| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.FiniteScheme |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.CurveFinite |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| variable {K : Type} [Field K] {X : Scheme} |
|
|
| theorem finite_subscheme_of_curve [IsIntegral X] |
| (f : X ⟶ Spec (CommRingCat.of K)) [IsProper f] |
| (hd : topologicalKrullDim X ≤ 1) (I : X.IdealSheafData) (hI : I ≠ ⊥) : |
| Finite I.subscheme := by |
| let : CompactSpace X := QuasiCompact.compactSpace_of_compactSpace f |
| let : IsLocallyNoetherian X := LocallyOfFiniteType.isLocallyNoetherian f |
| let : IsNoetherian X := ⟨⟩ |
| have hp : (I.support : Set X) ≠ Set.univ := by |
| intro he |
| apply hI |
| exact Scheme.IdealSheafData.support_eq_top_iff.mp (SetLike.coe_injective he) |
| have hf := proper_closed_finite_of_dimension_one hd I.support.isClosed hp |
| have hr : (Set.range I.subschemeι).Finite := I.range_subschemeι ▸ hf |
| let : Finite (Set.range I.subschemeι) := hr.to_subtype |
| exact Finite.of_injective (fun x : I.subscheme => |
| (⟨I.subschemeι x, ⟨x,rfl⟩⟩ : Set.range I.subschemeι)) |
| (fun a b h => I.subschemeι.isEmbedding.injective (congrArg (fun z : Set.range I.subschemeι => (z : X)) h)) |
|
|
| theorem curve_subscheme_isFinite [IsIntegral X] |
| (f : X ⟶ Spec (CommRingCat.of K)) [IsProper f] |
| (hd : topologicalKrullDim X ≤ 1) (I : X.IdealSheafData) (hI : I ≠ ⊥) : |
| IsFinite (I.subschemeι ≫ f) := by |
| let : Finite I.subscheme := finite_subscheme_of_curve f hd I hI |
| exact proper_finite_scheme_isFinite (I.subschemeι ≫ f) |
|
|
| end |
| end MaximalSeshadri.Geometry |
| end |
|
|
|
|
| end |
| end |
|
|
| end OAI |
|
|
