| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Cohomology.ProjectiveCurveCohomology |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.FiniteEuler |
| import OAI.AlgebraicGeometry.Seshadri.Intersection.EulerExact |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.SectionCokernel |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.SectionMono |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| |
| section |
|
|
| namespace MaximalSeshadri.Cohomology |
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian |
|
|
| lemma finite_middle_of_exact {K U V W : Type*} [Field K] |
| [AddCommGroup U] [Module K U] [AddCommGroup V] [Module K V] |
| [AddCommGroup W] [Module K W] [Module.Finite K U] [Module.Finite K W] |
| (f : U →ₗ[K] V) (g : V →ₗ[K] W) (h : Function.Exact f g) : |
| Module.Finite K V := by |
| have he : Function.Exact f g.rangeRestrict := by |
| intro x |
| rw [Subtype.ext_iff] |
| exact h x |
| exact Module.Finite.of_exact he g.surjective_rangeRestrict |
| end |
| end MaximalSeshadri.Cohomology |
|
|
| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits AlgebraicGeometry TopologicalSpace |
| open MaximalSeshadri.Projective MaximalSeshadri.Frames |
|
|
| variable {X : Scheme.{0}} [IsIntegral X] [IsNoetherian X] |
|
|
| omit [AlgebraicGeometry.IsNoetherian X] in |
| lemma LineBundle.section_shortExact (L : LineBundle X) (s : O X ⟶ L.sheaf) |
| (hs : s ≠ 0) : (ShortComplex.mk s (cokernel.π s) (cokernel.condition s)).ShortExact := by |
| exact { |
| exact := ShortComplex.exact_of_g_is_cokernel _ (cokernelIsCokernel s) |
| mono_f := L.mono_section s hs } |
|
|
| lemma LineBundle.cokernel_ext_zero (hd : topologicalKrullDim X ≤ 1) |
| (L : LineBundle X) (s : O X ⟶ L.sheaf) (hs : s ≠ 0) |
| (n : ℕ) (z : cohomology (cokernel s) (n+1)) : z = 0 := by |
| let := L.cokernel_flasque hd s hs |
| exact FlasqueCohomology.flasque_ext_zero X.ringCatSheaf n (cokernel s) z |
|
|
| theorem projective_section_cokernel_finite {σ : Type} [Fintype σ] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X = 1) {M : X.Modules} |
| (a : σ → (O X ⟶ M)) (ha : (⨆ i, SectionOpens.isoOpen (a i)) = ⊤) |
| [IsClosedImmersion (sectionsMorphism (baseScalars p) a ha)] |
| (L : LineBundle X) (s : O X ⟶ L.sheaf) (hs : s ≠ 0) (n : ℕ) : |
| letI := Module.compHom (cohomology (cokernel s) n) (baseScalars p) |
| Module.Finite ℂ (cohomology (cokernel s) n) := by |
| let : IsClosedImmersion (sectionsMorphism |
| (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom) a ha) := |
| inferInstanceAs (IsClosedImmersion (sectionsMorphism (baseScalars p) a ha)) |
| cases n with |
| | succ n => |
| let := Module.compHom (cohomology (cokernel s) (n+1)) (baseScalars p) |
| let : Subsingleton (cohomology (cokernel s) (n+1)) := ⟨fun x y => |
| (L.cokernel_ext_zero hd.le s hs n x).trans (L.cokernel_ext_zero hd.le s hs n y).symm⟩ |
| infer_instance |
| | zero => |
| let := sheafComplexLinear p |
| let B : LineBundle X := ⟨O X, fun x => |
| ⟨⊤,trivial,⟨Scheme.Modules.restrictUnitIso (⊤ : X.Opens).ι⟩⟩⟩ |
| let S := ShortComplex.mk s (cokernel.π s) (cokernel.condition s) |
| have hS := L.section_shortExact s hs |
