Cyclic cohomology - Imported Knowlpedia Content

Definition

Let $A$ be a unital algebra over a field of characteristic zero. A cyclic $n$ -cochain is a multilinear map $\varphi:A^{n+1}\to k$ satisfying

\varphi(a_n,a_0,\ldots,a_{n-1}) =(-1)^n\varphi(a_0,\ldots,a_n).

The Hochschild coboundary is

\begin{aligned} (b\varphi)(a_0,\ldots,a_{n+1}) ={}&\sum_{j=0}^{n}(-1)^j \varphi(a_0,\ldots,a_ja_{j+1},\ldots,a_{n+1})\\ &+(-1)^{n+1}\varphi(a_{n+1}a_0,a_1,\ldots,a_n). \end{aligned}

It preserves cyclic cochains and satisfies $b^2=0$ . The cyclic cohomology $HC^n(A)$ is the $n$ th cohomology of this subcomplex of the dual Hochschild complex.

Mixed-complex formulation

Over a characteristic-zero field, the cyclic-cochain definition agrees with the cohomology obtained from Connes's $(b,B)$ -bicomplex, where $B$ is the cyclic operator that shifts degree oppositely to $b$ . The mixed-complex viewpoint naturally produces periodic cyclic cohomology and the long exact SBI sequence relating Hochschild and cyclic theories Loday, chapters 2–5.

Pairings and geometric role

Cyclic cocycles pair with algebraic $K$ -theory classes: even cocycles pair with idempotents and odd cocycles with invertibles. In noncommutative geometry, these pairings generalize integration of differential forms and express index pairings. The cyclic Chern character of a Fredholm module is a central example Connes, §§II.1–II.3.

Conventions and scope

References

A. Connes, “Non-Commutative Differential Geometry,” Publications Mathématiques de l'IHÉS 62 (1985), 41–144. DOI record. Relevant: §§II.1–II.3 on cyclic cohomology, the bicomplex, and Chern characters.
J.-L. Loday, Cyclic Homology, 2nd ed., Springer, 1998. Publisher record. Relevant: chapters 2–5 on cyclic modules, the $(b,B)$ -bicomplex, and cyclic cohomology.