The Kummer isomorphism on a fixing subgroup, and its Galois equivariance #
Let K be a field, Kˢ a separable closure, G_K = AbsoluteGaloisGroup K, n a natural number
invertible in K, and σ : L →ₐ[K] Kˢ an embedding of an extension L/K. The subgroup
N = Gal(Kˢ/σ(L)) of G_K fixing σ(L) is a copy of G_L, and the Kummer isomorphism of L
read on it is
fixingSubgroupKummerEquiv σ hn : Lˣ ⧸ (Lˣ)ⁿ ≃ H¹(N, μₙ),
with μₙ = μₙ(Kˢ) the Kummer coefficient module of K. It sends the power class of b ∈ Lˣ to
the class of the cocycle h ↦ h α / α, for any nth root α ∈ Kˢ of σ b
(fixingSubgroupKummerEquiv_ofMul_mk); the cocycle is subgroupKummerCocycle.
When N is normal, G_K acts on H¹(N, μₙ) by conjugation on N together with its action on
μₙ (TauCeti.ContCohomology.explicitConj1), and the subgroup N acts trivially, so this is an
action of G_K ⧸ N. It acts on Lˣ ⧸ (Lˣ)ⁿ through the K-automorphisms of L: g ∈ G_K
acts as the automorphism τ with σ ∘ τ = g ∘ σ. The Kummer isomorphism is equivariant for
these two actions (smul_fixingSubgroupKummerEquiv). This is the Galois-module structure on
H¹(L, μₙ) that is used to compute with Kummer theory over a finite Galois extension L/K.
If moreover σ(L) contains the nth roots of unity, that is, N acts trivially on μₙ, then an
identification e : μₙ ≃ M with a module M on which G_K acts trivially, such as ℤ/n, gives
the Kummer isomorphism with trivial coefficients
Ψ = fixingSubgroupKummerEquivOfTrivial σ hn e htriv hN : Lˣ ⧸ (Lˣ)ⁿ ≃ H¹(N, M).
It is equivariant only up to a twist: G_K acts on μₙ through the cyclotomic character, and
if g acts on μₙ as the kth power map then k • (g • Ψ x) = Ψ (τ x)
(nsmul_smul_fixingSubgroupKummerEquivOfTrivial). So, as modules over G_K ⧸ N, which is
Gal(L/K) for L/K Galois, H¹(N, M) ≅ μₙ^{⊗ -1} ⊗ Lˣ ⧸ (Lˣ)ⁿ.
Main definitions #
TauCeti.subgroupKummerCocycle: the cocycleh ↦ h α / αon a subgroup fixingαⁿ.TauCeti.fixingSubgroupKummerEquiv: the Kummer isomorphismLˣ ⧸ (Lˣ)ⁿ ≃ H¹(N, μₙ).TauCeti.fixingSubgroupKummerEquivOfTrivial: the Kummer isomorphismLˣ ⧸ (Lˣ)ⁿ ≃ H¹(N, M)with trivial coefficientsM ≃ μₙ, whenσ(L)contains thenth roots of unity.
Main results #
TauCeti.fixingSubgroupKummerEquiv_ofMul_mk: the Kummer class ofbis the class ofh ↦ h α / αfor everynth rootαofσ b.TauCeti.smul_fixingSubgroupKummerEquiv: the Kummer isomorphism intertwines conjugation byg ∈ G_Kwith the action of the automorphism ofLthatginduces.TauCeti.nsmul_smul_fixingSubgroupKummerEquivOfTrivial: with trivial coefficients the same holds up to the cyclotomic twist.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (6.2.1), and the proof of (7.3.1).
- J. S. Milne, Arithmetic Duality Theorems, 2nd ed., I, proof of Theorem 2.8.
The Kummer cocycle on a subgroup fixing an nth power #
The ratio h α / α is an nth root of unity when h fixes αⁿ.
The Kummer cocycle on a subgroup N of G_K fixing αⁿ: the continuous 1-cocycle
h ↦ h α / α with values in μₙ, the counterpart for N of TauCeti.kummerCocycle. On the
subgroup fixing σ(L) and for αⁿ = σ b, its class is the Kummer class of b ∈ Lˣ
(fixingSubgroupKummerEquiv_ofMul_mk).
Equations
Instances For
The value of subgroupKummerCocycle at h is the ratio h α / α.
The Kummer isomorphism on the subgroup fixing σ(L) #
An element fixing σ(L) fixes every nth root of σ b, raised to the nth power.
The Kummer isomorphism of L on the subgroup of G_K fixing σ(L),
Lˣ ⧸ (Lˣ)ⁿ ≃ H¹(Gal(Kˢ/σ(L)), μₙ(Kˢ)), for n invertible in K: the Kummer isomorphism
TauCeti.kummerIso of L, transported along absoluteGaloisGroupEquivFixingSubgroup K L σ and
the identification of roots of unity kummerCoeffMapSymm K n L σ. The class of b is
represented by h ↦ h α / α for any nth root α of σ b
(fixingSubgroupKummerEquiv_ofMul_mk).
Equations
- One or more equations did not get rendered due to their size.
Instances For
fixingSubgroupKummerEquiv is the Kummer isomorphism of L followed by the transport to the
subgroup fixing σ(L).
The Kummer class on the subgroup fixing σ(L) is represented by h ↦ h α / α, for every
nth root α ∈ Kˢ of σ b.
The Kummer isomorphism is Galois equivariant. Let the subgroup N of G_K fixing σ(L)
be normal, so that G_K acts on H¹(N, μₙ) by conjugation. If g ∈ G_K and the
K-endomorphism τ of L satisfy σ ∘ τ = g ∘ σ, then conjugation by g corresponds under the
Kummer isomorphism to the map of power classes Lˣ ⧸ (Lˣ)ⁿ → Lˣ ⧸ (Lˣ)ⁿ induced by τ.
Trivial coefficients and the cyclotomic twist #
The Kummer isomorphism with trivial coefficients, Lˣ ⧸ (Lˣ)ⁿ ≃ H¹(N, M) on the subgroup
N of G_K fixing σ(L), when N acts trivially on μₙ, that is, when σ(L) contains the
nth roots of unity, and e : μₙ ≃ M identifies μₙ with a module on which G_K acts trivially:
fixingSubgroupKummerEquiv followed by the coefficient isomorphism induced by e, which is
N-equivariant because N acts trivially on both sides.
Equations
- One or more equations did not get rendered due to their size.
Instances For
fixingSubgroupKummerEquivOfTrivial is the Kummer isomorphism followed by the coefficient map
induced by e.
The Kummer isomorphism with trivial coefficients is equivariant up to the cyclotomic
twist. If g ∈ G_K acts on μₙ as the kth power map and the K-endomorphism τ of L
satisfies σ ∘ τ = g ∘ σ, then k times the conjugate by g of the class of x is the class of
τ x. As k is the value at g of the cyclotomic character modulo n, this identifies H¹(N, M)
with μₙ^{⊗ -1} ⊗ Lˣ ⧸ (Lˣ)ⁿ as a module over G_K ⧸ N.