Periodic Cyclic CohomologyEdit

Periodic cyclic cohomology is a two-periodic cohomology theory for associative algebras that plays a central role in noncommutative geometry. It generalizes classical invariants such as de Rham cohomology to settings where the underlying “space” need not be a manifold, and where algebras of operators or functions encode geometric data in a noncommutative framework. The theory emerges from cyclic cohomology, which was developed by Alain Connes as a natural cohomological framework for algebras, building on the ideas of [Hochschild cohomology] and encoding cyclic symmetry. The periodic version, often denoted HP^*(A) for an algebra A, is 2-periodic and comes equipped with pairings to K-theory that yield index-type numerical invariants for noncommutative spaces.

In the landscape of noncommutative geometry, periodic cyclic cohomology serves as a bridge between algebraic invariants and analytical data. It pairs with [K-theory] to produce index-type numbers for operators in noncommutative settings, mirroring the classical Chern character from K-theory to de Rham cohomology in the commutative case. This perspective connects the algebraic side with geometry and analysis, and it provides a robust framework for formulating and proving index theorems in contexts where traditional geometric methods do not apply. For a smooth manifold M, the theory recovers familiar invariants: HP^*(C^∞(M)) reproduces the de Rham cohomology groups, tying the noncommutative theory back to classical topology when the algebra is commutative.

This article surveys the main ideas, constructions, and consequences of periodic cyclic cohomology, without assuming prior exposure to the subject. Along the way, it highlights how the theory interacts with communities around [noncommutative geometry], [K-theory], and index theory, and it points to key results that have shaped the field.

Definition and construction

Periodic cyclic cohomology is defined for associative algebras A over a field of characteristic zero. The starting point is the cyclic cochain complex, built from sequences of multilinear maps on A that respect cyclic symmetry. The ordinary cyclic complex is equipped with two fundamental operators: the Hochschild coboundary b and Connes’ operator B. Together, (C^*(A), b, B) forms a mixed complex, from which cyclic and Hochschild theories are derived. The periodic cyclic complex is obtained by a formal periodicization of this mixed complex, which produces a 2-periodic object. The resulting cohomology groups are denoted HP^0(A), HP^1(A), and in general HP^{n+2}(A) ≅ HP^n(A).

Key features in this construction include: - The b operator encodes Hochschild-type coboundaries, reflecting the associative structure of A. - The B operator encodes the cyclic symmetry and provides the mechanism that links Hochschild and cyclic theories. - The periodic structure arises from the existence of a canonical periodicity operator that shifts degree by two, yielding the 2-periodic nature of HP^*. - The theory generalizes and unifies several other cohomology theories, and it specializes to de Rham cohomology in the commutative smooth case.

For more on the foundational elements, see [cyclic cohomology] and [Hochschild cohomology], and the role of mixed complexes in homological algebra. In detailed treatments, one often discusses the long exact sequence relating Hochschild, cyclic, and negative cyclic cohomology and how the periodic theory sits inside this framework.

Periodic cyclic cohomology and the commutative case

When A is the algebra of smooth functions on a manifold M, A = C^∞(M), periodic cyclic cohomology recovers classical differential-topological information. Concretely, HP^*(C^∞(M)) is 2-periodic and is isomorphic to the de Rham cohomology groups of M: - HP^0(C^∞(M)) corresponds to the even de Rham cohomology H_{dR}^{even}(M), - HP^1(C^∞(M)) corresponds to the odd de Rham cohomology H_{dR}^{odd}(M).

This identification rests on the Hochschild–Kostant–Rosenberg theorem, which relates Hochschild cohomology of smooth commutative algebras to differential forms, together with the cyclic structure that collapses to de Rham data in the commutative setting. For those who want a precise bridge, the HKR theorem provides the canonical isomorphism HH^(C^∞(M)) ≅ Ω^(M) in the appropriate degrees, and cyclic theory then translates into the familiar de Rham invariants.

