Merge pull request #83 from GTBarkley/main

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import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height
import Mathlib.RingTheory.Polynomial.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import CommAlg.krull
section AddToOrder
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type _}
variable [LT α] [LT β] (s t : Set α)
namespace Set
theorem append_mem_subchain_iff :
l ++ l' ∈ s.subchain ↔ l ∈ s.subchain ∧ l' ∈ s.subchain ∧ ∀ a ∈ l.getLast?, ∀ b ∈ l'.head?, a < b := by
simp [subchain, chain'_append]
aesop
end Set
namespace List
#check Option
theorem getLast?_map (l : List α) (f : α → β) :
(l.map f).getLast? = Option.map f (l.getLast?) := by
cases' l with a l
. rfl
induction' l with b l ih
. rfl
. simp [List.getLast?_cons_cons, ih]
end List
end AddToOrder
--trying and failing to prove ht p = dim R_p
section Localization
variable {R : Type _} [CommRing R] (I : PrimeSpectrum R)
variable {S : Type _} [CommRing S] [Algebra R S] [IsLocalization.AtPrime S I.asIdeal]
open Ideal
open LocalRing
open PrimeSpectrum
#check algebraMap R (Localization.AtPrime I.asIdeal)
#check PrimeSpectrum.comap (algebraMap R (Localization.AtPrime I.asIdeal))
#check krullDim
#check dim_eq_bot_iff
#check height_le_krullDim
variable (J₁ J₂ : PrimeSpectrum (Localization.AtPrime I.asIdeal))
example (h : J₁ ≤ J₂) : PrimeSpectrum.comap (algebraMap R (Localization.AtPrime I.asIdeal)) J₁ ≤
PrimeSpectrum.comap (algebraMap R (Localization.AtPrime I.asIdeal)) J₂ := by
intro x hx
exact h hx
def gadslfasd' : Ideal S := (IsLocalization.AtPrime.localRing S I.asIdeal).maximalIdeal
-- instance gadslfasd : LocalRing S := IsLocalization.AtPrime.localRing S I.asIdeal
example (f : α → β) (hf : Function.Injective f) (h : a₁ ≠ a₂) : f a₁ ≠ f a₂ := by library_search
instance map_prime (J : PrimeSpectrum R) (hJ : J ≤ I) :
(Ideal.map (algebraMap R S) J.asIdeal : Ideal S).IsPrime where
ne_top' := by
intro h
rw [eq_top_iff_one, map, mem_span] at h
mem_or_mem' := sorry
lemma comap_lt_of_lt (J₁ J₂ : PrimeSpectrum S) (J_lt : J₁ < J₂) :
PrimeSpectrum.comap (algebraMap R S) J₁ < PrimeSpectrum.comap (algebraMap R S) J₂ := by
apply lt_of_le_of_ne
apply comap_mono (le_of_lt J_lt)
sorry
lemma lt_of_comap_lt (J₁ J₂ : PrimeSpectrum S)
(hJ : PrimeSpectrum.comap (algebraMap R S) J₁ < PrimeSpectrum.comap (algebraMap R S) J₂) :
J₁ < J₂ := by
apply lt_of_le_of_ne
sorry
/- If S = R_p, then height p = dim S -/
lemma height_eq_height_comap (J : PrimeSpectrum S) :
height (PrimeSpectrum.comap (algebraMap R S) J) = height J := by
simp [height]
have H : {J_1 | J_1 < (PrimeSpectrum.comap (algebraMap R S)) J} =
(PrimeSpectrum.comap (algebraMap R S))'' {J_2 | J_2 < J}
. sorry
rw [H]
apply Set.chainHeight_image
intro x y
constructor
apply comap_lt_of_lt
apply lt_of_comap_lt
lemma disjoint_primeCompl (I : PrimeSpectrum R) :
{ p | Disjoint (I.asIdeal.primeCompl : Set R) p.asIdeal} = {p | p ≤ I} := by
ext p; apply Set.disjoint_compl_left_iff_subset
theorem localizationPrime_comap_range [Algebra R S] (I : PrimeSpectrum R) [IsLocalization.AtPrime S I.asIdeal] :
Set.range (PrimeSpectrum.comap (algebraMap R S)) = { p | p ≤ I} := by
rw [← disjoint_primeCompl]
apply localization_comap_range
#check Set.chainHeight_image
lemma height_eq_dim_localization : height I = krullDim S := by
--first show height I = height gadslfasd'
simp [@krullDim_eq_height _ _ (IsLocalization.AtPrime.localRing S I.asIdeal)]
simp [height]
let f := (PrimeSpectrum.comap (algebraMap R S))
have H : {J | J < I} = f '' {J | J < closedPoint S}
lemma height_eq_dim_localization' :
height I = krullDim (Localization.AtPrime I.asIdeal) := Ideal.height_eq_dim_localization I
end Localization
section Polynomial
open Ideal Polynomial
variables {R : Type _} [CommRing R]
variable (J : Ideal R[X])
#check Ideal.comap C J
--given ideals I J, I ⊔ J is their sum
--given a in R, span {a} is the ideal generated by a
--the map R → R[X] is C →+*
--to show p[x] is prime, show p[x] is the kernel of the map R[x] → R → R/p
#check RingHom.ker_isPrime
def adj_x_map (I : Ideal R) : R[X] →+* R I := (Ideal.Quotient.mk I).comp constantCoeff
--def adj_x_map' (I : Ideal R) : R[X] →+* R I := (Ideal.Quotient.mk I).comp (evalRingHom 0)
def adjoin_x (I : Ideal R) : Ideal (Polynomial R) := RingHom.ker (adj_x_map I)
def adjoin_x' (I : PrimeSpectrum R) : PrimeSpectrum (Polynomial R) where
asIdeal := adjoin_x I.asIdeal
IsPrime := RingHom.ker_isPrime _
/- This somehow isn't in Mathlib? -/
@[simp]
theorem span_singleton_one : span ({0} : Set R) = ⊥ := by simp only [span_singleton_eq_bot]
theorem coeff_C_eq : RingHom.comp constantCoeff C = RingHom.id R := by ext; simp
variable (I : PrimeSpectrum R)
#check RingHom.ker (C.comp (Ideal.Quotient.mk I.asIdeal))
--#check Ideal.Quotient.mk I.asIdeal
def map_prime' (I : PrimeSpectrum R) : IsPrime (I.asIdeal.map C) := Ideal.isPrime_map_C_of_isPrime I.IsPrime
def map_prime'' (I : PrimeSpectrum R) : PrimeSpectrum R[X] := ⟨I.asIdeal.map C, map_prime' I⟩
