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leopoldmayer 2023-06-15 22:13:11 -07:00
parent 4b9f84f7c1
commit c702c535b5
1 changed files with 8 additions and 18 deletions
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CommAlg/polynomial.lean
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@ -98,33 +98,23 @@ lemma map_lt_adjoin_x (I : PrimeSpectrum R) : map_prime I < adjoin_x I := by
simp simp
rw [←Ideal.eq_top_iff_one] rw [←Ideal.eq_top_iff_one]
exact I.IsPrime.ne_top' exact I.IsPrime.ne_top'
/- Given an ideal p in R, define the ideal p[x] in R[x] -/
lemma ht_adjoin_x_eq_ht_add_one [Nontrivial R] (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
suffices H : height I + (1 : ℕ) ≤ height (adjoin_x I) + (0 : ℕ) suffices H : height I + (1 : ℕ) ≤ height (adjoin_x I) + (0 : ℕ)
. norm_cast at H; rw [add_zero] at H; exact H . norm_cast at H; rw [add_zero] at H; exact H
rw [height, height, Set.chainHeight_add_le_chainHeight_add {J | J < I} _ 1 0] rw [height, height, Set.chainHeight_add_le_chainHeight_add {J | J < I} _ 1 0]
intro l hl intro l hl
use ((l.map map_prime) ++ [map_prime I]) use ((l.map map_prime) ++ [map_prime I])
constructor refine' ⟨_, by simp⟩
. simp [Set.append_mem_subchain_iff] . simp [Set.append_mem_subchain_iff]
refine' ⟨_,_,_⟩ refine' ⟨_, map_lt_adjoin_x I, fun a ha => _⟩
. show (List.map map_prime l).Chain' (· < ·) ∧ ∀ i ∈ _, i ∈ _ . refine' ⟨_, fun i hi => _⟩
constructor . apply List.chain'_map_of_chain' map_prime (fun a b hab => map_strictmono hab) hl.1
. apply List.chain'_map_of_chain' map_prime . rw [List.mem_map] at hi
intro a b hab
apply map_strictmono
exact hab
exact hl.1
. intro i hi
rw [List.mem_map] at hi
obtain ⟨a, ha, rfl⟩ := 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 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 _ < adjoin_x I := by apply map_lt_adjoin_x
. apply map_lt_adjoin_x . have H : ∃ b : PrimeSpectrum R, b ∈ l ∧ map_prime b = a
. intro a ha
have H : ∃ b : PrimeSpectrum R, b ∈ l ∧ map_prime b = a
. have H2 : l ≠ [] . have H2 : l ≠ []
. intro h . intro h
rw [h] at ha rw [h] at ha
@ -141,7 +131,7 @@ lemma ht_adjoin_x_eq_ht_add_one [Nontrivial R] (I : PrimeSpectrum R) : height I
apply map_strictmono apply map_strictmono
apply hl.2 apply hl.2
exact hb exact hb
. simp
/- /-
dim R + 1 ≤ dim R[X] dim R + 1 ≤ dim R[X]