The Tate module of a finite layer #
Let L/K be a finite Galois extension of p-adic fields with group G = Gal(L/K), and let
A(L) = lim_m Lˣ/(Lˣ)^(p^m) be the p-adic completion of Lˣ, a ℤ_p[G]-module. The module
Y = I_{G_K} / I_{G_L} I_{G_K} of NSW (5.6.5) is a finitely generated ℤ_p[G]-module of
projective dimension at most one which is an extension
0 → A(L) → Y → I_G → 0
of the augmentation ideal I_G of ℤ_p[G] by A(L). The integral decomposition
Y ≃ M₀ ⊕ ℤ_p[G]^N of the proof of NSW (7.4.1)
(TauCeti.nonempty_linearEquiv_tameFrameModule_prod) uses only these properties, so a Tate
module of the layer, TauCeti.LayerTateModule, is any module with exactly them; it is not
identified here with the quotient I_{G_K} / I_{G_L} I_{G_K}. (The name TauCeti.TateModule is
the p-adic Tate module lim_n A[p^n] of an abelian group.)
Projective dimension at most one is the theorem of Nakayama and Rim
(Rep.projective_ker_of_isZero_res): a cohomologically trivial representation has a projective
resolution of length one. So any cohomologically trivial extension of I_G by A(L) is a Tate
module of the layer (TauCeti.LayerTateModule.nonempty_of_isZero). The splitting module of a
class u ∈ H²(G, A(L)) is such an extension once u satisfies Tate's hypotheses on the
p-subgroups of G (TauCeti.LayerTateModule.nonempty_of_tateHypotheses); the class of the
layer, carried to A(L) along local reciprocity, is one.
Main definitions #
TauCeti.LayerTateModule: a finitely generatedℤ_p[Gal(L/K)]-module of projective dimension at most one which is an extension of the augmentation ideal byA(L).
Main results #
TauCeti.LayerTateModule.nonempty_of_isZero: a cohomologically trivial extension of the augmentation ideal byA(L)is a Tate module of the layer.TauCeti.LayerTateModule.nonempty_of_tateHypotheses: a class inH²(Gal(L/K), A(L))satisfying Tate's hypotheses on thep-subgroups gives a Tate module of the layer.TauCeti.LayerTateModule.nonempty_of_equiv: a Tate module transports along an isomorphism of layers.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (5.6.5) and the proof of (7.4.1).
- D. S. Rim, Modules over finite groups, Ann. of Math. 69 (1959).
A Tate module of a layer L/K: the properties of the module Y = I_{G_K}/I_{G_L} I_{G_K}
of NSW (5.6.5) that the integral decomposition in the proof of (7.4.1) consumes, without an
identification with that quotient. It is a finitely generated ℤ_p[Gal(L/K)]-module of
projective dimension at most one, together with an extension 0 → A(L) → Y → I_G → 0 of the
augmentation ideal I_G by the p-adic completion A(L) of Lˣ.
- carrier : Type u
The carrier
Y. - addCommGroup : AddCommGroup self.carrier
- module : Module (MonoidAlgebra ℤ_[p] Gal(L/K)) self.carrier
The inclusion of
A(L).- π : self.carrier →ₗ[MonoidAlgebra ℤ_[p] Gal(L/K)] ↥(RingHom.ker (MonoidAlgebra.augmentation ℤ_[p] Gal(L/K)))
The projection onto the augmentation ideal.
- ι_injective : Function.Injective ⇑self.ι
- π_surjective : Function.Surjective ⇑self.π
- finite : Module.Finite (MonoidAlgebra ℤ_[p] Gal(L/K)) self.carrier
- projdim : ∃ (n : ℕ) (f : (Fin n → MonoidAlgebra ℤ_[p] Gal(L/K)) →ₗ[MonoidAlgebra ℤ_[p] Gal(L/K)] self.carrier), Function.Surjective ⇑f ∧ Module.Projective (MonoidAlgebra ℤ_[p] Gal(L/K)) ↥f.ker
Projective dimension at most one: a quotient of a free module by a projective kernel.
Instances For
The inclusion of A(L) and the projection onto I_G form an exact sequence.
A cohomologically trivial extension of I_G by A(L) is a Tate module. Let L be a
finite extension of ℚ_p and K a subfield with Gal(L/K) finite. Let Y be a representation
of Gal(L/K) over ℤ_p whose Tate cohomology vanishes on every subgroup in every degree, and
0 → A(L) → Y → I_G → 0 an exact sequence of ℤ_p[Gal(L/K)]-modules. Then Y is finitely
generated, and by the theorem of Nakayama and Rim (Rep.projective_ker_of_isZero_res) it has
projective dimension at most one, so it is a Tate module of the layer.
A degree-two class of A(L) satisfying Tate's hypotheses gives a Tate module. Let L be
a finite extension of ℚ_p and K a subfield with Gal(L/K) finite, and let
u ∈ H²(Gal(L/K), A(L)). Suppose that for every p-subgroup S of Gal(L/K),
H¹(S, A(L)) = 0, the restriction of u generates H²(S, A(L)) as a ℤ_p-module, and
H²(S, A(L)) has order #S. Then the splitting module of u, an extension
0 → A(L) → Y → I_G → 0, is cohomologically trivial
(TauCeti.TateCohomology.isZero_res_splittingModule; on ℓ-subgroups for ℓ ≠ p all
positive-degree group cohomology of a ℤ_p-module vanishes), so it is a Tate module of the
layer.
Transport of the Tate module along an isomorphism of layers. Let L/K and L'/K' be
layers, L', K' in Type, with an isomorphism φ of their automorphism groups and a
ℤ_p-linear isomorphism A(L) ≃ A(L') intertwining the actions of σ and φ σ. Then a Tate
module of L'/K' gives one of L/K: its carrier, lifted to the universe of L, is a module over
ℤ_p[Gal(L/K)] through the induced isomorphism of group algebras.