File size: 4,324 Bytes
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 | import OAI.NumberTheory.PiExponent.Approximation.RegularComponentSection
import OAI.NumberTheory.PiExponent.Geometry.CurveComponentLocalLengths
import OAI.NumberTheory.PiExponent.Geometry.CurveComponentZeroPoints
import OAI.NumberTheory.PiExponent.Geometry.CurveCutQuotient
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
open PiExponent.SectionZeroIdeal PiExponent.SectionZeroStalk
theorem componentLocalCutLength_eq_zero_stalk {X : Scheme.{0}}
(L : LineBundle X) (s : GlobalSections X L.sheaf) (C : irreducibleComponents X)
(z : (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme)
(U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme)
(hy : (zeroIdeal L s).subschemeι (componentZeroToGlobal L s C z) ∈ U.1) :
componentLocalCutLength X ((zeroIdeal L s).subschemeι (componentZeroToGlobal L s C z))
⟨C,componentZeroToGlobal_mem L s C z⟩
(sectionGerm L s U e _ hy) =
Module.length ((zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk z)
((zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk z) := by
let i := reducedComponentι X C
let J := zeroIdeal (L.pullback i) (pullbackSection i s)
let y := J.subschemeι z
let a := sectionGerm L s U e (i y) hy
obtain ⟨eF,heF⟩ := exists_pullback_frame_sectionGerm i L s U e
let V : (reducedComponent X C).affineOpens := ⟨i ⁻¹ᵁ U.1,U.2.preimage i⟩
let b := sectionGerm (L.pullback i) (pullbackSection i s) V eF y hy
have hb : b = i.stalkMap y a := heF y hy
have hcut := reducedComponent_cut_length X C y a
change componentLocalCutLength X (i y) ⟨C,reducedComponentι_mem X C y⟩ a =
Module.length (J.subscheme.presheaf.stalk z) (J.subscheme.presheaf.stalk z)
change Module.length (X.presheaf.stalk (i y))
((X.presheaf.stalk (i y)) ⧸ ((reducedComponentStalkPrime X C y).val ⊔ Ideal.span {a})) = _
rw [hcut,← hb]
have hsc : Module.length ((reducedComponent X C).presheaf.stalk y)
(((reducedComponent X C).presheaf.stalk y) ⧸ Ideal.span {b}) =
Module.length (((reducedComponent X C).presheaf.stalk y) ⧸ Ideal.span {b})
(((reducedComponent X C).presheaf.stalk y) ⧸ Ideal.span {b}) :=
Module.length_eq_of_surjective (M := ((reducedComponent X C).presheaf.stalk y) ⧸
Ideal.span {b}) Ideal.Quotient.mk_surjective
exact hsc.trans (intrinsic_length_eq_of_ringEquiv
(zeroStalkQuotientEquiv (L.pullback i) (pullbackSection i s) V eF z hy))
theorem componentLocalCutLength_eq_zero_stalk_at {X : Scheme.{0}}
(L : LineBundle X) (s : GlobalSections X L.sheaf) (C : irreducibleComponents X)
(t : {y : (zeroIdeal L s).subscheme // (zeroIdeal L s).subschemeι y ∈ C.val})
(U : X.affineOpens) (e : L.sheaf.restrict U.1.ι ≅ O U.1.toScheme)
(hy : (zeroIdeal L s).subschemeι t.val ∈ U.1) :
componentLocalCutLength X ((zeroIdeal L s).subschemeι t.val) ⟨C,t.property⟩
(sectionGerm L s U e _ hy) =
Module.length ((zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk
((componentZeroPointEquiv L s C).symm t))
((zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk
((componentZeroPointEquiv L s C).symm t)) := by
obtain ⟨z,rfl⟩ := (componentZeroPointEquiv L s C).surjective t
exact (componentLocalCutLength_eq_zero_stalk L s C z U e hy).trans
(congrArg (fun w : (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme =>
Module.length ((zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk w)
((zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme.presheaf.stalk w))
((componentZeroPointEquiv L s C).symm_apply_apply z)).symm
end
end PiExponent.CurveCycle
end OAI
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