mirror of
https://github.com/GTBarkley/comm_alg.git
synced 2024-10-16 19:43:55 -05:00
342 lines
No EOL
13 KiB
Text
342 lines
No EOL
13 KiB
Text
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
|
||
|
||
/- This file contains the definitions of height of an ideal, and the krull
|
||
dimension of a commutative ring.
|
||
There are also sorried statements of many of the theorems that would be
|
||
really nice to prove.
|
||
I'm imagining for this file to ultimately contain basic API for height and
|
||
krull dimension, and the theorems will probably end up other files,
|
||
depending on how long the proofs are, and what extra API needs to be
|
||
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
|
||
open LocalRing
|
||
|
||
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 height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := by
|
||
apply Set.chainHeight_mono
|
||
intro J' hJ'
|
||
show J' < J
|
||
exact lt_of_lt_of_le hJ' I_le_J
|
||
|
||
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 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∞)
|
||
|
||
@[simp]
|
||
lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
|
||
le_iSup (λ I : PrimeSpectrum R => (height I : WithBot ℕ∞)) I
|
||
|
||
lemma le_krullDim_iff (R : Type _) [CommRing R] (n : 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) : 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
|
||
apply le_antisymm
|
||
. rw [krullDim_le_iff']
|
||
intro I
|
||
apply WithBot.coe_mono
|
||
apply height_le_of_le
|
||
apply le_maximalIdeal
|
||
exact I.2.1
|
||
. 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∞} :
|
||
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) = (_: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∞} :
|
||
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'
|
||
|
||
#check height_le_krullDim
|
||
--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 :=
|
||
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
|
||
constructor
|
||
. contrapose
|
||
rw [not_isEmpty_iff, ←not_nontrivial_iff_subsingleton, not_not]
|
||
apply PrimeSpectrum.instNonemptyPrimeSpectrum
|
||
. intro h
|
||
by_contra hneg
|
||
rw [not_isEmpty_iff] at hneg
|
||
rcases hneg with ⟨a, ha⟩
|
||
exact primeSpectrum_empty_of_subsingleton ⟨a, ha⟩
|
||
|
||
/-- A ring has Krull dimension -∞ if and only if it is the zero ring -/
|
||
lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
|
||
unfold Ideal.krullDim
|
||
rw [←primeSpectrum_empty_iff, iSup_eq_bot]
|
||
constructor <;> intro h
|
||
. rw [←not_nonempty_iff]
|
||
rintro ⟨a, ha⟩
|
||
specialize h ⟨a, ha⟩
|
||
tauto
|
||
. rw [h.forall_iff]
|
||
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
|
||
have h := dim_eq_bot_iff.not.mpr (not_subsingleton R)
|
||
lift (Ideal.krullDim R) to ℕ∞ using h with 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 : ℕ)) ↔ _
|
||
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 [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 [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
|
||
rw [←dim_le_zero_iff]
|
||
obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial R
|
||
have : n ≥ 0 := zero_le n
|
||
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]
|
||
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
|
||
|
||
/-- In a field, all primes have height 0. -/
|
||
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
|
||
|
||
/-- The Krull dimension of a field is 0. -/
|
||
lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by
|
||
unfold krullDim
|
||
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
|
||
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 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 (ENat.one_le_iff_pos.mp this)
|
||
have nonpos_height : height P' ≤ 0 := by
|
||
have := height_le_krullDim P'
|
||
rw [h] at this
|
||
aesop
|
||
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
|
||
constructor
|
||
· exact domain_dim_zero.isField
|
||
· intro fieldD
|
||
let h : Field D := fieldD.toField
|
||
exact dim_field_eq_zero
|
||
|
||
#check Ring.DimensionLEOne
|
||
-- This lemma is false!
|
||
lemma dim_le_one_iff : krullDim R ≤ 1 ↔ Ring.DimensionLEOne R := sorry
|
||
|
||
/-- 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
|
||
show _ → ((_ : WithBot ℕ∞) ≤ (1 : ℕ))
|
||
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
|
||
exact (not_maximal_of_lt_prime p.IsPrime hc2) H
|
||
|
||
/-- The Krull dimension of a PID is at most 1. -/
|
||
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 + 1 ≤ krullDim (Polynomial R) := sorry
|
||
|
||
-- lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
|
||
-- 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 : ℕ) : krullDim (MvPolynomial (Fin n) K) = n := sorry
|
||
|
||
lemma height_eq_dim_localization :
|
||
height I = krullDim (Localization.AtPrime I.asIdeal) := sorry
|
||
|
||
lemma dim_quotient_le_dim : height I + krullDim (R ⧸ I.asIdeal) ≤ krullDim R := sorry
|
||
|
||
lemma height_add_dim_quotient_le_dim : height I + krullDim (R ⧸ I.asIdeal) ≤ krullDim R := sorry |