openAI-math-dataset / lean /OAI /AlgebraicGeometry /Seshadri /Intersection /TwoChartIntersection.lean
| import Mathlib | |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.ChartPolynomial | |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.LaurentH0 | |
| namespace OAI | |
| section | |
| noncomputable section | |
| section | |
| namespace MaximalSeshadri.Geometry.BaseSections | |
| noncomputable section | |
| open AlgebraicGeometry CategoryTheory TopologicalSpace | |
| open scoped Polynomial | |
| variable {K : Type} [Field K] {U V W : Scheme.{0}} [IsAffine U] [IsAffine V] | |
| theorem two_chart_intersection_finite | |
| (kU : K →+* Γ(U,⊤)) (kV : K →+* Γ(V,⊤)) (kW : K →+* Γ(W,⊤)) | |
| (a : W ⟶ U) (b : W ⟶ V) [IsOpenImmersion a] [IsOpenImmersion b] | |
| (hka : a.appTop.hom.comp kU = kW) (hkb : b.appTop.hom.comp kV = kW) | |
| (u : Γ(U,⊤)) (v : Γ(V,⊤)) | |
| (ha : a.opensRange = U.basicOpen u) (_hb : b.opensRange = V.basicOpen v) | |
| (huv : a.appTop u * b.appTop v = 1) | |
| (hu : (Polynomial.eval₂RingHom kU u).Finite) | |
| (hv : (Polynomial.eval₂RingHom kV v).Finite) | |
| (L : LineBundle U) (P : LineBundle V) (N : W.Modules) | |
| (e : L.sheaf.restrict a ≅ N) (d : P.sheaf.restrict b ≅ N) : | |
| Module.Finite K ↥((chartMap kU kW a hka L.sheaf e).range ⊓ | |
| (chartMap kV kW b hkb P.sheaf d).range) := by | |
| let := polynomialModule kU u L.sheaf ⊤ | |
| let := polynomialModule kV v P.sheaf ⊤ | |
| let := polynomialModule kW (a.appTop u) N ⊤ | |
| let := polynomialTower kU u L.sheaf ⊤ | |
| let := polynomialTower kV v P.sheaf ⊤ | |
| let := polynomialTower kW (a.appTop u) N ⊤ | |
| let : Module.Finite K[X] (Sections kU L.sheaf ⊤) := polynomial_finite kU u L hu | |
| let : Module.Finite K[X] (Sections kV P.sheaf ⊤) := polynomial_finite kV v P hv | |
| let F := chartPolynomialMap kU kW a hka L.sheaf e u | |
| let G := chartMap kV kW b hkb P.sheaf d | |
| let : IsLocalizedModule.Away (Polynomial.X : K[X]) F := | |
| chartPolynomialMap_localize kU kW a hka L.sheaf e u ha | |
| have hInv (y : Sections kV P.sheaf ⊤) : | |
| (Polynomial.X : K[X]) • G ((Polynomial.X : K[X]) • y) = G y := by | |
| change (Polynomial.eval₂RingHom kW (a.appTop u)) Polynomial.X • | |
| G ((Polynomial.eval₂RingHom kV v) Polynomial.X • y) = G y | |
| simp only [Polynomial.coe_eval₂RingHom, Polynomial.eval₂_X] | |
| rw [chartMap_smul, ← mul_smul, huv, one_smul] | |
| exact LaurentCech.intersection_finite F G hInv | |
| end | |
| end MaximalSeshadri.Geometry.BaseSections | |
| end | |
| end | |
| end | |
| end OAI | |