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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 | 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
variable {k σ : Type*} [Field k] [Fintype σ]
theorem component_multiplicity_bound
(I Q : Ideal (MvPolynomial σ k)) [Q.IsPrime]
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(hQhom : Q.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(hQ : Q ∈ I.minimalPrimes) (v : σ) (hv : MvPolynomial.X v ∉ Q)
(hdegree : (actualHP Q hQhom).natDegree = (actualHP I hI).natDegree) :
∃ m : ℕ,
Module.length (Localization.AtPrime Q)
(Localization.AtPrime Q ⧸ I.map
(algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) = m ∧
1 ≤ actualMultiplicity Q hQhom (actualHP I hI).natDegree ∧
(m : ℚ) * actualMultiplicity Q hQhom (actualHP I hI).natDegree ≤
actualMultiplicity I hI (actualHP I hI).natDegree := by
classical
obtain ⟨n, J, f, degrees, hmono, hstart, hend, hdata, hlength⟩ :=
exists_homogeneous_component_length_count I Q hI hQ
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 : ℕ := ∑ i ∈ Finset.range n, if (J i).colon {f i} = Q then 1 else 0
have hm : Module.length (Localization.AtPrime Q)
(Localization.AtPrime Q ⧸ I.map
(algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) = m := by
rw [hlength]
simp only [m, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero]
have hpositive : 1 ≤ actualMultiplicity Q hQhom D := by
obtain ⟨N, hN⟩ := actualHP_eventually Q hQhom
exact W27.prime_eventual_hilbert_coefficient_ge_one Q v hv (actualHP Q hQhom)
(N+1) D (fun t ht => (hN t (by omega)).symm) hdegree
refine ⟨m, hm, hpositive, ?_⟩
rw [hsum]
have hmcast : (m : ℚ) =
∑ i ∈ Finset.range n, if (J i).colon {f i} = Q then (1 : ℚ) else 0 := by
simp only [m, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero]
rw [hmcast, Finset.sum_mul]
apply Finset.sum_le_sum
intro i hi
have hin : i < n := Finset.mem_range.mp hi
simp only [V, dite_eq_left hin]
by_cases heq : (J i).colon {f i} = Q
· rw [ite_eq_left heq, one_mul]
exact le_of_eq (actualMultiplicity_congr (hP ⟨i, hin⟩) hQhom heq D).symm
· rw [ite_eq_right heq, zero_mul]
exact actualMultiplicity_nonneg (P ⟨i, hin⟩) (hP ⟨i, hin⟩) D (hPdeg ⟨i, hin⟩)
end PiExponentJets.W22
end OAI
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