nLab
cyclic cohomology

Idea

The graded subspace of Hochschild complex of an qassociative algebra consisting of invariant chains under certain actions of symmetric groups, is in fact a subcomplex, the cyclic subcomplex. Cyclic homology is Hochschild homology on cyclically invariant chains.

There are many versions (e.g. periodic cyclic homology) and generalities of the notion (e.g. for schemes, for algebras over cyclic operads). Topological cyclic homology? is an adaptation to ring spectra.

We may understand Hochschild homology as the cohomology of free loop space objects (as described there). These free loop space objects are canonically equipped with a circle group-action that rotates the loops. Cyclic homology is the corresponding $S^1$-equivariant cohomology of free loop space objects.

Definition

The chain complex for cyclic homology

Let $A$ be an associative algebra over a ring $k$. Write $C_{•}(A,A)$ for the Hochschild homology chain complex of $A$ with coefficients in $A$.

For each $n\in \mathbb{N}$ let $\lambda :C_n(A,A)\to C_n(A,A)$ be the $k$-linear map that cyclically permutes the elements and introduces a sign:

$$\lambda :(a_0,a_1,\cdots ,a_{n-1},a_n)\mapsto (-1{)}^n(a_n,a_0,\cdots ,a_{n-1}).$$\lambda : (a_0, a_1, \cdots, a_{n-1}, a_n) \mapsto (-1)^n (a_n, a_0 , \cdots, a_{n-1}) \,.

Definition

The cyclic homology complex $C_{•}^{\lambda }(A)$ of $A$ is the quotient of the Hochschild homology complex of $A$ by cyclic permutations:

$$C_{•}^{\lambda }(A):=C_{•}(A,A)/\mathrm{im}(\mathrm{Id}-\lambda ).$$C_\bullet^\lambda(A) := C_\bullet(A,A)/im(Id-\lambda) \,.

The homology of the cyclic complex, denoted

$${\mathrm{HC}}_n(A):=H_n(C_{•}^{\lambda }(A))$$HC_n(A) := H_n( C_\bullet^\lambda(A) )

is called the cyclic homology of $A$.

If $I\subset A$ is an ideal, then the relative cyclic homology groups ${\mathrm{HC}}_n(A,I)$ are the homology groups of the complex $C_{•}(A,I)=\ker (C_{•}(A)\to C_{•}(A/I))$.

Related entries and literature

• $n$Lab: cycle category, cyclic object, Hochschild homology

The complex for cyclic homology was considered by Alain Connes and Boris Tsygan around 1981, 1982.

Monographs:

• J-L. Loday, Cyclic homology, Grundleheren Math.Wiss. 301, Springer, 2nd ed.
• Alain Connes, Noncommutative geometry, Acad. Press 1994, 661 p. PDF
• Max Karoubi, Homologie cyclique et K-théorie, Astérique 149, Société Mathématique de France (1987).

Quick lecture notes:

• D. Kaledin, Tokyo lectures “Homological methods in non-commutative geometry”, (pdf, TeX); and related but different Seoul lectures

• Masoud Khalkhali, A short survey of cyclic cohomology, arxiv/1008.1212

Some influential original references from 1980s:

• A. Connes, Noncommutative differential geometry, Part I, the Chern character in $K$-homology, Preprint, Inst. Hautes Études Sci., Bures-sur-Yvette, 1982; Part II, de Rham homology and noncommutative algebra, Preprint, IHÉS 1983; Cohomologie cyclique et foncteurs ${\mathrm{Ext}}^n$, C. R. Acad. Sci. Paris 296, (1983), pp. 953–958, MR86d:18007
• Boris Tsygan, Boris Feigin, Additive K-theory, in LNM 1289 (1987), edited by Yu. I. Manin, pp. 67–209, seminar 1984-1986 in Moscow), MR89a:18017; Аддитивная K-теория и кристальные когомологии, Функц. анализ и его прил., 19:2 (1985), 52–-62, pdf, MR88e:18008; Engl. transl. in B. L. Feĭgin, B. L. Tsygan, Additive $K$-theory and crystalline cohomology, Functional Analysis and Its Applications, 1985, 19:2, 124–132.
• T. Goodwillie, Cyclic homology, derivations, and the free loopspace, Topology 24 (1985), no. 2, 187–215, MR87c:18009, doi
• John D.S. Jones, Cyclic homology and equivariant homology, Invent. Math. 87, 403-423 (1987) pdf

Some modern illuminating references:

• D. Kaledin, Cyclic homology with coefficients, math.KT/0702068, to appear in Yu. Manin’s 70th anniversary volume.
• E. Getzler, M. Kapranov, Cyclic operads and cyclic homology, in: “Geometry, Topology and Physics for R. Bott”, ed. S.-T. Yau, p. 167-201, International Press, Cambridge MA, 1995, pdf
• Teimuraz Pirashvili, Birgit Richter, Hochschild and cyclic homology via functor homology, K-Theory 25 (2002), no. 1, 39–49, MR2003c:16011, doi
• Jolanta Słomińska, Decompositions of the category of noncommutative sets and Hochschild and cyclic homology, Cent. Eur. J. Math. 1 (2003), no. 3, 327–331, MR2004f:16011, doi