# 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 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 *A ^{e}*=

*A*⊗

*A*

^{o}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

*A*-modules. Cartan & Eilenberg (1956) defined the Hochschild homology and cohomology group of

^{e}*A*with coefficients in

*M*in terms of the Tor functor and Ext functor by

### 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 *A*^{⊗n } for the *n*-fold tensor product of *A* over *k*. The chain complex that gives rise to Hochschild homology is given by

with boundary operator *d*_{i} defined by

Here *a*_{i} is in *A* for all 1 ≤ *i* ≤ *n* and *m* ∈ *M*. If we let

then *b* ° *b* = 0, so (*C*_{n}(*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 *d*_{i} are face maps making the family of modules *C*_{n}(*A*,*M*) a simplicial object in the category of *k*-modules, i.e. a functor Δ^{o} → *k*-mod, where *Δ* is the simplex category and *k*-mod is the category of *k*-modules. Here Δ^{o} is the opposite category of Δ. The degeneracy maps are defined by *s*_{i}(*a*_{0} ⊗ ··· ⊗ *a*_{n}) = *a*_{0} ⊗ ··· *a*_{i} ⊗ 1 ⊗ *a*_{i+1} ⊗ ··· ⊗ *a*_{n}. Hochschild homology is the homology of this simplicial module.

## Hochschild homology of functors

The simplicial circle *S*^{1} is a simplicial object in the category *Fin _{*}* of finite pointed sets, i.e. a functor Δ

^{o}→

*Fin*. Thus, if F is a functor

_{*}*F*:

*Fin*→

*k*-mod, we get a simplicial module by composing F with

*S*

^{1}

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

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. The Loday functor *L(A,M)* is given on objects in *Fin _{*}* by

A morphism

is sent to the morphism f_{*} given by

where

and *b*_{j} = 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

and this definition agrees with the one above.

## See also

## References

- Cartan, Henri; Eilenberg, Samuel (1956),
*Homological algebra*, Princeton Mathematical Series,**19**, Princeton University Press, ISBN 978-0-691-04991-5, MR 0077480 - Govorov, V.E.; Mikhalev, A.V. (2001), "Cohomology of algebras", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4 - Hochschild, G. (1945), "On the cohomology groups of an associative algebra",
*Annals of Mathematics. Second Series*,**46**: 58–67, ISSN 0003-486X, JSTOR 1969145, MR 0011076 - Jean-Louis Loday,
*Cyclic Homology*, Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0 - Richard S. Pierce,
*Associative Algebras*, Graduate Texts in Mathematics (88), Springer, 1982. - Teimuraz Pirashvili, Hodge decomposition for higher order Hochschild homology

## External links

- Dylan G.L. Allegretti,
*Differential Forms on Noncommutative Spaces*. An elementary introduction to noncommutative geometry which uses Hochschild homology to generalize differential forms). - Hochschild cohomology in
*nLab*