Cyclic homology

Cyclic homology
Name	Cyclic homology
Field	Algebraic topology; Homological algebra; Noncommutative geometry
Introduced	1980s
Introduced by	Alain Connes; Boris Tsygan

Cyclic homology

Cyclic homology is an invariant for associative algebras introduced in the 1980s that extends Hochschild homology and interacts with de Rham cohomology, K-theory, and techniques from Homological algebra. It was developed in the context of ideas from Alain Connes and Boris Tsygan and has since played a central role in Noncommutative geometry and the study of operator algebras such as C*-algebra and von Neumann algebra. The theory links methods from Category theory, Algebraic topology, and Representation theory to produce computable invariants for rings, schemes, and operator algebras.

Introduction

Cyclic homology arose as a refinement of Hochschild homology motivated by questions in Noncommutative geometry and index theory associated with the Atiyah–Singer index theorem, Alain Connes's work on the Novikov conjecture, and developments in K-theory by Daniel Quillen and Max Karoubi. Early contributors included Jean-Louis Loday, Michael Cuntz, Dennis Sullivan, André Weil, and Thierry Lambre, while formal categorical foundations were clarified using ideas related to the Eilenberg–MacLane space and the Simplicial set framework. Cyclic homology connects to invariants studied by Gerstenhaber and Murray Gerstenhaber and underpins formulas used in the Atiyah–Bott localization technique and the study of characteristic classes by Bott and Chern.

Definitions and basic constructions

One starts with an associative algebra A over a field (often Complex numbers), its Hochschild complex (built from tensor powers of A), and the Connes boundary operator B introduced by Alain Connes; these assemble into the mixed complex formalism developed in parallel by Boris Tsygan and Jean-Louis Loday. The cyclic bicomplex involves the Hochschild differential b and Connes' operator B, producing total complexes whose homology yields cyclic, negative cyclic, and periodic cyclic homologies; related constructions appear in work of Daniel Quillen and Max Karoubi. The formalism uses categorical language from Category theory and examples draw on modules over Ring (mathematics) such as matrix algebras studied by Issai Schur and Claude Chevalley.

Connes' cyclic category and cyclic objects

Connes introduced the cyclic category Λ to organize cyclic symmetries; this category refines the Simplicial category Δ and has objects indexed by finite cyclically ordered sets, connecting to the concept of cyclic objects in any Category theory-valued functor category. Cyclic objects studied by Jean-Louis Loday, Alain Connes, and Boris Tsygan allow passage from algebraic data to homological invariants, paralleling constructions in the theory of Simplicial sets and the Dold–Kan correspondence. The cyclic category underlies applications to the Chern character from K-theory to cyclic homology and interacts with fixed-point ideas in the work of Gottlieb and fixed-point indices used by Lefschetz.

Computation and examples

Concrete computations include cyclic homology of polynomial algebras, matrix algebras, group algebras of discrete groups like the Free group and Finite groups, and smooth function algebras on manifolds such as those appearing in Differential topology and Symplectic geometry. Results by Jean-Louis Loday, Boris Tsygan, Max Karoubi, and Michael Cuntz compute cyclic homology of commutative algebras and relate it to de Rham cohomology of smooth varieties studied by Alexander Grothendieck and Jean-Pierre Serre. Examples from operator algebras use techniques from C*-algebra theory developed by Irving Kaplansky and John von Neumann and connect to invariants in Index theory explored by Atiyah and Singer.

Relations to Hochschild and de Rham homology

Cyclic homology sits in long exact sequences with Hochschild homology and negative cyclic homology via Connes' SBI (or periodicity) exact sequence introduced by Alain Connes and elaborated by Boris Tsygan and Jean-Louis Loday. For smooth commutative algebras over Complex numbers and schemes studied by Alexander Grothendieck and Grothendieck–Serre duality phenomena, periodic cyclic homology recovers de Rham cohomology and the algebraic de Rham theorem connects to work by Bernard Malgrange and Pierre Deligne. These relationships underpin comparisons used in results by C. Weibel and André relating algebraic K-theory to cyclic invariants.

Periodic cyclic homology and spectral sequences

Periodic cyclic homology, introduced by Alain Connes and developed by Boris Tsygan and Jean-Louis Loday, is obtained by inverting the periodicity operator in the mixed complex and admits spectral sequences from the Hochschild-to-cyclic filtration; such spectral sequences parallel constructions by Jean Leray and techniques used in the Serre spectral sequence. Convergence and E2-term identifications follow methods from Homological algebra and Spectral sequence theory as formulated by Jean Leray, Jean-Pierre Serre, and Henri Cartan. Computational tools draw on ideas from Koszul duality explored by J. L. Loday and Vladimir Drinfeld.

Applications and connections to K-theory and noncommutative geometry

Cyclic homology provides the target for Chern character maps from Algebraic K-theory developed by Daniel Quillen and Max Karoubi, giving bridges between algebraic invariants and analytic indices in the work of Atiyah and Singer. In Noncommutative geometry pioneered by Alain Connes, cyclic homology plays a role analogous to de Rham homology for noncommutative spaces, contributing to formulations of the Baum–Connes conjecture and index pairing calculations used by researchers such as Gennadi Kasparov and Nigel Higson. Applications extend to deformation quantization studied by Maxim Kontsevich and to invariants of foliations and groupoids treated by Jean Renault and André Haefliger.

Category:Homological algebra