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Merge branch 'monalisa' of github.com:GTBarkley/comm_alg into monalisa

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chelseaandmadrid 2023-06-16 12:20:32 -07:00
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52
CommAlg/final_poly_type.lean
View file

@ -54,8 +54,6 @@ noncomputable section
def PolyType (f : ℤ → ℤ) (d : ℕ) := ∃ Poly : Polynomial ℚ, ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → f n = Polynomial.eval (n : ℚ) Poly) ∧ d = Polynomial.degree Poly
section
#check PolyType
example (f : ℤ → ℤ) (hf : ∀ x, f x = x ^ 2) : PolyType f 2 := by
unfold PolyType
sorry
@ -139,8 +137,8 @@ lemma PolyType_0 (f : ℤ → ℤ) : (PolyType f 0) ↔ (∃ (c : ℤ), ∃ (N :
-- Δ of 0 times preserves the function
lemma Δ_0 (f : ℤ → ℤ) : (Δ f 0) = f := by rfl
--simp only [Δ]
-- Δ of 1 times decreaes the polynomial type by one
-- Δ of 1 times decreaes the polynomial type by one --can be golfed
lemma Δ_1 (f : ℤ → ℤ) (d : ℕ) : PolyType f (d + 1) → PolyType (Δ f 1) d := by
intro h
simp only [PolyType, Δ, Int.cast_sub, exists_and_right]
@ -193,53 +191,21 @@ lemma Δ_d_PolyType_d_to_PolyType_0 (f : ℤ → ℤ) (d : ℕ): PolyType f d
-- The "reverse" of Δ of 1 times increases the polynomial type by one
lemma Δ_1_ (f : ℤ → ℤ) (d : ℕ) : PolyType (Δ f 1) d → PolyType f (d + 1) := by
intro h
rintro ⟨P, N, ⟨h1, h2⟩⟩
simp only [PolyType, Nat.cast_add, Nat.cast_one, exists_and_right]
rcases h with ⟨P, N, h⟩
rcases h with ⟨h1, h2⟩
let G := fun (q : ℤ) => f (N)
sorry
lemma foo (d : ℕ) : (f : ℤ → ℤ) → (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0) → (PolyType f d) := by
lemma foo (d : ℕ) : (f : ℤ → ℤ) → (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n →
(Δ f d) (n) = c) ∧ c ≠ 0) → (PolyType f d) := by
induction' d with d hd
-- Base case
· intro f
intro h
rcases h with ⟨c, N, hh⟩
rw [PolyType_0]
use c
use N
tauto
-- Induction step
· intro f
intro h
rcases h with ⟨c, N, h⟩
have this : PolyType f (d + 1) := by
rcases h with ⟨H,c0⟩
let g := (Δ f 1)
have this1 : (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → (Δ g d) (n) = c) ∧ c ≠ 0) := by
use c; use N
constructor
· intro n
specialize H n
intro h
have this : Δ f (d + 1) n = c := by tauto
rw [←this]
rw [Δ_1_s_equiv_Δ_s_1]
· tauto
have this2 : PolyType g d := by
apply hd
tauto
exact Δ_1_ f d this2
exact this
· rintro f ⟨c, N, hh⟩; rw [PolyType_0 f]; exact ⟨c, N, hh⟩
· exact fun f ⟨c, N, ⟨H, c0⟩⟩ =>
Δ_1_ f d (hd (Δ f 1) ⟨c, N, fun n h => by rw [← H n h, Δ_1_s_equiv_Δ_s_1], c0⟩)
-- [BH, 4.1.2] (a) => (b)
-- Δ^d f (n) = c for some nonzero integer c for n >> 0 → f is of polynomial type d
lemma a_to_b (f : ℤ → ℤ) (d : ℕ) : (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0) → PolyType f d := by
sorry
lemma a_to_b (f : ℤ → ℤ) (d : ℕ) : (∃ (c : ℤ), ∃ (N : ℤ), (∀ (n : ℤ), N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0) → PolyType f d := fun h => (foo d f) h
-- [BH, 4.1.2] (a) <= (b)
-- f is of polynomial type d → Δ^d f (n) = c for some nonzero integer c for n >> 0

