Namespaces
Variants
Actions

Difference between revisions of "Cyclic cohomology"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
m (tex encoded by computer)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
Cyclic cohomology was developed as a replacement of the de Rham theory in the context of non-commutative algebras. It was discovered independently by B. Feigin and B. Tsygan as a non-commutative analogue of [[De Rham cohomology|de Rham cohomology]] and by A. Connes as the cohomological structure involved in the computation of indices of elliptic operators (cf. [[Index formulas|Index formulas]]) and the range of a [[Chern character|Chern character]] defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105001.png" />-homology (cf. also [[K-theory|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105002.png" />-theory]]). It plays a fundamental role in various generalizations of index theorems to foliations and coverings (higher indices of Connes and H. Moscovici). Among the more classical applications there are the proof of the Novikov conjecture for a large class of groups and local formulas for Pontryagin classes.
+
<!--
 +
c1105001.png
 +
$#A+1 = 76 n = 0
 +
$#C+1 = 76 : ~/encyclopedia/old_files/data/C110/C.1100500 Cyclic cohomology
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105003.png" /> is a unital [[Algebra|algebra]] over a [[Field|field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105004.png" /> of characteristic zero. Let
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105005.png" /></td> </tr></table>
+
Cyclic cohomology was developed as a replacement of the de Rham theory in the context of non-commutative algebras. It was discovered independently by B. Feigin and B. Tsygan as a non-commutative analogue of [[De Rham cohomology|de Rham cohomology]] and by A. Connes as the cohomological structure involved in the computation of indices of elliptic operators (cf. [[Index formulas|Index formulas]]) and the range of a [[Chern character|Chern character]] defined on  $  K $-
 +
homology (cf. also [[K-theory| $  K $-
 +
theory]]). It plays a fundamental role in various generalizations of index theorems to foliations and coverings (higher indices of Connes and H. Moscovici). Among the more classical applications there are the proof of the Novikov conjecture for a large class of groups and local formulas for Pontryagin classes.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105006.png" /></td> </tr></table>
+
Suppose that  $  A $
 +
is a [[Unital algebra|unital]] [[Algebra|algebra]] over a [[Field|field]]  $  k $
 +
of characteristic zero. Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105007.png" /></td> </tr></table>
+
$$
 +
C  ^ {n} ( A,A  ^ {*} ) = { \mathop{\rm Hom} } _ {k} ( A ^ {\otimes n } ,A  ^ {*} ) ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105008.png" /></td> </tr></table>
+
$$
 +
A  ^ {*} = { \mathop{\rm Hom} } _ {k} ( A,k ) ;  
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c1105009.png" /></td> </tr></table>
+
$$
 +
b \phi ( x _ {0} \dots x _ {n + 1 }  ) =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050010.png" /></td> </tr></table>
+
$$
 +
=  
 +
\sum _ {i = 0 } ^ { n }  ( - 1 )  ^ {i} \phi ( x _ {0} \dots x _ {i} x _ {i + 1 }  \dots x _ {n + 1 }  )  +
 +
$$
  
The complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050011.png" /> computes the Hochschild cohomology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050012.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050013.png" /> with values in the dual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050014.png" />-bimodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050015.png" />. The subspace of cyclic cochains,
+
$$
 +
+
 +
( - 1 ) ^ {n + 1 } \phi ( x _ {n + 1 }  x _ {0} \dots x _ {n} ) ;
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050016.png" /></td> </tr></table>
+
$$
 +
T \phi ( x _ {0} \dots x _ {n} ) = ( - 1 )  ^ {n} \phi ( x _ {n} ,x _ {0} \dots x _ {n - 1 }  ) .
 +
$$
  
is closed under the Hochschild coboundary mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050017.png" /> and the cohomology of the induced complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050018.png" /> is called the cyclic cohomology of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050019.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050020.png" />. This is a contravariant [[Functor|functor]] from the [[Category|category]] of associative algebras to the category of linear spaces. The inclusion of complexes
+
The complex  $  ( C  ^ {*} ( A,A  ^ {*} ) ,b ) $
 +
computes the Hochschild cohomology $  HH  ^ {*} ( A ) $
 +
of $  A $
 +
with values in the dual  $  A $-
 +
bimodule  $  A  ^ {*} $.  
 +
The subspace of cyclic cochains,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050021.png" /></td> </tr></table>
+
$$
 +
CC  ^ {*} ( A ) = \left \{ {\phi \in C  ^ {*} ( A,A  ^ {*} ) } : {T \phi = \phi } \right \}
 +
$$
 +
 
