Milnor K-theory

In mathematics, Milnor K-theory is an invariant of fields defined by Milnor (1970). Originally viewed as an approximation to algebraic K-theory, Milnor K-theory has turned out to be an important invariant in its own right.

The calculation of K₂ of a field by Matsumoto led Milnor to the following, seemingly naive, definition of the "higher" K-groups of a field F:

the quotient of the tensor algebra over the integers of the multiplicative group F^× by the two-sided ideal generated by the elements

for a ≠ 0, 1 in F. The nth Milnor K-group K_n^M(F) is the nth graded piece of this graded ring; for example, K₀^M(F) = Z and K₁^M(F) = F*. There is a natural homomorphism

from the Milnor K-groups of a field to the Quillen K-groups, which is an isomorphism for n ≤ 2 but not for larger n, in general. For nonzero elements a₁, ..., a_n in F, the symbol {a₁, ..., a_n} in K_n^M(F) means the image of a₁ ⊗ ... ⊗ a_n in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that {a, 1−a} = 0 in K₂^M(F) for a in F − {0,1} is sometimes called the Steinberg relation.

The ring K_*^M(F) is graded-commutative.

We have ${\displaystyle K_{n}^{M}(\mathbb {F} _{q})=0}$ for n ≧ 2, while ${\displaystyle K_{2}^{M}(\mathbb {C} )}$ is an uncountable uniquely divisible group. (An abelian group is uniquely divisible if it is a vector space over the rational numbers.) Also, ${\displaystyle K_{2}^{M}(\mathbb {R} )}$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; ${\displaystyle K_{2}^{M}(\mathbb {Q} _{p})}$ is the direct sum of the multiplicative group of ${\displaystyle \mathbb {F} _{p}}$ and an uncountable uniquely divisible group; ${\displaystyle K_{2}^{M}(\mathbb {Q} )}$ is the direct sum of the cyclic group of order 2 and cyclic groups of order ${\displaystyle p-1}$ for all odd prime ${\displaystyle p}$.

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