Milnor conjecture

Short description: Theorem describing the Milnor K-theory (mod 2) by means of the Galois cohomology

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

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

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

for all n ≥ 0, where K^M denotes the Milnor ring.

About the proof

The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Alexander Merkurjev, Andrei Suslin, Markus Rost, 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 and Markus Rost 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, http://math.rutgers.edu/~weibel/motiviclectures.html 
• Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9 (4): 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, Bibcode: 1970InMat...9..318M 
• Voevodsky, Vladimir (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 (98): 1–57, doi:10.1007/s10240-003-0009-z, ISSN 0073-8301, 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 (98): 59–104, doi:10.1007/s10240-003-0010-6, ISSN 0073-8301, http://www.numdam.org/item?id=PMIHES_2003__98__59_0 

Further reading

• Kahn, Bruno (2005), "La conjecture de Milnor (d'après V. Voevodsky)", in Friedlander, Eric M.; Grayson, D.R. (in French), Handbook of K-theory, 2, Springer-Verlag, pp. 1105–1149, ISBN 3-540-23019-X