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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cyclic homology in nLab
Hochschild cohomologycyclic cohomology. Hodge theoryHodge theorem.
Hochschild homology may be understood as the cohomology of free loop space object s as described there. These free loop space objects are canonically equipped with a circle group – action that rotates the loops.
Like Hochschild homologycyclic homology is an additive invariant of dg-categories or stable infinity-categoriesin the sense of noncommutative motives.
It also admits a Dennis trace map from algebraic K-theoryand has been successful in allowing lodsy of the latter.
Cyclic Homology – Jean-Louis Loday – Google Books
There are several definitions for the cyclic homology of an associative algebra A A over a commutative ring k k. Alain Connes originally defined cyclic homology over fields cyclci characteristic zeroas the homology groups of a cyclic variant of the chain complex computing Hochschild homology.
Jean-Louis Loday and Daniel Quillen gave a definition via a certain double complex for arbitrary commutative rings.
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. Following Alexandre GrothendieckCharles Weibel gave a definition of cyclic homology and Hochschild homology for schemesusing hypercohomology.
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? There is a version for ring spectra called topological cyclic homology.
Let A A be an associative algebra over a ring k k. The homology of the cyclic complex, denoted. Let X X be a simply connected topological 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. 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.
This is known as Jones’ theorem Jones Sullivan model of free loop space. Alain ConnesNoncommutative geometryAcad. Pressp.
Bernhard KellerOn the cyclic homology of ringed spaces and schemesDoc. DMV 3, pdf. Bernhard KellerOn the cyclic homology of exact categoriesJournal of Pure and Applied Algebra, pdf. KaledinCyclic homology with coefficientsmath. KapranovCyclic operads and cyclic homologyin: The relation to cyclic loop spaces:.
JonesCyclic homology and equivariant homologyInvent. Jean-Louis LodayFree loop space and homology arXiv: The Loday-Quillen-Tsygan theorem is originally due, independently, to. Last revised on March 27, at See the history of this page for a list of all contributions to it. This cycluc is running on Instiki 0.