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Merge pull request #24 from GTBarkley/grant

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stated some equivalences for bounding krull dim
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GTBarkley 2023-06-12 11:03:07 -07:00 committed by GitHub
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@ -2,6 +2,7 @@ import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import CommAlg.krull
#check (p q : PrimeSpectrum _) → (p ≤ q)
@ -39,22 +40,14 @@ lemma twoHeights : s ≠ ∅ → (some (Set.chainHeight s) : WithBot (WithTop
-- norm_cast
sorry
namespace Ideal
noncomputable def krullDim (R : Type _) [CommRing R] :=
Set.chainHeight (Set.univ : Set (PrimeSpectrum R))
def krullDimGE (R : Type _) [CommRing R] (n : ℕ) :=
∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1
def krullDimLE (R : Type _) [CommRing R] (n : ℕ) :=
∀ c : List (PrimeSpectrum R), c.Chain' (· < ·) → c.length ≤ n + 1
end Ideal
open Ideal
lemma krullDim_le (R : Type _) [CommRing R] : krullDimLE R n ↔ Ideal.krullDim R ≤ n := sorry
lemma krullDim_ge (R : Type _) [CommRing R] : krullDimGE R n ↔ Ideal.krullDim R ≥ n := sorry
lemma krullDim_le_iff' (R : Type _) [CommRing R] :
Ideal.krullDim R ≤ n ↔ (∀ c : List (PrimeSpectrum R), c.Chain' (· < ·) → c.length ≤ n + 1) := by
sorry
lemma krullDim_ge_iff' (R : Type _) [CommRing R] :
Ideal.krullDim R ≥ n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1 := sorry
-- #check ((4 : ℕ∞) : WithBot (WithTop ℕ))