Merge pull request #87 from SinTan1729/main

Almost completed polynomial_over_field_dim_one

CommAlg/krull.lean

@@ -19,6 +19,7 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
   developed.
 -/
 
+/-- If something is smaller that Bot of a PartialOrder after attaching another Bot, it must be Bot. -/
 lemma lt_bot_eq_WithBot_bot [PartialOrder α] [OrderBot α] {a : WithBot α} (h : a < (⊥ : α)) : a = ⊥ := by
   cases a
   . rfl
@@ -29,18 +30,19 @@ open LocalRing
 
 variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
 
+/-- Definitions -/
 noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
-
 noncomputable def krullDim (R : Type _) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
-
 noncomputable def codimension (J : Ideal R) : WithBot ℕ∞ := ⨅ I ∈ {I : PrimeSpectrum R | J ≤ I.asIdeal}, height I
 
 lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
 lemma krullDim_def (R : Type _) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
 lemma krullDim_def' (R : Type _) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
 
+/-- A lattice structure on WithBot ℕ∞. -/
 noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice
 
+/-- Height of ideals is monotonic. -/
 lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := by
   apply Set.chainHeight_mono
   intro J' hJ'
@@ -57,6 +59,34 @@ lemma krullDim_le_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
 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. -/
+@[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
+    rw [bot_eq_zero] at h
+    simp only [Set.chainHeight_eq_zero_iff] at h
+    apply eq_bot_of_minimal
+    intro I
+    by_contra x
+    have : I ∈ {J | J < P} := x
+    rw [h] at this
+    contradiction
+  · intro h
+    unfold height
+    simp only [bot_eq_zero', Set.chainHeight_eq_zero_iff]
+    by_contra spec
+    change _ ≠ _ at spec
+    rw [← Set.nonempty_iff_ne_empty] at spec
+    obtain ⟨J, JlP : J < P⟩ := spec
+    have JneP : J ≠ P := ne_of_lt JlP
+    rw [h, ←bot_lt_iff_ne_bot, ←h] at JneP
+    have := not_lt_of_lt JneP
+    contradiction
+
+/-- The Krull dimension of a ring being ≥ n is equivalent to there being an
+    ideal of height ≥ n. -/
 lemma le_krullDim_iff (R : Type _) [CommRing R] (n : ℕ) :
   n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by
   constructor
@@ -230,9 +260,9 @@ lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal
     . exact List.chain'_singleton _
     . constructor
       . intro I' hI'
-        simp at hI'
+        simp only [List.mem_singleton] at hI' 
         rwa [hI']
-      . simp
+      . simp only [List.length_singleton, Nat.cast_one, zero_add]
   . contrapose! h
     change (0 : ℕ∞) < (_ : WithBot ℕ∞) at h
     rw [lt_height_iff''] at h
@@ -259,7 +289,7 @@ 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
@@ -270,9 +300,16 @@ 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_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
+lemma field_prime_height_bot {K: Type _} [Nontrivial K] [Field K] {P : PrimeSpectrum K} : height P = ⊥ := by
+    -- This should be doable by
+    -- 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
+    simp only [Set.chainHeight_eq_zero_iff]
     by_contra spec
     change _ ≠ _ at spec
     rw [← Set.nonempty_iff_ne_empty] at spec
@@ -288,7 +325,7 @@ lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : heig
 /-- The Krull dimension of a field is 0. -/
 lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
   unfold krullDim
-  simp [field_prime_height_zero]
+  simp only [field_prime_height_bot, ciSup_unique]
 
 /-- A domain with Krull dimension 0 is a field. -/
 lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
@@ -335,7 +372,7 @@ lemma dim_le_one_of_dimLEOne :  Ring.DimensionLEOne R → krullDim R ≤ 1 := by
   rcases (lt_height_iff''.mp h) with ⟨c, ⟨hc1, hc2, hc3⟩⟩
   norm_cast at hc3
   rw [List.chain'_iff_get] at hc1
-  specialize hc1 0 (by rw [hc3]; simp)
+  specialize hc1 0 (by rw [hc3]; simp only)
   set q0 : PrimeSpectrum R := List.get c { val := 0, isLt := _ }
   set q1 : PrimeSpectrum R := List.get c { val := 1, isLt := _ } 
   specialize hc2 q1 (List.get_mem _ _ _)

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