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67 lines
2.1 KiB
Text
67 lines
2.1 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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-- import Mathlib.Data.ENat.Lattice
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-- import Mathlib.Order.OrderIsoNat
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-- import Mathlib.Tactic.TFAE
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namespace Ideal
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-- def foo : List Nat := [1, 2, 3, 4, 5]
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-- #check List.Chain'
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-- example : List.Chain' (· < ·) foo := by
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-- repeat { constructor; norm_num }
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example (x : Nat) : List.Chain' (· < ·) [x] := by
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constructor
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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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variable {K : Type _} [Field K]
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lemma dim_field_eq_zero : krullDim K = 0 := by
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have prime_bot (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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unfold krullDim
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have height_zero : ∀ P : PrimeSpectrum K, height P = 0 := by
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intro P
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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 [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 JeqP : J = P := PrimeSpectrum.ext J P this
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have JneqP : J ≠ P := ne_of_lt JlP
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contradiction
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simp [height_zero]
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