| import Mathlib |
| import OAI.RingTheory.Multiplicity.NormalizationModuleLimit |
| import OAI.RingTheory.Multiplicity.QuotientExtension |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open CategoryTheory |
| open scoped TensorProduct |
| namespace Lech.CharP |
| open Lech.RootTower Lech.PerfectDomainStages Lech.Koszul IsLocalRing Filter |
| open scoped ENNReal Topology nonZeroDivisors |
| universe u |
| variable (D : Type u) [CommRing D] [IsDomain D] [IsLocalRing D] |
| [IsNoetherianRing D] [IsAdicComplete (maximalIdeal D) D] |
| (p : ℕ) [Fact p.Prime] [CharP D p] [PerfectRing (ResidueField D) p] |
|
|
| theorem exists_length_with_module_limit (hh : 0 < Lech.dimension D) : |
| ∃ ell : Lech.AllModuleLength (PerfectClosure D p), |
| (∀ (M : ModuleCat.{u} D), IsFiniteLength D M → |
| Tendsto (fun n : ℕ => ((p : ℝ≥0∞)^(n*Lech.dimension D))⁻¹ * |
| (Module.length D (Lech.FrobeniusModule D p n M)).toENNReal) |
| atTop (𝓝 (ell.value ((ModuleCat.extendScalars (PerfectClosure.of D p)).obj M)))) ∧ |
| (∀ (H : Ideal D), H.radical = maximalIdeal D → |
| Tendsto (fun n : ℕ => ((p : ℝ≥0∞)^(n*Lech.dimension D))⁻¹ * |
| (Module.length D (D ⧸ H.map (iterateFrobenius D p n))).toENNReal) |
| atTop (𝓝 (ell.value (ModuleCat.of (PerfectClosure D p) |
| ((PerfectClosure D p) ⧸ H.map (PerfectClosure.of D p)))))) ∧ |
| (∀ (z : Fin (Lech.dimension D) → D), |
| (Ideal.span (Set.range z)).radical = maximalIdeal D → |
| ∀ a : ℕ, 0 < a → |
| ell.value (ModuleCat.of (PerfectClosure D p) ((PerfectClosure D p) ⧸ |
| (parameterIdeal D z a).map (PerfectClosure.of D p))) = |
| (a : ℝ≥0∞)^(Lech.dimension D) * |
| ENNReal.ofReal (Lech.Primary.multiplicity (Ideal.span (Set.range z)))) ∧ |
| (∀ (ys : List D), ys.length = Lech.dimension D → |
| (entryIdeal ys).radical = maximalIdeal D → ∀ i : ℤ, i < 0 → |
| ell.value ((unit (ys.map (PerfectClosure.of D p))).homology i) = 0) ∧ |
| (∀ (z : Fin (Lech.dimension D) → D), |
| (Ideal.span (Set.range z)).radical = maximalIdeal D → |
| 0 < Lech.Primary.multiplicity (Ideal.span (Set.range z))) := by |
| obtain ⟨h,ψ,hψ,hfin,hloc,hres,hd⟩ := |
| Lech.Normalization.exists_finite_powerSeries_normalization D p |
| have hdim : Lech.dimension D = h := by |
| have hd' := Lech.dimension_cast D |
| rw [hd] at hd' |
| exact_mod_cast hd' |
| let k := ResidueField D |
| let : CharP k p := CharP.of_ringHom_of_ne_zero (residue D) p |
| (Nat.Prime.ne_zero Fact.out) |
| let A := MvPowerSeries (Fin h) k |
| let C := PerfectClosure D p |
| let P := PerfectClosure A p |
| let : IsDomain A := NoZeroDivisors.to_isDomain _ |
| let : Algebra A D := ψ.toAlgebra |
| let : Module.Finite A D := hfin |
| let : IsLocalHom (algebraMap A D) := hloc |
| have hf : Function.Injective (algebraMap A D) := hψ |
| have hres' : Function.Surjective (algebraMap (ResidueField A) (ResidueField D)) := by |
| intro b |
| obtain ⟨a,ha⟩ := hres b |
| refine ⟨residue A a,?_⟩ |
| rw [ResidueField.algebraMap_residue] |
| exact ha |
| let : FaithfulSMul A D := (faithfulSMul_iff_algebraMap_injective A D).mpr hf |
