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\section*{cyclic homology}

[[!redirects cyclic cohomology]]

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{the_chain_complex_for_cyclic_homology}{The chain complex for cyclic homology}\dotfill \pageref*{the_chain_complex_for_cyclic_homology} \linebreak
\noindent\hyperlink{ordinary_cohomology_of__and_cyclic_homology_of_}{Ordinary cohomology of $\mathcal{L}X/S^1$ and cyclic homology of $X$}\dotfill \pageref*{ordinary_cohomology_of__and_cyclic_homology_of_} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{lodayquillentsygan_theorem}{Loday-Quillen-Tsygan theorem}\dotfill \pageref*{lodayquillentsygan_theorem} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

[[Hochschild homology]] may be understood as the [[cohomology]] of [[free loop space object]]s (as described there). These free loop space objects are canonically equipped with a [[circle group]]-[[action]] that rotates the loops. \emph{Cyclic homology} is the corresponding $S^1$-[[equivariant cohomology]] of free loop space objects.

Like [[Hochschild homology]], [[cyclic homology]] is an \emph{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). [[Alain Connes|Connes]] gave another definition by associating to $A$ a [[cyclic object|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 [[Charles Weibel|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]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{the_chain_complex_for_cyclic_homology}{}\subsubsection*{{The chain complex for cyclic homology}}\label{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:

\begin{displaymath}
\lambda : (a_0, a_1, \cdots, a_{n-1}, a_n)
  \mapsto 
   (-1)^n
   (a_n, a_0  , \cdots, a_{n-1})
  \,.
\end{displaymath}
\begin{udefn}
The \textbf{cyclic homology complex} $C^\lambda_\bullet(A)$ of $A$ is the quotient of the Hochschild homology complex of $A$ by cyclic permutations:

\begin{displaymath}
C_\bullet^\lambda(A) := C_\bullet(A,A)/im(Id-\lambda)
  \,.
\end{displaymath}
The [[homology]] of the cyclic complex, denoted

\begin{displaymath}
HC_n(A) := H_n( C_\bullet^\lambda(A) )
\end{displaymath}
is called the \textbf{cyclic homology} of $A$.

\end{udefn}
\begin{udefn}
The \textbf{cyclic cohomology} groups of $A$ Are the [[cohomology]] groups of the dual [[cochain complex]], denoted $HC^n(A)$.

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

\end{udefn}
\hypertarget{ordinary_cohomology_of__and_cyclic_homology_of_}{}\subsubsection*{{Ordinary cohomology of $\mathcal{L}X/S^1$ and cyclic homology of $X$}}\label{ordinary_cohomology_of__and_cyclic_homology_of_}

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

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

\begin{displaymath}
H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
\end{displaymath}
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:

\begin{displaymath}
H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
\end{displaymath}
(\hyperlink{Loday11}{Loday 11})

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then this may be computed by the \emph{[[Sullivan model for free loop spaces]]}, see there the section on \emph{\href{Sullivan+model+of+free+loop+space#RelationToHochschildHomology}{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

\begin{displaymath}
H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
\end{displaymath}
\begin{displaymath}
H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
\end{displaymath}
This is known as \emph{[[Jones' theorem]]} (\hyperlink{Jones87}{Jones 87})

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at \emph{\href{Hochschild+cohomology#JonesTheorem}{Hochschild cohomology -- Jones' theorem}}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{lodayquillentsygan_theorem}{}\subsubsection*{{Loday-Quillen-Tsygan theorem}}\label{lodayquillentsygan_theorem}

The \emph{[[Loday-Quillen-Tsygan theorem]]} (\hyperlink{LodayQuillen84}{Loday-Quillen 84}, \hyperlink{Tsygan83}{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 $A$:

\begin{displaymath}
H_\bullet(\mathfrak{gl}(A))
  \;\simeq\;
  \wedge( HC_{\bullet - 1}(A) )
\end{displaymath}
(see e.g \hyperlink{Loday07}{Loday 07, theorem 1.1}).

