Hochschild cycle - Imported Knowlpedia Content

Definition

Let $A$ be a unital algebra over a commutative ring $k$ , let $M$ be an $A$ -bimodule, and let

\bigl(C_\bullet(A,M),b\bigr)

be the Hochschild chain complex. A Hochschild $n$ -cycle is an element $c\in C_n(A,M)$ satisfying $b(c)=0$ . The module of such cycles is

Z_n(A,M)=\ker\!\left(b:C_n(A,M)\to C_{n-1}(A,M)\right).

A cycle determines a class $[c]$ in the Hochschild homology module

HH_n(A,M)=Z_n(A,M)/bC_{n+1}(A,M).

Accordingly, a cycle is a representative chain, not the homology class itself; two cycles represent the same class exactly when their difference is a Hochschild boundary.

Low-degree examples

Every zero-chain is a zero-cycle because the boundary out of degree zero vanishes. When $M=A$ , a one-chain $a_0\otimes a_1$ has boundary

b(a_0\otimes a_1)=a_0a_1-a_1a_0.

It is therefore a cycle precisely when $a_0$ and $a_1$ commute. Sums of one-chains can be cycles even when their individual summands are not.

Orientation in spectral geometry

The orientation axiom for a spectral triple uses a Hochschild cycle with coefficients in a bimodule involving the opposite algebra. Its represented image under

a_0\otimes\cdots\otimes a_n \longmapsto a_0[D,a_1]\cdots[D,a_n]

is required to recover the grading in the even case, or the identity in the odd case, subject to the selected convention. This is extra geometric data, not a property of every Hochschild cycle Connes, chapter VI.

Conventions and scope

References

J.-L. Loday, Cyclic Homology, 2nd ed., Springer, 1998. Publisher record. Relevant: §1.1 on Hochschild chains, cycles, boundaries, and homology.
A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: chapters III and VI on Hochschild homology and the orientation axiom.