Milnor conjecture

For Milnor's conjecture about the slice genus of torus knots, see Milnor conjecture (topology).

In mathematics, the Milnor conjecture was a proposal by John Milnor (1970) of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky (1996, 2003a, 2003b).

Statement of the theorem

Let F be a field of characteristic different from 2. Then there is an isomorphism

$K_n^M(F)/2 \cong H_{\acute{e}t}^n(F, \mathbb{Z}/2\mathbb{Z})$

for all n ≥ 0.

About the proof

The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Andrei Suslin, Fabien Morel, Eric Friedlander, and others, including the newly-minted theory of motivic cohomology (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra.

Generalizations

The analogue of this result for primes other than 2 was known as the Bloch–Kato conjecture. Work of Voevodsky, Markus Rost, and Charles Weibel yielded a complete proof of this conjecture in 2009; the result is now called the norm residue isomorphism theorem.

References

• Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1; 978-0-8218-3847-1, MR2242284, http://math.rutgers.edu/~weibel/motiviclectures.html 
• Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR0260844 
• Voevodsky, V. (1996), The Milnor Conjecture, Preprint, http://www.math.uiuc.edu/K-theory/0170 
• Voevodsky, Vladimir (2003a), "Reduced power operations in motivic cohomology", Institut des Hautes Études Scientifiques. Publications Mathématiques (98): 1–57, doi:10.1007/s10240-003-0009-z, ISSN 0073-8301, MR2031198, http://www.numdam.org/item?id=PMIHES_2003__98__1_0 
• Voevodsky, Vladimir (2003b), "Motivic cohomology with Z/2-coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques (98): 59–104, doi:10.1007/s10240-003-0010-6, ISSN 0073-8301, MR2031199, http://www.numdam.org/item?id=PMIHES_2003__98__59_0