Cohomological triviality of the splitting module

Let G be a finite group, k a commutative ring without additive torsion, A a representation of G over k and u ∈ H²(G, A). Suppose that for every subgroup S of G of prime-power order

• H¹(S, A) = 0,
• the restriction of u generates H²(S, A), and
• H²(S, A) has as many elements as k ⧸ |S|k.

Then the splitting module A(u) of u (Rep.splittingModule), the extension 0 → A → A(u) → I_G → 0 of the augmentation ideal by A attached to u, is cohomologically trivial: its Tate cohomology vanishes on every subgroup in every degree (TauCeti.TateCohomology.isZero_res_splittingModule). For k = ℤ these are Tate's hypotheses (J. Tate, The higher dimensional cohomology groups of class field theory, 1952), with ℤ ⧸ |S|ℤ of order |S|; for k = ℤ_p the order of ℤ_p ⧸ |S|ℤ_p is the p-part of |S|.

The proof is the second dimension shift of Tate's theorem (Milne, Class Field Theory, II, proof of Theorem 3.11). On a subgroup S of prime-power order, the long exact sequence of 0 → A → A(u) → I_G → 0 reads

H¹(S, A) → H¹(S, A(u)) → H¹(S, I_G) → H²(S, A) → H²(S, A(u)) → H²(S, I_G).

The connecting map is onto, because u restricts to a generator of H²(S, A) and dies in H²(S, A(u)) (Rep.map_splittingModuleIncl_res_eq_zero); its source H¹(S, I_G) is, by the augmentation dimension shift, H-hat^0(S, k) = k ⧸ |S|k (TauCeti.TateCohomology.H0LinearEquivTrivial), of the same finite order as its target, so it is a bijection. With H¹(S, A) = 0 and H²(S, I_G) = 0 this makes H¹(S, A(u)) and H²(S, A(u)) vanish, and Tate's cohomological triviality criterion (TauCeti.TateCohomology.isZero_of_forall_isPGroup) concludes.

Main statements

• TauCeti.TateCohomology.isZero_res_splittingModule: the splitting module of a class satisfying Tate's hypotheses on the subgroups of prime-power order is cohomologically trivial.

References

• J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56 (1952), 294–297.
• J. S. Milne, Class Field Theory, Chapter II, Theorems 3.10 and 3.11.

theorem TauCeti.TateCohomology.isZero_res_splittingModule {k G : Type u} [CommRing k] [Group G] (A : Rep.{u, u, u} k G) (u : ↑(groupCohomology A 2)) [IsAddTorsionFree k] [Finite G] (h1 : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G), IsPGroup p ↥S → CategoryTheory.Limits.IsZero (groupCohomology (Rep.res S.subtype A) 1)) (hgen : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G), IsPGroup p ↥S → ∀ (x : ↑(groupCohomology (Rep.res S.subtype A) 2)), ∃ (r : k), r • (CategoryTheory.ConcreteCategory.hom (groupCohomology.map S.subtype (CategoryTheory.CategoryStruct.id (Rep.res S.subtype A)) 2)) u = x) (hfin : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G), IsPGroup p ↥S → Finite (k ⧸ Ideal.span {↑(Nat.card ↥S)})) (hcard : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G), IsPGroup p ↥S → Nat.card ↑(groupCohomology (Rep.res S.subtype A) 2) = Nat.card (k ⧸ Ideal.span {↑(Nat.card ↥S)})) (S : Subgroup G) [Fintype ↥S] (n : ℤ) : CategoryTheory.Limits.IsZero (tateCohomology (Rep.res S.subtype (A.splittingModule u)) n)

The splitting module is cohomologically trivial (Tate; Milne, Class Field Theory, II, proof of Theorem 3.11). Let G be a finite group, k a commutative ring without additive torsion, A a representation of G over k and u ∈ H²(G, A). Suppose that for every subgroup S of G of prime-power order, H¹(S, A) = 0, the restriction of u generates H²(S, A), and H²(S, A) has the finite order of k ⧸ |S|k. Then the Tate cohomology of the splitting module A(u) vanishes on every subgroup of G in every degree.