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  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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Like Hochschild homologycyclic homology is an additive invariant of dg-categories or stable infinity-categoriesin the sense of noncommutative motives.
There are closely related variants called periodic cyclic homology? 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. The Loday-Quillen-Tsygan theorem is originally due, independently, to. Alain ConnesNoncommutative geometryAcad. Jean-Louis Loday and Daniel Quillen gave a definition via a certain double complex for arbitrary commutative rings.
JonesCyclic homology and equivariant homologyInvent.
In the special case that the topological space X X homoligy 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 are several definitions for the cyclic homology of homoloogy associative algebra Lody A over a commutative ring k k. Hochschild cohomologycyclic cohomology. It also admits a Dennis trace map from algebraic K-theoryand has been successful in allowing computations of the latter.
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. Bernhard KellerOn the cyclic homology of ringed spaces and schemesDoc.
This is known as Jones’ theorem Jones Following Alexandre GrothendieckCharles Weibel gave a definition of cyclic homology and Hochschild homology for schemesusing hypercohomology. KapranovCyclic operads and cyclic homologyin: Pressp. Last revised on March 27, at KaledinCyclic homology with coefficientsmath.
cyclic homology in nLab
See the history of this page for a list of all contributions to it. The relation to cyclic loop spaces:.
Hochschild homology may be understood as the cohomology of free loop space object s as described there. 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. The homology of the cyclic complex, denoted. Hodge theoryHodge theorem. These free loop space objects are canonically equipped with a circle group – action that rotates the loops.
Cyclic Homology – Jean-Louis Loday – Google Books
There is a version for ring spectra called topological cyclic homology. Let A A be an associative algebra over a ring k k. Bernhard KellerOn the cyclic homology of exact categoriesJournal of Pure and Applied Algebra, pdf.
This site is running on Instiki 0. Let X X be a simply connected topological space.
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 locay perfect complexes on a nice scheme X X coincides with the cyclic homology of X X in the sense of Weibel.
Sullivan model of free loop space. Jean-Louis LodayFree loop space and homology arXiv: