Question

When Milnor introduced in "Algebraic K-Theory and Quadratic Forms" the Milnor K-groups he said that his definition is motivated by Matsumoto's presentation of algebraic for a field but is in the end purely ad hoc for . My questions are:

1. What exactly could Milnor prove with these -groups? What was his motivation except for Matsumoto's theorem?
2. Why did this ad hoc definition become so important? Why is it so natural?

Answer 1

Milnor K-theory gives a way to compute étale cohomology of fields (i.e. Galois cohomology): if E is a field of characteristic different from a prime l, there is a residue map from the nth Milnor K-group of E mod l to the nth étale cohomology group of E with coefficients in the sheaf of lth roots of unity to the n (i.e. tensored with itself n times). There is the Bloch-Kato conjecture, which predicts that these residue maps are bijectvive. It happens that the case l=2 was conjectured by Milnor (up to a reformulation I guess). The Milnor conjecture has been proved by Voevodsky (and it was the first great achievements of homotopy theory of schemes, which he initiated with Morel during the 90's), and he got his Fields medal in 2002 for this. Now Rost and Voevodsky claimed they have a proof of the full Bloch-Kato conjecture for any prime l (which should appear some day, thanks to the work of quite a few people, among which Charles Weibel is not the least). Note also that the Bloch-Kato conjecture makes sense for l=p=char(E), but then, you have to replace étale cohomology by de Rham-Witt cohomology (and this has also been proved by Bloch and Kato). Suslin and Voevodsky also proved that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture, which predicts the precise relationship between torsion motivic cohomology of varieties with torsion étale cohomology.

Milnor K-theory is related to motivic cohomology (i.e. higher Chow groups) in degree n and weight n H^n(X,Z(n)): for X=Spec(E), H^n(X,Z(n)) is the nth Milnor K-group. This is how homotopy theory of schemes enters in the picture (one of the main feature introduced by Voevodsky to study motivic cohomology with finite coefficients is the theory of motivic Steenrod operations). On the other hand, Rost studied Milnor K-theory for itself: among a lot of other things, he proved that, if you consider it as a functor from the category of fields, with all its extra structures (residue maps interacting well), you can reconstruct higher Chow groups of schemes (over a field), via some Gersten complex.

Milnor K-theory is also a crucial ingredient in Kato's higher class field theory.

Answer 2

To help answer Question 1, Milnor proved a local-global theorem for Witt rings of global fields. Recall that The Grothendieck-Witt ring $\widehat{W}(k)$ of a field $k$ is the ring obtained by starting with the free abelian group on isomorphism classes of quadratic modules and moding out by the ideal generated by symbols of the form $[M]+[N]-[M']-[N']$, whenever $[M]\oplus[N]\simeq [M']+[N']$. The multiplication comes from tensor product of quadratic modules. There is a special quadratic module $H$ given by $x^2-y^2=0$. This is the hyperbolic module. The Witt ring $W(k)$ of a field $k$ is the quotient of $\widehat{W}(k)$ by the ideal generated by $[H]$.

Now, the main theorem of Milnor's paper is that there is a split exact sequence $$0\rightarrow W(k)\rightarrow W(k(t))\rightarrow \oplus_\pi W(\overline{k(t)}_\pi)\rightarrow 0,$$ where $\pi$ runs over all irreducible monic polynomials in $k[t]$, and $\overline{k(t)}_\pi$ denotes the residue field of the completion of $k(t)$ at $\pi$.

The morphisms $W(k(t))\rightarrow W(\overline{k(t)}_\pi)$ come from first the map $W(k(t))\rightarrow W(k(t)_\pi)$. Then, there is a map $W(k(t)_\pi)\rightarrow W(\overline(k(t))_\pi)$ that sends the quadratic module $u\pi x^2=0$ to $ux^2=0$, where $u$ is any unit of the local field.

Interestingly, Milnor $K$-theory is not used in the proof. However, the proof for Witt rings closely models the proof of a similar fact for Milnor $K$-theory: the sequence $$0\rightarrow K_n^M(k)\rightarrow K_n^M(k(t))\rightarrow\oplus_\pi K_{n-1}^M(\overline{k(t)}_\pi)\rightarrow 0.$$

The important new perspective is the formal symbolic perspective, which was already existent for lower $K$-groups, but is very fruitful for studying the Witt ring as well.

Answer 3

To see a few places where $K_2$ shows up, consult arXiv:math/0311099v4 [math.HO].

Answer 4

As already mentioned above by Denis-Charles Cisinski, Rost has shown (see "Chow Groups with Coefficients") that some version of higher Chow groups can be constructed via Milnor K groups.

In fact, Gillet in his survey "K Theory and Intersection Theory" (googleable, I believe originally in the K-Theory Handbook) explains on page 24 and most importantly page 25 (middle) how one may even motivate the defining relations of Milnor K (i.e. the Steinberg relation) by intersection-theoric ideas. Whether you find this explanation natural or not is your free choice, but there is some beauty in it.

Answer 5

Another application of Milnor K-groups:

1. The following are equivalent:

${a_1, \ldots, a_n} = 0 \in K^M_n(K)/2$

$\langle\kern-0.2em\langle{a_1, \ldots, a_n}\rangle\kern-0.2em\rangle$ (Pfister form) is totally hyperbolic

$\langle\kern-0.2em\langle{a_1, \ldots, a_n}\rangle\kern-0.2em\rangle$ is isotropic

$a_n$ is represented by $\langle\kern-0.2em\langle{a_1, \ldots, a_{n-1}\rangle\kern-0.2em}\rangle$

1. higher local class field theory: The class formation of an $n$-dimensional local field is $K^M_n(K)$. http://www.emis.de/journals/GT/ftp/main/m3/