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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 | import OAI.NumberTheory.PiExponent.Geometry.CurveComponentCutLengths
import OAI.NumberTheory.PiExponent.Geometry.CurveComponentIncidence
namespace OAI
namespace PiExponent.CurveCycle
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames
open PiExponent.SectionZeroIdeal
def stalkLength (Y : Scheme.{0}) (y : Y) : ℕ∞ :=
Module.length (Y.presheaf.stalk y) (Y.presheaf.stalk y)
theorem zero_stalk_lengthNat_eq_component_sum {X : Scheme.{0}} [IsLocallyNoetherian X]
(hd : topologicalKrullDim X ≤ 1)
(L : LineBundle X) (s : GlobalSections X L.sheaf) [Mono s]
(y : (zeroIdeal L s).subscheme) :
letI : Fintype {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val} :=
Fintype.ofFinite _
(stalkLength (zeroIdeal L s).subscheme y).toNat =
∑ C : {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val},
componentMultiplicity X C.val *
(stalkLength (zeroIdeal (L.pullback (reducedComponentι X C.val))
(pullbackSection (reducedComponentι X C.val) s)).subscheme
((componentZeroPointEquiv L s C.val).symm ⟨y,C.property⟩)).toNat := by
classical
let : Fintype {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val} :=
Fintype.ofFinite _
let f := fun C : {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val} =>
(componentMultiplicity X C.val : ℕ∞) *
stalkLength (zeroIdeal (L.pullback (reducedComponentι X C.val))
(pullbackSection (reducedComponentι X C.val) s)).subscheme
((componentZeroPointEquiv L s C.val).symm ⟨y,C.property⟩)
have hlocal : stalkLength (zeroIdeal L s).subscheme y = ∑ C, f C := by
obtain ⟨U,hy,⟨e⟩,_⟩ := common_affine_frames L L ((zeroIdeal L s).subschemeι y)
refine (zero_stalk_length_eq_component_sum hd L s U e y hy).trans ?_
apply Finset.sum_congr rfl
intro C _
exact congrArg (fun n : ℕ∞ => (componentMultiplicity X C.val : ℕ∞) * n)
(componentLocalCutLength_eq_zero_stalk_at L s C.val ⟨y,C.property⟩ U e hy)
have hsum : ∑ C, f C ≠ ⊤ := by
rw [← hlocal]
exact zero_stalk_length_ne_top hd L s y
have hn := congrArg ENat.toNat hlocal
rw [ENat.toNat_sum (ENat.sum_ne_top.mp hsum)] at hn
simpa only [f,ENat.toNat_mul,ENat.toNat_natCast] using hn
theorem zero_lengthNat_eq_component_sum {X : Scheme.{0}} [IsLocallyNoetherian X]
[Fintype (irreducibleComponents X)] (hd : topologicalKrullDim X ≤ 1)
(L : LineBundle X) (s : GlobalSections X L.sheaf) [Mono s]
[Fintype (zeroIdeal L s).subscheme] :
letI : ∀ C : irreducibleComponents X,
Fintype (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme := fun _ => Fintype.ofFinite _
(∑ y : (zeroIdeal L s).subscheme, (stalkLength (zeroIdeal L s).subscheme y).toNat) =
∑ C : irreducibleComponents X, componentMultiplicity X C *
∑ z, (stalkLength (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme z).toNat := by
classical
let : ∀ C : irreducibleComponents X,
Fintype (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme := fun _ => Fintype.ofFinite _
let : ∀ y : (zeroIdeal L s).subscheme,
Fintype {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val} :=
fun _ => Fintype.ofFinite _
calc
(∑ y : (zeroIdeal L s).subscheme, (stalkLength (zeroIdeal L s).subscheme y).toNat) =
∑ y : (zeroIdeal L s).subscheme,
∑ C : {C : irreducibleComponents X // (zeroIdeal L s).subschemeι y ∈ C.val},
componentMultiplicity X C.val *
(stalkLength (zeroIdeal (L.pullback (reducedComponentι X C.val))
(pullbackSection (reducedComponentι X C.val) s)).subscheme
((componentZeroPointEquiv L s C.val).symm ⟨y,C.property⟩)).toNat :=
Finset.sum_congr rfl (fun y _ => zero_stalk_lengthNat_eq_component_sum hd L s y)
_ = ∑ C : irreducibleComponents X, ∑ z,
componentMultiplicity X C *
(stalkLength (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme z).toNat :=
sum_component_zero_incidence L s (fun C z => componentMultiplicity X C *
(stalkLength (zeroIdeal (L.pullback (reducedComponentι X C))
(pullbackSection (reducedComponentι X C) s)).subscheme z).toNat)
_ = _ := Finset.sum_congr rfl (fun C _ => (Finset.mul_sum _ _ _).symm)
end
end PiExponent.CurveCycle
end OAI
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