Tate's duality maps along a short exact sequence #
Let G be a topological group, let N be a discrete G-module, and for a finite discrete
G-module M write M' = InternalHom G M N for its dual and
αᵢ : Hⁱ(G, M) → Hom(H²⁻ⁱ(G, M'), H²(G, N)), i = 0, 1, 2, for Tate's duality maps
(TauCeti.ContCohomology.dualityMap0, dualityMap1, dualityMap2). This file compares the
duality maps of the three terms of a short exact sequence 0 → A → B → C → 0 of finite discrete
G-modules. The compatibilities with the connecting maps (dualityMap1_explicitDelta0,
dualityMap2_explicitDelta1) need only the dual sequence DiscreteShortExact.dual, hence only the
extension hypothesis that precomposition with the inclusion is surjective; the four lemmas assume
in addition that the sequence is killed by a natural number n and that H²(G, N) satisfies
Baer's criterion over ℤ/nℤ, which makes the Hom(-, H²(G, N))-dual bottom row exact
(Function.Exact.compHom'_of_baer). For the coefficient system N = I(χ)/pⁱ of a Demushkin group
and n = pⁱ, the group H²(G, N) is ℤ/pⁱ, which is self-injective; for n = p prime and
N = 𝔽_p, it is an 𝔽_p-vector space.
The long exact cohomology sequence of S and the Hom(-, H²(G, N))-dual of the long exact
sequence of the dual sequence 0 → C' → B' → A' → 0 form a ladder
H⁰(A) → H⁰(B) → H⁰(C) → H¹(A) → H¹(B) → H¹(C) → H²(A) → H²(B) → H²(C)
↓α₀ ↓α₀ ↓α₀ ↓α₁ ↓α₁ ↓α₁ ↓α₂ ↓α₂ ↓α₂
H²(A')ᵛ → H²(B')ᵛ → H²(C')ᵛ → H¹(A')ᵛ → H¹(B')ᵛ → H¹(C')ᵛ → H⁰(A')ᵛ → H⁰(B')ᵛ → H⁰(C')ᵛ
whose squares commute, up to the Leibniz sign in the square at δ⁰, by the naturality of the
duality maps in the module (dualityMap0_explicitCoeff0, dualityMap1_explicitCoeff1,
dualityMap2_explicitCoeff2 in
TauCeti/RepresentationTheory/Homological/ContCohomology/Cup/Duality/Basic.lean) and their
compatibility with the connecting maps
(DiscreteShortExact.dualityMap1_explicitDelta0, DiscreteShortExact.dualityMap2_explicitDelta1).
The bottom row is exact because Hom(-, H²(G, N)) is exact on groups killed by n when
H²(G, N) is a Baer ℤ/nℤ-module (Function.Exact.compHom'_of_baer). The four lemmas then give
the dévissage steps of Tate's duality
argument (Serre's exposé, §9.1): from α₀ surjective, α₁ bijective and α₂ injective on A
and on C, the same follows on B
(DiscreteShortExact.dualityMap0_surjective, DiscreteShortExact.dualityMap1_injective,
DiscreteShortExact.dualityMap1_surjective, DiscreteShortExact.dualityMap2_injective). Read the
other way, they are the steps of dimension shifting: for i = 0, 1, if αᵢ is surjective on B
and αᵢ₊₁ is injective on B, injectivity of αᵢ on C gives injectivity of αᵢ₊₁ on A
(DiscreteShortExact.dualityMap1_injective_left, DiscreteShortExact.dualityMap2_injective_left).
Finally, the injectivity of α₀ on C follows from that of α₁ on A when H⁰(G, B) → H⁰(G, C)
vanishes (DiscreteShortExact.dualityMap0_injective_of_explicitCoeff0_eq_zero), with no hypothesis
on H²(G, N) beyond the extension hypothesis of the dual sequence.
Main results #
TauCeti.ContCohomology.DiscreteShortExact.dualityMap1_explicitDelta0,DiscreteShortExact.dualityMap2_explicitDelta1: the duality maps against the connecting maps of a short exact sequence and of its dual.TauCeti.ContCohomology.DiscreteShortExact.dualityMap0_surjective,DiscreteShortExact.dualityMap1_injective,DiscreteShortExact.dualityMap1_surjective,DiscreteShortExact.dualityMap2_injective: the four lemmas along a short exact sequence killed byn, forH²(G, N)a Baerℤ/nℤ-module.TauCeti.ContCohomology.DiscreteShortExact.dualityMap1_injective_left,DiscreteShortExact.dualityMap2_injective_left: the four lemmas concluding injectivity on the submoduleA, the steps of dimension shifting.TauCeti.ContCohomology.DiscreteShortExact.dualityMap0_injective_of_explicitCoeff0_eq_zero: injectivity ofα₀from a sequence whose map on invariants vanishes.
References #
- J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962/63), exposé 252, §9.1.
α₁ (δ⁰ x) b = - α₀ x (δ¹ b) for the connecting maps of S and of its dual sequence.
α₂ (δ¹ x) b = α₁ x (δ⁰ b) for the connecting maps of S and of its dual sequence.
The four lemmas #
The sign in the square at δ⁰ is absorbed by negating the vertical map at the end of the
four-term ladder that contains it, which changes neither injectivity nor surjectivity.
The first four lemma for α₁. If α₀(C) is surjective and α₁(A), α₁(C) are injective,
then α₁(B) is injective.
The second four lemma for α₁. If α₁(A), α₁(C) are surjective and α₂(A) is injective,
then α₁(B) is surjective.
The four lemma for α₀. If α₀(A), α₀(C) are surjective and α₁(A) is injective, then
α₀(B) is surjective.
The four lemma for α₂. If α₁(C) is surjective and α₂(A), α₂(C) are injective, then
α₂(B) is injective.
The four lemma for α₁ on the submodule. If α₀(B) is surjective and α₀(C), α₁(B) are
injective, then α₁(A) is injective. This is the step of dimension shifting that reads the duality
of A off that of a module B containing it.
The four lemma for α₂ on the submodule. If α₁(B) is surjective and α₁(C), α₂(B) are
injective, then α₂(A) is injective. This is the step of dimension shifting that reads the duality
of A off that of a module B containing it.
Injectivity of α₀ from a co-effacing sequence. If H⁰(G, B) → H⁰(G, C) is zero and
α₁(A) is injective, then α₀(C) is injective: a class of H⁰(G, C) killed by α₀ has a
connecting image killed by α₁, hence zero, and δ⁰ is injective on H⁰(G, C).