Excision in Hopf cyclic homology

Integral Excision for K-Theory

If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace from K(A) to TC(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision. The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the T-Tate spectrum of topological Hochschild homology, where T is the circle group

Hopf-cyclic homology with contramodule coefficients

A new class of coefficients for the Hopf-cyclic homology of module algebras and coalgebras is introduced. These coefficients, termed stable anti-Yetter-Drinfeld contramodules, are both modules and contramodules of a Hopf algebra that satisfy certain compatibility conditions.

Cyclic Homology of Hopf Comodule Algebras and Hopf Module Coalgebras

In this paper we construct a cylindrical module $A \natural \mathcal{H}$ for an $\mathcal{H}$-comodule algebra $A$, where the antipode of the Hopf algebra $\mathcal{H}$ is bijective. We show that the cyclic module associated to the diagonal of $A \natural \mathcal{H}$ is isomorphic with the cyclic module of the crossed product algebra $A \rtimes \mathcal{H}$. This enables us to derive a spectral sequence for the cyclic homology of the crossed product algebra. We also construct a cocylindrical module for Hopf module coalgebras and establish a similar spectral sequence to compute the cyclic coho…

Hopf-dihedral (co)homology and $L$-theory

We develop an appropriate dihedral extension of the Connes-Moscovici characteristic map for Hopf *-algebras. We then observe that one can use this extension together with the dihedral Chern character to detect non-trivial $L$-theory classes of a *-algebra that carry a Hopf symmetry over a Hopf *-algebra. Using our machinery we detect a previously unknown $L$-class of the standard Podleś sphere.

Jacobi-Zariski Exact Sequence for Hochschild Homology and Cyclic (Co)Homology

We prove that for an inclusion of unital associative but not necessarily commutative algebras $B\subseteq A$ we have long exact sequences in Hochschild homology and cyclic (co)homology akin to the Jacobi-Zariski sequence in André-Quillen homology, provided that the quotient $B$-bimodule $A/B$ is flat. We also prove that for an arbitrary r-flat morphism $f:B\to A$ with an H-unital kernel, one can express the Wodzicki excision sequence and the corresponding Jacobi-Zariski sequence in Hochschild homology and cyclic (co)homology as a single long exact sequence.

ample群oid作用的等变周期性循环同调

Equivariant periodic cyclic homology for ample groupoids

本文研究了ample群oid作用的双变等变周期性循环同调，证明了其在双变量中的同调不变性、稳定性与消去性，并证明了proper群oid作用的Green-Julg定理类比。

Hopf Algebra Equivariant Cyclic Homology and Cyclic Homology of Crossed Product Algebras

We introduce the cylindrical module $A \natural \mathcal{H}$, where $\mathcal{H}$ is a Hopf algebra and $A$ is a Hopf module algebra over $\mathcal{H}$. We show that there exists an isomorphism between $\mathsf{C}_{\bullet}(A^{op} \rtimes \mathcal{H}^{cop})$ the cyclic module of the crossed product algebra $A^{op} \rtimes \mathcal{H}^{cop} $, and $Δ(A \natural \mathcal{H}) $, the cyclic module related to the diagonal of $A \natural \mathcal{H}$. If $S$, the antipode of $\mathcal{H}$, is invertible it follows that $\mathsf{C}_{\bullet}(A \rtimes \mathcal{H}) \simeq Δ(A^{op} \natural \mathcal{H}…

Cyclic homology of braided Hopf crossed products

Let k be a field, A a unitary associative k-algebra and V a k-vector space endowed with a distinguished element 1_V. We obtain a mixed complex, simpler that the canonical one, that gives the Hochschild, cyclic, negative and periodic homology of a crossed product E:=A#_f V, in the sense of Brzezinski. We actually work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A that satisfies suitable hypothesis and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. Then, when E is a cleft braide…

Cyclic Homologies of Crossed Modules of Algebras

The Hochschild and (cotriple) cyclic homologies of crossed modules of (not-necessarily-unital) associative algebras are investigated. Wodzicki's excision theorem is extended for inclusion crossed modules in the category of crossed modules of algebras. The cyclic and cotriple cyclic homologies of crossed modules are compared in terms of long exact homology sequence, generalising the relative cyclic homology exact sequence.

