Finite-quotient Galois coefficient representations #
For an open normal subgroup V of G_F, galRepOfQuotient equips a representation of G_F / V
with the discrete topology and pulls its action back to G_F. Its values are smooth, and finite
modules remain finite. Conversely, every finite smooth discrete Galois representation is
isomorphic to a value of this functor for some V.
This is the coefficient reduction used to apply finite-group representation theory to local Euler characteristics. Neither a local-field hypothesis nor a prime exponent is needed for the reduction itself.
The quotient can be enlarged so that it also sees the roots of unity: for n invertible in F,
every V contains an open normal subgroup acting trivially on μₙ, of index dividing
[G_F : V] · φ n and with commutative quotient when G_F ⧸ V is commutative
(exists_openNormalSubgroup_le_muNRep_ρ_eq_self). On fixed fields this replaces a finite Galois
extension L by L(μₙ); for a prime n = ℓ it keeps the index prime to ℓ
(exists_openNormalSubgroup_le_muNRep_ρ_eq_self_of_coprime), which is what descent of
cohomology along the quotient requires. A subgroup acting trivially on μₙ makes μₙ the
inflation of a finite-dimensional representation of the quotient
(exists_fdGalRepOfQuotient_iso_muNRep).
The functor uses discreteTopRepFunctor and Mathlib's TopRep.resFunctor. The converse uses the
existing finite-set open-stabilizer theorem, not a second finite-quotient or Galois carrier.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, §VII.3, proof of (7.3.1).
- J. S. Milne, Arithmetic Duality Theorems, I, proof of Theorem 2.8.
Read a ZMod n-representation of a finite Galois quotient as a discrete representation of
G_F, by restriction along the canonical quotient map. The body is exposed so the pointwise
action and morphism equations can use the original algebraic carriers.
Instances For
Inflation preserves addition of coefficient morphisms, allowing short complexes to be mapped.
The coefficient object is the discrete representation restricted along the quotient map.
Inflation does not change the underlying algebraic coefficient module.
The inflated action is the quotient action evaluated on the class of the automorphism.
The open normal subgroup V acts trivially on every inflated representation.
The functor leaves the underlying coefficient map unchanged.
Inflation is faithful: it leaves the underlying coefficient maps unchanged.
Inflation is full: a map of inflated representations commuting with G_F commutes with its
quotient G_F ⧸ V, every class of which lifts to G_F.
The inflated coefficient object carries the discrete topology.
Inflation preserves finiteness of the coefficient module.
The open kernel of the quotient action makes the inflated representation smooth.
Supply smoothness as the instance hypothesis used by local cohomology and duality.
Inflation of finite-dimensional representations of a finite Galois quotient: the
restriction of galRepOfQuotient along Mathlib's forgetful functor FDRep → Rep. Its values are
finite for n ≠ 0.
Instances For
The inflation of a finite-dimensional representation is galRepOfQuotient of its underlying
representation.
The inflation of FDRep.of A.ρ is galRepOfQuotient of A.
The inflated action is the quotient action evaluated on the class of the automorphism.
The inflation leaves the underlying coefficient map unchanged.
The inflated coefficient object carries the discrete topology.
The inflation of a finite-dimensional representation is smooth.
For n ≠ 0, the inflation of a finite-dimensional representation has finite coefficients.
The open normal subgroup acts trivially on an inflated finite-dimensional representation.
Inflation preserves and reflects injectivity, exactness and surjectivity of a composable pair of morphisms of finite-dimensional representations.
A discrete Galois representation killed by V is isomorphic to an inflated representation
of the quotient by that particular subgroup.
Every finite smooth discrete Galois representation comes from a finite module over a finite Galois quotient. Thus finite-group representation theory applies without changing coefficients.
Adjoining μₙ to a finite Galois layer. For n invertible in F, every open normal
subgroup V of G_F contains an open normal subgroup W acting trivially on μₙ, of index
dividing [G_F : V] · φ n, and with G_F ⧸ W commutative when G_F ⧸ V is. On fixed fields this
replaces the layer L of V by L(μₙ). The subgroup is V intersected with the kernel of the
action on μₙ, a cyclic group of order n.
Adjoining μₗ keeps the index prime to ℓ. For a prime ℓ invertible in F and an
open normal subgroup V of G_F of index prime to ℓ, some open normal subgroup W ≤ V acts
trivially on μₗ, still has index prime to ℓ, and has commutative quotient when V has: its
index divides [G_F : V] · (ℓ - 1).
μₙ is inflated from the quotient by an open normal subgroup V acting trivially on it:
some finite-dimensional representation of G_F ⧸ V inflates to μₙ
(exists_galRepOfQuotient_iso_of_trivial).