The index-two exact sequence of restriction and corestriction

Let U be an open subgroup of index two in a profinite group G, and let M be a discrete G-module killed by two. The coinduction unit and the trace form the short exact sequence 0 → M → Coind_U^G M → M → 0 (TauCeti.DiscreteCoind.indexTwoShortExact). Shapiro's isomorphism Hⁿ(G, Coind_U^G M) ≅ Hⁿ(U, M) turns the coefficient map of the unit into restriction and the coefficient map of the trace into corestriction, so the long exact sequence of continuous cohomology becomes

⋯ → Hⁿ(G, M) --res--> Hⁿ(U, M) --cor--> Hⁿ(G, M) --δ--> Hⁿ⁺¹(G, M) --res--> Hⁿ⁺¹(U, M) → ⋯

in every degree, where δ is the connecting map of the coefficient sequence. This file proves exactness at its three repeating nodes and specializes exactness at Hⁿ(U, M) to trivial 𝔽₂ coefficients. On the explicit low-degree model, the degree-zero connecting map of the same sequence sends 1 ∈ H⁰(G, 𝔽₂) to the character of G with kernel U (TauCeti.ContCohomology.indexTwoShortExact_explicitDelta0).

Index exactly two is needed: at index two the kernel of the trace on Coind_U^G M is the image of the unit (TauCeti.DiscreteCoind.trace_eq_zero_iff_exists_unit_of_index_two), which fails at larger index. For S₃ ⊇ C₂ with trivial 𝔽₂ coefficients, restriction and corestriction in degree one are both surjective, so the composite sequence cannot be exact at H¹(C₂, 𝔽₂).

Main results

• TauCeti.ContinuousCohomology.exact_res_corestriction_of_index_two: exactness at Hⁿ(U, M), the image of restriction is the kernel of corestriction.
• TauCeti.ContinuousCohomology.exact_corestriction_delta_of_index_two: exactness at Hⁿ(G, M), the image of corestriction is the kernel of δ.
• TauCeti.ContinuousCohomology.exact_delta_res_of_index_two: exactness at Hⁿ⁺¹(G, M), the image of δ is the kernel of restriction.
• TauCeti.exact_trivialF2ResMap_trivialF2CorMap_of_index_two: exactness at Hⁿ(U, 𝔽₂) for trivial 𝔽₂ coefficients.

References

• J. Kr. Arason, Cohomologische Invarianten quadratischer Formen, J. Algebra 36 (1975), 448–491, the exact sequence of a quadratic extension.
• A. Kozlowski, The Evens–Kahn formula for the total Stiefel–Whitney class, Proc. Amer. Math. Soc. 91 (1984), 309–313, Lemma 2.4.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.3.2).

theorem TauCeti.ContinuousCohomology.exact_res_corestriction_of_index_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (U : Subgroup G) (hU : IsOpen ↑U) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] (hU2 : U.index = 2) (hM : ∀ (m : M), 2 • m = 0) (n : ℕ) : Function.Exact ⇑(CategoryTheory.ConcreteCategory.hom (res U (ofDiscreteModule ℤ G M) n)) ⇑(CategoryTheory.ConcreteCategory.hom (corestriction U M hU n))

Exactness at Hⁿ(U, M) for an open subgroup U of index two and a discrete G-module M killed by two: a class of Hⁿ(U, M) has zero corestriction exactly when it is a restriction.

theorem TauCeti.ContinuousCohomology.exact_corestriction_delta_of_index_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (U : Subgroup G) (hU : IsOpen ↑U) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] (hU2 : U.index = 2) (hM : ∀ (m : M), 2 • m = 0) (n : ℕ) : Function.Exact ⇑(CategoryTheory.ConcreteCategory.hom (corestriction U M hU n)) ⇑(CategoryTheory.ConcreteCategory.hom ((DiscreteCoind.indexTwoShortExact G U M hU2 hU hM).delta n))

Exactness at Hⁿ(G, M) for an open subgroup U of index two and a discrete G-module M killed by two: a class of Hⁿ(G, M) is a corestriction exactly when the connecting map δ of the coefficient sequence 0 → M → Coind_U^G M → M → 0 kills it.

theorem TauCeti.ContinuousCohomology.exact_delta_res_of_index_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (U : Subgroup G) (hU : IsOpen ↑U) (M : Type u) [AddCommGroup M] [TopologicalSpace M] [DiscreteTopology M] [DistribMulAction G M] [ContinuousSMul G M] (hU2 : U.index = 2) (hM : ∀ (m : M), 2 • m = 0) (n : ℕ) : Function.Exact ⇑(CategoryTheory.ConcreteCategory.hom ((DiscreteCoind.indexTwoShortExact G U M hU2 hU hM).delta n)) ⇑(CategoryTheory.ConcreteCategory.hom (res U (ofDiscreteModule ℤ G M) (n + 1)))

Exactness at Hⁿ⁺¹(G, M) for an open subgroup U of index two and a discrete G-module M killed by two: a class of Hⁿ⁺¹(G, M) restricts to zero on U exactly when it is in the image of the connecting map δ of the coefficient sequence 0 → M → Coind_U^G M → M → 0.

theorem TauCeti.exact_trivialF2ResMap_trivialF2CorMap_of_index_two (G : Type u) [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (U : Subgroup G) (hU : IsOpen ↑U) (hU2 : U.index = 2) (n : ℕ) : Function.Exact ⇑(CategoryTheory.ConcreteCategory.hom (trivialF2ResMap G U n)) ⇑(CategoryTheory.ConcreteCategory.hom (trivialF2CorMap G U hU n))

Exactness at Hⁿ(U, 𝔽₂) for an open subgroup U of index two and trivial 𝔽₂ coefficients: a class of Hⁿ(U, 𝔽₂) has zero corestriction exactly when it is a restriction.