mirror of
https://github.com/GTBarkley/comm_alg.git
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369 lines
No EOL
14 KiB
Text
369 lines
No EOL
14 KiB
Text
import Mathlib.RingTheory.Ideal.Operations
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import Mathlib.RingTheory.FiniteType
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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.Ideal.MinimalPrime
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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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/- This file contains the definitions of height of an ideal, and the krull
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dimension of a commutative ring.
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There are also sorried statements of many of the theorems that would be
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really nice to prove.
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I'm imagining for this file to ultimately contain basic API for height and
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krull dimension, and the theorems will probably end up other files,
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depending on how long the proofs are, and what extra API needs to be
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developed.
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-/
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lemma lt_bot_eq_WithBot_bot [PartialOrder α] [OrderBot α] {a : WithBot α} (h : a < (⊥ : α)) : a = ⊥ := by
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cases a
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. rfl
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. cases h.not_le (WithBot.coe_le_coe.2 bot_le)
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namespace Ideal
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open LocalRing
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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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noncomputable def codimension (J : Ideal R) : WithBot ℕ∞ := ⨅ I ∈ {I : PrimeSpectrum R | J ≤ I.asIdeal}, 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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noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice
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lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := by
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apply Set.chainHeight_mono
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intro J' hJ'
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show J' < J
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exact lt_of_lt_of_le hJ' I_le_J
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lemma krullDim_le_iff (R : Type _) [CommRing R] (n : ℕ) :
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krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
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lemma krullDim_le_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
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krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
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@[simp]
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lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
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le_iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) I
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lemma le_krullDim_iff (R : Type _) [CommRing R] (n : ℕ) :
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n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by
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constructor
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· unfold krullDim
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intro H
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by_contra H1
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push_neg at H1
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by_cases n ≤ 0
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· rw [Nat.le_zero] at h
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rw [h] at H H1
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have : ∀ (I : PrimeSpectrum R), ↑(height I) = (⊥ : WithBot ℕ∞) := by
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intro I
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specialize H1 I
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exact lt_bot_eq_WithBot_bot H1
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rw [←iSup_eq_bot] at this
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have := le_of_le_of_eq H this
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rw [le_bot_iff] at this
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exact WithBot.coe_ne_bot this
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· push_neg at h
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have : (n: ℕ∞) > 0 := Nat.cast_pos.mpr h
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replace H1 : ∀ (I : PrimeSpectrum R), height I ≤ n - 1 := by
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intro I
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specialize H1 I
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apply ENat.le_of_lt_add_one
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rw [←ENat.coe_one, ←ENat.coe_sub, ←ENat.coe_add, tsub_add_cancel_of_le]
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exact WithBot.coe_lt_coe.mp H1
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exact h
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replace H1 : ∀ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) ≤ ↑(n - 1):=
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fun _ ↦ (WithBot.coe_le rfl).mpr (H1 _)
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rw [←iSup_le_iff] at H1
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have : ((n : ℕ∞) : WithBot ℕ∞) ≤ (((n - 1 : ℕ) : ℕ∞) : WithBot ℕ∞) := le_trans H H1
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norm_cast at this
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have that : n - 1 < n := by refine Nat.sub_lt h (by norm_num)
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apply lt_irrefl (n-1) (trans that this)
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· rintro ⟨I, h⟩
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have : height I ≤ krullDim R := by apply height_le_krullDim
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exact le_trans h this
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#check ENat.recTopCoe
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/- terrible place for this lemma. Also this probably exists somewhere
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Also this is a terrible proof
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-/
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lemma eq_top_iff (n : WithBot ℕ∞) : n = ⊤ ↔ ∀ m : ℕ, m ≤ n := by
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aesop
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induction' n using WithBot.recBotCoe with n
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. exfalso
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have := (a 0)
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simp [not_lt_of_ge] at this
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induction' n using ENat.recTopCoe with n
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. rfl
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. have := a (n + 1)
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exfalso
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change (((n + 1) : ℕ∞) : WithBot ℕ∞) ≤ _ at this
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simp [WithBot.coe_le_coe] at this
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change ((n + 1) : ℕ∞) ≤ (n : ℕ∞) at this
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simp [ENat.add_one_le_iff] at this
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lemma krullDim_eq_top_iff (R : Type _) [CommRing R] :
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krullDim R = ⊤ ↔ ∀ (n : ℕ), ∃ I : PrimeSpectrum R, n ≤ height I := by
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simp [eq_top_iff, le_krullDim_iff]
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change (∀ (m : ℕ), ∃ I, ((m : ℕ∞) : WithBot ℕ∞) ≤ height I) ↔ _
