# Hochschild homology

In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by Gerhard Hochschild (1945) for algebras over a field, and extended to algebras over more general rings by Henri Cartan and Samuel Eilenberg (1956).

## Definition of Hochschild homology of algebras

Let k be a field, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product Ae=AAo of A with its opposite algebra. Bimodules over A are essentially the same as modules over the enveloping algebra of A, so in particular A and M can be considered as Ae-modules. Cartan & Eilenberg (1956) defined the Hochschild homology and cohomology group of A with coefficients in M in terms of the Tor functor and Ext functor by

${\displaystyle HH_{n}(A,M)={\text{Tor}}_{n}^{A^{e}}(A,M)}$
${\displaystyle HH^{n}(A,M)={\text{Ext}}_{A^{e}}^{n}(A,M)}$

### Hochschild complex

Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write An for the n-fold tensor product of A over k. The chain complex that gives rise to Hochschild homology is given by

${\displaystyle C_{n}(A,M):=M\otimes A^{\otimes n}}$

with boundary operator di defined by

{\displaystyle {\begin{aligned}d_{0}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=ma_{1}\otimes a_{2}\cdots \otimes a_{n}\\d_{i}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=m\otimes a_{1}\otimes \cdots \otimes a_{i}a_{i+1}\otimes \cdots \otimes a_{n}\\d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=a_{n}m\otimes a_{1}\otimes \cdots \otimes a_{n-1}\end{aligned}}}

where ai is in A for all 1 ≤ in and mM. If we let

${\displaystyle b=\sum _{i=0}^{n}(-1)^{i}d_{i},}$

then ${\displaystyle b\circ b=0,}$ so (Cn(A,M), b) is a chain complex called the Hochschild complex, and its homology is the Hochschild homology of A with coefficients in M.

### Remark

The maps di are face maps making the family of modules Cn(A,M) a simplicial object in the category of k-modules, i.e. a functor Δok-mod, where Δ is the simplex category and k-mod is the category of k-modules. Here Δo is the opposite category of Δ. The degeneracy maps are defined by

${\displaystyle s_{i}(a_{0}\otimes \cdots \otimes a_{n})=a_{0}\otimes \cdots \otimes a_{i}\otimes 1\otimes a_{i+1}\otimes \cdots \otimes a_{n}.}$

Hochschild homology is the homology of this simplicial module.

## Hochschild homology of functors

The simplicial circle S1 is a simplicial object in the category ${\displaystyle {\text{Fin}}_{*}}$ of finite pointed sets, i.e. a functor ${\displaystyle \Delta ^{o}\to {\text{Fin}}_{*}.}$ Thus, if F is a functor F: Fin → k-mod, we get a simplicial module by composing F with S1

${\displaystyle \Delta ^{o}{\overset {S^{1}}{\longrightarrow }}{\text{Fin}}_{*}{\overset {F}{\longrightarrow }}k{\text{-mod}}.}$

The homology of this simplicial module is the Hochschild homology of the functor F. The above definition of Hochschild homology of commutative algebras is the special case where F is the Loday functor.

### Loday functor

A skeleton for the category of finite pointed sets is given by the objects

${\displaystyle n_{+}=\{0,1,\ldots ,n\},}$

where 0 is the basepoint, and the morphisms are the basepoint preserving set maps. Let A be a commutative k-algebra and M be a symmetric A-bimodule[further explanation needed]. The Loday functor L(A,M) is given on objects in ${\displaystyle {\text{Fin}}_{*}}$ by

${\displaystyle n_{+}\mapsto M\otimes A^{\otimes n}.}$

A morphism

${\displaystyle f:m_{+}\to n_{+}}$

is sent to the morphism ${\displaystyle f_{*}}$ given by

${\displaystyle f_{*}(a_{0}\otimes \cdots \otimes a_{n})=b_{0}\otimes \cdots \otimes b_{m}}$

where

${\displaystyle \forall j\in \{0,\ldots ,n\}:\qquad b_{j}={\begin{cases}\prod _{i\in f^{-1}(j)}a_{i}&f^{-1}(j)\neq \emptyset \\1&f^{-1}(j)=\emptyset \end{cases}}}$

### Another description of Hochschild homology of algebras

The Hochschild homology of a commutative algebra A with coefficients in a symmetric A-bimodule M is the homology associated to the composition

${\displaystyle \Delta ^{o}{\overset {S^{1}}{\longrightarrow }}{\text{Fin}}_{*}{\overset {{\mathcal {L}}(A,M)}{\longrightarrow }}k{\text{-mod}},}$

and this definition agrees with the one above.

## Topological Hochschild homology

The above construction of the Hochschild complex can be adapted to more general situations, namely by replacing the category of (complexes of) k-modules by an ∞-category (equipped with a tensor product) C, and A by an associative algebra in this category. Applying this to the category C = Sp of spectra, and A being the Eilenberg–MacLane spectrum associated to an ordinary ring R yields topological Hochschild homology, denoted THH(R). The (non-topological) Hochschild homology introduced above can be reinterpreted along these lines, by taking for C the derived category of Z-modules (as an ∞-category).

Replacing tensor products over the sphere spectrum by tensor products over Z (or the Eilenberg–MacLane-spectrum HZ) leads to a natural comparison map THH(R) → HH(R). It induces an isomorphism on homotopy groups in degrees 0, 1, and 2. In general, however, they are different, and THH tends to yield simpler groups than HH. For example,

${\displaystyle THH(\mathbf {F} _{p})=\mathbf {F} _{p}[x],}$
${\displaystyle HH(\mathbf {F} _{p})=\mathbf {F} _{p}\langle x\rangle }$

is the polynomial ring (with x in degree 2), compared to the ring of divided powers in one variable.

Hesselholt (2016) showed that the Hasse-Weil zeta-function of a smooth proper variety over Fp can be expressed using regularized determinants involving topological Hochschild homology.