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Some minor renames and changes

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This commit is contained in:
Sayantan Santra 2023-06-16 13:02:25 -07:00
parent c5f0eb2081
commit db3bf05878
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 7 additions and 12 deletions
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CommAlg/krull.lean
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@ -63,11 +63,10 @@ lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
/-- In a domain, the height of a prime ideal is Bot (0 in this case) iff it's the Bot ideal. -/
@[simp]
lemma height_bot_iff_bot {D: Type _} [CommRing D] [IsDomain D] {P : PrimeSpectrum D} : height P = ⊥ ↔ P = ⊥ := by
lemma height_zero_iff_bot {D: Type _} [CommRing D] [IsDomain D] {P : PrimeSpectrum D} : height P = 0 ↔ P = ⊥ := by
constructor
· intro h
unfold height at h
rw [bot_eq_zero] at h
simp only [Set.chainHeight_eq_zero_iff] at h
apply eq_bot_of_minimal
intro I
@ -87,10 +86,6 @@ lemma height_bot_iff_bot {D: Type _} [CommRing D] [IsDomain D] {P : PrimeSpectru
have := not_lt_of_lt JneP
contradiction
@[simp]
lemma height_bot_eq {D: Type _} [CommRing D] [IsDomain D] : height (⊥ : PrimeSpectrum D) = ⊥ := by
rw [height_bot_iff_bot]
/-- The Krull dimension of a ring being ≥ n is equivalent to there being an
ideal of height ≥ n. -/
@[simp]
@ -307,16 +302,16 @@ lemma field_prime_bot {K: Type _} [Field K] {P : Ideal K} : IsPrime P ↔ P =
exact bot_prime
/-- In a field, all primes have height 0. -/
lemma field_prime_height_bot {K: Type _} [Nontrivial K] [Field K] (P : PrimeSpectrum K) : height P = ⊥ := by
lemma field_prime_height_zero {K: Type _} [Nontrivial K] [Field K] (P : PrimeSpectrum K) : height P = 0 := by
have : IsPrime P.asIdeal := P.IsPrime
rw [field_prime_bot] at this
have : P = ⊥ := PrimeSpectrum.ext P ⊥ this
rwa [height_bot_iff_bot]
rwa [height_zero_iff_bot]
/-- The Krull dimension of a field is 0. -/
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
unfold krullDim
simp only [field_prime_height_bot, ciSup_unique]
simp only [field_prime_height_zero, ciSup_unique]
/-- A domain with Krull dimension 0 is a field. -/
lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
@ -377,7 +372,7 @@ lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1
rw [dim_le_one_iff]
exact Ring.DimensionLEOne.principal_ideal_ring R
private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
private lemma singleton_chainHeight_le_one {α : Type _} {x : α} [Preorder α] : Set.chainHeight {x} ≤ 1 := by
unfold Set.chainHeight
simp only [iSup_le_iff, Nat.cast_le_one]
intro L h
@ -401,7 +396,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
intro I
have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
by_cases I = ⊥
· rw [← height_bot_iff_bot] at h
· rw [← height_zero_iff_bot] at h
simp only [WithBot.coe_le_one, ge_iff_le]
rw [h]
exact bot_le
@ -437,7 +432,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
unfold height
rw [sngletn]
simp only [WithBot.coe_le_one, ge_iff_le]
exact singleton_bot_chainHeight_one
exact singleton_chainHeight_le_one
· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
· obtain ⟨I, h⟩ := this
have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by