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e3e0a0d | 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 | 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
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