openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Geometry /CurveComponentMultiplicity.lean
| import Mathlib | |
| import OAI.NumberTheory.PiExponent.Geometry.IntegralCurveExistence | |
| namespace OAI | |
| namespace PiExponent.CurveCycle | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace Topology | |
| theorem generic_stalk_isArtinian {X : Scheme.{0}} [IsLocallyNoetherian X] | |
| (x : X) (hx : closure {x} ∈ irreducibleComponents X) : | |
| IsArtinianRing (X.presheaf.stalk x) := by | |
| wlog hX : ∃ R : CommRingCat, X = Spec R | |
| · obtain ⟨i,x,rfl⟩ := X.affineCover.exists_eq x | |
| let hNoeth : IsLocallyNoetherian (X.affineCover.X i) := | |
| LocallyOfFiniteType.isLocallyNoetherian (X.affineCover.f i) | |
| have hloc := this x (by | |
| rw [(X.affineCover.f i).isOpenEmbedding.closure_eq_preimage_closure_image, | |
| Set.image_singleton] | |
| exact preimage_mem_irreducibleComponents hx (X.affineCover.f i).isOpenEmbedding | |
| ⟨X.affineCover.f i x, subset_closure rfl, _, rfl⟩) ⟨_,rfl⟩ | |
| let := hloc | |
| exact (asIso ((X.affineCover.f i).stalkMap x)).commRingCatIsoToRingEquiv.symm.isArtinianRing | |
| obtain ⟨R,rfl⟩ := hX | |
| let : IsNoetherianRing ((Spec R).presheaf.stalk x) := inferInstance | |
| change PrimeSpectrum R at x | |
| change closure {x} ∈ irreducibleComponents (PrimeSpectrum R) at hx | |
| have hp : x.asIdeal ∈ minimalPrimes R := by | |
| rwa [← PrimeSpectrum.vanishingIdeal_singleton, | |
| PrimeSpectrum.vanishingIdeal_mem_minimalPrimes] | |
| let : Ring.KrullDimLE 0 ((Spec R).presheaf.stalk x) := by | |
| exact Ring.KrullDimLE.of_isLocalization x.asIdeal hp ((Spec.structureSheaf R).presheaf.stalk x) | |
| exact IsNoetherianRing.isArtinianRing_of_krullDimLE_zero | |
| def componentMultiplicity (X : Scheme.{0}) (C : irreducibleComponents X) : ℕ := | |
| (Module.length (X.presheaf.stalk (genericPoints.ofComponent C).val) | |
| (X.presheaf.stalk (genericPoints.ofComponent C).val)).toNat | |
| theorem componentMultiplicity_pos {X : Scheme.{0}} [IsLocallyNoetherian X] | |
| (C : irreducibleComponents X) : 0 < componentMultiplicity X C := by | |
| let x := (genericPoints.ofComponent C).val | |
| let : IsArtinianRing (X.presheaf.stalk x) := | |
| generic_stalk_isArtinian x (genericPoints.ofComponent C).property | |
| exact ENat.toNat_pos (ne_of_gt Module.length_pos) Module.length_ne_top | |
| def reducedComponent (X : Scheme.{0}) (C : irreducibleComponents X) : Scheme := | |
| (Scheme.IdealSheafData.vanishingIdeal | |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X)).subscheme | |
| def reducedComponentι (X : Scheme.{0}) (C : irreducibleComponents X) : | |
| reducedComponent X C ⟶ X := | |
| (Scheme.IdealSheafData.vanishingIdeal | |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X)).subschemeι | |
| instance reducedComponent_closed (X : Scheme.{0}) (C : irreducibleComponents X) : | |
| IsClosedImmersion (reducedComponentι X C) := by | |
| let I : X.IdealSheafData := Scheme.IdealSheafData.vanishingIdeal | |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X) | |
| change IsClosedImmersion I.subschemeι | |
| infer_instance | |
| instance reducedComponent_integral (X : Scheme.{0}) (C : irreducibleComponents X) : | |
| IsIntegral (reducedComponent X C) := by | |
| let Z : Closeds X := ⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ | |
| let I := Scheme.IdealSheafData.vanishingIdeal Z | |
| have hI : (I.support : Set X) = C.val := by | |
| simp only [I,Scheme.IdealSheafData.coe_support_vanishingIdeal] | |
| rfl | |
| let : IsReduced I.subscheme := NumericalAmpleness.isReduced_vanishingIdeal_subscheme Z | |
| let : IrreducibleSpace I.subscheme := by | |
| change IrreducibleSpace (I.support : Set X) | |
| rw [hI] | |
| exact Subtype.irreducibleSpace C.property.1 | |
| change IsIntegral I.subscheme | |
| exact isIntegral_of_irreducibleSpace_of_isReduced _ | |
| def componentCurve {X : Scheme.{0}} (C : irreducibleComponents X) | |
| (hdim : topologicalKrullDim (C.val : Set X) = 1) : | |
| NumericalAmpleness.IntegralCurve X := by | |
| refine { | |
| scheme := reducedComponent X C | |
| embedding := reducedComponentι X C | |
| closedImmersion := inferInstance | |
| integral := inferInstance | |
| dimension := ?_ } | |
| change topologicalKrullDim | |
| ((Scheme.IdealSheafData.vanishingIdeal | |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X)).support : Set X) = 1 | |
| have he : ((Scheme.IdealSheafData.vanishingIdeal | |
| (⟨C.val,isClosed_of_mem_irreducibleComponents C.val C.property⟩ : Closeds X)).support : Set X) = C.val := by | |
| simp only [Scheme.IdealSheafData.coe_support_vanishingIdeal] | |
| rfl | |
| rw [he] | |
| exact hdim | |
| end | |
| end PiExponent.CurveCycle | |
| end OAI | |