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Completed polynomial_over_field_dim_one

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Sayantan Santra 2023-06-16 11:11:13 -07:00
parent 3cf5665126
commit efbeadc4ce
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 2 additions and 2 deletions
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CommAlg/sayantan(poly_over_field).lean
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@ -7,7 +7,7 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
namespace Ideal namespace Ideal
private lemma singleton_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
unfold Set.chainHeight unfold Set.chainHeight
simp only [iSup_le_iff, Nat.cast_le_one] simp only [iSup_le_iff, Nat.cast_le_one]
intro L h intro L h
@ -67,7 +67,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
unfold height unfold height
rw [sngletn] rw [sngletn]
simp only [WithBot.coe_le_one, ge_iff_le] simp only [WithBot.coe_le_one, ge_iff_le]
exact singleton_chainHeight_one exact singleton_bot_chainHeight_one
· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞) · suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
· obtain ⟨I, h⟩ := this · obtain ⟨I, h⟩ := this
have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by