Cyclic cohomology

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