openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /LocalAlgebra /AffinePrimeDimensionBridge.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.AffineResidueTranscendence | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.ClosedPointLocalDimension | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentJets.W24 | |
| theorem affine_prime_dimension_of_closedPoint_presentation | |
| (k B β C : Type*) [Field k] [CommRing B] [IsDomain B] | |
| [Algebra k B] [Algebra.FiniteType k B] [Finite β] | |
| [CommRing C] [IsDomain C] [Algebra k C] [Algebra B C] | |
| [Algebra (FractionRing (MvPolynomial β k)) C] | |
| [IsScalarTower k B C] | |
| [IsScalarTower k (FractionRing (MvPolynomial β k)) C] | |
| [FaithfulSMul B C] | |
| [Algebra.FiniteType (FractionRing (MvPolynomial β k)) C] | |
| (q : Ideal B) [q.IsPrime] | |
| (b : β → q.ResidueField) (hb : IsTranscendenceBasis k b) | |
| (M : Submonoid B) [IsLocalization M C] | |
| (m : Ideal C) [m.IsMaximal] | |
| (e : Localization.AtPrime q ≃+* Localization.AtPrime m) : | |
| (q.height : WithBot ℕ∞) + ringKrullDim (B ⧸ q) = ringKrullDim B := by | |
| have he : (q.height : WithBot ℕ∞) = ringKrullDim C := by | |
| rw [← IsLocalization.AtPrime.ringKrullDim_eq_height q (Localization.AtPrime q), | |
| e.ringKrullDim, W29.closedPoint_local_dimension (FractionRing (MvPolynomial β k)) C m] | |
| have hd := coefficient_localization_dimension k (FractionRing (MvPolynomial β k)) B C M | |
| (rational_coefficient_trdeg_lt_aleph0 k β) | |
| rw [coefficient_trdeg_eq_prime_quotient_dimension k B β q b hb] at hd | |
| exact he.symm ▸ hd | |
| end PiExponentJets.W24 | |
| end | |
| end OAI | |