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52 lines
1.9 KiB
Text
52 lines
1.9 KiB
Text
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import CommAlg.krull
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variable {R : Type _} [CommRing R] (I : Ideal R)
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open Ideal
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open PrimeSpectrum
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lemma comap_strictmono {𝔭 𝔮 : PrimeSpectrum (R ⧸ I)} (h : 𝔭 < 𝔮) :
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PrimeSpectrum.comap (Ideal.Quotient.mk I) 𝔭 < PrimeSpectrum.comap (Ideal.Quotient.mk I) 𝔮 := by
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rw [lt_iff_le_and_ne] at h ⊢
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refine' ⟨Ideal.comap_mono h.1, fun H => h.2 _⟩
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. apply PrimeSpectrum.comap_injective_of_surjective (Ideal.Quotient.mk I)
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. exact Quotient.mk_surjective
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. exact H
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lemma ht_comap_eq_ht (𝔭 : PrimeSpectrum (R ⧸ I)) :
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height 𝔭 ≤ height (comap (Ideal.Quotient.mk I) 𝔭) := by
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rw [height, height, Set.chainHeight_le_chainHeight_iff]
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rintro l ⟨l_ch, l_lt⟩
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use l.map (comap <| Ideal.Quotient.mk I)
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refine' ⟨⟨_, _⟩, by simp⟩
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. apply List.chain'_map_of_chain' (PrimeSpectrum.comap (Ideal.Quotient.mk I)) _ l_ch
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intro a b hab; apply comap_strictmono; apply hab
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. rintro i hi
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rw [List.mem_map] at hi
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obtain ⟨a, a_mem, rfl⟩ := hi
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apply comap_strictmono
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apply l_lt a a_mem
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/- TODO: find a better lemma to avoid repeated code -/
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lemma dim_quotient_le_dim : krullDim (R ⧸ I) ≤ krullDim R := by
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by_cases H : Nontrivial (R ⧸ I)
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. obtain ⟨n, hn⟩ := krullDim_nonneg_of_nontrivial (R ⧸ I)
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rw [hn]
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induction' n using ENat.recTopCoe with n
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. have H := (krullDim_eq_top_iff _).mp hn
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show ⊤ ≤ _
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rw [top_le_iff, krullDim_eq_top_iff]
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intro n
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obtain ⟨𝔭, hI⟩ := H n
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use comap (Ideal.Quotient.mk I) 𝔭
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apply le_trans hI (ht_comap_eq_ht I _)
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. show n ≤ krullDim _
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rw [le_krullDim_iff]
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obtain ⟨𝔭, hI⟩ := le_krullDim_iff.mp <| le_of_eq hn.symm
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use comap (Ideal.Quotient.mk I) 𝔭
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norm_cast at hI ⊢
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apply le_trans hI (ht_comap_eq_ht I _)
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. suffices : krullDim (R ⧸ I) = ⊥
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. rw [this]; apply bot_le
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rw [dim_eq_bot_iff, ←not_nontrivial_iff_subsingleton]
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exact H
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