No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Abstract
There is a close connection between Hochschild homology groups of a k-algebra and cycles of the Gabriel quiver associated to the k-algebra. In this paper, based on the minimal projective bimodule resolution of a self-injective Koszul four-point algebra constructed by Furuya, we calculate the dimensions of Hochschild homology spaces of the algebra by using combinatorial methods, and give a k-basis of every Hochschild homology space in terms of cycles. Moreover, we obtain the dimensions of cyclic homology groups of the algebra when the base field k is of zero characteristic.
ResearchGate has not been able to resolve any citations for this publication.
In 1989 Happel conjectured that for a finite-dimensional algebra A over an algebraically closed field k, \gl A< \infty if and only if \hch A < \infty. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to Happel's conjecture. They found a family of pathological algebra for which \gl A_q = \infty but \hch A_q=2. These algebras are pathological in many aspects, however their Hochschild homology behaviors are not pathological any more, indeed one has \hh A_q = \infty=\gl A_q. This suggests to pose a seemingly more reasonable conjecture by replacing Hochschild cohomology dimension in Happel's conjecture with Hochschild homology dimension: \gl A < \infty if and only if \hh A < \infty if and only if \hh A = 0. The conjecture holds for commutative algebras and monomial algebras. In case A is a truncated quiver algebras these conditions are equivalent to the quiver of A has no oriented cycles. Moreover, an algorithm for computing the Hochschild homology of any monomial algebra is provided. Thus the cyclic homology of any monomial algebra can be read off in case the underlying field is characteristic 0.
In this paper, we construct an explicit minimal projective bimodule
resolution of a self-injective special biserial algebra ()
whose Grothendieck group is of rank 4. As a main result, we determine the
dimension of the Hochschild cohomology group of
for , completely. Moreover we give a presentation of the Hochschild
cohomology ring modulo nilpotence of
by generators and relations in the case where T=0.
Let Λ = kQ/I be a finite-dimensional Nakayama algebra, where Q is an Euclidean diagram
for some n with cyclic orientation, and I is an admissible ideal generated by a single monomial relation. In this note we determine explicitly all the Hochschild homology and cohomology groups of Λ based on a detailed description of the Bardzell complex. Moreover, the cyclic homology of Λ can be calculated in the case that the underlying field is of characteristic zero.
Hochschild cohomology of finite dimensional algebras
- D Happel
Happel D., Hochschild cohomology of finite dimensional algebras, Lecture Notes in Math., 1989, 1404:
108-126.