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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Hodge theoryHodge theorem. Jean-Louis LodayFree loop homologu and homology arXiv: Bernhard KellerOn the cyclic homology of exact categoriesJournal of Pure and Applied Algebra, pdf.

### Cyclic Homology – Jean-Louis Loday – Google Books

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. There homolgy several definitions for the cyclic homology of an associative algebra A A over a commutative ring k k. JonesCyclic homology and equivariant homologyInvent. This is known as Jones’ theorem Jones The relation to cyclic loop spaces:.

## CYCLIC HOMOLOGY

If the coefficients are rationaland X X is of finite type bomology this may be computed by the Sullivan model for free loop spacessee there the section on Relation to Hochschild homology. It also admits a Dennis trace map from algebraic K-theoryand has been successful in allowing computations of the latter.

Hochschild homology may be understood as the cohomology of free loop space object s as described there.

Jean-Louis Loday and Daniel Quillen gave a definition via a certain double complex for arbitrary commutative rings. Following Alexandre GrothendieckCharles Weibel gave a definition of cyclic homology and Hochschild homology for schemesusing hypercohomology. Last revised on March 27, at The homology of the cyclic complex, denoted. Like Hochschild homologycyclic homology is an additive invariant of dg-categories or stable infinity-categoriesin the sense of noncommutative motives.

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These free loop space objects are canonically cycclic with a circle group – action that rotates the loops. Sullivan model of free loop space. Pressp. Alain ConnesNoncommutative geometryAcad. DMV 3, pdf. On the other hand, the definition of Christian Kassel via mixed complexes was extended by Bernhard Keller to linear categories and dg-categoriesand 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.

There are closely related variants called periodic cyclic homology? Hochschild cohomologycyclic cohomology. There is a version for ring spectra called topological cyclic homology. See the hlmology of this page for a list of all contributions to it.

Bernhard KellerOn the cyclic homology of ringed spaces and schemesDoc. The Loday-Quillen-Tsygan theorem is originally due, independently, to. Let A A be an associative algebra over a ring k k. 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. KaledinCyclic homology with coefficientsmath.

In the special case that the topological space Lodah X carries the structure of a smooth manifoldthen the singular cochains on X X are equivalent to the homolkgy of differential forms the de Rham algebra and hence in this case the statement becomes that.

Let X X be a simply connected topological space. KapranovCyclic operads and cyclic homologyin: