File size: 2,530 Bytes
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 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | 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
|