View source for 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|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.

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

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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,

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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

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gives a long [[Exact sequence|exact sequence]] of cohomology which, in this case, has the form

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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

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

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Some examples are:

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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" />.

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

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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

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

==Properties of periodic cyclic cohomology.==


Stability. The mapping

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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" />.

Excision. Given an exact sequence of algebras

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there exists an associated six-term exact sequence

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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.

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

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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.

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

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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

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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.

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

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The mapping

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is an isomorphism.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Connes, "Noncommutative geometry" , Acad. Press (1994) {{MR|1303779}} {{ZBL|0818.46076}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. Cuntz, D. Quillen, "Algebra extensions and nonsingularity" ''J. Amer. Math. Soc.'' , '''8''' , Amer. Math. Soc. (1995) pp. 251–289 {{MR|1303029}} {{ZBL|0838.19001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J-L. Loday, "Cyclic homology" , Springer (1991) {{MR|1715877}} {{MR|1616336}} {{MR|1600246}} {{MR|1217970}} {{MR|1045851}} {{MR|0981743}} {{MR|0958781}} {{MR|0925871}} {{MR|0780077}} {{MR|0695381}} {{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}} </TD></TR></table>