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^e-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

$HH_n(A,M) = \text{Tor}_n^{A^e}(A, M)$

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

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

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

with boundary operator d_i defined by

$d_0(m\otimes a_1 \otimes \cdots \otimes a_n) = ma_1 \otimes a_2 \cdots \otimes a_n$

$d_i(m\otimes a_1 \otimes \cdots \otimes a_n) = m\otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n$

$d_n(m\otimes a_1 \otimes \cdots \otimes a_n) = a_n m\otimes a_1 \otimes \cdots \otimes a_{n-1}$

Here a_i is in A for all 1 ≤ i ≤ n and m ∈ M. If we let

$b=\sum_{i=0}^n (-1)^i d_i,$

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(a₀ ⊗ ··· ⊗ a_n) = a₀ ⊗ ··· a_i ⊗ 1 ⊗ a_i+1 ⊗ ··· ⊗ 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 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 S¹

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

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

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

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. The Loday functor L(A,M) is given on objects in Fin_* by

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

A morphism

$f:m_+ \rightarrow n_+$

is sent to the morphism f_* given by

$f_*(a_0 \otimes \cdots \otimes a_n) = (b_0 \otimes \cdots \otimes b_m)$

where

$b_j = \prod_{f(i)=j} a_i, \,\, j=0,\dots,n,$

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

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

and this definition agrees with the one above.

See also

• Cyclic homology

References

• 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