Cup Coproducts in Hopf Cyclic Cohomology

We define cup coproducts for Hopf cyclic cohomology of Hopf algebras and for its dual theory. We show that for universal enveloping algebras and group algebras our coproduct recovers the standard coproducts on Lie algebra homology and group homology, respectively.

Equivariant periodic cyclic homology

We define and study equivariant periodic cyclic homology for locally compact groups. This can be viewed as a noncommutative generalization of equivariant de Rham cohomology. Although the construction resembles the Cuntz-Quillen approach to ordinary cyclic homology, a completely new feature in the equivariant setting is the fact that the basic ingredient in the theory is not a complex in the usual sense. As a consequence, in the equivariant context only the periodic cyclic theory can be defined in complete generality. Our definition recovers particular cases studied previously by various author…

(Co)homology of Crossed Products by Weak Hopf Algebras

We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle $A\times_ρ^f H$, of a weak module algebra $A$ by a weak Hopf algebra $H$. This complex is provided with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in \cite{CGG}. When $f$ takes its values in a separable subalgebra of $A$ that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.

Equivariant cyclic homology for quantum groups

We define equivariant periodic cyclic homology for bornological quantum groups. Generalizing corresponding results from the group case, we show that the theory is homotopy invariant, stable and satisfies excision in both variables. Along the way we prove Radfords formula for the antipode of a bornological quantum group. Moreover we discuss anti-Yetter-Drinfeld modules and establish an analogue of the Takesaki-Takai duality theorem in the setting of bornological quantum groups.

Hopf--Hochschild (co)homology of module algebras

We define a version of Hochschild homology and cohomology suitable for a class of algebras admitting compatible actions of bialgebras, called module algebras. We show this (co)homology, called Hopf--Hochschild (co)homology, can also be defined as a derived functor on the category of representations of a crossed product algebra. We investigate the relationship of our theory with Hopf cyclic cohomology and also prove Morita invariance of the Hopf--Hochschild (co)homology.

The lazy homology of a Hopf algebra

To any Hopf algebra H we associate two commutative Hopf algebras, which we call the first and second lazy homology Hopf algebras of H. These algebras are related to the lazy cohomology groups based on the so-called lazy cocycles of H by universal coefficient theorems. When H is a group algebra, then its lazy homology can be expressed in terms of the 1- and 2-homology of the group. When H is a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero, then its first lazy homology is the Hopf algebra of the universal abelian grading group of the category of corepresenta…

Cyclic homology of Hopf Galois extensions and Hopf algebras

Let H be a Hopf algebra. By definition a modular crossed H-module is a vector space M on which H acts and coacts in a compatible way. To every modular crossed H-module M we associate a cyclic object Z(H,M). The cyclic homology of Z(H,M) extends the usual cyclic homology of the algebra structure of H, and the relative cyclic homology of an H-Galois extension. For a Hopf subalgebra K we compute, under some assumptions, the cyclic homology of an induced modular crossed module. As a direct application of this computation, we describe the relative cyclic homology of strongly graded algebras. In par…

The cyclic homology of monogenic extensions in the noncommutative setting

We study the Hochschild and cyclic homologies of noncommutative monogenic extensions. As an aplication we compute the Hochschild and cyclic homologies of the rank~1 Hopf algebras introduced by L. Krop and D. Radford in [Finite dimensional Hopf algebras of rank 1 in characteristic 0, Journal of Algebra 302, no. 1, 214-230} (2006)].

On the Cyclic Homology of Hopf Crossed Products

We consider Hopf crossed products of the the type $A#_σ\mathcal{H}$, where $\mathcal{H}$ is a cocommutative Hopf algebra, $A$ is an $\mathcal{H}$-module algebra and $σ$ is a "numerical" convolution invertible 2-cocycle on $\mathcal{H}$. we give an spectral sequence that converges to the cyclic homology of $A#_σ\mathcal{H}$ and identify the $E^1$ and $E^2$ terms of the spectral sequence.

Morita base change in Hopf-cyclic (co)homology

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that…

The universal Hopf cyclic theory

We define a Hopf cyclic (co)homology theory in an arbitrary symmetric strict monoidal category. Thus we unify all different types of Hopf cyclic (co)homologies under one single universal theory. We recover Hopf cyclic (co)homology of module algebras, comodule algebras and module coalgebras along with Hopf-Hochschild (co)homology of module algebras. This universal theory also allows us to define the missing Hopf cyclic theory for comodule coalgebras.

