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. -/ /-- If something is smaller that Bot of a PartialOrder after attaching another Bot, it must be Bot. -/ 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) /-- Definitions -/ 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 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 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 /-- A lattice structure on WithBot ℕ∞. -/ noncomputable instance : CompleteLattice (WithBot (ℕ∞)) := WithBot.WithTop.completeLattice /-- Height of ideals is monotonic. -/ 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 /-- In a domain, the height of a prime ideal is Bot (0 in this case) iff it's the Bot ideal. -/ @[simp] lemma height_bot_iff_bot {D: Type _} [CommRing D] [IsDomain D] {P : PrimeSpectrum D} : height P = ⊥ ↔ P = ⊥ := by constructor · intro h unfold height at h rw [bot_eq_zero] at h simp only [Set.chainHeight_eq_zero_iff] at h apply eq_bot_of_minimal intro I by_contra x have : I ∈ {J | J < P} := x rw [h] at this contradiction · intro h unfold height simp only [bot_eq_zero', Set.chainHeight_eq_zero_iff] by_contra spec change _ ≠ _ at spec rw [← Set.nonempty_iff_ne_empty] at spec obtain ⟨J, JlP : J < P⟩ := spec have JneP : J ≠ P := ne_of_lt JlP rw [h, ←bot_lt_iff_ne_bot, ←h] at JneP have := not_lt_of_lt JneP contradiction @[simp] lemma height_bot_eq {D: Type _} [CommRing D] [IsDomain D] : height (⊥ : PrimeSpectrum D) = ⊥ := by rw [height_bot_iff_bot] /-- The Krull dimension of a ring being ≥ n is equivalent to there being an ideal of height ≥ n. -/ 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 #check ENat.recTopCoe /- terrible place for this lemma. Also this probably exists somewhere Also this is a terrible proof -/ lemma eq_top_iff (n : WithBot ℕ∞) : n = ⊤ ↔ ∀ m : ℕ, m ≤ n := by aesop induction' n using WithBot.recBotCoe with n . exfalso have := (a 0) simp [not_lt_of_ge] at this induction' n using ENat.recTopCoe with n . rfl . have := a (n + 1) exfalso change (((n + 1) : ℕ∞) : WithBot ℕ∞) ≤ _ at this simp [WithBot.coe_le_coe] at this change ((n + 1) : ℕ∞) ≤ (n : ℕ∞) at this simp [ENat.add_one_le_iff] at this lemma krullDim_eq_top_iff (R : Type _) [CommRing R] : krullDim R = ⊤ ↔ ∀ (n : ℕ), ∃ I : PrimeSpectrum R, n ≤ height I := by simp [eq_top_iff, le_krullDim_iff] change (∀ (m : ℕ), ∃ I, ((m : ℕ∞) : WithBot ℕ∞) ≤ height I) ↔ _ simp [WithBot.coe_le_coe] /-- 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∞} : 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 : 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) = (_: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 --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 (0 : ℕ∞) < (_ : WithBot ℕ∞) rw [lt_height_iff''] use [I] constructor . exact List.chain'_singleton _ . constructor . intro I' hI' simp only [List.mem_singleton] at hI' rwa [hI'] . simp only [List.length_singleton, Nat.cast_one, zero_add] . 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 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_bot {K: Type _} [Nontrivial K] [Field K] (P : PrimeSpectrum K) : height P = ⊥ := by have : IsPrime P.asIdeal := P.IsPrime rw [field_prime_bot] at this have : P = ⊥ := PrimeSpectrum.ext P ⊥ this rwa [height_bot_iff_bot] /-- The Krull dimension of a field is 0. -/ lemma dim_field_eq_zero {K : Type _} [Field K] : krullDim K = 0 := by unfold krullDim simp only [field_prime_height_bot, ciSup_unique] /-- 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 only) 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 /-- The ring of polynomials over a field has dimension one. -/ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by rw [le_antisymm_iff] let X := @Polynomial.X K _ constructor · exact dim_le_one_of_pid · 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 _ _ have : IsPrime (span {X}) := by refine (span_singleton_prime ?hp).mpr 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, 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 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