We propose a category which can serve as the category of coefficients for the cyclic homology HC_*(A) of an associative algebra A over a field k. The construction is categorical in nature, and essentially uses only the tensor category A-bimod of A-bimodules; objects of our category are A-bimodules with an additional structure. We also generalize the construction to a more general tensor k-linear tensor category C instead of A-bimod. We add some remarks about Getzler's version of the Gauss-Manin connection for the periodic cyclic homology HP_*(A).
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 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.
We use the Beilinson $t$-structure on filtered complexes and the Hochschild-Kostant-Rosenberg theorem to construct filtrations on the negative cyclic and periodic cyclic homologies of a scheme $X$ with graded pieces given by the Hodge-completion of the derived de Rham cohomology of $X$. Such filtrations have previously been constructed by Loday in characteristic zero and by Bhatt-Morrow-Scholze for $p$-complete negative cyclic and periodic cyclic homology in the quasisyntomic case.
In this paper we prove that cyclic homology, topological cyclic homology, and algebraic $K$-theory satisfy a pro Mayer--Vietoris property with respect to abstract blow-up squares of varieties, in both zero and finite characteristic. This may be interpreted as the well-definedness of $K$-theory with compact support.
We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies $cdh$-descent. Combining that result with some computations of the $cdh$-cohomology of the sheaf of regular functions, we verify a conjecture of Weibel predicting the vanishing of algebraic $K$-theory of a scheme in degrees less than minus the dimension of the scheme, for schemes essentially of finite type over a field of characteristic zero.
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.
Let $V$ be a complete discrete valuation ring with residue field $k$ and with fraction field $K$ of characteristic 0. We clarify the analysis behind the Monsky--Washnitzer completion of a commutative $V$-algebra using spectral radius estimates for bounded subsets in complete bornological $V$-algebras. This leads us to a functorial chain complex for commutative $k$-algebras that computes Berthelot's rigid cohomology. This chain complex is related to the periodic cyclic homology of certain complete bornological $V$-algebras.
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 show that for a smooth, projective variety X defined over a number field K, cyclic homology with coefficients in the ring of infinite adeles of K, provides the right theory to obtain, using the lambda-operations, Serre's archimedean local factors of the complex L-function of X as regularized determinants.
Making use of the recent theory of noncommutative motives, we prove that every additive invariant satisfies Galois descent. Examples include mixed complexes, Hochschild homology, cyclic homology, periodic cyclic homology, negative cyclic homology, connective algebraic K-theory, mod-l algebraic K-theory, nonconnective algebraic K-theory, homotopy K-theory, topological Hochschild homology, and topological cyclic homology.
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 prove that for a homologically smooth and proper DG algebra over a field of characteristic 0, the Hodge-to-de Rham spectral sequence degenerates. This has been conjectured by M. Kontsevich and Y. Soibelman arXiv:math/0606241 and proved in arXiv:math/0611623 under a technical assumption. In this paper, the assumption is removed, and the argument is considerably simplified (in particular, it no longer uses Dold-Kan equivalence and simplicial methods). We also analyse the degeneration of the conjugate spectral sequence in positive caracteristic constructed in arXiv:1509.08784.
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.
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 $V$ 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…
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 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 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.
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…
There is a Chern character from K-theory to negative cyclic homology. We show that it preserves the decomposition coming from Adams operations, at least in characteristic 0. This is done by using infinitesimal cohomology to reduce to the case of a nilpotent ideal (which had been established by Cathelineau some time ago).
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…
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 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}…
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 $G$ .
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.
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.
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.
The goal of this paper is to establish fundamental properties of the Hochschild, topological Hochschild, and topological cyclic homologies of commutative, Noetherian rings, which are assumed only to be F-finite in the majority of our results. This mild hypothesis is satisfied in all cases of interest in finite and mixed characteristic algebraic geometry. We prove firstly that the topological Hochschild homology groups, and the homotopy groups of the fixed point spectra $TR^r$, are finitely generated modules. We use this to establish the continuity of these homology theories for any given ideal…
It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such a structure seems to be out of reach at the moment its counterpart in finite characteristic is much better understood thanks to recent groundbreaking works of Kaledin. In particular, it is proven by Kaledin that under some assumptions on a DG algebra $A$ over a perfect field …
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 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)].
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 $k$.
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.
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…
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.
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…
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-…
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…
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…
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…
A central result here is the computation of the entire cyclic homology of canonical smooth subalgebras of stable continuous trace C*-algebras having smooth manifolds M as their spectrum. More precisely, the entire cyclic homology is shown to be canonically isomorphic to the continuous periodic cyclic homology for these algebras. By an earlier result of the authors, one concludes that the entire cyclic homology of the algebra is canonically isomorphic to the twisted de Rham cohomology of M.
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.
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 study the periodic cyclic homology groups of the cross-product of a finite type algebra $A$ by a discrete group $Γ$. In case $A$ is commutative and $Γ$ is finite, our results are complete and given in terms of the singular cohomology of the strata of fixed points. These groups identify our cyclic homology groups with the \dlp orbifold cohomology\drp\ of the underlying (algebraic) orbifold. The proof is based on a careful study of localization at fixed points and of the resulting Koszul complexes. We provide examples of Azumaya algebras for which this identification is, however, no longer va…
