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 introduce the cylindrical module A\H, where H is a Hopf algebra and A is a Hopf module algebra over H. We show that there exists an isomorphism between C ðA op z H cop Þ the cyclic module of the crossed product algebra A op z H cop , and DðA\HÞ, the cyclic module related to the diagonal of A\H. If S, the antipode of H, is invertible it follows that C ðA z HÞ F DðA op \H cop Þ. When S is invertible, we approximate HC ðA z HÞ by a spectral sequence and give an interpretation of E 0 ; E 1 and E 2 terms of this spectral sequence.
Abstract. The scalar curvature for the noncommutative four torus T 4 Θ , where its flat geometry is conformally perturbed by a Weyl factor, is computed by making the use of a noncommutative residue that involves integration over the 3-sphere. This method is more convenient since it does not require the rearrangement lemma and it is advantageous as it explains the simplicity of the final functions of one and two variables, which describe the curvature with the help of a modular automorphism. In particular, it readily allows to write the function of two variables as the sum of a finite difference and a finite product of the one variable function. The curvature formula is simplified for dilatons of the form sp, where s is a real parameter and p ∈ C ∞ (T 4 Θ ) is an arbitrary projection, and it is observed that, in contrast to the two dimensional case studied by A. Connes and H. Moscovici, unbounded functions of the parameter s appear in the final formula. An explicit formula for the gradient of the analog of the Einstein-Hilbert action is also calculated.
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).
Abstract. In this paper we give a proof of the Gauss-Bonnet theorem of Connes and Tretkoff for noncommutative two tori T 2 Â equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number in the upper half plane, representing the conformal class of a metric on T 2 Â , and a Weyl factor given by a positive invertible element k 2 C 1 .T 2 Â /, the value at the origin, .0/, of the spectral zeta function of the Laplacian 4 0 attached to .T 2 Â ; ; k/ is independent of and k.Mathematics Subject Classification (2010). 58B34.
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