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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 | 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
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