2013/11/30 by Oliver Röndigs, Paul Arne Østvær · 45 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Algebraic Geometry and Number Theory #Cohomology #Conjecture #De Rham cohomology #Equivariant cohomology #Geometry #Hermitian matrix #Homotopy and Cohomology in Algebraic Topology #Ideal (ethics) #K-theory (physics) #Mathematics #Motivic cohomology #Pure mathematics #Quadratic equation #Ring (chemistry) #Spectral sequence #math.AG #math.AT #math.KT #msc:11E04 #msc:14F42 #msc:19G38 #msc:55P42
paper · pdf · doi:10.2140/gt.2016.20.1157
published in Geometry & Topology 20(2), 1157-1212 (Mathematical Sciences Publishers) · Version closer to the published paper
openalex publication_date 2016/04/28 · arxiv created 2017/05/30 · arxiv updated 2017/05/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
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.Suppose that F is a field of characteristic char(F ) = 2.In [33] the Milnor K-theory of F is defined in terms of generators and relations byHere T * F is the tensor algebra of the multiplicative group of units F .In degrees zero, one and two these groups agree with Quillen's K-groups, but for higher degrees they differ in general.Milnor [33] proposed two conjectures relating k M * (F ) = K M * (F )/2K M * (F ) to the mod-2 Galois cohomology ring H * (F ; Z/2) and the graded Witt ring GW * (F ) = q0 I(F ) q /I(F ) q+1 for the fundamental ideal I(F ) of even dimensional forms, via the two homomorphisms:GW * (F )