Milnor K-theory

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In mathematics, Milnor K-theory^[1] is an algebraic invariant (denoted ${\displaystyle K_{*}(F)}${\displaystyle K_{*}(F)} for a field ${\displaystyle F}${\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 ${\displaystyle K_{1}}${\displaystyle K_{1}} and ${\displaystyle K_{2}}${\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.^[2]

After the definition of the Grothendieck group ${\displaystyle K(R)}${\displaystyle K(R)} of a commutative ring, it was expected there should be an infinite set of invariants ${\displaystyle K_{i}(R)}${\displaystyle K_{i}(R)} called higher K-theory groups, from the fact there exists a short exact sequence

${\displaystyle K(R,I)\to K(R)\to K(R/I)\to 0}${\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 ${\displaystyle 1}${\displaystyle 1} and ${\displaystyle 2}${\displaystyle 2}. Then, if in a later generalization of algebraic K-theory was given, if the generators of ${\displaystyle K_{*}(R)}${\displaystyle K_{*}(R)} lived in degree ${\displaystyle 1}${\displaystyle 1} and the relations in degree ${\displaystyle 2}${\displaystyle 2}, then the constructions in degrees ${\displaystyle 1}${\displaystyle 1} and ${\displaystyle 2}${\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 ${\displaystyle K_{*}(R)}${\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 ${\displaystyle \mathbb {Q} }${\displaystyle \mathbb {Q} }, i.e. ${\displaystyle K_{n}^{M}(F)\otimes \mathbb {Q} \subseteq K_{n}(F)\otimes \mathbb {Q} }${\displaystyle K_{n}^{M}(F)\otimes \mathbb {Q} \subseteq K_{n}(F)\otimes \mathbb {Q} }.^[3] It turns out the natural map ${\displaystyle \lambda :K_{4}^{M}(F)\to K_{4}(F)}${\displaystyle \lambda :K_{4}^{M}(F)\to K_{4}(F)} fails to be injective for a global field ${\displaystyle F}${\displaystyle F}^{[3] pg 96}.

Note for fields the Grothendieck group can be readily computed as ${\displaystyle K_{0}(F)=\mathbb {Z} }${\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

${\displaystyle l\colon K_{1}(F)\to F^{*}}${\displaystyle l\colon K_{1}(F)\to F^{*}}

(the group of units of ${\displaystyle F}${\displaystyle F}) and observing the calculation of K₂ of a field by Hideya Matsumoto, which gave the simple presentation

${\displaystyle K_{2}(F)={\frac {F^{*}\otimes F^{*}}{\{l(a)\otimes l(1-a):a\neq 0,1\}}}}${\displaystyle K_{2}(F)={\frac {F^{*}\otimes F^{*}}{\{l(a)\otimes l(1-a):a\neq 0,1\}}}}

for a two-sided ideal generated by elements ${\displaystyle l(a)\otimes l(a-1)}${\displaystyle l(a)\otimes l(a-1)}, called Steinberg relations. Milnor took the hypothesis that these were the only relations, hence he gave the following "ad-hoc" definition of Milnor K-theory as

${\displaystyle K_{n}^{M}(F)={\frac {K_{1}(F)\otimes \cdots \otimes K_{1}(F)}{\{l(a_{1})\otimes \cdots \otimes l(a_{n}):a_{i}+a_{i+1}=1\}}}.}${\displaystyle K_{n}^{M}(F)={\frac {K_{1}(F)\otimes \cdots \otimes K_{1}(F)}{\{l(a_{1})\otimes \cdots \otimes l(a_{n}):a_{i}+a_{i+1}=1\}}}.}

The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group ${\displaystyle K_{1}(F)\cong F^{*}}${\displaystyle K_{1}(F)\cong F^{*}} modded out by the two-sided ideal generated by:

${\displaystyle \left\{l(a)\otimes l(1-a):0,1\neq a\in F\right\}}${\displaystyle \left\{l(a)\otimes l(1-a):0,1\neq a\in F\right\}}

so

${\displaystyle \bigoplus _{n=0}^{\infty }K_{n}^{M}(F)\cong {\frac {T^{*}(K_{1}^{M}(F))}{\{l(a)\otimes l(1-a):a\neq 0,1\}}}}${\displaystyle \bigoplus _{n=0}^{\infty }K_{n}^{M}(F)\cong {\frac {T^{*}(K_{1}^{M}(F))}{\{l(a)\otimes l(1-a):a\neq 0,1\}}}}

showing his definition is a direct extension of the Steinberg relations.

