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52 lines
2.6 KiB
Text
52 lines
2.6 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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noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice
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lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
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krullDim R + 1 ≤ krullDim (Polynomial R) := sorry -- Others are working on it
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-- private lemma sum_succ_of_succ_sum {ι : Type} (a : ℕ∞) [inst : Nonempty ι] :
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-- (⨆ (x : ι), a + 1) = (⨆ (x : ι), a) + 1 := by
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-- have : a + 1 = (⨆ (x : ι), a) + 1 := by rw [ciSup_const]
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-- have : a + 1 = (⨆ (x : ι), a + 1) := Eq.symm ciSup_const
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-- simp
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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) := by
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rw [le_antisymm_iff]
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constructor
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· exact dim_le_dim_polynomial_add_one
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· unfold krullDim
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have htPBdd : ∀ (P : PrimeSpectrum (Polynomial R)), (height P : WithBot ℕ∞) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
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intro P
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have : ∃ (I : PrimeSpectrum R), (height P : WithBot ℕ∞) ≤ ↑(height I + 1) := by
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sorry
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obtain ⟨I, IP⟩ := this
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have : (↑(height I + 1) : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum R), ↑(height I + 1) := by
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apply @le_iSup (WithBot ℕ∞) _ _ _ I
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apply ge_trans this IP
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have oneOut : (⨆ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) + 1) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by
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have : ∀ P : PrimeSpectrum R, (height P : WithBot ℕ∞) + 1 ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 :=
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fun P ↦ (by apply add_le_add_right (@le_iSup (WithBot ℕ∞) _ _ _ P) 1)
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apply iSup_le
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apply this
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simp
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intro P
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exact ge_trans oneOut (htPBdd P)
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