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-Drinfeld condition. We also compute this particular homology for the quantum deformation of an arbitrary semi-simple Lie algebra and the Hopf algebra of foliations of codimension $N$ with stable but non-anti-Yetter-Drinfeld coefficients.
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…
We present an approach to cyclic homology of A_{\infty} algebras. Our main technical tool is the concept of X-complex due to Cuntz and Quillen. This, in particular, enables us to compute the periodic cyclic homology of an A_{\infty} algebra in terms of the periodic cyclic homology of its homology.
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…
We define Getzler's Gauss-Manin connection in cyclic homology at the level of chains and outline some relations of this construction to noncommutative calculus.
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.
We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Ger…
We view the space of cotraces in the structural coalgebra of a principal coaction as a noncommutative counterpart of the classical Cartan model. Then we define the cyclic-homology Chern-Weil homomorphism by extending the Chern-Galois character from the characters of finite-dimensional comodules to arbitrary cotraces. To reduce the cyclic-homology Chern-Weil homomorphism to a tautological natural transformation, we replace the unital coaction-invariant subalgebra by its certain natural H-unital nilpotent extension (row extension), and prove that their cyclic-homology groups are isomorphic. In t…
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-…
We prove in this paper that the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of connected and simply connected Lie group, are isomorphic to the periodic cyclic homology of the quantized algebras of functions on coadjoint orbits of compact maximal subgroups, without localization. Some noncompact quantum groups and algebras were constructed and their irreducible representations were classified in recent works of Do Ngoc Diep and Nguyen Viet Hai, and Do Duc Hanh by using deformation quantization. In this paper we compute their K-groups, periodic cyclic homol…
The goal of this article is to explicitly compute the Hochschild (co)homology of the Fomin-Kirillov algebra on three generators over a field of characteristic different from 2 and 3. We also obtain the cyclic (co)homology of the Fomin-Kirillov algebra in case the characteristic of the field is zero. Moreover, we compute the algebra structure of the Hochschild cohomology.
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…
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…
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…
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.
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.
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.
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…
The spectral theory of the Karoubi operator due to Cuntz and Quillen is extended to general mixed (duchain) complexes, that is, chain complexes which are simultaneously cochain complexes. Connes' coboundary map B can be viewed as a perturbation of the noncommutative De Rham differential d by a polynomial in the Karoubi operator. The homological impact of such perturbations is expressed in terms of two short exact sequences.
We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.
Using a smooth version of the Connes--Thom isomorphism in Grensing's bivariant K-theory for locally convex algebras, we prove an equivariant version of the Connes--Thom isomorphism in periodic cyclic homology. As an application, we prove that periodic cyclic homology is invariant with respect to equivariant strict deformation quantization.
We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss-Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.
In this paper we show that both variants of the Hopf cyclic homology has excision under some natural homological conditions on the objects and the coefficient module.
We study the curved Koszul duality theory for associative algebras presented by quadratic-linear-constant (QLC) relations. As an application, we investigate the cyclic (co)homology of a QLC algebra and its Koszul dual curved DG algebra, and extend a result due to Feigin and Tsygan. We also study a commutative/Lie analog of this curved Koszul duality theory.
We calculate the bivariant local cyclic cohomology of the Banach convolution algebra of summable functions on a word-hyperbolic group. Our result implies that the Banach algebraic assembly map in local cyclic homology is an isomorphism for such a group.
We define and study equivariant analytic and local cyclic homology for smooth actions of totally disconnected groups on bornological algebras. Our approach contains equivariant entire cyclic cohomology in the sense of Klimek, Kondracki and Lesniewski as a special case and provides an equivariant extension of the local cyclic theory developped by Puschnigg. As a main result we construct a multiplicative Chern-Connes character for equivariant KK-theory with values in equivariant local cyclic homology.