| let hL : Module.Finite ℂ (cohomology L.sheaf 0) := by |
| change @Module.Finite ℂ (cohomology L.sheaf 0) _ _ (complexExtModule p L.sheaf 0) |
| rw [complexExtModule_eq] |
| exact projective_curve_cohomology_finite p hd a ha L 0 |
| let hO : Module.Finite ℂ (cohomology (O X) 1) := by |
| change @Module.Finite ℂ (cohomology (O X) 1) _ _ (complexExtModule p (O X) 1) |
| rw [complexExtModule_eq] |
| exact projective_curve_cohomology_finite p hd a ha B 1 |
| let : Module.Finite ℂ (Ext (O X) S.X₂ 0) := hL |
| let : Module.Finite ℂ (Ext (O X) S.X₁ (0+1)) := hO |
| have he := Cohomology.finite_middle_of_exact |
| (Cohomology.cohomologyMap₂ (K := ℂ) (O X) (S := S) 0) |
| (Cohomology.cohomologyBoundary (K := ℂ) (O X) hS 0) |
| (Cohomology.cohomology_exact₃ (K := ℂ) (O X) hS 0) |
| change @Module.Finite ℂ (cohomology (cokernel s) 0) _ _ |
| (complexExtModule p (cokernel s) 0) at he |
| rw [complexExtModule_eq] at he |
| exact he |
|
|
| theorem projective_section_euler_difference {σ : Type} [Fintype σ] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) [IsProper p] |
| (hd : topologicalKrullDim X = 1) {M : X.Modules} |
| (a : σ → (O X ⟶ M)) (ha : (⨆ i, SectionOpens.isoOpen (a i)) = ⊤) |
| [IsClosedImmersion (sectionsMorphism (baseScalars p) a ha)] |
| (L : LineBundle X) (s : O X ⟶ L.sheaf) (hs : s ≠ 0) : |
| eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (O X) = |
| letI := Module.compHom Γ(cokernel s, ⊤) (baseScalars p) |
| (Module.finrank ℂ Γ(cokernel s,⊤) : ℤ) := by |
| let : IsClosedImmersion (sectionsMorphism |
| (p.appTop.hom.comp (Scheme.ΓSpecIso (CommRingCat.of ℂ)).inv.hom) a ha) := |
| inferInstanceAs (IsClosedImmersion (sectionsMorphism (baseScalars p) a ha)) |
| let B : LineBundle X := ⟨O X, fun x => |
| ⟨⊤,trivial,⟨Scheme.Modules.restrictUnitIso (⊤ : X.Opens).ι⟩⟩⟩ |
| let : (O X).IsQuasicoherent := B.quasicoherent |
| let : Subsingleton (cohomology (O X) (1+1)) := ⟨fun x y => |
| (projective_curve_ext_zero p hd a ha (O X) 0 x).trans |
| (projective_curve_ext_zero p hd a ha (O X) 0 y).symm⟩ |
| have he := eulerCharacteristic_add p (L.section_shortExact s hs) 1 |
| (fun n _ => projective_curve_cohomology_finite p hd a ha B n) |
| (fun n _ => projective_curve_cohomology_finite p hd a ha L n) |
| (fun n _ => projective_section_cokernel_finite p hd a ha L s hs n) |
| change eulerCharacteristic p 1 L.sheaf = |
| eulerCharacteristic p 1 (O X) + eulerCharacteristic p 1 (cokernel s) at he |
| rw [he, add_sub_cancel_left] |
| rw [eulerCharacteristic, Finset.sum_range_succ, Finset.sum_range_succ] |
| let : Subsingleton (cohomology (cokernel s) 1) := ⟨fun x y => |
| (L.cokernel_ext_zero hd.le s hs 0 x).trans (L.cokernel_ext_zero hd.le s hs 0 y).symm⟩ |
| have hz : cohomologyDimension p (cokernel s) 1 = 0 := by |
| let := Module.compHom (cohomology (cokernel s) 1) (baseScalars p) |
| exact Module.finrank_zero_of_subsingleton |
| simp only [Finset.sum_range_zero, zero_add, pow_zero, one_mul, |
| hz, Nat.cast_zero, mul_zero, add_zero] |
| rw [cohomologyDimension_zero] |
| end |
| end MaximalSeshadri.Geometry |
| end |
|
|
|
|
| end |
| end |
|
|
| end OAI |
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|
