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b797a5b | 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 | import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertMultiplicityFiltration
import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousComponentLength
namespace OAI
namespace PiExponentJets.W22
open PiExponentJets.W64
open scoped BigOperators Classical
attribute [local instance] MvPolynomial.gradedAlgebra
theorem distinct_selected_factor_weight_le
{τ β : Type*} [Fintype τ] (Q : τ → β) (hQinj : Function.Injective Q)
(P : β) (W : τ → ℚ) (V : ℚ) (hV : 0 ≤ V)
(hmatch : ∀ a, P = Q a → W a = V) :
(∑ a, if P = Q a then W a else 0) ≤ V := by
classical
by_cases hex : ∃ a, P = Q a
· obtain ⟨a, ha⟩ := hex
have hsum : (∑ b, if P = Q b then W b else 0) = W a := by
rw [Finset.sum_eq_single a]
· simp only [ite_eq_left ha]
· intro b _ hba
have hb : P ≠ Q b := by
intro hb
exact hba (hQinj (hb.symm.trans ha))
exact ite_eq_right hb
· intro hnot
exact False.elim (hnot (Finset.mem_univ a))
rw [hsum, hmatch a ha]
· have hzero : (∑ a, if P = Q a then W a else 0) = 0 := by
apply Finset.sum_eq_zero
intro a _
exact ite_eq_right (fun ha => hex ⟨a, ha⟩)
rw [hzero]
exact hV
theorem grouped_count_weighted_sum_eq
{τ β : Type*} [Fintype τ] (n : ℕ) (P : ℕ → β) (Q : τ → β) (W : τ → ℚ) :
(∑ a, ((∑ i ∈ Finset.range n, if P i = Q a then 1 else 0 : ℕ) : ℚ) * W a) =
∑ i ∈ Finset.range n, ∑ a, if P i = Q a then W a else 0 := by
classical
simp only [Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero,
Finset.sum_mul, ite_mul, one_mul, zero_mul]
exact Finset.sum_comm
variable {k σ τ : Type*} [Field k] [Fintype σ] [Fintype τ]
theorem component_multiplicity_sum_bound
(I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(Q : τ → Ideal (MvPolynomial σ k)) [∀ a, (Q a).IsPrime]
(hQinj : Function.Injective Q)
(hQhom : ∀ a, (Q a).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(hQ : ∀ a, Q a ∈ I.minimalPrimes)
(v : τ → σ) (hv : ∀ a, MvPolynomial.X (v a) ∉ Q a)
(hdegree : ∀ a, (actualHP (Q a) (hQhom a)).natDegree = (actualHP I hI).natDegree) :
∃ m : τ → ℕ,
(∀ a, Module.length (Localization.AtPrime (Q a))
(Localization.AtPrime (Q a) ⧸ I.map
(algebraMap (MvPolynomial σ k) (Localization.AtPrime (Q a)))) = m a) ∧
(∀ a, 1 ≤ actualMultiplicity (Q a) (hQhom a) (actualHP I hI).natDegree) ∧
(∑ a, (m a : ℚ) * actualMultiplicity (Q a) (hQhom a)
(actualHP I hI).natDegree) ≤ actualMultiplicity I hI (actualHP I hI).natDegree := by
classical
obtain ⟨n, J, f, degrees, hmono, hstart, hend, hdata⟩ :=
exists_homogeneous_cyclic_filtration_nat I hI
let D := (actualHP I hI).natDegree
have hJ : ∀ i, (J i).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) := by
intro i
by_cases hi : i < n
· exact (hdata i hi).1
· have heq : J i = ⊤ := by
apply top_unique
rw [← hend]
exact (monotone_nat_of_le_succ hmono) (Nat.le_of_not_gt hi)
rw [heq]
exact Ideal.IsHomogeneous.top (MvPolynomial.homogeneousSubmodule σ k)
have hIle : ∀ i, I ≤ J i := fun i =>
hstart.symm.le.trans ((monotone_nat_of_le_succ hmono) (Nat.zero_le i))