For readers exploring this topic, the connection between de Rham cohomology and periodic cyclic cohomology offers a concrete anchor: the noncommutative theory extends these classical invariants to algebras that do not arise from spaces in the ordinary sense, while still reproducing the familiar topological information when the algebra is commutative.

The Chern character and index theory

A central feature of periodic cyclic cohomology is its natural pairing with K-theory. There is a Chern character map - ch: K_(A) → HP_(A), which generalizes the classical Chern character from topological K-theory to de Rham cohomology. This pairing allows the computation of numerical invariants associated with projections and unitary elements in matrix algebras over A, yielding index-type formulas that extend the Atiyah–Singer paradigm to noncommutative spaces.

In the noncommutative setting, the index theorem takes shape as a formula expressing the index of a suitably defined elliptic operator in terms of the pairing between K-theory and HP^*(A). Connes organized much of this framework, showing how cyclic cohomology supplies the natural receptacle for index data in the absence of a traditional geometric cycle. The interplay among [K-theory], [cyclic cohomology], and analytical tools continues to be a fertile ground for both abstract theory and concrete calculations, with applications ranging from operator algebras to mathematical physics.

Examples and computations

Classical case: For a compact smooth manifold M, the HP groups reproduce the de Rham cohomology of M, with HP^0 capturing the even forms and HP^1 capturing the odd forms. This confirms that the noncommutative theory specializes correctly to familiar geometry in the commutative limit.
Noncommutative examples: The theory yields computable invariants for noncommutative tori and other quantum spaces studied in [noncommutative geometry]. In these examples, HP^*(A) encodes geometric-like data (such as dimensions and symmetries) in an algebraic framework.
Index-type calculations: For an elliptic operator modeled in a noncommutative setting (e.g., a pseudo-differential operator on a noncommutative space), the index is recovered by pairing its K-theory class with HP^*(A), illustrating the power of the framework to handle generalized spaces.

Variations and related theories

Cyclic cohomology: The periodic theory sits inside the broader family of cyclic cohomology theories, which also include Hochschild cohomology, cyclic homology, and negative cyclic cohomology. The periodic version is particularly well-suited for index theory due to its 2-periodicity.
Mixed complexes and the long exact sequence: The machinery of mixed complexes provides a unifying language for these theories, and the Connes long exact sequence links Hochschild, cyclic, and negative cyclic theories.
Bivariant and topological refinements: Bivariant theories extend the framework to morphisms between algebras, and topological cyclic homology (TC) and related constructions offer robust homotopy-theoretic refinements that interact with modern algebraic K-theory.
HKR and de Rham connections: The HKR theorem is a key bridge between algebraic and differential-geometric viewpoints, showing how smooth commutative algebras connect with differential forms and de Rham cohomology.

Controversies and debates (mathematical context)

As with many powerful, abstract frameworks, periodic cyclic cohomology has its debates within the mathematical community. Some lines of discussion include: - Practical computability: In broad noncommutative settings, computing HP^(A) explicitly can be challenging. Critics note that some invariants are difficult to extract in concrete terms, while supporters emphasize the structural insight and the existence of index-type formulas that drive calculations in particular classes of algebras. - Generality versus concreteness: Periodic cyclic cohomology provides a unifying language for index phenomena across diverse noncommutative spaces, but some mathematicians argue for more concrete, geometric models in specific contexts. Proponents of the cyclic framework counter that the general theory reveals invariants invisible to classical approaches. - Role in physics: The noncommutative geometry program has found resonance in mathematical physics, especially in models where space-time may be noncommutative. The use of HP^(A) to formulate invariants in such theories is debated, with viewpoints ranging from strong endorsement for its unifying potential to caution about overreliance on formal invariants without physical interpretation. - Evolution of the field: The rise of homotopy-theoretic methods and topological cyclic homology has spurred discussion about the relative merits of different frameworks for capturing K-theoretic and index data in a robust, computable way. This debate reflects broader tensions between operator-algebraic approaches and more homotopy-theoretic, categorical ones, each offering distinct advantages.