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simp [WithBot.coe_le_coe]
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/-- The Krull dimension of a local ring is the height of its maximal ideal. -/
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lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) := by
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apply le_antisymm
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. rw [krullDim_le_iff']
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intro I
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apply WithBot.coe_mono
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apply height_le_of_le
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apply le_maximalIdeal
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exact I.2.1
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. simp only [height_le_krullDim]
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/-- The height of a prime `𝔭` is greater than `n` if and only if there is a chain of primes less than `𝔭`
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with length `n + 1`. -/
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lemma lt_height_iff' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
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n < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
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match n with
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| ⊤ =>
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constructor <;> intro h <;> exfalso
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. exact (not_le.mpr h) le_top
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. tauto
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| (n : ℕ) =>
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have (m : ℕ∞) : n < m ↔ (n + 1 : ℕ∞) ≤ m := by
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symm
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show (n + 1 ≤ m ↔ _ )
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apply ENat.add_one_le_iff
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exact ENat.coe_ne_top _
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rw [this]
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unfold Ideal.height
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show ((↑(n + 1):ℕ∞) ≤ _) ↔ ∃c, _ ∧ _ ∧ ((_ : WithTop ℕ) = (_:ℕ∞))
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rw [{J | J < 𝔭}.le_chainHeight_iff]
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show (∃ c, (List.Chain' _ c ∧ ∀𝔮, 𝔮 ∈ c → 𝔮 < 𝔭) ∧ _) ↔ _
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constructor <;> rintro ⟨c, hc⟩ <;> use c
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. tauto
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. change _ ∧ _ ∧ (List.length c : ℕ∞) = n + 1 at hc
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norm_cast at hc
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tauto
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/-- Form of `lt_height_iff''` for rewriting with the height coerced to `WithBot ℕ∞`. -/
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lemma lt_height_iff'' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
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(n : WithBot ℕ∞) < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
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rw [WithBot.coe_lt_coe]
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exact lt_height_iff'
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#check height_le_krullDim
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--some propositions that would be nice to be able to eventually
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/-- The prime spectrum of the zero ring is empty. -/
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lemma primeSpectrum_empty_of_subsingleton (x : PrimeSpectrum R) [Subsingleton R] : False :=
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x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem
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/-- A CommRing has empty prime spectrum if and only if it is the zero ring. -/
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lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by
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constructor
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. contrapose
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rw [not_isEmpty_iff, ←not_nontrivial_iff_subsingleton, not_not]
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apply PrimeSpectrum.instNonemptyPrimeSpectrum
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. intro h
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by_contra hneg
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rw [not_isEmpty_iff] at hneg
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rcases hneg with ⟨a, ha⟩
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exact primeSpectrum_empty_of_subsingleton ⟨a, ha⟩
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/-- A ring has Krull dimension -∞ if and only if it is the zero ring -/
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lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
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unfold Ideal.krullDim
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rw [←primeSpectrum_empty_iff, iSup_eq_bot]
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constructor <;> intro h
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. rw [←not_nonempty_iff]
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rintro ⟨a, ha⟩
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specialize h ⟨a, ha⟩
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tauto
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. rw [h.forall_iff]
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trivial
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/-- Nonzero rings have Krull dimension in ℕ∞ -/
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lemma krullDim_nonneg_of_nontrivial (R : Type _) [CommRing R] [Nontrivial R] : ∃ n : ℕ∞, Ideal.krullDim R = n := by
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have h := dim_eq_bot_iff.not.mpr (not_subsingleton R)
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lift (Ideal.krullDim R) to ℕ∞ using h with k
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use k
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/-- An ideal which is less than a prime is not a maximal ideal. -/
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lemma not_maximal_of_lt_prime {p : Ideal R} {q : Ideal R} (hq : IsPrime q) (h : p < q) : ¬IsMaximal p := by
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intro hp
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apply IsPrime.ne_top hq
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apply (IsCoatom.lt_iff hp.out).mp
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exact h
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/-- Krull dimension is ≤ 0 if and only if all primes are maximal. -/
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lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
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show ((_ : WithBot ℕ∞) ≤ (0 : ℕ)) ↔ _
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rw [krullDim_le_iff R 0]
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constructor <;> intro h I
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. contrapose! h
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have ⟨𝔪, h𝔪⟩ := I.asIdeal.exists_le_maximal (IsPrime.ne_top I.IsPrime)
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let 𝔪p := (⟨𝔪, IsMaximal.isPrime h𝔪.1⟩ : PrimeSpectrum R)
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have hstrct : I < 𝔪p := by
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apply lt_of_le_of_ne h𝔪.2
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intro hcontr
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rw [hcontr] at h
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exact h h𝔪.1
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use 𝔪p
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show (0 : ℕ∞) < (_ : WithBot ℕ∞)
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rw [lt_height_iff'']
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use [I]
