proved principle ideal theorem mod sorries

CommAlg/grant2.lean

@@ -5,7 +5,7 @@ import Mathlib.Order.Height
 import Mathlib.RingTheory.Noetherian
 import CommAlg.krull
 
-variable (R : Type _) [CommRing R] [IsNoetherianRing R]
+variable {R : Type _} [CommRing R] [IsNoetherianRing R]
 
 lemma height_le_of_gt_height_lt {n : ℕ∞} (q : PrimeSpectrum R)
   (h : ∀(p : PrimeSpectrum R), p < q → Ideal.height p ≤ n - 1) : Ideal.height q ≤ n := by
@@ -15,18 +15,18 @@ lemma height_le_of_gt_height_lt {n : ℕ∞} (q : PrimeSpectrum R)
 theorem height_le_one_of_minimal_over_principle (p : PrimeSpectrum R) (x : R):
   p ∈ minimals (· < ·) {p | x ∈ p.asIdeal} → Ideal.height p ≤ 1 := by
   intro h
-  apply height_le_of_gt_height_lt _ p
+  -- apply height_le_of_gt_height_lt _ p
-  intro q qlep
+  -- intro q qlep
-  by_contra hcontr
+  -- by_contra hcontr
-  push_neg at hcontr
+  -- push_neg at hcontr
-  simp only [le_refl, tsub_eq_zero_of_le] at hcontr 
+  -- simp only [le_refl, tsub_eq_zero_of_le] at hcontr 
   
   sorry
 
 #check (_ : Ideal R) ^ (_ : ℕ)
 #synth Pow (Ideal R) (ℕ)
 
-def symbolicIdeal(Q : Ideal R) {hin : Q.IsPrime} (I : Ideal R) : Ideal R where
+def symbolicIdeal (Q : Ideal R) [hin : Q.IsPrime] (I : Ideal R) : Ideal R where
   carrier := {r : R | ∃ s : R, s ∉ Q ∧ s * r ∈ I}
   zero_mem' := by
     simp only [Set.mem_setOf_eq, mul_zero, Submodule.zero_mem, and_true]
@@ -54,11 +54,22 @@ def symbolicIdeal(Q : Ideal R) {hin : Q.IsPrime} (I : Ideal R) : Ideal R where
     rw [←mul_assoc, mul_comm s, mul_assoc]
     exact Ideal.mul_mem_left _ _ hs2
 
+theorem Noetherian.height_zero_iff_symbolicPower_eq [IsNoetherianRing R] (P : Ideal R) [P.IsPrime] :
+  (∃ n : ℕ, symbolicIdeal P (P ^ n) = symbolicIdeal P (P ^ n.succ)) ↔ Ideal.height ⟨P, inferInstance⟩ = 0 := sorry
 
 theorem WF_interval_le_prime [IsNoetherianRing R] (I : Ideal R) (P : Ideal R) [P.IsPrime] 
   (h : ∀ J ∈ (Set.Icc I P), J.IsPrime → J = P ):
   WellFounded ((· < ·) : (Set.Icc I P) → (Set.Icc I P) → Prop ) := sorry 
 
+-- theorem smul_sup_eq_smul_sup_of_le_smul_of_le_jacobson {I J : Ideal R} {N N' : Submodule R M}
+--     (hN' : N'.FG) (hIJ : I ≤ jacobson J) (hNN : N ⊔ N' ≤ N ⊔ I • N') : N ⊔ I • N' = N ⊔ J • N' := sorry
+
+lemma nakaka {N N' I P : Ideal R} [P.IsPrime] [IsNoetherianRing R]
+  (hIP : I ≤ P) (hN : N ≤ P) (hNN : N ⊔ N' ≤ N ⊔ I • N') : N ⊔ I • N' = N := sorry 
+
+lemma symbolicPower_one (Q : Ideal R) [Q.IsPrime] : symbolicIdeal Q (Q ^ 1) = Q := sorry
+lemma symbolicPower_subset (Q : Ideal R) [Q.IsPrime] {n m : ℕ} (h : m ≤ n) : symbolicIdeal Q (Q ^ n) ≤ symbolicIdeal Q (Q ^ m) := sorry
+
 protected lemma LocalRing.height_le_one_of_minimal_over_principle
   [LocalRing R] {x : R} 
   (h : (closedPoint R).asIdeal ∈ (Ideal.span {x}).minimalPrimes) :
@@ -68,6 +79,33 @@ protected lemma LocalRing.height_le_one_of_minimal_over_principle
   -- rw [Ideal.lt_height_iff'] at hcont
   -- rcases hcont with ⟨c, hc1, hc2, hc3⟩
   apply height_le_of_gt_height_lt
-  intro p hp
+  intro Q hQ
-
+  let I := Ideal.span {x}
-  sorry
+  let P := (closedPoint R).asIdeal
+  have artint : WellFounded ((· < ·) : (Set.Icc I P) → (Set.Icc I P) → Prop ) := by
+    apply WF_interval_le_prime I P
+    intro J hJ hJPr
+    symm
+    apply eq_of_mem_minimals h
+    . exact ⟨hJPr, hJ.1⟩ 
+    . exact hJ.2
+  let fQ (n : ℕ) : Ideal R := symbolicIdeal Q.asIdeal (Q.asIdeal ^ n)
+  have : ∃ n, I ⊔ fQ n = I ⊔ fQ (n.succ) := sorry
+  simp only [le_refl, tsub_eq_zero_of_le, nonpos_iff_eq_zero]
+  apply (Noetherian.height_zero_iff_symbolicPower_eq _).mp
+  obtain ⟨n, hn⟩ := this
+  use n
+  have : fQ n.succ ⊔ I • fQ n = fQ n := sorry
+  show fQ n = fQ n.succ
+  rw [←this]
+  apply nakaka (P := P)-- (N := symbolicIdeal Q.asIdeal (Q.asIdeal ^ n.succ)) (N' := symbolicIdeal Q.asIdeal (Q.asIdeal ^ n)) (I := I) (P := P)
+  . exact h.1.2
+  . calc
+    _ ≤ fQ 1 := symbolicPower_subset Q.asIdeal (by show 1 ≤ n + 1; simp only [le_add_iff_nonneg_left, zero_le] : 1 ≤ n.succ)
+    _ = Q.asIdeal := symbolicPower_one _
+    _ ≤ P := le_of_lt hQ
+  . suffices fQ n = fQ n.succ ⊔ I • fQ n by
+      rw [←this, sup_eq_right.mpr]
+      exact symbolicPower_subset Q.asIdeal (by show _ ≤ n + 1; simp : n ≤ n.succ)
+    symm
+    assumption