15
CommAlg/grant2.lean
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@ -54,9 +54,20 @@ def symbolicIdeal(Q : Ideal R) {hin : Q.IsPrime} (I : Ideal R) : Ideal R where
rw [←mul_assoc, mul_comm s, mul_assoc]
exact Ideal.mul_mem_left _ _ hs2
theorem WF_interval_le_prime (I : Ideal R) (P : Ideal R) [P.IsPrime]
(h : ∀ J ∈ (Set.Icc I P), J.IsPrime → J = P ):
WellFounded ((· < ·) : (Set.Icc I P) → (Set.Icc I P) → Prop ) := sorry
protected lemma LocalRing.height_le_one_of_minimal_over_principle
[LocalRing R] (q : PrimeSpectrum R) {x : R}
[LocalRing R] {x : R}
(h : (closedPoint R).asIdeal ∈ (Ideal.span {x}).minimalPrimes) :
q = closedPoint R ∨ Ideal.height q = 0 := by
Ideal.height (closedPoint R) ≤ 1 := by
-- by_contra hcont
-- push_neg at hcont
-- rw [Ideal.lt_height_iff'] at hcont
-- rcases hcont with ⟨c, hc1, hc2, hc3⟩
apply height_le_of_gt_height_lt
intro p hp
sorry

126
CommAlg/krull.lean
View file

@ -19,6 +19,7 @@ import Mathlib.Order.ConditionallyCompleteLattice.Basic
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
@ -29,18 +30,19 @@ 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'
@ -57,6 +59,38 @@ lemma krullDim_le_iff' (R : Type _) [CommRing R] (n : ℕ∞) :
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
@ -95,8 +129,32 @@ lemma le_krullDim_iff (R : Type _) [CommRing R] (n : ℕ) :
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
#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
@ -206,9 +264,9 @@ lemma dim_le_zero_iff : krullDim R ≤ 0 ↔ ∀ I : PrimeSpectrum R, IsMaximal
. exact List.chain'_singleton _
. constructor
. intro I' hI'
simp at hI'
simp only [List.mem_singleton] at hI'
rwa [hI']
. simp
. 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
@ -235,7 +293,7 @@ lemma dim_eq_zero_iff [Nontrivial R] : krullDim R = 0 ↔ ∀ I : PrimeSpectrum
/-- 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
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
@ -246,25 +304,16 @@ lemma field_prime_bot {K: Type _} [Field K] (P : Ideal K) : IsPrime P ↔ P =
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
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 [field_prime_height_zero]
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
@ -311,7 +360,7 @@ lemma dim_le_one_of_dimLEOne : Ring.DimensionLEOne R → krullDim R ≤ 1 := by
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)
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 _ _ _)
@ -325,6 +374,37 @@ lemma dim_le_one_of_pid [IsDomain R] [IsPrincipalIdealRing R] : krullDim R ≤ 1
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

167
CommAlg/polynomial.lean Normal file
View file

@ -0,0 +1,167 @@
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 ChainLemma
variable {α β : Type _}
variable [LT α] [LT β] (s t : Set α)
namespace Set
open List
/-
Sorry for using aesop, but it doesn't take that long
-/
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
end ChainLemma
variable {R : Type _} [CommRing R]
open Ideal Polynomial
namespace Polynomial
/-
The composition R → R[X] → R is the identity
-/
theorem coeff_C_eq : RingHom.comp constantCoeff C = RingHom.id R := by ext; simp
end Polynomial
/-
Given an ideal I in R, we define the ideal adjoin_x' I to be the kernel
of R[X] → R → R/I
-/
def adj_x_map (I : Ideal R) : R[X] →+* R ⧸ I := (Ideal.Quotient.mk I).comp constantCoeff
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 _
@[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]
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]
/-
If I is prime in R, the pushforward I*R[X] is prime in R[X]
-/
def map_prime (I : PrimeSpectrum R) : PrimeSpectrum R[X] :=
⟨I.asIdeal.map C, isPrime_map_C_of_isPrime I.IsPrime⟩
/-
The pushforward map (Ideal R) → (Ideal R[X]) is injective
-/
lemma map_inj {I J : Ideal R} (h : I.map C = J.map C) : I = J := by
have H : map constantCoeff (I.map C) = map constantCoeff (J.map C) := by rw [h]
simp [Ideal.map_map, coeff_C_eq] at H
exact H
/-
The pushforward map (Ideal R) → (Ideal R[X]) is strictly monotone
-/
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 ⊢
exact ⟨map_mono h.1, fun H => h.2 (map_inj H)⟩
lemma map_lt_adjoin_x (I : PrimeSpectrum R) : map_prime I < adjoin_x I := by
simp [adjoin_x, adjoin_x_eq]
show I.asIdeal.map C < I.asIdeal.map C ⊔ span {X}
simp [Ideal.span_le, mem_map_C_iff]
use 1
simp
rw [←Ideal.eq_top_iff_one]
exact I.IsPrime.ne_top'
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 : ℕ) ≤ height (adjoin_x I) + (0 : ℕ)
. 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])
refine' ⟨_, by simp⟩
. simp [Set.append_mem_subchain_iff]
refine' ⟨_, map_lt_adjoin_x I, fun a ha => _⟩
. refine' ⟨_, fun i hi => _⟩
. apply List.chain'_map_of_chain' map_prime (fun a b hab => map_strictmono hab) hl.1
. rw [List.mem_map] at hi
obtain ⟨a, ha, rfl⟩ := hi
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
. 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
#check (⊤ : ℕ∞)
/-
dim R + 1 ≤ dim R[X]
-/
lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
krullDim R + (1 : ℕ∞) ≤ krullDim R[X] := by
obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial R
rw [hn]
change ↑(n + 1) ≤ krullDim R[X]
have := le_of_eq hn.symm
induction' n using ENat.recTopCoe with n
. change krullDim R = ⊤ at hn
change ⊤ ≤ krullDim R[X]
rw [krullDim_eq_top_iff] at hn
rw [top_le_iff, krullDim_eq_top_iff]
intro n
obtain ⟨I, hI⟩ := hn n
use adjoin_x I
calc n ≤ height I := hI
_ ≤ height I + 1 := le_self_add
_ ≤ height (adjoin_x I) := ht_adjoin_x_eq_ht_add_one I
change n ≤ krullDim R at this
change (n + 1 : ℕ) ≤ krullDim R[X]
rw [le_krullDim_iff] at this ⊢
obtain ⟨I, hI⟩ := this
use adjoin_x I
apply WithBot.coe_mono
calc n + 1 ≤ height I + 1 := by
apply add_le_add_right
change ((n : ℕ∞) : WithBot ℕ∞) ≤ (height I) at hI
rw [WithBot.coe_le_coe] at hI
exact hI
_ ≤ height (adjoin_x I) := ht_adjoin_x_eq_ht_add_one I

