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82 lines
3.2 KiB
Text
82 lines
3.2 KiB
Text
import Mathlib.RingTheory.Ideal.Basic
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import Mathlib.Order.Height
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import Mathlib.RingTheory.PrincipalIdealDomain
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import Mathlib.RingTheory.DedekindDomain.Basic
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import Mathlib.RingTheory.Ideal.Quotient
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import Mathlib.RingTheory.Localization.AtPrime
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import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.Order.ConditionallyCompleteLattice.Basic
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namespace Ideal
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variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
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noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
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noncomputable def krullDim (R : Type) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
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lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
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lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
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lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
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@[simp]
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lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
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constructor
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· intro primeP
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obtain T := eq_bot_or_top P
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have : ¬P = ⊤ := IsPrime.ne_top primeP
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tauto
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· intro botP
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rw [botP]
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exact bot_prime
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lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
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unfold height
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simp
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by_contra spec
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change _ ≠ _ at spec
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rw [← Set.nonempty_iff_ne_empty] at spec
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obtain ⟨J, JlP : J < P⟩ := spec
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have P0 : IsPrime P.asIdeal := P.IsPrime
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have J0 : IsPrime J.asIdeal := J.IsPrime
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rw [field_prime_bot] at P0 J0
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have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0)
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have : J = P := PrimeSpectrum.ext J P this
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have : J ≠ P := ne_of_lt JlP
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contradiction
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lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
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unfold krullDim
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simp [field_prime_height_zero]
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noncomputable
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instance : CompleteLattice (WithBot ℕ∞) :=
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inferInstanceAs <| CompleteLattice (WithBot (WithTop ℕ))
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lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
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unfold krullDim at h
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simp [height] at h
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by_contra x
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rw [Ring.not_isField_iff_exists_prime] at x
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obtain ⟨P, ⟨h1, primeP⟩⟩ := x
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let P' : PrimeSpectrum D := PrimeSpectrum.mk P primeP
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have h2 : P' ≠ ⊥ := by
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by_contra a
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have : P = ⊥ := by rwa [PrimeSpectrum.ext_iff] at a
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contradiction
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have PgtBot : P' > ⊥ := Ne.bot_lt h2
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have pos_height : ¬ ↑(Set.chainHeight {J | J < P'}) ≤ 0 := by
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have : ⊥ ∈ {J | J < P'} := PgtBot
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have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
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rw [←Set.one_le_chainHeight_iff] at this
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exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this)
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have nonpos_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
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have : (⨆ (I : PrimeSpectrum D), (Set.chainHeight {J | J < I} : WithBot ℕ∞)) ≤ 0 := h.le
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rw [iSup_le_iff] at this
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exact Iff.mp WithBot.coe_le_zero (this P')
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contradiction
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lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
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constructor
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· exact isField.dim_zero
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· intro fieldD
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let h : Field D := IsField.toField fieldD
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exact dim_field_eq_zero
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