openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /LocalAlgebra /LocalCurvePrincipal.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| namespace MaximalSeshadri.LocalCurve | |
| noncomputable section | |
| open IsLocalRing | |
| theorem principal_of_principal_maximal {R : Type*} [CommRing R] | |
| [IsNoetherianRing R] [IsLocalRing R] | |
| (hm : (maximalIdeal R).IsPrincipal) : IsPrincipalIdealRing R := by | |
| classical | |
| obtain ⟨t,ht⟩ := hm | |
| constructor | |
| intro I | |
| by_cases hI : I = ⊥ | |
| · rw [hI] | |
| infer_instance | |
| have hex : ∃ n : ℕ, ¬ I ≤ maximalIdeal R ^ (n+1) := by | |
| by_contra! H | |
| apply hI | |
| apply le_antisymm | |
| · rw [← Ideal.iInf_pow_eq_bot_of_isLocalRing (maximalIdeal R) (maximalIdeal.isMaximal R).ne_top] | |
| apply le_iInf | |
| rintro (_ | n) | |
| · simp | |
| · exact H n | |
| · exact bot_le | |
| let n := Nat.find hex | |
| have hle : I ≤ maximalIdeal R ^ n := by | |
| cases hn : n with | |
| | zero => simp | |
| | succ j => | |
| have hj : j < Nat.find hex := by change j < n; omega | |
| exact not_not.mp (Nat.find_min hex hj) | |
| obtain ⟨a,ha,han⟩ := SetLike.not_le_iff_exists.mp (Nat.find_spec hex) | |
| have hat : t^n ∣ a := by | |
| have H := hle ha | |
| rw [ht,Ideal.span_singleton_pow,Ideal.mem_span_singleton] at H | |
| exact H | |
| obtain ⟨b,hb⟩ := hat | |
| have hbu : IsUnit b := by | |
| by_contra h | |
| have hbmem : b ∈ maximalIdeal R := h | |
| apply han | |
| change a ∉ maximalIdeal R ^ (n+1) at han | |
| have htn : t^n ∈ maximalIdeal R^n := by | |
| exact Ideal.pow_mem_pow (by rw [ht]; exact Ideal.subset_span (by simp)) n | |
| rw [hb] | |
| exact Ideal.mul_mem_mul htn hbmem | |
| have he : I = maximalIdeal R^n := by | |
| apply le_antisymm hle | |
| rw [ht,Ideal.span_singleton_pow,Ideal.span_le,Set.singleton_subset_iff] | |
| obtain ⟨u,hu⟩ := hbu | |
| have H := I.mul_mem_left (↑(u⁻¹) : R) ha | |
| rw [hb,← hu] at H | |
| simpa [mul_left_comm,mul_comm,mul_assoc] using H | |
| rw [he,ht,Ideal.span_singleton_pow] | |
| infer_instance | |
| end | |
| end MaximalSeshadri.LocalCurve | |
| end | |
| end OAI | |