驯顺子扩张的Hopf拓扑不变量

Hopfological invariants for tame subextensions

该文针对域上有限维Hopf代数的扩张驯顺性问题，通过计算Hopf拓扑同调与Hopf循环同调，结合迹映射满射性与循环模性质，刻画了驯顺子扩张的同调行为，为Hopf代数扩张的驯顺性研究提供了拓扑不变量视角。

Hochschild (co)homology of Hopf crossed products

Let A a k-algebra, H a Hopf algebra, E = A#H a general crossed product and M an E-bimodule. We obtain a complex simpler than the canonical one, giving the Hochschild homology of E with coefficients in M. This complex is eqquiped with a natural filtration. We prove that the associated spectral sequence coincides with that obtained by either, the Hochschild-Serre direct method or the Cartan-Leray-Grothendieck method. We also get similar results for the cohomology.

Steinberg同调上的Quillen与sharbly Hopf代数结构一致

The Quillen and sharbly Hopf algebra structures on Steinberg homology coincide

本文证明Ash-Miller-Patzt与Brown-Chan-Galatius-Payne独立构造的Steinberg同调上的双分次交换Hopf代数结构相同，二者均由Steinberg模上的join与抛物限制运算诱导。

A New Cyclic Module for Hopf Algebras

We define a new cyclic module, dual to the Connes-Moscovici cyclic module, for Hopf algebras, and give a characteristric map for the coaction of Hopf algebras. We also compute the resulting cyclic homology for cocommutative Hopf algebras, and some quantum groups.

Extensions and renormalized traces

It has been shown by Nistor that given any extension of associative algebras over C, the connecting morphism in periodic cyclic homology is compatible, under the Chern-Connes character, with the index morphism in lower algebraic K-theory. The proof relies on the abstract properties of cyclic theory, essentially excision, which does not provide explicit formulas a priori. Avoiding the use of excision, we explain in this article how to get explicit formulas in a wide range of situations. The method is connected to the renormalization procedure introduced in our previous work on the bivariant Che…

Inertial Hopf-cyclic homology

We construct, study, and apply a characteristic map from the relative periodic cyclic homology of the quotient map for a group action to the periodic Hopf-cyclic homology with coefficients associated with inertia of the action. This result admits, and in fact, comes from, its noncommutative-geometric, or quantized, counterpart. The crucial ingredient is the construction of the appropriate quantization of the cyclic nerve of the action groupoid, the cyclic object related to inertia, as the Connes-cyclic dual of a Hopf-cyclic object with coefficients in some stable anti-Yetter--Drinfeld module q…

Bialgebra Cyclic Homology with Coefficients, Part I

The cyclic (co)homology of Hopf algebras is defined by Connes and Moscovici [math.DG/9806109] and later extended by Khalkhali et.al [math.KT/0306288] to admit stable anti-Yetter-Drinfeld coefficient module/comodules. In this paper we will show that one can further extend the cyclic homology of Hopf algebras with coefficients non-trivially. The new homology, called the bialgebra cyclic homology, admits stable coefficient module/comodules, dropping the anti-Yetter-Drinfeld condition. This fact allows the new homology to use bialgebras, not just Hopf algebras. We will also give computations for b…

Non-Archimedean analytic cyclic homology

Let $V$ be a complete discrete valuation ring with fraction field $F$ of characteristic zero and with residue field $\mathbb{F}$. We introduce analytic cyclic homology of complete torsion-free bornological algebras over $V$. We prove that it is homotopy invariant, stable, invariant under certain nilpotent extensions, and satisfies excision. We use these properties to compute it for tensor products with dagger completions of Leavitt path algebras. If $R$ is a smooth commutative $V$-algebra of relative dimension $1$, then we identify its analytic cyclic homology with Berthelot's rigid cohomology…

Coactions on Hochschild Homology of Hopf-Galois Extensions and Their Coinvariants

Let A be an H-Galois extension of B. If M is a Hopf bimodule then HH.(A,M), the Hochschild homology of A with coefficients in M, is a right comodule over the coalgebra C:=H/[H,H]. Given an injective left C-comodule V, we denote the cotensor product of M and V by N. Our aim is to investigate the relationship between the cotensor product of HH.(A,M) and V, on the one hand, and HH.(B,N) on the other hand. The roots of this problem can be found in Lorenz's work on the descent of Hochschild homology of centrally Galois extensions, where HH.(A)^G, the subspace of invariant cycles with respect to the…

