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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 | 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
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