openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /HomogeneousPrimeFiltration.lean
| 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 | |