# Hochschild homology

(Redirected from Hochschild cohomology)

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 Cartan & Eilenberg (1956).

## Definition of Hochschild homology of algebras

Let k be a ring, 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

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

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

For a commutative algebra, there is a nice derived-geometric interpretation of hochschield homology: it is the derived intersection of the diagonal in the derived product; i.e.

${\displaystyle HH_{k}(A,A)=H^{-k}\left(A\otimes _{A\otimes _{k}^{\mathbb {L} }A}^{\mathbb {L} }A\right)}$

### 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

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

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

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

then b ° 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 si(a0 ⊗ ··· ⊗ an) = a0 ⊗ ··· ai ⊗ 1 ⊗ ai+1 ⊗ ··· ⊗ an. Hochschild homology is the homology of this simplicial module.

## Hochschild homology of functors

The simplicial circle S1 is a simplicial object in the category Fin* of finite pointed sets, i.e. a functor ΔoFin*. Thus, if F is a functor F: Fink-mod, we get a simplicial module by composing F with S1

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

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

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

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 Fin* by

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

A morphism

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

is sent to the morphism f* given by

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

where

${\displaystyle b_{j}=\prod _{f(i)=j}a_{i},\,\,j=0,\dots ,n,}$

and bj = 1 if f −1(j) = ∅.

### 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

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

and this definition agrees with the one above.