The graded module ${\displaystyle K_{*}^{M}(F)}${\displaystyle K_{*}^{M}(F)} is a graded-commutative ring^{[1] pg 1-3}.^[4] If we write

${\displaystyle (l(a_{1})\otimes \cdots \otimes l(a_{n}))\cdot (l(b_{1})\otimes \cdots \otimes l(b_{m}))}${\displaystyle (l(a_{1})\otimes \cdots \otimes l(a_{n}))\cdot (l(b_{1})\otimes \cdots \otimes l(b_{m}))}

as

${\displaystyle l(a_{1})\otimes \cdots \otimes l(a_{n})\otimes l(b_{1})\otimes \cdots \otimes l(b_{m})}${\displaystyle l(a_{1})\otimes \cdots \otimes l(a_{n})\otimes l(b_{1})\otimes \cdots \otimes l(b_{m})}

then for ${\displaystyle \xi \in K_{i}^{M}(F)}${\displaystyle \xi \in K_{i}^{M}(F)} and ${\displaystyle \eta \in K_{j}^{M}(F)}${\displaystyle \eta \in K_{j}^{M}(F)} we have

${\displaystyle \xi \cdot \eta =(-1)^{i\cdot j}\eta \cdot \xi .}${\displaystyle \xi \cdot \eta =(-1)^{i\cdot j}\eta \cdot \xi .}

From the proof of this property, there are some additional properties which fall out, like $${\displaystyle l(a)^{2}=l(a)l(-1)}$${\displaystyle l(a)^{2}=l(a)l(-1)} for ${\displaystyle l(a)\in K_{1}(F)}${\displaystyle l(a)\in K_{1}(F)} since ${\displaystyle l(a)l(-a)=0}${\displaystyle l(a)l(-a)=0}. Also, if ${\displaystyle a_{1}+\cdots +a_{n}}${\displaystyle a_{1}+\cdots +a_{n}} of non-zero fields elements equals ${\displaystyle 0,1}${\displaystyle 0,1}, then $${\displaystyle l(a_{1})\cdots l(a_{n})=0}$${\displaystyle l(a_{1})\cdots l(a_{n})=0} There's a direct arithmetic application: ${\displaystyle -1\in F}${\displaystyle -1\in F} is a sum of squares if and only if every positive dimensional ${\displaystyle K_{n}^{M}(F)}${\displaystyle K_{n}^{M}(F)} is nilpotent, which is a powerful statement about the structure of Milnor K-groups. In particular, for the fields ${\displaystyle \mathbb {Q} (i)}${\displaystyle \mathbb {Q} (i)}, ${\displaystyle \mathbb {Q} _{p}(i)}${\displaystyle \mathbb {Q} _{p}(i)} with ${\displaystyle {\sqrt {-1}}\not \in \mathbb {Q} _{p}}${\displaystyle {\sqrt {-1}}\not \in \mathbb {Q} _{p}}, all of its Milnor K-groups are nilpotent. In the converse case, the field ${\displaystyle F}${\displaystyle F} can be embedded into a real closed field, which gives a total ordering on the field.

