Contents

cohomology

# Contents

## Idea

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 $S^1$-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 $A$ (over a commutative ring $k$). 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 $A$ a cyclic vector space $A^\sharp$, and showing that the cyclic homology of $A$ may be computed as via Ext-groups $Ext^*(A^\sharp, k^\sharp)$. 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 $A$.

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 $X$ coincides with the cyclic homology of $X$ 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.

## Definition

### The chain complex for cyclic homology

Let $A$ be an associative algebra over a ring $k$. Write $C_\bullet(A,A)$ 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:

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

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

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

The homology of the cyclic complex, denoted

$HC_n(A) := H_n( C_\bullet^\lambda(A) )$

is called the cyclic homology of $A$.

###### Definition

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

###### Definition

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

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

Let $X$ be a simply connected topological space.

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

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

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

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

(Jones 87, Thm. A, review in Loday 92, Cor. 7.3.14, Loday 11, Sec 4)

If the coefficients are rational, and $X$ 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 $X$ carries the structure of a smooth manifold, then the singular cochains on $X$ are equivalent to the dgc-algebra of differential forms (the de Rham algebra) and hence in this case the statement becomes that

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

This is known as Jones' theorem (Jones 87)

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

## Properties

### Loday-Quillen-Tsygan theorem

The Loday-Quillen-Tsygan theorem (Loday-Quillen 84, Tsygan 83) states that for any associative algebra, $A$ in characteristic zero, the Lie algebra homology $H_\bullet(\mathfrak{gl}(A))$ of the infinite general linear Lie algebra $\mathfrak{gl}(A)$ with coefficients in $A$ is, up to a degree shift, the exterior algebra $\wedge(HC_{\bullet - 1}(A))$ on the cyclic homology $HC_{\bullet - 1}(A)$ of $A$:

$H_\bullet(\mathfrak{gl}(A)) \;\simeq\; \wedge( HC_{\bullet - 1}(A) )$

(see e.g Loday 07, theorem 1.1).

## References

Foundational articles:

• 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 $Ext^n$, C. R. Acad. Sci. Paris 296, (1983), pp. 953–958, MR86d:18007

• Alain Connes, Cohomologie cyclique et foncteurs $Ext^n$, C.R.A.S. 296 (1983), Série I, 953-958 (pdf, pdf).

• 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 (doi:10.1007/978-3-662-21739-9)

• Christian Kassel, Cyclic homology, comodules and mixed complexes, J. Alg. 107 (1987), 195–216 (doi:10.1016/0021-8693(87)90086-X)

Monographs:

• 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.

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 (doi:10.2478/BF02475213, MR2004f:16011)

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 $K$-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

The relation to cyclic loop spaces:

reviewed in:

On the ordinary cohomology of cyclic loop spaces (“string cohomology”) computed directly (instead of via the isomorphism to cyclic homology):

and specifically for complex projective spaces:

On the cyclic homology of the groupoid convolution algebra of étale Lie groupoids, hence of orbifolds:

The Loday-Quillen-Tsygan theorem is originally due, independently, to

and

• Boris Tsygan, Homology of matrix algebras over rings and the Hochschild homology, Uspeki Math. Nauk., 38:217–218, 1983.

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