@[simp]
lemma adj_x_comp_C (I : Ideal R) : RingHom.comp (adj_x_map I) C = Ideal.Quotient.mk I := by
ext x; simp [adj_x_map]
-- ideal.mem_quotient_iff_mem_sup
lemma adjoin_x_eq (I : Ideal R) : adjoin_x I = I.map C ⊔ Ideal.span {X} := by
apply le_antisymm
. rintro p hp
have h : ∃ q r, p = C r + X * q := ⟨p.divX, p.coeff 0, p.divX_mul_X_add.symm.trans $ by ring⟩
obtain ⟨q, r, rfl⟩ := h
suffices : r ∈ I
. simp only [Submodule.mem_sup, Ideal.mem_span_singleton]
refine' ⟨C r, Ideal.mem_map_of_mem C this, X * q, ⟨q, rfl⟩, rfl⟩
rw [adjoin_x, adj_x_map, RingHom.mem_ker, RingHom.comp_apply] at hp
rw [constantCoeff_apply, coeff_add, coeff_C_zero, coeff_X_mul_zero, add_zero] at hp
rwa [←RingHom.mem_ker, Ideal.mk_ker] at hp
. rw [sup_le_iff]
constructor
. simp [adjoin_x, RingHom.ker, ←map_le_iff_le_comap, Ideal.map_map]
. simp [span_le, adjoin_x, RingHom.mem_ker, adj_x_map]
lemma adjoin_x_inj {I J : Ideal R} (h : adjoin_x I = adjoin_x J) : I = J := by
simp [adjoin_x_eq] at h
have H : Ideal.map constantCoeff (Ideal.map C I ⊔ span {X}) =
Ideal.map constantCoeff (Ideal.map C J ⊔ span {X}) := by rw [h]
simp [Ideal.map_sup, Ideal.map_span, Ideal.map_map, coeff_C_eq] at H
exact H
lemma map_lt_adjoin_x (I : PrimeSpectrum R) : map_prime'' I < adjoin_x' I := by
simp [map_prime'', adjoin_x', adjoin_x_eq]
show Ideal.map C I.asIdeal < Ideal.map C I.asIdeal ⊔ span {X}
simp [Ideal.span_le, mem_map_C_iff]
use 1
simp
intro h
apply I.IsPrime.ne_top'
rw [Ideal.eq_top_iff_one]
exact h
lemma map_inj {I J : Ideal R} (h : I.map C = J.map C) : I = J := by
have H : Ideal.map constantCoeff (Ideal.map C I) =
Ideal.map constantCoeff (Ideal.map C J) := by rw [h]
simp [Ideal.map_map, coeff_C_eq] at H
exact H
lemma map_strictmono (I J : Ideal R) (h : I < J) : I.map C < J.map C := by
rw [lt_iff_le_and_ne] at h ⊢
constructor
. apply map_mono h.1
. intro H
exact h.2 (map_inj H)
lemma adjoin_x_strictmono (I J : Ideal R) (h : I < J) : adjoin_x I < adjoin_x J := by
rw [lt_iff_le_and_ne] at h ⊢
constructor
. rw [adjoin_x_eq, adjoin_x_eq]
apply sup_le_sup_right
apply map_mono h.1
. intro H
exact h.2 (adjoin_x_inj H)
example (n : ℕ∞) : n + 0 = n := by simp?
#eval List.Chain' (· < ·) [2,3]
example : [4,5] ++ [2] = [4,5,2] := rfl
#eval [2,4,5].map (λ n => n + n)
/- Given an ideal p in R, define the ideal p[x] in R[x] -/
lemma ht_adjoin_x_eq_ht_add_one [Nontrivial R] (I : PrimeSpectrum R) : height I + 1 ≤ height (adjoin_x' I) := by
suffices H : height I + (1 : ) ≤ height (adjoin_x' I) + (0 : )
. norm_cast at H; rw [add_zero] at H; exact H
rw [height, height, Set.chainHeight_add_le_chainHeight_add {J | J < I} _ 1 0]
intro l hl
use ((l.map map_prime'') ++ [map_prime'' I])
constructor
. simp [Set.append_mem_subchain_iff]
refine' ⟨_,_,_⟩
. show (List.map map_prime'' l).Chain' (· < ·) ∧ ∀ i ∈ _, i ∈ _
constructor
. apply List.chain'_map_of_chain' map_prime''
intro a b hab
apply map_strictmono a.asIdeal b.asIdeal
exact hab
exact hl.1
. intro i hi
rw [List.mem_map] at hi
obtain ⟨a, ha, rfl⟩ := hi
show map_prime'' a < adjoin_x' I
calc map_prime'' a < map_prime'' I := by apply map_strictmono; apply hl.2; apply ha
_ < adjoin_x' I := by apply map_lt_adjoin_x
. apply map_lt_adjoin_x
. intro a ha
have H : ∃ b : PrimeSpectrum R, b ∈ l ∧ map_prime'' b = a
. have H2 : l ≠ []
. intro h
rw [h] at ha
tauto
use l.getLast H2
refine' ⟨List.getLast_mem H2, _⟩
have H3 : l.map map_prime'' ≠ []
. intro hl
apply H2
apply List.eq_nil_of_map_eq_nil hl
rw [List.getLast?_eq_getLast _ H3, Option.some_inj] at ha
simp [←ha, List.getLast_map _ H2]
obtain ⟨b, hb, rfl⟩ := H
apply map_strictmono
apply hl.2
exact hb
. simp
lemma ne_bot_iff_exists' (n : WithBot ℕ∞) : n ≠ ⊥ ↔ ∃ m : ℕ∞, n = m := by
convert WithBot.ne_bot_iff_exists using 3
exact comm
lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krullDim R + (1 : ℕ∞) ≤ krullDim (Polynomial R) := by
cases' krullDim_nonneg_of_nontrivial R with n hn
rw [hn]
change ↑(n + 1) ≤ krullDim R[X]
have hn' := le_of_eq hn.symm
rw [le_krullDim_iff'] at hn' ⊢
cases' hn' with I hI
use adjoin_x' I
apply WithBot.coe_mono
calc n + 1 ≤ height I + 1 := by
apply add_le_add_right
rw [WithBot.coe_le_coe] at hI
exact hI
_ ≤ height (adjoin_x' I) := ht_adjoin_x_eq_ht_add_one I
end Polynomial
open Ideal
variable {R : Type _} [CommRing R]
lemma height_le_top_iff_exists {I : PrimeSpectrum R} (hI : height I ≤ ) :
∃ n : , true := by
sorry
lemma eq_of_height_eq_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) (hJ : height J < )
(ht_eq : height I = height J) : I = J := by
by_cases h : I = J
. exact h
. have I_lt_J := lt_of_le_of_ne I_le_J h
exfalso
sorry
section Quotient
variables {R : Type _} [CommRing R] (I : Ideal R)
#check List.map <| PrimeSpectrum.comap <| Ideal.Quotient.mk I
lemma comap_chain {l : List (PrimeSpectrum (R I))} (hl : l.Chain' (· < ·)) :
List.Chain' (. < .) ((List.map <| PrimeSpectrum.comap <| Ideal.Quotient.mk I) l) := by