One of the core properties relating Milnor K-theory to higher algebraic K-theory is the fact there exists natural isomorphisms $${\displaystyle K_{n}^{M}(F)\to {\text{CH}}^{n}(F,n)}$${\displaystyle K_{n}^{M}(F)\to {\text{CH}}^{n}(F,n)} to Bloch's Higher chow groups which induces a morphism of graded rings $${\displaystyle K_{*}^{M}(F)\to {\text{CH}}^{*}(F,*)}$${\displaystyle K_{*}^{M}(F)\to {\text{CH}}^{*}(F,*)} This can be verified using an explicit morphism^{[2] pg 181} $${\displaystyle \phi :F^{*}\to {\text{CH}}^{1}(F,1)}$${\displaystyle \phi :F^{*}\to {\text{CH}}^{1}(F,1)} where $${\displaystyle \phi (a)\phi (1-a)=0~{\text{in}}~{\text{CH}}^{2}(F,2)~{\text{for}}~a,1-a\in F^{*}}$${\displaystyle \phi (a)\phi (1-a)=0~{\text{in}}~{\text{CH}}^{2}(F,2)~{\text{for}}~a,1-a\in F^{*}} This map is given by $${\displaystyle {\begin{aligned}\{1\}&\mapsto 0\in {\text{CH}}^{1}(F,1)\\\{a\}&\mapsto [a]\in {\text{CH}}^{1}(F,1)\end{aligned}}}$${\displaystyle {\begin{aligned}\{1\}&\mapsto 0\in {\text{CH}}^{1}(F,1)\\\{a\}&\mapsto [a]\in {\text{CH}}^{1}(F,1)\end{aligned}}} for ${\displaystyle [a]}${\displaystyle [a]} the class of the point ${\displaystyle [a:1]\in \mathbb {P} _{F}^{1}-\{0,1,\infty \}}${\displaystyle [a:1]\in \mathbb {P} _{F}^{1}-\{0,1,\infty \}} with ${\displaystyle a\in F^{*}-\{1\}}${\displaystyle a\in F^{*}-\{1\}}. The main property to check is that ${\displaystyle [a]+[1/a]=0}${\displaystyle [a]+[1/a]=0} for ${\displaystyle a\in F^{*}-\{1\}}${\displaystyle a\in F^{*}-\{1\}} and ${\displaystyle [a]+[b]=[ab]}${\displaystyle [a]+[b]=[ab]}. Note this is distinct from ${\displaystyle [a]\cdot [b]}${\displaystyle [a]\cdot [b]} since this is an element in ${\displaystyle {\text{CH}}^{2}(F,2)}${\displaystyle {\text{CH}}^{2}(F,2)}. Also, the second property implies the first for ${\displaystyle b=1/a}${\displaystyle b=1/a}. This check can be done using a rational curve defining a cycle in ${\displaystyle C^{1}(F,2)}${\displaystyle C^{1}(F,2)} whose image under the boundary map ${\displaystyle \partial }${\displaystyle \partial } is the sum ${\displaystyle [a]+[b]-[ab]}${\displaystyle [a]+[b]-[ab]}for ${\displaystyle ab\neq 1}${\displaystyle ab\neq 1}, showing they differ by a boundary. Similarly, if ${\displaystyle ab=1}${\displaystyle ab=1} the boundary map sends this cycle to ${\displaystyle [a]-[1/a]}${\displaystyle [a]-[1/a]}, showing they differ by a boundary. The second main property to show is the Steinberg relations. With these, and the fact the higher Chow groups have a ring structure $${\displaystyle {\text{CH}}^{p}(F,q)\otimes {\text{CH}}^{r}(F,s)\to {\text{CH}}^{p+r}(F,q+s)}$${\displaystyle {\text{CH}}^{p}(F,q)\otimes {\text{CH}}^{r}(F,s)\to {\text{CH}}^{p+r}(F,q+s)} we get an explicit map $${\displaystyle K_{*}^{M}(F)\to {\text{CH}}^{*}(F,*)}$${\displaystyle K_{*}^{M}(F)\to {\text{CH}}^{*}(F,*)} Showing the map in the reverse direction is an isomorphism is more work, but we get the isomorphisms $${\displaystyle K_{n}^{M}(F)\to {\text{CH}}^{n}(F,n)}$${\displaystyle K_{n}^{M}(F)\to {\text{CH}}^{n}(F,n)} We can then relate the higher Chow groups to higher algebraic K-theory using the fact there are isomorphisms $${\displaystyle K_{n}(X)\otimes \mathbb {Q} \cong \bigoplus _{p}{\text{CH}}^{p}(X,n)\otimes \mathbb {Q} }$${\displaystyle K_{n}(X)\otimes \mathbb {Q} \cong \bigoplus _{p}{\text{CH}}^{p}(X,n)\otimes \mathbb {Q} } giving the relation to Quillen's higher algebraic K-theory. Note that the maps

