mirror of
https://github.com/GTBarkley/comm_alg.git
synced 2024-12-26 07:38:36 -06:00
82 lines
3.2 KiB
Text
82 lines
3.2 KiB
Text
import Mathlib.RingTheory.Ideal.Basic
|
||
import Mathlib.Order.Height
|
||
import Mathlib.RingTheory.PrincipalIdealDomain
|
||
import Mathlib.RingTheory.DedekindDomain.Basic
|
||
import Mathlib.RingTheory.Ideal.Quotient
|
||
import Mathlib.RingTheory.Localization.AtPrime
|
||
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
|
||
import Mathlib.Order.ConditionallyCompleteLattice.Basic
|
||
|
||
namespace Ideal
|
||
|
||
variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
|
||
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
|
||
|
||
lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
|
||
lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
|
||
lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
|
||
|
||
@[simp]
|
||
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
|
||
constructor
|
||
· intro primeP
|
||
obtain T := eq_bot_or_top P
|
||
have : ¬P = ⊤ := IsPrime.ne_top primeP
|
||
tauto
|
||
· intro botP
|
||
rw [botP]
|
||
exact bot_prime
|
||
|
||
lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
|
||
unfold height
|
||
simp
|
||
by_contra spec
|
||
change _ ≠ _ at spec
|
||
rw [← Set.nonempty_iff_ne_empty] at spec
|
||
obtain ⟨J, JlP : J < P⟩ := spec
|
||
have P0 : IsPrime P.asIdeal := P.IsPrime
|
||
have J0 : IsPrime J.asIdeal := J.IsPrime
|
||
rw [field_prime_bot] at P0 J0
|
||
have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0)
|
||
have : J = P := PrimeSpectrum.ext J P this
|
||
have : J ≠ P := ne_of_lt JlP
|
||
contradiction
|
||
|
||
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
|
||
unfold krullDim
|
||
simp [field_prime_height_zero]
|
||
|
||
noncomputable
|
||
instance : CompleteLattice (WithBot ℕ∞) :=
|
||
inferInstanceAs <| CompleteLattice (WithBot (WithTop ℕ))
|
||
|
||
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, ⟨h1, primeP⟩⟩ := x
|
||
let P' : PrimeSpectrum D := PrimeSpectrum.mk P primeP
|
||
have h2 : P' ≠ ⊥ := by
|
||
by_contra a
|
||
have : P = ⊥ := by rwa [PrimeSpectrum.ext_iff] at a
|
||
contradiction
|
||
have PgtBot : P' > ⊥ := Ne.bot_lt h2
|
||
have pos_height : ¬ ↑(Set.chainHeight {J | J < P'}) ≤ 0 := by
|
||
have : ⊥ ∈ {J | J < P'} := PgtBot
|
||
have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
|
||
rw [←Set.one_le_chainHeight_iff] at this
|
||
exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this)
|
||
have nonpos_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
|
||
have : (⨆ (I : PrimeSpectrum D), (Set.chainHeight {J | J < I} : WithBot ℕ∞)) ≤ 0 := h.le
|
||
rw [iSup_le_iff] at this
|
||
exact Iff.mp WithBot.coe_le_zero (this P')
|
||
contradiction
|
||
|
||
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
|
||
let h : Field D := IsField.toField fieldD
|
||
exact dim_field_eq_zero
|