# Hochschild and cyclic homology (briefly)

###### 15th July, 2020

- Unmotivated definitions
- Some motivation
- Circles (and the free loop space)
- HKR
- A glance at cyclic homology

Once again I feel the urge to type something nice, but have nothing new of my own to share. I did recently find, however, some notes I once wrote after hearing the phrase “Hochschild homology” for what felt like the hundredth time, so I thought I’d share them here. They’re not particularly enlightening, and I can’t claim to add any insight, but I often use my own blog as a reference for definitions that I once knew but later forgot, so this will at least serve that purpose!

# Unmotivated definitions

↟There are a bunch of different definitions of the *Hochschild
complex* C_\bullet(A,N) for a
bimodule N over a (commutative) algebra
A. Let’s have a look at them.
Throughout, we make use of the *enveloping algebra* A^e of A,
which is defined by A^e=A\otimes
A^\mathrm{op} (which makes it look a bit like the endomorphism
algebra, but with \mathrm{op} instead
of duals). Useful to know is the fact that a bimodule over A is exactly the same thing as a module over
A^e.

Note that, when we’ve defined the Hochschild complex C_\bullet(A,N), we can define *Hochschild
homology* of an algebra A as simply
\mathrm{HH}_\bullet(A) =
\mathrm{H}_\bullet(C_\bullet(A,A)) using the fact that A is a bimodule over itself.

## Definition 1 (Bar complex)

We define C_\bullet(A,N)=N\otimes_{A^e}\mathrm{B}_\bullet
A, where B_\bullet A is the
*Bar construction* applied to A.
For nice details about what this means, I would probably pick up my copy
of Emily Riehl’s Categorical Homotopy
Theory. Either way, we can then define the differential by the
omnipresent formula \partial_n = \sum_{i=0}^n
(-1)^i\mathrm{d}_i \colon C_n(A,N) \to C_{n-1}(A,n) where, here,
for 0\leqslant i<n, \mathrm{d}_i\colon n\otimes a_1\otimes\ldots\otimes
a_n \mapsto n\otimes a_1\otimes\ldots\otimes
a_ia_{i+1}\otimes\ldots\otimes a_n (so for i=0 we get na_1\otimes a_2\otimes\ldots\otimes a_n), and
\mathrm{d}_n\colon n\otimes
a_1\otimes\ldots\otimes a_n \mapsto na_n\otimes a_1\otimes\ldots\otimes
a_{n-1}.

## Definition 2 (Tor)

If we already know about \operatorname{Tor}, then we can simply define C_\bullet(A,N) = \operatorname{Tor}_{A^e}^\bullet(N,A).

## Definition 3 (Infinity things)

Even cheekier is the definition using (\infty,1)-categories: C_\bullet(A,N) = N\otimes_{A^e} A, where the tensor product is taken in the (\infty,1)-category of chain complexes.

# Some motivation

↟So why do we care? Well here’s a nice fact that, for the moment, might still seem rather unmotivated.

**Fact.** Let X be a
simply connected topological space. Then \mathrm{H}^\bullet(\mathcal{L}X;\mathbb{R}) \simeq
\mathrm{HH}_\bullet(C^\bullet(X;\mathbb{R})), where C^\bullet(X;\mathbb{R}) is the (usual)
singular cochain complex of X with
coefficients in \mathbb{R}, and \mathcal{L}X is the free loop space of X.

Why do we care about this? Well, because of the de Rham theorem: on a smooth manifold, real cohomology is isomorphic to de Rham cohomology. This shows itself by the following lovely theorem:

**Jones’ theorem.** Let X be a simply connected smooth manifold. Then
\mathrm{H}^\bullet(\mathcal{L}X;\mathbb{R})
\simeq \mathrm{HH}_\bullet(\Omega_X^\bullet).

(Another nice fact about Hochschild homology (that we also won’t prove) is that \mathrm{HH}_\bullet(k[G]) \simeq \mathrm{HH}_\bullet(\mathcal{L}\mathcal{B}G).)

OK, so this is all well and good, but can we something a bit more concrete? Yes, we can.

# Circles (and the free loop space)

↟If we want to realise the circle S^1 as the geometric realisation of a simplicial set, then we have to define some simplicial set S_\bullet such that S_0 = \{*\} and S_1 = \{\tau\} with face maps f_0(\tau)=f_1(\tau)=*. That is, our 1-simplex \tau is being glued together at its two edges to make a circle.

