G. Cortiñas, C. Haesemeyer, C. A. Weibel
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we prove that infinitesimal equivariant Chern-Connes characters are well-defined. We decompose an equivariant index as a pairing of infinitesimal equivariant Chern-Connes characters with the Chern character of an idempotent matrix. We compute the limit of infinitesimal equivariant Chern- Connes characters when the time goes to zero by using the Getzler symbol calculus and then extend these theorems to the family case. We also prove that infinitesimal equivariant eta cochains are well-defined and prove the noncommutative infinitesimal equivariant index formula for manifolds with …
We study a noncommutative version of the infinitesimal site of Grothendieck. A theorem of Grothendieck establishes that the cohomology of the structure sheaf on the infinitesimal topology of a scheme of characteristic zero is de Rham cohomology. We prove that, for the noncommutative infinitesimal topology of an associative algebra over a field of characteristic zero, the cohomology of the structure sheaf modulo commutators is periodic cyclic cohomology. We also compute the noncommutative infinitesimal cohomology of other sheaves. For example we show that hypercohomology with coefficients in $K…
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 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…
We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate th…
Transgressions and Chern characters in coarse homotopy theory
本文探讨了粗同调论中超越现象与 Chern 特征类之间的关系，研究了它们之间的可交换性及数学结构。
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 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.
Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology, which allows us to reconstruct the semi-regularity map and the infinitesimal Abel-Jacobi map. Combining that construction of the semi-regularity map with obstruction theory of functors of Artin rings, we give a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection.
Let $k$ be a field of characteristic 0 and $\mathcal{A}$ a curved $k$-algebra. We obtain a Chern-Weil-type formula for the Chern character of a perfect $\mathcal{A}$-module taking values in $HN_0^{II}(\mathcal{A})$, the negative cyclic homology of the second kind associated to $\mathcal{A}$, when $\mathcal{A}$ satisfies a certain smoothness condition.
In this paper we study the fiber F of the rational Jones-Goodwillie character $$ F:=\hofiber(ch:K^\rat(A)@>>>HN^\rat(A)) $$ going from K-theory to negative cyclic homology of associative rings. We describe this fiber F in terms of sheaf cohomology. We prove that, for $n\ge 1$, there is an isomorphism: $$ π_n(F)\cong H^{-n}_{inf}(A,K^\rat) $$ between the homotopy of the fiber and the hypercohomology groups of $K^\rat$ on a non-commutative version of Grothendieck's infinitesimal site.
Using similarities between topological $K$-theory and periodic cyclic homology we show that, after tensoring with $\mathbb C$, for certain Fréchet algebras the Chern character provides an isomorphism between these functors. This is applied to prove that the Hecke algebra and the Schwartz algebra of a reductive $p$-adic group have isomorphic periodic cyclic homology. The main theorem in the first version was incorrect for algebras related to noncompact manifolds. This has no effect on the results concerning p-adic groups. In the appendix we show that an analogous cohomological result does hold …
Given a compact manifold $M$ and $g\in C^{\infty}(M,U(l;\mathbb{C}))$ we construct a Chern character $\mathrm{Ch}^-(g)$ which lives in the odd part of the equivariant (entire) cyclic Chen-normalized bar complex $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{T}))$ of $M$, and which is mapped to the odd Bismut-Chern character under the equivariant Chen integral map. It is also shown that the assignment $g\mapsto \mathrm{Ch}^-(g)$ induces a well-defined group homomorphism from the $K^{-1}$ theory of $M$ to the odd homology group of $\underline{\mathscr{C}}(Ω_{\mathbb{T}}(M\times \mathbb{…
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.
Dévissage for periodic cyclic homology of complete intersections
本文证明局部完全交嵌入光滑概形时周期循环同调的dévissage性质，并由此证明其导出范畴和奇点范畴的复化拓扑Chern特征映射为同构，验证了格猜想的新情形。
For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ring…
Chern character for torsion-free ample groupoids
本文构建了从 Baum-Connes 组合映射域到 Crainic-Moerdijk 群oids 带有有理系数的同调群的 Chern 符号映射，验证了 Matui 的 HK 猜想变体，并应用于超几何动力系统、拓扑满群同调等。
The Chern character maps are one of the most important working tools in mathematics. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this note we eliminate this gap by characterizing the Chern character maps, from the Grothendieck group to the (negative) cyclic homology groups, in terms of simple universal properties.