${\displaystyle K_{n}^{M}(F)\to K_{n}(F)}${\displaystyle K_{n}^{M}(F)\to K_{n}(F)}

from the Milnor K-groups of a field to the Quillen K-groups, which is an isomorphism for ${\displaystyle n\leq 2}${\displaystyle n\leq 2} but not for larger n, in general. For nonzero elements ${\displaystyle a_{1},\ldots ,a_{n}}${\displaystyle a_{1},\ldots ,a_{n}} in F, the symbol ${\displaystyle \{a_{1},\ldots ,a_{n}\}}${\displaystyle \{a_{1},\ldots ,a_{n}\}} in ${\displaystyle K_{n}^{M}(F)}${\displaystyle K_{n}^{M}(F)} means the image of ${\displaystyle a_{1}\otimes \cdots \otimes a_{n}}${\displaystyle a_{1}\otimes \cdots \otimes a_{n}} in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that ${\displaystyle \{a,1-a\}=0}${\displaystyle \{a,1-a\}=0} in ${\displaystyle K_{2}^{M}(F)}${\displaystyle K_{2}^{M}(F)} for ${\displaystyle a\in F\setminus \{0,1\}}${\displaystyle a\in F\setminus \{0,1\}} is sometimes called the Steinberg relation.

In motivic cohomology, specifically motivic homotopy theory, there is a sheaf ${\displaystyle K_{n,A}}${\displaystyle K_{n,A}} representing a generalization of Milnor K-theory with coefficients in an abelian group ${\displaystyle A}${\displaystyle A}. If we denote ${\displaystyle A_{tr}(X)=\mathbb {Z} _{tr}(X)\otimes A}${\displaystyle A_{tr}(X)=\mathbb {Z} _{tr}(X)\otimes A} then we define the sheaf ${\displaystyle K_{n,A}}${\displaystyle K_{n,A}} as the sheafification of the following pre-sheaf^{[5] pg 4} $${\displaystyle K_{n,A}^{pre}:U\mapsto A_{tr}(\mathbb {A} ^{n})(U)/A_{tr}(\mathbb {A} ^{n}-\{0\})(U)}$${\displaystyle K_{n,A}^{pre}:U\mapsto A_{tr}(\mathbb {A} ^{n})(U)/A_{tr}(\mathbb {A} ^{n}-\{0\})(U)} Note that sections of this pre-sheaf are equivalent classes of cycles on ${\displaystyle U\times \mathbb {A} ^{n}}${\displaystyle U\times \mathbb {A} ^{n}} with coefficients in ${\displaystyle A}${\displaystyle A} which are equidimensional and finite over ${\displaystyle U}${\displaystyle U} (which follows straight from the definition of ${\displaystyle \mathbb {Z} _{tr}(X)}${\displaystyle \mathbb {Z} _{tr}(X)}). It can be shown there is an ${\displaystyle \mathbb {A} ^{1}}${\displaystyle \mathbb {A} ^{1}}-weak equivalence with the motivic Eilenberg-Maclane sheaves ${\displaystyle K(A,2n,n)}${\displaystyle K(A,2n,n)} (depending on the grading convention).

For a finite field ${\displaystyle F=\mathbb {F} _{q}}${\displaystyle F=\mathbb {F} _{q}}, ${\displaystyle K_{1}^{M}(F)}${\displaystyle K_{1}^{M}(F)} is a cyclic group of order ${\displaystyle q-1}${\displaystyle q-1} (since is it isomorphic to ${\displaystyle \mathbb {F} _{q}^{*}}${\displaystyle \mathbb {F} _{q}^{*}}), so graded commutativity gives $${\displaystyle l(a)\cdot l(b)=-l(b)\cdot l(a)}$${\displaystyle l(a)\cdot l(b)=-l(b)\cdot l(a)} hence $${\displaystyle l(a)^{2}=-l(a)^{2}}$${\displaystyle l(a)^{2}=-l(a)^{2}} Because ${\displaystyle K_{2}^{M}(F)}${\displaystyle K_{2}^{M}(F)} is a finite group, this implies it must have order ${\displaystyle \leq 2}${\displaystyle \leq 2}. Looking further, ${\displaystyle 1}${\displaystyle 1} can always be expressed as a sum of quadratic non-residues, i.e. elements ${\displaystyle a,b\in F}${\displaystyle a,b\in F} such that ${\displaystyle [a],[b]\in F/F^{\times 2}}${\displaystyle [a],[b]\in F/F^{\times 2}} are not equal to ${\displaystyle 0}${\displaystyle 0}, hence ${\displaystyle a+b=1}${\displaystyle a+b=1} showing ${\displaystyle K_{2}^{M}(F)=0}${\displaystyle K_{2}^{M}(F)=0}. Because the Steinberg relations generate all relations in the Milnor K-theory ring, we have ${\displaystyle K_{n}^{M}(F)=0}${\displaystyle K_{n}^{M}(F)=0} for ${\displaystyle n>2}${\displaystyle n>2}.

