Karoubi's relative Chern character, the rigid syntomic regulator, and the Bloch-Kato exponential map

We construct a variant of Karoubi's relative Chern character for smooth, separated schemes over the ring of integers in a p-adic field and prove a comparison with the rigid syntomic regulator. For smooth projective schemes we further relate the relative Chern character to the etale p-adic regulator via the Bloch-Kato exponential map. This reproves a result of Huber and Kings for the spectrum of the ring of integers and generalizes it to all smooth projective schemes as above.Comment: v1:33 pages; v2:major revision (28 pages); v3:minor changes; v4:minor changes following suggestions by a refere

Stabilization of the Witt group

Using an idea due to R.Thomason, we define a "homology theory" on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find P. Balmer's higher Witt groups. For more general rings, this homology isomorphic to the KT-theory of J. Hornbostel, inspired by the work of B. Williams. For real or complex C*-algebras, we recover - up to 2 torsion - topological K-theory.Comment: 6 pages ; see also http://www.math.jussieu.fr/~karoubi

Hermitian K-theory of the integers

The 2-primary torsion of the higher algebraic K-theory of the integers has been computed by Rognes and Weibel. In this paper we prove analogous results for the Hermitian K-theory of the integers with 2 inverted (denoted by Z'). We also prove in this case the analog of the Lichtenbaum conjecture for the hermitian K-theory of Z' : the homotopy fixed point set of a suitable Z/2 action on the classifying space of the algebraic K-theory of Z' is the hermitian K-theory of Z' after 2-adic completion.Comment: 36 pages ; see also http://www.math.jussieu.fr/~karoubi/ and http://www.math.nus.edu.sg/~matberic

Generalized differential spaces with dN=0d^N=0 and the qq-differential calculus

We present some results concerning the generalized homologies associated with nilpotent endomorphisms $d$ such that $d^N=0$ for some integer $N\geq 2$. We then introduce the notion of graded $q$-differential algebra and describe some examples. In particular we construct the $q$-analog of the simplicial differential on forms, the $q$-analog of the Hochschild differential and the $q$-analog of the universal differential envelope of an associative unital algebra.Comment: 8 pages, Latex2e, uses pb-diagram, available at http://qcd.th.u-psud.fr, to be published in the Proceedings of the 5th Colloquium ``Quantum Groups and Integrable Systems", Prague, June 199