openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Projective /ProjectiveCoordinates.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Projective.PolynomialCharts | |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.GeneratedEtaleCoordinates | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace MaximalSeshadri.ProjectiveBertini | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open MaximalSeshadri.Projective | |
| attribute [local instance] MvPolynomial.gradedAlgebra | |
| variable {K : Type} [Field K] {X : Scheme} {σ : Type} | |
| lemma affine_restriction_etale (U V : X.affineOpens) (e : U.1 ≤ V.1) : | |
| (X.presheaf.map (homOfLE e).op).hom.Etale := by | |
| have hU : IsAffineOpen U.1 := U.2 | |
| have hV : IsAffineOpen V.1 := V.2 | |
| have he := hV.map_fromSpec hU (homOfLE e).op | |
| have : IsOpenImmersion (Spec.map (X.presheaf.map (homOfLE e).op) ≫ hV.fromSpec) := by | |
| rw [he] | |
| infer_instance | |
| let : IsOpenImmersion (Spec.map (X.presheaf.map (homOfLE e).op)) := | |
| IsOpenImmersion.of_comp _ hV.fromSpec | |
| exact (HasRingHomProperty.Spec_iff (P := @Etale)).mp inferInstance | |
| lemma awayToSection_awayι (i : σ) : | |
| Spec.map (Proj.awayToSection (PolyGrade K σ) (MvPolynomial.X i)) ≫ | |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X i) (poly_X_mem i) (by decide) = | |
| (Proj.isAffineOpen_basicOpen (PolyGrade K σ) _ (poly_X_mem i) (by decide)).fromSpec := by | |
| let hV := Proj.isAffineOpen_basicOpen (PolyGrade K σ) _ (poly_X_mem i) (by decide) | |
| rw [← cancel_epi hV.isoSpec.hom, ← Category.assoc, hV.isoSpec_hom] | |
| change Proj.basicOpenToSpec _ _ ≫ _ = _ | |
| rw [← Proj.basicOpenIsoSpec_hom _ _ (poly_X_mem i) (by decide), | |
| Proj.awayι, Iso.hom_inv_id_assoc] | |
| exact hV.toSpecΓ_fromSpec.symm | |
| theorem closed_projective_affine_surjective (h : X ⟶ Proj (PolyGrade K σ)) | |
| [IsClosedImmersion h] (i : σ) : | |
| ∃ (U : X.affineOpens) (φ : PolyChart (R := K) i →+* Γ(X, U.1)), | |
| U.1 = h ⁻¹ᵁ Proj.basicOpen (PolyGrade K σ) (MvPolynomial.X i) ∧ | |
| Function.Surjective φ ∧ | |
| Spec.map (CommRingCat.ofHom φ) ≫ | |
| Proj.awayι (PolyGrade K σ) (MvPolynomial.X i) (poly_X_mem i) (by decide) = | |
| U.2.fromSpec ≫ h := by | |
| let V := Proj.basicOpen (PolyGrade K σ) (MvPolynomial.X i) | |
| have hV : IsAffineOpen V := Proj.isAffineOpen_basicOpen _ _ (poly_X_mem i) (by decide) | |
| let U : X.affineOpens := ⟨h ⁻¹ᵁ V, hV.preimage h⟩ | |
| let e := Proj.basicOpenIsoAway (PolyGrade K σ) _ (poly_X_mem i) (by decide) | |
| let φ : PolyChart (R := K) i →+* Γ(X, U.1) := (h.app V).hom.comp e.hom.hom | |
| refine ⟨U, φ, rfl, ?_, ?_⟩ | |
| · exact (h.app_surjective V hV).comp (ConcreteCategory.bijective_of_isIso e.hom).surjective | |
| · change Spec.map (e.hom ≫ h.app V) ≫ _ = _ | |
| rw [Spec.map_comp, Category.assoc] | |
| have he : e.hom = Proj.awayToSection (PolyGrade K σ) (MvPolynomial.X i) := rfl | |
| rw [he, awayToSection_awayι] | |
| rw [Scheme.Hom.app_eq_appLE] | |
| exact hV.SpecMap_appLE_fromSpec h U.2 le_rfl | |
| open KaehlerDifferential MaximalSeshadri.AlgebraicJets | |
| lemma differentials_span_projective_chart {A : Type} [CommRing A] [Algebra K A] | |
| (i : σ) (φ : PolyChart (R := K) i →+* A) (hφ : Function.Surjective φ) | |
| (hconst : ∀ c, φ (chartConstants i c) = algebraMap K A c) : | |
| Submodule.span A (Set.range (fun j : ChartVariables i => | |
| D K A (-φ (chartCoordinate i j.1)))) = ⊤ := by | |
| have he : (MvPolynomial.aeval (R := K) | |
| (fun j : ChartVariables i => φ (chartCoordinate i j.1))).toRingHom = | |
| φ.comp (polyToChart i) := by | |
| apply MvPolynomial.ringHom_ext | |
| · intro c | |
| change MvPolynomial.aeval _ (MvPolynomial.C c) = φ (polyToChart i (MvPolynomial.C c)) | |
| rw [MvPolynomial.aeval_C] | |
| simpa only [polyToChart, MvPolynomial.eval₂Hom_C] using (hconst c).symm | |
| · intro j | |
| change MvPolynomial.aeval _ (MvPolynomial.X j) = φ (polyToChart i (MvPolynomial.X j)) | |
| rw [MvPolynomial.aeval_X, polyToChart, MvPolynomial.eval₂Hom_X'] | |
| have hp : Function.Surjective (MvPolynomial.aeval (R := K) | |
| (fun j : ChartVariables i => φ (chartCoordinate i j.1))) := by | |
| change Function.Surjective (MvPolynomial.aeval _).toRingHom | |
| rw [he] | |
| exact hφ.comp (polynomialChartEquiv (R := K) i).symm.surjective | |
| have hs := differentials_span_of_generators _ hp | |
| have hr : Set.range (fun j : ChartVariables i => D K A (-φ (chartCoordinate i j.1))) = | |
| -(Set.range (fun j : ChartVariables i => D K A (φ (chartCoordinate i j.1)))) := by | |
| ext x | |
| simp only [Set.mem_range, Set.mem_neg, map_neg, neg_eq_iff_eq_neg] | |
| rw [hr, Submodule.span_neg] | |
| exact hs | |
| end | |
| end MaximalSeshadri.ProjectiveBertini | |
| end | |
| end | |
| end OAI | |