Derived Equivalence Classification of the Gentle Two-Cycle Algebras

We complete a derived equivalence classification of the gentle two-cycle algebras initiated in earlier papers by Avella-Alaminos and Bobinski–Malicki.


Introduction and the Main Result
Throughout the paper k denotes a fixed algebraically closed field. For a (finite-dimensional basic connected) algebra one considers its (bounded) derived category D b ( ), which has a structure of a triangulated category. Derived categories seem to be a proper setup to do homological algebra. Derived categories appearing in representation theory of algebras have connections with derived categories studied in algebraic geometry (see for example [11,24,31]). Moreover, these categories serve as a source for constructions of categorifications of cluster algebras (this line of research was initiated by a fundamental paper by Buan, Marsh, Reineke, Reiten and Todorov [20]) and have links to theoretical physics (including famous Orlov's theorem [36]).
Algebras and are said to be derived equivalent if the categories D b ( ) and D b ( ) are triangle equivalent. A study of derived categories (in particular derived equivalences) in the representation theory of algebras was initiated by papers of Happel [ 29] and motivated by tilting theory, and is now an important direction of research (see for example [3,10,13,15,16,18,19,21,25,30,32,34,35,37,38]).
Gentle algebras were introduced by Assem and Skowroński [6] in their study of the algebras derived equivalent to the hereditary algebras of Euclidean typeÃ. Namely, they have proved that the algebras derived equivalent to the hereditary algebras of Euclidean typeÃ are precisely the gentle one-cycle algebras which satisfy the clock condition. On the other hand, the algebras derived equivalent to the hereditary algebras of Dynkin type A are precisely the gentle tree algebras [4]. Moreover, the gentle one-cycle algebras which do not satisfy the clock condition are precisely the discrete derived algebras, which are not locally finite [42]. The above motivates study of a derived equivalence classification for the gentle algebras. One should note that the class of gentle algebras is closed with respect to the derived equivalence [40].
By the above results the derived equivalence classes of the gentle algebras with at most one-cycle are known and they are distinguished by the invariant of Avella-Alaminos and Geiss [8]. It is natural to study as the next step a derived equivalence classification of the gentle two-cycle algebras. Here a gentle algebra is called two-cycle if the number of edges in the Gabriel quiver of exceeds by one the number of vertices in this quiver. Before formulating the main result of the paper we define some families of gentle two-cycle algebras.
Theorem A The above defined algebras are representatives of the derived equivalence classes of the gentle two-cycle algebras. More precisely, (1) if is a gentle two-cycle algebra, then is derived equivalent to one of the above defined algebras, and (2) the above defined algebras are pairwise not derived equivalent.
Parts of Theorem A have been already proved in [17] (see also [7]). More precisely, the following claims have been proved there: (1) If is a gentle two-cycle algebra, then is derived equivalent to an algebra from one of the families 0 , 1 and 2 . (2) The algebras from different families are not derived equivalent.
Thus in order to prove Theorem A, we have to show the following.
Partial versions of Theorem B have been obtained independently by Amiot [1] and Kalck [33]. In particular, Amiot has proved this result in the case when r's are "small" relative to p's (see Proposition 2.3 for a precise statement) by refining her earlier joint results with Grimeland on surface algebras [2]. The new ingredient of the paper is Corollary 3.2, which says that if (p, r ) and (p, r ) are derived equivalent, then (p +1, r ) and (p +1, r ) are derived equivalent. Using this and induction we reduce the situation to the setup of Amiot's result.
We note that one can replace derived equivalence by tilting-cotilting equivalence (see for example [6]) in Theorems A and B. Indeed, obviously if algebras are not derived equivalent, then they are not tilting-cotilting equivalent. On the other hand, every derived equivalence obtained in [17] is realized via a tilting-cotilting equivalence.
The paper consists of two sections. In Section 2 we recall necessary tools, including the invariant of Avella-Alaminos and Geiss, Auslander-Reiten quivers, (generalized APR) reflections, and behavior of derived equivalences under one-point coextensions. Next in Section 3 we prove Theorem B. In the paper we use a formalism of bound quivers introduced by Gabriel [23]. For related background see for example [5].
The author would like to thank the referee for the remarks, which helped improve the paper significantly. The author was supported by the National Science Center Grant No. 2015/17/B/ST1/01731.