 +
is closed under the Hochschild coboundary mapping  $  b $
 +
and the cohomology of the induced complex  $  ( CC  ^ {*} ( A ) ,b ) $
 +
is called the cyclic cohomology of the algebra  $  A $,
 +
denoted by  $  HC  ^ {*} ( A ) $.  
 +
This is a contravariant [[Functor|functor]] from the [[Category|category]] of associative algebras to the category of linear spaces. The inclusion of complexes
 +
 
 +
$$
 +
( CC  ^ {*} ( A ) ,b ) \subset  ( C  ^ {*} ( A,A  ^ {*} ) ,b )
 +
$$
  
 
gives a long [[Exact sequence|exact sequence]] of cohomology which, in this case, has the form
 
gives a long [[Exact sequence|exact sequence]] of cohomology which, in this case, has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050022.png" /></td> </tr></table>
+
$$
 +
\dots {\rightarrow ^ { B }  } HC ^ {n - 2 } {\rightarrow ^ { S }  } HC  ^ {n} {\rightarrow ^ { I }  } HH  ^ {n} {\rightarrow ^ { B }  } HC ^ {n - 1 } {\rightarrow ^ { S }  } \dots,
 +
$$
  
and is usually called the Connes–Gysin exact sequence. The induced spectral sequence is one of the main tools for the computation of cyclic cohomology. The periodicity operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050023.png" /> gives rise to the stabilized version (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050024.png" />-graded) of cyclic cohomology given by the direct limit
+
and is usually called the Connes–Gysin exact sequence. The induced spectral sequence is one of the main tools for the computation of cyclic cohomology. The periodicity operator $  S $
 +
gives rise to the stabilized version ( $  \mathbf Z/2 \mathbf Z $-
 +
graded) of cyclic cohomology given by the direct limit
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050025.png" /></td> </tr></table>
+
$$
 +
HC _ {\textrm{ per  }  }  ^ {*} ( A ) = {\lim\limits } _  \rightarrow  ( HC  ^ {*} ( A ) , S )
 +
$$
  
 
and called periodic cyclic cohomology. Both cyclic and periodic cyclic cohomologies admit external products:
 
and called periodic cyclic cohomology. Both cyclic and periodic cyclic cohomologies admit external products:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050026.png" /></td> </tr></table>
+
$$
 +
HC _ {\textrm{ per  }  }  ^ {k} ( A ) \otimes HC _ {\textrm{ per  }  }  ^ {k} ( B ) { \mathop \rightarrow \limits ^ {\#}  } HC _ {\textrm{ per  }  }  ^ {k} ( A \otimes B ) .
 +
$$
  
 
Some examples are:
 
Some examples are:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050027.png" /></td> </tr></table>
+
$$
 +
HC  ^ {n} ( k ) = \left \{
 +
\begin{array}{l}
 +
{k \  \textrm{ if  }  n  \textrm{ is  even  } , } \\
 +
{0 \  \textrm{ if  }  n  \textrm{ is  odd  } . }
 +
\end{array}
 +
\right .
 +
$$
  
