| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentMultiplicity |
| import OAI.NumberTheory.PiExponent.Geometry.CurveComponentStalks |
| import OAI.NumberTheory.PiExponent.Geometry.FromSpecStalkLocalIso |
| import OAI.NumberTheory.PiExponent.Geometry.SectionZeroStalk |
|
|
| namespace OAI |
|
|
| namespace PiExponent.CurveCycle |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
|
|
| theorem componentMultiplicity_cast {X : Scheme.{0}} [IsLocallyNoetherian X] |
| (C : irreducibleComponents X) : |
| (componentMultiplicity X C : ℕ∞) = |
| Module.length (X.presheaf.stalk (genericPoints.ofComponent C).val) |
| (X.presheaf.stalk (genericPoints.ofComponent C).val) := by |
| let x := (genericPoints.ofComponent C).val |
| let : IsArtinianRing (X.presheaf.stalk x) := |
| generic_stalk_isArtinian x (genericPoints.ofComponent C).property |
| exact ENat.natCast_toNat Module.length_ne_top |
|
|
| def localMinimalPrimeGenericStalkIso (X : Scheme.{0}) (x : X) |
| (P : minimalPrimes (X.presheaf.stalk x)) : |
| letI : P.val.IsPrime := P.property.1.1 |
| X.presheaf.stalk (genericPoints.ofComponent |
| (stalkMinimalPrimesEquivComponentsThrough X x P).val).val ≅ |
| CommRingCat.of (Localization.AtPrime P.val) := by |
| let y : Spec (X.presheaf.stalk x) := ⟨P.val,P.property.1.1⟩ |
| let e := (PiExponent.fromSpecStalkStalkIso X x y) ≪≫ |
| (Spec.stalkIso (X.presheaf.stalk x) y) |
| simpa only [y,stalkMinimalPrimesEquivComponentsThrough_point] using e |
|
|
| theorem localMinimalPrime_length_eq_componentMultiplicity |
| {X : Scheme.{0}} [IsLocallyNoetherian X] (x : X) |
| (P : minimalPrimes (X.presheaf.stalk x)) : |
| letI : P.val.IsPrime := P.property.1.1 |
| Module.length (Localization.AtPrime P.val) (Localization.AtPrime P.val) = |
| (componentMultiplicity X (stalkMinimalPrimesEquivComponentsThrough X x P).val : ℕ∞) := by |
| let : P.val.IsPrime := P.property.1.1 |
| rw [componentMultiplicity_cast] |
| exact (SectionZeroStalk.intrinsic_length_eq_of_ringEquiv |
| (localMinimalPrimeGenericStalkIso X x P).commRingCatIsoToRingEquiv).symm |
|
|
| end |
| end PiExponent.CurveCycle |
|
|
| end OAI |
|
|
