29 votes

intuition for hochschild homology

Slogan: Hochschild homology is a (derived) categorification of the trace. This means the identity at the end of John Pardon's answer is a categorification of the identity $\text{tr}(AB) = \text{tr}(...
Qiaochu Yuan's user avatar
24 votes
Accepted

revisiting $THH(\mathbb{F}_p)$

Alright, here is the promised answer. First, the Hopkins-Mahowald theorem states that $\mathbb{F}_p$ is the free $\mathbb{E}_2$-ring with $p=0$, i.e. it is the homotopy pushout in $\mathbb{E}_2$-rings ...
Dylan Wilson's user avatar
  • 13.2k
22 votes
Accepted

intuition for hochschild homology

If you have a right $A$-module $M_A$ and a left $A$-module $_AN$, then you can form their tensor product $$M\otimes_AN:=\operatorname{coker}(M\otimes_kA\otimes_kN\xrightarrow{(m,a,n)\mapsto(ma,n)-(m,...
John Pardon's user avatar
  • 18.3k
12 votes

revisiting $THH(\mathbb{F}_p)$

This computation was given a totally different proof, not using any of the ideas being discussed above by Franjou, Lannes, Schwartz in Autour de la cohomologie de Mac Lane des corps finis. [On the Mac ...
Nicholas Kuhn's user avatar
12 votes

What is the negative cyclic homology of a smooth projective variety?

There are conceptually simple definitions, but they require a more symmetric definition of Hochschild homology. The Hochschild homology of $X/k$ (with coefficients in $\mathcal O_X$) is the homology ...
Marc Hoyois's user avatar
  • 8,692
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 ...
Manuel Rivera's user avatar
8 votes
Accepted

An exact sequence involving THH

Let's extract a clear question (about spectra in general) from your question, and then answer it. Let $E$ be any spectrum. There is the degree $p$ map $p:S\to S$ from the sphere spectrum to itself. ...
Tom Goodwillie's user avatar
8 votes
Accepted

Hochschild homology with coefficients in a certain bimodule

(I'm assuming in the following that the base ring $k$ was a field.) First, we note that for any $A$-bimodule of the form $M \otimes N$, where $M$ is a left $A$-module and $N$ is a right $A$-module, ...
Tyler Lawson's user avatar
  • 51.5k
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. ...
Harrison Chen's user avatar
7 votes
Accepted

String cobracket from TFT

The string cobracket you are referring to is not part of the TQFT structure of string topology given by the smooth Calabi Yau algebra structure on $C_*(\Omega M)$ but rather associated to an action of ...
Manuel Rivera's user avatar
7 votes

Book on Hochschild (co)homology

Sarah Witherspoon's book on Hochschild cohomology for Algebras is now published. See https://www.ams.org/publications/authors/books/postpub/gsm-204
7 votes

Hochschild homology of a category of modules over an algebra

This is merely a couple of examples showing how $HH_*(\mathbf{C}_A)$ may behave. Let first $A$ be the polynomial algebra $\Bbb C[x]$. Then $\mathbf{C}_A$ is the category of coherent sheaves with ...
Grisha Papayanov's user avatar
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)\...
Dave Benson's user avatar
  • 11.6k
6 votes
Accepted

Defining Hochschild homology of non-commutative DG-algebras with animated rings with a circle action

No, this does not work in the non-commutative case. In general we have $HH(A)=A\otimes_{A\otimes A^{\mathrm{op}}} A$, and this is only a $k$-module, not an algebra. If $A$ is commutative, the tensor ...
Marc Hoyois's user avatar
  • 8,692
6 votes
Accepted

How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?

The first reduction is probably due to Akhil Mathew, cf. [Bhatt–Morrow–Scholze: Topological Hochschild homology and integral $p$-adic Hodge theory, Proof of Thm 8.17]. We have an equivalence \begin{...
Z. M's user avatar
  • 2,003
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 ...
Bertram Arnold's user avatar
5 votes

How do you prove that Hochschild cohomology is Morita invariant?

