Milnor conjecture

From Wikipedia, the free encyclopedia

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 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 around 1996 by Vladimir Voevodsky, having been open for some two decades.

Contents

[edit] 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.

[edit] 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.

[edit] Generalizations

The analogue of this result for primes other than 2 is known as the Bloch-Kato conjecture. Work of Voevodsky and Markus Rost has reportedly resulted in a complete proof of this conjecture also, although the proof still awaits publication.

[edit] References