non-zero path. If zn+1^{′} is a zero path, then the reduced walk δβzn+1^{′} contains a consecutive-zero having two zero-relations which are not overlapping. If zn+1^{′} is a non-zero path, then δβzn+1^{′} contains two overlapping zero-relations. If y_{n} is a zero path, then the reduced walk y_{n}z_{n+1} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{n} is a non-zero path, then (Q, I) contains a full subcategory of the form (i).

(b) The case where p_{1} is trivial and q_{n} is non-trivial is clearly the dual of the case where p_{1} is non-trivial and q_{n} is trivial done in (a).

(c) Now, we consider the case where p_{1} and q_{n} are both trivial paths. It follows from [18, (2.1)] that one of the following holds:

(D) There is a zero path z_{1} = αz1^{′}, with α an arrow, such that z_{1}^{−1}w is a reduced walk;

(B_{j}) There is a non-trivial path z_{j}, with 1 < j ≤ n, such that p_{j}z_{j} and q_{j−1}z_{j} are both zero paths;

(C) There is a zero path z_{n+1} = βzn+1^{′}, with β an arrow, such that wz_{n+1} is a reduced walk.

Observe that we can write w = q_{1}p_{2}^{−1} ⋯ p_{n}^{−1}. It follows from [18, (2.1)] that one of the following holds:

(D′) There is a non-trivial path y_{1} = y1^{′}γ, with γ an arrow, such that y_{1}q_{1} is a zero path, while γq_{1} is a non-zero path;

(Bi^{′}) There is a non-trivial path y_{i} with 1 < i < n, such that y_{i}q_{i} and y_{i}p_{i} are zero paths;

(C′) There is a non-trivial path y_{n} = yn^{′}δ, with δ an arrow, such that y_{n}p_{n} is a zero path, while δp_{n} is a non-zero path.

Clearly, we only have to consider the following cases:

- The case where (D) and (D′) hold. We obtain that γα is a zero-relation, because γq_{1} is a non-zero path. If z1^{′} is a zero path, then the reduced walk γαz1^{′} contains a consecutive-zero having two zero-relations which are not overlapping. If z1^{′} is a non-zero path, then γαz1^{′} contains two overlapping zero-relations. If y_{1} is a zero path, then the reduced walk y_{1}z_{1} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{1} is a non-zero path, then (Q, I) contains a full subcategory of the form (i).

- The case where (D) and (Bi^{′}) hold. The reduced walk y_{i}p_{i}q_{i−1}^{−1} ⋯ p_{2}q_{1}^{−1}z_{1} contains a consecutive-zero having two zero-relations which are not overlapping.

- The case where (D) and (C′) hold. The reduced walk y_{n}p_{n}q_{n−1}^{−1} ⋯ p_{2}q_{1}^{−1}z_{1} contains a consecutive-zero having two zerorelations which are not overlapping.

- The case where (B_{j}) and (D′) hold. The reduced walk y_{1}q_{1} ⋯ q_{j−1}z_{j} contains a consecutive-zero having two zero-relations which, from (1), must be overlapping. Hence, j = 2. From (1), the path containing these two overlapping zero-relations contained in y_{1}q_{1}z_{2} is either start-tight or end-tight. Let u_{1}u_{2}u_{3} be this path. If y_{1} is a zero path, then the reduced walk y_{1}q_{1}z_{2} contains a consecutive-zero having two zero-relations which are not overlapping. Suppose y_{1} is a non-zero path. Since γq_{1} is a nonzero path, the path u_{1}u_{2}u_{3} cannot be start-tight. Hence, it is end-tight and u_{3} is an arrow. Let α_{t} be this arrow. Since u_{1}u_{2}α_{t} is contained in y_{1}q_{1}z_{2}, we obtain that z_{2} is a path with initial arrow α_{t}. Recall that p_{2}z_{2} is a zero path. Suppose that p_{2}α_{t} is a non-zero path. If z_{2} is a zero path, then the reduced walk y_{1}q_{1}z_{2} contains a consecutive-zero having two zero-relations which are not overlapping. If z_{2} is not a zero path, then (Q, I) contains a full subcategory of the form (i′). Now, suppose that p_{2}α_{t} is a zero path. Since the path u_{1}u_{2}u_{3} cannot also be start-tight, (Q, I) contains a full subcategory of the form (ii′).

