import CommAlg.krull import Mathlib.RingTheory.Ideal.Operations import Mathlib.Order.Height import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.RingTheory.DedekindDomain.Basic import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic namespace Ideal /-- The ring of polynomials over a field has dimension one. -/ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by rw [le_antisymm_iff] let X := @Polynomial.X K _ constructor · unfold krullDim apply @iSup_le (WithBot ℕ∞) _ _ _ _ intro I have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance by_cases I = ⊥ · rw [← height_bot_iff_bot] at h simp only [WithBot.coe_le_one, ge_iff_le] rw [h] exact bot_le · push_neg at h have : I.asIdeal ≠ ⊥ := by by_contra a have : I = ⊥ := PrimeSpectrum.ext I ⊥ a contradiction have maxI := IsPrime.to_maximal_ideal this have singleton : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by intro P constructor · intro H simp only [Set.mem_setOf_eq] at H by_contra x push_neg at x have : P.asIdeal ≠ ⊥ := by by_contra a have : P = ⊥ := PrimeSpectrum.ext P ⊥ a contradiction have maxP := IsPrime.to_maximal_ideal this have IneTop := IsMaximal.ne_top maxI have : P ≤ I := le_of_lt H rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this have : P = I := PrimeSpectrum.ext P I this replace H : P ≠ I := ne_of_lt H contradiction · intro pBot simp only [Set.mem_setOf_eq, pBot] exact lt_of_le_of_ne bot_le h.symm replace singleton : {J | J < I} = {⊥} := Set.ext singleton unfold height sorry · suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞) · obtain ⟨I, h⟩ := this have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by apply @le_iSup (WithBot ℕ∞) _ _ _ I exact le_trans h this have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _ have : IsPrime (span {X}) := by refine (span_singleton_prime ?hp).mpr primeX exact Polynomial.X_ne_zero let P := PrimeSpectrum.mk (span {X}) this unfold height use P have : ⊥ ∈ {J | J < P} := by simp only [Set.mem_setOf_eq, bot_lt_iff_ne_bot] suffices : P.asIdeal ≠ ⊥ · by_contra x rw [PrimeSpectrum.ext_iff] at x contradiction by_contra x simp only [span_singleton_eq_bot] at x have := @Polynomial.X_ne_zero K _ _ contradiction have : {J | J < P}.Nonempty := Set.nonempty_of_mem this rwa [←Set.one_le_chainHeight_iff, ←WithBot.one_le_coe] at this