Almost completed polynomial_over_field_dim_one

CommAlg/krull.lean

@@ -60,7 +60,8 @@ lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
   le_iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) I
 
 /-- In a domain, the height of a prime ideal is Bot (0 in this case) iff it's the Bot ideal. -/
-lemma height_bot_iff_bot {D: Type} [CommRing D] [IsDomain D] (P : PrimeSpectrum D) : height P = ⊥ ↔ P = ⊥ := by
+@[simp]
+lemma height_bot_iff_bot {D: Type} [CommRing D] [IsDomain D] {P : PrimeSpectrum D} : height P = ⊥ ↔ P = ⊥ := by
   constructor
   · intro h
     unfold height at h
@@ -263,7 +264,7 @@ lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum
 
 /-- In a field, the unique prime ideal is the zero ideal. -/
 @[simp]
-lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
+lemma field_prime_bot {K: Type _} [Field K] {P : Ideal K} : IsPrime P ↔ P = ⊥ := by
       constructor
       · intro primeP
         obtain T := eq_bot_or_top P
@@ -274,9 +275,13 @@ lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P =
         exact bot_prime
 
 /-- In a field, all primes have height 0. -/
-lemma field_prime_height_bot {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = ⊥ := by
+lemma field_prime_height_bot {K: Type _} [Nontrivial K] [Field K] {P : PrimeSpectrum K} : height P = ⊥ := by
-    -- This should be doable by using field_prime_height_bot
+    -- This should be doable by
-    -- and height_bot_iff_bot
+    -- have : IsPrime P.asIdeal := P.IsPrime
+    -- rw [field_prime_bot] at this
+    -- have : P = ⊥ := PrimeSpectrum.ext P ⊥ this
+    -- rw [height_bot_iff_bot]
+    -- Need to check what's happening
     rw [bot_eq_zero]
     unfold height
     simp only [Set.chainHeight_eq_zero_iff]

CommAlg/sayantan(poly_over_field).lean

@@ -1,22 +1,56 @@
 import CommAlg.krull
 import Mathlib.RingTheory.Ideal.Operations
-import Mathlib.RingTheory.FiniteType
 import Mathlib.Order.Height
 import Mathlib.RingTheory.PrincipalIdealDomain
 import Mathlib.RingTheory.DedekindDomain.Basic
-import Mathlib.RingTheory.Ideal.Quotient
-import Mathlib.RingTheory.Ideal.MinimalPrime
-import Mathlib.RingTheory.Localization.AtPrime
 import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
-import Mathlib.Order.ConditionallyCompleteLattice.Basic
 
 namespace Ideal
 
+/-- The ring of polynomials over a field has dimension one. -/
 lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
-  -- unfold krullDim
   rw [le_antisymm_iff]
+  let X := @Polynomial.X K _
   constructor
-  · 
+  · unfold krullDim
+    apply @iSup_le (WithBot ℕ∞) _ _ _ _
+    intro I
+    have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
+    by_cases I = ⊥
+    · rw [← height_bot_iff_bot] at h
+      simp only [WithBot.coe_le_one, ge_iff_le]
+      rw [h]
+      exact bot_le
+    · push_neg at h
+      have : I.asIdeal ≠ ⊥ := by
+        by_contra a
+        have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
+        contradiction
+      have maxI := IsPrime.to_maximal_ideal this
+      have singleton : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
+          intro P
+          constructor
+          · intro H
+            simp only [Set.mem_setOf_eq] at H
+            by_contra x
+            push_neg at x
+            have : P.asIdeal ≠ ⊥ := by
+              by_contra a
+              have : P = ⊥ := PrimeSpectrum.ext P ⊥ a
+              contradiction 
+            have maxP := IsPrime.to_maximal_ideal this
+            have IneTop := IsMaximal.ne_top maxI
+            have : P ≤ I := le_of_lt H
+            rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this
+            have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this
+            have : P = I := PrimeSpectrum.ext P I this
+            replace H : P ≠ I := ne_of_lt H
+            contradiction
+          · intro pBot
+            simp only [Set.mem_setOf_eq, pBot]
+            exact lt_of_le_of_ne bot_le h.symm
+      replace singleton : {J | J < I} = {⊥} := Set.ext singleton
+      unfold height
       sorry
   · suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
     · obtain ⟨I, h⟩ := this
@@ -24,16 +58,14 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
         apply @le_iSup (WithBot ℕ∞) _ _ _ I
       exact le_trans h this
     have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _
-    let X := @Polynomial.X K _
     have : IsPrime (span {X}) := by
-      refine Iff.mpr (span_singleton_prime ?hp) primeX
+      refine (span_singleton_prime ?hp).mpr primeX
       exact Polynomial.X_ne_zero
     let P := PrimeSpectrum.mk (span {X}) this
     unfold height
     use P
     have : ⊥ ∈ {J | J < P} := by
-      simp only [Set.mem_setOf_eq]
+      simp only [Set.mem_setOf_eq, bot_lt_iff_ne_bot]
-      rw [bot_lt_iff_ne_bot]
       suffices : P.asIdeal ≠ ⊥
       · by_contra x
         rw [PrimeSpectrum.ext_iff] at x