Quivers and Their Representations
By a quiver we mean a set 0 of vertices and a set 1 of arrows together with two maps s = s , t = t : 1 → 0 , which assign to α ∈ 1 the starting vertex sα and the terminating vertex tα, respectively. We assume that all considered quivers are locally finite, i.e. for each x ∈ 0 there is only a finite number of α ∈ 1 such that either sα = x or tα = x. A quiver is called finite if 0 (and, consequently, also 1 ) is a finite set. For technical reasons we assume that if is a quiver, then 0 = ∅ and has no isolated vertices, i.e. there is no x ∈ 0 such that sα = x = tα for each α ∈ 1 . In particular, Let be a quiver. If l ∈ N + , then by a path in of length l we mean every sequence In the above situation we put sσ := sα l and tσ := tα 1 . Moreover, we call α 1 and α l the terminating and the starting arrows of σ , respectively. Observe that each α ∈ is a path in of length 1. Moreover, for each x ∈ 0 we introduce the path 1 x in of length 0 such that s1 x := x =: t1 x . We denote the length of a path σ by (σ ). If σ and σ are two paths in such that sσ = tσ , then we define the composition σ σ of σ and σ , which is a path in of length (σ ) + (σ ), in the obvious way (in particular, σ 1 sσ = σ = 1 tσ σ for each path σ ). A path σ 0 is called a subpath of a path σ , if there exist paths σ and σ such that σ = σ σ 0 σ . By a (monomial) bound quiver we mean a pair = ( , R) consisting of a finite quiver and a set R of paths in , such that: (1) (ρ) > 1 for each ρ ∈ R, and (2) there exists n ∈ N + such that every path σ in with (σ ) = n has a subpath which belongs to R.
is a bound quiver, then by a path in we mean a path in which does not have a subpath from R. A path σ in is said to be maximal in if σ is not a subpath of a longer path in . The lack of isolated vertices in implies that (σ ) > 0 for each maximal path σ in . By a representation V of a bound quiver = ( , R) we mean a collection of finitedimensional vector spaces V x , x ∈ 0 , and linear maps V α : (1) R consists of paths of length 2, (2) for each x ∈ 0 there are at most two α ∈ 1 such that sα = x and at most two α ∈ 1 such that tα = x, (3) for each α ∈ 1 there is at most one α ∈ 1 such that sα = tα and α α ∈ R, and at most one α ∈ 1 such that tα = sα and αα ∈ R, (4) for each α ∈ 1 there is at most one α ∈ 1 such that sα = tα and α α ∈ R, and at most one α ∈ 1 such that tα = sα and αα ∈ R.
Let = ( , R) be a gentle bound quiver. Note that by condition (1) above a path In particular, every path of length at most 1 is an antipath. Again we call an antipath ω maximal if ω is not a subpath of a longer antipath in .

