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Merge pull request #42 from GTBarkley/grant

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added krullDim_nonneg_of_nontrivial to krull
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GTBarkley 2023-06-13 14:45:17 -07:00 committed by GitHub
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1 changed files with 6 additions and 1 deletions
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CommAlg/krull.lean
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@ -39,7 +39,7 @@ lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤
show J' < J show J' < J
exact lt_of_lt_of_le hJ' I_le_J exact lt_of_lt_of_le hJ' I_le_J
lemma krullDim_le_iff (R : Type) [CommRing R] (n : ℕ) : lemma krullDim_le_iff (R : Type _) [CommRing R] (n : ℕ) :
krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞) krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
lemma krullDim_le_iff' (R : Type) [CommRing R] (n : ℕ∞) : lemma krullDim_le_iff' (R : Type) [CommRing R] (n : ℕ∞) :
@ -94,6 +94,11 @@ lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
. rw [h.forall_iff] . rw [h.forall_iff]
trivial trivial
lemma krullDim_nonneg_of_nontrivial [Nontrivial R] : ∃ n : ℕ∞, Ideal.krullDim R = n := by
have h := dim_eq_bot_iff.not.mpr (not_subsingleton R)
lift (Ideal.krullDim R) to ℕ∞ using h with k
use k
@[simp] @[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 constructor