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 simplicial 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.

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