openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /FullSupportNilpotence.lean
| import Mathlib.AlgebraicGeometry.IdealSheaf.Basic | |
| import Mathlib.AlgebraicGeometry.Noetherian | |
| import Mathlib.RingTheory.Noetherian.Nilpotent | |
| namespace OAI | |
| namespace PiExponentSeshadri.Geometry | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| variable {X : Scheme} | |
| lemma idealSheaf_le_nilradical_of_fullSupport (I : X.IdealSheafData) | |
| (hI : I.support = ⊤) : I ≤ X.nilradical := by | |
| have h : (⊤ : Closeds X) ≤ I.support := by rw [hI] | |
| simpa only [Scheme.IdealSheafData.vanishingIdeal_top] using | |
| Scheme.IdealSheafData.le_support_iff_le_vanishingIdeal.mp h | |
| lemma affineIdeal_isNilpotent_of_fullSupport [IsLocallyNoetherian X] | |
| (I : X.IdealSheafData) (hI : I.support = ⊤) (U : X.affineOpens) : | |
| IsNilpotent (I.ideal U) := by | |
| let := IsLocallyNoetherian.component_noetherian U | |
| apply (Ideal.FG.isNilpotent_iff_le_nilradical (IsNoetherian.noetherian _)).mpr | |
| exact idealSheaf_le_nilradical_of_fullSupport I hI U | |
| theorem exists_pos_pow_eq_bot_of_fullSupport [IsNoetherian X] | |
| (I : X.IdealSheafData) (hI : I.support = ⊤) : | |
| ∃ n : ℕ, 0 < n ∧ I ^ n = ⊥ := by | |
| classical | |
| have hlocal : ∀ U : X.affineOpens, ∃ n : ℕ, I.ideal U ^ n = ⊥ := | |
| fun U => affineIdeal_isNilpotent_of_fullSupport I hI U | |
| choose n hn using hlocal | |
| obtain ⟨S, hS⟩ := isCompact_univ.elim_finite_subcover | |
| (fun U : X.affineOpens => (U.1 : Set X)) (fun U => U.1.isOpen) | |
| (by | |
| intro x _ | |
| have hx : x ∈ (⨆ U : X.affineOpens, U.1) := by | |
| rw [iSup_affineOpens_eq_top X] | |
| trivial | |
| obtain ⟨U, hU⟩ := Opens.mem_iSup.mp hx | |
| exact Set.mem_iUnion.mpr ⟨U, hU⟩) | |
| refine ⟨S.sup n + 1, Nat.succ_pos _, ?_⟩ | |
| apply Scheme.IdealSheafData.ext_of_iSup_eq_top (fun U : S => U.val) | |
| · apply top_unique | |
| intro x _ | |
| obtain ⟨U, hU⟩ := Set.mem_iUnion.mp (hS (show x ∈ Set.univ from trivial)) | |
| obtain ⟨hUS, hxU⟩ := Set.mem_iUnion.mp hU | |
| exact Opens.mem_iSup.mpr ⟨⟨U, hUS⟩, hxU⟩ | |
| · intro U | |
| change I.ideal U.val ^ (S.sup n + 1) = ⊥ | |
| apply le_bot_iff.mp | |
| calc | |
| I.ideal U.val ^ (S.sup n + 1) ≤ I.ideal U.val ^ n U.val := | |
| Ideal.pow_le_pow_right ((Finset.le_sup U.property).trans (Nat.le_succ _)) | |
| _ = ⊥ := hn U.val | |
| theorem exists_pos_pow_eq_bot_of_support_univ [IsNoetherian X] | |
| (I : X.IdealSheafData) (hI : (I.support : Set X) = Set.univ) : | |
| ∃ n : ℕ, 0 < n ∧ I ^ n = ⊥ := | |
| exists_pos_pow_eq_bot_of_fullSupport I (SetLike.coe_injective hI) | |
| theorem exists_pos_nilradical_pow_eq_bot [IsNoetherian X] : | |
| ∃ n : ℕ, 0 < n ∧ X.nilradical ^ n = ⊥ := | |
| exists_pos_pow_eq_bot_of_fullSupport X.nilradical X.support_nilradical | |
| end | |
| end PiExponentSeshadri.Geometry | |
| end OAI | |