- The case where (C) and (D′) hold. The reduced walk y_{1}q_{1}p_{2}^{−1} ⋯ q_{n}p_{n}^{−1}z_{n+1} contains a consecutive-zero having two zerorelations which are not overlapping.

(d) Now, we consider the case where p_{1} and q_{n} are both non-trivial paths. It follows

from [18, (2.1)] that one of the following holds:

(A) There is a non-trivial path z_{1} = αz1^{′}, with α an arrow, such that p_{1}z_{1} is a zero path, while p_{1}α is a non-zero path;

(B_{j}) There is a non-trivial path z_{j}, with 1 < j ≤ n, such that p_{j}z_{j} and q_{j−1}z_{j} are both zero paths;

(E) There is a non-trivial path z_{n+1} = βzn+1^{′}, with β an arrow, such that q_{n}z_{n+1} is a zero path, while q_{n}β is a non-zero path.

Observe that we can write w = q_{0}p_{1}^{−1} ⋯ q_{n}p_{n+1}^{−1}, where q_{0} and p_{n+1} are both trivial paths. It follows from [18, (2.1)] that one of the following holds:

(A′) There is a zero path y_{0} = y0^{′}γ, with γ an arrow, such that y_{0}w is a reduced walk;

(Bi^{′}) There is a non-trivial path y_{i} with 1 ≤ i ≤ n, such that y_{i}q_{i} and y_{i}p_{i} are zero paths;

(E′) There is a zero path y_{n+1} = yn+1^{′}δ, with δ an arrow, such that wy_{n+1}^{−1} is a reduced walk.

Clearly, we only have to consider the following cases:

- The case where (A) and (E′) hold. The reduced walk y_{n+1}q_{n}^{−1} ⋯ p_{1}z_{1} contains a consecutive-zero having two zero-relations which are not overlapping.

- The case where (B_{j}) and (E′) hold. The reduced walk y_{n+1}q_{n}^{−1} ⋯ p_{j}z_{j} contains a consecutive-zero having two zero-relations which are not overlapping.

- The case where (E) and (A′) hold. The reduced walk y_{0}p_{1}^{−1} ⋯ q_{n}z_{n+1} contains a consecutive-zero having two zero-relations which are not overlapping.

- The case where (E) and (Bi^{′}) hold. The reduced walk y_{i}q_{i} ⋯ q_{n}z_{n+1} contains a consecutive-zero having two zero-relations which, from (1), must be overlapping. Hence, i = n. From (1), the path containing these two overlapping zero-relations contained in y_{n}q_{n}z_{n+1} is either starttight or end-tight. Let u_{1}u_{2}u_{3} be this path. If zn+1^{′} is a zero path, then the reduced walk y_{n}q_{n}z_{n+1} contains a consecutive-zero having two zero-relations which are not overlapping. Suppose that zn+1^{′} is a nonzero path. The path u_{1}u_{2}u_{3} cannot be end-tight, because q_{n}β is a nonzero path. Hence, it is start-tight and u_{1} is an arrow. Let α_{1} be this arrow. Since α_{1}u_{2}u_{3} is contained in y_{n}q_{n}z_{n+1}, we obtain that y_{n} is a path with terminal arrow α_{1}. Recall that y_{n}p_{n} is a zero path. Suppose that α_{1}p_{n} is not a zero path. If y_{n} is a zero path, then the reduced walk y_{n}q_{n}z_{n+1} contains a consecutive-zero having two zero-relations which are not overlapping. If y_{n} is not a zero path, then (Q, I) contains a full subcategory of the form (i). Now, suppose that α_{1}p_{n} is a zero path. Since u_{1}u_{2}u_{3} cannot also be end-tight, (Q, I) contains a full subcategory of the form (ii).

- The case where (E) and (E′) hold. Then δβ is a zero-relation, because q_{n}β is a non-zero path. If yn+1^{′} is a zero path, then the reduced walk yn+1^{′}γβ contains a consecutive-zero having two zero-relations which are not overlapping. Suppose that yn+1^{′} is a non-zero path, then yn+1^{′}γβ contains two overlapping zero-relations. If zn+1^{′} is a zero path, then the reduced walk y_{n+1}z_{n+1} contains a consecutive-zero having two zerorelations which are not overlapping. If zn+1^{′} is a non-zero path, then (Q, I) contains a full subcategory of the form (i′).

