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ae00908 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 | import Mathlib
namespace OAI
/-!
# Compact p-adic modules
The map `1 - shift` is surjective on the product of a tower of finite abelian groups,
without any surjectivity assumption on the transition maps. A compact Hausdorff module
vanishes if a topologically nilpotent scalar acts surjectively. This implies compact
p-adic Nakayama: a compact Hausdorff `ℤ_[p]`-module with continuous operations is finitely
generated exactly when its quotient modulo `p` is finite. Finitely generated
`ℤ_[p]`-modules also have finite `p`-torsion.
These are algebraic support results. The `K(n)`-local sphere, its homotopy groups,
their p-adic topology and module structure, and their ranks are not constructed here.
-/
open Filter Topology CategoryTheory
open scoped Pointwise
namespace KnLocalSphere.IntegralRecovery
universe u v
/-- The map `1 - shift` is surjective for any tower of finite abelian groups. -/
theorem one_sub_shift_surjective (A : ℕ → Type u)
[∀ n, AddCommGroup (A n)] [∀ n, Finite (A n)]
(f : ∀ n, A (n + 1) →+ A n) :
Function.Surjective (fun x : (n : ℕ) → A n => fun n => x n - f n (x (n + 1))) := by
intro y
let D : ℕᵒᵖ ⥤ Type u := CategoryTheory.Functor.ofOpSequence
(X := A) (fun n => ↾(fun z => y n + f n z))
have : ∀ n, Finite (D.obj n) := fun n => inferInstanceAs (Finite (A n.unop))
have : ∀ n, Nonempty (D.obj n) := fun n => ⟨(0 : A n.unop)⟩
obtain ⟨x, hx⟩ := nonempty_sections_of_finite_inverse_system D
refine ⟨fun n => x (Opposite.op n), ?_⟩
funext n
apply sub_eq_iff_eq_add.mpr
have h := hx (CategoryTheory.homOfLE (Nat.le_add_right n 1)).op
simp only [D, CategoryTheory.Functor.ofOpSequence_map_homOfLE_succ] at h
change y n + f n (x (Opposite.op (n + 1))) = x (Opposite.op n) at h
exact h.symm
/-- A surjective topologically nilpotent scalar annihilates a compact Hausdorff module. -/
theorem eq_zero_of_surjective_smul {R : Type u} {M : Type v}
[Semiring R] [TopologicalSpace R] [AddCommMonoid M] [Module R M]
[TopologicalSpace M] [CompactSpace M] [T2Space M] [ContinuousSMul R M]
(a : R) (ha : Tendsto (fun n : ℕ => a ^ n) atTop (𝓝 0))
(hs : Function.Surjective (fun x : M => a • x)) (x : M) : x = 0 := by
have hpow : ∀ n : ℕ, Function.Surjective (fun y : M => a ^ n • y) := by
intro n
induction n with
| zero =>
intro z
exact ⟨z, by change a ^ 0 • z = z; simp⟩
| succ n ih =>
intro z
obtain ⟨w, hw⟩ := hs z
obtain ⟨y, hy⟩ := ih w
refine ⟨y, ?_⟩
change a ^ (n + 1) • y = z
change a • w = z at hw
change a ^ n • y = w at hy
rw [pow_succ', mul_smul, hy, hw]
have hx : Tendsto (fun _ : ℕ => x) atTop (𝓝 (0 : M)) := by
intro U hU
have hV : ∀ᶠ r in 𝓝 (0 : R), ∀ y : M, r • y ∈ U := by
have h := (isCompact_univ : IsCompact (Set.univ : Set M)).eventually_forall_of_forall_eventually (x₀ := (0 : R))
(P := fun r y => r • y ∈ U) (fun y _ => by
have hy : Tendsto (fun z : R × M => z.1 • z.2) (𝓝 (0, y)) (𝓝 (0 : M)) := by
simpa only [ContinuousAt, zero_smul] using
(continuous_smul : Continuous (fun z : R × M => z.1 • z.2)).continuousAt (x := (0, y))
exact hy hU)
simpa only [Set.mem_univ, forall_true_left] using h
obtain ⟨n, hn⟩ := (ha.eventually hV).exists
obtain ⟨y, hy⟩ := hpow n x
change a ^ n • y = x at hy
have hxu : x ∈ U := by rw [← hy]; exact hn y
change ∀ᶠ _ : ℕ in atTop, x ∈ U
exact Filter.Eventually.of_forall (fun _ => hxu)
exact tendsto_nhds_unique tendsto_const_nhds hx
/-- Compact p-adic Nakayama: finiteness modulo `p` implies finite generation over `ℤ_[p]`. -/
theorem finite_of_finite_mod_p (p : ℕ) [Fact p.Prime]
(M : Type u) [AddCommGroup M] [Module ℤ_[p] M]
[TopologicalSpace M] [IsTopologicalAddGroup M] [ContinuousSMul ℤ_[p] M]
[CompactSpace M] [T2Space M]
[Finite (M ⧸ LinearMap.range (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p])))] :
Module.Finite ℤ_[p] M := by
classical
let P := LinearMap.range (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p]))
