Hochschild homology

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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[edit]

Let k be a ring, A an associative k-algebra, and M an A-bimodule. The enveloping algebra of A is the tensor product Ae=AAo 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 Ae-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[edit]

Let k be a ring, A an associative k-algebra that is a projective k-module, and M an A-bimodule. We will write An 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 di 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 ai is in A for all 1 ≤ in and mM. If we let

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

then b ° b = 0, so (Cn(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[edit]

The maps di are face maps making the family of modules Cn(A,M) a simplicial object in the category of k-modules, i.e. a functor Δok-mod, where Δ is the simplicial category and k-mod is the category of k-modules. Here Δo is the opposite category of Δ. The degeneracy maps are defined by si(a0 ⊗ ··· ⊗ an) = a0 ⊗ ··· ai ⊗ 1 ⊗ ai+1 ⊗ ··· ⊗ an. Hochschild homology is the homology of this simplicial module.

Hochschild homology of functors[edit]

The simplicial circle S1 is a simplicial object in the category Fin* of finite pointed sets, i.e. a functor ΔoFin*. Thus, if F is a functor F: Fink-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[edit]

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[further explanation needed]. 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 bj = 1 if f −1(j) = ∅.

Another description of Hochschild homology of algebras[edit]

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.

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