We prove that continuous Hochschild and cyclic homology satisfy excision for extensions of nuclear H-unital Frechet algebras and use this to compute them for the algebra of Whitney functions on an arbitrary closed subset of a smooth manifold. Using a similar excision result for periodic cyclic homology, we also compute the periodic cyclic homology of algebras of smooth functions and Whitney functions on closed subsets of smooth manifolds.
Let V be a complete discrete valuation ring with uniformiser p. We introduce an invariant of Banach V-algebras called local cyclic homology. This invariant is related to analytic cyclic homology for complete, bornologically torsionfree V-algebras. It is shown that local cyclic homology only depends on the reduction mod p of a Banach V-algebra and that it is homotopy invariant, matricially stable, and excisive.
We analyze the relationship between Bott periodicity in topological $K$-theory and the natural periodicity of cyclic homology. This is a basis for understanding the multiplicativity, in odd dimensions, of a bivariant Chern-Connes character from $K$-theory to cyclic theory.
The purpose of this article is to embed the string topology bracket developed by Chas-Sullivan and Menichi on negative cyclic cohomology groups as well as the dual bracket found by de Thanhoffer de Voelcsey-Van den Bergh on negative cyclic homology groups into the global picture of a noncommutative differential (or Cartan) calculus up to homotopy on the (co)cyclic bicomplex in general, in case a certain Poincare' duality is given. For negative cyclic cohomology, this in particular leads to a Batalin-Vilkovisky algebra structure on the underlying Hochschild cohomology. In the special case in wh…
Based on the ideas of Cuntz and Quillen, we give a simple construction of cyclic homology of unital algebras in terms of the noncommutative de Rham complex and a certain differential similar to the equivariant de Rham differential. We describe the Connes exact sequence in this setting. We define equivariant Deligne cohomology and construct, for each n > 0, a natural map from cyclic homology of an algebra to the GL_n-equivariant Deligne cohomology of the variety of n-dimensional representations of that algebra. The bridge between cyclic homology and equivariant Deligne cohomology is provided by…
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.
We study graded connected algebras over a field of characteristic zero and give an explicit formula for the cyclic homology of a tensor algebra. By means of a slightly new definition of David Anick's notion "strongly free" we are able to prove that cyclic homology of an algebra of global dimension two is zero in homological degree greater than one and is zero also in homological degree equal to one in case the relations are monomials. We give also explicit formulas for the cyclic homology of a tensor algebra modulo one symmetric quadratic form.
We define the notion of a relative matrad and realize the free relative matrad as a free H_\infty-bimodule structure on cellular chains of bimultiplihedra JJ={JJ_{n,m} = JJ_{m,n}}. We define a morphism G:A => B of A_\infty-bialgebras as a bimodule over H_\infty and prove that the homology of every A_\infty-bialgebra over a commutative ring with unity admits an induced A_\infty-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of A_\infty-bialgebras and identify the A_\infty-bialgebra structure of H_*(ΩΣX; Q). For each n>1, we construct a space X_n and identify an i…
The multiplicative group of a global field acts on its adele ring by multiplication. We consider the crossed product algebra of the resulting action on the space of Schwartz functions on the adele ring and compute its Hochschild, cyclic and periodic cyclic homology. We also compute the topological K-theory of the C*-algebra crossed product.
In the present paper we investigate the noncommutative geometry of a class of algebras, called the Hom-associative algebras, whose associativity is twisted by a homomorphism. We define the Hochschild, cyclic, and periodic cyclic homology and cohomology for this class of algebras generalizing these theories from the associative to the Hom-associative setting.
Assuming local one-sided units exist, I give an elementary proof of Wodzicki excision for cyclic homology. The proof is also constructive and provides an explicit inverse excision map. As far as I know, the latter is new.
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…
Certain cocycles constructed by Connes are characters of $p$-summable Fredholm modules. In this article, we establish some consequences of the universal properties which these characters enjoy. Our main technical result is that the entire cyclic cohomology of the p-th Schatten ideal L^p (respectively, homology) is independent of p and isomorphic to the entire cyclic cohomology (respectively, homology) of the ideal of trace class operators L^1.
