File size: 2,767 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 68 69 70 | 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
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