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Merge pull request #14 from GTBarkley/HEAD

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Leo Mayer 2023-06-11 21:06:57 -07:00 committed by GitHub
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2 changed files with 25 additions and 10 deletions
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21
comm_alg/krull.lean
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@ -4,6 +4,7 @@ import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
/- This file contains the definitions of height of an ideal, and the krull
dimension of a commutative ring.
@ -15,22 +16,24 @@ import Mathlib.RingTheory.Localization.AtPrime
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
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
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] :
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] :
krull_dim R = krull_dim (Polynomial R) + 1 := sorry
lemma height_eq_dim_localization [Ideal.IsPrime I] :
height I = krull_dim (Localization.AtPrime I) := sorry
lemma height_eq_dim_localization :
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

12
comm_alg/resources.lean
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@ -9,6 +9,8 @@ import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Order.Height
import Mathlib.RingTheory.MvPolynomial.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
variable {R M : Type _} [CommRing R] [AddCommGroup M] [Module R M]
@ -45,3 +47,13 @@ variable (I : Ideal R)
--this is the polynomial ring with variables indexed by ℕ
#check MvPolynomial ℕ R
--hopefully there's good communication between them
--There's a preliminary version of the going up theorem
#check Ideal.exists_ideal_over_prime_of_isIntegral
--Theorems relating primes of a ring to primes of its localization
#check PrimeSpectrum.localization_comap_injective
#check PrimeSpectrum.localization_comap_range
--Theorems relating primes of a ring to primes of a quotient
#check PrimeSpectrum.range_comap_of_surjective