For the field of real numbers ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} } the Milnor K-theory groups can be readily computed. In degree ${\displaystyle n}${\displaystyle n} the group is generated by $${\displaystyle K_{n}^{M}(\mathbb {R} )=\{(-1)^{n},l(a_{1})\cdots l(a_{n}):a_{1},\ldots ,a_{n}>0\}}$${\displaystyle K_{n}^{M}(\mathbb {R} )=\{(-1)^{n},l(a_{1})\cdots l(a_{n}):a_{1},\ldots ,a_{n}>0\}} where ${\displaystyle (-1)^{n}}${\displaystyle (-1)^{n}} gives a group of order ${\displaystyle 2}${\displaystyle 2} and the subgroup generated by the ${\displaystyle l(a_{1})\cdots l(a_{n})}${\displaystyle l(a_{1})\cdots l(a_{n})} is divisible. The subgroup generated by ${\displaystyle (-1)^{n}}${\displaystyle (-1)^{n}} is not divisible because otherwise it could be expressed as a sum of squares. The Milnor K-theory ring is important in the study of motivic homotopy theory because it gives generators for part of the motivic Steenrod algebra.^[6] The others are lifts from the classical Steenrod operations to motivic cohomology.

${\displaystyle K_{2}^{M}(\mathbb {C} )}${\displaystyle K_{2}^{M}(\mathbb {C} )} is an uncountable uniquely divisible group.^[7] Also, ${\displaystyle K_{2}^{M}(\mathbb {R} )}${\displaystyle K_{2}^{M}(\mathbb {R} )} is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; ${\displaystyle K_{2}^{M}(\mathbb {Q} _{p})}${\displaystyle K_{2}^{M}(\mathbb {Q} _{p})} is the direct sum of the multiplicative group of ${\displaystyle \mathbb {F} _{p}}${\displaystyle \mathbb {F} _{p}} and an uncountable uniquely divisible group; ${\displaystyle K_{2}^{M}(\mathbb {Q} )}${\displaystyle K_{2}^{M}(\mathbb {Q} )} is the direct sum of the cyclic group of order 2 and cyclic groups of order ${\displaystyle p-1}${\displaystyle p-1} for all odd prime ${\displaystyle p}${\displaystyle p}. For ${\displaystyle n\geq 3}${\displaystyle n\geq 3}, ${\displaystyle K_{n}^{M}(\mathbb {Q} )\cong \mathbb {Z} /2}${\displaystyle K_{n}^{M}(\mathbb {Q} )\cong \mathbb {Z} /2}. The full proof is in the appendix of Milnor's original paper.^[1] Some of the computation can be seen by looking at a map on ${\displaystyle K_{2}^{M}(F)}${\displaystyle K_{2}^{M}(F)} induced from the inclusion of a global field ${\displaystyle F}${\displaystyle F} to its completions ${\displaystyle F_{v}}${\displaystyle F_{v}}, so there is a morphism$${\displaystyle K_{2}^{M}(F)\to \bigoplus _{v}K_{2}^{M}(F_{v})/({\text{max. divis. subgr.}})}$${\displaystyle K_{2}^{M}(F)\to \bigoplus _{v}K_{2}^{M}(F_{v})/({\text{max. divis. subgr.}})} whose kernel finitely generated. In addition, the cokernel is isomorphic to the roots of unity in ${\displaystyle F}${\displaystyle F}.

