This commit is contained in:
Sameer Savkar 2023-06-14 11:50:35 -07:00
commit c3472c9ccd
8 changed files with 650 additions and 169 deletions

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.DS_Store vendored

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@ -83,6 +83,16 @@ height 𝔭 > n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ (∀
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' _
lemma height_le_iff {𝔭 : PrimeSpectrum R} {n : ℕ∞} :
height 𝔭 ≤ n ↔ ∀ c : List (PrimeSpectrum R), c ∈ {I : PrimeSpectrum R | I < 𝔭}.subchain ∧ c.length = n + 1 := by
sorry
lemma krullDim_nonneg_of_nontrivial [Nontrivial R] : ∃ n : ℕ∞, Ideal.krullDim R = n := by
have h := dim_eq_bot_iff.not.mpr (not_subsingleton R)
lift (Ideal.krullDim R) to ℕ∞ using h with k
@ -116,19 +126,103 @@ lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
constructor <;> intro h
. rw [←not_nonempty_iff]
rintro ⟨a, ha⟩
-- specialize h ⟨a, ha⟩
specialize h ⟨a, ha⟩
tauto
. rw [h.forall_iff]
trivial
#check (sorry : False)
#check (sorryAx)
#check (4 : WithBot ℕ∞)
#check List.sum
#check (_ ∈ (_ : List _))
variable (α : Type )
#synth Membership α (List α)
#check bot_lt_iff_ne_bot
-- #check ((4 : ℕ∞) : WithBot (WithTop ))
-- #check ( (Set.chainHeight s) : WithBot (ℕ∞))
variable (P : PrimeSpectrum R)
/-- 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. -/
lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ (1 : ) := by
rw [krullDim_le_iff R 1]
-- unfold Ring.DimensionLEOne
intro H p
-- have Hp := H p.asIdeal
-- have Hp := fun h => (Hp h) p.IsPrime
apply le_of_not_gt
intro h
rcases ((lt_height_iff'' R).mp h) with ⟨c, ⟨hc1, hc2, hc3⟩⟩
norm_cast at hc3
rw [List.chain'_iff_get] at hc1
specialize hc1 0 (by rw [hc3]; simp)
-- generalize hq0 : List.get _ _ = q0 at hc1
set q0 : PrimeSpectrum R := List.get c { val := 0, isLt := _ }
set q1 : PrimeSpectrum R := List.get c { val := 1, isLt := _ }
-- have hq0 : q0 ∈ c := List.get_mem _ _ _
-- have hq1 : q1 ∈ c := List.get_mem _ _ _
specialize hc2 q1 (List.get_mem _ _ _)
-- have aa := (bot_le : (⊥ : Ideal R) ≤ q0.asIdeal)
change q0.asIdeal < q1.asIdeal at hc1
have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1
specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime
-- change q1.asIdeal < p.asIdeal at hc2
apply IsPrime.ne_top p.IsPrime
apply (IsCoatom.lt_iff H.out).mp
exact hc2
--refine Iff.mp radical_eq_top (?_ (id (Eq.symm hc3)))
end Krull
#check {J | J < P}.le_chainHeight_iff (n := 4)
section iSupWithBot
variable {α : Type _} [CompleteSemilatticeSup α] {I : Type _} (f : I → α)
#synth SupSet (WithBot ℕ∞)
protected lemma WithBot.iSup_ge_coe_iff {a : α} :
(a ≤ ⨆ i : I, (f i : WithBot α) ) ↔ ∃ i : I, f i ≥ a := by
rw [WithBot.coe_le_iff]
sorry
end iSupWithBot
section myGreatElab
open Lean Meta Elab
syntax (name := lhsStx) "lhs% " term:max : term
syntax (name := rhsStx) "rhs% " term:max : term
@[term_elab lhsStx, term_elab rhsStx]
def elabLhsStx : Term.TermElab := fun stx expectedType? =>
match stx with
| `(lhs% $t) => do
let (lhs, _, eq) ← mkExpected expectedType?
