Dans cet article, nous définissons des modules de (co)-homologie ${ℌ}_0(𝔊,𝔄)$, ${ℌ}_1(𝔊,𝔄)$, ${ℌ}^{\circ }(𝔊$, $𝔄)$, ${ℌ}^1(𝔊,𝔄)$ où $𝔊$ et $𝔄$ sont des algèbres de Lie munies d’une structure supplémentaire (algèbres de Lie croisées), qui satisfont les propriétés usuelles des foncteurs cohomologiques. Si $A$ est une $k$-algèbre, nous utilisons ces modules d’homologie pour comparer le groupe d’homologie cyclique $H{C}_1\left(A\right)$ avec un analogue additif du groupe de $K$-théorie de Milnor $K_2^{\mathrm{Madd}}\left(A\right)$.

In the article we propose a construction of bivariant cohomology of a couple of chain complexes with periodicities. In this way we obtain definitions of bivariant dihedral and bivariant reflective cohomology of an algebra $A$. Bivariant cyclic and quaternionic cohomologies appear as particular cases of this construction. The case of $2$ invertible in the ground ring is studied particulary.Dans cet article nous proposons une définition de la cohomologie bivariante pour une paire de complexes de chaînes...

We study cohomology algebras of graded differential algebras which are models for Hochschild homology of some classes of topological spaces (e.g. homogeneous spaces of compact Lie groups). Explicit formulae are obtained. Some applications to cyclic homology are given.

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain $$\widehat{\otimes }$$ -algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ϕ: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H...

We review recent progress in the study of cyclic cohomology of Hopf algebras, extended Hopf algebras, invariant cyclic homology, and Hopf-cyclic homology with coefficients, starting with the pioneering work of Connes-Moscovici.