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ON A LOCAL ANALOGUE OF THE GROTHENDIECK CONJECTURE
 

Summary: ON A LOCAL ANALOGUE OF THE
GROTHENDIECK CONJECTURE
Victor A. Abrashkin
We prove that the functor from the category of all complete discrete valuation
fields with finite residue fields of characteristic #= 2 to the category of profinite fil­
tered groups given by taking the Galois group of corresponding field together with
its filtration by higher ramification subgroups is fully faithful. If [K; Qp ] < # we
also study the opportunity to recover K from the knowledge of the filtered group
#K (p)/# K (p) (a) , where a > 0, #K (p) is the absolute Galois group of the maximal
p­extension of K and filtration is induced by ramification filtration.
0. Introduction.
Recall the following result known as the Neukirch­Iwasawa­Uchida­Ikeda The­
orem, cf. [11]. If K 1 and K 2 are algebraic number fields in some fixed alge­
braic closure •
Q, then any continuous group isomorphism between Gal( •
Q/K 1 ) and
Gal( •
Q/K 2 ) comes from conjugation by some element of #Q = Gal( •
Q/Q). In partic­
ular, any algebraic number field can be uniquely recovered from its Galois group.

  

Source: Abrashkin, Victor - Department of Mathematical Sciences, University of Durham

 

Collections: Mathematics