lemma dim_quotient_le_dim : krullDim (R I) ≤ krullDim R := by
end Quotient

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@ -171,6 +171,48 @@ lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ (1 :
apply (IsCoatom.lt_iff H.out).mp apply (IsCoatom.lt_iff H.out).mp
exact hc2 exact hc2
--refine Iff.mp radical_eq_top (?_ (id (Eq.symm hc3))) --refine Iff.mp radical_eq_top (?_ (id (Eq.symm hc3)))
lemma not_maximal_of_lt_prime {p : Ideal R} {q : Ideal R} (hq : IsPrime q) (h : p < q) : ¬IsMaximal p := by
intro hp
apply IsPrime.ne_top hq
apply (IsCoatom.lt_iff hp.out).mp
exact h
lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
show ((_ : WithBot ℕ∞) ≤ (0 : )) ↔ _
rw [krullDim_le_iff R 0]
constructor <;> intro h I
. contrapose! h
have ⟨𝔪, h𝔪⟩ := I.asIdeal.exists_le_maximal (IsPrime.ne_top I.IsPrime)
let 𝔪p := (⟨𝔪, IsMaximal.isPrime h𝔪.1⟩ : PrimeSpectrum R)
have hstrct : I < 𝔪p := by
apply lt_of_le_of_ne h𝔪.2
intro hcontr
rw [hcontr] at h
exact h h𝔪.1
use 𝔪p
show (_ : WithBot ℕ∞) > (0 : ℕ∞)
rw [_root_.lt_height_iff'']
use [I]
constructor
. exact List.chain'_singleton _
. constructor
. intro I' hI'
simp at hI'
rwa [hI']
. simp
. contrapose! h
change (_ : WithBot ℕ∞) > (0 : ℕ∞) at h
rw [_root_.lt_height_iff''] at h
obtain ⟨c, _, hc2, hc3⟩ := h
norm_cast at hc3
rw [List.length_eq_one] at hc3
obtain ⟨𝔮, h𝔮⟩ := hc3
use 𝔮
specialize hc2 𝔮 (h𝔮 ▸ (List.mem_singleton.mpr rfl))
apply not_maximal_of_lt_prime _ I.IsPrime
exact hc2
end Krull end Krull
section iSupWithBot section iSupWithBot

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import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Noetherian
import CommAlg.krull
variable (R : Type _) [CommRing R] [IsNoetherianRing R]
lemma height_le_of_gt_height_lt {n : ℕ∞} (q : PrimeSpectrum R)
(h : ∀(p : PrimeSpectrum R), p < q → Ideal.height p ≤ n - 1) : Ideal.height q ≤ n := by
sorry
theorem height_le_one_of_minimal_over_principle (p : PrimeSpectrum R) (x : R):
p ∈ minimals (· < ·) {p | x ∈ p.asIdeal} → Ideal.height p ≤ 1 := by
intro h
apply height_le_of_gt_height_lt _ p
intro q qlep
by_contra hcontr
push_neg at hcontr
simp only [le_refl, tsub_eq_zero_of_le] at hcontr
sorry
#check (_ : Ideal R) ^ (_ : )
#synth Pow (Ideal R) ()
def symbolicIdeal(Q : Ideal R) {hin : Q.IsPrime} (I : Ideal R) : Ideal R where
carrier := {r : R | ∃ s : R, s ∉ Q ∧ s * r ∈ I}
zero_mem' := by
simp only [Set.mem_setOf_eq, mul_zero, Submodule.zero_mem, and_true]
use 1
rw [←Q.ne_top_iff_one]
exact hin.ne_top
add_mem' := by
rintro a b ⟨sa, hsa1, hsa2⟩ ⟨sb, hsb1, hsb2⟩
use sa * sb
constructor
. intro h
cases hin.mem_or_mem h <;> contradiction
ring_nf
apply I.add_mem --<;> simp only [I.mul_mem_left, hsa2, hsb2]
. rw [mul_comm sa, mul_assoc]
exact I.mul_mem_left sb hsa2
. rw [mul_assoc]
exact I.mul_mem_left sa hsb2
smul_mem' := by
intro c x
dsimp
rintro ⟨s, hs1, hs2⟩
use s
constructor; exact hs1
rw [←mul_assoc, mul_comm s, mul_assoc]
exact Ideal.mul_mem_left _ _ hs2
protected lemma LocalRing.height_le_one_of_minimal_over_principle
[LocalRing R] (q : PrimeSpectrum R) {x : R}
(h : (closedPoint R).asIdeal ∈ (Ideal.span {x}).minimalPrimes) :
q = closedPoint R Ideal.height q = 0 := by
sorry

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@ -16,18 +16,18 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Algebra.Ring.Pi import Mathlib.Algebra.Ring.Pi
import Mathlib.RingTheory.Finiteness import Mathlib.RingTheory.Finiteness
import Mathlib.Util.PiNotation import Mathlib.Util.PiNotation
import CommAlg.krull
open PiNotation open PiNotation
namespace Ideal namespace Ideal
variable (R : Type _) [CommRing R] (P : PrimeSpectrum R) variable (R : Type _) [CommRing R] (P : PrimeSpectrum R)
noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < P} -- noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < P}
noncomputable def krullDim (R : Type) [CommRing R] : -- noncomputable def krullDim (R : Type) [CommRing R] :
WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height R I -- WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height R I
--variable {R} --variable {R}
@ -43,7 +43,6 @@ class IsLocallyNilpotent {R : Type _} [CommRing R] (I : Ideal R) : Prop :=
#check Ideal.IsLocallyNilpotent #check Ideal.IsLocallyNilpotent
end Ideal end Ideal
-- Repeats the definition of the length of a module by Monalisa -- Repeats the definition of the length of a module by Monalisa
variable (R : Type _) [CommRing R] (I J : Ideal R) variable (R : Type _) [CommRing R] (I J : Ideal R)
variable (M : Type _) [AddCommMonoid M] [Module R M] variable (M : Type _) [AddCommMonoid M] [Module R M]
@ -67,9 +66,11 @@ lemma ring_Noetherian_iff_spec_Noetherian : IsNoetherianRing R
↔ TopologicalSpace.NoetherianSpace (PrimeSpectrum R) := by ↔ TopologicalSpace.NoetherianSpace (PrimeSpectrum R) := by
constructor constructor
intro RisNoetherian intro RisNoetherian
sorry
sorry
-- how do I apply an instance to prove one direction? -- how do I apply an instance to prove one direction?
-- Stacks Lemma 5.9.2:
