| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentCutLengths |
| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentIncidence |
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|
| namespace OAI |
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| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames |
| open PiExponent.SectionZeroIdeal |
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| def stalkLength (Y : Scheme.{0}) (y : Y) : ℕ∞ := |
| Module.length (Y.presheaf.stalk y) (Y.presheaf.stalk y) |
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| 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 |
|
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| 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 |
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| end OAI |
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