Slices of hermitian K-theory and Milnor's conjecture on quadratic forms

Abstract

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.

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References in corpus (10)

• An exact sequence for Milnor's K-theory with applications to quadratic forms
• The homotopy coniveau tower
• Motivic cell structures
• Hermitian K-theory, derived equivalences and Karoubi's Fundamental Theorem
• A comparison of motivic and classical homotopy theories
• On the zero slice of the sphere spectrum
• Motivic slices and colored operads
• Motivic connective K-theories and the cohomology of A(1)
• Cellularity of hermitian K-theory and Witt theory
• On the $\eta$-inverted sphere

Cited by in corpus (18)

• The Generalized Slices of Hermitian K-Theory
• The homotopy groups of the η-periodic motivic sphere spectrum
• On equivariant and motivic slices
• Cellularity of hermitian K-theory and Witt theory
• The slice spectral sequence for singular schemes and applications
• The $C_2$-effective spectral sequence for $C_2$-equivariant connective real $K$-theory
• Algebraic Cobordism and Étale Cohomology
• The homotopy limit problem and the cellular Picard group of Hermitian $K$-theory
• Topological models for stable motivic invariants of regular number rings
• Borel isomorphism and absolute purity
• Algebraic cobordism of number fields
• Recursive formulas for the motivic Milnor basis
• Combing a hedgehog over a field
• The motivic lambda algebra and motivic Hopf invariant one problem
• The geometric diagonal of the special linear algebraic cobordism
• The second stable homotopy groups of motivic spheres
• Embedding of category of twisted Chow-Witt correspondences into geometric stable $\A^1$-derived category over a field
• On the logarithmic slice filtration