discard <| Term.elabTermEnsuringType t eq
return lhs
| `(rhs% $t) => do
let (_, rhs, eq) ← mkExpected expectedType?
discard <| Term.elabTermEnsuringType t eq
return rhs
| _ => throwUnsupportedSyntax
where
mkExpected expectedType? := do
let α
if let some expectedType := expectedType? then
pure expectedType
else
mkFreshTypeMVar
let lhs ← mkFreshExprMVar α
let rhs ← mkFreshExprMVar α
let u ← getLevel α
let eq := mkAppN (.const ``Eq [u]) #[α, lhs, rhs]
return (lhs, rhs, eq)
#check lhs% (add_comm 1 2)
#check rhs% (add_comm 1 2)
#check lhs% (add_comm _ _ : _ = 1 + 2)
example (x y : α) (h : x = y) : lhs% h = rhs% h := h
def lhsOf {α : Sort _} {x y : α} (h : x = y) : α := x
#check lhsOf (add_comm 1 2)

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@ -0,0 +1,137 @@
import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Finiteness
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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.Localization.AtPrime
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Algebra.DirectSum.Ring
import Mathlib.RingTheory.Ideal.LocalRing
-- Setting for "library_search"
set_option maxHeartbeats 0
macro "ls" : tactic => `(tactic|library_search)
-- New tactic "obviously"
macro "obviously" : tactic =>
`(tactic| (
first
| dsimp; simp; done; dbg_trace "it was dsimp simp"
| simp; done; dbg_trace "it was simp"
| tauto; done; dbg_trace "it was tauto"
| simp; tauto; done; dbg_trace "it was simp tauto"
| rfl; done; dbg_trace "it was rfl"
| norm_num; done; dbg_trace "it was norm_num"
| /-change (@Eq _ _);-/ linarith; done; dbg_trace "it was linarith"
-- | gcongr; done
| ring; done; dbg_trace "it was ring"
| trivial; done; dbg_trace "it was trivial"
-- | nlinarith; done
| fail "No, this is not obvious."))
-- @[BH, 1.5.6 (b)(ii)]
lemma ss (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (p : associatedPrimes (⨁ i, 𝒜 i) (⨁ i, 𝓜 i)) : true := by
sorry
-- Ideal.IsHomogeneous 𝒜 p
noncomputable def length ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < }
def HomogeneousPrime { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous 𝒜 I)
def HomogeneousMax { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous 𝒜 I)
--theorem monotone_stabilizes_iff_noetherian :
-- (∀ f : →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
open GradedMonoid.GSmul
open DirectSum
instance tada1 (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (i : ) : SMul (𝒜 0) (𝓜 i)
where smul x y := @Eq.rec (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
lemma mylem (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ) (a : 𝒜 0) (m : 𝓜 i) :
of _ _ (a • m) = of _ _ a • of _ _ m := by
refine' Eq.trans _ (Gmodule.of_smul_of 𝒜 𝓜 a m).symm
refine' of_eq_of_gradedMonoid_eq _
exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
instance tada2 (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ) : SMulWithZero (𝒜 0) (𝓜 i) := by
letI := SMulWithZero.compHom (⨁ i, 𝓜 i) (of 𝒜 0).toZeroHom
exact Function.Injective.smulWithZero (of 𝓜 i).toZeroHom Dfinsupp.single_injective (mylem 𝒜 𝓜 i)
instance tada3 (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ): Module (𝒜 0) (𝓜 i) := by
letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)
noncomputable def hilbert_function (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (hilb : ) := ∀ i, hilb i = (ENat.toNat (length (𝒜 0) (𝓜 i)))
noncomputable def dimensionring { A: Type _}
[CommRing A] := krullDim (PrimeSpectrum A)
noncomputable def dimensionmodule ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A (( : Submodule A M).annihilator)) )
-- lemma graded_local (𝒜 : → Type _) [SetLike (⨁ i, 𝒜 i)] (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
-- [DirectSum.GCommRing 𝒜]
-- [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry
@[simp]
def PolyType (f : ) (d : ) := ∃ Poly : Polynomial , ∃ (N : ), ∀ (n : ), N ≤ n → f n = Polynomial.eval (n : ) Poly ∧ d = Polynomial.degree Poly
-- @[BH, 4.1.3]
theorem hilbert_polynomial (d : ) (d1 : 1 ≤ d) (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
(fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d) (hilb : )
(Hhilb: hilbert_function 𝒜 𝓜 hilb)
: PolyType hilb (d - 1) := by
sorry
-- @
lemma Graded_quotient (𝒜 : → Type _) [∀ i, AddCommGroup (𝒜 i)][DirectSum.GCommRing 𝒜]
