| 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 |
|
|
