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Mathlib.RingTheory.Valuation.RamificationGroup

Ramification groups #

The decomposition subgroup and inertia subgroups.

TODO: Define higher ramification groups in lower numbering

@[reducible, inline]
abbrev ValuationSubring.decompositionSubgroup (K : Type u_1) {L : Type u_2} [Field K] [Field L] [Algebra K L] (A : ValuationSubring L) :

The decomposition subgroup defined as the stabilizer of the action on the type of all valuation subrings of the field.

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    The valuation subring A (considered as a subset of L) is stable under the action of the decomposition group.

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      The multiplicative action of the decomposition subgroup on A.

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      The inertia subgroup defined as the kernel of the group homomorphism from the decomposition subgroup to the group of automorphisms of the residue field of A.

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