Is there a simple relationship between K-theory and Galois theory?

Question

I can (barely) understand the definition of the higher algebraic K-groups a la the plus construction right now (I have some past familiarity with K-theory for C*-algebras and can recall the rudiments of the situation for vector bundles). So by "simple", I mean to a mathematical layman. If you have a complicated answer, feel free to answer as well, but I probably won't be able to understand much.

Answer 1

Perhaps the other Bloch-Kato conjecture is more relevant; it relates Milnor's higher $K$-groups and Galois cohomology.

The following text is lifted from the expository account on the arXiv.

Let $F$ be a field, $n>0$ an integer which is invertible in $F$, $\bar F$ a separable closure of $F$ and $\Gamma=\operatorname{Gal}(\bar F|F)$. There is an exact sequence $$ \{1\}\to \mathbb{Z}/n\mathbb{Z}(1)\to {\bar F}^\times\to {\bar F}^\times\to \{1\} $$ of discrete $\Gamma$-modules, where $\mathbb{Z}/n\mathbb{Z}(1)$ is the group of $n$-th roots of $1$ in $\bar F$. The associated long exact cohomology sequence and Hilbert's theorem 90 furnish an isomorphism $\delta_1:F^\times/F^{\times n}\to H^1(\Gamma,\mathbb{Z}/n\mathbb{Z}(1))$. Cup product on cohomology $$ \smile\;:H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)) \times H^s(\Gamma,\mathbb{Z}/n\mathbb{Z}(s))\to H^{r+s}(\Gamma,\mathbb{Z}/n\mathbb{Z}(r+s)) $$ then provides a bilinear map $ \delta_2:F^\times/F^{\times n}\times F^\times/F^{\times n}\to H^2(\Gamma,\mathbb{Z}/n\mathbb{Z}(2)). $

Lemma (Tate, 1970) The map $\delta_2(x,y)=\delta_1(x)\smile\delta_1(y)$ is a symbol on $F$.

A symbol is a bilinear map $s:F^\times\times F^\times\to A$ to a commutative group such that $s(x,y)=0$ whenever $x+y=1$ in $F^\times$. There is a universal symbol $F^\times\times F^\times\to K_2(F)$, giving rise to Milnor's theory of higher $K$-groups $K_r(F)$ for every $r\in\mathbb{N}$, as explained in Milnor's book.

This symbol also gives rise to a homomorphism $$ \delta_r:K_r(F)/nK_r(F)\to H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r)). $$

Conjecture (Bloch-Kato, 1986) The map $\delta_r$ is an isomorphism for all fields $F$, all integers $n>0$ (invertible in $F$) and all indices $r\in\mathbb{N}$.

The main theorem of Merkurjev-Suslin (1982) says that the map $\delta_2$ is always an isomorphism ; Tate had proved this earlier (1976) for global fields. Bloch-Gabber-Kato prove this conjecture when $F$ is a field of characteristic $0$ endowed with a henselian discrete valuation of residual characteristic $p\neq0$ and $n$ is a power of $p$.

Somebody should ask a qustion about the current status of the Bloch-Kato conjecture and get some experts (such as Weibel) to answer. My impression is that it is now a theorem by the work of Rost and Voevodsky, but that a proof with all the details is not available in one place.

The Bloch-Kato conjecture makes the remarkable prediction that the graded algebra $\oplus_r H^r(\Gamma,\mathbb{Z}/n\mathbb{Z}(r))$ is generated by elements of degree 1. Galois groups should thus be very special among profinite groups in this respect.

Answer 2

A simpler statement may be that if $F \to E$ is a $G$-Galois extension, then there is a map $K(F) \to K(E)^{hG}$ from the algebraic $K$-theory space of $F$ to the $G$-homotopy fixed points of the algebraic $K$-theory space of $E$. The Lichtenbaum--Quillen conjecture identifies situations where this map is ``almost'' a weak equivalence, in the sense that it induces an isomorphism on homotopy groups with finite coefficients and in sufficiently high degrees.

There is a spectral sequence $E^2_{s,t} = H^{-s}(G; K_t(E))$ (group cohomology) converging to $\pi_{s+t}(K(E)^{hG})$, and similarly with finite coefficients $Z/p$, so when this is equivalent to $\pi_{s+t} K(F) = K_{s+t}(F)$ you can recover the algebraic $K$-theory of $F$ from the algebraic $K$-theory of $E$. This is a form of Galois descent.

Letting $E$ grow to the separable closure of $F$, you can use Suslin's theorem about the algebraic $K$-theory of separably closed fields to get at the algebraic $K$-theory of other fields $F$ by way of the cohomology of their absolute Galois groups, i.e., their Galois cohomology.

If you work with more general commutative rings, or schemes, and replace Galois extensions by etale covers, your answer will involve etale cohomology instead of Galois cohomology, but the idea is much the same.

The Lichtenbaum--Quillen conjecture (e.g. for number fields) is known to follow from the Milnor conjecture (at p=2) and the Bloch--Kato conjecture (at odd primes p). The argument compares etale cohomology to motivic cohomology, and uses the Atiyah--Hirzebruch type spectral sequence from motivic cohomology to algebraic $K$-theory. Voevodsky proved the Milnor conjecture in 1996. The (presumed confirmed) status of the full Bloch--Kato conjecture is discussed in other posts.

Given these results, the algebraic $K$-groups of (rings of integers in) number fields or local fields of characteristic zero are computed in terms of the corresponding etale cohomology groups.

If you work with brave new rings, or structured ring spectra, the Galois descent analogue of the Lichtenbaum--Quillen conjectures is open, the etale descent version is still in need of an optimal formulation, and I think there is no known definition of motivic cohomology. In these cases the best computational results are obtained using the cyclotomic trace map from algebraic $K$-theory to topological cyclic homology, not by descent.

• John

Answer 3

There are lots of maps from $K$ groups to various other things, including Galois groups. Here's an example, due to Kato, which generalizes local class field theory.

Classical local class field theory says that if $F$ is a local field (by which I mean, in this case, a field complete with respect to a discrete valuation and with a finite residue field), then there is a natural map $F^\times\to Gal(F^{ab}/F)$ which behaves well on the finite levels: if $L/F$ is a finite extension, then $F^\times/N_{L/F}L^\times\to Gal(L/F)$ is an isomorphism. But $F^\times=K_1(F)$, so we have a map $K_1(F)\to Gal(F^{ab}/F)$ that behaves in the same way on finite levels.

We'll generalize this by defining an $r$-dimensional local field inductively. A 0-dimensional local field is a finite field, and an $r$-dimensional local field is a field complete with respect to a discrete valuation whose residue field is an $(r-1)$-dimensional local field. Thus a classical local field is a 1-dimensional local field.

Kato found maps $K_r^M(F)\to Gal(F^{ab}/F)$ which are isomorphisms on the finite levels, where $K_r^M$ are the Milnor (not Quillen!) $K$-groups. They also behave similarly well on the finite levels.