openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /ComponentMultiplicityBound.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.HilbertMultiplicityFiltration | |
| import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousComponentLength | |
| namespace OAI | |
| namespace PiExponentJets.W22 | |
| open PiExponentJets.W64 | |
| open scoped BigOperators Classical | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k σ : Type*} [Field k] [Fintype σ] | |
| theorem component_multiplicity_bound | |
| (I Q : Ideal (MvPolynomial σ k)) [Q.IsPrime] | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| (hQhom : Q.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| (hQ : Q ∈ I.minimalPrimes) (v : σ) (hv : MvPolynomial.X v ∉ Q) | |
| (hdegree : (actualHP Q hQhom).natDegree = (actualHP I hI).natDegree) : | |
| ∃ m : ℕ, | |
| Module.length (Localization.AtPrime Q) | |
| (Localization.AtPrime Q ⧸ I.map | |
| (algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) = m ∧ | |
| 1 ≤ actualMultiplicity Q hQhom (actualHP I hI).natDegree ∧ | |
| (m : ℚ) * actualMultiplicity Q hQhom (actualHP I hI).natDegree ≤ | |
| actualMultiplicity I hI (actualHP I hI).natDegree := by | |
| classical | |
| obtain ⟨n, J, f, degrees, hmono, hstart, hend, hdata, hlength⟩ := | |
| exists_homogeneous_component_length_count I Q hI hQ | |
| let D := (actualHP I hI).natDegree | |
| have hJ : ∀ i, (J i).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) := by | |
| intro i | |
| by_cases hi : i < n | |
| · exact (hdata i hi).1 | |
| · have heq : J i = ⊤ := by | |
| apply top_unique | |
| rw [← hend] | |
| exact (monotone_nat_of_le_succ hmono) (Nat.le_of_not_gt hi) | |
| rw [heq] | |
| exact Ideal.IsHomogeneous.top (MvPolynomial.homogeneousSubmodule σ k) | |
| have hIle : ∀ i, I ≤ J i := fun i => | |
| hstart.symm.le.trans ((monotone_nat_of_le_succ hmono) (Nat.zero_le i)) | |
| let P : Fin n → Ideal (MvPolynomial σ k) := fun i => (J i).colon {f i} | |
| have hP : ∀ i : Fin n, (P i).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k) := | |
| fun i => homogeneous_colon_singleton (J i) (hJ i) (hdata i i.isLt).2.1 | |
| have hIP : ∀ i : Fin n, I ≤ P i := fun i => | |
| (hIle i).trans (ideal_le_colon_singleton (J i) (f i)) | |
| have hPdeg : ∀ i : Fin n, (actualHP (P i) (hP i)).natDegree ≤ D := | |
| fun i => actualHP_natDegree_antitone hI (hP i) (hIP i) | |
| let U : ℕ → ℚ := fun i => actualMultiplicity (J i) (hJ i) D | |
| let V : ℕ → ℚ := fun i => | |
| if hi : i < n then actualMultiplicity (P ⟨i, hi⟩) (hP ⟨i, hi⟩) D else 0 | |
| have hUstep : ∀ i < n, U i = U (i+1) + V i := by | |
| intro i hi | |
| simp only [U, V, dite_eq_left hi] | |
| exact actualMultiplicity_cyclic_step (J i) (J (i+1)) (hJ i) (hJ (i+1)) | |
| (f i) (hdata i hi).2.1 (hdata i hi).2.2.2.2 (hP ⟨i, hi⟩) D (hPdeg ⟨i, hi⟩) | |
| have hUzero : U 0 = actualMultiplicity I hI D := | |
| actualMultiplicity_congr (hJ 0) hI hstart D | |
| have hUend : U n = 0 := by | |
| have htop := Ideal.IsHomogeneous.top (MvPolynomial.homogeneousSubmodule σ k) | |
| exact (actualMultiplicity_congr (hJ n) htop hend D).trans (actualMultiplicity_top htop D) | |
| have hsum : actualMultiplicity I hI D = ∑ i ∈ Finset.range n, V i := by | |
| have ht := rational_filtration_telescope U V n hUstep | |
| rw [hUzero, hUend, add_zero] at ht | |
| exact ht | |
| let m : ℕ := ∑ i ∈ Finset.range n, if (J i).colon {f i} = Q then 1 else 0 | |
| have hm : Module.length (Localization.AtPrime Q) | |
| (Localization.AtPrime Q ⧸ I.map | |
| (algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) = m := by | |
| rw [hlength] | |
| simp only [m, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero] | |
| have hpositive : 1 ≤ actualMultiplicity Q hQhom D := by | |
| obtain ⟨N, hN⟩ := actualHP_eventually Q hQhom | |
| exact W27.prime_eventual_hilbert_coefficient_ge_one Q v hv (actualHP Q hQhom) | |
| (N+1) D (fun t ht => (hN t (by omega)).symm) hdegree | |
| refine ⟨m, hm, hpositive, ?_⟩ | |
| rw [hsum] | |
| have hmcast : (m : ℚ) = | |
| ∑ i ∈ Finset.range n, if (J i).colon {f i} = Q then (1 : ℚ) else 0 := by | |
| simp only [m, Nat.cast_sum, Nat.cast_ite, Nat.cast_one, Nat.cast_zero] | |
| rw [hmcast, Finset.sum_mul] | |
| apply Finset.sum_le_sum | |
| intro i hi | |
| have hin : i < n := Finset.mem_range.mp hi | |
| simp only [V, dite_eq_left hin] | |
| by_cases heq : (J i).colon {f i} = Q | |
| · rw [ite_eq_left heq, one_mul] | |
| exact le_of_eq (actualMultiplicity_congr (hP ⟨i, hin⟩) hQhom heq D).symm | |
| · rw [ite_eq_right heq, zero_mul] | |
| exact actualMultiplicity_nonneg (P ⟨i, hin⟩) (hP ⟨i, hin⟩) D (hPdeg ⟨i, hin⟩) | |
| end PiExponentJets.W22 | |
| end OAI | |