Tuesday, 5 February 2008

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

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, barF a
separable closure of F and Gamma=operatornameGal(barF|F). There
is an exact sequence
1tomathbbZ/nmathbbZ(1)tobarFtimestobarFtimesto1


of discrete Gamma-modules, where mathbbZ/nmathbbZ(1) is the group of n-th roots of 1 in barF. The associated long exact cohomology sequence and
Hilbert's theorem 90 furnish an isomorphism delta1:Ftimes/FtimesntoH1(Gamma,mathbbZ/nmathbbZ(1)). Cup product on cohomology
smile;:Hr(Gamma,mathbbZ/nmathbbZ(r))timesHs(Gamma,mathbbZ/nmathbbZ(s))toHr+s(Gamma,mathbbZ/nmathbbZ(r+s))

then provides a bilinear map
delta2:Ftimes/FtimesntimesFtimes/FtimesntoH2(Gamma,mathbbZ/nmathbbZ(2)).



Lemma (Tate, 1970)
The map delta2(x,y)=delta1(x)smiledelta1(y) is a
symbol on
F.



A symbol is a bilinear map s:FtimestimesFtimestoA to a commutative
group such that s(x,y)=0 whenever x+y=1 in Ftimes.
There is a universal symbol FtimestimesFtimestoK2(F), giving rise
to Milnor's theory of higher K-groups Kr(F) for every rinmathbbN,
as explained in Milnor's book.



This symbol also gives rise to a homomorphism
deltar:Kr(F)/nKr(F)toHr(Gamma,mathbbZ/nmathbbZ(r)).



Conjecture (Bloch-Kato, 1986)
The map
deltar is an isomorphism for all fields F, all integers n>0
(invertible in F) and all indices rinmathbbN.



The main theorem of Merkurjev-Suslin (1982) says that the map
delta2 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 pneq0 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 oplusrHr(Gamma,mathbbZ/nmathbbZ(r)) is generated by
elements of degree 1. Galois groups should thus be very special among
profinite groups in this respect.

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