Questions tagged [cyclic-homology]

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

Filter by
Sorted by
Tagged with
4 votes
1 answer
233 views

On the initiality of the inclusion from the simplex category to the paracycle category

Thm B.3 of Nikolaus and Scholze shows that the natural inclusion $\Delta \to \Lambda_\infty$, from the simplex category to the paracycle category, is an initial functor, i.e. satisfies the hypotheses ...
Tim Campion's user avatar
  • 60.6k
8 votes
2 answers
637 views

How to compute the periodic cyclic homology of this algebra

Let $k=\mathbb{C}$ be the field of complex numbers. I consider the (DG) algebra $A:=k[x]/(x^2)$ such that $\deg(x)=-1$. My question is how to compute the periodic cyclic homology, Hochschild homology ...
user41650's user avatar
  • 1,922
3 votes
0 answers
115 views

Cyclic K-theory as cyclic nerve in a letter of Goodwillie

Kaledin mentioned in https://arxiv.org/abs/2004.04279 Remark 11.5 that, in a letter to Waldhausen by Goodwillie in 1988, Goodwillie showed that the cyclic K-theory can be computed by the geometric ...
Z. M's user avatar
  • 1,948
7 votes
0 answers
80 views

Can cyclic and simplicial objects be related in a similar way to how the species of linear orders is the derivative of the species of cyclic orders?

The derivative of a combinatorial species $S: core(FinSet) \to core(FinSet)$ is given by $S^\prime [N] = S[N\sqcup 1]$. Intuitively, an $S^\prime$-structure is built by introducing a "hole" ...
Mathemologist's user avatar
7 votes
2 answers
439 views

Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?

A well-known theorem of Atiyah and Bott states that given a finite dimensional oriented manifold $M$ with circle action, the $S^1$-equivariant cohomology of $M$ (with $\mathbb{Q}$ coefficients) is ...
Eugenio Landi's user avatar
3 votes
1 answer
360 views

Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?

Note added by YC: the definition below of the cyclic sub-complex is incorrect; and the "higher order derivations" referred to here are traditionally known (since the 1940s) as n-cocycles. $\...
Emily's user avatar
  • 10.3k
1 vote
1 answer
267 views

Even and odd part of the Hochschild and cyclic homology of a super-algebra

Let $A$ be a $\mathbb Z_2$-graded $k$-algebra, where $k$ is a field of characteristic $0$. Then we know that the tensor product of $A$ with itself is also $\mathbb Z_2$-graded by $$(A\otimes_k A)_0:=...
Flavius Aetius's user avatar
9 votes
2 answers
455 views

Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$

Connes showed in Cohomologie cyclique et foncteurs $Ext^n$ (1983) that the classifying space of his cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty = B(S^1) = K(\mathbb Z,2)$. Connes' proof is ...
Tim Campion's user avatar
  • 60.6k
5 votes
1 answer
295 views

Morphisms of Hochschild (or cyclic) homology induced by homotopic maps

Let $f$ and $g$ be two maps between DG algebras $A$ and $B$, and assume that $f$ and $g$ are homotopic as chain maps, hence they induce the same map on the level of homology. Moreover, $f$ and $g$ ...
Yining Zhang's user avatar
5 votes
0 answers
138 views

Negative cyclic homology of the group algebra of discrete groups

I am looking for a reference for the calculation of the negative cyclic homology of the group algebra $\mathbb{K}[\Gamma]$ of a discrete group $\Gamma$ over a field $\mathbb{K}$ of characteristic 0. (...
user188722's user avatar
3 votes
0 answers
73 views

The Loday-Quillen-Tsygan theorem for topological (Fréchet) algebras

In "Additive K-theory" by Tsygan and Feigin, Section 0.4, a statement is given which seems to generalize (cohomological version of) the well-known Loday-Quillen-Tsygan theorem $$H_{\text{CE}}...
Lukas Miaskiwskyi's user avatar
2 votes
0 answers
118 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} \...
Sunny's user avatar
  • 609
8 votes
0 answers
193 views

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 ...
Zhaoting Wei's user avatar
  • 8,657
7 votes
0 answers
191 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 ...
kyler's user avatar
  • 71
6 votes
0 answers
246 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,...
Pedro's user avatar
  • 1,534
1 vote
0 answers
61 views

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 ...
Ali Taghavi's user avatar
8 votes
1 answer
373 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 ...
Yining Zhang's user avatar
3 votes
0 answers
104 views

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) \...
Mykola Pochekai's user avatar
5 votes
1 answer
499 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 ...
Matthias Ludewig's user avatar
3 votes
0 answers
117 views

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)}}...
truebaran's user avatar
  • 9,140
3 votes
0 answers
241 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 ...
Harrison Chen's user avatar
3 votes
0 answers
192 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 \# ...
truebaran's user avatar
  • 9,140
5 votes
1 answer
327 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 ...
truebaran's user avatar
  • 9,140
2 votes
0 answers
80 views

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 ...
truebaran's user avatar
  • 9,140
6 votes
1 answer
423 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 $B-B$ ...
truebaran's user avatar
  • 9,140
4 votes
0 answers
234 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 ...
vap's user avatar
  • 330
2 votes
1 answer
302 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 ...
truebaran's user avatar
  • 9,140
4 votes
0 answers
222 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-...
Harrison Chen's user avatar
4 votes
0 answers
95 views

$\lambda$-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 ...
Yining Zhang's user avatar
7 votes
0 answers
401 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,\,...
Yining Zhang's user avatar
3 votes
0 answers
95 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) ...
truebaran's user avatar
  • 9,140
7 votes
0 answers
288 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 $a \in A$ ...
truebaran's user avatar
  • 9,140
8 votes
1 answer
596 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(...
truebaran's user avatar
  • 9,140
2 votes
1 answer
209 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 ...
lks8271's user avatar
  • 165
1 vote
0 answers
154 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 ...
Arthur's user avatar
  • 61
7 votes
1 answer
441 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 ...
4 votes
0 answers
441 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 ...
2 votes
0 answers
304 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 ...
user36931's user avatar
  • 1,331
2 votes
0 answers
161 views

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 $...
Zhaoting Wei's user avatar
  • 8,657
3 votes
1 answer
383 views

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 ...
Felix Y.'s user avatar
  • 297
5 votes
1 answer
471 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 ...
Felix Y.'s user avatar
  • 297
4 votes
0 answers
468 views

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 ...
ABIM's user avatar
  • 5,019
2 votes
0 answers
82 views

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}^...
user36075's user avatar
  • 131
5 votes
2 answers
1k views

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 ...
user36075's user avatar
  • 131
10 votes
0 answers
637 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 ...
Yuji Tachikawa's user avatar
8 votes
0 answers
251 views

(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 ...
Yemon Choi's user avatar
  • 25.5k
17 votes
2 answers
1k 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 ...
gshar's user avatar
  • 291
12 votes
2 answers
3k views

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?
Thomas Riepe's user avatar
  • 10.7k
3 votes
2 answers
348 views

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.
ren l's user avatar
  • 31
1 vote
2 answers
240 views

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 ...
darij grinberg's user avatar