Any tracial functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050028.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050029.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050030.png" />) on the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050031.png" /> defines an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050032.png" />.
+
Any tracial functional $  \tau $(
 +
i.e. $  \tau ( ab ) = \tau ( ba ) $
 +
for all $  a,b \in A $)  
 +
on the algebra $  A $
 +
defines an element of $  HC  ^ {0} ( A ) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050033.png" /> be a smooth, closed [[Manifold|manifold]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050034.png" />. Restricting to the continuous (in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050035.png" />-topology) cochains, one finds
+
Let $  M $
 +
be a smooth, closed [[Manifold|manifold]] and $  A = C  ^  \infty  ( M ) $.  
 +
Restricting to the continuous (in the $  C  ^  \infty  $-
 +
topology) cochains, one finds
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050036.png" /></td> </tr></table>
+
$$
 +
HC  ^ {k} ( A ) \simeq \oplus _ {i \leq  1 }  H _ {k - 2i,DR }  ( M ) \oplus { \mathop{\rm Ker} } d  ^ {t} \mid  _ {( \Omega  ^ {k}  ( M ) )  ^ {*} } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050037.png" /> is the space of de Rham <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050038.png" />-currents on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050039.png" /> and the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050040.png" /> coincides with the transpose of the de Rham differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050041.png" />. In particular, the periodic cyclic cohomology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050042.png" /> coincides with de Rham homology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050043.png" />. An explicit mapping is given by
+
where $  ( \Omega  ^ {k} ( M ) )  ^ {*} $
 +
is the space of de Rham $  k $-
 +
currents on $  M $
 +
and the operator $  B $
 +
coincides with the transpose of the de Rham differential $  d  ^ {t} $.  
 +
In particular, the periodic cyclic cohomology of $  A $
 +
coincides with de Rham homology of $  M $.  
 +
An explicit mapping is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050044.png" /></td> </tr></table>
+
$$
 +
{ \mathop{\rm Ker} } d  ^ {t} \mid  _ {( \Omega  ^ {k}  ( M ) )  ^ {*} } \ni T ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050045.png" /></td> </tr></table>
+
$$
 +
( f _ {0} \dots f _ {k} ) \mapsto T ( f _ {0} ,df _ {1} \dots df _ {k} ) .
 +
$$
  
 
The definitions given above can be easily extended to non-unital algebras.
 
The definitions given above can be easily extended to non-unital algebras.
  
 
==Properties of periodic cyclic cohomology.==
 
==Properties of periodic cyclic cohomology.==
 
  
 
Stability. The mapping
 
Stability. The mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050046.png" /></td> </tr></table>
+
$$
 +
HC _ {\textrm{ per  }  }  ^ {*} ( A ) \ni \phi \rightarrow { \mathop{\rm Tr} } \# \phi \in HC _ {\textrm{ per  }  }  ^ {*} ( M _ {n} ( A ) )
 +
$$
  
is an isomorphism. This holds also for (non-periodic)) cyclic cohomology. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050047.png" /> is the mapping induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050048.png" /> from the algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050049.png" />-matrices over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050050.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050051.png" />.
+
is an isomorphism. This holds also for (non-periodic)) cyclic cohomology. Here, $  { \mathop{\rm Tr} } \# $
 +
is the mapping induced by $  { { \mathop{\rm Tr} } } : {M _ {n} ( A ) } \rightarrow A $
 +
from the algebra of $  ( n \times n ) $-
 +
matrices over $  A $
 +
to $  A $.
  
 
Excision. Given an exact sequence of algebras
 
Excision. Given an exact sequence of algebras
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050052.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow I { \mathop \rightarrow \limits ^  \iota  } A { \mathop \rightarrow \limits ^  \pi  } A/I \rightarrow 0,
 +
$$
  
 
there exists an associated six-term exact sequence
 
there exists an associated six-term exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050053.png" /></td> </tr></table>
+
$$
 +
 
