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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 | import Mathlib
namespace OAI
section
namespace MaximalSeshadri.Geometry
noncomputable section
open CategoryTheory AlgebraicGeometry TopologicalSpace
variable {X Y : Scheme.{0}}
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 _
lemma residue_image_integral (X : Scheme.{0}) (x : X) :
IsIntegral (X.fromSpecResidueField x).image := integral_scheme_image _
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]
end
end MaximalSeshadri.Geometry
end
end OAI
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