| let K := FractionRing A |
| let L := FractionRing D |
| let : Algebra K L := FractionRing.liftAlgebra A L |
| let : IsScalarTower A K L := FractionRing.isScalarTower_liftAlgebra A L |
| let : FiniteDimensional K L := Module.Finite.of_isLocalization A D A⁰ |
| obtain ⟨r,hr,hlimmod⟩ := |
| Lech.SeparableOrder.exists_normalization_frobenius_module_limit h k D p K L hf hres' |
| have hlim (H : Ideal D) (hH : H.radical = maximalIdeal D) : |
| Tendsto (fun n : ℕ => ((p : ℝ≥0∞)^(n*h))⁻¹ * |
| (Module.length D (D ⧸ H.map (iterateFrobenius D p n))).toENNReal) |
| atTop (𝓝 ((r : ℝ≥0∞) * normalizedLength (Fin h) k p |
| ((PerfectClosure D p) ⧸ H.map (PerfectClosure.of D p)))) := by |
| have ht := hlimmod (ModuleCat.of D (D ⧸ H)) (Lech.finiteLength_primary_quotient H hH) |
| have heq := (regularTower (Fin h) k p).length_eq_of_equiv |
| ((ModuleCat.restrictScalars (algebraMap P C)).mapIso |
| (Lech.quotientExtensionEquiv (PerfectClosure.of D p) H).toModuleIso).toLinearEquiv |
| change normalizedLength (Fin h) k p ((PerfectClosure D p) ⊗[D] (D ⧸ H)) = |
| normalizedLength (Fin h) k p ((PerfectClosure D p) ⧸ H.map (PerfectClosure.of D p)) at heq |
| rw [heq] at ht |
| apply ht.congr |
| intro n |
| congr 1 |
| exact congrArg ENat.toENNReal (Lech.quotientExtensionEquiv (iterateFrobenius D p n) H).length_eq |
| let ell := (regularTower (Fin h) k p).allModuleLength (algebraMap P C) r |
| (by exact_mod_cast (Nat.ne_of_gt hr)) |
| let φ := Lech.CoefficientField.localSection D p |
| have hφ : Function.Surjective ((residue D).comp φ) := |
| fun a => ⟨a,Lech.CoefficientField.localSection_rightInverse D p a⟩ |
| refine ⟨ell,?_,?_,?_,?_,?_⟩ |
| · intro M hM |
| rw [hdim] |
| exact hlimmod M hM |
| · intro H hH |
| rw [hdim] |
| exact hlim H hH |
| · intro z hz a ha |
| have hprim : (parameterIdeal D z a).radical = maximalIdeal D := |
| (Lech.Normalization.radical_span_powers z a (Nat.ne_of_gt ha)).trans hz |
| have ht := hlim (parameterIdeal D z a) hprim |
| have hp := Lech.Normalization.parameter_frobenius_multiplicity_enn |
| (Lech.dimension D) hh rfl φ hφ z hz a ha p |
| have hp' : Tendsto (fun n : ℕ => ((p : ℝ≥0∞)^(n*h))⁻¹ * |
| (Module.length D (D ⧸ (parameterIdeal D z a).map (iterateFrobenius D p n))).toENNReal) |
| atTop (𝓝 ((a : ℝ≥0∞)^(Lech.dimension D) * |
| ENNReal.ofReal (Lech.Primary.multiplicity (Ideal.span (Set.range z))))) := by |
| convert hp using 1 |
| ext n |
| apply congrArg (fun b : ℝ≥0∞ => b * |
| (Module.length D (D ⧸ (parameterIdeal D z a).map (iterateFrobenius D p n))).toENNReal) |
| rw [hdim] |
| exact tendsto_nhds_unique ht hp' |
| · intro ys hlen hys i hi |
| have hz := Lech.SeparableOrder.normalizedLength_parameter_koszul |
| h k D p K L ys (hlen.trans hdim) hys i hi |
| change (r : ℝ≥0∞) * _ = 0 |
| change (regularTower (Fin h) k p).length _ = 0 at hz |
| rw [hz,mul_zero] |
| · intro z hz |
| exact Lech.Normalization.parameter_multiplicity_pos |
| (Lech.dimension D) hh rfl φ hφ z hz |
| end Lech.CharP |
|
|
| end |
|
|
| end OAI |
|
|
