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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The Loday-Quillen-Tsygan theorem is originally due, independently, to. Jean-Louis LodayFree loop space and homology arXiv: Hochschild cohomologycyclic cohomology. The relation to cyclic loop spaces:. There is a version for ring spectra called topological cyclic homology.

Sullivan model of free loop space. DMV 3, pdf.

There are several definitions for the cyclic homology of an associative algebra A A over a commutative ring k k. KapranovCyclic operads and cyclic homologyin: 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 homooogy becomes that.

Pressp. Following Alexandre GrothendieckCtclic Weibel gave a definition of cyclic homology and Hochschild homology for schemesusing hypercohomology.

Bernhard KellerOn the cyclic homology of ringed spaces and schemesDoc. Bernhard KellerOn the cyclic homology of jomology categoriesJournal of Pure and Applied Algebra, 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.

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.

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

There are closely related variants called periodic cyclic homology? Last revised on March 27, at Let A A be an associative algebra over a ring k k.

Let X X be a simply connected topological space. Alain ConnesNoncommutative geometryAcad.

## CYCLIC HOMOLOGY

This is known as Jones’ theorem Jones KaledinCyclic homology with coefficientsmath. 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.

See the history of this page for a list of all contributions to it. 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.

Hmoology site is running on Instiki 0. It also admits a Dennis trace map from algebraic K-theoryand has been successful in allowing computations of the latter. The homology of the cyclic complex, denoted. JonesCyclic homology and equivariant homologyInvent.

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

These free loop space objects are canonically equipped with a circle group – action that rotates the loops. Jean-Louis Loday and Daniel Quillen gave a definition via a cycliv double complex for arbitrary commutative rings.

Like Hochschild homologycyclic homology is an additive invariant of dg-categories or stable infinity-categoriesin the sense of noncommutative motives.