Finally, suppose that M = M(w, ϕ) is a band module determined by w and an indecomposable automorphism ϕ of a finite-dimensional k-vector space. The definition of bands implies that w cannot be an oriented cycle. Then, we can write w = p_{1}^{−1}q_{1} ⋯ p_{n}^{−1}q_{n} where p_{i} and q_{i} are paths which are non-trivial for 1 ≤ i ≤ n such that t(p_{1}) = t(q_{n}). From [20] (see also [18]), we get the following case:

(B_{j}) There is a non-trivial path z_{j}, with 1 ≤ j ≤ n, such that p_{j}z_{j} and q_{j−1}z_{j} are both zero paths, and

(Bi^{′}) There is a non-trivial path y_{i} with 1 ≤ i ≤ n, such that y_{i}q_{i} and y_{i}p_{i} are both zero paths;

where q_{j−1} = q_{n} if j = 1.

Clearly, the proof in this case is the same as in (a). This completes the proof of the theorem. □

Example 3.8. (a) Consider the algebra A = kQ/I where (Q, I) is the bound quiver shown in the example 3.4. Clearly, A is a string algebra. Notice that α_{1}α_{2}α_{3}α_{4}α_{5} is a consecutive-zero containing three zero-relations. Therefore A is not shod, since (Q, I) does not satisfy the condition (1) stated in 1.1.

(b) Let (Q, I) be the following bound quiver:

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘]

Then, A = kQ/I is a string algebra. Since there exists a consecutive-zero, A is not quasi-tilted. But, (Q, I) satisfies the conditions (1) and (2) as stated in 1.1. Therefore, A is shod.

As an easy consequence of our classification, we also get the classification of the shod gentle algebras.

Definition 3.9. ([4], [22]) Let A = kQ/I be a string algebra. Then A is called gentle if (Q, I) is a bound quiver satisfying the following conditions:

(1) Every zero-relation is of length two.

(2) For an arrow α, there exist at most one arrow δ and at most one arrow ϵ such that αδ and ϵα belong to I.

Example 3.10. Let A = kQ/I where Θ = (Q, I) is the following bound quiver:

[figure: ∘; ∘; ∘; ∘]

Then, A is a gentle algebra.

Corollary 3.11. Let A = kQ/I be a gentle algebra. Then A is shod if and only if every consecutive-zero in (Q, I) contains at most two zero-relations, and in case there are two, they are overlapping and the path containing them is isomorphic to the bound quiver Θ in 3.10.

Proof: Since A is a string algebra, it suffices to verify that (Q, I) satisfies the conditions (1) and (2) as stated in 1.1. Clearly, A satifies the condition (2), since every zerorelation is of length two. □

Corollary 3.12. An algebra A = kQ/I is shod derived hereditary of type 𝔸_{n} if and only if A is gentle, Q is a tree and every consecutive-zero in (Q, I) contains at most two zero-relations, and in case there are two, they are overlapping and the path containing them is isomorphic to the bound quiver Θ in 3.10.

Proof: This follows from the classification of algebras derived hereditary of type 𝔸_{n} given in [3] and corollary 3.11. □

Corollary 3.13. An algebra A = kQ/I is shod derived hereditary of type 𝔸̃_{n} if and only if A is gentle and (Q, I) satisfies the following three conditions:

(i) Q contains a unique (non-oriented) cycle C.

(ii) On C, the number of clockwise oriented relations equals the number of counter-

clockwise oriented relations. (iii) Every consecutive-zero in (Q, I) contains at most two zero-relations, and in case

there are two, they are overlapping and the path containing them is isomorphic to the bound quiver Θ in 3.10.

Proof: This follows from the classification of algebras derived hereditary of type 𝔸̃_{n} given in [4] and corollary 3.11. □

4. Discrete derived category

In this section, we first classify the shod algebras with discrete derived category not of Dynkin type. Then, after stating some previous results, we show which algebras with discrete derived category are derived equivalent to a shod algebra.

Following [23], if A is a finite dimensional k-algebra, D^{b}(A) is said to be a discrete derived category if for every vector n = (n_{i})_{i∈ℤ} of natural numbers there are only finitely many isomorphism classes of indecomposable objects in D^{b}(A) of homology dimension vector n. From the description of the derived category of hereditary algebras [12], it is clear that every algebra derived hereditary of Dynkin type has discrete derived category.

Corollary 4.1. Let A = kQ/I be an algebra. The following conditions are equivalent:

(a) A is shod with discrete derived category not of Dynkin type.

(b) A is gentle and (Q, I) satisfies:

(b1) Q contains a unique (non-oriented) cycle C.

(b2) On C, the numbers of clockwise oriented relations and of counterclockwise

oriented relations are different. (b3) Every consecutive-zero in (Q, I) contains at most two zero-relations, and in case there are two, they are overlapping and the path containing them is isomorphic to the bound quiver Θ in 3.10.

Proof: In [23], it is proved that a derived category D^{b}(A) is discrete but not of Dynkin type if and only if A = kQ/I is gentle, Q contains a unique (non-oriented) cycle C and on C the numbers of clockwise and counterclockwise oriented relations are different. So, the corollary follows from this description and corollary 3.11. □

Observe that the bound quiver (Q, I) of a gentle algebra A = kQ/I containing a unique cycle has also some branches which are joined to the cycle at a single vertex. We call such vertex the root of the branch. A quiver of type 𝔸_{n}, with n > 0, such that all arrows point to the same direction is called an equioriented linear quiver. So, a branch which is an equioriented linear quiver oriented toward the cycle has its root in a sink.

We now give an example of a class of algebras with discrete derived category not of Dynkin type. Denote by Ω the set {(r, n, m), r, n, m ∈ ℤ | n ⩾ r > 0 and m ⩾ 0}. For each (r, n, m) ∈ Ω, the algebra Λ(r, n, m) is defined as the quotient of a path algebra kQ(r, n, m)/I(r, n, m), where Q(r, n, m) is the following quiver:

\xymatrix{ &  &  &  & 1 \ar[r]^{α_{1}} & ⋯ \ar[r]^{α_{n−r−2}} & n − r − 1 \ar[dr]^{α_{n−r−1}} \\ (−m) \ar[r]^{α_{−m}} & ⋯ \ar[r]^{α_{−2}} & (−1) \ar[r]^{α_{−1}} & 0 \ar[ur]^{α_{0}} &  &  &  & n − r \ar[dl]^{α_{n−r}} \\  &  &  &  & n − 1 \ar[ul]^{α_{n−1}} & ⋯ \ar[l]^{α_{n−2}} & n − r + 1 \ar[l]^{α_{n−r+1}}}

and I(r, n, m) is the ideal generated by the paths α_{n−r}α_{n−r+1}, ⋯, α_{n−2}α_{n−1}, α_{n−1}α_{0}. Recently, Bobinski, Geiss and Skowronski obtained the following result [5].

Theorem 4.2. [5] For an algebra A the following conditions are equivalent:

(i) D^{b}(A) is discrete not of Dynkin type.

(ii) D^{b}(A) ≅ D^{b}(Λ(r, n, m)) for some (r, n, m) ∈ Ω.

(iii) A is tilting-cotilting equivalent to Λ(r, n, m) for some (r, n, m) ∈ Ω.

Moreover, for (r, n, m), (r′, n′, m′) ∈ Ω, we have D^{b}(Λ(r, n, m)) ≅ D^{b}(Λ(r′, n′, m′)) if and only if (r, n, m) = (r′, n′, m′).

Remark 4.3. As a consequence of the above theorem, if A is an algebra with discrete derived category, then D^{b}(A) ≅ D^{b}(B) if and only if A and B are tilting-cotilting equivalent. Observe also, that if D^{b}(A) is discrete not of Dynkin type, then A is not piecewise hereditary. This follows from the fact that every piecewise hereditary algebra is triangular, whereas Λ(r, n, m) is not triangular for any (r, n, m) ∈ Ω.

The proof of theorem 4.2, given in [5], is done in several steps. We use these steps in the proof of our next result, and for convenience, we present them in the following four lemmata. Note that in 4.5 and 4.6 we give more specific results than what is given in [5].

