# Questions tagged [cyclic-homology]

For questions about cyclic homology of associative algebras and related concepts.

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questions

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### A cyclic-like chain complex

Let $V$ be a $\mathbb{Z}$-graded vector space over an algebraically closed field $k$ of characteristic zero.
Let $\overline{TV}= \bigoplus_{n=1}^{\infty} V^{\otimes n}$ be the reduced tensor algebra....

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59 views

### Relative de Rham Cohomology groups of k-algebra

Let $A$ be a commutative unital $k$-algebra. Then we have de Rham complex given as:
$C_{\ast}(A)$ : $ 0 \rightarrow A \rightarrow \Omega_{A \lvert k}^{1} \rightarrow \Omega_{A \lvert k}^{2} \...

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### Does a morphism which induces an isomorphism between Hochschild homology also induce an isomorphism between cyclic homology?

In a 1998 paper by B. Keller, the author consider the following problem in Section 1.4: Let $k$ be a commutative ring and $X$ a scheme over $k$. We can consider the cyclic homology as well as the ...

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127 views

### Equivariant L-infinity structure associated to a DGBV algebra

Let $V$ be a differential graded Batalin-Vilkovisky (DGBV) algebra over say $\mathbb{C}$. This means that $V$ is a commutative differential graded algebra (CDGA), with differential $\partial$ of ...

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198 views

### $\mathrm{HH}$ and $\mathrm{HC}$ as two different Taylor expansions at the same point

Let us consider the functors from dg algebras to graded spaces given by Hochshcild homology and cyclic homology. Then it is well known that in degree $0$, both functors are given by $A\longmapsto A/[A,...

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### A 2- cocycle $\tau$ which is not cyclic but it still satisfies the stability of $\tau(e,e,e)$ for idempotent $e$

I learned the following statement from page $20$ of the book Noncommutative Geometry by Alain Connes:
Let $\tau$ be a $2$-cyclic cocyle on a $C^*$ algebra. Then for every smooth curve $e(t)$ of ...

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260 views

### Algebraic models of non-simply connected spaces in string topology

I want to find some algebraic models relating string topology to Hochschild and cyclic homologies. If the space $X$ is simply-connected and we are working over rational numbers, we can use Sullivan ...

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### Explicit description of periodic map $S : HC_{i} \to HC_{i-2}$ for dg and $A_\infty$ algebras

Let $A$ some associative unital $k$-algebra, let $HC_*(A)$ is cyclic homology of $A$ and $HH_*(A)$ is hochschild homology of $A$. Then we have Connes exact sequence:
$$ ... \xrightarrow[]{} HH_n(A) \...

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315 views

### Definition of Non-commutative de-Rham-Cohomology

Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to ...

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### Connes-Chern pairing, compatibility with periodicity operator in the odd case

Let $A$ be an algebra (say unital). For an odd (say $2n-1$) cyclic cocycle $\varphi$ and a class in $K_1(A)$ represented by invertible $u$ we define
$$\langle [\varphi],[u] \rangle:=\frac{2^{-(2n+1)}}...

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130 views

### Failure of periodic cyclic homology to be a localizing invariant

A localizing invariant $E: dgcat_{sm} \rightarrow Ch_k$ from small dg categories to chain complexes over a field $k$ (say $k = \mathbb{C}$) is a dg functor which sends localization sequences to exact ...

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154 views

### Pairing between cyclic cohomology and $K$-theory: the odd case

I would like to understand the proof of Proposition 15 (see page 70 in this link ). More precisely: I would like to understand a particular step in the proof namely:
Why $\frac{d}{dt}(\varphi \# ...

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206 views

### Two approaches to periodic cyclic cohomology

Cyclic cohomology may be defined in several ways: the easiest way to define it is via a subcomplex $C^*_{\lambda}$of Hochschild complex consisting from cyclic cochains. There are also other ...

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### Relative version of Hopf cyclic cohomology

In this paper Connes and Moscovici introduced the Hopf algebra of transverse differential operators in order to compute the index formula for the diffeomorphism invariant geometry. They developed the ...

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272 views

### Morita equivalence and isomorphisms in cohomology theories

Let $A,B$ be two unital algebras. We say that $A,B$ are Morita equivalent if there are $A-B$ and $B-A$ bimodules $P,Q$ such that
$$P \otimes_{B} Q \cong A, Q \otimes_A P \cong B$$
(as $A-A$ and $B-B$ ...

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182 views

### Dense subalgebra of continuous functions with same K -theory

Suppose $X$ is a compact metric space. Is there a good candidate for a dense subalgebra $A\subseteq C(X)$, such that the inclusion induces an isomorphism in $K$-theory?
For example, if $X$ was a ...

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222 views

### Normalization of cyclic cocycles

This question is a continuation of the discussion
Normalization of Hochschild cocycles
but this time in the cyclic context. I would like to ask whether the following is true:
The inclusion of ...

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169 views

### Topological invariance of periodic cyclic homology of stacks

Goodwillie proved (in Cyclic homology, derivations, and the free loopspace) that the periodic cyclic homology of a connective dg algebra is that of its reduced classical ring. Preygel proved (in Ind-...

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### $\lambda$-Decomposition for Connes' Cyclic Complex

Let $k$ be a field of characteristic zero, and $A$ be a commutative unital $k$-algebra. Then the cyclic homology of $A$ has a $\lambda$-decomposition:
$$HC_{n}(A)=HC_{n}^{(1)}(A)\oplus \cdots \oplus ...

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336 views

### Definitions of Hochschild Cohomology $HH^{\bullet}(A)$

Let $A$ be an associative unital $k$-algebra, and let $M$ be a bimodule of $A$. The Hochschild cohomology of $A$ with coefficients in $M$ can be defined as
$$HH^{n}(A,\,M)=\mathrm{Ext}^{n}_{A^{e}}(A,\,...

