In mathematics, Milnor K-theory is an algebraic invariant (denoted K ∗ ( F ) {\displaystyle K_{*}(F)} for a field F {\displaystyle F} ) defined by John Milnor (1970) as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for K 1 {\displaystyle K_{1}} and K 2 {\displaystyle K_{2}} . Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute.
Definition
Motivation After the definition of the Grothendieck group K ( R ) {\displaystyle K(R)} of a commutative ring, it was expected there should be a sequence of invariants K i ( R ) {\displaystyle K_{i}(R)} called higher K-theory groups, from the fact that there exists a short exact sequence
K ( R , I ) → K ( R ) → K ( R / I ) → 0 {\displaystyle K(R,I)\to K(R)\to K(R/I)\to 0}
which should have a continuation by a long exact sequence. Note the group on the left is relative K-theory. This led to much study and as a first guess for what this theory would look like, Milnor gave a definition for fields. His definition is based upon two calculations of what higher K-theory "should" look like in degrees 1 {\displaystyle 1} and 2 {\displaystyle 2} . Then, if in a later generalization of algebraic K-theory was given, if the generators of K ∗ ( R ) {\displaystyle K_{*}(R)} lived in degree 1 {\displaystyle 1} and the relations in degree 2 {\displaystyle 2} , then the constructions in degrees 1 {\displaystyle 1} and 2 {\displaystyle 2} would give the structure for the rest of the K-theory ring. Under this assumption, Milnor gave his "ad-hoc" definition. It turns out algebraic K-theory K ∗ ( R ) {\displaystyle K_{*}(R)} in general has a more complex structure, but for fields the Milnor K-theory groups are contained in the general algebraic K-theory groups after tensoring with Q {\displaystyle \mathbb {Q} } , i.e. K n M ( F ) ⊗ Q ⊆ K n ( F ) ⊗ Q {\displaystyle K_{n}^{M}(F)\otimes \mathbb {Q} \subseteq K_{n}(F)\otimes \mathbb {Q} } . It turns out the natural map λ : K 4 M ( F ) → K 4 ( F ) {\displaystyle \lambda :K_{4}^{M}(F)\to K_{4}(F)} fails to be injective for a global field F {\displaystyle F} pg 96.
Definition Note for fields the Grothendieck group can be readily computed as K 0 ( F ) = Z {\displaystyle K_{0}(F)=\mathbb {Z} } since the only finitely generated modules are finite-dimensional vector spaces. Also, Milnor's definition of higher K-groups depends upon the canonical isomorphism
l : K 1 ( F ) → F ∗ {\displaystyle l\colon K_{1}(F)\to F^{*}}
(the group of units of F {\displaystyle F} ) and observing the calculation of K2 of a field by Hideya Matsumoto, which gave the simple presentation
K 2 ( F ) = F ∗ ⊗ F ∗ { l ( a ) ⊗ l ( 1 − a ) : a ≠ 0 , 1 } {\displaystyle K_{2}(F)={\frac {F^{*}\otimes F^{*}}{\{l(a)\otimes l(1-a):a\neq 0,1\}}}}
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