An exact sequence for $K_\ast^M/2$ with applications to quadratic forms

By Dmitri Orlov, A. Vishik, Vladimir Voevodsky. 2001

241 citations recorded by the source corpus.

We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnor's K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring, the Khan-Rost-Sujatha conjecture and the J-filtration conjecture. The first version of this paper was written in the spring of 1996.

Subfields: Artificial Intelligence, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/65bf9a4a8ef32028034126cd5b238aa97f74738d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Alexander S. Merkurjev
2. M. Kerz
3. N. Karpenko
4. R. Elman
5. Vladimir Voevodsky

In Geometry and Topology

1. P. Ostvaer
2. O. Rondigs
3. N. Karpenko
4. Alexander S. Merkurjev
5. K. Ormsby
6. J. Heller
7. M. Kerz
8. M. Levine

In Mathematical Physics

1. P. Ostvaer
2. N. Karpenko
3. O. Rondigs
4. Alexander S. Merkurjev
5. K. Ormsby
6. J. Heller
7. M. Kerz
8. M. Levine

Citations in the corpus

• The Algebraic and Geometric Theory of Quadratic Forms (2008, methodology): “In this chapter we shall assume the results of Voevodsky in [136] and Orlov-Vishik-Voevodsky in [107] for fields of characteristic not 2, and Kato in [78] for fields of characteristic 2 on the solution for the analog of the Milnor Conjecture in algebraic K-theory.”
• The Gersten conjecture for Milnor K-theory (2008, background): “This is because the left vertical map is surjective for elementary reasons and the other vertical maps are isomorphisms by Voevodsky’s theorem [ 21 ].”
• Motivic cohomology with Z/2-coefficients (2003, background): “For the proof of this result together with more detailed computations of motivic cohomology groups of norm quadrics see [ 25 ].”
• The special fiber of the motivic deformation of the stable homotopy category is algebraic (2018, background): “Motivic homotopy theory, introduced by Voevodsky and Morel [52,53,55,69–71,73], is a successful application of abstract homotopy theory to solve problems in number theory and algebraic geometry (see [57, 67, 72] for example).”
• Galois equivariance and stable motivic homotopy theory (2014, background): “ology [49, 52]. Its stable version plays an essential role in the theory of motives and motivic cohomology [53]. This circle of ideas led to the resolution of the Milnor conjecture on quadratic forms [41] and the Quillen-Lichtenbaum conjecture, a powerful result linking algebraic K-theory and values”
• Milnor -theory of local rings with finite residue fields (2010, background): “[14] this result has some interesting application to quadratic forms over local rings.”
• Slices of hermitian $K$–theory and Milnor's conjecture on quadratic forms (2013, background): “mensional forms, via the two homomorphisms: kM ∗ (F) GW∗(F) ≺ sF ∗ H∗(F;Z/2) hF ∗ ≻ The solutions of the Milnor conjectures on Galois cohomology [62] – hF ∗ is an isomorphism – and on quadratic forms [39] – sF ∗ is an isomorphism – are two striking applications of motivic homotopy theory. For backgr”