 +
\begin{array}{ccccc}
 +
HC _ {\textrm{ per  }  } ^ {\textrm{ ev  } } ( A/I )  &  \mathop \rightarrow \limits ^ { {HC ( \pi )  }}    &HC _ {\textrm{ per  }  } ^ {\textrm{ ev  } } ( A )  &  \mathop \rightarrow \limits ^ { {HC ( \iota )  }}    &HC _ {\textrm{ per  }  } ^ {\textrm{ ev  } } ( I )  \\
 +
\uparrow  &{}  &{}  &{}  &\uparrow  \\
 +
HC _ {\textrm{ per  }  } ^ {\textrm{ odd  } } ( I )  & \leftarrow _ {HC ( \iota ) }  &HC _ {\textrm{ per  }  } ^ {\textrm{ odd  } } ( A )  & \leftarrow _ {HC ( \pi ) }  &HC _ {\textrm{ per  }  } ^ {\textrm{ odd  } } ( A/I ) . \\
 +
\end{array}
 +
 
 +
$$
  
The excision holds also in the case of Fréchet algebras (cf. also [[Fréchet algebra|Fréchet algebra]]) if the quotient mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050054.png" /> has a continuous (linear) lifting. For cyclic cohomology, under a certain condition on the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050055.png" /> (called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050057.png" />-unitality) there exists a corresponding long exact sequence.
+
The excision holds also in the case of Fréchet algebras (cf. also [[Fréchet algebra|Fréchet algebra]]) if the quotient mapping $  \pi $
 +
has a continuous (linear) lifting. For cyclic cohomology, under a certain condition on the ideal $  I $(
 +
called $  H $-
 +
unitality) there exists a corresponding long exact sequence.
  
 
Homotopy invariance. Suppose that, in the context of topological algebras and continuous cochains, a differentiable family of homomorphisms is given:
 
Homotopy invariance. Suppose that, in the context of topological algebras and continuous cochains, a differentiable family of homomorphisms is given:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050058.png" /></td> </tr></table>
+
$$
 +
[ 0,1 ] \ni t \rightarrow {\phi _ {t} } : A \rightarrow B .
 +
$$
  
The corresponding mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050059.png" /> on the periodic cyclic cohomology are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050060.png" />-independent. In the context of cyclic cohomology the corresponding result says that inner derivations act trivially.
+
The corresponding mappings $  HC _ {\textrm{ per  }  }  ^ {*} ( \phi _ {t} ) $
 +
on the periodic cyclic cohomology are $  t $-
 +
independent. In the context of cyclic cohomology the corresponding result says that inner derivations act trivially.
  
Pairing with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050062.png" />-theory. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050063.png" /> is a [[Banach algebra|Banach algebra]] and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050064.png" /> is a dense subalgebra closed under holomorphic [[Functional calculus|functional calculus]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050065.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050066.png" /> has been given a [[Fréchet topology|Fréchet topology]] under which its imbedding into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050067.png" /> is continuous. Restricting to continuous cochains, there exists a pairing
+
Pairing with $  K $-
 +
theory. Suppose that $  {\mathcal A} $
 +
is a [[Banach algebra|Banach algebra]] and that $  A $
 +
is a dense subalgebra closed under holomorphic [[Functional calculus|functional calculus]] in $  {\mathcal A} $.  
 +
Suppose that $  A $
 +
has been given a [[Fréchet topology|Fréchet topology]] under which its imbedding into $  {\mathcal A} $
 +
is continuous. Restricting to continuous cochains, there exists a pairing
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050068.png" /></td> </tr></table>
+
$$
 +
K _ {*} ( {\mathcal A} ) \times HC _ {\textrm{ per  }  }  ^ {*} ( A ) \rightarrow \mathbf C.
 +
$$
  
On the level of idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050069.png" />, this pairing is given by
+
On the level of idempotents $  e \in {\mathcal M} _ {n} ( A ) $,  
 +
this pairing is given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050070.png" /></td> </tr></table>
+
$$
 +
\left \langle  {[ e ] , \phi } \right \rangle = { \mathop{\rm Tr} } \# \phi ( e \dots e ) ,  \phi \in HC _ {\textrm{ per  }  } ^ {\textrm{ ev  } } ( A ) .
 +
$$
  
This pairing is consistent with the six-term exact sequences in periodic cyclic cohomology and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050071.png" />-theory.
+
This pairing is consistent with the six-term exact sequences in periodic cyclic cohomology and $  K $-
 +
theory.
  
