| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.LaurentCech |
| import OAI.AlgebraicGeometry.Seshadri.Geometry.ChartPolynomial |
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| namespace OAI |
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| section |
| noncomputable section |
| |
| section |
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| namespace MaximalSeshadri.Geometry.BaseSections |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory TopologicalSpace |
| open scoped Polynomial |
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| variable {K : Type} [Field K] {U V W : Scheme.{0}} [IsAffine U] [IsAffine V] |
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| theorem two_chart_cokernel_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) |
| (L : LineBundle U) (P : LineBundle V) (N : W.Modules) |
| (e : L.sheaf.restrict a ≅ N) (d : P.sheaf.restrict b ≅ N) : |
| Module.Finite K (Sections kW N ⊤ ⧸ |
| ((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 ⊤ |
| have := polynomialTower kU u L.sheaf ⊤ |
| have := polynomialTower kV v P.sheaf ⊤ |
| have := polynomialTower kW (a.appTop u) N ⊤ |
| have : Module.Finite K[X] (Sections kU L.sheaf ⊤) := polynomial_finite kU u L hu |
| let F := chartPolynomialMap kU kW a hka L.sheaf e u |
| let G := chartMap kV kW b hkb P.sheaf d |
| have : 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] |
| have hDen (x : OpenSections N ⊤) : ∃ n : ℕ, ∃ y : Sections kV P.sheaf ⊤, |
| (Polynomial.X : K[X])^n • G y = x := by |
| let : Module Γ(V,⊤) (OpenSections N ⊤) := |
| Module.compHom (OpenSections N ⊤) b.appTop.hom |
| let G' := chartModuleMap P.sheaf b d |
| have : IsLocalizedModule.Away v G' := chartModuleMap_localize P.sheaf b d v hb |
| obtain ⟨n,y,hy⟩ := IsLocalizedModule.Away.surj G' v x |
| refine ⟨n,y,?_⟩ |
| change (Polynomial.eval₂RingHom kW (a.appTop u)) (Polynomial.X^n) • G y = x |
| simp only [map_pow, Polynomial.coe_eval₂RingHom, Polynomial.eval₂_X] |
| change b.appTop (v^n) • (x : OpenSections N ⊤) = G' y at hy |
| change (a.appTop u)^n • (G' y : OpenSections N ⊤) = x |
| rw [← hy, map_pow, ← mul_smul, ← mul_pow, huv, one_pow, one_smul] |
| exact LaurentCech.cokernel_finite F G hInv hDen |
| end |
| end MaximalSeshadri.Geometry.BaseSections |
| end |
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| end |
| end |
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| end OAI |
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