openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /ClosedPointLocalDimension.lean
| import Mathlib.RingTheory.IntegralClosure.GoingDown | |
| import Mathlib.RingTheory.Polynomial.RationalRoot | |
| import Mathlib.RingTheory.Polynomial.UniqueFactorization | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.PrimeHilbertDimension | |
| import OAI.NumberTheory.PiExponent.Polynomials.PolynomialMaximalHeight | |
| namespace OAI | |
| namespace PiExponentJets.W29 | |
| section GoingDownHeight | |
| variable {A B : Type*} [CommRing A] [CommRing B] [Algebra A B] | |
| [Algebra.HasGoingDown A B] | |
| theorem under_height_le_of_goingDown (P : Ideal B) [P.IsPrime] : | |
| (P.under A).height ≤ P.height := by | |
| rw [PrimeSpectrum.height_eq_orderHeight (⟨P.under A, inferInstance⟩ : PrimeSpectrum A), | |
| PrimeSpectrum.height_eq_orderHeight (⟨P, inferInstance⟩ : PrimeSpectrum B)] | |
| apply Order.height_le | |
| intro l hl | |
| let : P.LiesOver l.last.asIdeal := ⟨congrArg PrimeSpectrum.asIdeal hl⟩ | |
| obtain ⟨L, hlen, hlast, _⟩ := Ideal.exists_ltSeries_of_hasGoingDown l P | |
| have h : (L.length : ℕ∞) ≤ Order.height (⟨P, inferInstance⟩ : PrimeSpectrum B) := | |
| Order.length_le_height (le_of_eq hlast) | |
| simpa only [hlen] using h | |
| end GoingDownHeight | |
| variable (k B : Type*) [Field k] [CommRing B] [IsDomain B] | |
| [Algebra k B] [Algebra.FiniteType k B] | |
| include k | |
| theorem affineDomain_maximal_height (m : Ideal B) [m.IsMaximal] : | |
| (m.height : WithBot ℕ∞) = ringKrullDim B := by | |
| obtain ⟨d, g, hg, hfin, hdim⟩ := W24.exists_finite_normalization_with_dimension k B | |
| let A := MvPolynomial (Fin d) k | |
| let : Algebra A B := g.toRingHom.toAlgebra | |
| have : Algebra.IsIntegral A B := ⟨hfin.to_isIntegral⟩ | |
| have : FaithfulSMul A B := (faithfulSMul_iff_algebraMap_injective A B).mpr hg | |
| have : IsIntegrallyClosed A := inferInstance | |
| have : Algebra.HasGoingDown A B := inferInstance | |
| let p : Ideal A := m.under A | |
| have : p.IsMaximal := by | |
| dsimp [p] | |
| infer_instance | |
| have hp : p.height = (d : ℕ∞) := | |
| PiExponentSiegelAux.W09.polynomialMaximal_height k d p | |
| have hlo : (d : ℕ∞) ≤ m.height := by | |
| rw [← hp] | |
| exact under_height_le_of_goingDown m | |
| apply le_antisymm | |
| · exact Ideal.height_le_ringKrullDim_of_ne_top | |
| (Ideal.IsMaximal.ne_top (inferInstance : m.IsMaximal)) | |
| · rw [hdim] | |
| exact WithBot.coe_le_coe.mpr hlo | |
| theorem closedPoint_local_dimension (m : Ideal B) [m.IsMaximal] : | |
| ringKrullDim (Localization.AtPrime m) = ringKrullDim B := by | |
| rw [IsLocalization.AtPrime.ringKrullDim_eq_height m (Localization.AtPrime m)] | |
| exact affineDomain_maximal_height k B m | |
| end PiExponentJets.W29 | |
| end OAI | |