openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Polynomials /PolynomialMaximalHeight.lean
| 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 | |