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Merge pull request #108 from SinTan1729/main

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Some refinements
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Sayantan Santra 2023-06-23 11:33:16 -07:00 committed by GitHub
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1 changed files with 30 additions and 15 deletions
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CommAlg/krull.lean
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@ -190,6 +190,14 @@ lemma lt_height_iff'' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
rw [WithBot.coe_lt_coe]
exact lt_height_iff'
/-- Convert elements in Ideal.minimalPrimes to PrimeSpectrum -/
lemma minimalPrimes.toPrimeSpectrum {R : Type _} [CommRing R] {I P : Ideal R} : P ∈ Ideal.minimalPrimes I → PrimeSpectrum R := by
unfold Ideal.minimalPrimes
intro Pmin
obtain ⟨L, _⟩ := Pmin
simp only [Set.mem_setOf_eq] at L
exact PrimeSpectrum.mk P L.1
#check height_le_krullDim
--some propositions that would be nice to be able to eventually
@ -356,19 +364,27 @@ 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_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
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 _)
/-- Singleton sets have chainHeight 1 -/
lemma singleton_chainHeight_one {α : Type _} {x : α} [Preorder α] : Set.chainHeight {x} = 1 := by
have le : Set.chainHeight {x} ≤ 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 _)
suffices : Set.chainHeight {x} > 0
· change _ < _ at this
rw [←ENat.one_le_iff_pos] at this
apply le_antisymm <;> trivial
by_contra x
simp only [gt_iff_lt, not_lt, nonpos_iff_eq_zero, Set.chainHeight_eq_zero_iff, Set.singleton_ne_empty] at x
/-- 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
@ -378,7 +394,6 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
· unfold krullDim
apply @iSup_le (WithBot ℕ∞) _ _ _ _
intro I
have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
by_cases I = ⊥
· rw [← height_zero_iff_bot] at h
simp only [WithBot.coe_le_one, ge_iff_le]
@ -416,7 +431,7 @@ 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_chainHeight_le_one
exact le_of_eq singleton_chainHeight_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