Hochschild homology

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

1 Definition of Hochschild homology of algebras
- 1.1 Remark
2 Hochschild homology of functors
- 2.1 Loday functor
- 2.2 Another description of Hochschild homology of algebras
3 References
4 See also

[edit] 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

$d_0(m\otimes a_0 \otimes \cdots \otimes a_n) = ma_1 \otimes a_2 \cdots \otimes a_n$

$d_i(m\otimes a_0 \otimes \cdots \otimes a_n) = m\otimes a_1 \otimes \cdots \otimes a_i a_{i+1} \otimes \cdots \otimes a_n$

$d_n(m\otimes a_0 \otimes \cdots \otimes a_n) = a_n m\otimes a_1 \otimes \cdots \otimes a_{n-1}$

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.

[edit] 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(a₀ ⊗ ··· ⊗ a_n) = a₀ ⊗ ··· a_i ⊗ 1 ⊗ a_i+1 ⊗ ··· ⊗ a_n. Hochschild homology is the homology of this simplicial module.

[edit] Hochschild homology of functors

The simplicial circle S¹ 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 S¹

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

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.

[edit] 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

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