openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /PolynomialLocalRegularParameters.lean
| import OAI.NumberTheory.PiExponent.LocalAlgebra.CoordinateBasisSplitting | |
| import OAI.NumberTheory.PiExponent.LocalAlgebra.TriangularLocalParametersTransport | |
| import OAI.NumberTheory.PiExponent.Polynomials.PolynomialMaximalHeight | |
| namespace OAI | |
| noncomputable section | |
| namespace PiExponentJets.PolynomialLocalResidueResolution | |
| variable (K : Type*) [Field K] (n : ℕ) | |
| variable (Q : Ideal (MvPolynomial (Fin n) K)) [Q.IsPrime] | |
| theorem exists_regularParameters_actual_height : | |
| ∃ h : ℕ, Q.height = (h : ℕ∞) ∧ | |
| ∃ xs : List (Localization.AtPrime Q), | |
| xs.length = h ∧ RingTheory.Sequence.IsRegular (Localization.AtPrime Q) xs ∧ | |
| Ideal.ofList xs = IsLocalRing.maximalIdeal (Localization.AtPrime Q) := by | |
| classical | |
| obtain ⟨t, j, htfinite, hj, hcoord, ht⟩ := exists_coordinate_basis_indices K n Q | |
| let A := RemainingCoordinate j | |
| let F := SplitCoefficientField K t | |
| let Qs := coordinateSplitPrime K Q t j hj | |
| let hB := coordinateSplitPrime_basis K Q t j hj hcoord ht | |
| let m₀ := closedPolynomialIdeal K A t Qs hB | |
| let : m₀.IsMaximal := closedPolynomialIdeal_isMaximal K A t Qs hB | |
| let e₀ : Localization.AtPrime Q ≃+* Localization.AtPrime m₀ := | |
| coordinateBasisClosedPointEquiv K Q t j hj hcoord ht | |
| let h := Fintype.card A | |
| let eFin : MvPolynomial A F ≃ₐ[F] MvPolynomial (Fin h) F := | |
| MvPolynomial.renameEquiv F (Fintype.equivFin A) | |
| let m : Ideal (MvPolynomial (Fin h) F) := m₀.map eFin.toRingHom | |
| let : m.IsMaximal := (inferInstance : m₀.IsMaximal).map_bijective eFin.toRingHom eFin.bijective | |
| let e : Localization.AtPrime Q ≃+* Localization.AtPrime m := | |
| e₀.trans (PiExponentSiegel.W23.atPrimeEquivOfRingEquiv eFin.toRingEquiv m₀) | |
| have hheight : Q.height = (h : ℕ∞) := by | |
| have he := ringKrullDim_eq_of_ringEquiv e | |
| rw [IsLocalization.AtPrime.ringKrullDim_eq_height Q (Localization.AtPrime Q), | |
| IsLocalization.AtPrime.ringKrullDim_eq_height m (Localization.AtPrime m), | |
| PiExponentSiegelAux.W09.polynomialMaximal_height F h m] at he | |
| exact WithBot.coe_injective he | |
| refine ⟨h, hheight, ?_⟩ | |
| exact PiExponentSiegel.W10.TriangularLocalParameters.polynomialMaximal_regularParameters_of_ringEquiv | |
| h F m e | |
| end PiExponentJets.PolynomialLocalResidueResolution | |
| end | |
| end OAI | |