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.
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}…
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…
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.
We discuss the cyclic homology of crossed product algebras from the Cuntz-Quillen point of view. The periodic cyclic homology of a crossed product algebra $A\rtimes G$ is described in terms of the $G$-action on periodic cyclic bicomplexes of crossed products of $A$ by the cyclic subgroups of $G$ .
Following an old suggestion of M. Kontsevich, and inspired by recent work of A. Beilinson and B. Bhatt, we introduce a new version of periodic cyclic homology for DG agebras and DG categories. We call it co-periodic cyclic homology. It is always torsion, so that it vanishes in char 0. However, we show that co-periodic cyclic homology is derived-Morita invariant, and that it coincides with the usual periodic cyclic homology for smooth cohomologically bounded DG algebras over a torsion ring. For DG categories over a field of odd positive characteristic, we also establish a non-commutative genera…
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)].
Relying of properties of the inductive tensor product, we construct cyclic type homology theories for certain nuclear algebras. In this context we establish continuity theorems. We compute the periodic cyclic homology of the Schwartz algebra of p-adic GL(n) in terms of compactly supported de Rham cohomology of the tempered dual of GL(n).
We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.
Algebraic K-theory has applications in a broad range of mathematical subjects, from number theory to functional analysis. It is also fiendishly hard to calculate. Presently there are two main inroads: motivic and cyclic homology. I've been asked to present an overview of the applications of topological cyclic homology to algebraic K-theory "from a historical perspective". The timeline spans from the very early days of algebraic K-theory to the present, starting with ideas in the seventies around the "tangent space" of algebraic K-theory all the way to the current state of affair where we see a…
We describe the derived functor DRep_V(A) of the affine representation scheme Rep_V(A), parametrizing the representations of an associative k-algebra A on a finite-dimensional vector space V. We construct the characteristic maps Tr_V(A)_n: HC_n(A) \to H_n[DRep_V(A)], extending the canonical trace Tr_V(A): HC_0(A) \to k[Rep_V(A)] to the higher cyclic homology of the algebra A, and describe a related derived version of the representation functor introduced recently by M. Van den Bergh. We study various operations on the homology of DRep_V(A) induced by known operations on cyclic and Hochschild h…
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…
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 Hochschild homology of the ring $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$ has been known and calculated several ways. This paper uses those calculations to calculate cyclic, negative cyclic, and periodic cyclic homology of $k[x_1,x_2,\ldots,x_d]/(x_1,x_2,\ldots,x_d)^2$ over $k$.
Cuntz and Quillen have shown that for algebras over a field $k$ with $char(k)=0$, periodic cyclic homology may be regarded, in some sense, as the derived functor of (non-commutative) de Rham (co-)homology. The purpose of this paper is to formalize this derived functor analogy. We show that the localization ${Def}^{-1}\Cal{PA}$ of the category $\Cal{PA}$ of countable pro-algebras at the class of (infinitesimal) deformations exists (in any characteristic) (Theorem 3.2) and that, in characteristic zero, periodic cyclic homology is the derived functor of de Rham cohomology with respect to this loc…
Topological periodic cyclic homology (i.e., $\mathbb{T}$-Tate fixed points of $THH$) has the structure of a strong symmetric monoidal functor of smooth and proper dg categories over a perfect field of finite characteristic.
In the present paper the cyclic homology functor from the category of $A_\infty$-algebras over any commutative unital ring $K$ to the category of graded $K$-modules is constructed. Further, it is showed that this functor sends homotopy equivalences of $A_\infty$-algebras into isomorphisms of graded modules. As a corollary, it is obtained that the cyclic homology of an $A_\infty$-algebra over any field is isomorphic to the cyclic homology of the $A_\infty$-algebra of homologies for the source $A_\infty$-algebra.
This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of Bökstedt periodicity that closely resembles Bökstedt's original unpublished proof. We explain the extension of Bökstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic gr…