-- Use TopologicalSpace.NoetherianSpace.exists_finset_irreducible : -- Use TopologicalSpace.NoetherianSpace.exists_finset_irreducible :
-- Every closed subset of a noetherian space is a finite union -- Every closed subset of a noetherian space is a finite union
-- of irreducible closed subsets. -- of irreducible closed subsets.
@ -100,9 +101,9 @@ lemma containment_radical_power_containment :
-- Stacks Lemma 10.52.6: I is a maximal ideal and IM = 0. Then length of M is -- Stacks Lemma 10.52.6: I is a maximal ideal and IM = 0. Then length of M is
-- the same as the dimension as a vector space over R/I, -- the same as the dimension as a vector space over R/I,
lemma length_eq_dim_if_maximal_annihilates {I : Ideal R} [Ideal.IsMaximal I] -- lemma length_eq_dim_if_maximal_annihilates {I : Ideal R} [Ideal.IsMaximal I]
: I • ( : Submodule R M) = 0 -- : I • ( : Submodule R M) = 0
→ Module.length R M = Module.rank RI M(I • ( : Submodule R M)) := by sorry -- → Module.length R M = Module.rank RI M(I • ( : Submodule R M)) := by sorry
-- Does lean know that M/IM is a R/I module? -- Does lean know that M/IM is a R/I module?
-- Use 10.52.5 -- Use 10.52.5
@ -116,19 +117,44 @@ lemma power_zero_finite_length [Ideal.IsMaximal I] (h₁ : Ideal.FG I) [Module.F
-- rcases power with ⟨n, npower⟩ -- rcases power with ⟨n, npower⟩
-- how do I get a generating set? -- how do I get a generating set?
open Finset
-- Stacks Lemma 10.53.3: R is Artinian iff R has finitely many maximal ideals -- Stacks Lemma 10.53.3: R is Artinian iff R has finitely many maximal ideals
lemma Artinian_has_finite_max_ideal lemma Artinian_has_finite_max_ideal
[IsArtinianRing R] : Finite (MaximalSpectrum R) := by [IsArtinianRing R] : Finite (MaximalSpectrum R) := by
by_contra infinite by_contra infinite
simp only [not_finite_iff_infinite] at infinite simp only [not_finite_iff_infinite] at infinite
let m' : ↪ MaximalSpectrum R := Infinite.natEmbedding (MaximalSpectrum R)
have m'inj := m'.injective
let m'' : → Ideal R := fun n : ↦ ⨅ k ∈ range n, (m' k).asIdeal
-- let f : → MaximalSpectrum R := fun n : ↦ m' n
-- let F : (n : ) → Fin n → MaximalSpectrum R := fun n k ↦ m' k
have DCC : ∃ n : , ∀ k : , n ≤ k → m'' n = m'' k := by
apply IsArtinian.monotone_stabilizes {
toFun := m''
monotone' := sorry
}
cases' DCC with n DCCn
specialize DCCn (n+1)
specialize DCCn (Nat.le_succ n)
have containment1 : m'' n < (m' (n + 1)).asIdeal := by sorry
have : ∀ (j : ), (j ≠ n + 1) → ∃ x, x ∈ (m' j).asIdeal ∧ x ∉ (m' (n+1)).asIdeal := by
intro j jnotn
have notcontain : ¬ (m' j).asIdeal ≤ (m' (n+1)).asIdeal := by
-- apply Ideal.IsMaximal (m' j).asIdeal
sorry
sorry
sorry
-- have distinct : (m' j).asIdeal ≠ (m' (n+1)).asIdeal := by
-- intro h
-- apply Function.Injective.ne m'inj jnotn
-- exact MaximalSpectrum.ext _ _ h
-- simp
-- unfold
-- Stacks Lemma 10.53.4: R Artinian => Jacobson ideal of R is nilpotent -- Stacks Lemma 10.53.4: R Artinian => Jacobson ideal of R is nilpotent
lemma Jacobson_of_Artinian_is_nilpotent -- This is in mathlib: IsArtinianRing.isNilpotent_jacobson_bot
[IsArtinianRing R] : IsNilpotent (Ideal.jacobson (⊥ : Ideal R)) := by
have J := Ideal.jacobson (⊥ : Ideal R)
-- Stacks Lemma 10.53.5: If R has finitely many maximal ideals and -- 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 -- locally nilpotent Jacobson radical, then R is the product of its localizations at
@ -142,7 +168,7 @@ abbrev Prod_of_localization :=
def foo : Prod_of_localization R →+* R where def foo : Prod_of_localization R →+* R where
toFun := sorry toFun := sorry
invFun := sorry -- invFun := sorry
left_inv := sorry left_inv := sorry
right_inv := sorry right_inv := sorry
map_mul' := sorry map_mul' := sorry
@ -167,23 +193,27 @@ lemma primes_of_Artinian_are_maximal
-- Lemma: Krull dimension of a ring is the supremum of height of maximal ideals -- 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 -- 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] : lemma dim_le_zero_Noetherian_iff_Artinian (R : Type _) [CommRing R] :
IsNoetherianRing R ∧ Ideal.krullDim R = 0 ↔ IsArtinianRing R := by IsNoetherianRing R ∧ Ideal.krullDim R 0 ↔ IsArtinianRing R := by
constructor constructor
rintro ⟨RisNoetherian, dimzero⟩
rw [ring_Noetherian_iff_spec_Noetherian] at RisNoetherian
let Z := irreducibleComponents (PrimeSpectrum R)
have Zfinite : Set.Finite Z := by
-- apply TopologicalSpace.NoetherianSpace.finite_irreducibleComponents ?_
sorry
sorry sorry
intro RisArtinian intro RisArtinian
constructor constructor
apply finite_length_is_Noetherian apply finite_length_is_Noetherian
rwa [IsArtinian_iff_finite_length] at RisArtinian rwa [IsArtinian_iff_finite_length] at RisArtinian
sorry rw [Ideal.dim_le_zero_iff]
intro I
apply primes_of_Artinian_are_maximal
-- Use TopologicalSpace.NoetherianSpace.exists_finset_irreducible :

