Abstract. Following the idea of an invariant differential complex, we construct general-type cyclic modules that provide the common denominator of known cyclic theories. The cyclicity of these modules is governed by Hopfalgebraic structures. We prove that the existence of a cyclic operator forces a modification of the Yetter-Drinfeld compatibility condition leading to the concept of a stable anti-Yetter-Drinfeld module. This module plays the role of the space of coefficients in the thus obtained cyclic cohomology of module algebras and coalgebras, and the cyclic homology and cohomology of comodule algebras. Along the lines of Connes and Moscovici, we show that there is a pairing between the cyclic cohomology of a module coalgebra acting on a module algebra and closed 0-cocycles on the latter. The pairing takes values in the usual cyclic cohomology of the algebra. Similarly, we argue that there is an analogous pairing between closed 0-cocycles of a module coalgebra and the cyclic cohomology of a module algebra. Homologie et cohomologie Hopf-cycliquesà coefficientsRésumé. Suivant l'idée d'un complexe différentiel invariant, nous construisons des modules cycliques de type général qui fournissent un dénominateur commun aux théories cycliques connues. Le caractère cyclique de ces modules est gouverné par des structures Hopf-algébriques. Nous montrons que l'existence d'un opérateur cyclique obligeà une modification de la condition de compatibilité de Yetter-Drinfeld et mène au concept de module antiYetter-Drinfeld stable. Ce module joue le rôle d'espace de coefficients pour la cohomologie de modules algèbres et de modules cogèbres ainsi obtenue, ainsi que pour l'homologie et la cohomologie cycliques de comodules algèbres. Comme l'ont fait Connes et Moscovici pour leur cohomologie, nous montrons qu'il existe un appariement entre la cohomologie cyclique d'un module cogèbre agissant sur un module algèbre et les 0-cycles fermés sur ce dernier. Cet appariement prend ses valeurs dans la cohomologie cyclique usuelle de l'algèbre. De façon similaire, nousétablissons un appariement analogue entre les 0-cycles fermés d'un module cogèbre et la cohomologie cyclique d'un module algèbre.

We define and study a class of entwined modules (stable anti-Yetter-Drinfeld modules) that serve as coefficients for the Hopf-cyclic homology and cohomology. In particular, we explain their relationship with Yetter-Drinfeld modules and Drinfeld doubles. Among sources of examples of stable anti-Yetter-Drinfeld modules, we find HopfGalois extensions with a flipped version of the Miyashita-Ulbrich action.To cite this article: P. M. Hajac et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). RésuméModules Paris, Ser. I 336 (2003).

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We associate to each infinite primitive Lie pseudogroup a Hopf algebra of 'transverse symmetries', by refining a procedure due to Connes and the first author in the case of the general pseudogroup. The affiliated Hopf algebra can be viewed as a 'quantum group' counterpart of the infinite-dimensional primitive Lie algebra of the pseudogroup. It is first constructed via its action on theétale groupoid associated to the pseudogroup, and then realized as a bicrossed product of a universal enveloping algebra by a Hopf algebra of regular functions on a formal group. The bicrossed product structure allows to express its Hopf cyclic cohomology in terms of a bicocyclic bicomplex analogous to the Chevalley-Eilenberg complex. As an application, we compute the relative Hopf cyclic cohomology modulo the linear isotropy for the Hopf algebra of the general pseudogroup, and find explicit cocycle representatives for the universal Chern classes in Hopf cyclic cohomology. As another application, we determine all Hopf cyclic cohomology groups for the Hopf algebra associated to the pseudogroup of local diffeomorphisms of the line.

Abstract. We construct cup products of two different kinds for Hopf-cyclic cohomology. When the Hopf algebra reduces to the ground field our first cup product reduces to Connes' cup product in ordinary cyclic cohomology. The second cup product generalizes Connes-Moscovici's characteristic map for actions of Hopf algebras on algebras.Résumé. Cup-produits dans la cohomologie Hopf-cyclique Nous construisons deux types de cupproduits pour la cohomologie Hopf-cyclique. Lorsque l'algébre de Hopf se réduit au corps de base, notre premier cup-produit se réduit au cup-produit de Connes en cohomologie cyclique ordinaire. Le deuxième cup-produit généralise l'application caractéristique de Connes-Moscovici pour l'action des algèbres de Hopf sur les algèbres.