Cyclic structures in algebraic (co)homology theories

This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-…

Hopfological algebra for infinite dimensional Hopf algebras

We consider "Hopfological" techniques as in \cite{Ko} but for infinite dimensional Hopf algebras, under the assumption of being co-Frobenius. In particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose corepresentations category is d.g. vector spaces. Motivated by this example we define the "Homology functor" (we prove it is homological) for any co-Frobenius algebra, with coefficients in $H$-comodules, that recover usual homology of a complex when $H=k[{\mathbb Z}]\#k[x]/x^2$. Another easy example of co-Frobenius Hopf algebra gives the category of "mixed complexes" and we see (by c…

On the Hochschild (co)homology of Quantum Homogeneous Spaces

The recent result of Brown and Zhang establishing Poincare duality in the Hochschild (co)homology of a large class of Hopf algebras is extended to right coideal subalgebras over which the Hopf algebra is faithfully flat, and applied to the standard Podles quantum 2-sphere.

The tangent complex of K-theory

We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $λ$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\infty \to K$. The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory à …

Hopf-cyclic homology and cohomology with coefficients

Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopf-algebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule alge…

An algorithm for low dimensional group homology

Given a finitely presented group $G$, Hopf's formula expresses the second integral homology of $G$ in terms of generators and relators. We give an algorithm that exploits Hopf's formula to estimate $H_2(G;k)$, with coefficients in a finite field k, and give examples using $G=SL_2$ over specific rings of integers. These examples are related to a conjecture of Quillen.

Cyclic Cohomology of Hopf algebras and Hopf Algebroids

We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.

Higher colimits, derived functors and homology

A theory of higher colimits over categories of free presentations is developed. It is shown that different homology functors such as Hoshcshild and cyclic homology of algebras over a field of characteristic zero, simplicial derived functors, and group homology can be obtained as higher colimits of simply defined functors. Connes' exact sequence linking Hochschild and cyclic homology was obtained using this approach as a corollary of a simple short exact sequence. As an application of the developed theory it is shown that the third reduced $K$-functor can be defined as the colimit of the second…

Hochschild homology of Hopf algebras and free Yetter-Drinfeld resolutions of the counit

We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter-Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to get a finite free resolution of the counit of $\mathcal B(E)$, the Hopf algebra of the bilinear form associated to an invertible matrix $E$, generalizing an ealier construction of Collins, Hartel and Thom in the orthogonal case $E=I_n$. It follows tha…

Invariant Cyclic Homology

We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple $(A,\mathcal{H},M)$ consisting of a Hopf algebra $\mathcal{H}$, an $\mathcal{H}$-comodule algebra $A$, an $\mathcal{H}$-module $M$, and a compatible grouplike element $σ$ in $\mathcal{H}$, we define the cyclic module of invariant chains on $A$ with coefficients in $M$ and call its cyclic homology the invariant cyclic homology of $A$ with coefficients in $M$. We also develop a dual theory for coalgebras. Examples include cyclic cohomology of Hopf algebras defined by Connes-Moscovici and its dual th…

Homology of L_{\infty}-Algebras and Cyclic Homology

A classical result of Loday-Quillen and Tsygan states that the Lie algebra homology of the algebra of stable matrices over an associative algebra is isomorphic, as a Hopf algebra, to the exterior algebra of the cyclic homology of the algebra. In this paper we develop the necessary tools needed to extend extend this result to the category of L_{\infty} algebras.

Bialgebra Cyclic Homology with Coefficients, Part II

This is the second part of the article [math.KT/0408094]. In the first paper, we used the underlying coalgebra structure to develop a cyclic theory. In this paper we define a dual theory by using the algebra structure. We define a cyclic homology theory for triples $(X,B,Y)$ where $B$ is a bialgebra, $X$ is a $B$--comodule algebra and $Y$ is just a stable $B$--module/comodule. We recover the main result of [math.KT/0310088] that these homology theories are dual to each other in the appropriate sense when the bialgebra is a Hopf algebra and the stable coefficient module satisfies anti-Yetter-Dr…