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@ -4,6 +4,7 @@ import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain 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.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.AtPrime import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.ConditionallyCompleteLattice.Basic
@ -18,6 +19,11 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
developed. developed.
-/ -/
lemma lt_bot_eq_WithBot_bot [PartialOrder α] [OrderBot α] {a : WithBot α} (h : a < (⊥ : α)) : a = ⊥ := by
cases a
. rfl
. cases h.not_le (WithBot.coe_le_coe.2 bot_le)
namespace Ideal namespace Ideal
open LocalRing open LocalRing
@ -27,6 +33,8 @@ noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J <
noncomputable def krullDim (R : Type _) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I noncomputable def krullDim (R : Type _) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
noncomputable def codimension (J : Ideal R) : WithBot ℕ∞ := ⨅ I ∈ {I : PrimeSpectrum R | J ≤ I.asIdeal}, height I
lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
lemma krullDim_def (R : Type _) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl lemma krullDim_def (R : Type _) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
lemma krullDim_def' (R : Type _) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl lemma krullDim_def' (R : Type _) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
@ -45,16 +53,52 @@ lemma krullDim_le_iff (R : Type _) [CommRing R] (n : ) :
lemma krullDim_le_iff' (R : Type _) [CommRing R] (n : ℕ∞) : lemma krullDim_le_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞) krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot ℕ∞)
lemma le_krullDim_iff (R : Type _) [CommRing R] (n : ) :
n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by sorry
lemma le_krullDim_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by sorry
@[simp] @[simp]
lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R := lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
le_iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) I le_iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) I
lemma le_krullDim_iff (R : Type _) [CommRing R] (n : ) :
n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by
constructor
· unfold krullDim
intro H
by_contra H1
push_neg at H1
by_cases n ≤ 0
· rw [Nat.le_zero] at h
rw [h] at H H1
have : ∀ (I : PrimeSpectrum R), ↑(height I) = (⊥ : WithBot ℕ∞) := by
intro I
specialize H1 I
exact lt_bot_eq_WithBot_bot H1
rw [←iSup_eq_bot] at this
have := le_of_le_of_eq H this
rw [le_bot_iff] at this
exact WithBot.coe_ne_bot this
· push_neg at h
have : (n: ℕ∞) > 0 := Nat.cast_pos.mpr h
replace H1 : ∀ (I : PrimeSpectrum R), height I ≤ n - 1 := by
intro I
specialize H1 I
apply ENat.le_of_lt_add_one
rw [←ENat.coe_one, ←ENat.coe_sub, ←ENat.coe_add, tsub_add_cancel_of_le]
exact WithBot.coe_lt_coe.mp H1
exact h
replace H1 : ∀ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) ≤ ↑(n - 1):=
fun _ ↦ (WithBot.coe_le rfl).mpr (H1 _)
rw [←iSup_le_iff] at H1
have : ((n : ℕ∞) : WithBot ℕ∞) ≤ (((n - 1 : ) : ℕ∞) : WithBot ℕ∞) := le_trans H H1
norm_cast at this
have that : n - 1 < n := by refine Nat.sub_lt h (by norm_num)
apply lt_irrefl (n-1) (trans that this)
· rintro ⟨I, h⟩
have : height I ≤ krullDim R := by apply height_le_krullDim
exact le_trans h this
lemma le_krullDim_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by sorry
/-- The Krull dimension of a local ring is the height of its maximal ideal. -/
lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) := by lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) := by
apply le_antisymm apply le_antisymm
. rw [krullDim_le_iff'] . rw [krullDim_le_iff']
@ -65,12 +109,46 @@ lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) :=
exact I.2.1 exact I.2.1
. simp only [height_le_krullDim] . simp only [height_le_krullDim]
/-- The height of a prime `𝔭` is greater than `n` if and only if there is a chain of primes less than `𝔭`
with length `n + 1`. -/
lemma lt_height_iff' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
n < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
match n with
| =>
constructor <;> intro h <;> exfalso
. exact (not_le.mpr h) le_top
. tauto
| (n : ) =>
have (m : ℕ∞) : n < m ↔ (n + 1 : ℕ∞) ≤ m := by
symm
show (n + 1 ≤ m ↔ _ )
apply ENat.add_one_le_iff
exact ENat.coe_ne_top _
rw [this]
unfold Ideal.height
show ((↑(n + 1):ℕ∞) ≤ _) ↔ ∃c, _ ∧ _ ∧ ((_ : WithTop ) = (_:ℕ∞))
rw [{J | J < 𝔭}.le_chainHeight_iff]
show (∃ c, (List.Chain' _ c ∧ ∀𝔮, 𝔮 ∈ c → 𝔮 < 𝔭) ∧ _) ↔ _
constructor <;> rintro ⟨c, hc⟩ <;> use c
. tauto
. change _ ∧ _ ∧ (List.length c : ℕ∞) = n + 1 at hc
norm_cast at hc
tauto
/-- Form of `lt_height_iff''` for rewriting with the height coerced to `WithBot ℕ∞`. -/
lemma lt_height_iff'' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
(n : WithBot ℕ∞) < height 𝔭 ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
rw [WithBot.coe_lt_coe]
exact lt_height_iff'
#check height_le_krullDim #check height_le_krullDim