In addition, for a general local field ${\displaystyle F}${\displaystyle F} (such as a finite extension ${\displaystyle K/\mathbb {Q} _{p}}${\displaystyle K/\mathbb {Q} _{p}}), the Milnor K-groups ${\displaystyle K_{n}^{M}(F)}${\displaystyle K_{n}^{M}(F)} are divisible.

There is a general structure theorem computing ${\displaystyle K_{n}^{M}(F(t))}${\displaystyle K_{n}^{M}(F(t))} for a field ${\displaystyle F}${\displaystyle F} in relation to the Milnor K-theory of ${\displaystyle F}${\displaystyle F} and extensions ${\displaystyle F[t]/(\pi )}${\displaystyle F[t]/(\pi )} for non-zero primes ideals ${\displaystyle (\pi )\in {\text{Spec}}(F[t])}${\displaystyle (\pi )\in {\text{Spec}}(F[t])}. This is given by an exact sequence $${\displaystyle 0\to K_{n}^{M}(F)\to K_{n}^{M}(F(t))\xrightarrow {\partial _{\pi }} \bigoplus _{(\pi )\in {\text{Spec}}(F[t])}K_{n-1}F[t]/(\pi )\to 0}$${\displaystyle 0\to K_{n}^{M}(F)\to K_{n}^{M}(F(t))\xrightarrow {\partial _{\pi }} \bigoplus _{(\pi )\in {\text{Spec}}(F[t])}K_{n-1}F[t]/(\pi )\to 0} where ${\displaystyle \partial _{\pi }:K_{n}^{M}(F(t))\to K_{n-1}F[t]/(\pi )}${\displaystyle \partial _{\pi }:K_{n}^{M}(F(t))\to K_{n-1}F[t]/(\pi )} is a morphism constructed from a reduction of ${\displaystyle F}${\displaystyle F} to ${\displaystyle {\overline {F}}_{v}}${\displaystyle {\overline {F}}_{v}} for a discrete valuation ${\displaystyle v}${\displaystyle v}. This follows from the theorem there exists only one homomorphism $${\displaystyle \partial :K_{n}^{M}(F)\to K_{n-1}^{M}({\overline {F}})}$${\displaystyle \partial :K_{n}^{M}(F)\to K_{n-1}^{M}({\overline {F}})} which for the group of units ${\displaystyle U\subset F}${\displaystyle U\subset F} which are elements have valuation ${\displaystyle 0}${\displaystyle 0}, having a natural morphism $${\displaystyle U\to {\overline {F}}_{v}^{*}}$${\displaystyle U\to {\overline {F}}_{v}^{*}} where ${\displaystyle u\mapsto {\overline {u}}}${\displaystyle u\mapsto {\overline {u}}} we have $${\displaystyle \partial (l(\pi )l(u_{2})\cdots l(u_{n}))=l({\overline {u}}_{2})\cdots l({\overline {u}}_{n})}$${\displaystyle \partial (l(\pi )l(u_{2})\cdots l(u_{n}))=l({\overline {u}}_{2})\cdots l({\overline {u}}_{n})} where ${\displaystyle \pi }${\displaystyle \pi } a prime element, meaning ${\displaystyle {\text{Ord}}_{v}(\pi )=1}${\displaystyle {\text{Ord}}_{v}(\pi )=1}, and $${\displaystyle \partial (l(u_{1})\cdots l(u_{n}))=0}$${\displaystyle \partial (l(u_{1})\cdots l(u_{n}))=0} Since every non-zero prime ideal ${\displaystyle (\pi )\in {\text{Spec}}(F[t])}${\displaystyle (\pi )\in {\text{Spec}}(F[t])} gives a valuation ${\displaystyle v_{\pi }:F(t)\to F[t]/(\pi )}${\displaystyle v_{\pi }:F(t)\to F[t]/(\pi )}, we get the map ${\displaystyle \partial _{\pi }}${\displaystyle \partial _{\pi }} on the Milnor K-groups.

Milnor K-theory plays a fundamental role in higher class field theory, replacing ${\displaystyle K_{1}^{M}(F)=F^{\times }\!}${\displaystyle K_{1}^{M}(F)=F^{\times }\!} in the one-dimensional class field theory.