let r : M ⧸ P → M := Function.surjInv P.mkQ_surjective
have hr : ∀ z, P.mkQ (r z) = z := Function.surjInv_eq P.mkQ_surjective
let N : Submodule ℤ_[p] M := Submodule.span ℤ_[p] (Set.range r)
have hN : N.FG := Submodule.fg_span (Set.finite_range r)
have : CompactSpace (M ⧸ N) := N.mkQ_surjective.compactSpace N.continuous_mkQ
have : IsClosed (N : Set M) := (Submodule.isCompact_of_fg hN).isClosed
have hsurj : Function.Surjective (fun x : M ⧸ N => (p : ℤ_[p]) • x) := by
intro z
obtain ⟨x, rfl⟩ := N.mkQ_surjective z
have hmem : x - r (P.mkQ x) ∈ P := by
apply (Submodule.Quotient.mk_eq_zero _).mp
change P.mkQ (x - r (P.mkQ x)) = 0
rw [map_sub, hr, sub_self]
obtain ⟨y, hy⟩ := hmem
refine ⟨N.mkQ y, ?_⟩
have hn : N.mkQ (r (P.mkQ x)) = 0 := by
apply (Submodule.Quotient.mk_eq_zero _).mpr
exact Submodule.subset_span (Set.mem_range_self _)
change (p : ℤ_[p]) • y = x - r (P.mkQ x) at hy
change (p : ℤ_[p]) • N.mkQ y = N.mkQ x
rw [← map_smul, hy, map_sub, hn, sub_zero]
have hp : Tendsto (fun n : ℕ => (p : ℤ_[p]) ^ n) atTop (𝓝 0) := by
apply tendsto_pow_atTop_nhds_zero_of_norm_lt_one
rw [PadicInt.norm_p]
apply inv_lt_one_of_one_lt₀
exact_mod_cast (Fact.out : p.Prime).one_lt
have htop : N = ⊤ := by
apply Submodule.eq_top_iff'.mpr
intro x
apply (Submodule.Quotient.mk_eq_zero _).mp
exact eq_zero_of_surjective_smul (p : ℤ_[p]) hp hsurj (N.mkQ x)
exact ⟨htop ▸ hN⟩
/-- A finitely generated `ℤ_[p]`-module has finite quotient modulo `p`. -/
theorem finite_mod_p_of_finite_module (p : ℕ) [Fact p.Prime]
(M : Type u) [AddCommGroup M] [Module ℤ_[p] M] [Module.Finite ℤ_[p] M] :
Finite (M ⧸ LinearMap.range (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p]))) := by
have : Finite (ℤ_[p] ⧸ Ideal.span {(p : ℤ_[p])}) := by
rw [← PadicInt.maximalIdeal_eq_span_p]
exact Finite.of_injective PadicInt.residueField PadicInt.residueField.injective
have h := Submodule.finite_quotient_smul (M := M) (Ideal.span {(p : ℤ_[p])})
(N := ⊤) (Module.Finite.fg_top (R := ℤ_[p]) (M := M))
have heq : Ideal.span {(p : ℤ_[p])} • (⊤ : Submodule ℤ_[p] M) =
LinearMap.range (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p])) := by
rw [Submodule.ideal_span_singleton_smul]
ext x
simp [Submodule.mem_smul_pointwise_iff_exists, LinearMap.mem_range]
exact heq ▸ h
/-- A finitely generated `ℤ_[p]`-module killed by `p` is finite. -/
theorem finite_of_finite_module_killed_by_p (p : ℕ) [Fact p.Prime]
(M : Type u) [AddCommGroup M] [Module ℤ_[p] M] [Module.Finite ℤ_[p] M]
(hp : ∀ x : M, (p : ℤ_[p]) • x = 0) : Finite M := by
let P := LinearMap.range (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p]))
have : Finite (M ⧸ P) := finite_mod_p_of_finite_module p M
apply Finite.of_injective P.mkQ
apply LinearMap.ker_eq_bot.mp
rw [Submodule.ker_mkQ]
apply LinearMap.range_eq_bot.mpr
ext x
exact hp x
/-- The `p`-torsion of a finitely generated `ℤ_[p]`-module is finite. -/
theorem finite_p_torsion (p : ℕ) [Fact p.Prime]
(M : Type u) [AddCommGroup M] [Module ℤ_[p] M] [Module.Finite ℤ_[p] M] :
Finite (LinearMap.ker (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p]))) := by
let T := LinearMap.ker (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p]))
have : Module.Finite ℤ_[p] T := inferInstance
apply finite_of_finite_module_killed_by_p p T
intro x
apply Subtype.ext
exact x.property
/-- A compact Hausdorff `ℤ_[p]`-module is finitely generated iff it is finite modulo `p`. -/
theorem module_finite_iff_finite_mod_p (p : ℕ) [Fact p.Prime]
(M : Type u) [AddCommGroup M] [Module ℤ_[p] M]
[TopologicalSpace M] [IsTopologicalAddGroup M] [ContinuousSMul ℤ_[p] M]
[CompactSpace M] [T2Space M] :
Module.Finite ℤ_[p] M ↔
Finite (M ⧸ LinearMap.range (LinearMap.lsmul ℤ_[p] M (p : ℤ_[p]))) := by
constructor
· intro _
exact finite_mod_p_of_finite_module p M
· intro _
exact finite_of_finite_mod_p p M
end KnLocalSphere.IntegralRecovery
end OAI
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