--some propositions that would be nice to be able to eventually --some propositions that would be nice to be able to eventually
/-- The prime spectrum of the zero ring is empty. -/
lemma primeSpectrum_empty_of_subsingleton (x : PrimeSpectrum R) [Subsingleton R] : False := lemma primeSpectrum_empty_of_subsingleton (x : PrimeSpectrum R) [Subsingleton R] : False :=
x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem
/-- A CommRing has empty prime spectrum if and only if it is the zero ring. -/
lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by
constructor constructor
. contrapose . contrapose
@ -94,17 +172,68 @@ lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
. rw [h.forall_iff] . rw [h.forall_iff]
trivial trivial
/-- Nonzero rings have Krull dimension in ℕ∞ -/
lemma krullDim_nonneg_of_nontrivial (R : Type _) [CommRing R] [Nontrivial R] : ∃ n : ℕ∞, Ideal.krullDim R = n := by lemma krullDim_nonneg_of_nontrivial (R : Type _) [CommRing R] [Nontrivial R] : ∃ n : ℕ∞, Ideal.krullDim R = n := by
have h := dim_eq_bot_iff.not.mpr (not_subsingleton R) have h := dim_eq_bot_iff.not.mpr (not_subsingleton R)
lift (Ideal.krullDim R) to ℕ∞ using h with k lift (Ideal.krullDim R) to ℕ∞ using h with k
use k use k
/-- An ideal which is less than a prime is not a maximal ideal. -/
lemma not_maximal_of_lt_prime {p : Ideal R} {q : Ideal R} (hq : IsPrime q) (h : p < q) : ¬IsMaximal p := by
intro hp
apply IsPrime.ne_top hq
apply (IsCoatom.lt_iff hp.out).mp
exact h
/-- Krull dimension is ≤ 0 if and only if all primes are maximal. -/
lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
show ((_ : WithBot ℕ∞) ≤ (0 : )) ↔ _
rw [krullDim_le_iff R 0]
constructor <;> intro h I
. contrapose! h
have ⟨𝔪, h𝔪⟩ := I.asIdeal.exists_le_maximal (IsPrime.ne_top I.IsPrime)
let 𝔪p := (⟨𝔪, IsMaximal.isPrime h𝔪.1⟩ : PrimeSpectrum R)
have hstrct : I < 𝔪p := by
apply lt_of_le_of_ne h𝔪.2
intro hcontr
rw [hcontr] at h
exact h h𝔪.1
use 𝔪p
show (0 : ℕ∞) < (_ : WithBot ℕ∞)
rw [lt_height_iff'']
use [I]
constructor
. exact List.chain'_singleton _
. constructor
. intro I' hI'
simp at hI'
rwa [hI']
. simp
. contrapose! h
change (0 : ℕ∞) < (_ : WithBot ℕ∞) at h
rw [lt_height_iff''] at h
obtain ⟨c, _, hc2, hc3⟩ := h
norm_cast at hc3
rw [List.length_eq_one] at hc3
obtain ⟨𝔮, h𝔮⟩ := hc3
use 𝔮
specialize hc2 𝔮 (h𝔮 ▸ (List.mem_singleton.mpr rfl))
apply not_maximal_of_lt_prime I.IsPrime
exact hc2
/-- For a nonzero ring, Krull dimension is 0 if and only if all primes are maximal. -/
lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
constructor <;> intro h rw [←dim_le_zero_iff]
. intro I obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial R
sorry have : n ≥ 0 := zero_le n
. sorry change _ ≤ _ at this
rw [←WithBot.coe_le_coe,←hn] at this
change (0 : WithBot ℕ∞) ≤ _ at this
constructor <;> intro h'
. rw [h']
. exact le_antisymm h' this
/-- In a field, the unique prime ideal is the zero ideal. -/
@[simp] @[simp]
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
constructor constructor
@ -116,6 +245,7 @@ lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P =
rw [botP] rw [botP]
exact bot_prime exact bot_prime
/-- In a field, all primes have height 0. -/
lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
unfold height unfold height
simp simp
@ -131,10 +261,12 @@ lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : heig
have : J ≠ P := ne_of_lt JlP have : J ≠ P := ne_of_lt JlP
contradiction contradiction
/-- The Krull dimension of a field is 0. -/
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
unfold krullDim unfold krullDim
simp [field_prime_height_zero] simp [field_prime_height_zero]
/-- A domain with Krull dimension 0 is a field. -/
lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
by_contra x by_contra x
rw [Ring.not_isField_iff_exists_prime] at x rw [Ring.not_isField_iff_exists_prime] at x
@ -149,55 +281,29 @@ lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim
have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
unfold height unfold height
rw [←Set.one_le_chainHeight_iff] at this rw [←Set.one_le_chainHeight_iff] at this
exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this) exact not_le_of_gt (ENat.one_le_iff_pos.mp this)
have nonpos_height : height P' ≤ 0 := by have nonpos_height : height P' ≤ 0 := by
have := height_le_krullDim P' have := height_le_krullDim P'
rw [h] at this rw [h] at this
aesop aesop
contradiction contradiction
/-- A domain has Krull dimension 0 if and only if it is a field. -/
lemma domain_dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by lemma domain_dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
constructor constructor
· exact domain_dim_zero.isField · exact domain_dim_zero.isField
· intro fieldD · intro fieldD
let h : Field D := IsField.toField fieldD let h : Field D := fieldD.toField
exact dim_field_eq_zero exact dim_field_eq_zero
#check Ring.DimensionLEOne #check Ring.DimensionLEOne
-- This lemma is false! -- This lemma is false!
