The Grayson spectral sequence for hermitian K-theory

Let R be a regular ring such that 2 is invertible. We construct a spectral sequence converging to the hermitian K-theory, alias the Grothendieck-Witt theory, of R. In particular, we construct a tower for the hermitian K-groups in even shifts, whose terms are given by the hermitian K-theory of automorphisms. The spectral sequence arises as the homotopy spectral sequence of this tower and is analogous to Grayson’s version of the motivic spectral sequence [Gra95]. Further, we construct similar towers for the hermitian K-theory 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) K-theory of automorphisms in terms of the general linear group, the orthogonal group, or in terms of e-symmetric matrices, respectively. The K-theory of automorphisms can be identified with motivic cohomology if R is local and of finite type over a field. Therefore the hermitian K-theory of automorphisms as presented in this thesis is a candidate for the analogue of motivic cohomology in the hermitian world