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A note on product structures on Hochschild homology of schemes

Abhishek Banerjee (2011)

Colloquium Mathematicae

We extend the definition of Hochschild and cyclic homologies of a scheme over a commutative ring k to define the Hochschild homologies HH⁎(X/S) and cyclic homologies HC⁎(X/S) of a scheme X with respect to an arbitrary base scheme S. Our main purpose is to study product structures on the Hochschild homology groups HH⁎(X/S). In particular, we show that H H ( X / S ) = n H H ( X / S ) carries the structure of a graded algebra.

Algebraic K -theory of the first Morava K -theory

Christian Ausoni, John Rognes (2012)

Journal of the European Mathematical Society

For a prime p 5 , we compute the algebraic K -theory modulo p and v 1 of the mod p Adams summand, using topological cyclic homology. On the way, we evaluate its modulo p and v 1 topological Hochschild homology. Using a localization sequence, we also compute the K -theory modulo p and v 1 of the first Morava K -theory.

Cohomologie des algèbres de Lie croisées et K -théorie de Milnor additive

Daniel Guin (1995)

Annales de l'institut Fourier

Dans cet article, nous définissons des modules de (co)-homologie 0 ( 𝔊 , 𝔄 ) , 1 ( 𝔊 , 𝔄 ) , ( 𝔊 , 𝔄 ) , 1 ( 𝔊 , 𝔄 ) 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 ( A ) avec un analogue additif du groupe de K -théorie de Milnor K 2 Madd ( A ) .

Cohomologies bivariantes de type cyclique

Nikolay V. Solodov (2005)

Annales mathématiques Blaise Pascal

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...

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

Zinaida Lykova (2008)

Open Mathematics

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain ^ -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...

Cyclic cohomology of (extended) Hopf algebras

M. Khalkhali, B. Rangipour (2003)

Banach Center Publications

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.

Decompositions of the category of noncommutative sets and Hochschild and cyclic homology

Jolanta Słomińska (2003)

Open Mathematics

In this note we show that the main results of the paper [PR] can be obtained as consequences of more general results concerning categories whose morphisms can be uniquely presented as compositions of morphisms of their two subcategories with the same objects. First we will prove these general results and then we will apply it to the case of finite noncommutative sets.

Excision in entire cyclic cohomology

Ralf Meyer (2001)

Journal of the European Mathematical Society

We prove that entire and periodic cyclic cohomology satisfy excision for extensions of bornological algebras with a bounded linear section. That is, for such an extension we obtain a six term exact sequence in cohomology.

Fixed point theory and the K-theoretic trace

Ross Geoghegan, Andrew Nicas (1999)

Banach Center Publications

The relationship between fixed point theory and K-theory is explained, both classical Nielsen theory (versus K 0 ) and 1-parameter fixed point theory (versus K 1 ). In particular, various zeta functions associated with suspension flows are shown to come in a natural way as “traces” of “torsions” of Whitehead and Reidemeister type.

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