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95 lines
3.7 KiB
Text
95 lines
3.7 KiB
Text
import CommAlg.krull
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import Mathlib.RingTheory.Ideal.Operations
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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.AlgebraicGeometry.PrimeSpectrum.Basic
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namespace Ideal
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private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
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unfold Set.chainHeight
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simp only [iSup_le_iff, Nat.cast_le_one]
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intro L h
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unfold Set.subchain at h
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simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at h
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rcases L with (_ | ⟨a,L⟩)
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. simp only [List.length_nil, zero_le]
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rcases L with (_ | ⟨b,L⟩)
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. simp only [List.length_singleton, le_refl]
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simp only [List.chain'_cons, List.find?, List.mem_cons, forall_eq_or_imp] at h
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rcases h with ⟨⟨h1, _⟩, ⟨rfl, rfl, _⟩⟩
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exact absurd h1 (lt_irrefl _)
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/-- The ring of polynomials over a field has dimension one. -/
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-- It's the exact same lemma as in krull.lean, added ' to avoid conflict
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lemma polynomial_over_field_dim_one' {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
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rw [le_antisymm_iff]
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let X := @Polynomial.X K _
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constructor
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· unfold krullDim
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apply @iSup_le (WithBot ℕ∞) _ _ _ _
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intro I
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have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
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by_cases I = ⊥
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· rw [← height_bot_iff_bot] at h
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simp only [WithBot.coe_le_one, ge_iff_le]
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rw [h]
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exact bot_le
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· push_neg at h
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have : I.asIdeal ≠ ⊥ := by
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by_contra a
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have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
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contradiction
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have maxI := IsPrime.to_maximal_ideal this
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have sngletn : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
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intro P
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constructor
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· intro H
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simp only [Set.mem_setOf_eq] at H
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by_contra x
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push_neg at x
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have : P.asIdeal ≠ ⊥ := by
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by_contra a
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have : P = ⊥ := PrimeSpectrum.ext P ⊥ a
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contradiction
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have maxP := IsPrime.to_maximal_ideal this
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have IneTop := IsMaximal.ne_top maxI
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have : P ≤ I := le_of_lt H
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rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this
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have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this
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have : P = I := PrimeSpectrum.ext P I this
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replace H : P ≠ I := ne_of_lt H
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contradiction
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· intro pBot
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simp only [Set.mem_setOf_eq, pBot]
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exact lt_of_le_of_ne bot_le h.symm
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replace sngletn : {J | J < I} = {⊥} := Set.ext sngletn
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unfold height
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rw [sngletn]
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simp only [WithBot.coe_le_one, ge_iff_le]
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exact singleton_bot_chainHeight_one
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· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
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· obtain ⟨I, h⟩ := this
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have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by
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apply @le_iSup (WithBot ℕ∞) _ _ _ I
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exact le_trans h this
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have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _
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have : IsPrime (span {X}) := by
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refine (span_singleton_prime ?hp).mpr primeX
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exact Polynomial.X_ne_zero
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let P := PrimeSpectrum.mk (span {X}) this
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unfold height
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use P
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have : ⊥ ∈ {J | J < P} := by
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simp only [Set.mem_setOf_eq, bot_lt_iff_ne_bot]
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suffices : P.asIdeal ≠ ⊥
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· by_contra x
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rw [PrimeSpectrum.ext_iff] at x
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contradiction
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by_contra x
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simp only [span_singleton_eq_bot] at x
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have := @Polynomial.X_ne_zero K _ _
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contradiction
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have : {J | J < P}.Nonempty := Set.nonempty_of_mem this
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rwa [←Set.one_le_chainHeight_iff, ←WithBot.one_le_coe] at this
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