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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 | import OAI.NumberTheory.PiExponent.Geometry.CurveLocalAssociativity
import OAI.NumberTheory.PiExponent.Geometry.CurveLocalMultiplicity
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentJets.W28.LocalIntersection
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
instance finite_componentsThrough {X : Scheme.{0}} [IsLocallyNoetherian X] (x : X) :
Finite {C : irreducibleComponents X // x ∈ C.val} := by
let : Finite (minimalPrimes (X.presheaf.stalk x)) :=
((⊥ : Ideal (X.presheaf.stalk x)).finite_minimalPrimes_of_isNoetherianRing _).to_subtype
exact Finite.of_equiv _ (stalkMinimalPrimesEquivComponentsThrough X x)
def componentLocalPrime (X : Scheme.{0}) (x : X)
(C : {C : irreducibleComponents X // x ∈ C.val}) :
Ideal (X.presheaf.stalk x) :=
((stalkMinimalPrimesEquivComponentsThrough X x).symm C).val
def componentLocalCutLength (X : Scheme.{0}) (x : X)
(C : {C : irreducibleComponents X // x ∈ C.val})
(c : X.presheaf.stalk x) : ℕ∞ :=
Module.length (X.presheaf.stalk x)
((X.presheaf.stalk x) ⧸ (componentLocalPrime X x C ⊔ Ideal.span {c}))
theorem minimalPrimeCutSum_eq_component_sum {X : Scheme.{0}} [IsLocallyNoetherian X]
(x : X) (c : X.presheaf.stalk x) :
letI : Fintype {C : irreducibleComponents X // x ∈ C.val} := Fintype.ofFinite _
minimalPrimeCutSum (⊥ : Ideal (X.presheaf.stalk x)) c =
∑ C : {C : irreducibleComponents X // x ∈ C.val},
(componentMultiplicity X C.val : ℕ∞) * componentLocalCutLength X x C c := by
classical
let : Fintype {C : irreducibleComponents X // x ∈ C.val} := Fintype.ofFinite _
let : Fintype (minimalPrimes (X.presheaf.stalk x)) :=
((⊥ : Ideal (X.presheaf.stalk x)).finite_minimalPrimes_of_isNoetherianRing _).fintype
unfold minimalPrimeCutSum
apply Fintype.sum_equiv (stalkMinimalPrimesEquivComponentsThrough X x)
intro P
let : P.val.IsPrime := P.property.1.1
rw [Ideal.map_bot]
have hlength := SectionZeroStalk.intrinsic_length_eq_of_ringEquiv
(RingEquiv.quotientBot (Localization.AtPrime P.val))
have hsc : Module.length (Localization.AtPrime P.val)
(Localization.AtPrime P.val ⧸ (⊥ : Ideal (Localization.AtPrime P.val))) =
Module.length (Localization.AtPrime P.val ⧸ (⊥ : Ideal (Localization.AtPrime P.val)))
(Localization.AtPrime P.val ⧸ (⊥ : Ideal (Localization.AtPrime P.val))) :=
Module.length_eq_of_surjective (M := Localization.AtPrime P.val ⧸
(⊥ : Ideal (Localization.AtPrime P.val))) Ideal.Quotient.mk_surjective
rw [hsc,hlength,localMinimalPrime_length_eq_componentMultiplicity x P]
congr 1
exact (congrArg (fun Q : minimalPrimes (X.presheaf.stalk x) =>
Module.length (X.presheaf.stalk x)
((X.presheaf.stalk x) ⧸ (Q.val ⊔ Ideal.span {c})))
((stalkMinimalPrimesEquivComponentsThrough X x).symm_apply_apply P)).symm
theorem zero_stalk_length_eq_component_sum {X : Scheme.{0}} [IsLocallyNoetherian X]
(hd : topologicalKrullDim X ≤ 1)
(L : LineBundle X) (s : GlobalSections X L.sheaf) [Mono s]
(U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme)
(y : (SectionZeroIdeal.zeroIdeal L s).subscheme)
(hy : (SectionZeroIdeal.zeroIdeal L s).subschemeι y ∈ U.1) :
let x := (SectionZeroIdeal.zeroIdeal L s).subschemeι y
letI : Fintype {C : irreducibleComponents X // x ∈ C.val} := Fintype.ofFinite _
Module.length ((SectionZeroIdeal.zeroIdeal L s).subscheme.presheaf.stalk y)
((SectionZeroIdeal.zeroIdeal L s).subscheme.presheaf.stalk y) =
∑ C : {C : irreducibleComponents X // x ∈ C.val},
(componentMultiplicity X C.val : ℕ∞) * componentLocalCutLength X x C
(SectionZeroStalk.sectionGerm L s U e x hy) := by
exact (zero_stalk_length_eq_minimalPrimeCutSum hd L s U e y hy).trans
(minimalPrimeCutSum_eq_component_sum _ _)
end
end PiExponent.CurveCycle
end OAI
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