72
CommAlg/sayantan(poly_over_field).lean
View file

@ -1,39 +1,87 @@
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
private lemma singleton_bot_chainHeight_one {α : Type} [Preorder α] [Bot α] : Set.chainHeight {(⊥ : α)} ≤ 1 := by
unfold Set.chainHeight
simp only [iSup_le_iff, Nat.cast_le_one]
intro L h
unfold Set.subchain at h
simp only [Set.mem_singleton_iff, Set.mem_setOf_eq] at h
rcases L with (_ | ⟨a,L⟩)
. simp only [List.length_nil, zero_le]
rcases L with (_ | ⟨b,L⟩)
. simp only [List.length_singleton, le_refl]
simp only [List.chain'_cons, List.find?, List.mem_cons, forall_eq_or_imp] at h
rcases h with ⟨⟨h1, _⟩, ⟨rfl, rfl, _⟩⟩
exact absurd h1 (lt_irrefl _)
/-- 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
-- unfold krullDim
rw [le_antisymm_iff]
let X := @Polynomial.X K _
constructor
·
sorry
· unfold krullDim
apply @iSup_le (WithBot ℕ∞) _ _ _ _
intro I
have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
by_cases I = ⊥
· rw [← height_bot_iff_bot] at h
simp only [WithBot.coe_le_one, ge_iff_le]
rw [h]
exact bot_le
· push_neg at h
have : I.asIdeal ≠ ⊥ := by
by_contra a
have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
contradiction
have maxI := IsPrime.to_maximal_ideal this
have sngletn : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
intro P
constructor
· intro H
simp only [Set.mem_setOf_eq] at H
by_contra x
push_neg at x
have : P.asIdeal ≠ ⊥ := by
by_contra a
have : P = ⊥ := PrimeSpectrum.ext P ⊥ a
contradiction
have maxP := IsPrime.to_maximal_ideal this
have IneTop := IsMaximal.ne_top maxI
have : P ≤ I := le_of_lt H
rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this
have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this
have : P = I := PrimeSpectrum.ext P I this
replace H : P ≠ I := ne_of_lt H
contradiction
· intro pBot
simp only [Set.mem_setOf_eq, pBot]
exact lt_of_le_of_ne bot_le h.symm
replace sngletn : {J | J < I} = {⊥} := Set.ext sngletn
unfold height
rw [sngletn]
simp only [WithBot.coe_le_one, ge_iff_le]
exact singleton_bot_chainHeight_one
· 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
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]
rw [bot_lt_iff_ne_bot]
simp only [Set.mem_setOf_eq, bot_lt_iff_ne_bot]
suffices : P.asIdeal ≠ ⊥
· by_contra x
rw [PrimeSpectrum.ext_iff] at x