Bott periodicity. In the topological context, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050072.png" /> be the projective tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050073.png" /> with the algebra of rapidly decreasing functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050074.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050075.png" /> denote the cyclic cocycle on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050076.png" /> given by the current
+
Bott periodicity. In the topological context, let $  SA = S ( \mathbf R ) \otimes  ^  \pi  A $
 +
be the projective tensor product of $  A $
 +
with the algebra of rapidly decreasing functions on $  \mathbf R $.  
 +
Let $  \tau $
 +
denote the cyclic cocycle on $  S ( \mathbf R ) $
 +
given by the current
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050077.png" /></td> </tr></table>
+
$$
 +
( f _ {0} ,f _ {1} ) \mapsto \int\limits {f _ {0} }  {df _ {1} } .
 +
$$
  
 
The mapping
 
The mapping
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c110/c110500/c11050078.png" /></td> </tr></table>
+
$$
 +
HC _ {\textrm{ per  }  }  ^ {*} ( A ) \ni \phi \mapsto \tau \# \phi \in HC _ {\textrm{ per  }  } ^ {* + 1 } ( SA )
 +
$$
  
 
is an isomorphism.
 
is an isomorphism.

Latest revision as of 17:31, 5 June 2020


Cyclic cohomology was developed as a replacement of the de Rham theory in the context of non-commutative algebras. It was discovered independently by B. Feigin and B. Tsygan as a non-commutative analogue of de Rham cohomology and by A. Connes as the cohomological structure involved in the computation of indices of elliptic operators (cf. Index formulas) and the range of a Chern character defined on $ K $- homology (cf. also $ K $- theory). It plays a fundamental role in various generalizations of index theorems to foliations and coverings (higher indices of Connes and H. Moscovici). Among the more classical applications there are the proof of the Novikov conjecture for a large class of groups and local formulas for Pontryagin classes.

Suppose that $ A $ is a unital algebra over a field $ k $ of characteristic zero. Let

$$ C ^ {n} ( A,A ^ {*} ) = { \mathop{\rm Hom} } _ {k} ( A ^ {\otimes n } ,A ^ {*} ) , $$

$$ A ^ {*} = { \mathop{\rm Hom} } _ {k} ( A,k ) ; $$

$$ b \phi ( x _ {0} \dots x _ {n + 1 } ) = $$

$$ = \sum _ {i = 0 } ^ { n } ( - 1 ) ^ {i} \phi ( x _ {0} \dots x _ {i} x _ {i + 1 } \dots x _ {n + 1 } ) + $$

$$ + ( - 1 ) ^ {n + 1 } \phi ( x _ {n + 1 } x _ {0} \dots x _ {n} ) ; $$

$$ T \phi ( x _ {0} \dots x _ {n} ) = ( - 1 ) ^ {n} \phi ( x _ {n} ,x _ {0} \dots x _ {n - 1 } ) . $$

The complex $ ( C ^ {*} ( A,A ^ {*} ) ,b ) $ computes the Hochschild cohomology $ HH ^ {*} ( A ) $ of $ A $ with values in the dual $ A $- bimodule $ A ^ {*} $. The subspace of cyclic cochains,

$$ CC ^ {*} ( A ) = \left \{ {\phi \in C ^ {*} ( A,A ^ {*} ) } : {T \phi = \phi } \right \} $$

is closed under the Hochschild coboundary mapping $ b $ and the cohomology of the induced complex $ ( CC ^ {*} ( A ) ,b ) $ is called the cyclic cohomology of the algebra $ A $, denoted by $ HC ^ {*} ( A ) $. This is a contravariant functor from the category of associative algebras to the category of linear spaces. The inclusion of complexes

