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new: Made some progress on the other side of dim_eq_zero_iff_field

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Sayantan Santra 2023-06-12 21:55:06 -07:00
parent dff6fb21d3
commit 6bbeb7e36d
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 21 additions and 7 deletions
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CommAlg/sayantan.lean
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@ -6,9 +6,7 @@ import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Localization.AtPrime import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.ConditionallyCompleteLattice.Basic
-- import Mathlib.Data.ENat.Lattice
-- import Mathlib.Order.OrderIsoNat
-- import Mathlib.Tactic.TFAE
namespace Ideal namespace Ideal
example (x : Nat) : List.Chain' (· < ·) [x] := by example (x : Nat) : List.Chain' (· < ·) [x] := by
@ -17,9 +15,7 @@ example (x : Nat) : List.Chain' (· < ·) [x] := by
variable {R : Type _} [CommRing R] (I : PrimeSpectrum R) variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I} noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
noncomputable def krullDim (R : Type) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I noncomputable def krullDim (R : Type) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
@ -48,10 +44,28 @@ lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : heig
have J0 : IsPrime J.asIdeal := J.IsPrime have J0 : IsPrime J.asIdeal := J.IsPrime
rw [field_prime_bot] at P0 J0 rw [field_prime_bot] at P0 J0
have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0) have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0)
have JeqP : J = P := PrimeSpectrum.ext J P this have : J = P := PrimeSpectrum.ext J P this
have JneqP : J ≠ P := ne_of_lt JlP have : J ≠ P := ne_of_lt JlP
contradiction contradiction
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
unfold krullDim unfold krullDim
simp [field_prime_height_zero] simp [field_prime_height_zero]
lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
unfold krullDim at h
simp [height] at h
by_contra x
rw [Ring.not_isField_iff_exists_prime] at x
obtain ⟨P, ⟨h, primeP⟩⟩ := x
have PgtBot : P > ⊥ := Ne.bot_lt h
sorry
lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
constructor
· exact isField.dim_zero
· intro fieldD
have : Field D := IsField.toField fieldD
-- Not exactly sure why this is failing
-- apply @dim_field_eq_zero D _
sorry