: true := by
sorry
-- @Existence of a chain of submodules of graded submoduels of f.g graded R-mod M
lemma Exist_chain_of_graded_submodules (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜] [DirectSum.Gmodule 𝒜 𝓜] (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
: true := by
sorry
#print Exist_chain_of_graded_submodules
#print Finset

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@ -1,97 +1,183 @@
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 : , 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 : , J ^ n ≤ I := by
rintro ⟨RisNoetherian, containment⟩
rw [isNoetherianRing_iff_ideal_fg] at RisNoetherian
specialize RisNoetherian (Ideal.radical I)
-- rcases RisNoetherian with ⟨S, Sgenerates⟩
have containment2 : ∃ n : , (Ideal.radical I) ^ n ≤ I := by
apply Ideal.exists_radical_pow_le_of_fg I RisNoetherian
cases' containment2 with n containment2'
have containment3 : J ^ n ≤ (Ideal.radical I) ^ n := by
apply Ideal.pow_mono containment
use n
apply le_trans containment3 containment2'
-- The above can be proven using the following quicker theorem that is in the wrong place.
-- Ideal.exists_pow_le_of_le_radical_of_fG
-- 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,
lemma length_eq_dim_if_maximal_annihilates {I : Ideal R} [Ideal.IsMaximal I]
: I • ( : Submodule R M) = 0
→ 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?
-- 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 : , (I ^ n) • ( : Submodule R M) = 0)
→ (∃ m : , 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 : , 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 : , 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

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@ -25,11 +25,11 @@ 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
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
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
@ -42,13 +42,13 @@ lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤
lemma krullDim_le_iff (R : Type _) [CommRing R] (n : ) :
krullDim R ≤ n ↔ ∀ I : PrimeSpectrum R, (height I : WithBot ℕ∞) ≤ ↑n := iSup_le_iff (α := WithBot 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∞)
lemma le_krullDim_iff (R : Type) [CommRing R] (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∞) :
lemma le_krullDim_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
n ≤ krullDim R ↔ ∃ I : PrimeSpectrum R, n ≤ (height I : WithBot ℕ∞) := by sorry
@[simp]
@ -94,10 +94,16 @@ lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
. rw [h.forall_iff]
trivial
lemma krullDim_nonneg_of_nontrivial [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)
lift (Ideal.krullDim R) to ℕ∞ using h with k
use k
lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal I.asIdeal := by
constructor <;> intro h
. intro I
sorry
. sorry
@[simp]
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
@ -130,8 +136,6 @@ lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
simp [field_prime_height_zero]
lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
unfold krullDim at h
simp [height] at h
by_contra x
rw [Ring.not_isField_iff_exists_prime] at x
obtain ⟨P, ⟨h1, primeP⟩⟩ := x
@ -140,16 +144,16 @@ lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0)
by_contra a
have : P = ⊥ := by rwa [PrimeSpectrum.ext_iff] at a
contradiction
have PgtBot : P' > ⊥ := Ne.bot_lt h2
have pos_height : ¬ ↑(Set.chainHeight {J | J < P'}) ≤ 0 := by
have : ⊥ ∈ {J | J < P'} := PgtBot
have pos_height : ¬ (height P') ≤ 0 := by
have : ⊥ ∈ {J | J < P'} := Ne.bot_lt h2
have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
unfold height
rw [←Set.one_le_chainHeight_iff] at this
exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this)
have nonpos_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
have : (⨆ (I : PrimeSpectrum D), (Set.chainHeight {J | J < I} : WithBot ℕ∞)) ≤ 0 := h.le
rw [iSup_le_iff] at this
exact Iff.mp WithBot.coe_le_zero (this P')
have nonpos_height : height P' ≤ 0 := by
have := height_le_krullDim P'
rw [h] at this
aesop
contradiction
lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
@ -160,17 +164,67 @@ lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D =
exact dim_field_eq_zero
#check Ring.DimensionLEOne
-- This lemma is false!
