openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Geometry /CurveFunctionFieldCompatibility.lean
| 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 | |