We give a survey of cyclic homology/cohomology theory including a detailed discussion of cyclic theories for various classes of topological algebras. We show how to associate cyclic classes with Fredholm modules and $K$-theory classes and how to construct a completely general bivariant Chern-Connes character from bivariant $K$-theory to bivariant cyclic theory.
In this paper we construct a bivariant Chern character for the equivariant KK-theory of a totally disconnected group with values in bivariant equivariant cohomology in the sense of Baum and Schneider. We prove in particular that the complexified left hand side of the Baum-Connes conjecture for a totally disconnected group is isomorphic to cosheaf homology. Moreover, it is shown that our transformation extends the Chern character defined by Baum and Schneider for profinite groups.
In this paper we compute the K-theory (algebraic and topological) and entire periodic cyclic homology of compact Lie group C*-algebras, define Chern characters between them and show that the Chern characters in both topological and algebraic cases are isomorphisms.
We construct a Hochschild-Kostant-Rosenberg-type quasi-isomorphism for the negative cyclic homology of the category of global matrix factorizations on a smooth separated scheme of finite type over a field. The map is explicit enough to yield a negative cyclic Chern character formula for global matrix factorizations. We also extend these results to the equivariant case of a finite group.
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…
The purpose of this work is to give a definition of a topological K-theory for dg-categories over C and to prove that the Chern character map from algebraic K-theory to periodic cyclic homology descends naturally to this new invariant. This topological Chern map provides a natural candidate for the existence of a rational structure on the periodic cylic homology of a smooth proper dg-algebra, within the theory of noncommutative Hodge structures. The definition of topological K-theory consists in two steps : taking the topological realization of algebraic K-theory, and inverting the Bott elemen…
If I is a nilpotent ideal in a $\mathbb{Q}$-algebra $A$, Goodwillie defined two isomorphisms from $K_*(A,I)$ to negative cyclic homology, $HN_*(A,I)$. One is the relative version of the absolute Chern character, and the other is defined using rational homotopy theory. The question of whether they agree was implicit in Goodwillie's 1986 Annals paper. In this paper, we show that the two isomorphisms agree. Here are three applications. 1.Cathelineau proved that the rational homotopy character is compatible with the $λ$-filtration. It follows that the relative Chern character is also compatible wi…
Filtrations and cohomology II: the Gauss-Manin connection
该研究用导出方法分析多类同调/上同调中的高斯-马宁联络，给出零幂不变性等问题的新方法，推广Bhatt特征零结果至混合特征，揭示与哈特肖恩复形的关联并解释特征零棱柱上同调特征。
We construct a Chern character map from the K-theory of the reduced C^* algebra of the p-adic GL(n) with values in the periodic cyclic homology of the Schwartz algebra of this group. We prove that this map is an isomorphism after tensoring with C by comparing an explicit formula, stated in the algebraic case by Cuntz and Quillen, with the classical Chern character. This Chern character is a crucial ingredient in the proof of the Baum-Connes conjecture for the p-adic GL(n) due to Baum, Higson and Plymen.
In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total λ-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation …
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…
We give details of the proof of the remark made in \cite{G2} that the Chern characters of the canonical generators on the K homology of the quantum group $SU_q(2)$ are not invariant under the natural $SU_q(2)$ coaction. Furthermore, the conjecture made in \cite{G2} about the nontriviality of the twisted Chern character coming from an odd equivariant spectral triple on $SU_q(2)$ is settled in the affirmative.
