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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 | import OAI.NumberTheory.PiExponent.LocalAlgebra.ComponentMultiplicityBound
import OAI.NumberTheory.PiExponent.LocalAlgebra.PositiveDimensionalHilbert
import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHilbertComparison
import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularDegreePositivity
namespace OAI
namespace PiExponentJets.W22
open PiExponentJets.W64
open scoped BigOperators
attribute [local instance] MvPolynomial.gradedAlgebra
variable {k σ : Type*} [Field k] [Fintype σ]
theorem actualHP_regular_sequence
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length < Fintype.card σ)
(hI : (Ideal.ofList rs).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) :
actualHP (Ideal.ofList rs) hI = regularSequenceHilbertPolynomial rs degrees := by
apply actualHP_unique _ hI _
(rationalDegreePolynomial (degreeAt degrees) rs.length).natDegree
intro n hn
exact regularSequenceHilbertPolynomial_eventually_exact
(lt_of_le_of_lt (Nat.zero_le _) hlen) rs degrees hhom hreg hlen.le n hn
omit [Fintype σ] in
theorem regular_degreeAt_product_ne_zero
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs) :
(∏ i ∈ Finset.range rs.length, degreeAt degrees i : ℕ) ≠ 0 := by
classical
apply Finset.prod_ne_zero_iff.mpr
intro i hi
have hi' := Finset.mem_range.mp hi
simp only [degreeAt, dite_eq_left hi']
exact Nat.ne_of_gt (PiExponentSiegel.W20.regular_homogeneous_degrees_pos
rs degrees hhom hreg ⟨i, hi'⟩)
theorem actualHP_regular_sequence_natDegree
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length < Fintype.card σ)
(hI : (Ideal.ofList rs).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) :
(actualHP (Ideal.ofList rs) hI).natDegree = Fintype.card σ - rs.length - 1 := by
rw [actualHP_regular_sequence rs degrees hhom hreg hlen hI]
have hp : (rationalDegreePolynomial (degreeAt degrees) rs.length).eval 1 ≠ 0 := by
rw [rationalDegreePolynomial_eval_one]
exact_mod_cast regular_degreeAt_product_ne_zero rs degrees hhom hreg
exact (hilbertPolynomial_degree_and_leading _ _ (Nat.sub_pos_of_lt hlen) hp).1
theorem actualMultiplicity_regular_sequence
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length < Fintype.card σ)
(hI : (Ideal.ofList rs).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) :
actualMultiplicity (Ideal.ofList rs) hI (actualHP (Ideal.ofList rs) hI).natDegree =
(∏ i ∈ Finset.range rs.length, degreeAt degrees i : ℕ) := by
rw [actualHP_regular_sequence_natDegree rs degrees hhom hreg hlen hI]
unfold actualMultiplicity
rw [actualHP_regular_sequence rs degrees hhom hreg hlen hI]
exact regularSequenceHilbertPolynomial_multiplicity rs degrees hlen
theorem regular_sequence_component_length_le_degree_product
(rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ)
(hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i))
(hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs)
(hlen : rs.length < Fintype.card σ)
(Q : Ideal (MvPolynomial σ k)) [Q.IsPrime]
(hQhom : Q.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(hQ : Q ∈ (Ideal.ofList rs).minimalPrimes)
(v : σ) (hv : MvPolynomial.X v ∉ Q)
(hdim : ringKrullDim (MvPolynomial σ k ⧸ Q) =
(Fintype.card σ - rs.length : ℕ)) :
∃ m : ℕ,
Module.length (Localization.AtPrime Q)
(Localization.AtPrime Q ⧸ (Ideal.ofList rs).map
(algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) = m ∧
m ≤ ∏ i ∈ Finset.range rs.length, degreeAt degrees i := by
have hI : (Ideal.ofList rs).IsHomogeneous
(MvPolynomial.homogeneousSubmodule σ k) := by
simpa only [List.take_length] using
PiExponentSiegel.W20.prefixIdeal_homogeneous rs degrees hhom rs.length
obtain ⟨N, hN⟩ := actualHP_eventually Q hQhom
have hQdim := W24.prime_hilbert_natDegree_add_one_eq_krullDim
Q hQhom v hv (actualHP Q hQhom) (N+1) (fun n hn => (hN n (by omega)).symm)
have hQdeg : (actualHP Q hQhom).natDegree + 1 = Fintype.card σ - rs.length := by
exact_mod_cast hQdim.trans hdim
have hdegree : (actualHP Q hQhom).natDegree =
(actualHP (Ideal.ofList rs) hI).natDegree := by
rw [actualHP_regular_sequence_natDegree rs degrees hhom hreg hlen hI]
omega
obtain ⟨m, hm, hpositive, hbound⟩ :=
component_multiplicity_bound (Ideal.ofList rs) Q hI hQhom hQ v hv hdegree
refine ⟨m, hm, ?_⟩
have hprod := actualMultiplicity_regular_sequence rs degrees hhom hreg hlen hI
rw [hprod] at hbound
have hsmall : (m : ℚ) ≤ (∏ i ∈ Finset.range rs.length, degreeAt degrees i : ℕ) := by
have hmpos : (0 : ℚ) ≤ m := Nat.cast_nonneg m
nlinarith
exact_mod_cast hsmall
end PiExponentJets.W22
end OAI
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