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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 | import OAI.NumberTheory.PiExponent.Polynomials.ConeHomogeneousEquations
namespace OAI
noncomputable section
attribute [local instance] Localization.AtPrime.algebraOfLiesOver
namespace PiExponentSiegel.W17.ConeLocalLength
variable (k σ : Type*) [CommRing k]
variable (P : Ideal (AffineRing k σ)) [P.IsPrime]
theorem conePrimeLocalizationEquiv_algebraMap (F : ConeRing k σ) :
conePrimeLocalizationEquiv k σ P
(algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P)) F) =
algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)) (coneToLaurent k σ F) := by
let a := IsLocalization.localizationLocalizationAtPrimeIsoLocalization
(Submonoid.powers (MvPolynomial.X none : ConeRing k σ)) (awayChartPrime k σ P)
let b := PiExponentJets.W22.primeLocalizationEquiv (coneLaurentEquiv k σ)
(awayChartPrime k σ P) (laurentPrime (AffineRing k σ) P) rfl
change b (a (algebraMap (ConeRing k σ) _ F)) = _
rw [a.commutes F]
change b (algebraMap (ConeAway k σ) _ (algebraMap (ConeRing k σ) (ConeAway k σ) F)) = _
rw [show b (algebraMap (ConeAway k σ) _
(algebraMap (ConeRing k σ) (ConeAway k σ) F)) =
algebraMap (LaurentChart k σ) _ ((coneLaurentEquiv k σ)
(algebraMap (ConeRing k σ) (ConeAway k σ) F)) from
IsLocalization.ringEquivOfRingEquiv_eq
(PiExponentJets.W22.primeCompl_map_eq_of_comap (coneLaurentEquiv k σ)
(awayChartPrime k σ P) (laurentPrime (AffineRing k σ) P) rfl) _]
exact congrArg (algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)))
(coneAwayToLaurent_algebraMap k σ F)
theorem conePrimeLocalizationEquiv_map_ideal (J : Ideal (ConeRing k σ)) :
(J.map (algebraMap (ConeRing k σ)
(Localization.AtPrime (conePrime k σ P)))).map
(conePrimeLocalizationEquiv k σ P).toRingHom =
(J.map (coneToLaurent k σ)).map (algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P))) := by
rw [Ideal.map_map, Ideal.map_map]
congr 1
apply RingHom.ext
intro F
exact conePrimeLocalizationEquiv_algebraMap k σ P F
theorem affine_ideal_map_to_laurent_local (J : Ideal (AffineRing k σ)) :
(J.map (algebraMap (AffineRing k σ) (Localization.AtPrime P))).map
(algebraMap (Localization.AtPrime P)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P))) =
(J.map (LaurentPolynomial.C : AffineRing k σ →+* LaurentChart k σ)).map
(algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P))) := by
rw [Ideal.map_map, Ideal.map_map]
congr 1
apply RingHom.ext
intro x
rw [RingHom.comp_apply, RingHom.comp_apply, ← IsScalarTower.algebraMap_apply]
rw [LaurentPolynomial.C_eq_algebraMap, ← IsScalarTower.algebraMap_apply]
theorem homogenized_cone_local_length_eq {ι : Type*} (f : ι → AffineRing k σ)
(hfinite : IsFiniteLength (Localization.AtPrime P)
(Localization.AtPrime P ⧸ (Ideal.span (Set.range f)).map
(algebraMap (AffineRing k σ) (Localization.AtPrime P)))) :
Module.length (Localization.AtPrime (conePrime k σ P))
(Localization.AtPrime (conePrime k σ P) ⧸
(Ideal.span (Set.range (fun i => PiExponentJets.W22.homogenize (f i)))).map
(algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P)))) =
Module.length (Localization.AtPrime P)
(Localization.AtPrime P ⧸ (Ideal.span (Set.range f)).map
(algebraMap (AffineRing k σ) (Localization.AtPrime P))) := by
let J := Ideal.span (Set.range f)
let H := Ideal.span (Set.range (fun i => PiExponentJets.W22.homogenize (f i)))
have hid := conePrimeLocalizationEquiv_map_ideal k σ P H
rw [map_homogenized_span_eq_laurent_span k σ f] at hid
have hlength := PiExponentJets.W22.quotient_length_eq_of_ringEquiv
(conePrimeLocalizationEquiv k σ P)
(H.map (algebraMap (ConeRing k σ) (Localization.AtPrime (conePrime k σ P))))
((J.map (LaurentPolynomial.C : AffineRing k σ →+* LaurentChart k σ)).map
(algebraMap (LaurentChart k σ)
(Localization.AtPrime (laurentPrime (AffineRing k σ) P)))) hid.symm
apply hlength.trans
have h := laurentLocal_quotient_length (AffineRing k σ) P
(J.map (algebraMap (AffineRing k σ) (Localization.AtPrime P))) hfinite
rw [affine_ideal_map_to_laurent_local k σ P J] at h
exact h
end PiExponentSiegel.W17.ConeLocalLength
end
end OAI
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