$Local^{3}$ Index Theorem means $Local(Local(Local \;Index \; Theorem)))$. $Local \; Index \; Theorem$ is the Connes-Moscovici local index theorem \cite{Connes-Moscovici1}, \cite{Connes-Moscovici2}. The second "Local" refers to the cyclic homology localised to a certain separable subring of the ground algebra, while the last one refers to Alexander-Spanier type cyclic homology. The Connes-Moscovici work is based on the operator $R(A) = \mathbf{P} - \mathbf{e}$ associated to the elliptic pseudo-differential operator $A$ on the smooth manifold $M$, where $\mathbf{P}$, $\mathbf{e}$ are idempotent…
To each algebra over the complex numbers we associate a sequence of abelian groups in a contravariant functorial way. In degree (m-1) we have the m-summable Fredholm modules over the algebra modulo stable m-summable perturbations. These new finitely summable K-homology groups pair with cyclic homology and algebraic K-theory. In the case of cyclic homology the pairing is induced by the Chern-Connes character. The pairing between algebraic K-theory and finitely summable K-homology is induced by the Connes-Karoubi multiplicative character.
In this paper we compute the K-theory (algebraic and topological) and entire periodic cyclic homology for compact quantum groups, define Chern characters between them and show that the Chern characters in both topological and algebraic cases are isomorphisms modulo torsion. The case of compact Lie group C*-algebras was treated in <A HREF="http://www.math.uiuc.edu/K-theory/0257">Non-Commutative Chern characters of compact Lie groups C*-algebras, 0257</A> , a revised version of which was accepted for publication in <I>K-Theory.</I>
The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing some fundamental results of Connes and Hochschild-Kostant-Rosenberg. The Connes-Chern character is also identified here with the twisted Chern character.
We observe that the von Neumann envelope of the quantum algebra of functions on the normalizer of thegroup $\SU(1,1)\cong \SL(2,\mathbb R)$ in $\SL(2,\mathbb C)$ via deformation quantization contains the von Neumann algebraic quantum normalizer of $\SU(1,1)$ in the frame work of Waronowicz-Korogodsky. We then use the technique of reduction to the maximal subgroup to compute the K-theory, the periodic cyclic homology and the corresponding Chern-Connes character.
This article is a tribute to the memory of Professor Enzo Martinelli, with deep respect and reconesance. Nicolae Teleman. The index formula is a local statement made on global and local data; for this reason we introduce local Alexander - Spanier co-homology, local periodic cyclic homology, local Chern character and local $T^{\ast}$-theory. Index theory should be done: Case 1: for arbitrary rings, Case 2: for rings of functions over topo- logical manifolds. Case 1 produces general index theorems, as for example, over pseudo-manifolds. Case 2 gives a general treatment of classical and non- comm…
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…
We study Chern characters and the assembly mapping for free actions using the framework of geometric $K$-homology. The focus is on the relative groups associated with a group homomorphism $ϕ:Γ_1\to Γ_2$ along with applications to Novikov type properties. In particular, we prove a relative strong Novikov property for homomorphisms of hyperbolic groups and a relative strong $\ell^1$-Novikov property for polynomially bounded homomorphisms of groups with polynomially bounded cohomology in $\C$. As a corollary, relative higher signatures on a manifold with boundary $W$, with $π_1(\partial W)\to π_1…
Goodwillie's rational isomorphism between relative algebraic K-theory and relative cyclic homology, together with the lambda decomposition of cyclic homology, illustrates the close relationships among algebraic K-theory, cyclic homology, and differential forms. In this paper, I prove a Goodwillie-type theorem for relative Milnor $K$-theory, working over a very general class of commutative rings, defined via the stability criterion of Van der Kallen. Early results of Van der Kallen and Bloch are special cases. The result likely generalizes in terms of de Rahm-Witt complexes, by weakening some i…
The C*-algebras called Quantum Heisenberg Manifolds (QHM) were introduced by Rieffel in 1989 as strict deformation quantizations of Heisenberg manifolds. In this article, we compute the pairings of K-theory and cyclic cohomology on the QHM. Combining these calculations with other results proved elsewhere, we also determine the periodic cyclic homology and cohomology of these algebras, and obtain explicit bases of the periodic cyclic cohomology of the QHM. We further isolate bases of periodic cyclic homology, expressed as Chern characters of the K-theory.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。