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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 | 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
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