% This is a Plain TeX file (1 page)

\overfullrule=0pt
\magnification=1200
\hsize=12.50cm    
\vsize=19cm
\nopagenumbers
\hoffset=0.5cm

\centerline {\bf Cyclic Homology}
\medskip
\noindent {\bf Hochschild homology}. For any associative algebra $A$ over a field $K$, and any $A$-bimodule $M$, there is
defined the Hochschild chain complex $(C_*(A,M), b)$ as follows: $C_n(A,M):= M\otimes 
A^{\otimes n}$ and the boundary map is given by
$$
b(a_0,a_1,\ldots ,a_n)=\sum^{n-1}_{i=0}(-1)^i(a_0,\ldots ,a_ia_{i+1},\ldots ,a_n)
+
(-1)^n(a_na_0,a_1,\ldots ,a_{n-1}),
$$
where $a_0\in M$ and $a_i\in A$ for $i\geq 1$. The homology of this complex is the
{\it Hochschild homology} of $A$ with coefficients in $M$. 

\noindent {\bf Cyclic homology}. When $M=A$ a nice phenomenon appears. Let $t$ be the cyclic
operator on 
$A^{\otimes n+1}, t(a_0,a_1,\ldots ,a_n):= (-1)^n(a_n,a_0,\ldots ,a_{n-1})$. Alain Connes
remarked that the operator $b$ is still well-defined on the quotient $C^{\lambda} _n(A)$ of
$C_n(A,A)$ by the action of $t$, and defines a chain complex $C^{\lambda} _*(A)$. Its homology is
denoted $H^{\lambda} _*(A)$.

There is another way of using the cyclic operator $t$. Let $b':A^{\otimes n+1}\to
A^{\otimes n}$ be given by $
b'(a_0,\ldots ,a_n):=\sum^{n-1}_{i=0}(-1)^i(a_0,\ldots ,a_ia_{i+1},\ldots ,a_n),$
and put $N=1+t+\cdots +t^n$ (observe that $t^{n+1}={\rm id}$). Then one can show that
the following relations hold : $(1-t)b'=b(1-t),\quad {\rm and}\quad b'N=Nb.$

As a consequence, one can build a bicomplex whose module of $(p,q)$-chains is
$A^{\otimes q}$, whose vertical differential is alternatively $b$ or $-b'$ and whose
horizontal differential is alternatively $1-t$ or $N$. The homology of the total complex
is, by definition, the {\it cyclic homology} of the algebra $A$,and denoted $HC_n(A)$.

When the field $K$ is of characteristic zero there is an isomorphism $H^{\lambda} _*(A)\cong 
HC_*(A)$. 

\noindent {\bf Connes periodicity sequence}. The relationship between Hochschild homology and cyclic homology is given by
the Connes periodicity exact sequence :
$$
\cdots \to HH_n(A)\buildrel{I}\over \to HC_n(A)\buildrel{S}\over\to HC_{n-2}(A)\buildrel
{B}\over \to HH_{n-1}(A)\to\cdots .
$$
 and generalization of
it to higher $K$-theory. It plays a prominent role in non-commutative geometry.

\noindent {\bf Chern character.} Let $e$ be a matrix with coefficients in $A$, such that $e^2=e$.
This idempotent defines an element in the Grothendieck group $K_0(A)$.  On the other hand, the
element  $e\otimes\cdots \otimes e$ ($2n+1$ factors) is a cyclic cycle and there is a
well-defined map $Ch : K_0(A)\to HC_{2n}(A)$.  It plays a prominent role in non-commutative geometry.
There is a generalization of it to higher $K$-theory.

 Another important feature of cyclic homology is its close relationship
with the homology of the Lie algebra of matrices, and with de Rham homology for smooth algebras.
 \smallskip

\noindent {\bf Ariane's thread:}

\noindent J.-L. Loday,   {\bf Cyclic homology}. Grund. math.
Wiss., 301. Springer-Verlag, Berlin, 2nd edition 1998.

\hfill ${}_{\rm JLL/mek/CyclicHomology/01.98}$
 \end