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Cohomologie cyclique et foncteurs $$Ext^ n$$. (French) Zbl 0534.18009
In his articles ”Non-commutative differential geometry. I, II” (I.H.E.S. preprints), the author has defined and studied the cyclic cohomology $$H^ n_{\lambda}({\mathcal A})$$ of a $$C^*$$-algebra. The purpose of this note is to show how that cohomology may be obtained in terms of the Ext functor, where the ”modules” in question are functors from a suitable small category.
To be specific, the author defines a category $$\Lambda$$ whose objects $$\Lambda_ n$$ are indexed by the non-negative integers and whose morphisms $$f\in Hom(\Lambda_ n,\Lambda_ m)$$ are homotopy classes of continuous increasing maps $$\phi$$ : $$S^ 1\to S^ 1$$ which map the $$(n+1)th$$ roots of unity into the $$(m+1)th$$ roots of unity. A $$\Lambda$$- module is then a covariant functor from $$\Lambda$$ into the category of Abelian groups, and, for a field k, a $$k(\Lambda)$$-module is a covariant functor from $$\Lambda$$ into the category of vector spaces over k. He associates to each unital algebra $${\mathcal A}$$ over k a $$k(\Lambda)$$-module $${\mathcal A}^{\diamond}$$ whose objects are the n-fold tensor products of $${\mathcal A}$$ with itself. A similar definition is made when $${\mathcal A}$$ is a ring; in this case $${\mathcal A}^{\diamond}$$ is a $$\Lambda$$-module. The author shows that for a k-algebra $$H^ n_{\lambda}({\mathcal A})$$ is just $$Ext^ n_{k(\Lambda)}({\mathcal A}^{\diamond},k^{\diamond}).$$
Reviewer: W.Moran

##### MSC:
 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) 46L05 General theory of $$C^*$$-algebras 46M05 Tensor products in functional analysis 16Exx Homological methods in associative algebras
##### Keywords:
cyclic cohomology of $$C^*$$-algebra; Ext