comm_alg/CommAlg/jayden(krull-dim-zero).lean
2023-06-12 14:32:20 -07:00

79 lines
2.4 KiB
Text
Raw Blame History

This file contains ambiguous Unicode characters

This file contains Unicode characters that might be confused with other characters. If you think that this is intentional, you can safely ignore this warning. Use the Escape button to reveal them.

import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.RingTheory.Noetherian
import Mathlib.Order.KrullDimension
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Data.Finite.Defs
import Mathlib.Order.Height
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.ConditionallyCompleteLattice.Basic
-- copy from krull.lean; the name of Krull dimension for rings is changed to krullDim' since krullDim already exists in the librrary
namespace Ideal
variable (R : Type _) [CommRing R] (I : PrimeSpectrum R)
noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
noncomputable def krullDim' (R : Type) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height R I
-- copy ends
-- Stacks Lemma 10.60.5: R is Artinian iff R is Noetherian of dimension 0
lemma dim_zero_Noetherian_iff_Artinian (R : Type _) [CommRing R] :
IsNoetherianRing R ∧ krullDim' R = 0 ↔ IsArtinianRing R := by
variable {R : Type _} [CommRing R]
-- Repeats the definition by Monalisa
noncomputable def length : krullDim (Submodule _ _)
-- The following is Stacks Lemma 10.60.5
lemma dim_zero_Noetherian_iff_Artinian (R : Type _) [CommRing R] :
IsNoetherianRing R ∧ krull_dim R = 0 ↔ IsArtinianRing R := by
sorry
#check IsNoetherianRing
#check krullDim
-- Repeats the definition of the length of a module by Monalisa
variable (M : Type _) [AddCommMonoid M] [Module R M]
noncomputable def length := krullDim (Submodule R M)
#check length
-- Stacks Lemma 10.53.6: R is Artinian iff R has finite length as an R-mod
lemma IsArtinian_iff_finite_length : IsArtinianRing R ↔ ∃ n : , length R R ≤ n := by sorry
-- Stacks Lemma 10.53.3: R is Artinian iff R has finitely many maximal ideals
lemma IsArtinian_iff_finite_max_ideal : IsArtinianRing R ↔ Finite (MaximalSpectrum R) := by sorry
-- Stacks Lemma 10.53.4: R Artinian => Jacobson ideal of R is nilpotent
lemma Jacobson_of_Artinian_is_nilpotent : Is
-- how to use namespace
namespace something
end something
open something
-- The following is Stacks Lemma 10.53.6
lemma IsArtinian_iff_finite_length : IsArtinianRing R ↔ ∃ n : , length R R ≤ n := by sorry