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}(...

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 ...

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,...

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 ...

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 ...

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 ...

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. ...

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, ...

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.
...

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 ...

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

Community wiki

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 ...

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)\...

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 ...

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{...

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 ...

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 ...

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 ...

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 $...

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 ...

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[...

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(...

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). ...

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

Community wiki

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 ...

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

#### Related Tags

hochschild-homology × 93homological-algebra × 31

hochschild-cohomology × 21

ag.algebraic-geometry × 14

ra.rings-and-algebras × 14

at.algebraic-topology × 13

homotopy-theory × 9

cyclic-homology × 9

rt.representation-theory × 8

noncommutative-algebra × 8

reference-request × 7

noncommutative-geometry × 7

ct.category-theory × 5

kt.k-theory-and-homology × 5

loop-spaces × 5

ac.commutative-algebra × 4

simplicial-stuff × 4

differential-graded-algebras × 4

string-topology × 4

sg.symplectic-geometry × 3

cohomology × 3

higher-category-theory × 3

derived-categories × 3

d-modules × 3

lie-algebras × 2