Merge pull request #76 from GTBarkley/leo

Leo
This commit is contained in:
Leo Mayer 2023-06-15 18:06:41 -07:00 committed by GitHub
commit 828bf44695
No known key found for this signature in database
GPG key ID: 4AEE18F83AFDEB23

View file

@ -1,6 +1,7 @@
import Mathlib.RingTheory.Ideal.Operations import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.FiniteType import Mathlib.RingTheory.FiniteType
import Mathlib.Order.Height import Mathlib.Order.Height
import Mathlib.RingTheory.Polynomial.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.RingTheory.Ideal.Quotient import Mathlib.RingTheory.Ideal.Quotient
@ -9,6 +10,40 @@ import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.ConditionallyCompleteLattice.Basic
import CommAlg.krull import CommAlg.krull
section AddToOrder
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type _}
variable [LT α] [LT β] (s t : Set α)
namespace Set
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
namespace List
#check Option
theorem getLast?_map (l : List α) (f : α → β) :
(l.map f).getLast? = Option.map f (l.getLast?) := by
cases' l with a l
. rfl
induction' l with b l ih
. rfl
. simp [List.getLast?_cons_cons, ih]
end List
end AddToOrder
--trying and failing to prove ht p = dim R_p --trying and failing to prove ht p = dim R_p
section Localization section Localization
@ -102,6 +137,8 @@ section Polynomial
open Ideal Polynomial open Ideal Polynomial
variables {R : Type _} [CommRing R] variables {R : Type _} [CommRing R]
variable (J : Ideal R[X])
#check Ideal.comap C J
--given ideals I J, I ⊔ J is their sum --given ideals I J, I ⊔ J is their sum
--given a in R, span {a} is the ideal generated by a --given a in R, span {a} is the ideal generated by a
@ -109,48 +146,138 @@ variables {R : Type _} [CommRing R]
--to show p[x] is prime, show p[x] is the kernel of the map R[x] → R → R/p --to show p[x] is prime, show p[x] is the kernel of the map R[x] → R → R/p
#check RingHom.ker_isPrime #check RingHom.ker_isPrime
def adj_x_map (I : Ideal R) : R[X] →+* R I := (Ideal.Quotient.mk I).comp (evalRingHom 0) def adj_x_map (I : Ideal R) : R[X] →+* R I := (Ideal.Quotient.mk I).comp constantCoeff
--def adj_x_map' (I : Ideal R) : R[X] →+* R I := (Ideal.Quotient.mk I).comp (evalRingHom 0)
def adjoin_x (I : Ideal R) : Ideal (Polynomial R) := RingHom.ker (adj_x_map I) 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 def adjoin_x' (I : PrimeSpectrum R) : PrimeSpectrum (Polynomial R) where
asIdeal := adjoin_x I.asIdeal asIdeal := adjoin_x I.asIdeal
IsPrime := RingHom.ker_isPrime _ IsPrime := RingHom.ker_isPrime _
/- This somehow isn't in Mathlib? -/
@[simp]
theorem span_singleton_one : span ({0} : Set R) = ⊥ := by simp only [span_singleton_eq_bot]
theorem coeff_C_eq : RingHom.comp constantCoeff C = RingHom.id R := by ext; simp
variable (I : PrimeSpectrum R)
#check RingHom.ker (C.comp (Ideal.Quotient.mk I.asIdeal))
--#check Ideal.Quotient.mk I.asIdeal
def map_prime' (I : PrimeSpectrum R) : IsPrime (I.asIdeal.map C) := Ideal.isPrime_map_C_of_isPrime I.IsPrime
def map_prime'' (I : PrimeSpectrum R) : PrimeSpectrum R[X] := ⟨I.asIdeal.map C, map_prime' I⟩
@[simp] @[simp]
lemma adj_x_comp_C (I : Ideal R) : RingHom.comp (adj_x_map I) C = Ideal.Quotient.mk I := by 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] ext x; simp [adj_x_map]
-- ideal.mem_quotient_iff_mem_sup
lemma adjoin_x_eq (I : Ideal R) : adjoin_x I = I.map C ⊔ Ideal.span {X} := by lemma adjoin_x_eq (I : Ideal R) : adjoin_x I = I.map C ⊔ Ideal.span {X} := by
