Motivic cohomology with Z/2 coefficients (Milnor conjecture)

Statement

Voevodsky develops motivic cohomology with ℤ/2 coefficients for smooth varieties over a field: operations, Bockstein, and comparison with étale cohomology. The main consequence is a proof of the Milnor conjecture (the mod‑2 case of the Bloch–Kato conjecture): for a field F of characteristic ≠ 2, the norm residue homomorphism induces an isomorphism Kₙᴹ(F)/2 ≅ Hⁿ(F, μ₂^{⊗n}), identifying Milnor K‑theory mod 2 with Galois cohomology. The paper is the published capstone of the motivic approach to quadratic forms and Galois cohomology; formalizable in ZFC.

Formula

Kₙᴹ(F)/2 ≅ Hⁿ(F, μ₂^{⊗n})

Equation

Motivic → étale; Milnor K₂ and quadratic forms (special cases)

Proof sketch

Voevodsky, Publ. Math. Inst. Hautes Études Sci. 98 (2003). Builds on the triangulated category of motives, motivic Steenrod operations, and the proof strategy for the Milnor conjecture completed in this work.

Source

Voevodsky, V. Motivic cohomology with Z/2-coefficients. Publications Mathématiques de l’Institut des Hautes Études Scientifiques 98 (2003), 59–104.

Domains

mathematics