It is even a derived invariant. Here is a proof in a special case, but Im not sure whether it works more general (with the same proof?) Let $A$ and $B$ two noetherian $K$-algebras for a commutative ...
Mare's user avatar
  • 25.9k
4 votes
Accepted

Topological Hochschild homology and Hochschild homology of dg algebras

The notion of Hochschild homology can be defined abstractly in any suitable homotopical context in which a tensor product exist (say, in any presentably symmetric monoidal $\infty$-category). Given an ...
Yonatan Harpaz's user avatar
4 votes
Accepted

A mysterious quasi-isomorphism in Kashiwara-Schapira's proof of HKR

First, $P_k$ is not a direct sum. It is, in fact, an extension (non-trivial!) corresponding to the Atiyah class. Of course, $\delta^*P_k$ is not isomorphic to $\Omega^k$, but there is a canonical map $...
Sasha's user avatar
  • 37.2k
4 votes

What is $TP(\mathbb{Z}_p)$?

The calculation of $\pi_* TP(\mathbb{F}_p) = \pi_* THH(\mathbb{F}_p)^{tS^1} = \pi_* \widehat{\mathbb{H}}(S^1, THH(\mathbb{F}_p))$ (the notation has changed over the years) was first published by ...
John Rognes's user avatar
  • 8,837
4 votes

Is the exterior algebra intrinsically formal?

It's not intrinsically formal. Let $A=\Lambda(V)$ be the exterior algebra on an $n$-dimensional vector space $V$, it follows from the Hochschild-Konstant-Rosenberg isomorphism that $\mathit{HH}^q(A,A[...
YHBKJ's user avatar
  • 3,157
3 votes
Accepted

Strict graded commutativity of $\pi_*(\operatorname{THH}(A))$?

To my knowledge, there is no such result for THH of a commutative ring. (Could be?) However, we need less for this result; we only need degree 1 elements to square to zero. Every element in $\pi_1 THH(...
Tyler Lawson's user avatar
  • 51.5k
3 votes

Measure the failure of colimit to commute with taking free loops (or Hochschild homology)?

If you consider Goodwillie calculus for functors ${\rm Top} \to Ch_{\mathbb Z}$, then a functor will be linear if and only if it preserves all small (homotopy) colimits (as in Maxime's comment). ...
Phil Tosteson's user avatar
3 votes

How do you prove that Hochschild cohomology is Morita invariant?

You are right that the ``obvious'' isomorphism $M_r(A^*)\cong M_r(A)^*$ does not preserve the $M_r(A)$-bimodule structure. However, there is another isomorphism that does, essentially given by taking ...
Bertram Arnold's user avatar
3 votes

Confusion about topological Hochschild homology and $\mathbb{Z}_p$-topological Hochschild homology

Remark 6.1. here seems to imply that the answer is no: "When the input ring $A$ is $p$-adically complete (but not killed by a power of p), then $HH(A/\mathbb{Z}_p)$, $THH(A)$, etc., as we have ...
rori's user avatar
  • 231
3 votes
Accepted

Slick construction of Hochschild complex

The category Δ_a of finite linearly ordered sets with disjoint union is a monoidal category. A monoid M in a monoidal category F is a strong monoidal functor Δ_a → F. Discarding the monoidal ...
Dmitri Pavlov's user avatar
3 votes

Does rational surface have exceptional collection of maximal length but not full?

One paper which does (edit: not, I missed the word rational) address your first question (by giving an example) is Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived ...
pbelmans's user avatar
  • 1,486
3 votes

Is the exterior algebra intrinsically formal?

Let me focus on the $\mathbb Z$-graded picture in characteristic $0$. (You can collapse the $\mathbb Z$-grading to a $\mathbb Z / 2$-grading if you like, but one should be a bit careful when ...
Severin Barmeier's user avatar
2 votes

Book on Hochschild (co)homology

There is so much on this subject coming from different perspectives. I would add to all of the above a complete, up to date, exposition regarding the relationship of Hochschild homology with loop ...
2 votes
Accepted

Variant of co-Tor in a bimodule category

I am not sure that I really understand what you want, but I'd say the relevant structure is that of a module category over a monoidal category. Given a monoidal category $\mathcal E$, one can ...
Leonid Positselski's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible