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111 lines
No EOL
4.3 KiB
Text
111 lines
No EOL
4.3 KiB
Text
import Mathlib.Order.KrullDimension
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import Mathlib.Order.JordanHolder
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import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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import Mathlib.Order.Height
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import Mathlib.RingTheory.Noetherian
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import CommAlg.krull
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variable {R : Type _} [CommRing R] [IsNoetherianRing R]
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lemma height_le_of_gt_height_lt {n : ℕ∞} (q : PrimeSpectrum R)
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(h : ∀(p : PrimeSpectrum R), p < q → Ideal.height p ≤ n - 1) : Ideal.height q ≤ n := by
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sorry
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theorem height_le_one_of_minimal_over_principle (p : PrimeSpectrum R) (x : R):
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p ∈ minimals (· < ·) {p | x ∈ p.asIdeal} → Ideal.height p ≤ 1 := by
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intro h
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-- apply height_le_of_gt_height_lt _ p
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-- intro q qlep
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-- by_contra hcontr
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-- push_neg at hcontr
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-- simp only [le_refl, tsub_eq_zero_of_le] at hcontr
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sorry
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#check (_ : Ideal R) ^ (_ : ℕ)
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#synth Pow (Ideal R) (ℕ)
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def symbolicIdeal (Q : Ideal R) [hin : Q.IsPrime] (I : Ideal R) : Ideal R where
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carrier := {r : R | ∃ s : R, s ∉ Q ∧ s * r ∈ I}
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zero_mem' := by
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simp only [Set.mem_setOf_eq, mul_zero, Submodule.zero_mem, and_true]
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use 1
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rw [←Q.ne_top_iff_one]
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exact hin.ne_top
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add_mem' := by
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rintro a b ⟨sa, hsa1, hsa2⟩ ⟨sb, hsb1, hsb2⟩
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use sa * sb
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constructor
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. intro h
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cases hin.mem_or_mem h <;> contradiction
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ring_nf
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apply I.add_mem --<;> simp only [I.mul_mem_left, hsa2, hsb2]
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. rw [mul_comm sa, mul_assoc]
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exact I.mul_mem_left sb hsa2
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. rw [mul_assoc]
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exact I.mul_mem_left sa hsb2
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smul_mem' := by
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intro c x
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dsimp
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rintro ⟨s, hs1, hs2⟩
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use s
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constructor; exact hs1
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rw [←mul_assoc, mul_comm s, mul_assoc]
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exact Ideal.mul_mem_left _ _ hs2
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theorem Noetherian.height_zero_iff_symbolicPower_eq [IsNoetherianRing R] (P : Ideal R) [P.IsPrime] :
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(∃ n : ℕ, symbolicIdeal P (P ^ n) = symbolicIdeal P (P ^ n.succ)) ↔ Ideal.height ⟨P, inferInstance⟩ = 0 := sorry
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theorem WF_interval_le_prime [IsNoetherianRing R] (I : Ideal R) (P : Ideal R) [P.IsPrime]
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(h : ∀ J ∈ (Set.Icc I P), J.IsPrime → J = P ):
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WellFounded ((· < ·) : (Set.Icc I P) → (Set.Icc I P) → Prop ) := sorry
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-- theorem smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
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-- (hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N ⊔ N' ≤ N ⊔ I • N') : N ⊔ I • N' = N ⊔ J • N' := sorry
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lemma nakaka {N N' I P : Ideal R} [P.IsPrime] [IsNoetherianRing R]
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(hIP : I ≤ P) (hN : N ≤ P) (hNN : N ⊔ N' ≤ N ⊔ I • N') : N ⊔ I • N' = N := sorry
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lemma symbolicPower_one (Q : Ideal R) [Q.IsPrime] : symbolicIdeal Q (Q ^ 1) = Q := sorry
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lemma symbolicPower_subset (Q : Ideal R) [Q.IsPrime] {n m : ℕ} (h : m ≤ n) : symbolicIdeal Q (Q ^ n) ≤ symbolicIdeal Q (Q ^ m) := sorry
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protected lemma LocalRing.height_le_one_of_minimal_over_principle
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[LocalRing R] {x : R}
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(h : (closedPoint R).asIdeal ∈ (Ideal.span {x}).minimalPrimes) :
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Ideal.height (closedPoint R) ≤ 1 := by
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-- by_contra hcont
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-- push_neg at hcont
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-- rw [Ideal.lt_height_iff'] at hcont
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-- rcases hcont with ⟨c, hc1, hc2, hc3⟩
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apply height_le_of_gt_height_lt
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intro Q hQ
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let I := Ideal.span {x}
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let P := (closedPoint R).asIdeal
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have artint : WellFounded ((· < ·) : (Set.Icc I P) → (Set.Icc I P) → Prop ) := by
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apply WF_interval_le_prime I P
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intro J hJ hJPr
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symm
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apply eq_of_mem_minimals h
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. exact ⟨hJPr, hJ.1⟩
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. exact hJ.2
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let fQ (n : ℕ) : Ideal R := symbolicIdeal Q.asIdeal (Q.asIdeal ^ n)
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have : ∃ n, I ⊔ fQ n = I ⊔ fQ (n.succ) := sorry
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simp only [le_refl, tsub_eq_zero_of_le, nonpos_iff_eq_zero]
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apply (Noetherian.height_zero_iff_symbolicPower_eq _).mp
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obtain ⟨n, hn⟩ := this
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use n
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have : fQ n.succ ⊔ I • fQ n = fQ n := sorry
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show fQ n = fQ n.succ
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rw [←this]
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apply nakaka (P := P)-- (N := symbolicIdeal Q.asIdeal (Q.asIdeal ^ n.succ)) (N' := symbolicIdeal Q.asIdeal (Q.asIdeal ^ n)) (I := I) (P := P)
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. exact h.1.2
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. calc
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_ ≤ fQ 1 := symbolicPower_subset Q.asIdeal (by show 1 ≤ n + 1; simp only [le_add_iff_nonneg_left, zero_le] : 1 ≤ n.succ)
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_ = Q.asIdeal := symbolicPower_one _
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_ ≤ P := le_of_lt hQ
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. suffices fQ n = fQ n.succ ⊔ I • fQ n by
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rw [←this, sup_eq_right.mpr]
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exact symbolicPower_subset Q.asIdeal (by show _ ≤ n + 1; simp : n ≤ n.succ)
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symm
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assumption |