apply le_antisymm apply le_antisymm
. sorry . 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] . rw [sup_le_iff]
constructor constructor
. simp [adjoin_x, RingHom.ker, ←map_le_iff_le_comap, Ideal.map_map] . simp [adjoin_x, RingHom.ker, ←map_le_iff_le_comap, Ideal.map_map]
. simp [span_le, adjoin_x, RingHom.mem_ker, adj_x_map] . simp [span_le, adjoin_x, RingHom.mem_ker, adj_x_map]
lemma adjoin_x_inj {I J : Ideal R} (h : adjoin_x I = adjoin_x J) : I = J := by
simp [adjoin_x_eq] at h
have H : Ideal.map constantCoeff (Ideal.map C I ⊔ span {X}) =
Ideal.map constantCoeff (Ideal.map C J ⊔ span {X}) := by rw [h]
simp [Ideal.map_sup, Ideal.map_span, Ideal.map_map, coeff_C_eq] at H
exact H
lemma map_lt_adjoin_x (I : PrimeSpectrum R) : map_prime'' I < adjoin_x' I := by
simp [map_prime'', adjoin_x', adjoin_x_eq]
show Ideal.map C I.asIdeal < Ideal.map C I.asIdeal ⊔ span {X}
simp [Ideal.span_le, mem_map_C_iff]
use 1
simp
intro h
apply I.IsPrime.ne_top'
rw [Ideal.eq_top_iff_one]
exact h
lemma map_inj {I J : Ideal R} (h : I.map C = J.map C) : I = J := by
have H : Ideal.map constantCoeff (Ideal.map C I) =
Ideal.map constantCoeff (Ideal.map C J) := by rw [h]
simp [Ideal.map_map, coeff_C_eq] at H
exact H
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 ⊢
constructor
. apply map_mono h.1
. intro H
exact h.2 (map_inj H)
lemma adjoin_x_strictmono (I J : Ideal R) (h : I < J) : adjoin_x I < adjoin_x J := by lemma adjoin_x_strictmono (I J : Ideal R) (h : I < J) : adjoin_x I < adjoin_x J := by
rw [lt_iff_le_and_ne] at h ⊢ rw [lt_iff_le_and_ne] at h ⊢
rw [adjoin_x_eq, adjoin_x_eq]
constructor constructor
. apply sup_le_sup_right . rw [adjoin_x_eq, adjoin_x_eq]
apply sup_le_sup_right
apply map_mono h.1 apply map_mono h.1
. intro H . intro H
have H' : Ideal.comap C (Ideal.map C I ⊔ span {X}) = Ideal.comap C (Ideal.map C J ⊔ span {X}) exact h.2 (adjoin_x_inj H)
. rw [H]
sorry
example (n : ℕ∞) : n + 0 = n := by simp?
#eval List.Chain' (· < ·) [2,3]
example : [4,5] ++ [2] = [4,5,2] := rfl
#eval [2,4,5].map (λ n => n + n)
/- Given an ideal p in R, define the ideal p[x] in R[x] -/ /- Given an ideal p in R, define the ideal p[x] in R[x] -/
lemma ht_adjoin_x_eq_ht_add_one (I : PrimeSpectrum R) : height I + 1 ≤ height (adjoin_x' I) := by lemma ht_adjoin_x_eq_ht_add_one [Nontrivial R] (I : PrimeSpectrum R) : height I + 1 ≤ height (adjoin_x' I) := by
have H : ∀ l ∈ {J : PrimeSpectrum R | J < I}.subchain, ∃ suffices H : height I + (1 : ) ≤ height (adjoin_x' I) + (0 : )
. norm_cast at H; rw [add_zero] at H; exact H
lemma ne_bot_iff_exists (n : WithBot ℕ∞) : n ≠ ⊥ ↔ ∃ m : ℕ∞, n = m := by rw [height, height, Set.chainHeight_add_le_chainHeight_add {J | J < I} _ 1 0]
cases' n with n; intro l hl
simp use ((l.map map_prime'') ++ [map_prime'' I])
intro x hx constructor
cases hx . simp [Set.append_mem_subchain_iff]
simp refine' ⟨_,_,_⟩
use n . show (List.map map_prime'' l).Chain' (· < ·) ∧ ∀ i ∈ _, i ∈ _
rfl constructor
. apply List.chain'_map_of_chain' map_prime''
intro a b hab
apply map_strictmono a.asIdeal b.asIdeal
exact hab
exact hl.1
. intro i hi
rw [List.mem_map] at hi
obtain ⟨a, ha, rfl⟩ := hi
show map_prime'' a < adjoin_x' I
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
. apply map_lt_adjoin_x
. intro a ha
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
. simp
lemma ne_bot_iff_exists' (n : WithBot ℕ∞) : n ≠ ⊥ ↔ ∃ m : ℕ∞, n = m := by lemma ne_bot_iff_exists' (n : WithBot ℕ∞) : n ≠ ⊥ ↔ ∃ m : ℕ∞, n = m := by
convert WithBot.ne_bot_iff_exists using 3 convert WithBot.ne_bot_iff_exists using 3