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Slices of Hermitian $$K$$-theory and Milnor’s conjecture on quadratic forms. (English) Zbl 1416.19001
Summary: We advance the understanding of $$K$$-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian $$K$$-groups and Witt groups. By an explicit computation of the slice spectral sequence for higher Witt theory, we prove Milnor’s conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian $$K$$-groups in terms of motivic cohomology.

##### MSC:
 19G38 Hermitian $$K$$-theory, relations with $$K$$-theory of rings 11E70 $$K$$-theory of quadratic and Hermitian forms 11E04 Quadratic forms over general fields 14F42 Motivic cohomology; motivic homotopy theory 55P42 Stable homotopy theory, spectra 55T05 General theory of spectral sequences in algebraic topology
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