File size: 2,969 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 | import OAI.NumberTheory.PiExponent.Geometry.CurveNormalizationLocalRing
namespace OAI
noncomputable section
namespace PiExponent.CurveNormalizationModel
open CategoryTheory AlgebraicGeometry
universe u
def functionFieldRestriction {U X : Scheme.{u}} [IsIntegral U] [IsIntegral X]
(i : U ⟶ X) [IsOpenImmersion i] : X.functionField ⟶ U.functionField :=
eqToHom (congrArg (fun x => X.presheaf.stalk x)
(genericPoint_eq_of_isOpenImmersion i).symm) ≫ i.stalkMap (genericPoint U)
private theorem stalkSpecializes_eqToHom (X : Scheme.{u}) {x y : X} (h : x = y) :
X.presheaf.stalkSpecializes (Inseparable.of_eq h).specializes =
eqToHom (congrArg (fun z => X.presheaf.stalk z) h.symm) := by
subst y
simp
theorem functionFieldRestriction_stalk {U X : Scheme.{u}} [IsIntegral U] [IsIntegral X]
(i : U ⟶ X) [IsOpenImmersion i] (q : U) :
X.presheaf.stalkSpecializes (genericPoint_specializes (i q)) ≫ functionFieldRestriction i =
i.stalkMap q ≫ U.presheaf.stalkSpecializes (genericPoint_specializes q) := by
unfold functionFieldRestriction
rw [← Category.assoc, ← stalkSpecializes_eqToHom X (genericPoint_eq_of_isOpenImmersion i),
TopCat.Presheaf.stalkSpecializes_comp]
exact i.stalkSpecializes_stalkMap (genericPoint U) q (genericPoint_specializes q)
private theorem stalkMap_eqToHom {U X : Scheme.{u}} (i : U ⟶ X)
(x y : U) (h : x = y) :
i.stalkMap x ≫ eqToHom (congrArg (fun z => U.presheaf.stalk z) h) =
eqToHom (congrArg (fun z => X.presheaf.stalk (i z)) h) ≫ i.stalkMap y := by
subst y
simp
theorem functionFieldRestriction_comp {V U X : Scheme.{u}}
[IsIntegral V] [IsIntegral U] [IsIntegral X]
(j : V ⟶ U) (i : U ⟶ X) [IsOpenImmersion j] [IsOpenImmersion i] :
functionFieldRestriction (j ≫ i) = functionFieldRestriction i ≫ functionFieldRestriction j := by
symm
unfold functionFieldRestriction
rw [Scheme.Hom.stalkMap_comp]
simp only [Category.assoc]
rw [← Category.assoc (i.stalkMap (genericPoint U)),
stalkMap_eqToHom i (genericPoint U) (j (genericPoint V))
(genericPoint_eq_of_isOpenImmersion j).symm]
simp
private theorem toStalk_eqToHom (A : CommRingCat.{u})
(x y : PrimeSpectrum A) (h : x = y) :
StructureSheaf.toStalk A x ≫ eqToHom (congrArg
(fun z => (Spec A).presheaf.stalk z) h) = StructureSheaf.toStalk A y := by
subst y
exact Category.comp_id _
theorem functionFieldRestriction_toStalk {A B : CommRingCat.{u}} [IsDomain A] [IsDomain B]
(f : A ⟶ B) [IsOpenImmersion (Spec.map f)] :
StructureSheaf.toStalk A (genericPoint (Spec A)) ≫
functionFieldRestriction (Spec.map f) =
f ≫ StructureSheaf.toStalk B (genericPoint (Spec B)) := by
unfold functionFieldRestriction
erw [← Category.assoc, toStalk_eqToHom A _ _
(genericPoint_eq_of_isOpenImmersion (Spec.map f)).symm]
exact AlgebraicGeometry.stalkMap_toStalk f _
end PiExponent.CurveNormalizationModel
end
end OAI
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