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constructor
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. exact List.chain'_singleton _
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. constructor
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. intro I' hI'
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simp at hI'
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rwa [hI']
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. simp
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. contrapose! h
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change (0 : ℕ∞) < (_ : WithBot ℕ∞) at h
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rw [lt_height_iff''] at h
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obtain ⟨c, _, hc2, hc3⟩ := h
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norm_cast at hc3
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rw [List.length_eq_one] at hc3
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obtain ⟨𝔮, h𝔮⟩ := hc3
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use 𝔮
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specialize hc2 𝔮 (h𝔮 ▸ (List.mem_singleton.mpr rfl))
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apply not_maximal_of_lt_prime I.IsPrime
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exact hc2
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/-- For a nonzero ring, Krull dimension is 0 if and only if all primes are maximal. -/
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lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
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rw [←dim_le_zero_iff]
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obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial R
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have : n ≥ 0 := zero_le n
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change _ ≤ _ at this
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rw [←WithBot.coe_le_coe,←hn] at this
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change (0 : WithBot ℕ∞) ≤ _ at this
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constructor <;> intro h'
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. rw [h']
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. exact le_antisymm h' this
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/-- In a field, the unique prime ideal is the zero ideal. -/
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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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/-- In a field, all primes have height 0. -/
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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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/-- The Krull dimension of a field is 0. -/
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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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/-- A domain with Krull dimension 0 is a field. -/
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lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
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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 pos_height : ¬ (height P') ≤ 0 := by
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have : ⊥ ∈ {J | J < P'} := Ne.bot_lt h2
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have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
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unfold height
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rw [←Set.one_le_chainHeight_iff] at this
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exact not_le_of_gt (ENat.one_le_iff_pos.mp this)
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have nonpos_height : height P' ≤ 0 := by
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have := height_le_krullDim P'
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rw [h] at this
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aesop
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contradiction
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/-- A domain has Krull dimension 0 if and only if it is a field. -/
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lemma domain_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 domain_dim_zero.isField
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· intro fieldD
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let h : Field D := fieldD.toField
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exact dim_field_eq_zero
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#check Ring.DimensionLEOne
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-- This lemma is false!
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lemma dim_le_one_iff : krullDim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry
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/-- The converse of this is false, because the definition of "dimension ≤ 1" in mathlib
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applies only to dimension zero rings and domains of dimension 1. -/
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lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ 1 := by
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show _ → ((_ : WithBot ℕ∞) ≤ (1 : ℕ))
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rw [krullDim_le_iff R 1]
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intro H p
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apply le_of_not_gt
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intro h
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rcases (lt_height_iff''.mp h) with ⟨c, ⟨hc1, hc2, hc3⟩⟩
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norm_cast at hc3
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rw [List.chain'_iff_get] at hc1
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specialize hc1 0 (by rw [hc3]; simp)
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set q0 : PrimeSpectrum R := List.get c { val := 0, isLt := _ }
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set q1 : PrimeSpectrum R := List.get c { val := 1, isLt := _ }
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specialize hc2 q1 (List.get_mem _ _ _)
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change q0.asIdeal < q1.asIdeal at hc1
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have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1
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specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime
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exact (not_maximal_of_lt_prime p.IsPrime hc2) H
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/-- The Krull dimension of a PID is at most 1. -/
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lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by
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rw [dim_le_one_iff]
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exact Ring.DimensionLEOne.principal_ideal_ring R
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lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
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krullDim R + 1 ≤ krullDim (Polynomial R) := sorry
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-- lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
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-- krullDim R + 1 = krullDim (Polynomial R) := sorry
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lemma krull_height_theorem [Nontrivial R] [IsNoetherianRing R] (P: PrimeSpectrum R) (S: Finset R)
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(h: P.asIdeal ∈ Ideal.minimalPrimes (Ideal.span S)) : height P ≤ S.card := by
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sorry
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lemma dim_mvPolynomial [Field K] (n : ℕ) : krullDim (MvPolynomial (Fin n) K) = n := sorry
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lemma height_eq_dim_localization :
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height I = krullDim (Localization.AtPrime I.asIdeal) := sorry
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lemma dim_quotient_le_dim : height I + krullDim (R ⧸ I.asIdeal) ≤ krullDim R := sorry
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lemma height_add_dim_quotient_le_dim : height I + krullDim (R ⧸ I.asIdeal) ≤ krullDim R := sorry |