openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /HomogeneousQuotientDimension.lean
| import Mathlib.LinearAlgebra.Dimension.Constructions | |
| import Mathlib.LinearAlgebra.DirectSum.Finite | |
| import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.QuotientSections | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentJets.W25 | |
| open MvPolynomial | |
| open PiExponentJets.W64 | |
| open DirectSum | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {k σ : Type*} [Field k] | |
| def quotientDegreeProjection (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (n : ℕ) : | |
| (MvPolynomial σ k ⧸ I) →ₗ[k] (MvPolynomial σ k ⧸ I) := | |
| (I.restrictScalars k).liftQ | |
| ((Ideal.Quotient.mkₐ k I).toLinearMap.comp | |
| (GradedAlgebra.proj (MvPolynomial.homogeneousSubmodule σ k) n)) (by | |
| intro p hp | |
| change Ideal.Quotient.mk I | |
| (GradedAlgebra.proj (MvPolynomial.homogeneousSubmodule σ k) n p) = 0 | |
| apply Ideal.Quotient.eq_zero_iff_mem.mpr | |
| exact (Ideal.IsHomogeneous.mem_iff | |
| (MvPolynomial.homogeneousSubmodule σ k) hI).mp hp n) | |
| @[simp] theorem quotientDegreeProjection_mk (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (n : ℕ) | |
| (p : MvPolynomial σ k) : | |
| quotientDegreeProjection I hI n (Ideal.Quotient.mk I p) = | |
| Ideal.Quotient.mk I | |
| (GradedAlgebra.proj (MvPolynomial.homogeneousSubmodule σ k) n p) := rfl | |
| theorem quotientDegreeProjection_of_mem (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) | |
| (n m : ℕ) (x : MvPolynomial σ k ⧸ I) (hx : x ∈ quotientSection I m) : | |
| quotientDegreeProjection I hI n x = if n = m then x else 0 := by | |
| obtain ⟨p, hp, rfl⟩ := hx | |
| change quotientDegreeProjection I hI n (Ideal.Quotient.mk I p) = _ | |
| rw [quotientDegreeProjection_mk, GradedAlgebra.proj_apply] | |
| by_cases hnm : n = m | |
| · subst n | |
| rw [DirectSum.decompose_of_mem_same (MvPolynomial.homogeneousSubmodule σ k) hp] | |
| simp | |
| · rw [DirectSum.decompose_of_mem_ne (MvPolynomial.homogeneousSubmodule σ k) hp | |
| (Ne.symm hnm)] | |
| simp [hnm] | |
| theorem quotientSection_iSupIndep (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| iSupIndep (quotientSection I) := by | |
| classical | |
| rw [iSupIndep_iff_finsetSum_eq_zero_imp_eq_zero] | |
| intro s v hv hsum i hi | |
| have hcalc : quotientDegreeProjection I hI i (∑ j ∈ s, v j) = v i := by | |
| rw [map_sum] | |
| calc | |
| ∑ j ∈ s, quotientDegreeProjection I hI i (v j) = | |
| ∑ j ∈ s, if i = j then v j else 0 := by | |
| apply Finset.sum_congr rfl | |
| intro j hj | |
| exact quotientDegreeProjection_of_mem I hI i j (v j) (hv j hj) | |
| _ = v i := by simp [hi] | |
| rw [hsum, map_zero] at hcalc | |
| exact hcalc.symm | |
| theorem quotientSection_iSup_eq_top (I : Ideal (MvPolynomial σ k)) : | |
| (⨆ n, quotientSection I n) = ⊤ := by | |
| simp only [quotientSection, ← Submodule.map_iSup] | |
| rw [(DirectSum.Decomposition.isInternal | |
| (MvPolynomial.homogeneousSubmodule σ k)).submodule_iSup_eq_top] | |
| rw [Submodule.map_top] | |
| exact LinearMap.range_eq_top.mpr (Ideal.Quotient.mkₐ_surjective k I) | |
| theorem quotientSection_isInternal (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| DirectSum.IsInternal (quotientSection I) := | |
| DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top | |
| (quotientSection_iSupIndep I hI) (quotientSection_iSup_eq_top I) | |
| def homogeneousQuotientDirectSumEquiv (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) : | |
| (⨁ n : ℕ, quotientSection I n) ≃ₗ[k] (MvPolynomial σ k ⧸ I) := | |
| LinearEquiv.ofBijective (DirectSum.coeLinearMap (quotientSection I)) | |
| (quotientSection_isInternal I hI) | |
| theorem quotientSection_bounded_iSup_eq_top (I : Ideal (MvPolynomial σ k)) (B : ℕ) | |
| (hzero : ∀ n, B < n → quotientSection I n = ⊥) : | |
| (⨆ n : Fin (B+1), quotientSection I n.val) = ⊤ := by | |
| rw [← quotientSection_iSup_eq_top I] | |
| apply le_antisymm | |
| · exact iSup_le fun n => le_iSup (quotientSection I) n.val | |
| · apply iSup_le | |
| intro n | |
| by_cases hn : n < B+1 | |
| · exact le_iSup (fun j : Fin (B+1) => quotientSection I j.val) ⟨n, hn⟩ | |
| · rw [hzero n (Nat.lt_of_succ_le (Nat.le_of_not_lt hn))] | |
| exact bot_le | |
| def boundedHomogeneousQuotientEquiv (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (B : ℕ) | |
| (hzero : ∀ n, B < n → quotientSection I n = ⊥) : | |
| (⨁ n : Fin (B+1), quotientSection I n.val) ≃ₗ[k] (MvPolynomial σ k ⧸ I) := | |
| LinearEquiv.ofBijective | |
| (DirectSum.coeLinearMap (fun n : Fin (B+1) => quotientSection I n.val)) | |
| (DirectSum.isInternal_submodule_of_iSupIndep_of_iSup_eq_top | |
| ((quotientSection_iSupIndep I hI).comp Fin.val_injective) | |
| (quotientSection_bounded_iSup_eq_top I B hzero)) | |
| variable [Finite σ] | |
| theorem homogeneousQuotient_finite (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (B : ℕ) | |
| (hzero : ∀ n, B < n → quotientSection I n = ⊥) : | |
| Module.Finite k (MvPolynomial σ k ⧸ I) := | |
| Module.Finite.equiv (boundedHomogeneousQuotientEquiv I hI B hzero) | |
| theorem homogeneousQuotient_finrank_eq_sum (I : Ideal (MvPolynomial σ k)) | |
| (hI : I.IsHomogeneous (MvPolynomial.homogeneousSubmodule σ k)) (B : ℕ) | |
| (hzero : ∀ n, B < n → quotientSection I n = ⊥) : | |
| Module.finrank k (MvPolynomial σ k ⧸ I) = | |
| ∑ n : Fin (B+1), Module.finrank k (quotientSection I n.val) := by | |
| rw [← (boundedHomogeneousQuotientEquiv I hI B hzero).finrank_eq, | |
| Module.finrank_directSum] | |
| end PiExponentJets.W25 | |
| end | |
| end OAI | |