Summary: ONE GROUP­THEORETIC PROPERTY
OF THE RAMIFICATION FILTRATION
Victor A. Abrashkin
Abstract. Let #(p) be the Galois group of a maximal p­extension of a complete
discrete valuation field with perfect residue field of characteristic p > 0. If v0 > -1
and #(p) (v 0 ) is the ramification subgroup of #(p) in upper numbering, we prove that
any closed but not open finitely generated subgroup of the quotient #(p)/#(p) (v 0 ) is
a free pro­p­group. In particular, this quotient does not have non­trivial torsion and
non­trivially commuting elements.
1. The statement of the main theorem.
Let K be a complete discrete valuation field with perfect residue field k of char­
acteristic p > 0. Choose a separable closure K sep of K and denote by K(p) the
maximal p­extension of K in K sep .
If # = Gal(K sep /K) and {# (v)
} v#0 is the ramification filtration of # in upper
numbering, cf. [Se, Ch.III], we have the induced filtration {#(p) (v)
} v#0 of the group
#(p) = Gal(K(p)/K). We note that for -1 < v # 1, #(p) (v) = I(p) is the inertia
subgroup of #(p), i.e. K(p) I(p) is the maximal unramified extension K(p) ur of K
in K(p).

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

Collections: Mathematics