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feat(ring_theory/valuation/valuation_subring): Add `valuation_subring…
….inertia_subgroup` (#17086) The decomposition and inertia subgroups. Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Michail Karatarakis <40603357+mkaratarakis@users.noreply.github.com>
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/- | ||
Copyright (c) 2022 Michail Karatarakis. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Michail Karatarakis | ||
-/ | ||
import ring_theory.ideal.local_ring | ||
import ring_theory.valuation.valuation_subring | ||
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/-! | ||
# Ramification groups | ||
The decomposition subgroup and inertia subgroups. | ||
TODO: Define higher ramification groups in lower numbering | ||
-/ | ||
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namespace valuation_subring | ||
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open_locale pointwise | ||
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variables (K : Type*) {L : Type*} [field K] [field L] [algebra K L] | ||
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/-- The decomposition subgroup defined as the stabilizer of the action | ||
on the type of all valuation subrings of the field. -/ | ||
@[reducible] def decomposition_subgroup (A : valuation_subring L) : | ||
subgroup (L ≃ₐ[K] L) := | ||
mul_action.stabilizer (L ≃ₐ[K] L) A | ||
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/-- The valuation subring `A` (considered as a subset of `L`) | ||
is stable under the action of the decomposition group. -/ | ||
def sub_mul_action (A : valuation_subring L) : | ||
sub_mul_action (A.decomposition_subgroup K) L := | ||
{ carrier := A, | ||
smul_mem' := λ g l h, set.mem_of_mem_of_subset (set.smul_mem_smul_set h) g.prop.le } | ||
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/-- The multiplicative action of the decomposition subgroup on `A`. -/ | ||
instance decomposition_subgroup_mul_semiring_action (A : valuation_subring L) : | ||
mul_semiring_action (A.decomposition_subgroup K) A := | ||
{ smul_add := λ g k l, subtype.ext $ smul_add g k l, | ||
smul_zero := λ g, subtype.ext $ smul_zero g, | ||
smul_one := λ g, subtype.ext $ smul_one g, | ||
smul_mul := λ g k l, subtype.ext $ smul_mul' g k l, | ||
..(sub_mul_action.mul_action (A.sub_mul_action K)) } | ||
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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`. -/ | ||
def inertia_subgroup (A : valuation_subring L) : | ||
subgroup (A.decomposition_subgroup K) := | ||
monoid_hom.ker $ | ||
mul_semiring_action.to_ring_aut (A.decomposition_subgroup K) (local_ring.residue_field A) | ||
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end valuation_subring |