### Citations

155 |
Derived equivalences as derived functors
- Rickard
- 1991
(Show Context)
Citation Context ...nce classes [2, 3, 7], but in general it is not complete and thus further derived invariants are needed. A possible candidate is the Hochschild cohomology, which is well-known to be derived invariant =-=[10, 12]-=-, defined as HH∗(Λ) = Ext∗Λop⊗KΛ(Λ,Λ) for an algebra Λ over a field K. Indeed, in [6] it was used, together with other derived invariants, to classify up to derived equivalence the gentle algebras wit... |

89 |
cohomology of finite-dimensional algebras
- Happel, Hochschild
- 1989
(Show Context)
Citation Context ...nce classes [2, 3, 7], but in general it is not complete and thus further derived invariants are needed. A possible candidate is the Hochschild cohomology, which is well-known to be derived invariant =-=[10, 12]-=-, defined as HH∗(Λ) = Ext∗Λop⊗KΛ(Λ,Λ) for an algebra Λ over a field K. Indeed, in [6] it was used, together with other derived invariants, to classify up to derived equivalence the gentle algebras wit... |

27 |
The alternating syzygy behavior of monomial algebras
- Bardzell
- 1997
(Show Context)
Citation Context ...sions of the Hochschild cohomology groups in terms of the Avella-Alaminos-Geiss derived invariant, based on the projective resolution of a monomial algebra as a bimodule over itself given by Bardzell =-=[4]-=- (see also [14]), observing that gentle algebras are quadratic monomial algebras. In order to formulate our results more precisely, we encode the dimensions of the Hochschild cohomology groups of a fi... |

20 | Stable endomorphism algebras of modules over special biserial algebras
- Schröer, Zimmermann
- 2003
(Show Context)
Citation Context ...s defined combinatorially in terms of quivers with relations. They have remarkable homological properties; they are Gorenstein [9] and the class of gentle algebras is closed under derived equivalence =-=[13]-=-. It is a long standing problem to classify gentle algebras up to derived equivalence. To this end, one searches for derived invariants that will allow to distinguish the different derived equivalence... |

16 |
Mutation classes of Ãn-quivers and derived equivalence classification of cluster tilted algebras of type Ãn, Algebra & Number Theory (to appear
- Bastian
- 901
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Citation Context ...4(g − 1) + 2b+ d ) f3(z). Corollary 4. HH2(Λ) = 0 for any gentle algebra Λ arising from surface triangulation. The above theorem applies in particular to cluster-tilted algebras of affine type Ã. In =-=[5]-=-, the quivers of these algebras have been explicitly described and their derived equivalence classes were characterized in terms of four parameters. Corollary 5. Let Λ be a cluster-tilted algebra of t... |

10 |
Gentle algebras are Gorenstein, Representations of algebras and related topics
- Geiß, Reiten
- 2005
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Citation Context ...laminos and Geiss. Gentle algebras are certain finite-dimensional algebras defined combinatorially in terms of quivers with relations. They have remarkable homological properties; they are Gorenstein =-=[9]-=- and the class of gentle algebras is closed under derived equivalence [13]. It is a long standing problem to classify gentle algebras up to derived equivalence. To this end, one searches for derived i... |

9 | q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras, preprint 2005
- Holm
(Show Context)
Citation Context ...ts are needed. A possible candidate is the Hochschild cohomology, which is well-known to be derived invariant [10, 12], defined as HH∗(Λ) = Ext∗Λop⊗KΛ(Λ,Λ) for an algebra Λ over a field K. Indeed, in =-=[6]-=- it was used, together with other derived invariants, to classify up to derived equivalence the gentle algebras with two vertices and those with three vertices and zero Cartan determinant. However, in... |

8 | Algebras from surfaces without punctures
- David-Roesler, Schiffler
(Show Context)
Citation Context ... each connected component HOCHSCHILD COHOMOLOGY OF GENTLE ALGEBRAS 3 of ∂S. In [1] the authors associate to each ideal triangulation T of (S,M) a gentle algebra ΛT . The invariant φΛT was computed in =-=[8]-=-. Notations. Denote by g the genus of S, by b ≥ 1 the number of connected components of its boundary ∂S and by c1 the number of those components with exactly one marked point. For an ideal triangulati... |

7 |
Gabrielle Charbonneau-Jodoin, and Pierre-Guy Plamondon, Gentle algebras arising from surface triangulations, Algebra Number Theory 4
- Assem, Brüstle
- 2010
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Citation Context ... > 1. As an application of our results, we compute the Hochschild cohomology groups of the gentle algebras arising from surface triangulations introduced by Assem, Brüstle, Charbonneau and Plamondon =-=[1]-=-. A marked bordered oriented surface without punctures is a pair (S,M) where S is a compact, connected Riemann surface with non-empty boundary ∂S and M ⊂ ∂S is a finite set of marked points containing... |

6 | Derived classification of gentle algebras with two cycles
- Avella-Alaminos
(Show Context)
Citation Context ...ion φΛ : N 2 → N that can be effectively computed from the quiver with relations of a gentle algebra Λ. In some cases this invariant is fully capable of distinguishing the derived equivalence classes =-=[2, 3, 7]-=-, but in general it is not complete and thus further derived invariants are needed. A possible candidate is the Hochschild cohomology, which is well-known to be derived invariant [10, 12], defined as ... |

5 |
Avella-Alaminos and Christof Geiss, Combinatorial derived invariants for gentle algebras
- Diana
(Show Context)
Citation Context ...d equivalence. To this end, one searches for derived invariants that will allow to distinguish the different derived equivalence classes. One such invariant was developed by Avella-Alaminos and Geiss =-=[3]-=-. It takes the form of a function φΛ : N 2 → N that can be effectively computed from the quiver with relations of a gentle algebra Λ. In some cases this invariant is fully capable of distinguishing th... |

5 |
The Hochschild homology of truncated and quadratic monomial algebras
- Sköldberg
- 1999
(Show Context)
Citation Context ...ochschild cohomology groups in terms of the Avella-Alaminos-Geiss derived invariant, based on the projective resolution of a monomial algebra as a bimodule over itself given by Bardzell [4] (see also =-=[14]-=-), observing that gentle algebras are quadratic monomial algebras. In order to formulate our results more precisely, we encode the dimensions of the Hochschild cohomology groups of a finite-dimensiona... |

4 | Hochschild cohomology of the cluster-tilted algebras of finite representation type
- Ladkani
(Show Context)
Citation Context ...ments are sides of the same triangle in T ; d – the number of triangles of T such that two of their sides are boundary segments. In addition we set f3(z) = z + g3(z) (this agrees with our notation in =-=[11]-=-). Theorem 2. Let T be an ideal triangulation of a marked bordered oriented surface without punctures. Then hΛT (z) = c0 + (2g + b− 1 + c1)z + ( 4(g − 1) + 2b+ d ) f3(z). Corollary 4. HH2(Λ) = 0 for a... |

3 | On derived equivalence classification of gentle twocycle algebras
- Bobiński, Malicki
(Show Context)
Citation Context ...ion φΛ : N 2 → N that can be effectively computed from the quiver with relations of a gentle algebra Λ. In some cases this invariant is fully capable of distinguishing the derived equivalence classes =-=[2, 3, 7]-=-, but in general it is not complete and thus further derived invariants are needed. A possible candidate is the Hochschild cohomology, which is well-known to be derived invariant [10, 12], defined as ... |