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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 | import OAI.NumberTheory.PiExponent.LocalAlgebra.CyclicPrimeFactor
import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousPrimeFactor
namespace OAI
namespace PiExponentJets.W22
attribute [local instance] MvPolynomial.gradedAlgebra
variable {k σ : Type*} [Field k]
def HomogeneousPrimeStep (I J : Ideal (MvPolynomial σ k)) : Prop :=
I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) ∧
∃ (d : ℕ) (f : MvPolynomial σ k), f.IsHomogeneous d ∧ f ∉ I ∧
(I.colon {f}).IsPrime ∧ J = I ⊔ Ideal.span {f}
theorem HomogeneousPrimeStep.homogeneous {I J : Ideal (MvPolynomial σ k)}
(h : HomogeneousPrimeStep I J) :
J.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) := by
obtain ⟨hI, d, f, hf, _, _, rfl⟩ := h
apply hI.sup
apply Ideal.homogeneous_span (MvPolynomial.homogeneousSubmodule σ k)
intro x hx
obtain rfl := Set.mem_singleton_iff.mp hx
exact ⟨d, hf⟩
theorem HomogeneousPrimeStep.lt {I J : Ideal (MvPolynomial σ k)}
(h : HomogeneousPrimeStep I J) : I < J := by
obtain ⟨_, d, f, _, hfI, _, rfl⟩ := h
refine lt_of_le_of_ne le_sup_left ?_
intro heq
have hm : f ∈ I ⊔ Ideal.span {f} :=
(show Ideal.span {f} ≤ I ⊔ Ideal.span {f} from le_sup_right)
(Ideal.subset_span (by simp))
exact hfI (by simpa only [← heq] using hm)
theorem HomogeneousPrimeStep.isQuotientEquivQuotientPrime
{I J : Ideal (MvPolynomial σ k)} (h : HomogeneousPrimeStep I J) :
I.IsQuotientEquivQuotientPrime J := by
obtain ⟨_, d, f, _, _, hp, rfl⟩ := h
exact ⟨le_sup_left, ⟨⟨I.colon {f}, hp⟩, ⟨(cyclicFactorEquiv I f).symm⟩⟩⟩
variable [Finite σ]
theorem exists_homogeneous_prime_filtration
(I : Ideal (MvPolynomial σ k))
(hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) :
∃ s : RelSeries {(A, B) | HomogeneousPrimeStep (k := k) (σ := σ) A B},
s.head = I ∧ s.last = ⊤ := by
let r : Set (Ideal (MvPolynomial σ k) × Ideal (MvPolynomial σ k)) :=
{(A, B) | HomogeneousPrimeStep A B}
have ht : (⊤ : Ideal (MvPolynomial σ k)).IsHomogeneous
(MvPolynomial.homogeneousSubmodule σ k) ∧
∃ s : RelSeries r, s.head = I ∧ s.last = ⊤ := by
refine WellFoundedGT.induction_top
⟨I, hI, RelSeries.singleton r I, rfl, rfl⟩ ?_
rintro J hJtop ⟨hJ, s, hshead, hslast⟩
obtain ⟨d, f, hf, hfJ, hp⟩ := exists_homogeneous_prime_colon J hJ hJtop
have hstep : HomogeneousPrimeStep J (J ⊔ Ideal.span {f}) :=
⟨hJ, d, f, hf, hfJ, hp, rfl⟩
refine ⟨J ⊔ Ideal.span {f}, hstep.lt, hstep.homogeneous,
s.snoc (J ⊔ Ideal.span {f}) (by
change HomogeneousPrimeStep s.last (J ⊔ Ideal.span {f})
simpa only [hslast] using hstep), ?_, by simp only [RelSeries.last_snoc]⟩
simpa using hshead
exact ht.2
end PiExponentJets.W22
end OAI
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