Cyclic Homology and Quantum Orbits

A natural isomorphism between the cyclic object computing the relative cyclic homology of a homogeneous quotient-coalgebra-Galois extension, and the cyclic object computing the cyclic homology of a Galois coalgebra with SAYD coefficients is presented. The isomorphism can be viewed as the cyclic-homological counterpart of the Takeuchi-Galois correspondence between the left coideal subalgebras and the quotient right module coalgebras of a Hopf algebra. A spectral sequence generalizing the classical computation of Hochschild homology of a Hopf algebra to the case of arbitrary homogeneous quotient…

双函子、下降范畴与广义Hopf模

Duplicial functors, descent categories and generalized Hopf modules

本文研究双函子与下降范畴，提出左χ-余代数可用模和余模描述的条件，并探讨广义Hopf模定理的应用。

Generators for Group Homology and a Vanishing Conjecture

Letting $G=F/R$ be a finitely-presented group, Hopf's formula expresses the second integral homology of $G$ in terms of $F$ and $R$. Expanding on previous work, we explain how to find generators of $H_2(G;\mathbb{F}_p)$. The context of the problem, which is related to a conjecture of Quillen, is presented, as well as example calculations.

The cyclic theory of Hopf algebroids

We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and étale groupoids.

Products in Hopf-Cyclic Cohomology

We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant cyclic (co)homology, and the Yoneda interpretation of Ext-groups. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category o…

$GL_n(F_2)$的同调中周期族的构造

Periodic families in the homology of $GL_n(F_2)$

本文通过构造$GL_n(F_2)$同调中的周期族，证明了斜率$\frac{2}{3}$稳定性的最优性，并改进了相关群的稳定性结果。

Cyclic Homology of Strong Smash Product Algebras

For any strong smash product algebra $A\#_{_R}B$ of two algebras $A$ and $B$ with a bijective morphism $R$ mapping from $B\ot A$ to $A\ot B$, we construct a cylindrical module $A\natural B$ whose diagonal cyclic module $Δ_{\bullet}(A\natural B)$ is graphically proven to be isomorphic to $C_{\bullet}(A\#_{_R}B)$ the cyclic module of the algebra. A spectral sequence is established to converge to the cyclic homology of $A\#_{_R}B$. Examples are provided to show how our results work. Particularly, the cyclic homology of the Pareigis' Hopf algebra is obtained in the way.

Cyclic homology of crossed products

We obtain a mixed complex, simpler that the canonical one, given the Hochschild, cyclic, negative and periodic homology of a crossed product E=A#fH, where H is an arbitrary Hopf algebra and f is a convolution invertible cocycle with values in A. Actually, we work in the more general context of relative cyclic homology. Specifically, we consider a subalgebra K of A which is stable under the action of H, and we find a mixed complex computing the Hochschild, cyclic, negative and periodic homology of E relative to K. As an application we obtain two spectral sequences converging to the cyclic homol…

Enveloping algebras of some quantum Lie algebras

We define a family of Hopf algebra objects, $H$, in the braided category of $\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $ψ^2_{H\otimes H}=id_{H\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $ψ^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analogous to the non-super and the super case, the well known relation between the periodic Hopf cyclic cohomology …

Hopf 代数、Steinberg 模与 $SL_n(\mathbb Z)$ 和 $GL_n(\mathbb Z)$ 的不稳定上同调

Hopf algebras, Steinberg modules, and the unstable cohomology of $SL_n(\mathbb Z)$ and $GL_n(\mathbb Z)$

本文证明整数一般线性群以 Steinberg 模为系数的同调直和构成交换 Hopf 代数，并由此构造 $SL_n(\mathbb Z)$ 不稳定上同调类的无限族。

Comparison between algebraic and topological K-theory of locally convex algebras

This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and annou…

A pairing between super Lie-Rinehart and periodic cyclic homology

We consider a pairing producing various cyclic Hochschild cocycles, which led Alain Connes to cyclic cohomology. We are interested in geometrical meaning and homological properties of this pairing. We define a non-trivial pairing between the homology of a Lie-Rinehart (super-)algebra with coefficients in some partial traces and relative periodic cyclic homology. This pairing generalizes the index formula for summable Fredholm modules, the Connes-Kubo formula for the Hall conductivity and the formula computing the K0-group of a smooth noncommutative torus. It also produces new homological invar…