lemma dim_le_one_iff : krullDim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry lemma dim_le_one_iff : krullDim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry
lemma lt_height_iff' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
height 𝔭 > n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
rcases n with _ | n
. constructor <;> intro h <;> exfalso
. exact (not_le.mpr h) le_top
. tauto
have (m : ℕ∞) : m > some n ↔ m ≥ some (n + 1) := by
symm
show (n + 1 ≤ m ↔ _ )
apply ENat.add_one_le_iff
exact ENat.coe_ne_top _
rw [this]
unfold Ideal.height
show ((↑(n + 1):ℕ∞) ≤ _) ↔ ∃c, _ ∧ _ ∧ ((_ : WithTop ) = (_:ℕ∞))
rw [{J | J < 𝔭}.le_chainHeight_iff]
show (∃ c, (List.Chain' _ c ∧ ∀𝔮, 𝔮 ∈ c → 𝔮 < 𝔭) ∧ _) ↔ _
constructor <;> rintro ⟨c, hc⟩ <;> use c
. tauto
. change _ ∧ _ ∧ (List.length c : ℕ∞) = n + 1 at hc
norm_cast at hc
tauto
lemma lt_height_iff'' {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
height 𝔭 > (n : WithBot ℕ∞) ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀ 𝔮 ∈ c, 𝔮 < 𝔭) ∧ c.length = n + 1 := by
show (_ < _) ↔ _
rw [WithBot.coe_lt_coe]
exact lt_height_iff'
/-- The converse of this is false, because the definition of "dimension ≤ 1" in mathlib /-- The converse of this is false, because the definition of "dimension ≤ 1" in mathlib
applies only to dimension zero rings and domains of dimension 1. -/ applies only to dimension zero rings and domains of dimension 1. -/
lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ (1 : ) := by lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ 1 := by
show _ → ((_ : WithBot ℕ∞) ≤ (1 : ))
rw [krullDim_le_iff R 1] rw [krullDim_le_iff R 1]
intro H p intro H p
apply le_of_not_gt apply le_of_not_gt
@ -212,10 +318,9 @@ lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ (1 :
change q0.asIdeal < q1.asIdeal at hc1 change q0.asIdeal < q1.asIdeal at hc1
have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1 have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1
specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime
apply IsPrime.ne_top p.IsPrime exact (not_maximal_of_lt_prime p.IsPrime hc2) H
apply (IsCoatom.lt_iff H.out).mp
exact hc2
/-- The Krull dimension of a PID is at most 1. -/
lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by
rw [dim_le_one_iff] rw [dim_le_one_iff]
exact Ring.DimensionLEOne.principal_ideal_ring R exact Ring.DimensionLEOne.principal_ideal_ring R
@ -223,8 +328,12 @@ lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1
lemma dim_le_dim_polynomial_add_one [Nontrivial R] : lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krullDim R + 1 ≤ krullDim (Polynomial R) := sorry krullDim R + 1 ≤ krullDim (Polynomial R) := sorry
lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] : -- lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
krullDim R + 1 = krullDim (Polynomial R) := sorry -- krullDim R + 1 = krullDim (Polynomial R) := sorry
lemma krull_height_theorem [Nontrivial R] [IsNoetherianRing R] (P: PrimeSpectrum R) (S: Finset R)
(h: P.asIdeal ∈ Ideal.minimalPrimes (Ideal.span S)) : height P ≤ S.card := by
sorry
lemma dim_mvPolynomial [Field K] (n : ) : krullDim (MvPolynomial (Fin n) K) = n := sorry lemma dim_mvPolynomial [Field K] (n : ) : krullDim (MvPolynomial (Fin n) K) = n := sorry

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@ -6,10 +6,20 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.DedekindDomain.DVR import Mathlib.RingTheory.DedekindDomain.DVR
lemma FieldisArtinian (R : Type _) [CommRing R] (h: IsField R) : lemma FieldisArtinian (R : Type _) [CommRing R] (h : IsField R) :
IsArtinianRing R := by sorry IsArtinianRing R := by
let inst := h.toField
rw [isArtinianRing_iff, isArtinian_iff_wellFounded]
apply WellFounded.intro
intro I
constructor
intro J hJI
constructor
intro K hKJ
exfalso
rcases Ideal.eq_bot_or_top J with rfl | rfl
. exact not_lt_bot hKJ
. exact not_top_lt hJI
lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R] lemma ArtinianDomainIsField (R : Type _) [CommRing R] [IsDomain R]
(IsArt : IsArtinianRing R) : IsField (R) := by (IsArt : IsArtinianRing R) : IsField (R) := by

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@ -3,34 +3,59 @@ import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain 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.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.AtPrime import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Ncard
import CommAlg.krull
namespace Ideal namespace Ideal
variable {R : Type _} [CommRing R] (I : PrimeSpectrum R) 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 I
lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl /--
lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl -- noncomputable def height : ℕ∞ := Set.chainHeight {J : PrimeSpectrum R | J < I}
lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl -- noncomputable def krullDim (R : Type) [CommRing R] : WithBot ℕ∞ := ⨆ (I : PrimeSpectrum R), height I