$$ ( CC ^ {*} ( A ) ,b ) \subset ( C ^ {*} ( A,A ^ {*} ) ,b ) $$

gives a long exact sequence of cohomology which, in this case, has the form

$$ \dots {\rightarrow ^ { B } } HC ^ {n - 2 } {\rightarrow ^ { S } } HC ^ {n} {\rightarrow ^ { I } } HH ^ {n} {\rightarrow ^ { B } } HC ^ {n - 1 } {\rightarrow ^ { S } } \dots, $$

and is usually called the Connes–Gysin exact sequence. The induced spectral sequence is one of the main tools for the computation of cyclic cohomology. The periodicity operator $ S $ gives rise to the stabilized version ( $ \mathbf Z/2 \mathbf Z $- graded) of cyclic cohomology given by the direct limit

$$ HC _ {\textrm{ per } } ^ {*} ( A ) = {\lim\limits } _ \rightarrow ( HC ^ {*} ( A ) , S ) $$

and called periodic cyclic cohomology. Both cyclic and periodic cyclic cohomologies admit external products:

$$ HC _ {\textrm{ per } } ^ {k} ( A ) \otimes HC _ {\textrm{ per } } ^ {k} ( B ) { \mathop \rightarrow \limits ^ {\#} } HC _ {\textrm{ per } } ^ {k} ( A \otimes B ) . $$

Some examples are:

$$ HC ^ {n} ( k ) = \left \{ \begin{array}{l} {k \ \textrm{ if } n \textrm{ is even } , } \\ {0 \ \textrm{ if } n \textrm{ is odd } . } \end{array} \right . $$

Any tracial functional $ \tau $( i.e. $ \tau ( ab ) = \tau ( ba ) $ for all $ a,b \in A $) on the algebra $ A $ defines an element of $ HC ^ {0} ( A ) $.

Let $ M $ be a smooth, closed manifold and $ A = C ^ \infty ( M ) $. Restricting to the continuous (in the $ C ^ \infty $- topology) cochains, one finds

$$ HC ^ {k} ( A ) \simeq \oplus _ {i \leq 1 } H _ {k - 2i,DR } ( M ) \oplus { \mathop{\rm Ker} } d ^ {t} \mid _ {( \Omega ^ {k} ( M ) ) ^ {*} } , $$

where $ ( \Omega ^ {k} ( M ) ) ^ {*} $ is the space of de Rham $ k $- currents on $ M $ and the operator $ B $ coincides with the transpose of the de Rham differential $ d ^ {t} $. In particular, the periodic cyclic cohomology of $ A $ coincides with de Rham homology of $ M $. An explicit mapping is given by

$$ { \mathop{\rm Ker} } d ^ {t} \mid _ {( \Omega ^ {k} ( M ) ) ^ {*} } \ni T , $$

$$ ( f _ {0} \dots f _ {k} ) \mapsto T ( f _ {0} ,df _ {1} \dots df _ {k} ) . $$

The definitions given above can be easily extended to non-unital algebras.

Properties of periodic cyclic cohomology.

Stability. The mapping

$$ HC _ {\textrm{ per } } ^ {*} ( A ) \ni \phi \rightarrow { \mathop{\rm Tr} } \# \phi \in HC _ {\textrm{ per } } ^ {*} ( M _ {n} ( A ) ) $$

is an isomorphism. This holds also for (non-periodic)) cyclic cohomology. Here, $ { \mathop{\rm Tr} } \# $ is the mapping induced by $ { { \mathop{\rm Tr} } } : {M _ {n} ( A ) } \rightarrow A $ from the algebra of $ ( n \times n ) $- matrices over $ A $ to $ A $.

