In
noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
and related branches of mathematics, cyclic homology and cyclic cohomology are certain (co)homology theories for
associative algebras which generalize the
de Rham (co)homology of manifolds. These notions were independently introduced by
Boris Tsygan (homology) and
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...
(cohomology) in the 1980s. These invariants have many interesting relationships with several older branches of mathematics, including de Rham theory, Hochschild (co)homology, group cohomology, and the
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
. Contributors to the development of the theory include
Max Karoubi
__NOTOC__
Max Karoubi () is a French mathematician, topologist, who works on K-theory, cyclic homology and noncommutative geometry and who founded the first European Congress of Mathematics.
In 1967, he received his Ph.D. in mathematics (Doct ...
, Yuri L. Daletskii,
Boris Feigin,
Jean-Luc Brylinski,
Mariusz Wodzicki
Mariusz Wodzicki (Count Wodzicki) (born 1956) is a Polish mathematician and nobleman, whose works primarily focus on analysis, algebraic k-theory, noncommutative geometry, and algebraic geometry.
Wodzicki was born in Bytom, Poland in 1956. He rec ...
,
Jean-Louis Loday
Jean-Louis Loday (12 January 1946 – 6 June 2012) was a French mathematician who worked on cyclic homology and who introduced Leibniz algebras (sometimes called Loday algebras) and Zinbiel algebras.
He occasionally used the pseudonym Guillaume Wil ...
, Victor Nistor,
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
,
Joachim Cuntz, Ryszard Nest, Ralf Meyer, and Michael Puschnigg.
Hints about definition
The first definition of the cyclic homology of a ring ''A'' over a field of
characteristic zero, denoted
:''HC''
''n''(''A'') or ''H''
''n''λ(''A''),
proceeded by the means of the following explicit
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
related to the
Hochschild homology complex of ''A'', called the Connes complex:
For any natural number ''n ≥ 0'', define the operator
which generates the natural cyclic action of
on the ''n''-th tensor product of ''A'':
:
Recall that the Hochschild complex groups of ''A'' with coefficients in ''A'' itself are given by setting
for all ''n ≥ 0''. Then the components of the Connes complex are defined as
, and the differential
is the restriction of the Hochschild differential to this quotient. One can check that the Hochschild differential does indeed factor through to this space of coinvariants.
Connes later found a more categorical approach to cyclic homology using a notion of cyclic object in an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
, which is analogous to the notion of
simplicial object
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
. In this way, cyclic homology (and cohomology) may be interpreted as a
derived functor
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.
Motivation
It was noted in vari ...
, which can be explicitly computed by the means of the (''b'', ''B'')-bicomplex. If the field ''k'' contains the rational numbers, the definition in terms of the Connes complex calculates the same homology.
One of the striking features of cyclic homology is the existence of a
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
connecting
Hochschild and cyclic homology. This long exact sequence is referred to as the periodicity sequence.
Case of commutative rings
Cyclic cohomology of the commutative algebra ''A'' of regular functions on an
affine algebraic variety
Affine may describe any of various topics concerned with connections or affinities.
It may refer to:
* Affine, a relative by marriage in law and anthropology
* Affine cipher, a special case of the more general substitution cipher
* Affine com ...
over a field ''k'' of characteristic zero can be computed in terms of
Grothendieck's
algebraic de Rham complex. In particular, if the variety ''V''=Spec ''A'' is smooth, cyclic cohomology of ''A'' are expressed in terms of the
de Rham cohomology of ''V'' as follows:
:
This formula suggests a way to define de Rham cohomology for a 'noncommutative spectrum' of a noncommutative algebra ''A'', which was extensively developed by Connes.
Variants of cyclic homology
One motivation of cyclic homology was the need for an approximation of
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
that is defined, unlike K-theory, as the homology of a
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of t ...
. Cyclic cohomology is in fact endowed with a pairing with K-theory, and one hopes this pairing to be non-degenerate.
There has been defined a number of variants whose purpose is to fit better with algebras with topology, such as
Fréchet algebra
In mathematics, especially functional analysis, a Fréchet algebra, named after Maurice René Fréchet, is an associative algebra A over the real or complex numbers that at the same time is also a (locally convex) Fréchet space. The multiplicati ...
s,
-algebras, etc. The reason is that K-theory behaves much better on topological algebras such as
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s or
C*-algebras than on algebras without additional structure. Since, on the other hand, cyclic homology degenerates on C*-algebras, there came up the need to define modified theories. Among them are entire cyclic homology due to
Alain Connes
Alain Connes (; born 1 April 1947) is a French mathematician, and a theoretical physicist, known for his contributions to the study of operator algebras and noncommutative geometry. He is a professor at the , , Ohio State University and Vand ...
, analytic cyclic homology due to Ralf Meyer or asymptotic and local cyclic homology due to Michael Puschnigg. The last one is very close to
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
as it is endowed with a bivariant
Chern character
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
from
KK-theory.
Applications
One of the applications of cyclic homology is to find new proofs and generalizations of the
Atiyah-Singer index theorem. Among these generalizations are index theorems based on spectral triples and
deformation quantization of
Poisson structures.
An
elliptic operator D on a compact smooth manifold defines a class in K homology. One invariant of this class is the analytic index of the operator. This is seen as the pairing of the class
with the element 1 in HC(C(M)). Cyclic cohomology can be seen as a way to get higher invariants of elliptic differential operators not only for smooth manifolds, but also for foliations,
orbifolds, and singular spaces that appear in noncommutative geometry.
Computations of algebraic K-theory
The
cyclotomic trace map is a map from
algebraic K-theory
Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
(of a ring ''A'', say), to cyclic homology:
:
In some situations, this map can be used to compute K-theory by means of this map. A pioneering result in this direction is a theorem of : it asserts that the map
:
between the relative K-theory of ''A'' with respect to a ''nilpotent'' two-sided ideal ''I'' to the relative cyclic homology (measuring the difference between K-theory or cyclic homology of ''A'' and of ''A''/''I'') is an isomorphism for ''n''≥1.
While Goodwillie's result holds for arbitrary rings, a quick reduction shows that it is in essence only a statement about
. For rings not containing Q, cyclic homology must be replaced by topological cyclic homology in order to keep a close connection to K-theory. (If Q is contained in ''A'', then cyclic homology and topological cyclic homology of ''A'' agree.) This is in line with the fact that (classical)
Hochschild homology
In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a field, ...
is less well-behaved than topological Hochschild homology for rings not containing Q. proved a far-reaching generalization of Goodwillie's result, stating that for a commutative ring ''A'' so that the
Henselian lemma holds with respect to the ideal ''I'', the relative K-theory is isomorphic to relative topological cyclic homology (without tensoring both with Q). Their result also encompasses a theorem of , asserting that in this situation the relative K-theory spectrum modulo an integer ''n'' which is invertible in ''A'' vanishes. used Gabber's result and
Suslin rigidity to reprove Quillen's computation of the K-theory of
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
s.
See also
*
Noncommutative geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some g ...
Notes
References
*
*
*
*
*
* .
Errata
External links
*
A personal note on Hochschild and Cyclic homology
{{DEFAULTSORT:Cyclic Homology
Homological algebra