Hochschild and Cyclic Homology

Hochschild homology and its variants (cyclic, negative cyclic, periodic cyclic) provide computable approximations to algebraic K-theory. The trace map from K-theory to these theories is the main tool for computing K-groups of many important rings and algebras.

Hochschild homology

Definition8.1Hochschild homology

For an associative algebra $A$ over a commutative ring $k$ , the Hochschild homology $HH_n(A/k)$ is defined as:

$HH_n(A/k) = \operatorname{Tor}_n^{A \otimes_k A^{\text{op}}}(A, A) = H_n(C_\bullet(A/k))$

where the Hochschild complex $C_\bullet(A/k)$ is:

$\cdots \to A^{\otimes (n+1)} \xrightarrow{b} A^{\otimes n} \to \cdots \to A \otimes A \xrightarrow{b} A$

with the Hochschild boundary $b: A^{\otimes(n+1)} \to A^{\otimes n}$ given by:

$b(a_0 \otimes \cdots \otimes a_n) = \sum_{i=0}^{n-1} (-1)^i a_0 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n + (-1)^n a_n a_0 \otimes a_1 \otimes \cdots \otimes a_{n-1}.$

ExampleHH of polynomial rings

For $A = k[x_1, \ldots, x_d]$ (polynomial ring over a field $k$ of characteristic 0):

$HH_n(A/k) \cong \Omega^n_{A/k} = \Lambda^n_A \Omega^1_{A/k}$

the module of Kahler $n$ -forms. This is the Hochschild--Kostant--Rosenberg (HKR) theorem:

$HH_n(k[x_1, \ldots, x_d]/k) \cong k[x_1, \ldots, x_d] \otimes \Lambda^n(k \cdot dx_1 \oplus \cdots \oplus k \cdot dx_d).$

In particular, $HH_0 = A$ , $HH_1 = \Omega^1 = \bigoplus A \, dx_i$ , $HH_n = 0$ for $n > d$ .

More generally, for a smooth algebra $A/k$ : $HH_n(A/k) \cong \Omega^n_{A/k}$ (smooth = formally smooth and finitely presented).

ExampleHH of group algebras

For the group algebra $A = k[G]$ of a discrete group $G$ :

$HH_n(k[G]/k) \cong \bigoplus_{[g] \in \langle G \rangle} H_n(C_G(g); k)$

where $\langle G \rangle$ denotes the set of conjugacy classes of $G$ and $C_G(g)$ is the centralizer of $g$ . The summand for $g = 1$ gives $H_n(G; k)$ (group homology), while the other summands are "twisted" contributions from non-identity conjugacy classes.

For $G = \mathbb{Z}$ : $k[\mathbb{Z}] = k[t, t^{-1}]$ and $HH_0 = k[t, t^{-1}]$ , $HH_1 = k[t, t^{-1}] \, dt$ , $HH_n = 0$ for $n \geq 2$ .

Cyclic homology

Definition8.2Cyclic homology

The Hochschild complex $C_\bullet(A)$ carries an action of the cyclic groups: the operator $t: A^{\otimes(n+1)} \to A^{\otimes(n+1)}$ defined by $t(a_0, \ldots, a_n) = (-1)^n(a_n, a_0, \ldots, a_{n-1})$ satisfies $t^{n+1} = 1$ .

Cyclic homology $HC_n(A/k)$ is defined using Connes' long exact sequence (the SBI sequence):

$\cdots \to HH_n(A) \xrightarrow{I} HC_n(A) \xrightarrow{S} HC_{n-2}(A) \xrightarrow{B} HH_{n-1}(A) \to \cdots$

where $B$ is Connes' boundary operator, $I$ is the natural inclusion, and $S$ is the "periodicity" operator.

Alternatively, using Connes' double complex or Tsygan's complex:

$HC_n(A) = H_n(C_\bullet(A) / (1 - t))$

where we take coinvariants under the cyclic action.

Definition8.3Variants of cyclic homology

Negative cyclic homology $HC_n^-(A)$ : uses homotopy fixed points instead of coinvariants. Defined via $HC_n^-(A) = H_n(\operatorname{holim}_{S^1} C_\bullet(A))$ .
Periodic cyclic homology $HP_n(A)$ : the 2-periodic theory $HP_n(A) = \varprojlim_S HC_{n+2k}(A)$ . For $A$ smooth over a field of characteristic 0: $HP_n(A) \cong \bigoplus_{i} H^{n-2i}_{\text{dR}}(A/k)$ (de Rham cohomology).

The three theories fit into an exact triangle:

$HC^-(A) \xrightarrow{} HP(A) \xrightarrow{} HC(A) \xrightarrow{[1]}$

or equivalently, a long exact sequence:

$\cdots \to HC_n^-(A) \to HP_n(A) \to HC_{n-1}(A) \to HC_{n-1}^-(A) \to \cdots$

The Dennis trace

Definition8.4Dennis trace map

The Dennis trace is a natural map

$\operatorname{tr}: K_n(A) \to HH_n(A)$

defined at the chain level by: for $\alpha \in GL_r(A) \subset K_1(A)$ , $\operatorname{tr}(\alpha) = \sum_{i=1}^r \alpha_{ii} \in A/[A, A] = HH_0(A)$ . More generally, for $K_n$ , the trace sends an element of $\pi_n(BGL(A)^+)$ to a class in $HH_n(A)$ via the map on classifying spaces.

The Dennis trace factors through cyclic homology:

$K_n(A) \xrightarrow{\operatorname{tr}} HH_n(A) \xrightarrow{I} HC_n(A)$

and more importantly, through negative cyclic homology:

$K_n(A) \xrightarrow{\operatorname{tr}^-} HC_n^-(A)$

(the "Chern character" map of Connes--Karoubi). This is an isomorphism rationally for smooth algebras:

$K_n(A) \otimes \mathbb{Q} \xrightarrow{\sim} HC_n^-(A) \otimes \mathbb{Q} \quad (\text{Goodwillie}).$

RemarkGoodwillie's theorem on rational K-theory

For a nilpotent ideal $I \trianglelefteq A$ (with $I^N = 0$ ), Goodwillie proved:

$K_n(A, I) \otimes \mathbb{Q} \cong HC_{n-1}^-(A, I) \otimes \mathbb{Q}$

where $K_n(A, I)$ is the relative K-group and $HC_{n-1}^-$ is the relative negative cyclic homology. This means that the "infinitesimal" part of K-theory (the part depending on nilpotent extensions) is completely captured by negative cyclic homology, at least rationally.

Integrally, the story is more subtle and leads to topological cyclic homology (TC).