openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /PolynomialCoefficientLocalization.lean
| import Mathlib.Algebra.MvPolynomial.Equiv | |
| import Mathlib.RingTheory.Localization.FractionRing | |
| import Mathlib.RingTheory.MvPolynomial.Localization | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentSiegelAux.W09 | |
| variable (K : Type*) [Field K] (α β : Type*) | |
| abbrev SplitPolynomial := MvPolynomial (α ⊕ β) K | |
| abbrev CoefficientPolynomial := MvPolynomial β K | |
| abbrev CoefficientFractionField := FractionRing (CoefficientPolynomial K β) | |
| abbrev FractionCoefficientPolynomial := MvPolynomial α (CoefficientFractionField K β) | |
| def coefficientDenominators : Submonoid (SplitPolynomial K α β) := | |
| (nonZeroDivisors (CoefficientPolynomial K β)).map | |
| (MvPolynomial.rename (Sum.inr : β → α ⊕ β)).toRingHom | |
| abbrev PolynomialCoefficientLocalization := Localization (coefficientDenominators K α β) | |
| theorem coefficientRename_injective : | |
| Function.Injective (MvPolynomial.rename (Sum.inr : β → α ⊕ β) : | |
| CoefficientPolynomial K β → SplitPolynomial K α β) := | |
| MvPolynomial.rename_injective _ Sum.inr_injective | |
| @[simp] theorem mem_coefficientDenominators (p : SplitPolynomial K α β) : | |
| p ∈ coefficientDenominators K α β ↔ | |
| ∃ f : CoefficientPolynomial K β, f ≠ 0 ∧ MvPolynomial.rename Sum.inr f = p := by | |
| simp [coefficientDenominators, Submonoid.mem_map, mem_nonZeroDivisors_iff_ne_zero] | |
| @[simp] theorem sumAlgEquiv_coefficientRename (f : CoefficientPolynomial K β) : | |
| MvPolynomial.sumAlgEquiv K α β (MvPolynomial.rename Sum.inr f) = MvPolynomial.C f := by | |
| exact congrArg (fun g : CoefficientPolynomial K β →ₐ[K] | |
| MvPolynomial α (CoefficientPolynomial K β) => g f) | |
| (MvPolynomial.sumAlgEquiv_comp_rename_inr K α β) | |
| theorem sumAlgEquiv_coefficientDenominators : | |
| (coefficientDenominators K α β).map (MvPolynomial.sumAlgEquiv K α β) = | |
| (nonZeroDivisors (CoefficientPolynomial K β)).map (MvPolynomial.C (σ := α)) := by | |
| ext y | |
| constructor | |
| · rintro ⟨x, ⟨f, hf, rfl⟩, rfl⟩ | |
| exact ⟨f, hf, (sumAlgEquiv_coefficientRename K α β f).symm⟩ | |
| · rintro ⟨f, hf, rfl⟩ | |
| exact ⟨MvPolynomial.rename Sum.inr f, ⟨f, hf, rfl⟩, | |
| sumAlgEquiv_coefficientRename K α β f⟩ | |
| attribute [local instance] MvPolynomial.algebraMvPolynomial | |
| local instance coefficientPolynomial_scalarTower : | |
| IsScalarTower K (MvPolynomial α (CoefficientPolynomial K β)) | |
| (FractionCoefficientPolynomial K α β) := | |
| IsScalarTower.of_algebraMap_eq (fun c => by | |
| change MvPolynomial.C (algebraMap K (CoefficientFractionField K β) c) = | |
| MvPolynomial.map | |
| (algebraMap (CoefficientPolynomial K β) (CoefficientFractionField K β)) | |
| (MvPolynomial.C (MvPolynomial.C c)) | |
| rw [MvPolynomial.map_C] | |
| exact congrArg (MvPolynomial.C (σ := α)) | |
| (IsScalarTower.algebraMap_apply K (CoefficientPolynomial K β) | |
| (CoefficientFractionField K β) c)) | |
| def polynomialCoefficientLocalizationEquiv : | |
| PolynomialCoefficientLocalization K α β ≃ₐ[K] FractionCoefficientPolynomial K α β := | |
