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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 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import Mathlib.AlgebraicGeometry.Stalk
import OAI.NumberTheory.PiExponent.Ampleness.AffineFunctionFieldCompatibility
import OAI.NumberTheory.PiExponent.Geometry.CurveNormalizationProper
namespace OAI
noncomputable section
namespace PiExponent.CurveNormalizationModel
open CategoryTheory AlgebraicGeometry
open scoped Polynomial
open PiExponent.CurveZeroPole
universe u
private theorem eqToHom_fromSpecStalk (X : Scheme.{u}) {x y : X} (h : x = y) :
Spec.map (eqToHom (congrArg (fun z => X.presheaf.stalk z) h)) ≫
X.fromSpecStalk x = X.fromSpecStalk y := by
subst y
simp
theorem functionFieldRestriction_fromSpecStalk {U X : Scheme.{u}}
[IsIntegral U] [IsIntegral X] (i : U ⟶ X) [IsOpenImmersion i] :
Spec.map (functionFieldRestriction i) ≫ X.fromSpecStalk (genericPoint X) =
U.fromSpecStalk (genericPoint U) ≫ i := by
unfold functionFieldRestriction
rw [Spec.map_comp, Category.assoc,
eqToHom_fromSpecStalk X (genericPoint_eq_of_isOpenImmersion i).symm]
exact Scheme.SpecMap_stalkMap_fromSpecStalk i
theorem affineFunctionFieldEquiv_fromSpecStalk
(A E : Type u) [CommRing A] [IsDomain A] [Field E]
[Algebra A E] [IsFractionRing A E] :
Spec.map (CommRingCat.ofHom (affineFunctionFieldEquiv A E).toRingHom) ≫
(Spec (.of A)).fromSpecStalk (genericPoint (Spec (.of A))) =
Spec.map (CommRingCat.ofHom (algebraMap A E)) := by
erw [Spec.fromSpecStalk_eq', ← Spec.map_comp]
congr 1
apply CommRingCat.hom_ext
apply RingHom.ext
exact affineFunctionFieldEquiv_toStalk A E
private theorem affineRestriction_fromSpecStalk
(A E : Type u) [CommRing A] [IsDomain A] [Field E]
[Algebra A E] [IsFractionRing A E] {X : Scheme.{u}} [IsIntegral X]
(i : Spec (.of A) ⟶ X) [IsOpenImmersion i] :
Spec.map (functionFieldRestriction i ≫
CommRingCat.ofHom (affineFunctionFieldEquiv A E).toRingHom) ≫
X.fromSpecStalk (genericPoint X) =
Spec.map (CommRingCat.ofHom (algebraMap A E)) ≫ i := by
let e := Spec.map (CommRingCat.ofHom (affineFunctionFieldEquiv A E).toRingHom)
let r := Spec.map (functionFieldRestriction i)
let q := X.fromSpecStalk (genericPoint X)
have hmap := congrArg (fun k => k ≫ q)
(Spec.map_comp (functionFieldRestriction i)
(CommRingCat.ofHom (affineFunctionFieldEquiv A E).toRingHom))
have hrestrict := congrArg (fun k => e ≫ k) (functionFieldRestriction_fromSpecStalk i)
have haffine := congrArg (fun k => k ≫ i) (affineFunctionFieldEquiv_fromSpecStalk A E)
have hassoc := Category.assoc e
((Spec (.of A)).fromSpecStalk (genericPoint (Spec (.of A)))) i
exact hmap.trans ((Category.assoc e r q).trans
(hrestrict.trans (hassoc.symm.trans haffine)))
variable {F E : Type u} [Field F] [Field E] [Algebra F E]
variable (f : E) (hf : Transcendental F f)
def parameterChartConstantMap : F →+* parameterChart f hf :=
letI := parameterPolynomialAlgebra f hf
(algebraMap F[X] (parameterChart f hf)).comp Polynomial.C
theorem parameterChartConstantMap_toField :
(algebraMap (parameterChart f hf) E).comp (parameterChartConstantMap f hf) =
algebraMap F E := by
let := parameterPolynomialAlgebra f hf
apply RingHom.ext
intro c
change algebraMap F[X] E (Polynomial.C c) = algebraMap F E c
rw [parameterPolynomialAlgebra_map]
simp
@[reassoc] theorem zeroChart_structureMap :
zeroChartInclusion f hf ≫ parameterCurveStructureMap f hf =
Spec.map (CommRingCat.ofHom (parameterChartConstantMap f hf)) := by
unfold parameterCurveStructureMap
rw [zeroChart_projection_assoc, ProjectiveLine.chartMap_structureMap,
parameterChartProjection, ← Spec.map_comp]
rfl
variable [FiniteDimensional (IntermediateField.adjoin F {f}) E]
def parameterCurveGenericPoint : Spec (.of E) ⟶ parameterCurve f hf :=
Spec.map (CommRingCat.ofHom (parameterCurveFunctionFieldEquiv f hf).toRingHom) ≫
(parameterCurve f hf).fromSpecStalk (genericPoint (parameterCurve f hf))
theorem parameterCurveGenericPoint_eq_zeroChart :
parameterCurveGenericPoint f hf =
Spec.map (CommRingCat.ofHom (algebraMap (parameterChart f hf) E)) ≫
zeroChartInclusion f hf := by
let := parameterAlgebra f hf
let := parameterPolynomialAlgebra f hf
let := parameter_scalarTower f hf
let := parameter_finite f hf
change Spec.map (functionFieldRestriction (zeroChartInclusion f hf) ≫
CommRingCat.ofHom (affineFunctionFieldEquiv (parameterChart f hf) E).toRingHom) ≫
(parameterCurve f hf).fromSpecStalk (genericPoint (parameterCurve f hf)) = _
exact affineRestriction_fromSpecStalk (parameterChart f hf) E (zeroChartInclusion f hf)
@[reassoc] theorem parameterCurveGenericPoint_over_base :
parameterCurveGenericPoint f hf ≫ parameterCurveStructureMap f hf =
Spec.map (CommRingCat.ofHom (algebraMap F E)) := by
rw [parameterCurveGenericPoint_eq_zeroChart, Category.assoc,
zeroChart_structureMap, ← Spec.map_comp]
congr 1
apply CommRingCat.hom_ext
exact parameterChartConstantMap_toField f hf
end PiExponent.CurveNormalizationModel
end
end OAI
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