We show that string algebras are `homologically tame' in the following sense: First, the syzygies of arbitrary representations of a finite dimensional string algebra $Λ$ are direct sums of cyclic representations, and the left finitistic dimensions, both little and big, of $Λ$ can be computed from a finite set of cyclic left ideals contained in the Jacobson radical. Second, our main result shows that the functorial finiteness status of the full subcategory $\Cal P$ consisting of the finitely generated left $Λ$-modules of finite projective dimension is completely determined by a finite number of, possibly infinite dimensional, string modules -- one for each simple $Λ$-module -- which are algorithmically constructible from quiver and relations of $Λ$. Namely, $\Cal P$ is contravariantly finite in $Λ$-mod precisely when all of these string modules are finite dimensional, in which case they coincide with the minimal $\Cal P$-approximations of the corresponding simple modules. Yet, even when $\Cal P$ fails to be contravariantly finite, these `characteristic' string modules encode, in an accessible format, all desirable homological information about $Λ$-mod.
The minimal projective bimodule resolutions of the exterior algebras are explicitly constructed. They are applied to calculate the Hochschild (co)homology of the exterior algebras. Thus the cyclic homology of the exterior algebras can be calculated in case the underlying field is of characteristic zero. Moreover, the Hochschild cohomology rings of the exterior algebras are determined by generators and relations.
We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally we also study the fundamental groups of the bound quivers involved in t…
In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative algebra $A$ and as applications, we study extensions, deformations and extensibility of finite order deformations of $A$. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.
We describe how the Hochschild (co)homology of a bound quiver algebra changes when adding or deleting arrows to the quiver. The main tools are relative Hochschild (co)homology, the Jacobi-Zariski long exact sequence obtained by A. Kaygun and a one step relative projective resolution of a tensor algebra.
In this paper, we compute the Hochschild and cyclic homologies of the Auslander algebras of the Taft algebras. We also describe the first Chern character for the Taft algebras and for their Auslander algebras.
In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type $B, C$ and $D$ over an associative unital $k$-algebra with $k$ being a field of characteristic $0$.
We revisit a recent paper of Fialowski and Iohara. They compute the homology of the Lie algebra $\mathfrak{gl}(\infty,R)$ for $R$ an associative unital algebra over a field of characteristic zero. We explain how to obtain essentially the same results by a completely different method.
We consider algebras over a field $k$ of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces \[ H(\gl(A))\iso\wedge (HC(A)[-1]) \] between the Lie algebra homology of matrices and the free graded commutative algebra on the cyclic homology of the $k$-algebra $A$, shifted down one degree. For unital algebras this isomorphism is a classical result obtained by Loday and Quillen and independently by Tsygan. For $H$-unital algebras, it is known to hold too, as is that the proof follows from results of Hanlon's. However, to our knowledge, the proof is not immedia…
We calculate the additive and multiplicative structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types T. We also compute the calculus structure which is formed by the Hochschild homology/cohomology pair.
We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups.
We compute the Hochschild cohomology and homology of a class of quantum exterior algebras, with coefficients in twisted bimodules. As a result we obtain several interesting examples of the homological behavior of these algebras.
We give a detailed calculation of the Hochschild and cyclic homology of the algebra $\CIc(G)$ of locally constant, compactly supported functions on a reductive p-adic group G. We use these calculations to extend to arbitrary elements the definition the higher orbital integrals introduced in \cite{Blanc-Brylinski} for regular semisimple elements. Then we extend to higher orbital integrals some results of Shalika. We also investigate the effect of the ``induction morphism'' on Hochschild homology.
Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz \cite{F} and was further developed in the work of S. Ault \cite{Au1, Au2}. In this paper, we show that, for algebras defined over a field of characteristic $0$, the symmetric homology theory is naturally equivalent to the (one-dimensional) representation homology theory introduced by the authors (jointly with G. Khachatryan) in \cite{BKR}. Using known results on representa…
We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky-Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky-Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold $M$ of dimension $k$, the Goresky-Hingston algebra on reduced Hochschild homology induces an algebra structure of degree $k-1$ on $\bar{H}^*(LM;\mathbb{Q})$, the reduced rational cohomology of the free loop space of $M$. As a con…
Let A be the central extension of the preprojective algebra of an ADE quiver introduced by P. Etingof and E. Rains in math/0503393. The paper math/0606403 computes the structure of the zeroth Hochschild (co)homology of A. We generalize the results of math/0606403 by calculating the additive structure of all the Hochschild homology and cohomology groups of A and the cyclic homology of A, and to describe the universal deformation of A. Namely, we show that the (co)homology is periodic with period 4, and compute the first four (co)homology groups in each case.
We calculate the additive structure (together with the grading) of the Hochschild homology and cohomology and the cyclic homology of preprojective algebras of types ADE.
We determine the periodic cyclic homology of the Iwahori-Hecke algebras $\Hecke_q$, for $q \in \CC^*$ not a ``proper root of unity.'' (In this paper, by a {\em proper root of unity} we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a ``weakly spectrum preserving'' morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazh…
