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

This commit is contained in:
leopoldmayer 2023-06-16 11:43:12 -07:00
commit 50ed3f1de9
3 changed files with 141 additions and 40 deletions

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@ -66,13 +66,6 @@ end section
def Δ : () → → ()
| f, 0 => f
| f, d + 1 => fun (n : ) ↦ (Δ f d) (n + 1) - (Δ f d) (n)
section
#check Δ
def f (n : ) := n
#eval (Δ f 1) 100
-- #check (by (show_term unfold Δ) : Δ f 0=0)
end section
-- (NO need to prove another direction) Constant polynomial function = constant function
lemma Poly_constant (F : Polynomial ) (c : ) :
@ -86,12 +79,28 @@ lemma Poly_constant (F : Polynomial ) (c : ) :
simp
· sorry
-- Get the polynomial G (X) = F (X + s) from the polynomial F(X)
lemma Polynomial_shifting (F : Polynomial ) (s : ) : ∃ (G : Polynomial ), (∀ (x : ), Polynomial.eval x G = Polynomial.eval (x + s) F) ∧ (Polynomial.degree G = Polynomial.degree F) := by
sorry
-- Shifting doesn't change the polynomial type
lemma Poly_shifting (f : ) (g : ) (hf : PolyType f d) (s : ) (hfg : ∀ (n : ), f (n + s) = g (n)) : PolyType g d := by
simp only [PolyType]
rcases hf with ⟨F, hh⟩
rcases hh with ⟨N,ss⟩
rcases hh with ⟨N,s1, s2⟩
have this : ∃ (G : Polynomial ), (∀ (x : ), Polynomial.eval x G = Polynomial.eval (x + s) F) ∧ (Polynomial.degree G = Polynomial.degree F) := by
exact Polynomial_shifting F s
rcases this with ⟨Poly, h1, h2⟩
use Poly
use N
constructor
· intro n
specialize s1 (n + s)
intro hN
have this1 : f (n + s) = Polynomial.eval (n + s : ) F := by
sorry
sorry
· rw [h2, s2]
-- PolyType 0 = constant function
lemma PolyType_0 (f : ) : (PolyType f 0) ↔ (∃ (c : ), ∃ (N : ), (∀ (n : ),
@ -125,12 +134,44 @@ lemma PolyType_0 (f : ) : (PolyType f 0) ↔ (∃ (c : ), ∃ (N :
lemma Δ_0 (f : ) : (Δ f 0) = f := by rfl
--simp only [Δ]
-- Δ of 1 times decreaes the polynomial type by one
lemma Δ_1 (f : ) (d : ): PolyType f (d + 1) → PolyType (Δ f 1) d := by
lemma Δ_1 (f : ) (d : ) : PolyType f (d + 1) → PolyType (Δ f 1) d := by
intro h
simp only [PolyType, Δ, Int.cast_sub, exists_and_right]
rcases h with ⟨F, N, h⟩
rcases h with ⟨h1, h2⟩
have this : ∃ (G : Polynomial ), (∀ (x : ), Polynomial.eval x G = Polynomial.eval (x + 1) F) ∧ (Polynomial.degree G = Polynomial.degree F) := by
exact Polynomial_shifting F 1
rcases this with ⟨G, hG, hGG⟩
let Poly := G - F
use Poly
constructor
· use N
intro n hn
specialize hG n
norm_num
rw [hG]
let h3 := h1
specialize h3 n
have this1 : f n = Polynomial.eval (n : ) F := by tauto
have this2 : f (n + 1) = Polynomial.eval ((n + 1) : ) F := by
specialize h1 (n + 1)
have this3 : N ≤ n + 1 := by linarith
aesop
rw [←this1, ←this2]
· have this1 : Polynomial.degree Poly = d := by
have this2 : Polynomial.degree Poly ≤ d := by
sorry
have this3 : Polynomial.degree Poly ≥ d := by
sorry
sorry
tauto
-- Δ of d times maps polynomial of degree d to polynomial of degree 0
lemma Δ_1_s_equiv_Δ_s_1 (f : ) (s : ) : Δ (Δ f 1) s = (Δ f (s + 1)) := by
sorry
induction' s with s hs
· norm_num
· aesop
lemma foofoo (d : ) : (f : ) → (PolyType f d) → (PolyType (Δ f d) 0):= by
induction' d with d hd
· intro f h
@ -143,7 +184,17 @@ lemma foofoo (d : ) : (f : ) → (PolyType f d) → (PolyType (Δ
lemma Δ_d_PolyType_d_to_PolyType_0 (f : ) (d : ): PolyType f d → PolyType (Δ f d) 0 :=
fun h => (foofoo d f) h
lemma foofoofoo (d : ) : (f : ) → (∃ (c : ), ∃ (N : ), (∀ (n : ), N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0) → (PolyType f d) := by
-- The "reverse" of Δ of 1 times increases the polynomial type by one
lemma Δ_1_ (f : ) (d : ) : PolyType (Δ f 1) d → PolyType f (d + 1) := by
intro h
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 : ) => f (N)
sorry
lemma foo (d : ) : (f : ) → (∃ (c : ), ∃ (N : ), (∀ (n : ), N ≤ n → (Δ f d) (n) = c) ∧ c ≠ 0) → (PolyType f d) := by
induction' d with d hd
-- Base case
@ -160,8 +211,23 @@ lemma foofoofoo (d : ) : (f : ) → (∃ (c : ), ∃ (N : ),
intro h
rcases h with ⟨c, N, h⟩
have this : PolyType f (d + 1) := by
sorry
rcases h with ⟨H,c0⟩
let g := (Δ f 1)
have this1 : (∃ (c : ), ∃ (N : ), (∀ (n : ), 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
-- [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

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@ -61,7 +61,7 @@ lemma height_le_krullDim (I : PrimeSpectrum R) : height I ≤ krullDim R :=
/-- 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
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
@ -85,6 +85,10 @@ lemma height_bot_iff_bot {D: Type} [CommRing D] [IsDomain D] {P : PrimeSpectrum
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 : ) :
@ -284,27 +288,11 @@ 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_bot {K: Type _} [Nontrivial K] [Field K] {P : PrimeSpectrum K} : height P = ⊥ := by
-- This should be doable by
-- have : IsPrime P.asIdeal := P.IsPrime
-- rw [field_prime_bot] at this
-- have : P = ⊥ := PrimeSpectrum.ext P ⊥ this
-- rw [height_bot_iff_bot]
-- Need to check what's happening
rw [bot_eq_zero]
unfold height
simp only [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 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
@ -370,6 +358,37 @@ 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

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@ -7,6 +7,20 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.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
rw [le_antisymm_iff]
@ -27,7 +41,7 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
contradiction
have maxI := IsPrime.to_maximal_ideal this
have singleton : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
have sngletn : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
intro P
constructor
· intro H
@ -49,9 +63,11 @@ lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullD
· intro pBot
simp only [Set.mem_setOf_eq, pBot]
exact lt_of_le_of_ne bot_le h.symm
replace singleton : {J | J < I} = {⊥} := Set.ext singleton
replace sngletn : {J | J < I} = {⊥} := Set.ext sngletn
unfold height
sorry
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