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7f5fab8 | 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 | 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
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