openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /SchematicIntersection.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.FiniteIntersection | |
| namespace OAI | |
| section | |
| noncomputable section | |
| section | |
| namespace MaximalSeshadri.Geometry | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| variable {K : Type} [Field K] {X : Scheme} | |
| theorem finite_schematic_intersection (g : X ⟶ Spec (CommRingCat.of K)) [IsProper g] | |
| (I J : X.IdealSheafData) [IsIntegral I.subscheme] | |
| (hd : topologicalKrullDim I.subscheme ≤ 1) (hJ : ¬ J ≤ I) : | |
| Finite (I ⊔ J).subscheme := by | |
| classical | |
| let Q := J.comap I.subschemeι | |
| have hQ : Q ≠ ⊥ := by | |
| intro he | |
| apply hJ | |
| have h := Scheme.IdealSheafData.le_map_comap J I.subschemeι | |
| simpa only [show J.comap I.subschemeι = ⊥ from he, | |
| Scheme.IdealSheafData.map_bot, Scheme.IdealSheafData.ker_subschemeι] using h | |
| let : Finite Q.subscheme := finite_subscheme_of_curve (I.subschemeι ≫ g) hd Q hQ | |
| have hfin : (Q.support : Set I.subscheme).Finite := by | |
| rw [← Q.range_subschemeι] | |
| exact Set.finite_range Q.subschemeι | |
| let u : (I ⊔ J).subscheme ⟶ I.subscheme := | |
| Scheme.IdealSheafData.inclusion le_sup_left | |
| have hu : u ≫ I.subschemeι = (I ⊔ J).subschemeι := | |
| Scheme.IdealSheafData.inclusion_subschemeι le_sup_left | |
| have hm (z : (I ⊔ J).subscheme) : u z ∈ Q.support := by | |
| rw [Scheme.IdealSheafData.support_comap] | |
| change I.subschemeι (u z) ∈ J.support | |
| rw [← Scheme.Hom.comp_apply, hu] | |
| have hh : (I ⊔ J).subschemeι z ∈ (I ⊔ J).support := by | |
| change (I ⊔ J).subschemeι z ∈ ((I ⊔ J).support : Set X) | |
| rw [← Scheme.IdealSheafData.range_subschemeι] | |
| exact ⟨z,rfl⟩ | |
| rw [Scheme.IdealSheafData.support_sup] at hh | |
| exact hh.2 | |
| let : Finite (Q.support : Set I.subscheme) := hfin.to_subtype | |
| apply Finite.of_injective (fun z => (⟨u z,hm z⟩ : (Q.support : Set I.subscheme))) | |
| intro a b hab | |
| apply (I ⊔ J).subschemeι.isEmbedding.injective | |
| have h := congrArg (fun z : (Q.support : Set I.subscheme) => I.subschemeι z.1) hab | |
| simpa only [← Scheme.Hom.comp_apply, hu] using h | |
| end | |
| end MaximalSeshadri.Geometry | |
| end | |
| end | |
| end | |
| end OAI | |