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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 | import OAI.NumberTheory.PiExponent.LocalAlgebra.LocalizedFiltrationLength
import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousPrimeFiltration
namespace OAI
namespace PiExponentJets.W22
open scoped BigOperators Classical
attribute [local instance] MvPolynomial.gradedAlgebra
variable {k σ : Type*} [Field k] [Finite σ]
theorem exists_homogeneous_cyclic_filtration_nat
(I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) :
∃ (n : ℕ) (J : ℕ → Ideal (MvPolynomial σ k))
(f : ℕ → MvPolynomial σ k) (d : ℕ → ℕ),
(∀ i, J i ≤ J (i + 1)) ∧ J 0 = I ∧ J n = ⊤ ∧
∀ i < n, (J i).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) ∧
(f i).IsHomogeneous (d i) ∧ f i ∉ J i ∧
((J i).colon {f i}).IsPrime ∧ J (i + 1) = J i ⊔ Ideal.span {f i} := by
classical
obtain ⟨s, hshead, hslast⟩ := exists_homogeneous_prime_filtration I hI
have hsmono : Monotone s :=
(Fin.strictMono_iff_lt_succ.mpr fun i => (s.step i).lt).monotone
let J : ℕ → Ideal (MvPolynomial σ k) := fun i =>
s ⟨min i s.length, Nat.lt_succ_of_le (min_le_right _ _)⟩
have hd : ∀ i : Fin s.length, ∃ (d : ℕ) (f : MvPolynomial σ k),
f.IsHomogeneous d ∧ f ∉ s i.castSucc ∧
((s i.castSucc).colon {f}).IsPrime ∧
s i.succ = s i.castSucc ⊔ Ideal.span {f} := fun i => (s.step i).2
let degrees : Fin s.length → ℕ := fun i => (hd i).choose
let generators : Fin s.length → MvPolynomial σ k := fun i => (hd i).choose_spec.choose
let f : ℕ → MvPolynomial σ k := fun i =>
if hi : i < s.length then generators ⟨i, hi⟩ else 0
let d : ℕ → ℕ := fun i => if hi : i < s.length then degrees ⟨i, hi⟩ else 0
refine ⟨s.length, J, f, d, ?_, ?_, ?_, ?_⟩
· intro i
apply hsmono
exact min_le_min_right s.length (Nat.le_succ i)
· change s.head = I
exact hshead
· simpa only [J, min_self, RelSeries.last, Fin.last] using hslast
· intro i hi
have hfi := (hd ⟨i, hi⟩).choose_spec.choose_spec
have hleft : J i = s (Fin.castSucc ⟨i, hi⟩) := by
apply congrArg s
apply Fin.ext
exact min_eq_left (Nat.le_of_lt hi)
have hright : J (i + 1) = s (Fin.succ ⟨i, hi⟩) := by
apply congrArg s
apply Fin.ext
exact min_eq_left (Nat.succ_le_of_lt hi)
simpa only [f, d, dite_eq_left hi, degrees, generators, hleft, hright] using
And.intro (s.step ⟨i, hi⟩).1 hfi
theorem exists_homogeneous_component_length_count
(I Q : Ideal (MvPolynomial σ k)) [Q.IsPrime]
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k))
(hQ : Q ∈ I.minimalPrimes) :
∃ (n : ℕ) (J : ℕ → Ideal (MvPolynomial σ k))
(f : ℕ → MvPolynomial σ k) (d : ℕ → ℕ),
(∀ i, J i ≤ J (i + 1)) ∧ J 0 = I ∧ J n = ⊤ ∧
(∀ i < n, (J i).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) ∧
(f i).IsHomogeneous (d i) ∧ f i ∉ J i ∧
((J i).colon {f i}).IsPrime ∧ J (i + 1) = J i ⊔ Ideal.span {f i}) ∧
Module.length (Localization.AtPrime Q)
(Localization.AtPrime Q ⧸ I.map
(algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) =
∑ i ∈ Finset.range n, if (J i).colon {f i} = Q then 1 else 0 := by
obtain ⟨n, J, f, d, hmono, hstart, hend, hstep⟩ :=
exists_homogeneous_cyclic_filtration_nat I hI
refine ⟨n, J, f, d, hmono, hstart, hend, hstep, ?_⟩
exact localized_cyclic_filtration_length I Q hQ J hmono hstart n hend f
(fun i hi => (hstep i hi).2.2.2.2) (fun i hi => (hstep i hi).2.2.2.1)
end PiExponentJets.W22
end OAI
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