Adams spectral sequence and cohomological invariants of quadratic forms

For any field k of characteristic 0 the Adams spectral sequence for the sphere spectrum based on Suslin-Voevodsky module 2 motivic cohomology [8] converges to the graded ring associated to the filtration of the Grothendieck-Witt ring of quadratic forms over k by powers of the ideal generated by even-dimensional forms. Moreover, some property of the module 2 motivic cohomology of k, which is a consequence of Voevodsky's proof of Milnor's conjecture on module 2 Galois cohomology of k [9], implies that the spectral sequence degenerates in the critical area. This allows Its to give a new proof of the Milnor conjecture on the graded ring of the Witt ring of k [4] which differs front [11]. (C) Academie des Sciences/Elsevier, Paris.