| import Mathlib |
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| namespace OAI |
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| section |
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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open CategoryTheory AlgebraicGeometry TopologicalSpace |
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| variable {X Y : Scheme.{0}} |
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| lemma integral_scheme_image (f : X ⟶ Y) [QuasiCompact f] [IsIntegral X] : IsIntegral f.image := by |
| let (chart : f.ker.subschemeCover.openCover.I₀) : |
| AlgebraicGeometry.IsReduced (f.ker.subschemeCover.openCover.X chart) := by |
| change Y.affineOpens at chart |
| let : _root_.IsReduced (Γ(Y,chart.1) ⧸ f.ker.ideal chart) := by |
| rw [Scheme.Hom.ker_apply] |
| exact isReduced_of_injective (RingHom.kerLift (f.app chart.1).hom) |
| (RingHom.kerLift_injective (f.app chart.1).hom) |
| change IsReduced (Spec (.of (Γ(Y,chart.1) ⧸ f.ker.ideal chart))) |
| infer_instance |
| let : IsReduced f.image := IsReduced.of_openCover f.image f.ker.subschemeCover.openCover |
| have hi : IsIrreducible (Set.range f.toImage) := by |
| simpa only [Set.image_univ] using |
| (IrreducibleSpace.isIrreducible_univ (X := X)).image f.toImage |
| f.toImage.continuous.continuousOn |
| let : IrreducibleSpace f.image := (irreducibleSpace_def _).mpr (by |
| change IsIrreducible (Set.univ : Set f.image) |
| rw [← f.toImage.denseRange.closure_range] |
| exact hi.closure) |
| exact isIntegral_of_irreducibleSpace_of_isReduced _ |
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| lemma residue_image_integral (X : Scheme.{0}) (x : X) : |
| IsIntegral (X.fromSpecResidueField x).image := integral_scheme_image _ |
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| lemma residue_image_range (X : Scheme.{0}) (x : X) : |
| Set.range (X.fromSpecResidueField x).imageι = closure {x} := by |
| rw [Scheme.Hom.imageι, Scheme.IdealSheafData.range_subschemeι, |
| Scheme.Hom.support_ker, Scheme.range_fromSpecResidueField] |
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| end |
| end MaximalSeshadri.Geometry |
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| end |
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| end OAI |
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