The Invariant of Avella-Alaminos and Geiss
Throughout this subsection = ( , R) is a fixed gentle bound quiver.
By a permitted thread in we mean either a maximal path in or 1 x , for x ∈ 0 , such that there is at most one arrow α with sα = x, there is at most one arrow β with tβ = x, and if such α and β exist, then αβ ∈ R. Similarly, by a forbidden thread we mean either a maximal antipath in or 1 x , for x ∈ 0 , such that there is at most one arrow α with sα = x, there is at most one arrow β with tβ = x, and if such α and β exist, then αβ ∈ R.
Denote by P and F the sets of the permitted and forbidden threads in , respectively. We define bijections 1 : P → F and 2 : F → P. First, if σ is a maximal path in , then we put 1 (σ ) := ω, where ω is the unique forbidden thread such that tω = tσ and either (ω) = 0 or (ω) > 0 and the terminating arrows of σ and ω differ. If 1 x , for x ∈ 0 , is a permitted thread, there are two cases to consider. If there is an arrow β such that tβ = x (note that such β is uniquely determined), then 1 (1 x ) is the (unique) forbidden thread whose terminating arrow is β. Otherwise we put 1 (1 x ) := 1 x . We define 2 dually. Namely, if ω is a maximal antipath, then 2 (ω) := σ , where σ is the permitted thread such that sσ = sω and either (σ ) = 0 or (σ ) > 0 and the starting arrows of ω and σ differ. Now, let x ∈ 0 and 1 x be a forbidden thread. If there is α ∈ 1 such that sα = x, then 2 (1 x ) is the permitted thread whose starting arrow is α. Otherwise, 2 (1 x ) := 1 x . Finally, we put := 1 2 : F → F.
Let F be the set of arrows in which are not subpaths of any maximal antipath in (i.e. every antipath containing α can be extended to a longer antipath). For every α ∈ F there exists uniquely determined α ∈ F such that αα ∈ R. We put (α) := α . In this way we get a bijection : F → F . In other words, F is the set of arrows which lie on oriented cycles with full relations. Moreover, two arrows in F belong to the same orbit with respect to the action of if and only if they lie on the same oriented cycle with full relations. The following result seems to be well-known, however we could not find a reference for it, hence we include its proof for completeness. Proof For a vertex x of we denote by S x and P x the simple and the projective representations of at x, respectively. For α ∈ 1 we denote by P α the corresponding map P tα → P sα .
Assume first F = ∅ and fix x ∈ 0 . Assume there are exactly two arrows α and β starting at x. Let α n · · · α 1 and β m · · · β 1 be the maximal antipaths, whose starting arrows are α and β, respectively (in particular, α 1 = α and β 1 = β) -such antipaths exist, since F = ∅. Then · · · → P tα 2 ⊕ P tβ 2 is a minimal projective presentation of S x , so pdim S x = max{n, m} < ∞. If there is only one arrow starting at x, then we have a degenerate version of the above. Finally, if there is no arrow starting at x, then S x = P x . Now assume F = ∅, choose α ∈ F , and put α i := −i (α), i ∈ N. Then · · · → P tα 1 is a minimal projective presentation of Coker P α , so pdim Coker P α = ∞.
Let Avella-Alaminos and Geiss have proved [8] that φ is a derived invariant, i.e. if and are derived equivalent gentle bound quivers, then φ = φ .
For a function φ : N 2 → N we put φ := (n,m)∈N 2 φ(n, m). If is a gentle bound quiver, then φ equals |F / | + |F / |. We will need the following observation.  (2) of the definition of a gentle bound quiver this means that for each x ∈ 0 there are exactly two arrows starting at x. Consequently, condition (4) of the definition implies that for each α ∈ 1 there exists α ∈ 1 such that sα = tα and α α ∈ R. Thus, there exist paths in of arbitrary length, which contradicts condition (2) of the definition of a bound quiver.

Let
= ( , R) be a gentle bound quiver. One defines the Auslander-Reiten quiver (D b ( )) of D b ( ) in the following way: the vertices of (D b ( )) are (representatives of) the isomorphism classes of the indecomposable complexes in D b ( ) and the number of arrows between vertices X and Y equals the dimension of the space of irreducible maps between X and Y .
Since the gentle bound quivers are Gorenstein (see [27]), the Auslander-Reiten translation τ (see [30]) is an autoequivalence on the subcategory of perfect complexes (i.e. complexes, which are quasi-isomorphic to bounded complexes of projective representations). In particular, if gldim < ∞, then τ is an automorphism of D b ( ).
An indecomposable complex X ∈ D b ( ) is called boundary if X is perfect and there is only one arrow in (D b ( )) terminating at X. Equivalently, X is perfect and in the Auslander-Reiten triangle (see [30]) terminating at X the middle term is indecomposable.
The invariant of Avella-Alaminos and Geiss describes the action of the shift on the components of (D b ( )) containing boundary complexes. We will use the following excerpt from their results in [ (1, 1), then |C/ | = |X |. In particular, if |X | = 1 and X and Y are boundary complexes, which do not lie in homogeneous tubes, then there exists p ∈ Z such that p X and Y belong to the same component. If φ = 1, we have even more.

