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Report on the ramification filtration of the Galois group of a local field (by Victor Abrashkin).
 

Summary: Report on the ramification filtration of the Galois group of a local field
(by Victor Abrashkin).
Let K be a complete discrete valuation field with residue field k # F p N 0 . Let
K(p) be its maximal p­extension, then #(p) = Gal(K(p)/K) is a pro­p­group.
Remark. We have:
#(p)/C 2 (#(p) = #(p) ab
# (cl­f. theory) “
K # = lim
#- K # /K #p m
;
char K = p # #(p) is pro­p­free; it has # many generators;
char K = 0, # p /
# K # #(p) is pro­p­free, it has finitely many generators;
char K = 0, # p # K # is a Demushkin group.
Let {#(p) (v)
} v#1 be the ramificatiom filtration of the #(p).
Remark. Recall that, if #(p) = lim
#- # L/K with respect to the filter of finite normal
subextensions L/K, then:
#x # 0, # L/K,x = {# # # L/K |v L (##L - #L ) # x + 1} --- the lower numbering;

  

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

 

Collections: Mathematics