Merge branch 'GTBarkley:main' into main

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Sayantan Santra 2023-06-14 00:55:38 -05:00 committed by GitHub
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3 changed files with 177 additions and 14 deletions

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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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@ -77,12 +77,25 @@ lemma containment_radical_power_containment :
rintro ⟨RisNoetherian, containment⟩
rw [isNoetherianRing_iff_ideal_fg] at RisNoetherian
specialize RisNoetherian (Ideal.radical I)
rcases RisNoetherian with ⟨S, Sgenerates⟩
-- 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
-- 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
--
-- 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.

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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]
@ -94,11 +94,17 @@ 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
constructor
@ -158,8 +164,58 @@ 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