File size: 3,508 Bytes
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 | import OAI.NumberTheory.PiExponent.Ampleness.AffineFunctionFieldCompatibility
namespace OAI
noncomputable section
namespace PiExponent.CurveNormalizationModel
open CategoryTheory AlgebraicGeometry
universe u
variable (A E : Type u) [CommRing A] [IsDomain A] [Field E]
[Algebra A E] [IsFractionRing A E]
def affineChartFunctionFieldMap {X : Scheme.{u}} [IsIntegral X]
(i : Spec (.of A) ⟶ X) [IsOpenImmersion i] : X.functionField →+* E :=
(affineFunctionFieldEquiv A E).toRingHom.comp (functionFieldRestriction i).hom
theorem affineChartFunctionFieldMap_stalk {X : Scheme.{u}} [IsIntegral X]
(i : Spec (.of A) ⟶ X) [IsOpenImmersion i] (q : PrimeSpectrum A)
[Algebra (Localization.AtPrime q.asIdeal) E]
[IsScalarTower A (Localization.AtPrime q.asIdeal) E]
(a : X.presheaf.stalk (i q)) :
affineChartFunctionFieldMap A E i
(X.presheaf.stalkSpecializes (genericPoint_specializes (i q)) a) =
algebraMap (Localization.AtPrime q.asIdeal) E
((Spec.stalkIso (.of A) q).hom (i.stalkMap q a)) := by
have h := congrArg (fun h => h a) (functionFieldRestriction_stalk i q)
change functionFieldRestriction i
(X.presheaf.stalkSpecializes (genericPoint_specializes (i q)) a) =
(Spec (.of A)).presheaf.stalkSpecializes (genericPoint_specializes q)
(i.stalkMap q a) at h
change affineFunctionFieldEquiv A E (_) = _
rw [h]
exact affineFunctionFieldEquiv_stalk A E q _
private theorem restriction_congr {U X : Scheme.{u}} [IsIntegral U] [IsIntegral X]
{i j : U ⟶ X} [IsOpenImmersion i] [IsOpenImmersion j] (h : i = j) :
functionFieldRestriction i = functionFieldRestriction j := by
subst j
rfl
theorem affineChartFunctionFieldMap_eq_of_overlap
(B G : Type u) [CommRing B] [IsDomain B] [CommRing G] [IsDomain G]
[Algebra B E] [IsFractionRing B E] [Algebra G E] [IsFractionRing G E]
[Algebra A G] [Algebra B G] [IsScalarTower A G E] [IsScalarTower B G E]
{X : Scheme.{u}} [IsIntegral X]
(iA : Spec (.of A) ⟶ X) (iB : Spec (.of B) ⟶ X)
[IsOpenImmersion iA] [IsOpenImmersion iB]
[IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap A G)))]
[IsOpenImmersion (Spec.map (CommRingCat.ofHom (algebraMap B G)))]
(h : Spec.map (CommRingCat.ofHom (algebraMap A G)) ≫ iA =
Spec.map (CommRingCat.ofHom (algebraMap B G)) ≫ iB) :
affineChartFunctionFieldMap A E iA = affineChartFunctionFieldMap B E iB := by
have hr : functionFieldRestriction iA ≫
functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A G))) =
functionFieldRestriction iB ≫
functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap B G))) := by
rw [← functionFieldRestriction_comp, ← functionFieldRestriction_comp]
exact restriction_congr h
apply RingHom.ext
intro z
have hz := congrArg (fun g => affineFunctionFieldEquiv G E (g z)) hr
change affineFunctionFieldEquiv G E
(functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap A G)))
(functionFieldRestriction iA z)) =
affineFunctionFieldEquiv G E
(functionFieldRestriction (Spec.map (CommRingCat.ofHom (algebraMap B G)))
(functionFieldRestriction iB z)) at hz
have hA := RingHom.congr_fun (affineFunctionFieldEquiv_naturality A E G)
(functionFieldRestriction iA z)
have hB := RingHom.congr_fun (affineFunctionFieldEquiv_naturality B E G)
(functionFieldRestriction iB z)
exact hA.symm.trans (hz.trans hB)
end PiExponent.CurveNormalizationModel
end
end OAI
|