openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /CumulativeHilbertSections.lean
| import Mathlib.Algebra.BigOperators.Fin | |
| import OAI.NumberTheory.PiExponent.Geometry.NormalizationGrowth | |
| import OAI.NumberTheory.PiExponent.Polynomials.HomogeneousQuotientDimension | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentJets.W25 | |
| open MvPolynomial DirectSum | |
| open PiExponentJets.W24 PiExponentJets.W64 | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k σ : Type*} [Field k] | |
| theorem restrictTotalDegree_eq_iSup_homogeneous (n : ℕ) : | |
| restrictTotalDegree σ k n = | |
| ⨆ j : Fin (n + 1), homogeneousSubmodule σ k j.val := by | |
| classical | |
| apply le_antisymm | |
| · intro p hp | |
| have hpdegree : p.totalDegree ≤ n := (mem_restrictTotalDegree σ _ _).mp hp | |
| have hsum : (∑ j ∈ Finset.range (p.totalDegree + 1), homogeneousComponent j p) ∈ | |
| ⨆ j : Fin (n + 1), homogeneousSubmodule σ k j.val := by | |
| apply Submodule.sum_mem | |
| intro j hj | |
| have hjdegree : j ≤ n := | |
| (Nat.le_of_lt_succ (Finset.mem_range.mp hj)).trans hpdegree | |
| exact (le_iSup (fun a : Fin (n + 1) => homogeneousSubmodule σ k a.val) | |
| ⟨j, Nat.lt_succ_of_le hjdegree⟩) (homogeneousComponent_mem j p) | |
| simpa only [sum_homogeneousComponent] using hsum | |
| · apply iSup_le | |
| intro j p hp | |
| apply (mem_restrictTotalDegree σ _ _).mpr | |
| exact (show p.IsHomogeneous j.val from hp).totalDegree_le.trans | |
| (Nat.le_of_lt_succ j.isLt) | |
| theorem quotientDegreeFiltration_eq_iSup_sections | |
| (I : Ideal (MvPolynomial σ k)) (n : ℕ) : | |
| quotientDegreeFiltration I n = | |
| ⨆ j : Fin (n + 1), quotientSection I j.val := by | |
| unfold quotientDegreeFiltration quotientSection | |
| rw [restrictTotalDegree_eq_iSup_homogeneous, Submodule.map_iSup] | |
| theorem cumulativeSectionMap_injective (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) : | |
| Function.Injective | |
| (DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val)) := | |
| ((quotientSection_iSupIndep I hI).comp Fin.val_injective).dfinsupp_lsum_injective | |
| theorem cumulativeSectionMap_range (I : Ideal (MvPolynomial σ k)) (n : ℕ) : | |
| LinearMap.range | |
| (DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val)) = | |
| quotientDegreeFiltration I n := by | |
| rw [DirectSum.range_coeLinearMap, quotientDegreeFiltration_eq_iSup_sections] | |
| def cumulativeHomogeneousQuotientEquiv (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) : | |
| (⨁ j : Fin (n + 1), quotientSection I j.val) ≃ₗ[k] | |
| quotientDegreeFiltration I n := | |
| (LinearEquiv.ofInjective | |
| (DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val)) | |
| (cumulativeSectionMap_injective I hI n)).trans | |
| (LinearEquiv.ofEq _ _ (cumulativeSectionMap_range I n)) | |
| @[simp] theorem cumulativeHomogeneousQuotientEquiv_apply | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) | |
| (x : ⨁ j : Fin (n + 1), quotientSection I j.val) : | |
| (cumulativeHomogeneousQuotientEquiv I hI n x : MvPolynomial σ k ⧸ I) = | |
| DirectSum.coeLinearMap (fun j : Fin (n + 1) => quotientSection I j.val) x := rfl | |
| variable [Finite σ] | |
| theorem quotientDegreeFiltration_finrank_eq_sum_fin | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) : | |
| Module.finrank k (quotientDegreeFiltration I n) = | |
| ∑ j : Fin (n + 1), Module.finrank k (quotientSection I j.val) := by | |
| rw [← (cumulativeHomogeneousQuotientEquiv I hI n).finrank_eq, Module.finrank_directSum] | |
| theorem quotientDegreeFiltration_finrank_eq_sum | |
| (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (homogeneousSubmodule σ k)) (n : ℕ) : | |
| Module.finrank k (quotientDegreeFiltration I n) = | |
| ∑ j ∈ Finset.range (n + 1), Module.finrank k (quotientSection I j) := by | |
| rw [quotientDegreeFiltration_finrank_eq_sum_fin I hI n] | |
| exact Fin.sum_univ_eq_sum_range | |
| (fun j : ℕ => Module.finrank k (quotientSection I j)) (n + 1) | |
| end PiExponentJets.W25 | |
| end | |
| end OAI | |