For a complex Grassmannian X, there is an isomorphism

between Balmer's Witt group and the quotient of topological K-theories.

The norm map on the Milnor K-groups of a finite extension of complete, discrete valuation fields is continuous with respect to the unit group filtrations. The only proof in the literature, due to K. Kato, uses semi-global methods. Here we present an elementary algebraic proof.

Starting from the Karoubi multiplicative K-theory, we construct a Chern-Weil theory adapted to isolated conical singularities. The Chern character takes its values in the intersection cohomology of Goresky-MacPherson. We also propose an integer intersection K-theory for such singularities.

In this paper we prove the p-adic analogue of a result of Hamida [11], namely that the p-adic Borel regulator introduced by Huber and Kings for the K-theory of a p-adic number field equals Karoubi's p-adic regulator up to an explicit rational factor.