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b797a5b | 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 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 | 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
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