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
applies only to dimension zero rings and domains of dimension 1. -/
lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ (1 : ) := by
rw [krullDim_le_iff R 1]
intro H p
apply le_of_not_gt
intro h
rcases (lt_height_iff''.mp h) with ⟨c, ⟨hc1, hc2, hc3⟩⟩
norm_cast at hc3
rw [List.chain'_iff_get] at hc1
specialize hc1 0 (by rw [hc3]; simp)
set q0 : PrimeSpectrum R := List.get c { val := 0, isLt := _ }
set q1 : PrimeSpectrum R := List.get c { val := 1, isLt := _ }
specialize hc2 q1 (List.get_mem _ _ _)
change q0.asIdeal < q1.asIdeal at hc1
have q1nbot := Trans.trans (bot_le : ⊥ ≤ q0.asIdeal) hc1
specialize H q1.asIdeal (bot_lt_iff_ne_bot.mp q1nbot) q1.IsPrime
apply IsPrime.ne_top p.IsPrime
apply (IsCoatom.lt_iff H.out).mp
exact hc2
lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1 := by
rw [dim_le_one_iff]
exact Ring.DimensionLEOne.principal_ideal_ring R
lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krullDim R ≤ krullDim (Polynomial R) + 1 := sorry
krullDim R + 1 ≤ krullDim (Polynomial R) := sorry
lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
krullDim R = krullDim (Polynomial R) + 1 := sorry
krullDim R + 1 = krullDim (Polynomial R) := sorry
lemma height_eq_dim_localization :
height I = krullDim (Localization.AtPrime I.asIdeal) := sorry

140
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import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Finiteness
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
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.Localization.AtPrime
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Algebra.DirectSum.Ring
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
import Mathlib.Order.Height
import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Ideal.AssociatedPrime
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
import Mathlib.Algebra.Module.GradedModule
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Finiteness
import Mathlib.RingTheory.Ideal.Operations
noncomputable def length ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := Set.chainHeight {M' : Submodule A M | M' < }
def HomogeneousPrime { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsPrime I) ∧ (Ideal.IsHomogeneous 𝒜 I)
def HomogeneousMax { A σ : Type _} [CommRing A] [SetLike σ A] [AddSubmonoidClass σ A] (𝒜 : σ) [GradedRing 𝒜] (I : Ideal A):= (Ideal.IsMaximal I) ∧ (Ideal.IsHomogeneous 𝒜 I)
--theorem monotone_stabilizes_iff_noetherian :
-- (∀ f : →o Submodule R M, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsNoetherian R M := by
-- rw [isNoetherian_iff_wellFounded, WellFounded.monotone_chain_condition]
open GradedMonoid.GSmul
open DirectSum
instance tada1 (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (i : ) : SMul (𝒜 0) (𝓜 i)
where smul x y := @Eq.rec (0+i) (fun a _ => 𝓜 a) (GradedMonoid.GSmul.smul x y) i (zero_add i)
lemma mylem (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ) (a : 𝒜 0) (m : 𝓜 i) :
of _ _ (a • m) = of _ _ a • of _ _ m := by
refine' Eq.trans _ (Gmodule.of_smul_of 𝒜 𝓜 a m).symm
