Questions tagged [cyclic-homology]
For questions about cyclic homology of associative algebras and related concepts.
52
questions
4
votes
1
answer
197
views
On Connes' fomulae of pairing between cyclic cohomology and K-theory
The following proposition comes from Connes' paper in IHES. See the link Non-commutative differential geometry.
On page 109, Proposition 15. of Part II, he claims that
(1) The following equality ...
- 41
1
vote
0
answers
105
views
Cyclic homology with coefficients in a bimodule
I've recently been trying to understand Hochschild and cyclic co/homology better, and I've noticed that while it's common to define the Hochschild homology $\mathrm{HH}_{\bullet}(A;M)$ of an $R$-...
- 11.5k
4
votes
1
answer
246
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 ...
- 63.2k
8
votes
2
answers
724
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 ...
- 1,982
3
votes
0
answers
129
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 ...
- 2,571
7
votes
0
answers
84
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" ...
- 713
7
votes
2
answers
470
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 ...
- 449
3
votes
1
answer
372
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.
$\...
- 11.5k
1
vote
1
answer
310
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
$$(A\otimes_k A)_0:=...
- 599
9
votes
2
answers
501
views
Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$
Connes showed in Cohomologie cyclique et foncteurs $Ext^n$ (1983) that the classifying space of his cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty = B(S^1) = K(\mathbb Z,2)$.
Connes' proof is ...
- 63.2k
5
votes
1
answer
313
views
Morphisms of Hochschild (or cyclic) homology induced by homotopic maps
Let $f$ and $g$ be two maps between DG algebras $A$ and $B$, and assume that $f$ and $g$ are homotopic as chain maps, hence they induce the same map on the level of homology. Moreover, $f$ and $g$ ...
- 661
5
votes
0
answers
144
views
Negative cyclic homology of the group algebra of discrete groups
I am looking for a reference for the calculation of the negative cyclic homology of the group algebra $\mathbb{K}[\Gamma]$ of a discrete group $\Gamma$ over a field $\mathbb{K}$ of characteristic 0. (...
- 173
3
votes
0
answers
76
views
The Loday-Quillen-Tsygan theorem for topological (Fréchet) algebras
In "Additive K-theory" by Tsygan and Feigin, Section 0.4, a statement is given which seems to generalize (cohomological version of) the well-known Loday-Quillen-Tsygan theorem
$$H_{\text{CE}}...
2
votes
0
answers
127
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} \...
- 609
9
votes
0
answers
196
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 ...
- 8,909
7
votes
0
answers
197
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 ...
- 71
6
votes
0
answers
248
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,...
- 1,554
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 ...
- 708
8
votes
1
answer
377
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 ...
- 661
3
votes
0
answers
105
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:
$$ ... \xrightarrow[]{} HH_n(A) \...
- 1,237
5
votes
1
answer
527
views
Definition of Non-commutative de-Rham-Cohomology
Let $A$ be a (not necessarily commutative, associative) $k$-algebra. The bimodule of non-commutative one-forms $\Omega^1_A$ is the free $A$-bimodule generated by symbols $da$, $a \in A$, subject to ...
- 9,681
5
votes
0
answers
178
views
Connes-Chern pairing, compatibility with periodicity operator in the odd case
Let $A$ be an algebra (say unital). For an odd (say $2n-1$) cyclic cocycle $\varphi$ and a class in $K_1(A)$ represented by invertible $u$ we define
$$\langle [\varphi],[u] \rangle:=\frac{2^{-(2n+1)}}...
- 9,300
3
votes
0
answers
257
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 ...
- 1,413
3
votes
0
answers
205
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 \# ...
- 9,300
5
votes
1
answer
347
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 ...
- 9,300
2
votes
0
answers
83
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 ...
- 9,300
6
votes
1
answer
443
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 $B-B$ ...
- 9,300
4
votes
0
answers
244
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 ...
- 411
2
votes
1
answer
313
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 ...
- 9,300
4
votes
0
answers
234
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-...
- 1,413
4
votes
0
answers
98
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:
$$HC_{n}(A)=HC_{n}^{(1)}(A)\oplus \cdots \oplus ...
- 661
7
votes
0
answers
409
views
Definitions of Hochschild Cohomology $HH^{\bullet}(A)$
Let $A$ be an associative unital $k$-algebra, and let $M$ be a bimodule of $A$. The Hochschild cohomology of $A$ with coefficients in $M$ can be defined as
$$HH^{n}(A,\,M)=\mathrm{Ext}^{n}_{A^{e}}(A,\,...
- 661
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) ...
- 9,300
7
votes
0
answers
294
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 $a \in A$ ...
- 9,300
8
votes
1
answer
600
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(...
- 9,300
2
votes
1
answer
221
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 ...
- 165
1
vote
0
answers
160
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 ...
- 61
7
votes
1
answer
454
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
449
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
314
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 ...
- 1,331
2
votes
0
answers
165
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 $...
- 8,909
3
votes
1
answer
394
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 ...
- 297
5
votes
1
answer
484
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 ...
- 297
4
votes
0
answers
481
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 ...
- 5,261
2
votes
0
answers
88
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}^...
- 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 ...
- 131
10
votes
0
answers
648
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 ...
- 6,113
8
votes
0
answers
255
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 ...
- 25.7k
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 ...
- 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?
- 10.8k