cyclic homology




Special and general types

Special notions


Extra structure





Hochschild homology may be understood 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 1S^1-equivariant cohomology of free loop space objects.

Like Hochschild homology, cyclic homology is an additive invariant of dg-categories or stable infinity-categories, in the sense of noncommutative motives. It also admits a Dennis trace map from algebraic K-theory, and has been successful in allowing computations of the latter.

There are several definitions for the cyclic homology of an associative algebra AA (over a commutative ring kk). Alain Connes originally defined cyclic homology over fields of characteristic zero, as the homology groups of a cyclic variant of the chain complex computing Hochschild homology. Jean-Louis Loday and Daniel Quillen gave a definition via a certain double complex (for arbitrary commutative rings). Connes gave another definition by associating to AA a cyclic vector space A A^\sharp, and showing that the cyclic homology of AA may be computed as via Ext-groups Ext *(A ,k )Ext^*(A^\sharp, k^\sharp). A fourth definition was given by Christian Kassel, who showed that the cyclic homology groups may be computed as the homology groups of a certain mixed complex associated to AA.

Following Alexandre Grothendieck, Charles Weibel gave a definition of cyclic homology (and Hochschild homology) for schemes, using hypercohomology. On the other hand, the definition of Christian Kassel via mixed complexes was extended by Bernhard Keller to linear categories and dg-categories, and he showed that the cyclic homology of the dg-category of perfect complexes on a (nice) scheme XX coincides with the cyclic homology of XX in the sense of Weibel.

There are closely related variants called periodic cyclic homology? and negative cyclic homology?. There is a version for ring spectra called topological cyclic homology.


The chain complex for cyclic homology

Let AA be an associative algebra over a ring kk. Write C (A,A)C_\bullet(A,A) for the Hochschild homology chain complex of AA with coefficients in AA.

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

λ:(a 0,a 1,,a n1,a n)(1) n(a n,a 0,,a n1). \lambda : (a_0, a_1, \cdots, a_{n-1}, a_n) \mapsto (-1)^n (a_n, a_0 , \cdots, a_{n-1}) \,.

The cyclic homology complex C λ(A)C^\lambda_\bullet(A) of AA is the quotient of the Hochschild homology complex of AA by cyclic permutations:

C λ(A):=C (A,A)/im(Idλ). C_\bullet^\lambda(A) := C_\bullet(A,A)/im(Id-\lambda) \,.

The homology of the cyclic complex, denoted

HC n(A):=H n(C λ(A)) HC_n(A) := H_n( C_\bullet^\lambda(A) )

is called the cyclic homology of AA.


The cyclic cohomology groups of AA Are the cohomology groups of the dual cochain complex, denoted HC n(A)HC^n(A).


If IAI\subset A is an ideal, then the relative cyclic homology groups HC n(A,I)HC_n(A,I) are the homology groups of the complex C (A,I)=ker(C (A)C (A/I))C_\bullet(A,I) = ker(C_\bullet(A)\to C_\bullet(A/I)).

Ordinary cohomology of X/S 1\mathcal{L}X/S^1 and cyclic homology of XX

Let XX be a simply connected topological space.

The ordinary cohomology H H^\bullet of its free loop space is the Hochschild homology HH HH_\bullet of its singular chains C (X)C^\bullet(X):

H (X)HH (C (X)). H^\bullet(\mathcal{L}X) \simeq HH_\bullet( C^\bullet(X) ) \,.

Moreover the S 1S^1-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space X/ hS 1\mathcal{L}X/^h S^1 is the cyclic homology HC HC_\bullet of the singular chains:

H (X/ hS 1)HC (C (X)) H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( C^\bullet(X) )

(Loday 11)

If the coefficients are rational, and XX is of finite type then this may be computed by the Sullivan model for free loop spaces, see there the section on Relation to Hochschild homology.

In the special case that the topological space XX carries the structure of a smooth manifold, then the singular cochains on XX are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

H (X)HH (Ω (X)). H^\bullet(\mathcal{L}X) \simeq HH_\bullet( \Omega^\bullet(X) ) \,.
H (X/ hS 1)HC (Ω (X)). H^\bullet(\mathcal{L}X/^h S^1) \simeq HC_\bullet( \Omega^\bullet(X) ) \,.

This is known as Jones' theorem (Jones 87)

An infinity-category theoretic proof of this fact is indicated at Hochschild cohomology – Jones’ theorem.


Foundational articles:

  • A. Connes, Noncommutative differential geometry, Part I, the Chern character in KK-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 Ext nExt^n, C. R. Acad. Sci. Paris 296, (1983), pp. 953–958, MR86d:18007
  • A. Connes, Cohomologie cyclique et foncteur Ext nExt^n, Comptes Rendues Ac. Sci. Paris Sér. A-B, 296 (1983), 953-958.
  • B. L. Tsygan, The homology of matrix Lie algebras over rings and the Hochschild homology, Uspekhi Mat. Nauk, 38:2(230) (1983), 217–218.
  • Jean-Louis Loday, Daniel Quillen, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helvetici 59 (1984) 565-591.
  • Christian Kassel, Cyclic homology, comodules and mixed complexes, J. Alg. 107 (1987), 195–216.


  • Jean-Louis Loday, Cyclic homology, Grundlehren Math.Wiss. 301, Springer (1998)

  • Alain Connes, Noncommutative geometry, Acad. Press 1994, 661 p. PDF

  • Max Karoubi, Homologie cyclique et K-théorie, Astérique 149, Société Mathématique de France (1987).

  • Ib Madsen, Algebraic K-theory and traces, Current Developments in Mathematics, 1995.

Quick lecture notes:

Some modern treatments:

  • Bernhard Keller, On the cyclic homology of ringed spaces and schemes, Doc. Math. J. DMV 3 (1998), 231-259, pdf.

  • Bernhard Keller, On the cyclic homology of exact categories, Journal of Pure and Applied Algebra 136 (1999), 1-56, pdf.

  • Bernhard Keller, Invariance and Localization for Cyclic Homology of DG algebras, Journal of Pure and Applied Algebra, 123 (1998), 223-273, pdf.

  • Charles Weibel, Cyclic homology for schemes, Proceedings of the AMS, 124 (1996), 1655-1662, web.

  • 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

Some influential original references from 1980s:

  • 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 KK-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

The relation to cyclic loop spaces:

Revised on February 23, 2017 13:07:34 by Urs Schreiber (