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Hochschild and cyclic homology (briefly)

15th of July, 2020
  1. Unmotivated definitions
  2. Some motivation
  3. Circles (and the free loop space)
  4. HKR
  5. 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).


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)).