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Completed polynomial_over_field_dim_one

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Sayantan Santra 2023-06-16 10:42:02 -07:00
parent bcb867258a
commit e735a5254f
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 19 additions and 3 deletions
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22
CommAlg/sayantan(poly_over_field).lean
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@ -7,6 +7,20 @@ 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
unfold Set.chainHeight
simp only [iSup_le_iff, Nat.cast_le_one]
intro L h
unfold Set.subchain at h
simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at h
rcases L with (_ | ⟨a,L⟩)
. simp only [List.length_nil, zero_le]
rcases L with (_ | ⟨b,L⟩)
. simp only [List.length_singleton, le_refl]
simp only [List.chain'_cons, List.find?, List.mem_cons, forall_eq_or_imp] at h
rcases h with ⟨⟨h1, _⟩, ⟨rfl, rfl, _⟩⟩
exact absurd h1 (lt_irrefl _)
/-- The ring of polynomials over a field has dimension one. -/ /-- The ring of polynomials over a field has dimension one. -/
lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
rw [le_antisymm_iff] rw [le_antisymm_iff]
@ -27,7 +41,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
have : I = ⊥ := PrimeSpectrum.ext I ⊥ a have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
contradiction contradiction
have maxI := IsPrime.to_maximal_ideal this have maxI := IsPrime.to_maximal_ideal this
have singleton : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by have sngletn : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
intro P intro P
constructor constructor
· intro H · intro H
@ -49,9 +63,11 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
· intro pBot · intro pBot
simp only [Set.mem_setOf_eq, pBot] simp only [Set.mem_setOf_eq, pBot]
exact lt_of_le_of_ne bot_le h.symm exact lt_of_le_of_ne bot_le h.symm
replace singleton : {J | J < I} = {⊥} := Set.ext singleton replace sngletn : {J | J < I} = {⊥} := Set.ext sngletn
unfold height unfold height
sorry rw [sngletn]
simp only [WithBot.coe_le_one, ge_iff_le]
exact singleton_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