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golf, added dim_eq_bot_iff

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leopoldmayer 2023-06-12 13:03:35 -07:00
parent 7a189c7b1d
commit fd5bd05f01
1 changed files with 5 additions and 3 deletions
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CommAlg/krull.lean
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@ -1,4 +1,5 @@
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
@ -33,11 +34,12 @@ noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.c
lemma krullDim_le_iff (R : Type) [CommRing R] (n : ℕ) :
iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) ≤ n ↔
∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := by
convert @iSup_le_iff (WithBot ℕ∞) (PrimeSpectrum R) inferInstance _ (↑n)
∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
--some propositions that would be nice to be able to eventually
lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := sorry
lemma dim_eq_zero_iff_field [IsDomain R] : krullDim R = 0 ↔ IsField R := by sorry
#check Ring.DimensionLEOne