Merge pull request #78 from SinTan1729/main

Proved one side of poly_over_field
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Sayantan Santra 2023-06-15 21:45:48 -05:00 committed by GitHub
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2 changed files with 48 additions and 2 deletions

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@ -88,7 +88,7 @@ lemma le_krullDim_iff (R : Type _) [CommRing R] (n : ) :
fun _ ↦ (WithBot.coe_le rfl).mpr (H1 _)
rw [←iSup_le_iff] at H1
have : ((n : ℕ∞) : WithBot ℕ∞) ≤ (((n - 1 : ) : ℕ∞) : WithBot ℕ∞) := le_trans H H1
norm_cast at this
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⟩
@ -280,7 +280,7 @@ lemma domain_dim_zero.isField {D: Type _} [CommRing D] [IsDomain D] (h: krullDim
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 (Iff.mp ENat.one_le_iff_pos 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

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@ -0,0 +1,46 @@
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
lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
-- unfold krullDim
rw [le_antisymm_iff]
constructor
·
sorry
· 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
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]
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