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Trying to break it down to smaller parts

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Sayantan Santra 2023-06-14 17:01:34 -07:00
parent c64025b1b4
commit 70007679cd
Signed by: SinTan1729
GPG key ID: EB3E68BFBA25C85F
1 changed files with 10 additions and 0 deletions
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CommAlg/sayantan(dim_eq_dim_polynomial_add_one).lean
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@ -3,9 +3,11 @@ import Mathlib.Order.Height
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
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.AtPrime import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Basic import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Set.Ncard
namespace Ideal namespace Ideal
@ -24,6 +26,13 @@ lemma dim_le_dim_polynomial_add_one [Nontrivial R] :
lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := sorry -- Already done in main file lemma height_le_of_le {I J : PrimeSpectrum R} (I_le_J : I ≤ J) : height I ≤ height J := sorry -- Already done in main file
lemma primeIdeal_finite_height_of_noetherianRing [Nontrivial R] [IsNoetherianRing R] (P: PrimeSpectrum R) : height P ≠ ⊤ := by
sorry
lemma exist_elts_MinimalOver_of_primeIdeal_of_noetherianRing [Nontrivial R] [IsNoetherianRing R] (P: PrimeSpectrum R) :
∃S : Set R, Set.ncard s = height P ∧ P.asIdeal ∈ Ideal.minimalPrimes (Ideal.span S) := by
sorry
lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] : lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
krullDim R + 1 = krullDim (Polynomial R) := by krullDim R + 1 = krullDim (Polynomial R) := by
rw [le_antisymm_iff] rw [le_antisymm_iff]
@ -43,6 +52,7 @@ lemma dim_eq_dim_polynomial_add_one [Nontrivial R] [IsNoetherianRing R] :
have : height P ≤ height P' := height_le_of_le PleP' have : height P ≤ height P' := height_le_of_le PleP'
simp only [WithBot.coe_le_coe] simp only [WithBot.coe_le_coe]
have : ∃ (I : PrimeSpectrum R), height P' ≤ height I + 1 := by have : ∃ (I : PrimeSpectrum R), height P' ≤ height I + 1 := by
-- Prime avoidance is called subset_union_prime
sorry sorry
obtain ⟨I, h⟩ := this obtain ⟨I, h⟩ := this