refine' of_eq_of_gradedMonoid_eq _
exact Sigma.ext (zero_add _).symm <| eq_rec_heq _ _
instance tada2 (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ) : SMulWithZero (𝒜 0) (𝓜 i) := by
letI := SMulWithZero.compHom (⨁ i, 𝓜 i) (of 𝒜 0).toZeroHom
exact Function.Injective.smulWithZero (of 𝓜 i).toZeroHom Dfinsupp.single_injective (mylem 𝒜 𝓜 i)
instance tada3 (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)] [DirectSum.GCommRing 𝒜]
[h : DirectSum.Gmodule 𝒜 𝓜] (i : ): Module (𝒜 0) (𝓜 i) := by
letI := Module.compHom (⨁ j, 𝓜 j) (ofZeroRingHom 𝒜)
exact Dfinsupp.single_injective.module (𝒜 0) (of 𝓜 i) (mylem 𝒜 𝓜 i)
-- (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0))
noncomputable def dummyhil_function (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (hilb : → ℕ∞) := ∀ i, hilb i = (length (𝒜 0) (𝓜 i))
lemma hilbertz (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜]
(finlen : ∀ i, (length (𝒜 0) (𝓜 i)) < ) : := by
intro i
let h := dummyhil_function 𝒜 𝓜
simp at h
let n : := fun i ↦ WithTop.untop _ (finlen i).ne
have hn : ∀ i, (n i : ℕ∞) = length (𝒜 0) (𝓜 i) := fun i ↦ WithTop.coe_untop _ _
have' := hn i
exact ((n i) : )
noncomputable def hilbert_function (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (hilb : ) := ∀ i, hilb i = (ENat.toNat (length (𝒜 0) (𝓜 i)))
noncomputable def dimensionring { A: Type _}
[CommRing A] := krullDim (PrimeSpectrum A)
noncomputable def dimensionmodule ( A : Type _) (M : Type _)
[CommRing A] [AddCommGroup M] [Module A M] := krullDim (PrimeSpectrum (A (( : Submodule A M).annihilator)) )
-- lemma graded_local (𝒜 : → Type _) [SetLike (⨁ i, 𝒜 i)] (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
-- [DirectSum.GCommRing 𝒜]
-- [DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) : ∃ ( I : Ideal ((⨁ i, 𝒜 i))),(HomogeneousMax 𝒜 I) := sorry
def PolyType (f : ) (d : ) := ∃ Poly : Polynomial , ∃ (N : ), ∀ (n : ), N ≤ n → f n = Polynomial.eval (n : ) Poly ∧ d = Polynomial.degree Poly
theorem hilbert_polynomial (𝒜 : → Type _) (𝓜 : → Type _) [∀ i, AddCommGroup (𝒜 i)] [∀ i, AddCommGroup (𝓜 i)]
[DirectSum.GCommRing 𝒜]
[DirectSum.Gmodule 𝒜 𝓜] (art: IsArtinianRing (𝒜 0)) (loc : LocalRing (𝒜 0)) (fingen : IsNoetherian (⨁ i, 𝒜 i) (⨁ i, 𝓜 i))
(findim : ∃ d : , dimensionmodule (⨁ i, 𝒜 i) (⨁ i, 𝓜 i) = d):True := sorry
-- Semiring A]
-- variable [SetLike σ A]

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@ -0,0 +1,52 @@
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Order.Height
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
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 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
lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krullDim R + 1 ≤ krullDim (Polynomial R) := sorry -- Others are working on it
-- private lemma sum_succ_of_succ_sum {ι : Type} (a : ℕ∞) [inst : Nonempty ι] :
-- (⨆ (x : ι), a + 1) = (⨆ (x : ι), a) + 1 := by
-- have : a + 1 = (⨆ (x : ι), a) + 1 := by rw [ciSup_const]
-- have : a + 1 = (⨆ (x : ι), a + 1) := Eq.symm ciSup_const
-- simp
lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