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[additive K-theory]]


\item [[cycle category]], [[cyclic object]], [[Hochschild homology]]


\item [[Sullivan model of free loop space]]


\item [[dihedral homology]]


\item [[topological cyclic homology]]


\item [[Loday-Quillen-Tsygan theorem]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Foundational articles:

\begin{itemize}%
\item [[A. Connes]], \emph{Noncommutative differential geometry, Part I, the Chern character in $K$-homology}, Preprint, Inst. Hautes \'E{}tudes Sci., Bures-sur-Yvette, 1982; \emph{Part II, de Rham homology and noncommutative algebra}, Preprint, IH\'E{}S 1983; \emph{Cohomologie cyclique et foncteurs $Ext^n$}, C. R. Acad. Sci. Paris \textbf{296}, (1983), pp. 953--958, \href{http://www.ams.org/mathscinet-getitem?mr=777584}{MR86d:18007}
\item [[A. Connes]], \emph{Cohomologie cyclique et foncteur $Ext^n$}, Comptes Rendues Ac. Sci. Paris S\'e{}r. A-B, 296 (1983), 953-958.
\item [[B. L. Tsygan]], \emph{The homology of matrix Lie algebras over rings and the Hochschild homology}, Uspekhi Mat. Nauk, 38:2(230) (1983), 217--218.
\item [[Jean-Louis Loday]], [[Daniel Quillen]], \emph{Cyclic homology and the Lie algebra homology of matrices}, Comment. Math. Helvetici 59 (1984) 565-591.
\item [[Christian Kassel]], \emph{Cyclic homology, comodules and mixed complexes}, J. Alg. 107 (1987), 195--216.

\end{itemize}
Monographs:

\begin{itemize}%
\item [[Jean-Louis Loday]], \emph{Cyclic homology}, Grundlehren Math.Wiss. \textbf{301}, Springer (1998)


\item [[Alain Connes]], \emph{Noncommutative geometry}, Acad. Press 1994, 661 p. \href{http://www.alainconnes.org/docs/book94bigpdf.pdf}{PDF}


\item [[Max Karoubi]], \emph{Homologie cyclique et K-th\'e{}orie}, Ast\'e{}rique \textbf{149}, Soci\'e{}t\'e{} Math\'e{}matique de France (1987).


\item [[Ib Madsen]], \emph{Algebraic K-theory and traces}, Current Developments in Mathematics, 1995.



\end{itemize}
Quick lecture notes:

\begin{itemize}%
\item [[D. Kaledin]], \emph{Tokyo lectures ``Homological methods in non-commutative geometry''}, (\href{http://imperium.lenin.ru/~kaledin/tokyo/final.pdf}{pdf}, \href{http://imperium.lenin.ru/~kaledin/tokyo/final.tex}{TeX}); and related but different \href{http://imperium.lenin.ru/~kaledin/seoul}{Seoul lectures}


\item [[Masoud Khalkhali]], \emph{A short survey of cyclic cohomology}, \href{http://arxiv.org/abs/1008.1212}{arxiv/1008.1212}



\end{itemize}
Some modern treatments:

\begin{itemize}%
\item [[Bernhard Keller]], \emph{On the cyclic homology of ringed spaces and schemes}, Doc. Math. J. DMV 3 (1998), 231-259, \href{http://www.math.jussieu.fr/~keller/publ/KellerHCSchemes.pdf}{pdf}.


\item [[Bernhard Keller]], \emph{On the cyclic homology of exact categories}, Journal of Pure and Applied Algebra 136 (1999), 1-56, \href{http://www.math.jussieu.fr/~keller/publ/cyex.pdf}{pdf}.


\item [[Bernhard Keller]], \emph{Invariance and Localization for Cyclic Homology of DG algebras}, Journal of Pure and Applied Algebra, 123 (1998), 223-273, \href{http://www.math.jussieu.fr/~keller/publ/ilc.pdf}{pdf}.


