Hochschild homology

In mathematics, Hochschild homology is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors.

Definition of Hochschild homology of algebras

Let k be a ring, A an associative k- algebra, 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 : $d\_0(motimes\; a\_0\; otimes\; cdots\; otimes\; a\_n)\; =\; ma\_1\; otimes\; a\_2\; cdots\; otimes\; a\_n$: $d\_i(motimes\; a\_0\; otimes\; cdots\; otimes\; a\_n)\; =\; motimes\; a\_1\; otimes\; cdots\; otimes\; a\_i\; a\_\{i+1\}\; otimes\; cdots\; otimes\; a\_n$: $d\_n(motimes\; a\_0\; otimes\; cdots\; otimes\; a\_n)\; =\; a\_n\; motimes\; a\_1\; otimes\; cdots\; otimes\; a\_\{n-1\}$

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, ie. 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("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, ie. 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 : $Delta^o\; overset\{S^1\}\{longrightarrow\}\; Fin\_*\; overset\{F\}\{longrightarrow\}\; k\; ext\{-\}operatorname\{mod\}$ .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 : $n\_+\; =\; \{0,1,...,n\}$, 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 :$f:m\_+\; ightarrow\; n\_+$ is sent to the morphism f_* given by : $f\_*(a\_0\; otimes\; cdots\; otimes\; a\_n)\; =\; (b\_0\; otimes\; cdots\; otimes\; b\_m)$ where : $b\_j\; =\; prod\_\{f(i)=j\}\; a\_i,\; ,,\; j=0,...,n,$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 : $Delta^o\; overset\{S^1\}\{longrightarrow\}\; Fin\_*\; overset\{mathcal\{L\}(A,M)\}\{longrightarrow\}\; k\; ext\{-\}operatorname\{mod\},$and this definition agrees with the one above.

References

*Jean-Louis Loday, "Cyclic Homology", Grundlehren der mathematischen Wissenschaften Vol. 301, Springer (1998) ISBN 3-540-63074-0
* [http://www.math.purdue.edu/~jinhyun/note/cyclic/cyclic.pdf A personal note on Hochschild and Cyclic homology]
* [http://www.numdam.org/item?id=ASENS_2000_4_33_2_151_0 Hodge decomposition for higher order Hochschild homology]

ee also

*Cyclic homology