CYCLIC HOMOLOGY. Jean-Louis LODAY. 2nd edition Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, xviii+ pp. The basic object of study in cyclic homology are algebras. We shall thus begin [9] Loday, J-L., Cyclic Homology, Grundlehren der math. Wissenschaften . Cyclic homology will be seen to be a natural generalization of de Rham Jean- Louis Loday. .. Hochschild, cyclic, dihedral and quaternionic homology.

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There are several definitions for the cyclic homology of an associative algebra A A over a commutative ring k k. Hochschild homology may be understood as the cohomology lpday free loop space object s as described there.

Sullivan model of free loop space. If the coefficients are rationaland X X is of finite type then this may be computed by the Sullivan model for free loop spacessee there the section on Relation to Hochschild homology.

## CYCLIC HOMOLOGY

The Loday-Quillen-Tsygan theorem is originally due, independently, to. Jean-Louis LodayFree loop space and homology arXiv: Bernhard KellerOn the cyclic homology of exact categoriesJournal of Pure and Applied Homologt, pdf.

JonesCyclic homology and equivariant homologyInvent. See the history of this page for a list of all contributions to it. The relation to cyclic loop spaces:. Let A A be an associative algebra over a ring k k.

In the special case that the topological space X X carries the structure of a smooth manifoldthen the singular cochains on X X are equivalent to the dgc-algebra of differential forms the de Rham algebra and hence in this case the statement becomes that. There is a version for ring spectra called topological cyclic homology.

It also admits a Dennis trace map homilogy algebraic K-theoryand has been successful in allowing computations of the latter. Bernhard KellerOn the cyclic homology of ringed spaces and cyclixDoc.

KaledinCyclic homology with coefficientsmath. Following Alexandre GrothendieckCharles Weibel gave a definition of cyclic homology and Hochschild homology for schemesusing hypercohomology. Like Hochschild homologycyclic homology is an additive invariant of dg-categories or stable infinity-categoriesin the sense of noncommutative motives.

### cyclic homology in nLab

Alain ConnesNoncommutative geometryAcad. Hodge theoryHodge theorem. A fourth definition was given by Christian Kasselwho showed that the cyclic homology groups may be computed as the homology groups of a certain mixed complex associated to A A. Let X X be a simply connected topological space. DMV 3, pdf. This is known as Jones’ theorem Jones Alain Connes originally defined cyclic homology over fields of characteristic zeroas the homology groups of a cyclic variant of the chain complex computing Hochschild homology.

Last revised on March 27, at This site ccyclic running on Instiki 0. There are closely related variants called periodic cyclic homology?

The homology of the cyclic complex, denoted. Jean-Louis Loday and Daniel Quillen gave a definition via a certain double complex for arbitrary homloogy rings.

On the other hand, the definition of Christian Kassel via mixed complexes was extended by Bernhard Keller to linear categories and dg-categories lloday, and he showed that the cyclic homology of the dg-category of perfect complexes on a nice scheme X X coincides with the cyclic homology of X X in the sense of Weibel.

Pressp. These free loop space objects are canonically equipped with a circle group – action that rotates the loops. KapranovCyclic operads and cyclic homologyin: Hochschild cohomologycyclic cohomology.