Dévissage of Tate's duality maps for a pro-p group #
Let G be a pro-p group and N a discrete G-module which is a ℤ/nℤ-module satisfying Baer's
criterion, such that H²(G, N) satisfies Baer's criterion over ℤ/nℤ as well. For a finite
discrete G-module M write M' = InternalHom G M N for its N-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 runs Tate's
dévissage argument: if α₀ is surjective and α₁ injective on every discrete G-module of order
p with trivial action killed by n, then the same holds on every finite p-primary discrete
G-module killed by n (TauCeti.IsProP.dualityMap0_surjective_dualityMap1_injective), and
likewise for α₁ surjective and α₂ injective
(TauCeti.IsProP.dualityMap1_surjective_dualityMap2_injective); the two dévissages run side by
side in TauCeti.IsProP.dualityMap0_surjective_dualityMap1_bijective_dualityMap2_injective, from
a base case stating all three properties at once. The induction is
TauCeti.IsProP.finite_pPrimary_induction along the trivial filtration, and the inductive step
is the four lemmas along a short exact sequence 0 → A → B → C → 0 killed by n
(TauCeti.ContCohomology.DiscreteShortExact.dualityMap0_surjective and its companions), whose
N-dual sequence exists because N is Baer, and whose Hom(-, H²(G, N))-dual bottom row is exact
because H²(G, N) is Baer. The four lemmas pair off: surjectivity of α₀ and injectivity of α₁
on the two ends of an extension give both on its middle, and so do surjectivity of α₁ and
injectivity of α₂.
For an infinite profinite pro-p group, α₀ is moreover injective
(TauCeti.IsProP.dualityMap0_injective): H⁰ is co-effaceable, the trace from a coinduced module
along a deep enough open subgroup vanishing on invariants
(TauCeti.IsProP.exists_isOpen_trace_eq_zero_of_mem_H0), and α₁ is injective on the kernel of
the trace by the dévissage of α₀ surjective and α₁ injective, so only those two base cases are
needed. Together with the dévissage this gives all three maps at once
(TauCeti.IsProP.dualityMap0_bijective_dualityMap1_bijective_dualityMap2_injective): α₀ and α₁
bijective, α₂ injective.
The two instances of this argument are the coefficient system of a Demushkin group: N = 𝔽_p
with n = p, where H²(G, 𝔽_p) is an 𝔽_p-vector space, and the twisted coefficients
N = I(χ)/pⁱ of the canonical character with n = pⁱ, where H²(G, I(χ)/pⁱ) ≅ ℤ/pⁱ is
self-injective. The remaining half of Tate's duality, the bijectivity of α₂, is a counting
argument on top of this dévissage that uses the double duality of M with respect to N, and is
not part of this file.
Main results #
TauCeti.IsProP.dualityMap0_surjective_dualityMap1_injective,TauCeti.IsProP.dualityMap1_surjective_dualityMap2_injective: the two dévissages of Tate's duality maps, from the trivial modules of orderpto every finitep-primary module killed byn.TauCeti.IsProP.dualityMap0_surjective_dualityMap1_bijective_dualityMap2_injective: the two dévissages combined, from a base case stating the three properties at once.TauCeti.IsProP.dualityMap0_injective: for an infinite profinite pro-pgroup,α₀is injective on every such module; andTauCeti.IsProP.dualityMap0_bijective_dualityMap1_bijective_dualityMap2_injective, the combined dévissage for such a group, whereα₀is bijective.
References #
- J.-P. Serre, Structure de certains pro-p-groupes (d'après Demuškin), Séminaire Bourbaki 8 (1962/63), exposé 252, §9.1.
The dévissage of α₀ surjective and α₁ injective. Let G be a pro-p group and N a
discrete G-module which is a Baer ℤ/nℤ-module with H²(G, N) a Baer ℤ/nℤ-module. If Tate's
duality maps α₀ and α₁ into H²(G, N) are respectively surjective and injective on every
discrete G-module of order p with trivial action killed by n, then they are so on every finite
p-primary discrete G-module M killed by n: the pair of properties passes through extensions
by the four lemmas DiscreteShortExact.dualityMap0_surjective and dualityMap1_injective.
Injectivity of α₀ for an infinite pro-p group. Under the hypotheses of the dévissage
of α₀ surjective and α₁ injective, if G is an infinite profinite pro-p group, then Tate's
duality map α₀ : H⁰(G, M) → Hom(H²(G, M'), H²(G, N)) is injective on every finite p-primary
discrete G-module M killed by n: the trace Coind_V^G M → M along a deep enough open
subgroup V vanishes on invariants, and α₁ is injective on its kernel.
The dévissage of α₁ surjective and α₂ injective. Under the hypotheses of the dévissage,
if Tate's duality maps α₁ and α₂ into H²(G, N) are respectively surjective and injective on
every discrete G-module of order p with trivial action killed by n, then they are so on every
finite p-primary discrete G-module M killed by n: the pair of properties passes through
extensions by the four lemmas DiscreteShortExact.dualityMap1_surjective and
dualityMap2_injective.
The two dévissages of Tate's duality maps, side by side. Under the hypotheses of the
dévissage, if on every discrete G-module of order p with trivial action killed by n Tate's
duality maps α₀, α₁, α₂ into H²(G, N) are respectively surjective, bijective and injective,
then they are so on every finite p-primary discrete G-module M killed by n.
The dévissage of Tate's duality maps for an infinite pro-p group. Under the hypotheses of
IsProP.dualityMap0_surjective_dualityMap1_bijective_dualityMap2_injective, if G is an infinite
profinite pro-p group, then on every finite p-primary discrete G-module M killed by n,
α₀ and α₁ are bijective and α₂ is injective: α₀ is moreover injective by
IsProP.dualityMap0_injective.