Milnor K-theory

In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by Milnor (1970).

The calculation of K₂ of a field k led Milnor to the following ad hoc definition of "higher" K-groups by

$K^M_*(k) := T^*k^\times/(a\otimes (1-a)), \,$

thus as graded parts of a quotient of the tensor algebra of the multiplicative group k^× by the two-sided ideal, generated by the

$a\otimes(1-a) \,$

for a ≠ 0, 1. For n = 0,1,2 these coincide with Quillen's K-groups of a field, but for n ≧ 3 they differ in general. For example, we have $K^M_n(\mathbb{F}_q) = 0$ for n ≧ 3.

Milnor K-theory modulo 2 is related to étale (or Galois) cohomology of the field by the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch–Kato conjecture, proved by Voevodsky, Rost, and others.

References

• Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR0260844