From 55ea06c1419d46c5399997a0a67c6d619d5b5d2c Mon Sep 17 00:00:00 2001 From: poincare-duality Date: Tue, 13 Jun 2023 17:21:42 -0700 Subject: [PATCH] add more stuff --- CommAlg/jayden(krull-dim-zero).lean | 203 +++++++++++++++++++--------- 1 file changed, 138 insertions(+), 65 deletions(-) diff --git a/CommAlg/jayden(krull-dim-zero).lean b/CommAlg/jayden(krull-dim-zero).lean index 3651102..6f282ba 100644 --- a/CommAlg/jayden(krull-dim-zero).lean +++ b/CommAlg/jayden(krull-dim-zero).lean @@ -1,97 +1,170 @@ import Mathlib.RingTheory.Ideal.Basic +import Mathlib.RingTheory.Ideal.Operations import Mathlib.RingTheory.JacobsonIdeal import Mathlib.RingTheory.Noetherian import Mathlib.Order.KrullDimension import Mathlib.RingTheory.Artinian import Mathlib.RingTheory.Ideal.Quotient import Mathlib.RingTheory.Nilpotent -import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal +import Mathlib.AlgebraicGeometry.PrimeSpectrum.Noetherian import Mathlib.Data.Finite.Defs import Mathlib.Order.Height import Mathlib.RingTheory.DedekindDomain.Basic import Mathlib.RingTheory.Localization.AtPrime import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Algebra.Ring.Pi -import Mathlib.Topology.NoetherianSpace +import Mathlib.RingTheory.Finiteness + --- 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) +variable (R : Type _) [CommRing R] (P : 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 sorry - - -#check IsNoetherianRing - -#check krullDim - --- Repeats the definition of the length of a module by Monalisa -variable (M : Type _) [AddCommMonoid M] [Module R M] - --- change the definition of length -noncomputable def length := Set.chainHeight {M' : Submodule R M | 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 : 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 : IsArtinianRing R → IsNilpotent (Ideal.jacobson (⊤ : Ideal R)) := by sorry - - --- Stacks Definition 10.32.1: An ideal is locally nilpotent --- if every element is nilpotent -namespace Ideal -class IsLocallyNilpotent (I : Ideal R) : Prop := - h : ∀ x ∈ I, IsNilpotent x - -end Ideal - -#check Ideal.IsLocallyNilpotent - --- Stacks Lemma 10.53.5: If R has finitely many maximal ideals and --- locally nilpotent Jacobson radical, then R is the product of its localizations at --- its maximal ideals. Also, all primes are maximal - -lemma product_of_localization_at_maximal_ideal : Finite (MaximalSpectrum R) - ∧ Ideal.IsLocallyNilpotent (Ideal.jacobson (⊤ : Ideal R)) → Pi.commRing (MaximalSpectrum R) Localization.AtPrime R I - := by sorry --- Haven't finished this. - --- Stacks Lemma 10.31.5: R is Noetherian iff Spec(R) is a Noetherian space -lemma ring_Noetherian_iff_spec_Noetherian : IsNoetherianRing R - ↔ TopologicalSpace.NoetherianSpace (PrimeSpectrum R) := by sorry --- Use TopologicalSpace.NoetherianSpace.exists_finset_irreducible : --- Every closed subset of a noetherian space is a finite union --- of irreducible closed subsets. +noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < P} +noncomputable def krullDim (R : Type) [CommRing R] : + WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height R I -- Stacks Lemma 10.26.1 (Should already exists) -- (1) The closure of a prime P is V(P) -- (2) the irreducible closed subsets are V(P) for P prime -- (3) the irreducible components are V(P) for P minimal prime --- Stacks Lemma 10.32.5: R Noetherian. I,J ideals. If J ⊂ √I, then J ^ n ⊂ I for some n +-- Stacks Definition 10.32.1: An ideal is locally nilpotent +-- if every element is nilpotent +class IsLocallyNilpotent (I : Ideal R) : Prop := + h : ∀ x ∈ I, IsNilpotent x +#check Ideal.IsLocallyNilpotent +end Ideal + + +-- Repeats the definition of the length of a module by Monalisa +variable (R : Type _) [CommRing R] (I J : Ideal R) +variable (M : Type _) [AddCommMonoid M] [Module R M] + +-- change the definition of length of a module +namespace Module +noncomputable def length := Set.chainHeight {M' : Submodule R M | M' < ⊤} +end Module + +-- Stacks Lemma 10.31.5: R is Noetherian iff Spec(R) is a Noetherian space +example [IsNoetherianRing R] : + TopologicalSpace.NoetherianSpace (PrimeSpectrum R) := + inferInstance + +instance ring_Noetherian_of_spec_Noetherian + [TopologicalSpace.NoetherianSpace (PrimeSpectrum R)] : + IsNoetherianRing R where + noetherian := by sorry + +lemma ring_Noetherian_iff_spec_Noetherian : IsNoetherianRing R + ↔ TopologicalSpace.NoetherianSpace (PrimeSpectrum R) := by + constructor + intro RisNoetherian + -- how do I apply an instance to prove one direction? + + +-- Use TopologicalSpace.NoetherianSpace.exists_finset_irreducible : +-- Every closed subset of a noetherian space is a finite union +-- of irreducible closed subsets. + +-- Stacks Lemma 10.32.5: R Noetherian. I,J ideals. +-- If J ⊂ √I, then J ^ n ⊂ I for some n. In particular, locally nilpotent +-- and nilpotent are the same for Noetherian rings +lemma containment_radical_power_containment : + IsNoetherianRing R ∧ J ≤ Ideal.radical I → ∃ n : ℕ, J ^ n ≤ I := by + rintro ⟨RisNoetherian, containment⟩ + rw [isNoetherianRing_iff_ideal_fg] at RisNoetherian + specialize RisNoetherian (Ideal.radical I) + rcases RisNoetherian with ⟨S, Sgenerates⟩ + + -- how to I get a generating set? + +-- Stacks Lemma 10.52.6: I is a maximal ideal and IM = 0. Then length of M is +-- + +-- Stacks Lemma 10.52.8: I is a finitely generated maximal ideal of R. +-- M is a finite R-mod and I^nM=0. Then length of M is finite. +lemma power_zero_finite_length : Ideal.FG I → Ideal.IsMaximal I → Module.Finite R M + → (∃ n : ℕ, (I ^ n) • (⊤ : Submodule R M) = 0) + → (∃ m : ℕ, Module.length R M ≤ m) := by + intro IisFG IisMaximal MisFinite power + rcases power with ⟨n, npower⟩ + -- how do I get a generating set? + + +-- 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 : + IsArtinianRing R → IsNilpotent (Ideal.jacobson (⊤ : Ideal R)) := by sorry + +-- Stacks Lemma 10.53.5: If R has finitely many maximal ideals and +-- locally nilpotent Jacobson radical, then R is the product of its localizations at +-- its maximal ideals. Also, all primes are maximal + +-- lemma product_of_localization_at_maximal_ideal : Finite (MaximalSpectrum R) +-- ∧ + +def jaydensRing : Type _ := sorry + -- ∀ I : MaximalSpectrum R, Localization.AtPrime R I + +instance : CommRing jaydensRing := sorry -- this should come for free, don't even need to state it + +def foo : jaydensRing ≃+* R where + toFun := _ + invFun := _ + left_inv := _ + right_inv := _ + map_mul' := _ + map_add' := _ + -- Ideal.IsLocallyNilpotent (Ideal.jacobson (⊤ : Ideal R)) → + -- Pi.commRing (MaximalSpectrum R) Localization.AtPrime R I + -- := by sorry +-- Haven't finished this. + +-- 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 : ℕ, Module.length R R ≤ n) := by sorry + +-- Lemma: if R has finite length as R-mod, then R is Noetherian +lemma finite_length_is_Noetherian : + (∃ n : ℕ, Module.length R R ≤ n) → IsNoetherianRing R := by sorry + +-- Lemma: if R is Artinian then all the prime ideals are maximal +lemma primes_of_Artinian_are_maximal : + IsArtinianRing R → Ideal.IsPrime I → Ideal.IsMaximal I := by sorry + +-- Lemma: Krull dimension of a ring is the supremum of height of maximal ideals + + +-- 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 ∧ Ideal.krullDim R = 0 ↔ IsArtinianRing R := by + constructor + sorry + intro RisArtinian + constructor + apply finite_length_is_Noetherian + rwa [IsArtinian_iff_finite_length] at RisArtinian + sorry + + + + + + + + + + --- how to use namespace -namespace something -end something -open something