Lemma 4.4. Let A = kQ/I be a gentle algebra containing a unique cycle C. Then, A is tilting-cotilting equivalent to a gentle algebra A′ = kQ′/I′ with a unique cycle C′ such that:

(i) All external branches of C′ are equioriented linear quivers oriented toward C′. (ii) C′ is obtained from C by adding arrows inside the strings of C. (iii) All zero-relations are those of C.

Proof: See [5], proposition 2.3, steps a), b), c) and d). □

Lemma 4.5. Let A = kQ/I be a gentle algebra containing a unique cycle C and having a full subquiver Σ of the form

[figure: b_{t+l}; ⋮; b_{1}; ⋯; b_{t−1}; b_{t}; e; a_{0} = b_{0}; a_{1}; ⋯; a_{s−1}; a_{s}; c; ⋮; a_{s+k}]

with the vertices c, a_{s}, …, a_{0} = b_{0}, …, b_{t}, e on the cycle, the string w with end-points a_{s} and b_{t} not being a path, s ⩾ 1, t ⩾ 1, and equioriented linear quivers rooted to the cycle in a_{s} and b_{t} of length k ⩾ 0 and l ⩾ 0, respectively. Then, A is tilting-cotilting equivalent to a gentle algebra A′ = kQ′/I′ where (Q′, I′) is obtained from (Q, I) by replacing Σ by the following quiver without zero-relations

\xymatrix{c \ar[r] & x_{1} \ar[r] & ⋯ \ar[r] & x_{i} \ar[r] & z & y_{j} \ar[l] & ⋯ \ar[l] & y_{1} \ar[l] & e \ar[l]}

where i is the number of arrows of the branch rooted to b_{t} plus the number of arrows in w oriented from a_{s} to b_{t}, and j is the number of arrows of the branch rooted to a_{s} plus the number of arrows in w oriented from b_{t} to a_{s}.

Proof: See [5], proposition 2.3, step e). It is not difficult to check the observation on the numbers i and j. □

Lemma 4.6. Let A = kQ/I be a gentle algebra containing a unique cycle C and having a full subquiver Σ of the form

[figure: e_{p}; ⋮; a; b; c = w_{0}; w_{1}; ⋯; w_{l−1}; d = w_{l}; e_{0}; f]

with the vertices a, b, c = w_{0}, ⋯, d = w_{l}, e_{0}, f on the cycle, the string w with end-points c and d not being a path, l ⩾ 2, and an equioriented linear quiver rooted to the cycle in e_{0}, p ⩾ 0. Then, A is tilting-cotilting equivalent to a gentle algebra A′ = kQ′/I′ where (Q′, I′) is obtained from (Q, I) by replacing Σ by the following quiver

[figure: a; b; c; v_{l}; ⋯; v_{t+1}; v_{t}; v_{t−1}; ⋯; v_{1}; e_{0}; f]

where t ⩾ 1 is the number of arrows in w oriented from d to c, and l − t ⩾ 1 is the number of arrows in w oriented from c to d, and the branch with root e_{0} is inserted inside a string of the cycle.

Proof: See [5], proposition 2.3, step g). It is not difficult to check the observation on the numbers l and t. □

Lemma 4.7. Let A = kQ/I be a gentle algebra containing a unique cycle C = su, where s is a clockwise oriented path with r overlapped zero-relations and u is a string with x clockwise oriented arrows and y counterclockwise oriented arrows, such that (Q, I) has at most one branch which is an equioriented linear quiver of length z oriented toward C, and the zero-relations of (Q, I) are those of s. Then, A is tilting-cotilting equivalent to Λ(r, r + 1 + x, y + z).

Proof: See [5], proposition 2.3, step h). □

We now give the proof of the theorem 1.2 stated in the introduction. Proof of the theorem 1.2:

i) ⇒ ii) From 4.2, A is of Dynkin type or D^{b}(A) ≅ D^{b}(Λ(r, n, m)) for some (r, n, m) ∈ Ω. If A is of Dynkin type, we have nothing to do. If D^{b}(A) ≅ D^{b}(Λ(r, n, m)), and assuming there exists a shod algebra B such that D^{b}(A) ≅ D^{b}(B), we will prove that n ⩾ 3r and m ⩾ 2r.

Notice that B is strictly shod. If B is not strictly shod, that is, if B is quasi-tilted, then A is piecewise hereditary. But this contradicts 4.3.

