11
votes
Accepted
Algebraic models of non-simply connected spaces in string topology
In my paper with Zeinalian (https://arxiv.org/abs/1612.04801) we prove that the coHochschild complex of the dg coalgebra of singular chains on a path connected (possibly non-simply connected) space ...
- 2,035
10
votes
Accepted
Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$
This statement appears as Corollary B.4 in Nikolaus-Scholze's On topological cyclic homology; essentially, $\Lambda$ is the quotient of the paracyclic category $\Lambda_\infty$ by a free $B\mathbb Z$-...
- 4,709
8
votes
Accepted
How to compute the periodic cyclic homology of this algebra
You can use a derived version of the HKR theorem, i.e.
$HH(A) = \text{Sym}_A^\bullet(\mathbb{L}_A[1])$ where $\mathbb{L}_A$ is the cotangent complex (over $k$). I'm not sure about a reference though.
...
- 1,423
7
votes
Accepted
Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
I don't know what you would mean by the normal bundle of $M$ in $LM$, but the answer to one question is "no": When the $\mathbb Q[u]$-module $H^\ast_{S^1}(LM)$ is localized by inverting $u$, ...
- 55.9k
6
votes
How to compute the periodic cyclic homology of this algebra
There's a spectral sequence starting with, for example, Hochschild cohomology of the cohomology of a DGA algebra $A$ and converging to the Hochschild cohomology of $A$, $E_2^{p,q}=H\!H^*H^*(A)\...
- 16.2k
6
votes
Accepted
Morita equivalence and isomorphisms in cohomology theories
The conceptual point is that all of these invariants are Morita invariant because they can be defined directly in terms of the category of modules. Explicitly:
Starting from the category of modules $\...
- 118k
5
votes
Accepted
Morphisms of Hochschild (or cyclic) homology induced by homotopic maps
$\newcommand{\dd}{\mathrm d}\DeclareMathOperator{\Sym}{Sym}\DeclareMathOperator{\id}{id}$If $f,g:A\to B$ are dga morphisms which are homotopic as chain maps, the maps induced by $f$ and $g$ on ...
- 4,709
4
votes
Proofs that the classifying space of Connes' cycle category $\Lambda$ is $\mathbb C \mathbb P^\infty$
Another proof appears as Lemma 8.1.1.15 and Remark 8.1.1.16 of Igusa's Higher Franz-Reidemeister torsion. It uses spaces of graphs to verify that $|\Lambda|$ classifies oriented circle bundles.
- 8,167
4
votes
On Connes' fomulae of pairing between cyclic cohomology and K-theory
Let me turn this long comment into an answer, which does not address the concrete computations, but give a conceptual explanation of this pairing, in the same time fixing some typos (while the ...
- 2,806
4
votes
Is the localised $S^1$-equivariant cohomology of the free loop space of a space $X$ isomorphic to that of $X$ itself?
As the example by Tom Goodwillie shows, the restriction to fixed points does not generally induce an isomorphism between localized equivariant cohomologies. Yet, as the question shows, there's good ...
- 6,777
3
votes
Accepted
Is there an analogue of the module of differentials for "higher order derivations" in the Hochschild/cyclic senses?
The answer is yes and it is very simple. It helps to understand the case $n=1$ first in the way I explained in my thesis in Prop. 4.5.3. Namely, $\Omega^1_{S/R}$ can be constructed as the quotient of ...
- 63.1k
2
votes
Accepted
On the initiality of the inclusion from the simplex category to the paracycle category
As predicted by Maxime, the argument in Nikolaus and Scholze is indeed incorrect, but it can be fixed as follows. Note that since each map $(1/k)\mathbb Z \to (1/n)\mathbb Z$ factors through some $\...
- 63.9k
2
votes
Morita equivalence of DG algebras? (reference needed)
Just a comment on "For example, every linear invariant that I know of (algebraic K-theory, cyclic (co)homology, Hochschild (co)homology,...) is not only invariant under Morita equivalence but also ...
- 1,028
2
votes
Accepted
Definition of Non-commutative de-Rham-Cohomology
I do not really know the reason, but if I would guess I think it might be $\Omega(A)$ is no longer an abelian category. The exact sequences you written down are not just exact sequences of abelian ...
- 6,249
2
votes
Two approaches to periodic cyclic cohomology
The original reference is Theorem 40, c) of http://www.alainconnes.org/docs/noncommutative_differential_geometry.pdf
A maybe more pedagogical version is Theorem 29 c) of http://www.alainconnes.org/...
- 6,920
2
votes
Accepted
Normalization of cyclic cocycles
If I'm not mistaken this is just the dual version of what Loday and Quillen proves in Proposition 4.4 (or Proposition 2.2.14 in Loday's book, with the same proof). And sorry for nitpicking, but it's ...
- 2,066
1
vote
Lie algebra (co)homology of the Lie algebra of differential operators
Lie algebra (co)homology of the Lie algebra of differential operators
was used in the Feigin-Tsygan works on the Riemann-Roch theorem.
See, for example, this paper and later papers of Tsygan with co-...
- 743
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