| IsLocalization.algEquivOfAlgEquiv | |
| (PolynomialCoefficientLocalization K α β) (FractionCoefficientPolynomial K α β) | |
| (MvPolynomial.sumAlgEquiv K α β) (sumAlgEquiv_coefficientDenominators K α β) | |
| @[simp] theorem polynomialCoefficientLocalizationEquiv_algebraMap | |
| (p : SplitPolynomial K α β) : | |
| polynomialCoefficientLocalizationEquiv K α β | |
| (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β) p) = | |
| MvPolynomial.map | |
| (algebraMap (CoefficientPolynomial K β) (CoefficientFractionField K β)) | |
| (MvPolynomial.sumAlgEquiv K α β p) := | |
| IsLocalization.algEquivOfAlgEquiv_eq (sumAlgEquiv_coefficientDenominators K α β) p | |
| @[simp] theorem polynomialCoefficientLocalizationEquiv_X_inl (i : α) : | |
| polynomialCoefficientLocalizationEquiv K α β | |
| (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β) | |
| (MvPolynomial.X (Sum.inl i))) = MvPolynomial.X i := by | |
| rw [polynomialCoefficientLocalizationEquiv_algebraMap] | |
| rw [MvPolynomial.sumAlgEquiv_X_inl, MvPolynomial.map_X] | |
| @[simp] theorem polynomialCoefficientLocalizationEquiv_X_inr (j : β) : | |
| polynomialCoefficientLocalizationEquiv K α β | |
| (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β) | |
| (MvPolynomial.X (Sum.inr j))) = | |
| MvPolynomial.C (algebraMap (CoefficientPolynomial K β) (CoefficientFractionField K β) | |
| (MvPolynomial.X j)) := by | |
| rw [polynomialCoefficientLocalizationEquiv_algebraMap] | |
| rw [MvPolynomial.sumAlgEquiv_X_inr, MvPolynomial.map_C] | |
| @[simp] theorem polynomialCoefficientLocalizationEquiv_coefficient | |
| (f : CoefficientPolynomial K β) : | |
| polynomialCoefficientLocalizationEquiv K α β | |
| (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β) | |
| (MvPolynomial.rename Sum.inr f)) = | |
| MvPolynomial.C (algebraMap (CoefficientPolynomial K β) (CoefficientFractionField K β) f) := by | |
| rw [polynomialCoefficientLocalizationEquiv_algebraMap, | |
| sumAlgEquiv_coefficientRename, MvPolynomial.map_C] | |
| theorem polynomialCoefficientLocalizationEquiv_mk' | |
| (p : SplitPolynomial K α β) (s : coefficientDenominators K α β) : | |
| polynomialCoefficientLocalizationEquiv K α β | |
| (IsLocalization.mk' (PolynomialCoefficientLocalization K α β) p s) = | |
| IsLocalization.mk' (FractionCoefficientPolynomial K α β) | |
| (MvPolynomial.sumAlgEquiv K α β p) | |
| ⟨MvPolynomial.sumAlgEquiv K α β s, | |
| (sumAlgEquiv_coefficientDenominators K α β) ▸ | |
| Submonoid.mem_map_of_mem (MvPolynomial.sumAlgEquiv K α β) s.property⟩ := | |
| IsLocalization.algEquivOfAlgEquiv_mk' (sumAlgEquiv_coefficientDenominators K α β) p s | |
| theorem coefficientDenominator_isUnit (f : CoefficientPolynomial K β) (hf : f ≠ 0) : | |
| IsUnit (algebraMap (SplitPolynomial K α β) (PolynomialCoefficientLocalization K α β) | |
| (MvPolynomial.rename Sum.inr f)) := by | |
| exact IsLocalization.map_units (PolynomialCoefficientLocalization K α β) | |
| ⟨MvPolynomial.rename Sum.inr f, (mem_coefficientDenominators K α β _).mpr ⟨f, hf, rfl⟩⟩ | |
| end PiExponentSiegelAux.W09 | |
| end | |
| end OAI | |