\item [[Charles Weibel]], \emph{Cyclic homology for schemes}, Proceedings of the AMS, 124 (1996), 1655-1662, \href{http://www.math.uiuc.edu/K-theory/0043/}{web}.


\item [[D. Kaledin]], \emph{Cyclic homology with coefficients}, \href{http://arxiv.org/abs/math.KT/0702068}{math.KT/0702068}, to appear in Yu. Manin's 70th anniversary volume.


\item [[E. Getzler]], [[M. Kapranov]], \emph{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, \href{http://www.math.northwestern.edu/~getzler/Papers/cyclic.pdf}{pdf}


\item [[Teimuraz Pirashvili]], [[Birgit Richter]], \emph{Hochschild and cyclic homology via functor homology}, K-Theory \textbf{25} (2002), no. 1, 39--49, \href{http://www.ams.org/mathscinet-getitem?mr=1899698}{MR2003c:16011}, \href{http://dx.doi.org/10.1023/A:1015064621329}{doi}


\item Jolanta Somiska, \emph{Decompositions of the category of noncommutative sets and Hochschild and cyclic homology}, Cent. Eur. J. Math. \textbf{1} (2003), no. 3, 327--331, \href{http://www.ams.org/mathscinet-getitem?mr=1992895}{MR2004f:16011}, \href{http://dx.doi.org/10.2478/BF02475213}{doi}



\end{itemize}
Some influential original references from 1980s:

\begin{itemize}%
\item [[Boris Tsygan]], [[Boris Feigin]], \emph{[[Additive K-theory]]}, in LNM 1289 (1987), edited by Yu. I. Manin, pp. 67--209, seminar 1984-1986 in Moscow), \href{http://www.ams.org/mathscinet-getitem?mr=923136}{MR89a:18017}; \emph{ K-   }, .    ., 19:2 (1985), 52---62, \href{http://www.mathnet.ru/php/getFT.phtml?jrnid=faa&paperid=1358&what=fullt&option_lang=rus}{pdf}, \href{http://www.ams.org/mathscinet-getitem?mr=800920}{MR88e:18008}; Engl. transl. in B. L. Fegin, B. L. Tsygan, \emph{Additive $K$-theory and crystalline cohomology}, Functional Analysis and Its Applications, 1985, 19:2, 124--132.
\item [[T. Goodwillie]], \emph{Cyclic homology, derivations, and the free loopspace}, Topology \textbf{24} (1985), no. 2, 187--215, \href{http://www.ams.org/mathscinet-getitem?mr=793184}{MR87c:18009}, 

\end{itemize}
The relation to [[cyclic loop spaces]]:

\begin{itemize}%
\item [[John D.S. Jones]], \emph{Cyclic homology and equivariant homology}, Invent. Math. \textbf{87}, 403-423 (1987) (\href{https://math.berkeley.edu/~nadler/jones.pdf}{pdf})


\item [[Jean-Louis Loday]], \emph{Free loop space and homology} (\href{https://arxiv.org/abs/1110.0405}{arXiv:1110.0405})



\end{itemize}
The [[Loday-Quillen-Tsygan theorem]] is originally due, independently, to

\begin{itemize}%
\item [[Jean-Louis Loday]], [[Daniel Quillen]], \emph{Cyclic homology and the Lie algebra homology of matrices} Comment. Math. Helv., 59(4):569–591, 1984.

\end{itemize}
and

\begin{itemize}%
\item [[Boris Tsygan]], \emph{Homology of matrix algebras over rings and the Hochschild homology}, Uspeki Math. Nauk., 38:217–218, 1983.

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Jean-Louis Loday]], \emph{Cyclic Homology Theory, Part II}, notes taken by Pawe l Witkowsk (2007) (\href{https://www.impan.pl/swiat-matematyki/notatki-z-wyklado~/loday_cht_2.pdf}{pdf})

\end{itemize}
[[!redirects cyclic cohomology]] [[!redirects cyclic (co)homology]]



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