Hochschild homology

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

${\displaystyle {\begin{aligned}d_{0}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=ma_{1}\otimes a_{2}\cdots \otimes a_{n}\\d_{i}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=m\otimes a_{1}\otimes \cdots \otimes a_{i}a_{i+1}\otimes \cdots \otimes a_{n}\\d_{n}(m\otimes a_{1}\otimes \cdots \otimes a_{n})&=a_{n}m\otimes a_{1}\otimes \cdots \otimes a_{n-1}\end{aligned}}}$

where 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 ${\displaystyle b\circ b=0,}$ 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.

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

${\displaystyle s_{i}(a_{0}\otimes \cdots \otimes a_{n})=a_{0}\otimes \cdots \otimes a_{i}\otimes 1\otimes a_{i+1}\otimes \cdots \otimes a_{n}.}$

Hochschild homology is the homology of this simplicial module.

A skeleton for the category of finite pointed sets is given by the objects

${\displaystyle n_{+}=\{0,1,\ldots ,n\},}$

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 ${\displaystyle {\text{Fin}}_{*}}$ by

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

A morphism

${\displaystyle f:m_{+}\to n_{+}}$

is sent to the morphism ${\displaystyle f_{*}}$ given by

${\displaystyle f_{*}(a_{0}\otimes \cdots \otimes a_{n})=b_{0}\otimes \cdots \otimes b_{m}}$

where

${\displaystyle \forall j\in \{0,\ldots ,n\}:\qquad b_{j}={\begin{cases}\prod _{i\in f^{-1}(j)}a_{i}&f^{-1}(j)\neq \emptyset \\1&f^{-1}(j)=\emptyset \end{cases}}}$

The Hochschild homology of a commutative algebra A with coefficients in a symmetric A-bimodule M is the homology associated to the composition

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

and this definition agrees with the one above.