The inertia subgroup of the absolute Galois group of a local field #
Let K be a nonarchimedean local field with residue field of cardinality q, let K^{alg} be
its algebraic closure, and let G_K = Field.absoluteGaloisGroup K = Gal(K^{alg}/K). This file
defines the inertia subgroup
TauCeti.inertiaSubgroup K ≤ G_K,
the automorphisms of K^{alg} fixing the maximal unramified extension K^{ur} of K inside
K^{alg}, so that IntermediateField.fixingSubgroupEquiv identifies it with Gal(K^{alg}/K^{ur}).
It is a closed normal subgroup, and it sits in the exact sequence
1 → I_K → G_K → Gal(K^{ur}/K) → 1
given by restriction TauCeti.restrictMaximalUnramifiedHom K, which is surjective with kernel
I_K; the unramified quotient G_K ⧸ I_K is identified with Gal(K^{ur}/K) as a topological
group. Inertia is not open in G_K, since Gal(K^{ur}/K) ≃ ℤ̂ is infinite. A finite separable
subextension of K^{alg}/K is unramified exactly when inertia fixes it, and on a finite normal
subextension L inertia restricts into the inertia group G_0 of L/K.
An arithmetic Frobenius lift is an element of G_K restricting to the arithmetic Frobenius
TauCeti.maximalUnramifiedFrobenius of K^{ur}/K; equivalently, it raises every root of every
polynomial X^{q^f} − X, f ≠ 0, to the q-th power. Lifts exist, they form a single left coset
of I_K, and each of them generates G_K topologically together with I_K.
Main definitions #
TauCeti.inertiaSubgroup K: the inertia subgroupI_KofG_K.TauCeti.restrictMaximalUnramifiedHom K: restrictionG_K →* Gal(K^{ur}/K).TauCeti.unramifiedQuotient K,TauCeti.unramifiedDegree K: the quotientG_K ⧸ I_Kand its canonical quotient map.TauCeti.quotientInertiaSubgroupEquiv K: the unramified quotientG_K ⧸ I_K ≃ₜ* Gal(K^{ur}/K).TauCeti.IsArithFrobeniusLift K σ:σ ∈ G_Krestricts to the arithmetic Frobenius ofK^{ur}.
Main results #
TauCeti.mem_inertiaSubgroup_iff_pow_natCard_pow_eq_self:σ ∈ I_Kexactly whenσfixes the roots of the polynomialsX^{q^f} − X.TauCeti.apply_eq_self_of_mem_inertiaSubgroup_of_pow_eq_one:I_Kfixes the roots of unity of order prime to the residue characteristic.TauCeti.isClosed_inertiaSubgroup,TauCeti.inertiaSubgroup_normal:I_Kis closed and normal.TauCeti.not_isOpen_inertiaSubgroup:I_Kis not open inG_K.TauCeti.restrictMaximalUnramifiedHom_surjective,TauCeti.ker_restrictMaximalUnramifiedHom: restrictionG_K → Gal(K^{ur}/K)is surjective with kernelI_K.TauCeti.unramifiedDegree_surjective,TauCeti.continuous_unramifiedDegree,TauCeti.ker_unramifiedDegree: the quotient mapG_K → G_K ⧸ I_Kis a continuous surjection with kernelI_K.TauCeti.inertiaSubgroup_le_fixingSubgroup_iff: a finite separable subextension is unramified exactly whenI_Kfixes it.TauCeti.restrictNormal_mem_lowerRamificationGroup_zero: the restriction of an element ofI_Kto a finite normal subextensionLlies in the inertia groupG_0ofL/K.TauCeti.isArithFrobeniusLift_iff,TauCeti.exists_isArithFrobeniusLift,TauCeti.IsArithFrobeniusLift.setOf_eq_leftCoset: the arithmetic Frobenius lifts are characterised by their action on the roots of the polynomialsX^{q^f} − X, exist, and form a left coset ofI_K.TauCeti.restrictMaximalUnramifiedHom_eq_frobenius_pow_iff: more generally,σacts onK^{ur}as then-th power of Frobenius exactly when it raises the roots of the polynomialsX^{q^f} − Xto theq^n-th power.TauCeti.IsArithFrobeniusLift.apply_of_pow_eq_one: a Frobenius lift acts on the roots of unity of order prime to the residue characteristic byζ ↦ ζ ^ q.TauCeti.IsArithFrobeniusLift.restrictNormal_smul_residueField_eq_pow: on a finite normal subextensionL, a Frobenius lift acts on the residue field ofLbyx ↦ x ^ q.TauCeti.IsArithFrobeniusLift.topologicalClosure_zpowers_sup_inertiaSubgroup: a Frobenius lift andI_KgenerateG_Ktopologically.
References #
- J.-P. Serre, Corps Locaux, Chapter III, §5.
- J. Neukirch, Algebraic Number Theory, Chapter II, §9.
The inertia subgroup #
The inertia subgroup I_K of the absolute Galois group of a nonarchimedean local field K:
the automorphisms of the algebraic closure fixing the maximal unramified extension K^{ur}. Through
IntermediateField.fixingSubgroupEquiv it is Gal(K^{alg}/K^{ur}).
Equations
Instances For
The inertia subgroup is the fixing subgroup of the maximal unramified extension.
An automorphism lies in the inertia subgroup exactly when it fixes every element of the maximal unramified extension.
