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corrected definition of height, krull dim

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leopoldmayer 2023-06-11 21:02:46 -07:00
parent 17bc8ade26
commit a8b295fa0e
1 changed files with 12 additions and 9 deletions
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comm_alg/krull.lean
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@ -4,6 +4,7 @@ import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Localization.AtPrime import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
/- This file contains the definitions of height of an ideal, and the krull /- This file contains the definitions of height of an ideal, and the krull
dimension of a commutative ring. dimension of a commutative ring.
@ -15,22 +16,24 @@ import Mathlib.RingTheory.Localization.AtPrime
developed. developed.
-/ -/
variable {R : Type _} [CommRing R] (I : Ideal R) namespace Ideal
namespace ideal variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
noncomputable def height : ℕ∞ := Set.chainHeight {J | J ≤ I ∧ J.IsPrime} noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J ≤ I} - 1
noncomputable def krull_dim (R : Type _) [CommRing R] := height (⊤ : Ideal R) noncomputable def krull_dim (R : Type) [CommRing R]: WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
--some propositions that would be nice to be able to eventually --some propositions that would be nice to be able to eventually
lemma dim_eq_zero_iff_field : krull_dim R = 0 ↔ IsField R := sorry lemma dim_eq_zero_iff_field : krull_dim R = 0 ↔ IsField R := by sorry
#check Ring.DimensionLEOne #check Ring.DimensionLEOne
lemma dim_le_one_iff : krull_dim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry lemma dim_le_one_iff : krull_dim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry
lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krull_dim R ≤ 1 := sorry lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krull_dim R ≤ 1 := by
rw [dim_le_one_iff]
exact Ring.DimensionLEOne.principal_ideal_ring R
lemma dim_le_dim_polynomial_add_one [Nontrivial R] : lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krull_dim R ≤ krull_dim (Polynomial R) + 1 := sorry krull_dim R ≤ krull_dim (Polynomial R) + 1 := sorry
@ -38,7 +41,7 @@ lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] : lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
krull_dim R = krull_dim (Polynomial R) + 1 := sorry krull_dim R = krull_dim (Polynomial R) + 1 := sorry
lemma height_eq_dim_localization [Ideal.IsPrime I] : lemma height_eq_dim_localization :
height I = krull_dim (Localization.AtPrime I) := sorry height I = krull_dim (Localization.AtPrime I.asIdeal) := sorry
lemma height_add_dim_quotient_le_dim : height I + krull_dim (R ⧸ I) ≤ krull_dim R := sorry lemma height_add_dim_quotient_le_dim : height I + krull_dim (R ⧸ I.asIdeal) ≤ krull_dim R := sorry