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85 views

### Quasi isomorphism for bicomplexes both defining cyclic cohomology

Let $A$ be a complex unital algebra. Consider the cyclic (cohomological) bicomplex $\mathcal{C}(A)$. This is a bicomplex
where in the $p$-th row one has $C^p(A)$ (the space of all $p+1$-linear forms) ...

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238 views

### Injectivity of the Chern character in $K$-homology

Let $(\pi,H,F)$ be a Fredholm module: here $\pi:A \to B(H)$ is a representation of an algebra on the Hilbert space $H$ and $F$ is a self adjoint operator with square one such that for each $a \in A$ ...

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536 views

### Isomorphism in cyclic cohomology vs isomorphism in Hochschild cohomology

Let $A$ be a unital algebra over a field $K$, $C^n(A)$ a space of all $n+1$ linear maps into scalar field $k$ (I'm interested in case $k=\mathbb{C}$) and
$$(bf)(a_0,...,a_{n+1})=\sum_{i=0}^n(-1)^if(...

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162 views

### Lie algebra (co)homology of the Lie algebra of differential operators

Let $X$ be an algebraic variety (or a complex manifold) over $\mathbb C$. Let $D(X)$ be its algebra of differential operators. Mostly I am interested in algebraic differential operators, but the case ...

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123 views

### Concerning the $SBI$-sequence for dihedral homology (Loday, Cyclic Homology, 5.2)

I was wondering about the signs of the $SBI$-sequence ("Connes' periodicity exact sequence") in equations $(5.2.7.2)$ and $(5.2.7.3)$ of 'Cyclic Homology' by Jean-Louis Loday. Why is the sequence ...

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326 views

### Errata for Getzler-Kapranov “Cyclic operads and Cyclic homology”

Do you know if anyone made an errata for the Getzler-Kapranov paper "Cyclic operads and Cyclic homology"? I was trying to read it, and I have found (at least I think I did) quite a few typos not all ...

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397 views

### Motivation for cyclic (co)homology

Question: What is the motivation for cyclic (co)homology?
Comment: There are two types of things which can motivate such notion. Natural construction in which they appear (for example Hochschild ...

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246 views

### Question about a theorem of Goodwillie on periodic cyclic homology

In his paper Cyclic homology, Derivations and the Free Loopsace, Goodwillie defines periodic cyclic homology for differential graded algebras (A,d) concentrated in non-negative degree.
Why does he ...

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### How to understand $\mathcal{L}BG \simeq G/^{\text{ad}}G$ in term of simplicial sets?

First let $G$ be a topological group and $BG$ its classifying space. Let $\mathcal{L}BG=\text{Map}(S^1, BG)$ be the free loop space of $BG$.
We can see that $\mathcal{L}BG$ has the homotopy type of $...

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### Are there analogs of String Homology structure in cyclic homology?

I was reading John D.S. Jones' paper "Cyclic homology and equivariant homology" where he introduces a variant of cyclic homology that is isomorphic (as modules over the ring $K[u]$) to equivariant ...

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373 views

### Advantage in Using Cyclic Homology to a compute Equivariant (Co)Homology of Loop Spaces

I am trying to compute equivariant (co)homology of the free loop space of a manifold $M$ that is not a Lie group, $H^{S^1}_*(LM)$ with the natural rotation action of $S^1$ on the loops of the free ...

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### Soft Question: What does periodic cyclic theory measure?

Ex1) The cyclic homology of $\mathbb{C}[X,Y]$ and that of the algebra of functions on the sphere $S^2$ have the same periodic cyclic homology. Clearly, however, these objects are topologically very ...

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### Equivariant version of cyclic versus de Rham (co)homology of commutative algebras?

Let $A$ be a commutative algebra that is a $g$-module, for some Abelian Lie algebra $g$. The primary example of my interest is when $A$ is the ring of functions over an affine variety, say $\mathbb{C}^...

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### Hochschild and cyclic cohomology of commutative algebra?

I'm wondering how to compute the Hochschild cohomology ${\rm HH}^n(A,A)$ and the cyclic cohomology ${\rm HC}^n(A)$ where $A = \mathbb{C}[t_i]$ is the algebra of polynomials in a finite set of ...

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573 views

### On cyclic homology of Ginzburg's DG algebra

Let $(Q,W)$ be a quiver with potential, and $D$ be Ginzburg's DG algebra associated to it (as explained in http://arxiv.org/abs/math/0612139 and other places), so that it is a 3-Calabi-Yau algebra. I ...

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### (Reduced) cyclic homology of a free product of unital algebras

Shameless upfloat of 1-year old question - the motivation is that in general the corresponding Banach version is false, so I am trying to see where the proof breaks down, and what (if anything) can be ...

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931 views

### Morita equivalence of DG algebras? (reference needed)

A stupid question: whether ther exists a universally recognized definition of when two differential graded algebras should be Morita equivalent? I mean a sort of equivalence which would incorporate ...

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### Hilbert's 3rd problem,number theory, motives, cyclic homology,…

This talk by Jinhyun Park connects a lot of interesting themes, making me curious to read more about that. Do you know where?

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### Is there some textbook for the details of the computation of the homology groups

Is there some results for the cyclic homology group $HC_1(A)$, for example, when it is zero, or which case we can compute out it explictly, here $A$ is a commutative algebra over the complex field.

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### Invariant space of lifted Chevalley automorphisms of the tensor algebra

Question. Let $k$ be a field of characteristic $0$. Let $L$ be a $k$-vector space. Consider the subspace $S$ of $L\otimes L\otimes L\otimes L$ spanned by all tensors of the form
$\left[a,\left\lbrace ...