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

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

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

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

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

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

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

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. 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 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

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

and this definition agrees with the one above.

See also

• Cyclic homology

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