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