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poincare-duality 2023-06-14 21:32:22 -07:00
parent b7aae64f76
commit 40e26d8636
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14
CommAlg/jayden(krull-dim-zero).lean
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@ -125,19 +125,23 @@ lemma Artinian_has_finite_max_ideal
let m' : ℕ ↪ MaximalSpectrum R := Infinite.natEmbedding (MaximalSpectrum R)
have m'inj := m'.injective
let m'' : ℕ → Ideal R := fun n : ℕ ↦ ⨅ k ∈ range n, (m' k).asIdeal
let f : ℕ → Ideal R := fun n : ℕ ↦ (m' n).asIdeal
let F : Fin n → Ideal R := fun k ↦ (m' k).asIdeal
have comaximal : ∀ i j : ℕ, i ≠ j → (m' i).asIdeal ⊔ (m' j).asIdeal =
(⊤ : Ideal R) := by
intro i j distinct
apply Ideal.IsMaximal.coprime_of_ne
sorry
sorry
-- by_contra equal
exact (m' i).IsMaximal
exact (m' j).IsMaximal
have : (m' i) ≠ (m' j) := by
exact Function.Injective.ne m'inj distinct
intro h
apply this
exact MaximalSpectrum.ext _ _ h
-- let g :`= Ideal.quotientInfRingEquivPiQuotient m' comaximal
have ∀ n : ℕ, (R ⧸ ⨅ (i : Fin n), (F n) i) ≃+* ((i : Fin n) → R ⧸ (F n) i) := by
sorry
-- (let F : Fin n → Ideal R := fun k : Fin n ↦ (m' k).asIdeal)
-- let g := Ideal.quotientInfRingEquivPiQuotient f comaximal
-- Stacks Lemma 10.53.4: R Artinian => Jacobson ideal of R is nilpotent
@ -193,7 +197,7 @@ lemma dim_zero_Noetherian_iff_Artinian (R : Type _) [CommRing R] :
constructor
apply finite_length_is_Noetherian
rwa [IsArtinian_iff_finite_length] at RisArtinian
sorry
sorry -- can use Grant's lemma dim_eq_zero_iff