Lemma 2.3 Let be a gentle bound quiver such that
Proof Assume first that is derived equivalent to a hereditary algebra of Dynkin type A, i.e. is a gentle tree. In this case (D b ( )) is ZA n for some n ∈ N + (see [29,Section I.5]), hence the boundary complexes form two orbits with respect to the action of τ , which is an autoequivalence of D b ( ), since gldim < ∞ by Lemma 2.2. Moreover, interchanges these orbits, hence the claim follows in this case.
If is one-cycle gentle bound quiver, then φ  (1, 1). In particular, there are no homogeneous tubes in (D b ( )). Consequently, by the discussion above we know there exists p ∈ Z such that p X and Y belong to the same component of (D b ( )). Moreover, [26,Theorem 2.6] implies that p X and Y belong to the same τorbit, i.e. there exists q ∈ Z such that τ q p X = Y . Finally, gldim < ∞ by Lemma 2.2, hence τ is an autoequivalence of D b ( ), and the claim follows.
If σ is a path in , then we have the corresponding (string) representation M(σ ) (see for example [22]). We have the following observation.

Lemma 2.4 Let be a gentle bound quiver. If σ is a maximal path in , then M(σ ) (viewed as a complex concentrated in degree 0) is a boundary complex in D b ( ).
Proof In the terminology of [14] (see also [12]) a projective presentation of M(σ ) is given by the complex which corresponds to the antipath −1 2 (σ ). In particular, this implies that M(σ ) is a perfect complex in D b ( ). Moreover, if one uses results of [14] in order to calculate the Auslander-Reiten triangle terminating at M(σ ), then one gets that its middle term is indecomposable. Alternatively, one may use the Happel functor [28,29] and wellknown formulas (see for example [22,41]) for calculating the Auslander-Reiten triangles in the stable category of the category of representations of the repetitive categoryˆ of . We leave details to the reader.
We formulate the following consequence.

Corollary 2.5 Let and be derived equivalent gentle bound quivers such that
. If σ and σ are maximal paths in and , respectively, then there exists a derived equivalence F :

One-point Coextensions
If is a bound quiver and M is a representation of , then one defines a bound quiver [M] , called the one-point coextension of by M (see for example [9]). However, usually Proof Exercise.
The following is a special version of the dual of Barot and Lenzing's [9, Theorem 1].

Proposition 2.7
Let σ and σ be maximal paths in gentle bound quivers and , respectively. If there exists a triangle equivalence F :
The following pictures, where the relations are indicated by dots, illustrate the situation: if locally (in a neighbourhood of x) has the form then locally has the form In the above situation we say that is obtained from by applying the (generalized APR) reflection at x. The bound quiver is derived equivalent to (see [17,Section 1]). We will need the following application of this operation, which is a special version of [17, Lemma 1.1]. Lemma 2.9 Let = ( , R) be a gentle bound quiver such that is of the form for p ∈ N + . Assume that α i−1 α i ∈ R and α i α i+1 ∈ R for some i ∈ [2, p − 1]. Then is derived equivalent to the gentle bound quiver := ( , R ), where Proof We obtain from by applying the reflection at tα i , hence and are derived equivalent by the discussion above.
In the above situation we say that is obtained from by a shift of the relation α i α i+1 .
We have the following consequence of Lemma 3.1.
An important role in our proof is played by the following result due to Amiot [1,Corollary 4.4]. Proposition 3.3 Let q ≥ 3 and −1 ≤ r , r ≤ q 2 −1. If r = r , then the algebras 0 (q, r ) and 0 (q, r ) are not derived equivalent. Now we are ready to prove Theorem B.
Proof of Theorem B Let p , p ∈ N, r ∈ [−1, p − 1] and r ∈ [−1, p − 1] be such that (p , r ) = (1, −1) = (p , r ). Obviously, 0 (p , r ) and 0 (p , r ) are not derived equivalent if p = p (e.g. they have different numbers of vertices). Thus assume that p = p and denote this common value by p. Choose q ≥ p such that r , r ≤ q 2 −1. If 0 (p, r ) and 0 (p, r ) are derived equivalent, then Corollary 3.2 implies that 0 (q, r ) and 0 (q, r ) are derived equivalent as well. Consequently, r = r according to Proposition 3.3 and the claim follows.
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