Then, B ≅ kQ/I where (Q, I) satisfies the conditions of 4.1.

Moreover, being B a strictly shod algebra, there exists a pair of overlapping zero-relations. From (b2), there exist zero-relations on C. From (b3), C = p_{1}w_{1}q_{1}^{−1}v_{1} ⋯ p_{n}w_{n}q_{n}^{−1}v_{n} where p_{i} and q_{i} are, each of them, a zero-relation or a path containing two overlapping zero-relations, w_{i} and v_{i} are strings, and all zero-relations on C are those of p_{i} and q_{i}, for 1 ⩽ i ⩽ n.

From 4.4, B is tilting-cotilting equivalent to a gentle algebra B_{1} = kQ_{1}/I_{1} with a unique cycle C_{1} such that all branches are equioriented linear quivers oriented toward C_{1}, C_{1} is obtained from C by adding arrows inside the strings of C, and all zerorelations of (Q_{1}, I_{1}) are the zero-relations of C. So, C_{1} = p_{1}w′_{1}q_{1}^{−1}v′_{1} ⋯ p_{n}w′_{n}q_{n}^{−1}v′_{n} with p_{i} and q_{i} the zero-relations or paths containing two overlapping zero-relations from C, for all 1 ⩽ i ⩽ n. Observe that the branches must be rooted in the midpoints of the zero-relations. Claim. The algebra B_{1} is tilting-cotilting equivalent to a gentle algebra B_{2} = kQ_{2}/I_{2} with a unique cycle C_{2} = s_{1}u_{1} ⋯ s_{r}u_{r} with s_{i} a clockwise oriented zero-relation and u_{i} a string with at least one arrow clockwise oriented and at least two arrows counterclockwise oriented, for 1 ⩽ i ⩽ r, all the branches are equioriented linear quivers oriented toward C_{2}, and s_{1}, ⋯, s_{r} are all the zero-relations in (Q_{2}, I_{2}). Applying 4.5, in a first step the consecutive zero-relations oriented in opposite directions of each pair (p_{i}, q_{i}) (with 1 ⩽ i ⩽ n) are removed. In a second step, the remaining consecutive zero-relations in opposite directions are removed. Then, we obtain r zerorelations, with the same orientation, left over on the cycle. From (b2), we have that r > 0. We can assume that these zero-relations are clockwise oriented. Each of them was a zero-relation of some pair (p_{i}, q_{i}) such that p_{i} = αβγ is a path containing two overlapping zero-relations and q_{i} = δϵ is a zero-relation. So, when 4.5 is applied to erase the zero-relations βγ and δϵ, we replace the following full subquiver of (Q_{1}, I_{1})

[figure: b_{t+l}; ⋮; b_{1}; ⋯; b_{t−1}; b_{t}; ϵ; e; δ; a_{0} = b_{0}; a_{1}; ⋯; a_{s−1}; a_{s}; γ; c_{0}; β; d; α; ⋮; ⋮; a_{s+k}; c_{p}]

with the vertices d, c_{0}, a_{s}, …, a_{0} = b_{0}, …, b_{t}, e on the cycle, s ⩾ 1, t ⩾ 1, k ⩾ 0, l ⩾ 0, p ⩾ 0, by a quiver of the form

[figure: y_{j}; ⋯; y_{1}; e; z; x_{i}; ⋯; x_{1}; c_{0}; β; d; α; ⋮; c_{p}]

with i ⩾ 1, j ⩾ 1. Hence, applying systematically 4.5, the required algebra B_{2} = kQ_{2}/I_{2} is obtained.

