The second cohomology of the twisted coefficients I(χ)/pⁱ #
Let G be a topological group and χ : G →ₜ* ℤ_pˣ a continuous character with the prescription
property (TauCeti.HasPrescriptionProperty), so that multiplication by p is injective on
H²(G, I(χ)/pⁱ) → H²(G, I(χ)/pⁱ⁺¹) for every i. Suppose moreover that H²(G, -) is right exact
on the tower of twisted coefficients, in the sense that every reduction
H²(G, I(χ)/pⁱ⁺¹) → H²(G, I(χ)/p) is surjective, and that H²(G, I(χ)/p) has order p. Then the
short exact sequences 0 → I(χ)/pⁱ → I(χ)/pⁱ⁺¹ → I(χ)/p → 0 give short exact sequences
0 → H²(G, I(χ)/pⁱ) → H²(G, I(χ)/pⁱ⁺¹) → H²(G, I(χ)/p) → 0,
so H²(G, I(χ)/pⁱ) has order pⁱ; and a class of H²(G, I(χ)/pⁱ⁺¹) whose reduction to
H²(G, I(χ)/p) is nonzero has order pⁱ⁺¹, because p times it is the multiplication by p of
its reduction to level i. Hence H²(G, I(χ)/pⁱ) is cyclic of order pⁱ, and isomorphic to
ℤ/pⁱ.
The right exactness hypothesis holds when G is compact with cd_p G ≤ 2, because the connecting
map H²(G, I(χ)/p) → H³(G, I(χ)/pⁱ) then vanishes
(TauCeti.ZModTwist.surjective_explicitCoeff2_reduce_of_cohomologicalDimensionAt_le_two, in the
module TauCeti.Topology.Algebra.Group.Profinite.ProP.CohomologicalDimension). The hypotheses are
met by the canonical character of an infinite Demushkin group, where the result is the finite-level
form of the statement that the dualizing module of such a group is ℚ_p/ℤ_p with G acting
through its orientation (Serre's exposé, §9); that specialization lives in
TauCeti.Topology.Algebra.Group.Profinite.Demushkin.TwistedCoefficients.
Main results #
TauCeti.HasPrescriptionProperty.natCard_H2_zModTwist:H²(G, I(χ)/pⁱ)has orderpⁱ.TauCeti.HasPrescriptionProperty.addOrderOf_eq_pow_of_explicitCoeff2_reduce_ne_zero: a class ofH²(G, I(χ)/pⁱ⁺¹)with nonzero reduction toH²(G, I(χ)/p)has orderpⁱ⁺¹.TauCeti.HasPrescriptionProperty.isAddCyclic_H2_zModTwist,TauCeti.HasPrescriptionProperty.nonempty_addEquiv_H2_zModTwist_zmod:H²(G, I(χ)/pⁱ)is cyclic, isomorphic toℤ/pⁱ.TauCeti.HasPrescriptionProperty.moduleBaer_H2_zModTwist:H²(G, I(χ)/pⁱ)satisfies Baer's criterion overℤ/pⁱ, so thatHom(-, H²(G, I(χ)/pⁱ))is exact on the groups killed bypⁱ.
References #
- J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962/63), exposé 252, §9.
- J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §2, Proposition 6.
The order and cyclicity of H²(G, I(χ)/pⁱ) #
Under the prescription property, a class of H²(G, I(χ)/pⁱ⁺¹) whose reduction to
H²(G, I(χ)/p) is nonzero is not killed by pⁱ: p times the class is the multiplication by p
of its reduction to level i, which is injective, and the reduction has the same image at the
bottom level.
Under the prescription property, a class of H²(G, I(χ)/pⁱ⁺¹) whose reduction to
H²(G, I(χ)/p) is nonzero has order exactly pⁱ⁺¹.
The order of H²(G, I(χ)/pⁱ). Under the prescription property, if every reduction
H²(G, I(χ)/pⁱ⁺¹) → H²(G, I(χ)/p) is surjective and H²(G, I(χ)/p) has order p, then
H²(G, I(χ)/pⁱ) has order pⁱ.
H²(G, I(χ)/pⁱ) is cyclic. Under the prescription property, if every reduction
H²(G, I(χ)/pⁱ⁺¹) → H²(G, I(χ)/p) is surjective and H²(G, I(χ)/p) has order p, then
H²(G, I(χ)/pⁱ) is cyclic: any lift of a nonzero class of H²(G, I(χ)/p) generates it.
H²(G, I(χ)/pⁱ) ≅ ℤ/pⁱ. Under the prescription property, if every reduction
H²(G, I(χ)/pⁱ⁺¹) → H²(G, I(χ)/p) is surjective and H²(G, I(χ)/p) has order p, then
H²(G, I(χ)/pⁱ) is isomorphic to ℤ/pⁱ as an additive group.
H²(G, I(χ)/pⁱ) is self-injective over ℤ/pⁱ. Under the prescription property, if every
reduction H²(G, I(χ)/pⁱ⁺¹) → H²(G, I(χ)/p) is surjective and H²(G, I(χ)/p) has order p, then
H²(G, I(χ)/pⁱ), a ℤ/pⁱ-module isomorphic to ℤ/pⁱ, satisfies Baer's criterion over ℤ/pⁱ:
Hom(-, H²(G, I(χ)/pⁱ)) is exact on the groups killed by pⁱ.