import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTheory.Jacobson.Ring
import Mathlib.RingTheory.KrullDimension.Polynomial

namespace OAI

noncomputable section
namespace PiExponentSiegelAux.W09

variable (K : Type*) [Field K]

theorem polynomialMaximal_height (n : ℕ)
    (m : Ideal (MvPolynomial (Fin n) K)) [m.IsMaximal] :
    m.height = (n : ℕ∞) := by
  induction n with
  | zero =>
      let e := MvPolynomial.isEmptyAlgEquiv K (Fin 0)
      let p : Ideal K := m.comap e.symm.toRingHom
      let : p.IsMaximal := Ideal.comap_isMaximal_of_equiv e.symm
      have hp : p = ⊥ := p.eq_bot_or_top.resolve_right (Ideal.IsMaximal.ne_top (inferInstance : p.IsMaximal))
      calc
        m.height = p.height := (e.symm.toRingEquiv.height_comap m).symm
        _ = (0 : ℕ∞) := by rw [hp, Ideal.height_bot]
  | succ n ih =>
      let e := MvPolynomial.finSuccEquiv K n
      let P : Ideal (Polynomial (MvPolynomial (Fin n) K)) :=
        m.comap e.symm.toRingHom
      let : P.IsMaximal := Ideal.comap_isMaximal_of_equiv e.symm
      let p : Ideal (MvPolynomial (Fin n) K) := P.comap Polynomial.C
      let : p.IsMaximal := Polynomial.isMaximal_comap_C_of_isJacobsonRing P
      let : P.LiesOver p := ⟨rfl⟩
      calc
        m.height = P.height := (e.symm.toRingEquiv.height_comap m).symm
        _ = p.height + 1 := Polynomial.height_eq_height_add_one p P
        _ = ((n + 1 : ℕ) : ℕ∞) := by rw [ih p]; simp

theorem maximal_height_of_polynomial_equiv {A : Type*} [CommRing A]
    (n : ℕ) (e : A ≃+* MvPolynomial (Fin n) K)
    (m : Ideal A) [m.IsMaximal] : m.height = (n : ℕ∞) := by
  let : (m.map e).IsMaximal := Ideal.map_isMaximal_of_equiv e
  calc
    m.height = (m.map e).height := (e.height_map m).symm
    _ = (n : ℕ∞) := polynomialMaximal_height K n (m.map e)

theorem localized_maximal_comap_height {R A : Type*} [CommRing R] [CommRing A]
    [Algebra R A] (S : Submonoid R) [IsLocalization S A]
    (n : ℕ) (e : A ≃+* MvPolynomial (Fin n) K)
    (m : Ideal A) [m.IsMaximal] :
    (m.comap (algebraMap R A)).height = (n : ℕ∞) := by
  rw [IsLocalization.height_under S m]
  exact maximal_height_of_polynomial_equiv K n e m

theorem prime_height_of_localized_polynomial_maximal {R A : Type*}
    [CommRing R] [CommRing A] [Algebra R A]
    (S : Submonoid R) [IsLocalization S A]
    (n : ℕ) (e : A ≃+* MvPolynomial (Fin n) K)
    (p : Ideal R) [p.IsPrime]
    (hdisj : Disjoint (S : Set R) (p : Set R))
    [(p.map (algebraMap R A)).IsMaximal] : p.height = (n : ℕ∞) := by
  calc
    p.height = (p.map (algebraMap R A)).height :=
      (IsLocalization.height_map_of_disjoint S p hdisj).symm
    _ = (n : ℕ∞) := maximal_height_of_polynomial_equiv K n e _

end PiExponentSiegelAux.W09

end

end OAI