But we also need to worry about degeneracies, e.g. s_0(*)\in S_1. Generally, in S_n we have the elements s_0^n(*) and (s_{n-1}\circ\ldots\circ s_i\circ s_{i-2}\circ\ldots\circ s_0)(\tau) for i=1,\ldots,n (and this, in fact, suffices, by the degeneracy relations). Thus |S_n|=n+1, and, in fact, S_n\simeq\mathbb{Z}/(n+1)\mathbb{Z}, where s_0^n(*)\leftrightarrow0 and (s_{n-1}\circ\ldots\circ s_i\circ s_{i-2}\circ\ldots\circ s_0)(\tau)\leftrightarrow i.

Now, the geometric realisation of the *cosimplicial* space
\operatorname{Hom}(S_\bullet,X), for
some topological space X, is \|\operatorname{Hom}(S_\bullet,X)\| =
\operatorname{Hom}_\mathsf{cosimpTop}(\Delta^\bullet,\operatorname{Hom}(S_\bullet,X)).
But, if X is simply connected, then
\|\operatorname{Hom}(S_\bullet,X)\| \simeq
\operatorname{Hom}_\mathsf{Top}(|S_\bullet|,X) = \mathcal{L}X
where the last equality can be taken as the *definition* of the
free loop space \mathcal{L}X of X.

**Claim.** C_\bullet(A,A)
\simeq C_\bullet(S_\bullet\otimes A).

In the above, C_\bullet on the
right-hand side denotes the *Moore complex* of a simplicial
group: C_n(G_\bullet) =
\bigcap_{i=1}^n\operatorname{Ker}f_i^n; \partial_n= f_0^n\colon C_n(G_\bullet)\to
C_{n-1}(G_\bullet), and S_\bullet\otimes A is the simplicial group
with A in degree 0, A\otimes_k
A in degree 1, A\otimes_k A\otimes_k A in degree 2, and so on, and where the (n+1) maps from (S_\bullet\times A)_n to (S_\bullet\times A)_{n-1} are given by
applying the multiplication \mu of
A, but after first applying a cyclic
permutation of the elements of the tensor product. So, for example, from
A\otimes A\otimes A to A\otimes A we have three maps: \operatorname{id}\otimes\mu and \mu\otimes\operatorname{id}, as well as (\mu\otimes\operatorname{id})\circ\sigma_{(231)},
where \sigma_{(231)} is the cyclic
permutation (123)\mapsto(231).

# HKR

↟It would be remiss of me to not even mention the
Hochschild-Kostant-Rosenberg theorem, so here we go: if R is a nice enough k-algebra, then we have an isomorphism of
graded R-algebras \Omega^\bullet(R/k)\simeq\mathrm{HH}_\bullet(R);
dually, there is an isomorphism between wedge products of derivations
and Hochschild *cohomology*, \bigwedge_R^\bullet\operatorname{Der}(R,R)\simeq\mathrm{HH}^\bullet(R).

There we go, I’ve mentioned it.

# A glance at cyclic homology

↟Jones’ theorem told us about the cohomology of the free loop space of
X, bu can we say anything about the
*cyclic* loop space of X? This
is the homotopy quotient \mathcal{L}X/^hS^1 by the canonical
action of S^1 on \mathcal{L}X. Then \mathrm{H}^\bullet(\mathcal{L}X/^hS^1)\simeq\mathrm{HC}_\bullet(C^\bullet(X))
where, again, C^\bullet(X) is the
complex of singular chains, and \mathrm{HC}_\bullet is the *cyclic
homology*. But how do we define cyclic homology?

Let \lambda\colon C_n(A,A)\to
C_n(A,A) be the automorphism of the Hochschild complex given by
cyclic permutation (and a sign), i.e. \lambda\colon a_0\otimes a_1\otimes\ldots\otimes a
\mapsto (-1)^n a_n\otimes a_0\otimes\ldots\otimes a_{n-1}. Then
we define the *cyclic homology complex* C_\bullet^\lambda(A,A)=C_\bullet(A,A)/\operatorname{Im}(\operatorname{id}-\lambda),
and the *cyclic homology* \mathrm{HC}_\bullet(A)=\mathrm{H}_\bullet(C_\bullet^\lambda(A)).