All Questions
Tagged with hochschild-homology at.algebraic-topology
13 questions
4
votes
1
answer
469
views
How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?
$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes:
https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf
https://...
- 959
7
votes
1
answer
468
views
An exact sequence involving THH
Disclaimer: I'm a relative beginner in this area. I'm trying to prove that if one has a commutative ring $R$ and a prime number $p$, then there is an exact sequence of the form
$$\DeclareMathOperator\...
- 73
4
votes
0
answers
317
views
What is the geometric interpretation of the first Hochschild homology group of path algebra constructed from a directed graph?
Let $\mathcal{G} = (V, E, s, t)$ is a directed graph, where $V$ - the set of its vertices, $E$ - the set of its edges, $s: E \rightarrow V, s((v_1, v_2)) = v_1$ and $t: E \rightarrow V, s((v_1, v_2)) =...
- 91
8
votes
1
answer
479
views
What is $TP(\mathbb{Z}_p)$?
Let $TP$ be periodic topological cyclic homology. What is $\pi_* TP(\mathbb{Z}_p)$?
(i) I know that $\pi_* TP(\mathbb{F}_p) \cong \mathbb{Z}_p[v^{\pm 1}]$ with $v$ in degree $-2$ by IV.4.8 of Nikolaus-...
- 81
20
votes
2
answers
2k
views
revisiting $THH(\mathbb{F}_p)$
Reading through Bhatt-Morrow-Scholze's "Topological Hochschild Homology and Integral p-adic Hodge Theory" I encountered the following statement.
We use only “formal” properties of THH throughout ...
- 858
11
votes
1
answer
556
views
Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?
For a space1 $X$, let $\mathcal{L}X = \mathrm{Maps}(S^1, X)$ be the free loop space.
Inclusion of constant loops gives a natural map $X \to \mathcal{L}X$. This is not a homotopy equivalence unless $X$...
- 8,723
14
votes
2
answers
2k
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Applications of topological chiral homology and factorization algebras (aka higher Hochschild cohomology)
I recently heard a talk about these topics and found them very interesting.
The talk was centered on the formal structure and didn't really focus on examples.
So my question is: what is your favorite ...
- 1,273
3
votes
1
answer
280
views
Hochschild homology of acyclic complex
Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic.
Question: Does this imply that the Hochschild homology $HH_*(A)$ also vanishes ...
- 177
8
votes
1
answer
378
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
4
votes
0
answers
158
views
Describing the THH of function spectra?
Are there any results describing the $THH$ of spectra of the form $F(X, E)$ where $X$ is a space (say, finite CW) and $E$ is a (nice enough) ring spectrum?
I'm happy to put various (further, or ...
- 1,150
10
votes
1
answer
548
views
String cobracket from TFT
Let $M$ be a closed oriented manifold. Chas and Sullivan (https://arxiv.org/abs/math/0212358) introduced a Lie bialgebra structure on $H_\bullet^{S^1}(LM, M)$, $S^1$-equivariant homology of the loop ...
- 3,357
5
votes
2
answers
944
views
What is $TC(\Sigma^\infty \Omega X)$?
I know that for $X$ a connected space, $THH(\Sigma^\infty \Omega X) = \Sigma^\infty \Lambda X$, the suspension spectrum of the free loop space of $X$. The computation can be carried out in spaces and ...
- 25.2k
3
votes
0
answers
300
views
Hochschild homology of a tensor algebra modulo a two-sided ideal
Let $V$ is a module over a field $k$, and $A=T(V)$ the tensor algebra generated by $V$. The Hochschild homology $HH_*(A)$ has been determined by Loday and Quillen in their paper "Cyclic homology and ...
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