Hochschild chain complex - Imported Knowlpedia Content

Definition

Let $k$ be a commutative ring, $A$ a unital associative $k$ -algebra, and $M$ an $A$ -bimodule. The Hochschild chain complex has

C_n(A,M)=M\otimes_k A^{\otimes_k n}\qquad(n\geq0)

and boundary $b:C_n\to C_{n-1}$ given by

\begin{aligned} b(m\otimes a_1\otimes\cdots\otimes a_n) ={}&ma_1\otimes a_2\otimes\cdots\otimes a_n\\ &+\sum_{i=1}^{n-1}(-1)^i m\otimes\cdots\otimes a_i a_{i+1}\otimes\cdots\otimes a_n\\ &+(-1)^n a_n m\otimes a_1\otimes\cdots\otimes a_{n-1}. \end{aligned}

The identity $b^2=0$ makes this an chain complex, whose homology is Hochschild homology.

Homology and low degrees

Its homology is Hochschild homology $HH_n(A,M)$ . In degree zero,

HH_0(A,M)=M/\langle ma-am:a\in A,\ m\in M\rangle.

For $M=A$ , one writes $C_n(A)=A^{\otimes(n+1)}$ and $HH_n(A)=HH_n(A,A)$ . The final term in $b$ is what closes the linear chain cyclically by using the left $A$ -action on $M$ .

Normalization and cyclic theory

For a unital algebra, quotienting by chains having some $a_i=1$ produces the normalized Hochschild complex, which has the same homology. The Hochschild boundary is also one of the two operators in Connes's cyclic bicomplex; the additional operator $B$ supplies the cyclic direction. These constructions and sign checks appear in Loday, §1.1.

Conventions and scope

References

J.-L. Loday, Cyclic Homology, 2nd ed., Springer, 1998. Publisher record. Relevant: §1.1, the Hochschild complex and boundary.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: chapter III, Hochschild and cyclic complexes.