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wrote some characterizations of bounding krull

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GTBarkley 2023-06-12 04:02:16 +00:00
parent dc034c2b09
commit dcc9925882
1 changed files with 20 additions and 1 deletions
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comm_alg/grant.lean
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@ -28,6 +28,7 @@ def series_to_chain : StrictSeries s → s.subchain
-- there should be a coercion from WithTop ℕ to WithBot (WithTop ℕ) but it doesn't seem to work
-- it looks like this might be because someone changed the instance from CoeCT to Coe during the port
-- actually it looks like we can coerce to WithBot (ℕ∞) fine
lemma twoHeights : s ≠ ∅ → (some (Set.chainHeight s) : WithBot (WithTop ℕ)) = krullDim s := by
intro hs
unfold Set.chainHeight
@ -38,5 +39,23 @@ 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
∃ 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
-- #check ((4 : ℕ∞) : WithBot (WithTop ℕ))
#check ( (Set.chainHeight s) : WithBot (ℕ∞))