Excision. Given an exact sequence of algebras

$$ 0 \rightarrow I { \mathop \rightarrow \limits ^ \iota } A { \mathop \rightarrow \limits ^ \pi } A/I \rightarrow 0, $$

there exists an associated six-term exact sequence

$$ \begin{array}{ccccc} HC _ {\textrm{ per } } ^ {\textrm{ ev } } ( A/I ) & \mathop \rightarrow \limits ^ { {HC ( \pi ) }} &HC _ {\textrm{ per } } ^ {\textrm{ ev } } ( A ) & \mathop \rightarrow \limits ^ { {HC ( \iota ) }} &HC _ {\textrm{ per } } ^ {\textrm{ ev } } ( I ) \\ \uparrow &{} &{} &{} &\uparrow \\ HC _ {\textrm{ per } } ^ {\textrm{ odd } } ( I ) & \leftarrow _ {HC ( \iota ) } &HC _ {\textrm{ per } } ^ {\textrm{ odd } } ( A ) & \leftarrow _ {HC ( \pi ) } &HC _ {\textrm{ per } } ^ {\textrm{ odd } } ( A/I ) . \\ \end{array} $$

The excision holds also in the case of Fréchet algebras (cf. also Fréchet algebra) if the quotient mapping $ \pi $ has a continuous (linear) lifting. For cyclic cohomology, under a certain condition on the ideal $ I $( called $ H $- unitality) there exists a corresponding long exact sequence.

Homotopy invariance. Suppose that, in the context of topological algebras and continuous cochains, a differentiable family of homomorphisms is given:

$$ [ 0,1 ] \ni t \rightarrow {\phi _ {t} } : A \rightarrow B . $$

The corresponding mappings $ HC _ {\textrm{ per } } ^ {*} ( \phi _ {t} ) $ on the periodic cyclic cohomology are $ t $- independent. In the context of cyclic cohomology the corresponding result says that inner derivations act trivially.

Pairing with $ K $- theory. Suppose that $ {\mathcal A} $ is a Banach algebra and that $ A $ is a dense subalgebra closed under holomorphic functional calculus in $ {\mathcal A} $. Suppose that $ A $ has been given a Fréchet topology under which its imbedding into $ {\mathcal A} $ is continuous. Restricting to continuous cochains, there exists a pairing

$$ K _ {*} ( {\mathcal A} ) \times HC _ {\textrm{ per } } ^ {*} ( A ) \rightarrow \mathbf C. $$

On the level of idempotents $ e \in {\mathcal M} _ {n} ( A ) $, this pairing is given by

$$ \left \langle {[ e ] , \phi } \right \rangle = { \mathop{\rm Tr} } \# \phi ( e \dots e ) , \phi \in HC _ {\textrm{ per } } ^ {\textrm{ ev } } ( A ) . $$

This pairing is consistent with the six-term exact sequences in periodic cyclic cohomology and $ K $- theory.

Bott periodicity. In the topological context, let $ SA = S ( \mathbf R ) \otimes ^ \pi A $ be the projective tensor product of $ A $ with the algebra of rapidly decreasing functions on $ \mathbf R $. Let $ \tau $ denote the cyclic cocycle on $ S ( \mathbf R ) $ given by the current

$$ ( f _ {0} ,f _ {1} ) \mapsto \int\limits {f _ {0} } {df _ {1} } . $$

The mapping

$$ HC _ {\textrm{ per } } ^ {*} ( A ) \ni \phi \mapsto \tau \# \phi \in HC _ {\textrm{ per } } ^ {* + 1 } ( SA ) $$

is an isomorphism.

References

[a1] A. Connes, "Noncommutative geometry" , Acad. Press (1994) MR1303779 Zbl 0818.46076
[a2] J. Cuntz, D. Quillen, "Algebra extensions and nonsingularity" J. Amer. Math. Soc. , 8 , Amer. Math. Soc. (1995) pp. 251–289 MR1303029 Zbl 0838.19001
[a3] J-L. Loday, "Cyclic homology" , Springer (1991) MR1715877 MR1616336 MR1600246 MR1217970 MR1045851 MR0981743 MR0958781 MR0925871 MR0780077 MR0695381 Zbl 0964.19003 Zbl 0928.19001 Zbl 0885.18007 Zbl 0780.18009 Zbl 0719.19002 Zbl 0686.18006 Zbl 0669.13006 Zbl 0637.16013 Zbl 0565.17006
How to Cite This Entry:
Cyclic cohomology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_cohomology&oldid=24061
This article was adapted from an original article by R. Nest (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article