Cyclic and Hochschild homology of 2-nilpotent algebras

We compute the cyclic homology of a 2-nilpotent algebra, using a reduced way of obtaining it for an algebra containing a separable subalgebra as a direct summand.Supported by the Swiss National Foundation, FNSRS.     

The Hochschild cohomology ring of a cyclic block

Suppose is a block of a group algebra with cyclic defect group. We calculate the Hochschild cohomology ring of , giving a complete set of generators and relations. We then show that if is the principal block, the canonical map from to the Hochschild cohomology ring of induces an isomorphism modulo radicals.     

Decompositions of the category of noncommutative sets and Hochschild and cyclic homology

In this note we show that the main results of the paper [PR] can be obtained as consequences of more general results concerning categories whose morphisms can be uniquely presented as compositions of morphisms of their two subcategories with the same objects. First we will prove these general results and then we will apply it to the case of finite noncommutative sets.     

Fibonacci代数的Hochschild同调群

设Λd是Fibonacci代数,基于对Bardzell极小投射双模分解的细致分析,用组合的方法清晰地计算了Fibonacci代数Λd的各阶Hochschild同调群的维数.     

Hochschild Homology of Singular Algebras

One of the main properties of Hochschild homology of the algebra of smooth functions on a smooth manifold is its local character. In this paper, we consider subalgebras of smooth functions which are significant for singular spaces such that simplicial complexes or cones over smooth manifolds. We compute their Hochschild homology and investigate the local character. Our computations show that, in opposition with the smooth case, the (local part of the) Hochschild homology is not always isomorphic to the corresponding de Rham complex of differential forms. The method we use is a slight modification of the localization procedure introduced in by Sullivan.     

Leibniz and Hochschild homology

We construct and study the map from Leibniz homology HL_?(𝔥) of an abelian extension 𝔥 of a simple real Lie algebra 𝔤 to the Hochschild homology HH_??1(U(𝔥)) of the universal envelopping algebra U(𝔥). To calculate some homology groups, we use the Hochschild-Serre spectral sequences and Pirashvili spectral sequences. The result shows what part of the non-commutative Leibniz theory is detected by classical Hochschild homology, which is of interest today in string theory.     

Hochschild Cohomology of the Algebra of Smooth Functions on the Torus

We calculate the Hochschild cohomology of the algebra of smooth functions on a finite-dimensional real torus with coefficients in the adjoint representation, generalizing the previously developed technique to the discrete case for this.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 3, pp. 435–452, September, 2005.     

Hochschild and cyclic (co)homology of superadditive categories

We define the Hochschild and cyclic (co)homology groups for superadditive categories and show that these (co)homology groups are graded Morita invariants. We also show that the Hochschild and cyclic homology are compatible with the tensor product of superadditive categories.     

Hochschild and Cyclic Homology of Ore Extensions and Some Examples of Quantum Algebras

For every Ore extension we construct a chain complex giving its Hochschild homology. As an application we compute the Hochschild and cyclic homology of an arbitrary multiparametric affine space and the Hochschild homology of the algebra of differential operators over this space, in the generic case.     

Local cyclic homology

Local (cyclic) homology of filtered associative algebras is defined and discussed. In the commutative case, it fits naturally in the setting of (formal) de Rham cohomology; we thus obtain a satisfactory complement to a result of Feigin and Tsygan. There is a spectral sequence from discrete (cyclic) homology of the associated graded algebra to local (cyclic) homology of the original filtered algebra; we exhibit manageable conditions on its convergence.     

Cyclic homology of algebras with one generator

We compute the cyclic homology of A = k[X]/X ⁿ for an arbitrary commutative ring k, and we apply this result to compute the cyclic homology of k[X]/f, when k is a field and f is an arbitrary polynomial.This work was presented at the IX-ELAM, Santigo de Chile, July 1988, and was partially supported by CONICET.The following pepole participated in this research: Jorge A. Guccione, juan José guccione, Maria Julia Redondo, Andrea Solotar, and Orlando E. Villamayor.     

Hochschild cohomology of special biserial algebras

Based on a four-term exact sequence,the formulae on the dimensions of the first and the second Hochschild cohomology groups of special biserial algebras with normed bases are obtained in terms of combinatorics.     

A relation between Hochschild homology and cohomology for Gorenstein rings

Let ``' stand for Hochschild (co)homology. In this note we show that for many rings there exists such that for an arbitrary -bimodule we have . Such a result may be viewed as an analog of Poincaré duality.

Combining this equality with a computation of Soergel allows one to compute the Hochschild homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo.

     

Koszul代数的Hochschild上同调环

基于Buchweitz等人对Koszul代数的Hochschild上同调环的乘法结构的细致分析,给出了Koszul代数的Hochschild上同调环的乘法本质上是平行路的毗连的一个充要条件,并由此重新证明了二次三角string代数的Hochschild上同调环的乘法是平凡的,从而改进了Bustamante的证明.     

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.     

Hochschild homology criteria for trivial algebra structures

We prove two similar results by quite different methods. The first one deals with augmented artinian algebras over a field: we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the Hochschild homology (or cyclic homology) groups. For the second result, let be a 1-connected finite CW-complex. We characterize the trivial algebra structure on the cohomology algebra of with coefficients in a fixed field in terms of the maximality of the Betti numbers of the free loop space.

     

一个Cluster-Tilted代数的Hochschild同调与循环同调

基于Furuya构造的一个Cluster-Tilted代数的极小投射双模分解,用组合的方法计算了Cluster-Tilted代数的Hochschild同调空间的维数与基.当基础域的特征为零时,也计算了代数的循环同调群的维数.     

Hochschild （Co）homology of Galois Coverings of Grassmann Algebras

Let ∧ be the Z2-Galois covering of the Grassmann algebra A over a field k of characteristic not equal to 2. In this paper, the dimensional formulae of Hochschild homology and cohomology groups of ∧ are calculated explicitly. And the cyclic homology of∧ can also be calculated when the underlying field is of characteristic zero. As a result, we prove that there is an isomorphism from i≥1 HH^i（∧） to i≥1 HH^i（∧）.     

Invariance of Hochschild and Cyclic (Co)homology of Superalgebras Under Graded Equivalences

It is shown that the Hochschild and cyclic (co)homology of superalgebras are graded equivalent invariants.