| 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 | |