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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 ).
Let F be a field of characteristic different from 2. Then there is an
isomorphism
K
n
M
(
F
)
/
2
≅
H
e
´
t
n
(
F
,
Z
/
2
Z
)
{\displaystyle K_{n}^{M}(F)/2\cong H_{{\acute {e}}t}^{n}(F,\mathbb {Z} /2\mathbb {Z} )}
for all n ≥ 0, where KM denotes the
Milnor ring .
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 .
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 .
Mazza, Carlo;
Voevodsky, Vladimir ;
Weibel, Charles (2006),
Lecture notes on motivic cohomology ,
Clay Mathematics Monographs , vol. 2, Providence, R.I.:
American Mathematical Society ,
ISBN
978-0-8218-3847-1 ,
MR
2242284
Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms",
Inventiones Mathematicae , 9 (4): 318–344,
Bibcode :
1970InMat...9..318M ,
doi :
10.1007/BF01425486 ,
ISSN
0020-9910 ,
MR
0260844 ,
S2CID
13549621
Voevodsky, Vladimir (1996),
The Milnor Conjecture , Preprint
Voevodsky, Vladimir (2003a),
"Reduced power operations in motivic cohomology" , Institut des Hautes Études Scientifiques. Publications Mathématiques , 98 (98): 1–57,
arXiv :
math/0107109 ,
doi :
10.1007/s10240-003-0009-z ,
ISSN
0073-8301 ,
MR
2031198 ,
S2CID
8172797
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 ,
MR
2031199 ,
S2CID
54823073
Kahn, Bruno (2005), "La conjecture de Milnor (d'après V. Voevodsky)", in Friedlander, Eric M.; Grayson, D.R. (eds.), Handbook of K-theory (in French), vol. 2,
Springer-Verlag , pp. 1105–1149,
ISBN
3-540-23019-X ,
Zbl
1101.19001