-- lemma height_def : height I = Set.chainHeight {J : PrimeSpectrum R | J < I} := rfl
-- lemma krullDim_def (R : Type) [CommRing R] : krullDim R = (⨆ (I : PrimeSpectrum R), height I : WithBot ℕ∞) := rfl
-- lemma krullDim_def' (R : Type) [CommRing R] : krullDim R = iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) := rfl
noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice
lemma dim_le_dim_polynomial_add_one [Nontrivial R] : -- lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krullDim R + 1 ≤ krullDim (Polynomial R) := sorry -- Others are working on it -- krullDim R + 1 ≤ krullDim (Polynomial R) := sorry -- Others are working on it
lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := sorry -- Already done in main file -- lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := sorry -- Already done in main file
lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] : -- lemma primeIdeal_finite_height_of_noetherianRing [Nontrivial R] [IsNoetherianRing R]
-- (P: PrimeSpectrum R) : height P ≠ := by
-- sorry
--/
lemma exist_elts_MinimalOver_of_primeIdeal_of_noetherianRing [Nontrivial R] [IsNoetherianRing R]
(P: PrimeSpectrum R) (h : height P < ) :
∃S : Set R, Set.ncard s = height P ∧ P.asIdeal ∈ Ideal.minimalPrimes (Ideal.span S) := by
sorry
lemma dim_eq_dim_polynomial_add_one [h1: Nontrivial R] [IsNoetherianRing R] :
krullDim R + 1 = krullDim (Polynomial R) := by krullDim R + 1 = krullDim (Polynomial R) := by
rw [le_antisymm_iff] rw [le_antisymm_iff]
constructor constructor
· exact dim_le_dim_polynomial_add_one · exact dim_le_dim_polynomial_add_one
· unfold krullDim · by_cases krullDim R =
have htPBdd : ∀ (P : PrimeSpectrum (Polynomial R)), (height P : WithBot ℕ∞) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by calc
krullDim (Polynomial R) ≤ := le_top
_ ≤ krullDim R := top_le_iff.mpr h
_ ≤ krullDim R + 1 := by
apply le_of_eq
rw [h]
rfl
have h:= Ne.lt_top h
unfold krullDim
have htPBdd : ∀ (P : PrimeSpectrum (Polynomial R)), (height P : WithBot ℕ∞)
≤ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
intro P intro P
have : ∃ (I : PrimeSpectrum R), (height P : WithBot ℕ∞) ≤ ↑(height I + 1) := by have : ∃ (I : PrimeSpectrum R), (height P : WithBot ℕ∞) ≤ ↑(height I + 1) := by
have : ∃ M, Ideal.IsMaximal M ∧ P.asIdeal ≤ M := by have : ∃ M, Ideal.IsMaximal M ∧ P.asIdeal ≤ M := by
@ -43,7 +68,7 @@ lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
have : height P ≤ height P' := height_le_of_le PleP' have : height P ≤ height P' := height_le_of_le PleP'
simp only [WithBot.coe_le_coe] simp only [WithBot.coe_le_coe]
have : ∃ (I : PrimeSpectrum R), height P' ≤ height I + 1 := by have : ∃ (I : PrimeSpectrum R), height P' ≤ height I + 1 := by
-- Prime avoidance is called subset_union_prime
sorry sorry
obtain ⟨I, h⟩ := this obtain ⟨I, h⟩ := this
use I use I
@ -52,7 +77,8 @@ lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
have : (↑(height I + 1) : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum R), ↑(height I + 1) := by have : (↑(height I + 1) : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum R), ↑(height I + 1) := by
apply @le_iSup (WithBot ℕ∞) _ _ _ I apply @le_iSup (WithBot ℕ∞) _ _ _ I
exact ge_trans this IP exact ge_trans this IP
have oneOut : (⨆ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) + 1) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by have oneOut : (⨆ (I : PrimeSpectrum R), (height I : WithBot ℕ∞) + 1)
≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := by
have : ∀ P : PrimeSpectrum R, (height P : WithBot ℕ∞) + 1 ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 := have : ∀ P : PrimeSpectrum R, (height P : WithBot ℕ∞) + 1 ≤ (⨆ (I : PrimeSpectrum R), ↑(height I)) + 1 :=
fun P ↦ (by apply add_le_add_right (@le_iSup (WithBot ℕ∞) _ _ _ P) 1) fun P ↦ (by apply add_le_add_right (@le_iSup (WithBot ℕ∞) _ _ _ P) 1)
apply iSup_le apply iSup_le

View file

@ -0,0 +1,46 @@
import CommAlg.krull
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic
namespace Ideal
lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
-- unfold krullDim
rw [le_antisymm_iff]
constructor
·
sorry
· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
· obtain ⟨I, h⟩ := this
have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by
apply @le_iSup (WithBot ℕ∞) _ _ _ I
exact le_trans h this
have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _
let X := @Polynomial.X K _
have : IsPrime (span {X}) := by
refine Iff.mpr (span_singleton_prime ?hp) primeX
exact Polynomial.X_ne_zero
let P := PrimeSpectrum.mk (span {X}) this
unfold height
use P
have : ⊥ ∈ {J | J < P} := by
simp only [Set.mem_setOf_eq]
rw [bot_lt_iff_ne_bot]
suffices : P.asIdeal ≠ ⊥
· by_contra x
rw [PrimeSpectrum.ext_iff] at x
contradiction
by_contra x
simp only [span_singleton_eq_bot] at x
have := @Polynomial.X_ne_zero K _ _
contradiction
have : {J | J < P}.Nonempty := Set.nonempty_of_mem this
rwa [←Set.one_le_chainHeight_iff, ←WithBot.one_le_coe] at this