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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import Mathlib.RingTheory.Flat.FaithfullyFlat.Algebra
import Mathlib.RingTheory.Regular.Flat
import OAI.NumberTheory.PiExponent.LocalAlgebra.ConeEquationLength
namespace OAI
noncomputable section
attribute [local instance] Localization.AtPrime.algebraOfLiesOver
namespace PiExponentSiegel.W17.ConeLocalLength
open RingTheory.Sequence
section Units
variable {R τ M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
theorem ofList_map_mul_units (xs : List τ) (a u : τ → R) (hu : ∀ x, IsUnit (u x)) :
Ideal.ofList (xs.map (fun x => a x * u x)) = Ideal.ofList (xs.map a) := by
apply le_antisymm
· apply Ideal.span_le.mpr
intro r hr
obtain ⟨x, hx, rfl⟩ := List.mem_map.mp hr
exact Ideal.mul_mem_right _ _ (Ideal.subset_span (List.mem_map.mpr ⟨x, hx, rfl⟩))
· apply Ideal.span_le.mpr
intro r hr
obtain ⟨x, hx, rfl⟩ := List.mem_map.mp hr
apply (Ideal.mul_unit_mem_iff_mem _ (hu x)).mp
exact Ideal.subset_span (List.mem_map.mpr ⟨x, hx, rfl⟩)
theorem isRegular_map_mul_units (xs : List τ) (a u : τ → R) (hu : ∀ x, IsUnit (u x))
(hreg : IsRegular M (xs.map a)) :
IsRegular M (xs.map (fun x => a x * u x)) := by
refine ⟨⟨?_⟩, ?_⟩
· intro i hi
have hi' : i < xs.length := by simpa only [List.length_map] using hi
have ha := hreg.regular_mod_prev i (by simpa only [List.length_map] using hi')
simp only [List.getElem_map] at ha ⊢
have hI : Ideal.ofList ((xs.map (fun x => a x * u x)).take i) =
Ideal.ofList ((xs.map a).take i) := by
simp only [← List.map_take]
exact ofList_map_mul_units (xs.take i) a u hu
have hN : (Ideal.ofList ((xs.map (fun x => a x * u x)).take i) •
(⊤ : Submodule R M)) = Ideal.ofList ((xs.map a).take i) • ⊤ :=
congrArg (fun J : Ideal R => J • (⊤ : Submodule R M)) hI
exact Eq.mp (congrArg (fun N : Submodule R M =>
IsSMulRegular (M ⧸ N) (a xs[i] * u xs[i])) hN.symm)
(ha.mul ((hu xs[i]).isSMulRegular _))
· rw [ofList_map_mul_units xs a u hu]
exact hreg.top_ne_smul
end Units
variable (k σ : Type*) [CommRing k]
variable (P : Ideal (AffineRing k σ)) [P.IsPrime]
def coneLocalCoordinate : Localization.AtPrime (conePrime k σ P) :=
algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P)) (MvPolynomial.X none)
theorem coneLocalCoordinate_isUnit : IsUnit (coneLocalCoordinate k σ P) :=
IsLocalization.map_units (Localization.AtPrime (conePrime k σ P))
(⟨MvPolynomial.X none, homogenizing_coordinate_not_mem_conePrime k σ P⟩ :
(conePrime k σ P).primeCompl)
theorem homogenized_local_eq_affine_mul_coordinate (f : AffineRing k σ) :
algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P))
(PiExponentJets.W22.homogenize f) =
affineLocalToConeLocal k σ P (algebraMap (AffineRing k σ) (Localization.AtPrime P) f) *
coneLocalCoordinate k σ P ^ f.totalDegree := by
let e := conePrimeLocalizationEquiv k σ P
have ha : e (affineLocalToConeLocal k σ P
(algebraMap (AffineRing k σ) (Localization.AtPrime P) f)) =
algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)) (LaurentPolynomial.C f) := by
change e (e.symm (algebraMap (Localization.AtPrime P)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P))
(algebraMap (AffineRing k σ) (Localization.AtPrime P) f))) = _
rw [e.apply_symm_apply]
exact (IsScalarTower.algebraMap_apply (AffineRing k σ) (Localization.AtPrime P)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)) f).symm.trans
(IsScalarTower.algebraMap_apply (AffineRing k σ) (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)) f)
apply e.injective
rw [e.map_mul, e.map_pow, ha]
change conePrimeLocalizationEquiv k σ P
(algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P))
(PiExponentJets.W22.homogenize f)) =
algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)) (LaurentPolynomial.C f) *
conePrimeLocalizationEquiv k σ P
(algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P))
(MvPolynomial.X none)) ^ f.totalDegree
rw [conePrimeLocalizationEquiv_algebraMap, conePrimeLocalizationEquiv_algebraMap,
coneToLaurent_homogenize, coneToLaurent_X_none, map_mul, map_pow]
theorem affineLocalToConeLocal_faithfullyFlat :
letI := affineLocalConeAlgebra k σ P
Module.FaithfullyFlat (Localization.AtPrime P)
(Localization.AtPrime (conePrime k σ P)) := by
let := affineLocalConeAlgebra k σ P
let := affineLocalToConeLocal_flat k σ P
let : IsLocalHom (algebraMap (Localization.AtPrime P)
(Localization.AtPrime (conePrime k σ P))) :=
((IsLocalRing.local_hom_TFAE _).out 1 3).mpr
(affineLocalToConeLocal_map_maximalIdeal k σ P).le
exact Module.FaithfullyFlat.of_flat_of_isLocalHom
theorem homogenized_list_regular_at_cone (fs : List (AffineRing k σ))
(hreg : IsRegular (Localization.AtPrime P)
(fs.map (algebraMap (AffineRing k σ) (Localization.AtPrime P)))) :
IsRegular (Localization.AtPrime (conePrime k σ P))
((fs.map PiExponentJets.W22.homogenize).map
(algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P)))) := by
let := affineLocalConeAlgebra k σ P
let := affineLocalToConeLocal_faithfullyFlat k σ P
have hbase : IsRegular (Localization.AtPrime (conePrime k σ P))
(fs.map (fun f => affineLocalToConeLocal k σ P
(algebraMap (AffineRing k σ) (Localization.AtPrime P) f))) := by
have hb := hreg.of_faithfullyFlat (S := Localization.AtPrime (conePrime k σ P))
change IsRegular (Localization.AtPrime (conePrime k σ P))
((fs.map (algebraMap (AffineRing k σ) (Localization.AtPrime P))).map
(affineLocalToConeLocal k σ P)) at hb
simpa only [List.map_map, Function.comp_def] using hb
have hscaled := isRegular_map_mul_units fs
(fun f => affineLocalToConeLocal k σ P
(algebraMap (AffineRing k σ) (Localization.AtPrime P) f))
(fun f => coneLocalCoordinate k σ P ^ f.totalDegree)
(fun f => (coneLocalCoordinate_isUnit k σ P).pow f.totalDegree) hbase
simpa only [List.map_map, Function.comp_def,
homogenized_local_eq_affine_mul_coordinate] using hscaled
end PiExponentSiegel.W17.ConeLocalLength
end
end OAI
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