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12dee50 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import Mathlib
import OAI.AlgebraicGeometry.Seshadri.LocalAlgebra.LocalCurvePrincipal
namespace OAI
section
namespace MaximalSeshadri.LocalCurve
noncomputable section
open IsLocalRing
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
end
end OAI
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