openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /CompleteIntersectionComponentBound.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.ComponentMultiplicityBound | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PositiveDimensionalHilbert | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHilbertComparison | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularDegreePositivity | |
| namespace OAI | |
| namespace PiExponentJets.W22 | |
| open PiExponentJets.W64 | |
| open scoped BigOperators | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k σ : Type*} [Field k] [Fintype σ] | |
| theorem actualHP_regular_sequence | |
| (rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ) | |
| (hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i)) | |
| (hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs) | |
| (hlen : rs.length < Fintype.card σ) | |
| (hI : (Ideal.ofList rs).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| actualHP (Ideal.ofList rs) hI = regularSequenceHilbertPolynomial rs degrees := by | |
| apply actualHP_unique _ hI _ | |
| (rationalDegreePolynomial (degreeAt degrees) rs.length).natDegree | |
| intro n hn | |
| exact regularSequenceHilbertPolynomial_eventually_exact | |
| (lt_of_le_of_lt (Nat.zero_le _) hlen) rs degrees hhom hreg hlen.le n hn | |
| omit [Fintype σ] in | |
| theorem regular_degreeAt_product_ne_zero | |
| (rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ) | |
| (hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i)) | |
| (hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs) : | |
| (∏ i ∈ Finset.range rs.length, degreeAt degrees i : ℕ) ≠ 0 := by | |
| classical | |
| apply Finset.prod_ne_zero_iff.mpr | |
| intro i hi | |
| have hi' := Finset.mem_range.mp hi | |
| simp only [degreeAt, dite_eq_left hi'] | |
| exact Nat.ne_of_gt (PiExponentSiegel.W20.regular_homogeneous_degrees_pos | |
| rs degrees hhom hreg ⟨i, hi'⟩) | |
| theorem actualHP_regular_sequence_natDegree | |
| (rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ) | |
| (hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i)) | |
| (hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs) | |
| (hlen : rs.length < Fintype.card σ) | |
| (hI : (Ideal.ofList rs).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| (actualHP (Ideal.ofList rs) hI).natDegree = Fintype.card σ - rs.length - 1 := by | |
| rw [actualHP_regular_sequence rs degrees hhom hreg hlen hI] | |
| have hp : (rationalDegreePolynomial (degreeAt degrees) rs.length).eval 1 ≠ 0 := by | |
| rw [rationalDegreePolynomial_eval_one] | |
| exact_mod_cast regular_degreeAt_product_ne_zero rs degrees hhom hreg | |
| exact (hilbertPolynomial_degree_and_leading _ _ (Nat.sub_pos_of_lt hlen) hp).1 | |
| theorem actualMultiplicity_regular_sequence | |
| (rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ) | |
| (hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i)) | |
| (hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs) | |
| (hlen : rs.length < Fintype.card σ) | |
| (hI : (Ideal.ofList rs).IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| actualMultiplicity (Ideal.ofList rs) hI (actualHP (Ideal.ofList rs) hI).natDegree = | |
| (∏ i ∈ Finset.range rs.length, degreeAt degrees i : ℕ) := by | |
| rw [actualHP_regular_sequence_natDegree rs degrees hhom hreg hlen hI] | |
| unfold actualMultiplicity | |
| rw [actualHP_regular_sequence rs degrees hhom hreg hlen hI] | |
| exact regularSequenceHilbertPolynomial_multiplicity rs degrees hlen | |
| theorem regular_sequence_component_length_le_degree_product | |
| (rs : List (MvPolynomial σ k)) (degrees : Fin rs.length → ℕ) | |
| (hhom : ∀ i : Fin rs.length, rs[i].IsHomogeneous (degrees i)) | |
| (hreg : RingTheory.Sequence.IsRegular (MvPolynomial σ k) rs) | |
| (hlen : rs.length < Fintype.card σ) | |
| (Q : Ideal (MvPolynomial σ k)) [Q.IsPrime] | |
| (hQhom : Q.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| (hQ : Q ∈ (Ideal.ofList rs).minimalPrimes) | |
| (v : σ) (hv : MvPolynomial.X v ∉ Q) | |
| (hdim : ringKrullDim (MvPolynomial σ k ⧸ Q) = | |
| (Fintype.card σ - rs.length : ℕ)) : | |
| ∃ m : ℕ, | |
| Module.length (Localization.AtPrime Q) | |
| (Localization.AtPrime Q ⧸ (Ideal.ofList rs).map | |
| (algebraMap (MvPolynomial σ k) (Localization.AtPrime Q))) = m ∧ | |
| m ≤ ∏ i ∈ Finset.range rs.length, degreeAt degrees i := by | |
| have hI : (Ideal.ofList rs).IsHomogeneous | |
| (MvPolynomial.homogeneousSubmodule σ k) := by | |
| simpa only [List.take_length] using | |
| PiExponentSiegel.W20.prefixIdeal_homogeneous rs degrees hhom rs.length | |
| obtain ⟨N, hN⟩ := actualHP_eventually Q hQhom | |
| have hQdim := W24.prime_hilbert_natDegree_add_one_eq_krullDim | |
| Q hQhom v hv (actualHP Q hQhom) (N+1) (fun n hn => (hN n (by omega)).symm) | |
| have hQdeg : (actualHP Q hQhom).natDegree + 1 = Fintype.card σ - rs.length := by | |
| exact_mod_cast hQdim.trans hdim | |
| have hdegree : (actualHP Q hQhom).natDegree = | |
| (actualHP (Ideal.ofList rs) hI).natDegree := by | |
| rw [actualHP_regular_sequence_natDegree rs degrees hhom hreg hlen hI] | |
| omega | |
| obtain ⟨m, hm, hpositive, hbound⟩ := | |
| component_multiplicity_bound (Ideal.ofList rs) Q hI hQhom hQ v hv hdegree | |
| refine ⟨m, hm, ?_⟩ | |
| have hprod := actualMultiplicity_regular_sequence rs degrees hhom hreg hlen hI | |
| rw [hprod] at hbound | |
| have hsmall : (m : ℚ) ≤ (∏ i ∈ Finset.range rs.length, degreeAt degrees i : ℕ) := by | |
| have hmpos : (0 : ℚ) ≤ m := Nat.cast_nonneg m | |
| nlinarith | |
| exact_mod_cast hsmall | |
| end PiExponentJets.W22 | |
| end OAI | |