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Merge pull request #47 from GTBarkley/jayden

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Jidong Wang 2023-06-13 21:18:44 -07:00 committed by GitHub
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CommAlg/jayden(krull-dim-zero).lean
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@ -77,12 +77,25 @@ lemma containment_radical_power_containment :
rintro ⟨RisNoetherian, containment⟩ rintro ⟨RisNoetherian, containment⟩
rw [isNoetherianRing_iff_ideal_fg] at RisNoetherian rw [isNoetherianRing_iff_ideal_fg] at RisNoetherian
specialize RisNoetherian (Ideal.radical I) specialize RisNoetherian (Ideal.radical I)
rcases RisNoetherian with ⟨S, Sgenerates⟩ -- rcases RisNoetherian with ⟨S, Sgenerates⟩
have containment2 : ∃ n : ℕ, (Ideal.radical I) ^ n ≤ I := by
apply Ideal.exists_radical_pow_le_of_fg I RisNoetherian
cases' containment2 with n containment2'
have containment3 : J ^ n ≤ (Ideal.radical I) ^ n := by
apply Ideal.pow_mono containment
use n
apply le_trans containment3 containment2'
-- The above can be proven using the following quicker theorem that is in the wrong place.
-- Ideal.exists_pow_le_of_le_radical_of_fG
-- how to I get a generating set?
-- Stacks Lemma 10.52.6: I is a maximal ideal and IM = 0. Then length of M is -- Stacks Lemma 10.52.6: I is a maximal ideal and IM = 0. Then length of M is
-- -- the same as the dimension as a vector space over R/I,
lemma length_eq_dim_if_maximal_annihilates {I : Ideal R} [Ideal.IsMaximal I]
: I • (⊤ : Submodule R M) = 0
→ Module.length R M = Module.rank R⧸I M⧸(I • (⊤ : Submodule R M)) := by sorry
-- Does lean know that M/IM is a R/I module?
-- Stacks Lemma 10.52.8: I is a finitely generated maximal ideal of R. -- Stacks Lemma 10.52.8: I is a finitely generated maximal ideal of R.
-- M is a finite R-mod and I^nM=0. Then length of M is finite. -- M is a finite R-mod and I^nM=0. Then length of M is finite.