Hochschild cohomology: some applications in representation theory of algebras

Bustamante, Juan Carlos

doi:10.1007/s40863-016-0047-3

Cited by 1 publication

(3 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which is similar to the equality (7). Computing Ker [∇ ′ 2m ] N leads to the analogous result, which is similar to (8). Computations equal to those in the proof of Theorem 7.5 show that f = g = 0, except when A = k and B = k. If M = BA and N = AB, we infer HH 0 (Λ) = k × k. Note that, as vector spaces, End B−A BA = BA and End A−B AB = AB.…”

Section: Corollary 74 Let λ =supporting

confidence: 61%

“…which is similar to the equality (7). Computing Ker [∇ ′ 2m ] N leads to the analogous result, which is similar to (8). Computations equal to those in the proof of Theorem 7.5 show that f = g = 0, except when A = k and B = k. ⋄ Corollary 7.7 Let A and B be finite dimensional algebras, and let Λ = A AB BA B be the square algebra where the corner bimodules are free of rank one.…”

Section: Proofsupporting

confidence: 61%

“…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%

See 2 more Smart Citations

Hochschild cohomology of algebras arising from categories and from bounded quivers

Cibils

Lanzilotta

Marcos

et al. 2019

J. Noncommut. Geom.

View full text Add to dashboard Cite

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.

show abstract

Section: Corollary 74 Let λ =supporting

confidence: 61%

“…which is similar to the equality (7). Computing Ker [∇ ′ 2m ] N leads to the analogous result, which is similar to (8). Computations equal to those in the proof of Theorem 7.5 show that f = g = 0, except when A = k and B = k. ⋄ Corollary 7.7 Let A and B be finite dimensional algebras, and let Λ = A AB BA B be the square algebra where the corner bimodules are free of rank one.…”

Section: Proofsupporting

confidence: 61%