krullDim R + 1 = krullDim (Polynomial R) := by
rw [le_antisymm_iff]
constructor
· exact dim_le_dim_polynomial_add_one
· unfold krullDim
have htPBdd : ∀ (P : PrimeSpectrum (Polynomial R)), (height P : WithBot ℕ∞) ≤ (⨆ (I : PrimeSpectrum R), ↑(height I + 1)) := by
intro P
have : ∃ (I : PrimeSpectrum R), (height P : WithBot ℕ∞) ≤ ↑(height I + 1) := by
sorry
obtain ⟨I, IP⟩ := this
have : (↑(height I + 1) : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum R), ↑(height I + 1) := by
apply @le_iSup (WithBot ℕ∞) _ _ _ I
apply ge_trans this IP
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 :=
fun P ↦ (by apply add_le_add_right (@le_iSup (WithBot ℕ∞) _ _ _ P) 1)
apply iSup_le
apply this
simp
intro P
exact ge_trans oneOut (htPBdd P)

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@ -1,82 +0,0 @@
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Order.Height
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
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 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
@[simp]
lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P = ⊥ := by
constructor
· intro primeP
obtain T := eq_bot_or_top P
have : ¬P = := IsPrime.ne_top primeP
tauto
· intro botP
rw [botP]
exact bot_prime
lemma field_prime_height_zero {K: Type _} [Field K] (P : PrimeSpectrum K) : height P = 0 := by
unfold height
simp
by_contra spec
change _ ≠ _ at spec
rw [← Set.nonempty_iff_ne_empty] at spec
obtain ⟨J, JlP : J < P⟩ := spec
have P0 : IsPrime P.asIdeal := P.IsPrime
have J0 : IsPrime J.asIdeal := J.IsPrime
rw [field_prime_bot] at P0 J0
have : J.asIdeal = P.asIdeal := Eq.trans J0 (Eq.symm P0)
have : J = P := PrimeSpectrum.ext J P this
have : J ≠ P := ne_of_lt JlP
contradiction
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
unfold krullDim
simp [field_prime_height_zero]
noncomputable
instance : CompleteLattice (WithBot ℕ∞) :=
inferInstanceAs <| CompleteLattice (WithBot (WithTop ))
lemma isField.dim_zero {D: Type _} [CommRing D] [IsDomain D] (h: krullDim D = 0) : IsField D := by
unfold krullDim at h
simp [height] at h
by_contra x
rw [Ring.not_isField_iff_exists_prime] at x
obtain ⟨P, ⟨h1, primeP⟩⟩ := x
let P' : PrimeSpectrum D := PrimeSpectrum.mk P primeP
have h2 : P' ≠ ⊥ := by
by_contra a
have : P = ⊥ := by rwa [PrimeSpectrum.ext_iff] at a
contradiction
have PgtBot : P' > ⊥ := Ne.bot_lt h2
have pos_height : ¬ ↑(Set.chainHeight {J | J < P'}) ≤ 0 := by
have : ⊥ ∈ {J | J < P'} := PgtBot
have : {J | J < P'}.Nonempty := Set.nonempty_of_mem this
rw [←Set.one_le_chainHeight_iff] at this
exact not_le_of_gt (Iff.mp ENat.one_le_iff_pos this)
have nonpos_height : (Set.chainHeight {J | J < P'}) ≤ 0 := by
have : (⨆ (I : PrimeSpectrum D), (Set.chainHeight {J | J < I} : WithBot ℕ∞)) ≤ 0 := h.le
rw [iSup_le_iff] at this
exact Iff.mp WithBot.coe_le_zero (this P')
contradiction
lemma dim_eq_zero_iff_field {D: Type _} [CommRing D] [IsDomain D] : krullDim D = 0 ↔ IsField D := by
constructor
· exact isField.dim_zero
· intro fieldD
let h : Field D := IsField.toField fieldD
exact dim_field_eq_zero