Tate's theorem: cup product with a degree-two class

Let G be a finite group, N a representation of G over ℤ and u ∈ H²(G, N) a class which generates H²(G, N), a group of order |G|. Suppose that for every subgroup S of G of prime-power order, H¹(S, N) = 0 and H²(S, N) has order |S|. Then cup product with u is a bijection H^r(G, ℤ) → H^{r+2}(G, ℤ ⊗ N) in every integer degree r (TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup). This is Tate's theorem (J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56, 1952), in the form used to derive the reciprocity isomorphisms of class field theory. Throughout this file, H^n(G, M) denotes Tate cohomology in the integer degree n.

The classical statement also asks that the restriction of u to each subgroup S generate H²(S, N). That hypothesis is not needed: corestriction after restriction is multiplication by the index, so the restricted class already has order |S| in a group of order |S|. The proof here makes exactly this observation, in degree zero after dimension shifting.

Proof

The cup product with a degree-two class is not induced by a morphism of representations, but after two upward dimension shifts it is: writing N₂ for the twice-shifted representation, u corresponds to the class of an invariant a ∈ H⁰(G, N₂), and cup product with u is, up to the dimension-shifting isomorphisms, the map induced on Tate cohomology by ℤ ⟶ ℤ ⊗ N₂, 1 ↦ 1 ⊗ a (TauCeti.TateCohomology.cup_dimensionShiftUpTwoIso_hom). Tate's isomorphism criterion for a morphism of representations (TauCeti.TateCohomology.map_bijective_of_forall_isPGroup) then reduces the theorem to the three degrees -1, 0, 1 on subgroups of prime-power order. In degree -1 the target vanishes because H¹(S, N) = 0, in degree 1 the source H¹(S, ℤ) vanishes (TauCeti.TateCohomology.isZero_tateCohomology_one_res_trivial_int), and in degree 0 the map H⁰(S, ℤ) → H⁰(S, ℤ ⊗ N₂) is a bijection between groups of order |S| by the corestriction argument above. On the whole group only the injectivity of the degree-zero cup product is used, so that is the hypothesis of the generic statement.

Main statements

• TauCeti.TateCohomology.map_res_zero_bijective_of_injective: the degree-zero step on a subgroup, by the corestriction argument.
• TauCeti.TateCohomology.cup_bijective_of_cupTrivialInt_injective, TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup: Tate's theorem, with the degree-zero injection respectively the generation and order of H²(G, N) as hypothesis.

References

• J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56 (1952), 294–297.
• E. Artin and J. Tate, Class Field Theory, Preliminaries §2, Theorem A, and Chapter XIV §4, Theorem 1.
• J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter IV (Atiyah–Wall), §10.

The degree-zero step on a subgroup

theorem TauCeti.TateCohomology.map_res_zero_bijective_of_injective {G : Type} [Group G] [Fintype G] {B : Rep.{0, 0, 0} ℤ G} (f : Rep.trivial ℤ G ℤ ⟶ B) (hinj : Function.Injective ⇑(CategoryTheory.ConcreteCategory.hom ((tateCohomologyFunctor 0).map f))) (S : Subgroup G) (hcard : Nat.card ↑(tateCohomology (Rep.res S.subtype B) 0) = Nat.card ↥S) : Function.Bijective ⇑(CategoryTheory.ConcreteCategory.hom ((tateCohomologyFunctor 0).map ((Rep.resFunctor S.subtype).map f)))

The degree-zero step of Tate's theorem on a subgroup S. Let f : ℤ ⟶ B be a morphism from the trivial integral representation which is injective on H⁰(G, ℤ) → H⁰(G, B), and suppose H⁰(S, B) has order |S|. Then f is bijective on H⁰(S, ℤ) → H⁰(S, B): the image of the class of 1 on S is the restriction of the image of the class of 1 on G, which has order |G|, and corestriction after restriction is multiplication by [G : S], so the restricted class has order |S| in a group of order |S|.

Tate's theorem

theorem TauCeti.TateCohomology.cup_bijective_of_cupTrivialInt_injective {G : Type} [Group G] [Fintype G] (N : Rep.{0, 0, 0} ℤ G) (u : ↑(tateCohomology N 2)) (hinj₀ : Function.Injective ⇑(cupTrivialInt N u)) (h1 : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G) [inst : Fintype ↥S], IsPGroup p ↥S → CategoryTheory.Limits.IsZero (tateCohomology (Rep.res S.subtype N) 1)) (hcard : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G) [inst : Fintype ↥S], IsPGroup p ↥S → Nat.card ↑(tateCohomology (Rep.res S.subtype N) 2) = Nat.card ↥S) (r r' : ℤ) (h : r + 2 = r') : Function.Bijective fun (x : ↑(tateCohomology (Rep.trivial ℤ G ℤ) r)) => ((cup (Rep.trivial ℤ G ℤ) N r 2 r' h) x) u

Tate's theorem, with the degree-zero injection as hypothesis. Let G be a finite group, N a representation of G over ℤ and u ∈ H²(G, N). Suppose that

• cup product with u is injective H⁰(G, ℤ) → H²(G, N);
• for every subgroup S of G of prime-power order, H¹(S, N) = 0 and H²(S, N) has order |S|.

Then cup product with u is a bijection H^r(G, ℤ) → H^{r+2}(G, ℤ ⊗ N) in every integer degree r.

theorem TauCeti.TateCohomology.cup_bijective_of_forall_isPGroup {G : Type} [Group G] [Fintype G] (N : Rep.{0, 0, 0} ℤ G) (u : ↑(tateCohomology N 2)) (hgen : ∀ (y : ↑(tateCohomology N 2)), ∃ (m : ℤ), m • u = y) (hcardG : Nat.card ↑(tateCohomology N 2) = Nat.card G) (h1 : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G) [inst : Fintype ↥S], IsPGroup p ↥S → CategoryTheory.Limits.IsZero (tateCohomology (Rep.res S.subtype N) 1)) (hcard : ∀ (p : ℕ) [Fact (Nat.Prime p)] (S : Subgroup G) [inst : Fintype ↥S], IsPGroup p ↥S → Nat.card ↑(tateCohomology (Rep.res S.subtype N) 2) = Nat.card ↥S) (r r' : ℤ) (h : r + 2 = r') : Function.Bijective fun (x : ↑(tateCohomology (Rep.trivial ℤ G ℤ) r)) => ((cup (Rep.trivial ℤ G ℤ) N r 2 r' h) x) u

Tate's theorem. Let G be a finite group, N a representation of G over ℤ and u ∈ H²(G, N). Suppose that

• u generates H²(G, N), a group of order |G|;
• for every subgroup S of G of prime-power order, H¹(S, N) = 0 and H²(S, N) has order |S|.

Then cup product with u is a bijection H^r(G, ℤ) → H^{r+2}(G, ℤ ⊗ N) in every integer degree r. The classical hypothesis that the restriction of u to each subgroup generates its H² is a consequence of these and is not assumed.