defined symbolicIdeal (generalized symbolic power)

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GTBarkley 2023-06-16 01:22:46 +00:00
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import Mathlib.Order.KrullDimension
import Mathlib.Order.JordanHolder
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.Height
import Mathlib.RingTheory.Noetherian
import CommAlg.krull
variable (R : Type _) [CommRing R] [IsNoetherianRing R]
lemma height_le_of_gt_height_lt {n : ℕ∞} (q : PrimeSpectrum R)
(h : ∀(p : PrimeSpectrum R), p < q → Ideal.height p ≤ n - 1) : Ideal.height q ≤ n := by
sorry
theorem height_le_one_of_minimal_over_principle (p : PrimeSpectrum R) (x : R):
p ∈ minimals (· < ·) {p | x ∈ p.asIdeal} → Ideal.height p ≤ 1 := by
intro h
apply height_le_of_gt_height_lt _ p
intro q qlep
by_contra hcontr
push_neg at hcontr
simp only [le_refl, tsub_eq_zero_of_le] at hcontr
sorry
#check (_ : Ideal R) ^ (_ : )
#synth Pow (Ideal R) ()
def symbolicIdeal(Q : Ideal R) {hin : Q.IsPrime} (I : Ideal R) : Ideal R where
carrier := {r : R | ∃ s : R, s ∉ Q ∧ s * r ∈ I}
zero_mem' := by
simp only [Set.mem_setOf_eq, mul_zero, Submodule.zero_mem, and_true]
use 1
rw [←Q.ne_top_iff_one]
exact hin.ne_top
add_mem' := by
rintro a b ⟨sa, hsa1, hsa2⟩ ⟨sb, hsb1, hsb2⟩
use sa * sb
constructor
. intro h
cases hin.mem_or_mem h <;> contradiction
ring_nf
apply I.add_mem --<;> simp only [I.mul_mem_left, hsa2, hsb2]
. rw [mul_comm sa, mul_assoc]
exact I.mul_mem_left sb hsa2
. rw [mul_assoc]
exact I.mul_mem_left sa hsb2
smul_mem' := by
intro c x
dsimp
rintro ⟨s, hs1, hs2⟩
use s
constructor; exact hs1
rw [←mul_assoc, mul_comm s, mul_assoc]
exact Ideal.mul_mem_left _ _ hs2
protected lemma LocalRing.height_le_one_of_minimal_over_principle
[LocalRing R] (q : PrimeSpectrum R) {x : R}
(h : (closedPoint R).asIdeal ∈ (Ideal.span {x}).minimalPrimes) :
q = closedPoint R Ideal.height q = 0 := by
sorry