Merge branch 'main' into grant

CommAlg/krull.lean

@@ -99,7 +99,65 @@ lemma krullDim_nonneg_of_nontrivial [Nontrivial R] : ∃ n : ℕ∞, Ideal.krull
   lift (Ideal.krullDim R) to ℕ∞ using h with k
   use k
   
-lemma dim_eq_zero_iff_field [IsDomain R] : krullDim R = 0 ↔ IsField R := by sorry
+@[simp]
+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
+        have : ¬P = ⊤ := IsPrime.ne_top primeP
+        tauto
+      · intro botP
+        rw [botP]
+        exact bot_prime
+
+lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
+    unfold height
+    simp
+    by_contra spec
+    change _ ≠ _ at spec
+    rw [← Set.nonempty_iff_ne_empty] at spec
+    obtain ⟨J, JlP : J < P⟩ := spec
+    have P0 : IsPrime P.asIdeal := P.IsPrime
+    have J0 : IsPrime J.asIdeal := J.IsPrime
+    rw [field_prime_bot] at P0 J0
+    have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0)
+    have : J = P := PrimeSpectrum.ext J P this
+    have : J ≠ P := ne_of_lt JlP
+    contradiction
+
+lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
+  unfold krullDim
+  simp [field_prime_height_zero]
+
+lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
+  unfold krullDim at h
+  simp [height] at h
+  by_contra x
+  rw [Ring.not_isField_iff_exists_prime] at x
+  obtain ⟨P, ⟨h1, primeP⟩⟩ := x
+  let P' : PrimeSpectrum D := PrimeSpectrum.mk P primeP
+  have h2 : P' ≠ ⊥ := by
+    by_contra a
+    have : P = ⊥ := by rwa [PrimeSpectrum.ext_iff] at a
+    contradiction
+  have PgtBot : P' > ⊥ := Ne.bot_lt h2
+  have pos_height : ¬ ↑(Set.chainHeight {J | J < P'}) ≤ 0  := by
+    have : ⊥ ∈ {J | J < P'} := PgtBot
+    have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
+    rw [←Set.one_le_chainHeight_iff] at this
+    exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this)
+  have nonpos_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
+    have : (⨆ (I : PrimeSpectrum D), (Set.chainHeight {J | J < I} : WithBot ℕ∞)) ≤ 0 := h.le
+    rw [iSup_le_iff] at this
+    exact Iff.mp WithBot.coe_le_zero (this P')
+  contradiction
+
+lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
+  constructor
+  · exact isField.dim_zero
+  · intro fieldD
+    let h : Field D := IsField.toField fieldD
+    exact dim_field_eq_zero
 
 #check Ring.DimensionLEOne
 lemma dim_le_one_iff : krullDim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry

CommAlg/sayantan(dim_eq_zero_iff_field).lean

@@ -68,7 +68,7 @@ lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0)
     have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
     rw [←Set.one_le_chainHeight_iff] at this
     exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this)
-  have zero_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
+  have nonpos_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
     have : (⨆ (I : PrimeSpectrum D), (Set.chainHeight {J | J < I} : WithBot ℕ∞)) ≤ 0 := h.le
     rw [iSup_le_iff] at this
     exact Iff.mp WithBot.coe_le_zero (this P')