| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.LocalCurvePrincipal |
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| namespace OAI |
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| section |
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| namespace MaximalSeshadri.LocalCurve |
| noncomputable section |
| open IsLocalRing |
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| theorem prime_principal_of_binary_maximal_of_not_mem {R : Type*} [CommRing R] |
| [IsNoetherianRing R] [IsLocalRing R] |
| (a b : R) (hm : maximalIdeal R = Ideal.span {a,b}) |
| (p : Ideal R) [hp : p.IsPrime] (ha : a ∉ p) : p.IsPrincipal := by |
| classical |
| let J : Ideal R := Ideal.span {a} |
| have hJm : J ≤ maximalIdeal R := by |
| rw [hm] |
| exact Ideal.span_mono (by simp) |
| have hJ : J ≠ ⊤ := ne_top_of_le_ne_top (maximalIdeal.isMaximal R).ne_top hJm |
| let : Nontrivial (R ⧸ J) := Ideal.Quotient.nontrivial_iff.mpr hJ |
| let q := Ideal.Quotient.mk J |
| let : IsLocalRing (R ⧸ J) := IsLocalRing.of_surjective' q Ideal.Quotient.mk_surjective |
| have hmQ : maximalIdeal (R ⧸ J) = Ideal.span {q b} := by |
| rw [← map_maximalIdeal_of_surjective q Ideal.Quotient.mk_surjective,hm,Ideal.map_span] |
| have hqa : q a = 0 := Ideal.Quotient.eq_zero_iff_mem.mpr (Ideal.subset_span (by simp)) |
| simp only [Set.image_insert_eq,Set.image_singleton,hqa,Ideal.span_insert_zero] |
| let : IsPrincipalIdealRing (R ⧸ J) := principal_of_principal_maximal ⟨q b,hmQ⟩ |
| obtain ⟨c₀,hc₀⟩ := (inferInstance : (p.map q).IsPrincipal) |
| change p.map q = Ideal.span {c₀} at hc₀ |
| have hc₀mem : c₀ ∈ p.map q := by rw [hc₀]; exact Ideal.subset_span (by simp) |
| obtain ⟨c,hcp,hc⟩ := (Ideal.mem_map_iff_of_surjective q Ideal.Quotient.mk_surjective).mp hc₀mem |
| have hp_le : p ≤ Ideal.span {c} ⊔ J • p := by |
| intro z hz |
| have hzq : q z ∈ Ideal.span {c₀} := by |
| rw [← hc₀] |
| exact Ideal.mem_map_of_mem q hz |
| obtain ⟨w₀,hw₀⟩ := Ideal.mem_span_singleton.mp hzq |
| obtain ⟨w,hw⟩ := Ideal.Quotient.mk_surjective w₀ |
| have hdiff : z-c*w ∈ J := by |
| apply Ideal.Quotient.eq_zero_iff_mem.mp |
| change q (z-c*w) = 0 |
| rw [map_sub,map_mul,hc,hw,hw₀,sub_self] |
| obtain ⟨t,ht⟩ := Ideal.mem_span_singleton.mp hdiff |
| have hat : a*t ∈ p := ht ▸ p.sub_mem hz (p.mul_mem_right w hcp) |
| have htp : t ∈ p := (hp.mem_or_mem hat).resolve_left ha |
| have hleft : c*w ∈ Ideal.span {c} := Ideal.mem_span_singleton.mpr ⟨w,rfl⟩ |
| have hright : a*t ∈ J • p := by |
| exact Submodule.smul_mem_smul (Ideal.subset_span (by simp)) htp |
| have H := Submodule.add_mem_sup hleft hright |
| have he : c*w+a*t = z := by linear_combination -ht |
| exact he ▸ H |
| have hple : p ≤ Ideal.span {c} := |
| Submodule.le_of_le_smul_of_le_jacobson_bot |
| (IsNoetherian.noetherian p) (by rw [jacobson_eq_maximalIdeal _ bot_ne_top]; exact hJm) hp_le |
| exact ⟨c,le_antisymm hple (Ideal.span_le.mpr (by simpa using hcp))⟩ |
| |
| theorem prime_principal_of_binary_maximal {R : Type*} [CommRing R] |
| [IsNoetherianRing R] [IsLocalRing R] |
| (a b : R) (hm : maximalIdeal R = Ideal.span {a,b}) |
| (p : Ideal R) [hp : p.IsPrime] (hne : p ≠ maximalIdeal R) : p.IsPrincipal := by |
| by_cases ha : a ∈ p |
| · have hb : b ∉ p := by |
| intro hb |
| apply hne |
| apply le_antisymm (le_maximalIdeal hp.ne_top) |
| rw [hm,Ideal.span_le,Set.insert_subset_iff,Set.singleton_subset_iff] |
| exact ⟨ha,hb⟩ |
| exact prime_principal_of_binary_maximal_of_not_mem b a |
| (hm.trans Ideal.span_pair_comm) p hb |
| · exact prime_principal_of_binary_maximal_of_not_mem a b hm p ha |
| end |
| end MaximalSeshadri.LocalCurve |
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| end |
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| end OAI |
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