Milnor K-theory fits into the broader context of motivic cohomology, via the isomorphism

${\displaystyle K_{n}^{M}(F)\cong H^{n}(F,\mathbb {Z} (n))}${\displaystyle K_{n}^{M}(F)\cong H^{n}(F,\mathbb {Z} (n))}

of the Milnor K-theory of a field with a certain motivic cohomology group.^[8] In this sense, the apparently ad hoc definition of Milnor K-theory becomes a theorem: certain motivic cohomology groups of a field can be explicitly computed by generators and relations.

A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or étale cohomology:

${\displaystyle K_{n}^{M}(F)/r\cong H_{\mathrm {et} }^{n}(F,\mathbb {Z} /r(n)),}${\displaystyle K_{n}^{M}(F)/r\cong H_{\mathrm {et} }^{n}(F,\mathbb {Z} /r(n)),}

for any positive integer r invertible in the field F. This conjecture was proved by Vladimir Voevodsky, with contributions by Markus Rost and others.^[9] This includes the theorem of Alexander Merkurjev and Andrei Suslin as well as the Milnor conjecture as special cases (the cases when ${\displaystyle n=2}${\displaystyle n=2} and ${\displaystyle r=2}${\displaystyle r=2}, respectively).

Finally, there is a relation between Milnor K-theory and quadratic forms. For a field F of characteristic not 2, define the fundamental ideal I in the Witt ring of quadratic forms over F to be the kernel of the homomorphism ${\displaystyle W(F)\to \mathbb {Z} /2}${\displaystyle W(F)\to \mathbb {Z} /2} given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:

${\displaystyle {\begin{cases}K_{n}^{M}(F)/2\to I^{n}/I^{n+1}\\\{a_{1},\ldots ,a_{n}\}\mapsto \langle \langle a_{1},\ldots ,a_{n}\rangle \rangle =\langle 1,-a_{1}\rangle \otimes \cdots \otimes \langle 1,-a_{n}\rangle \end{cases}}}${\displaystyle {\begin{cases}K_{n}^{M}(F)/2\to I^{n}/I^{n+1}\\\{a_{1},\ldots ,a_{n}\}\mapsto \langle \langle a_{1},\ldots ,a_{n}\rangle \rangle =\langle 1,-a_{1}\rangle \otimes \cdots \otimes \langle 1,-a_{n}\rangle \end{cases}}}

where ${\displaystyle \langle \langle a_{1},a_{2},\ldots ,a_{n}\rangle \rangle }${\displaystyle \langle \langle a_{1},a_{2},\ldots ,a_{n}\rangle \rangle } denotes the class of the n-fold Pfister form.^[10]

Dmitri Orlov, Alexander Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism ${\displaystyle K_{n}^{M}(F)/2\to I^{n}/I^{n+1}}${\displaystyle K_{n}^{M}(F)/2\to I^{n}/I^{n+1}} is an isomorphism.^[11]

• Azumaya algebra
• Motivic homotopy theory

• Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander (2008), Algebraic and geometric theory of quadratic forms, American Mathematical Society, ISBN 978-0-8218-4329-1, MR 2427530
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. MR 2266528. Zbl 1137.12001.
• Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lectures in Motivic Cohomology, Clay Mathematical Monographs, vol. 2, 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), With an appendix by John Tate: 318–344, Bibcode:1970InMat...9..318M, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, S2CID 13549621, Zbl 0199.55501
• Orlov, Dmitri; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for ${\displaystyle K_{*}^{M}/2}${\displaystyle K_{*}^{M}/2} with applications to quadratic forms", Annals of Mathematics , 165: 1–13, arXiv:math/0101023 , doi:10.4007/annals.2007.165.1, MR 2276765, S2CID 9504456
• Voevodsky, Vladimir (2011), "On motivic cohomology with ${\displaystyle \mathbb {Z} /\ell }${\displaystyle \mathbb {Z} /\ell }-coefficients", Annals of Mathematics , 174 (1): 401–438, arXiv:0805.4430 , doi:10.4007/annals.2011.174.1.11, MR 2811603, S2CID 15583705

• Some aspects of the functor ${\displaystyle K_{2}}${\displaystyle K_{2}} of fields
• About Tate's computation of ${\displaystyle K_{2}(\mathbb {Q} )}${\displaystyle K_{2}(\mathbb {Q} )}

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