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poincare-duality 2023-06-14 19:14:01 -07:00
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CommAlg/jayden(krull-dim-zero).lean
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@ -19,7 +19,6 @@ import Mathlib.Util.PiNotation
open PiNotation open PiNotation
namespace Ideal namespace Ideal
variable (R : Type _) [CommRing R] (P : PrimeSpectrum R) variable (R : Type _) [CommRing R] (P : PrimeSpectrum R)
@ -116,12 +115,29 @@ lemma power_zero_finite_length [Ideal.IsMaximal I] (h₁ : Ideal.FG I) [Module.F
-- rcases power with ⟨n, npower⟩ -- rcases power with ⟨n, npower⟩
-- how do I get a generating set? -- how do I get a generating set?
open Finset
-- Stacks Lemma 10.53.3: R is Artinian iff R has finitely many maximal ideals -- Stacks Lemma 10.53.3: R is Artinian iff R has finitely many maximal ideals
lemma Artinian_has_finite_max_ideal lemma Artinian_has_finite_max_ideal
[IsArtinianRing R] : Finite (MaximalSpectrum R) := by [IsArtinianRing R] : Finite (MaximalSpectrum R) := by
by_contra infinite by_contra infinite
simp only [not_finite_iff_infinite] at infinite simp only [not_finite_iff_infinite] at infinite
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
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
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
-- Stacks Lemma 10.53.4: R Artinian => Jacobson ideal of R is nilpotent -- Stacks Lemma 10.53.4: R Artinian => Jacobson ideal of R is nilpotent