Title:

The Grayson spectral sequence for hermitian Ktheory

Let R be a regular ring such that 2 is invertible. We construct a spectral sequence converging to the hermitian Ktheory, alias the GrothendieckWitt theory, of R. In particular, we construct a tower for the hermitian Kgroups in even shifts, whose terms are given by the hermitian Ktheory of automorphisms. The spectral sequence arises as the homotopy spectral sequence of this tower and is analogous to Graysonâ€™s version of the motivic spectral sequence [Gra95]. Further, we construct similar towers for the hermitian Ktheory in odd shifts if R is a field of characteristic different from 2. We show by a counter example that the arising spectral sequence does not behave as desired. We proceed by proposing an alternative version for the tower and verify its correctness in weight 1. Finally we give a geometric representation of the (hermitian) Ktheory of automorphisms in terms of the general linear group, the orthogonal group, or in terms of esymmetric matrices, respectively. The Ktheory of automorphisms can be identified with motivic cohomology if R is local and of finite type over a field. Therefore the hermitian Ktheory of automorphisms as presented in this thesis is a candidate for the analogue of motivic cohomology in the hermitian world.
