The almost split triangles for perfect complexes over gentle algebras

Abstract

a b s t r a c tThroughout the paper k denotes a fixed field. All vector spaces and linear maps are k-vector spaces and k-linear maps, respectively. By Z, N, and N + , we denote the sets of integers, nonnegative integers, and positive integers, respectively. For i, j ∈ Z, [i, j] IntroductionGiven an abelian category A one defines its bounded derived category D b (A) [24] (of bounded complexes of objects of A) having a structure of a triangulated category, which is an important homological invariant of A. In particular, given a finite dimensional algebra A one may study the bounded derived category D b (mod A) of the category mod A of finite dimensional A-modules, which one shortly calls the derived category of A and denotes D b (A). Since the observation of Happel [15] (generalized by Cline et al. [11]), which states that derived category is invariant under tilting process, an importance of derived categories in the representation theory of finite dimensional algebras became clear. This observation was supported by results connecting derived categories of finite dimensional algebras with derived categories of coherent sheaves over projective schemes [6,12]. Since that time a lot of results concerning derived categories of finite dimensional algebras were obtained (see for example [1,8,9,13,20]). In particular, Rickard [21] developed the Morita theory for derived categories of finite dimensional algebras. One of the consequences is that the derived categories of two finite dimensional algebras are equivalent as triangulated categories if and only if the subcategories of perfect complexes are equivalent as triangulated categories. Recall that if A is a finite dimensional algebra, then the subcategory of D b (A) formed by perfect complexes can be identified with the bounded homotopy category K b (proj A) of (bounded complexes of) projective A-modules.A class of finite dimensional algebras whose derived categories attract a lot of interest is the class of gentle algebras introduced by Assem and Skowroński [4]. An important feature of this class of algebras is that it is closed under derived equivalence, i.e. if A is a gentle algebra and D b (A) is equivalent as a triangulated category to D b (B) for a finite dimensional algebra B, then B is also gentle [23]. Next, this class of algebras appears naturally in many classification problems. Namely, the tree gentle algebras are precisely the piecewise hereditary algebras of type A [2] (i.e. the algebras derived equivalent to hereditary algebras of type A). Further, if A is a derived discrete algebra, then either A is piecewise hereditary of Dynkin type or A is a one-cycle gentle algebra which does not satisfy the clock condition [25]. Moreover, the one-cycle gentle algebras coincide with the piecewise hereditary algebras of typeà [4].If A is a gentle algebra, then it is possible to investigate D b (A) by means of the stable category mod of the module category mod over the repetitive algebra [22] (which is no longer finite dimensional) and the Happel functor [16] D b (...

Mapping cones in the bounded derived category of a gentle algebra

“…From now on let Λ be a gentle algebra and D b (Λ) be its bounded derived category with shift functor Σ. The indecomposable objects in D b (Λ) have been classified [8] in terms of string combinatorics: namely they are given in terms of homotopy strings and homotopy bands, the terminology originating in [9]; see also [15] for a similar approach in the context of nodal algebras. The corresponding indecomposable complexes are called string complexes and band complexes, respectively.…”

Indecomposables with smaller cohomological length in the derived category of gentle algebras

“…Thus, there is also no gaps in the sequence of cohomological ranges of indecomposable determined by a generalized band w. Thus we divide the proof of Theorem 2 into three theorems as follows and their proofs depend strongly on the description of the indecomposables in the bounded derived category of gentle algebras due to Bekkert and Merklen [2]. We should recall more notations for a gentle algebras A = kQ/I from [2,3], some of which are slightly different for our convenience. For any p ∈ Pa ≥1 , there is a unique maximal pathp = pp starting with p. Besides the pathp, there may be another maximal path, sayp, beginning with the starting point s(p) of p. If this is not the case, we write l(p) = 0.…”

Derived Equivalence Classification of the Gentle Two-Cycle Algebras

“…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 ( ).…”

“…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 .…”