Inertia, through the roots of X^{q^f} − X. An automorphism of K^{alg} lies in the
inertia subgroup exactly when it fixes every root of every polynomial X^{q^f} − X with f ≠ 0,
where q is the cardinality of the residue field of K.
Inertia fixes the roots of unity of order prime to p. If m is prime to the residue
characteristic of K, every m-th root of unity of K^{alg} lies in the maximal unramified
extension by TauCeti.mem_maximalUnramifiedExtension_of_pow_eq_one, and is therefore fixed by the
inertia subgroup.
The inertia subgroup is closed in the Krull topology.
The inertia subgroup is normal, since K^{ur}/K is normal.
The exact sequence 1 → I_K → G_K → Gal(K^{ur}/K) → 1 #
Restriction of automorphisms of K^{alg} to the maximal unramified extension, as a homomorphism
G_K →* Gal(K^{ur}/K). It is AlgEquiv.restrictNormalHom, typed at Field.absoluteGaloisGroup K,
whose group structure is not reducibly that of Gal(K^{alg}/K).
Equations
Instances For
Restricting σ to K^{ur} agrees with σ on underlying elements: its value at x ∈ K^{ur}
is σ x.
Restriction to K^{ur} is continuous.
Restriction to K^{ur} is surjective: every automorphism of K^{ur}/K extends to
K^{alg}.
The inertia subgroup is the kernel of restriction to K^{ur}. With
TauCeti.restrictMaximalUnramifiedHom_surjective, this is the exactness of
1 → I_K → G_K → Gal(K^{ur}/K) → 1.
The unramified quotient G_K ⧸ I_K of the absolute Galois group.
Equations
Instances For
The canonical quotient map from the absolute Galois group to its unramified quotient.
Equations
Instances For
The unramified degree map G_K → G_K ⧸ I_K is surjective.
The unramified degree map G_K → G_K ⧸ I_K is continuous.
The kernel of the unramified degree map is the inertia subgroup.
The unramified degree of σ is trivial exactly when σ lies in the inertia subgroup.
The unramified quotient of the absolute Galois group: restriction to the maximal unramified
extension induces an isomorphism of topological groups G_K ⧸ I_K ≃ₜ* Gal(K^{ur}/K).
Equations
Instances For
Identifying the unramified quotient with Gal(K^{ur}/K) carries the unramified degree of σ
to its restriction to K^{ur}.
The inverse identification of Gal(K^{ur}/K) with the unramified quotient sends the
restriction of σ to K^{ur} back to the unramified degree of σ; with
TauCeti.restrictMaximalUnramifiedHom_surjective this computes it on every element.
Inertia is not open in G_K. Otherwise G_K ⧸ I_K ≃ ℤ̂ would be finite, since G_K is
compact; but ℤ embeds into ℤ̂.
Unramified subextensions are those fixed by inertia. A finite separable subextension E of
K^{alg}/K, with a structure of nonarchimedean local field compatible with K, is unramified over
K exactly when every element of the inertia subgroup fixes it.
Separability cannot be dropped: in positive characteristic a purely inseparable extension of K
is fixed by all of G_K, but it is ramified.
Inertia at finite level #
Inertia restricts into G_0. The restriction of an element of the inertia subgroup I_K
to a finite normal subextension L of K^{alg}/K lies in the inertia group G_0 of L/K.
Arithmetic Frobenius lifts #
An arithmetic Frobenius lift is an element of the absolute Galois group of K whose
restriction to the maximal unramified extension is its arithmetic Frobenius
TauCeti.maximalUnramifiedFrobenius. By TauCeti.isArithFrobeniusLift_iff these are the
automorphisms of K^{alg} raising every root of every X^{q^f} − X, f ≠ 0, to the q-th
power.
Equations
Instances For
σ is an arithmetic Frobenius lift exactly when it restricts to the arithmetic Frobenius of
K^{ur}.
Powers of Frobenius, through the roots of X^{q^f} − X. An automorphism σ of K^{alg}
acts on the maximal unramified extension as the n-th power of arithmetic Frobenius exactly when it
raises every root of every polynomial X^{q^f} − X with f ≠ 0 to the q^n-th power, where q is
the cardinality of the residue field of K.
Frobenius lifts, through the roots of X^{q^f} − X. An automorphism of K^{alg} is an
arithmetic Frobenius lift exactly when it raises every root of every polynomial X^{q^f} − X with
f ≠ 0 to the q-th power, where q is the cardinality of the residue field of K.
Arithmetic Frobenius lifts exist, since restriction to K^{ur} is surjective.
Given one arithmetic Frobenius lift σ, an element τ is another exactly when σ⁻¹ τ lies in
the inertia subgroup.
A Frobenius lift raises roots of unity of order prime to p to the q-th power. If m
is prime to the residue characteristic of K, every m-th root of unity ζ of K^{alg} is a
root of X^{q^{φ(m)}} − X, so an arithmetic Frobenius lift sends it to ζ ^ q.
A Frobenius lift acts on finite residue fields as the q-th power map. The restriction of
an arithmetic Frobenius lift to a finite normal subextension L of K^{alg}/K acts on the residue
field of L by x ↦ x ^ q, where q is the cardinality of the residue field of K.
The arithmetic Frobenius lifts form a left coset of the inertia subgroup: they are the
elements of σ I_K, for any one of them σ.
A Frobenius lift and inertia generate the absolute Galois group topologically: the closure
of the subgroup generated by an arithmetic Frobenius lift and the inertia subgroup is G_K.