openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Geometry /CurveComponentLocalLengths.lean
| 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 | |