Classes killed by restriction to an open subgroup are killed by its index

Let G be a locally compact group, for instance a profinite group, U an open subgroup of finite index and X a topological representation of G. If a class x ∈ Hⁿ⁺¹(G, X) restricts to zero in Hⁿ⁺¹(U, X), then [G : U] • x = 0. Classically this is read off from the identity cor ∘ res = [G : U] (NSW (1.5.7)). Applied to an open subgroup of index prime to p, and to a class of p-power order, it makes restriction injective. For a profinite group with discrete p-primary torsion coefficients, every cohomology class has p-power order; this is how its cohomology is compared with that of a Sylow subgroup.

The proof uses no corestriction. It works directly on Mathlib's coinduced resolution C(G, C(G, …, X)). The right-coset factorization of an open subgroup provides a continuous map w : G → U with w (u * g) = u * w g. Pulling back along w in every variable turns a cochain of U into a U-invariant cochain of the resolution of G. The operators

h₀ = 0,   hₙ₊₁ F x = F (w x) x - hₙ (F (w x)),

form a U-equivariant homotopy between the identity and "restrict to U, then pull back along w". So if the restriction of a G-cocycle F is the coboundary d v of a homogeneous cochain v of U, then F = d W for the U-invariant cochain W = h F + w^* v. The coset sum ∑_{q ∈ G ⧸ U} q.out • W is then G-invariant, and its coboundary is [G : U] • F. Local compactness of G is what makes hₙ₊₁ F continuous, through the continuity of evaluation.

Degree zero is excluded: there restriction H⁰(G, X) = X^G → X^U = H⁰(U, X) is simply injective.

Main results

• TauCeti.ContinuousCohomology.index_nsmul_eq_zero_of_res_eq_zero: a class of Hⁿ⁺¹(G, X) whose restriction to an open finite-index subgroup U vanishes is killed by [G : U].

References

• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.5.7) and (1.6.1).
• K. S. Brown, Cohomology of Groups, Chapter III, (9.5) and (10.1).

Pulling cochains of U back along a map G → U

The homotopy

Equivariance for a U-equivariant w

The coset sum

The annihilation theorem

theorem TauCeti.ContinuousCohomology.index_nsmul_eq_zero_of_res_eq_zero {k : Type u} {G : Type v} [Ring k] [TopologicalSpace k] [Group G] [TopologicalSpace G] [IsTopologicalGroup G] {X : TopRep k G} {U : Subgroup G} [LocallyCompactSpace G] (hU : IsOpen ↑U) [U.FiniteIndex] {n : ℕ} {x : ↑(continuousCohomology (n + 1) X).toModuleCat} (hx : (TopModuleCat.Hom.hom (res U X (n + 1))) x = 0) : U.index • x = 0

A class killed by restriction to an open subgroup is killed by the index. Let G be a locally compact group, for instance a profinite group, and U an open subgroup of finite index. If the restriction to U of a class x ∈ Hⁿ⁺¹(G, X) vanishes, then [G : U] • x = 0.