Claim. The algebra B_{2} is tilting-cotilting equivalent to a gentle algebra B_{3} = kQ_{3}/I_{3} with a unique cycle C_{3} = su, where s is a clockwise oriented path of length r + 1 and with r overlapping zero-relations, and u is a string with at least 2r − 1 arrows clockwise oriented and at least 2r arrows counterclockwise oriented, all the branches are equioriented linear quivers oriented toward C_{3}, and the zero-relations of (Q_{3}, I_{3}) are those of s. In order to obtain the algebra B_{3}, 4.6 must be applied r − 1 times. We prove, by induction, that after h applications of 4.6, with 0 ⩽ h ⩽ r − 1, we obtain a cycle C′_{h} = s′_{h+1}u′_{h+1}s_{h+2}u_{h+2} ⋯ s_{r}u_{r} where s′_{h+1} is a clockwise oriented path of length h + 2 and with h + 1 overlapping zero-relations, u′_{h+1} is a string with at least 2h + 1 arrows clockwise oriented and at least 2h + 2 arrows counterclockwise oriented, and for h + 2 ⩽ i ⩽ r, s_{i}, u_{i} are those defined on C_{2}. If h = 0, that is, before applying 4.6, C′_{0} is just C_{2} and the statement of the induction is true. Suppose that the statement is true for 0 ⩽ h − 1 ⩽ r − 2. Then, C′_{h−1} = s′_{h}u′_{h}s_{h+1}u_{h+1} ⋯ s_{r}u_{r} and C′_{h−1} has the following full subquiver

[figure: a; α; b; β; c = w_{0}; w_{1} ⋯ w_{l−1}; d = w_{l}; γ; e; δ; f]

where αβ is the last of the h overlapping zero-relations of s′_{h}, the subquiver between c and d is u′_{h} and γδ = s_{h+1}. Applying 4.6, the last subquiver is replaced by the following one

[figure: a; b; c; v_{l}; ⋯; v_{t+1}; v_{t}; v_{t−1}; ⋯; v_{1}; e; δ; f]

with t ⩾ 2h and l − t ⩾ 2h − 1. If there is a branch rooted to the cycle in the vertice e, that branch is inserted into the cycle. So, we obtain C′_{h} = s′_{h+1}u′_{h+1}s_{h+2}u_{h+2} ⋯ s_{r}u_{r}, where s′_{h+1} has h+1 overlapping zero-relations. The clockwise oriented arrows of u′_{h+1} are obtained from the l − t of u′_{h}, δ, and those of u_{h+1}. The counterclockwise oriented arrows of u′_{h+1} are obtained from the t of u′_{h}, and those of u_{h+1}. So, u′_{h+1} has at least (2h − 1) + 2 = 2h + 1 arrows clockwise oriented and 2h + 2 arrows counterclockwise oriented.

Therefore, after applying 4.6 r−1 times, we obtain C′_{r−1} = C_{3} = su with the desired conditions.

Finally, from 4.7, B_{3} is tilting-cotilting equivalent to Λ(r, n, m), with n ⩾ (r + 1) + (2r − 1) = 3r and m ⩾ 2r.

ii) ⇒ i) Hereditary algebras are shod, so we only have to prove that if n ⩾ 3r and m ⩾ 2r, then Λ(r, n, m) is tilting-cotilting equivalent to a shod algebra.

Let B = kQ/I be a gentle algebra such that (Q, I) consists of a unique cycle C = p_{1}q_{1}^{−1} ⋯ p_{r}q_{r}^{−1}u^{−1}v where p_{i} is a path containing two overlapping zero-relations and q_{i} is a zero-relation for 1 ⩽ i ⩽ r, u is a path of length m − 2r and v is a path of length n − 3r, and the zero-relations of (Q, I) are those of p_{i} and q_{i}. Clearly, B is a strictly shod algebra with discrete derived category not of Dynkin type.

Applying the differents steps as above, we get that B is tilting-cotilting equivalent to Λ(r, n, m). □

Example 4.8. (a) The algebra A = kQ/I given by the following bound quiver (Q, I) is strictly shod with discrete derived category not of Dynkin type. In fact, A is derived equivalent to Λ(1, 3, 2).

[figure: ∘; ∘; ∘; ∘; ∘]

(b) Let (Q, I) the following bound quiver:

\xymatrix{ & ∘ \ar[dr] \\ ∘ \ar[ur] \ar[rr] &  & ∘}

Then, A = kQ/I is derived equivalent to Λ(1, 2, 1), so it is not derived equivalent to a shod algebra.

Acknowledgements. This paper is part of both authors Ph.D. thesis done respectively in the University of Sherbrooke under the direction of Ibrahim Assem and François Huard and in the University of São Paulo under the direction of Flavio Coelho. The first author gratefully thanks professors Ibrahim Assem, François Huard and Shiping Liu for their helpful comments. The second author acknowledges the financial support by FAPESP during her Ph.D..

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