On the First Hochschild Cohomology Group of a Cluster-Tilted Algebra

Assem, Ibrahim; Redondo, Maria Julia; Schiffler, Ralf

doi:10.1007/s10468-015-9551-x

Cited by 11 publications

(38 citation statements)

References 47 publications

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“…This generalises to arbitrary relation extensions the main results of each of [4,3,5], furnishing at the same time a homological interpretation of these results. As a nice consequence of this theorem, we get that, if C is tilted, so that B is cluster-tilted, then the Hochschild projection morphism ϕ * : HH * (B) → HH * (C) is a surjective morphism of algebras, see Theorem 5.8.…”

Section: Introductionsupporting

confidence: 66%

“…Of course, there exist relation extensions which are not cluster-tilted algebras. The first Hochschild cohomology groups of a tilted algebra C and the corresponding cluster-tilted algebra B were compared by means of a linear map ϕ : HH 1 (B) → HH 1 (C), see [4,3,5]. In each of these papers, it appeared that ϕ is surjective, but the proof in each case was long and combinatorial.The present paper arose from an attempt to produce a purely homological proof of the above mentioned results.…”

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confidence: 99%

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Hochschild cohomology of relation extension algebras

Assem

Gatica

Schiffler

et al. 2016

Journal of Pure and Applied Algebra

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Abstract. Let B be the split extension of a finite dimensional algebra C by a C-C-bimodule E. We define a morphism of associative graded algebras ϕ * : HH * (B) → HH * (C) from the Hochschild cohomology of B to that of C, extending similar constructions for the first cohomology groups made and studied by Assem, Bustamante, Igusa, Redondo and Schiffler.In the case of a trivial extension B = C ⋉E, we give necessary and sufficient conditions for each ϕ n to be surjective. We prove the surjectivity of ϕ 1 for a class of trivial extensions that includes relation extensions and hence cluster-tilted algebras. Finally, we study the kernel of ϕ 1 for any trivial extension, and give a more precise description of this kernel in the case of relation extensions.

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Section: Introductionsupporting

confidence: 66%

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confidence: 99%

Hochschild cohomology of relation extension algebras

Assem

Gatica

Schiffler

et al. 2016

Journal of Pure and Applied Algebra

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“…Recall that a cleft singular extension algebra (see [24, p. 284]) is a k-algebra Λ with a decomposition Λ = A ⊕ M, where A is a subalgebra and M is a two-sided ideal of Λ verifying M 2 = 0. These algebras are also called trivial extensions, see for instance [2] and [5,4] where the natural generalization for abelian categories is considered.…”

Section: Remark 212mentioning

confidence: 99%

“…It has been defined in 1945 by Hochschild in [21], it provides the theory of infinitesimal deformations and the deformation theory on the variety of algebras of a fixed dimension, see [16]. Moreover it is a Gerstenhaber algebra, and it is related with the representation theory of the given algebra, see for example [2,8]. Rephrasing the introduction of [23], observe that Hochschild cohomology is not functorial, there is no natural way to relate the Hochschild cohomology of an algebra to that of its quotient algebras or of its subalgebras.…”

Section: Introductionmentioning

confidence: 99%

Hochschild cohomology of algebras arising from categories and from bounded quivers

Cibils

Lanzilotta

Marcos

et al. 2019

J. Noncommut. Geom.

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The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver Q, with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that we obtain via the study of trajectories of the quiver. We introduce cohomology along paths, and we compute it under suitable Tor vanishing hypotheses. The cup product on Hochschild cohomology enables us to describe the connecting homomorphism of the long exact sequence.Algebras arising from a linear category where the quiver is the round trip one, provide square matrix algebras which have two algebras on the diagonal and two bimodules on the corners. If the bimodules are projective, we show that five-terms exact sequences arise. If the bimodules are free of rank one, we provide a complete computation of the Hochschild cohomology. On the other hand, if the corner bimodules are projective without producing new cycles, Hochschild cohomology in large enough degrees is that of the product of the algebras on the diagonal.As a by-product, we obtain some families of bound quiver algebras which are of infinite global dimension, and have Hochschild cohomology zero in large enough degrees.

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“…It is natural to ask how the module categories of C and B are related, and several results in this direction have been obtained, see for example [4,5,6,13,15,25]. The Hochschild cohomology of the algebras C and B has been compared in [7,9,10,31].…”

Section: Introductionmentioning

confidence: 99%

Induced and coinduced modules over cluster-tilted algebras

Schiffler

Serhiyenko

2017

Journal of Algebra

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We propose a new approach to study the relation between the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B = C E. This new approach consists of using the induction functor − ⊗ C B as well as the coinduction functor D(B ⊗ C D−). We give an explicit construction of injective resolutions of projective B-modules, and as a consequence, we obtain a new proof of the 1-Gorenstein property for cluster-tilted algebras. We show that DE is a partial tilting and a τ -rigid C-module and that the induced module DE ⊗ C B is a partial tilting and a τ -rigid B-module. Furthermore, if C = End A T for a tilting module T over a hereditary algebra A, we compare the induction and coinduction functors to the Buan-Marsh-Reiten functor Hom C A (T, −) from the cluster-category of A to the module category of B. We also study the question which B-modules are actually induced or coinduced from a module over a tilted algebra.

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On the First Hochschild Cohomology Group of a Cluster-Tilted Algebra

Cited by 11 publications

References 47 publications

Hochschild cohomology of relation extension algebras

Hochschild cohomology of relation extension algebras

Hochschild cohomology of algebras arising from categories and from bounded quivers

Induced and coinduced modules over cluster-tilted algebras

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