openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /CompleteIntersectionHilbert.lean
| import OAI.NumberTheory.PiExponent.Analysis.RegularSequenceSeries | |
| namespace OAI | |
| namespace PiExponentJets.W64 | |
| open scoped BigOperators | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| noncomputable def degreePolynomial (D : ℕ → ℕ) (m : ℕ) : Polynomial ℤ := | |
| ∏ i ∈ Finset.range m, ∑ j ∈ Finset.range (D i), (Polynomial.X : Polynomial ℤ) ^ j | |
| theorem degreePolynomial_eval_one (D : ℕ → ℕ) (m : ℕ) : | |
| (degreePolynomial D m).eval 1 = ∏ i ∈ Finset.range m, (D i : ℤ) := by | |
| simp [degreePolynomial, Polynomial.eval_prod, Polynomial.eval_finsetSum] | |
| variable {k σ : Type*} [Field k] [Fintype σ] | |
| theorem regular_sequence_series_polynomial | |
| (hσ : 0 < Fintype.card σ) | |
| (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) : | |
| sectionHilbertSeries (Ideal.ofList rs) = | |
| ((degreePolynomial (degreeAt degrees) rs.length) : PowerSeries ℤ) := by | |
| rw [regular_sequence_series_geometric hσ rs degrees hhom hreg hlen] | |
| change _ = (Polynomial.coeToPowerSeries.ringHom : Polynomial ℤ →+* PowerSeries ℤ) | |
| (∏ i ∈ Finset.range rs.length, ∑ j ∈ Finset.range (degreeAt degrees i), | |
| (Polynomial.X : Polynomial ℤ) ^ j) | |
| rw [map_prod] | |
| apply Finset.prod_congr rfl | |
| intro i hi | |
| rw [map_sum] | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| rw [map_pow] | |
| simp only [Polynomial.coeToPowerSeries.ringHom_apply, Polynomial.coe_X] | |
| theorem regular_sequence_section_dimension_coeff | |
| (hσ : 0 < Fintype.card σ) | |
| (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) (n : ℕ) : | |
| (Module.finrank k (quotientSection (Ideal.ofList rs) n) : ℤ) = | |
| (degreePolynomial (degreeAt degrees) rs.length).coeff n := by | |
| have h := congrArg (PowerSeries.coeff n) | |
| (regular_sequence_series_polynomial hσ rs degrees hhom hreg hlen) | |
| simpa only [coeff_sectionHilbertSeries, Polynomial.coeff_coe] using h | |
| theorem regular_sequence_section_dimension_eq_zero | |
| (hσ : 0 < Fintype.card σ) | |
| (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) (n : ℕ) | |
| (hn : n ∉ (degreePolynomial (degreeAt degrees) rs.length).support) : | |
| Module.finrank k (quotientSection (Ideal.ofList rs) n) = 0 := by | |
| have h := regular_sequence_section_dimension_coeff hσ rs degrees hhom hreg hlen n | |
| have hz : (degreePolynomial (degreeAt degrees) rs.length).coeff n = 0 := by | |
| simpa only [Polynomial.mem_support_iff, not_not] using hn | |
| rw [hz] at h | |
| exact_mod_cast h | |
| theorem regular_sequence_section_dimension_sum | |
| (hσ : 0 < Fintype.card σ) | |
| (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) : | |
| (∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support, | |
| Module.finrank k (quotientSection (Ideal.ofList rs) n)) = | |
| ∏ i ∈ Finset.range rs.length, degreeAt degrees i := by | |
| have h : | |
| (∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support, | |
| (Module.finrank k (quotientSection (Ideal.ofList rs) n) : ℤ)) = | |
| ∏ i ∈ Finset.range rs.length, (degreeAt degrees i : ℤ) := by | |
| calc | |
| _ = ∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support, | |
| (degreePolynomial (degreeAt degrees) rs.length).coeff n := by | |
| apply Finset.sum_congr rfl | |
| intro n _ | |
| exact regular_sequence_section_dimension_coeff hσ rs degrees hhom hreg hlen n | |
| _ = (degreePolynomial (degreeAt degrees) rs.length).eval 1 := by | |
| simp [Polynomial.eval_eq_sum, Polynomial.sum] | |
| _ = _ := degreePolynomial_eval_one _ _ | |
| exact_mod_cast h | |
| theorem regular_sequence_section_dimension_sum_le | |
| (hσ : 0 < Fintype.card σ) | |
| (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) (D : ℕ) | |
| (hD : ∀ i, degrees i ≤ D) : | |
| (∑ n ∈ (degreePolynomial (degreeAt degrees) rs.length).support, | |
| Module.finrank k (quotientSection (Ideal.ofList rs) n)) ≤ D ^ rs.length := by | |
| rw [regular_sequence_section_dimension_sum hσ rs degrees hhom hreg hlen] | |
| calc | |
| _ ≤ ∏ _i ∈ Finset.range rs.length, D := by | |
| apply Finset.prod_le_prod | |
| intro i hi | |
| simpa only [degreeAt, dite_eq_left (Finset.mem_range.mp hi)] using | |
| hD ⟨i, Finset.mem_range.mp hi⟩ | |
| _ = _ := by simp | |
| end PiExponentJets.W64 | |
| end OAI | |