let P : Fin n → Ideal (MvPolynomial σ k) := fun i => (J i).colon {f i}
have hP : ∀ i : Fin n, (P i).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) :=
fun i => homogeneous_colon_singleton (J i) (hJ i) (hdata i i.isLt).2.1
have hIP : ∀ i : Fin n, I ≤ P i := fun i =>
(hIle i).trans (ideal_le_colon_singleton (J i) (f i))
have hPdeg : ∀ i : Fin n, (actualHP (P i) (hP i)).natDegree ≤ D :=
fun i => actualHP_natDegree_antitone hI (hP i) (hIP i)
let U : ℕ → ℚ := fun i => actualMultiplicity (J i) (hJ i) D
let V : ℕ → ℚ := fun i =>
if hi : i < n then actualMultiplicity (P ⟨i, hi⟩) (hP ⟨i, hi⟩) D else 0
have hUstep : ∀ i < n, U i = U (i+1) + V i := by
intro i hi
simp only [U, V, dite_eq_left hi]
exact actualMultiplicity_cyclic_step (J i) (J (i+1)) (hJ i) (hJ (i+1))
(f i) (hdata i hi).2.1 (hdata i hi).2.2.2.2 (hP ⟨i, hi⟩) D (hPdeg ⟨i, hi⟩)
have hUzero : U 0 = actualMultiplicity I hI D :=
actualMultiplicity_congr (hJ 0) hI hstart D
have hUend : U n = 0 := by
have htop := Ideal.IsHomogeneous.top (MvPolynomial.homogeneousSubmodule σ k)
exact (actualMultiplicity_congr (hJ n) htop hend D).trans (actualMultiplicity_top htop D)
have hsum : actualMultiplicity I hI D = ∑ i ∈ Finset.range n, V i := by
have ht := rational_filtration_telescope U V n hUstep
rw [hUzero, hUend, add_zero] at ht
exact ht
let m : τ → ℕ := fun a =>
∑ i ∈ Finset.range n, if (J i).colon {f i} = Q a then 1 else 0
have hm : ∀ a, Module.length (Localization.AtPrime (Q a))
(Localization.AtPrime (Q a) ⧸ I.map
(algebraMap (MvPolynomial σ k) (Localization.AtPrime (Q a)))) = m a := by
intro a
have hlength := localized_cyclic_filtration_length I (Q a) (hQ a)
J hmono hstart n hend f
(fun i hi => (hdata i hi).2.2.2.2) (fun i hi => (hdata i hi).2.2.2.1)
rw [hlength]
simp only [m, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero]
have hpositive : ∀ a, 1 ≤ actualMultiplicity (Q a) (hQhom a) D := by
intro a
obtain ⟨N, hN⟩ := actualHP_eventually (Q a) (hQhom a)
exact W27.prime_eventual_hilbert_coefficient_ge_one (Q a) (v a) (hv a)
(actualHP (Q a) (hQhom a)) (N+1) D
(fun t ht => (hN t (by omega)).symm) (hdegree a)
refine ⟨m, hm, hpositive, ?_⟩
change (∑ a, (m a : ℚ) * actualMultiplicity (Q a) (hQhom a) D) ≤
actualMultiplicity I hI D
rw [hsum]
have hgroup := grouped_count_weighted_sum_eq n
(fun i => (J i).colon {f i}) Q (fun a => actualMultiplicity (Q a) (hQhom a) D)
change (∑ a, (m a : ℚ) * actualMultiplicity (Q a) (hQhom a) D) = _ at hgroup
rw [hgroup]
apply Finset.sum_le_sum
intro i hi
have hin : i < n := Finset.mem_range.mp hi
have hnonneg : 0 ≤ V i := by
simp only [V, dite_eq_left hin]
exact actualMultiplicity_nonneg (P ⟨i, hin⟩) (hP ⟨i, hin⟩) D (hPdeg ⟨i, hin⟩)
apply distinct_selected_factor_weight_le Q hQinj ((J i).colon {f i})
(fun a => actualMultiplicity (Q a) (hQhom a) D) (V i) hnonneg
intro a heq
simp only [V, dite_eq_left hin]
exact (actualMultiplicity_congr (hP ⟨i, hin⟩) (hQhom a) heq D).symm
end PiExponentJets.W22
end OAI
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