| import Mathlib |
| import OAI.AlgebraicGeometry.Seshadri.Divisors.AffineSectionQuotient |
|
|
| namespace OAI |
|
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| section |
| noncomputable section |
| |
| section |
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| namespace MaximalSeshadri.Geometry |
| noncomputable section |
| open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits AlgebraicGeometry TopologicalSpace |
| open MaximalSeshadri.Frames MaximalSeshadri.Projective |
|
|
| variable {X Y : Scheme.{0}} |
|
|
| def sectionRestrictionLinear (M : X.Modules) (f : Y ⟶ X) [IsOpenImmersion f] : |
| Γ(M,⊤) →ₛₗ[f.appTop.hom] Γ(M.restrict f,⊤) where |
| toFun m := (M.restrictAppIso f ⊤).inv |
| (M.presheaf.map (homOfLE (le_top : f ''ᵁ ⊤ ≤ ⊤)).op m) |
| map_add' := by intro a b; simp only [map_add] |
| map_smul' r m := by |
| change (M.restrictAppIso f ⊤).inv |
| (M.presheaf.map (homOfLE (le_top : f ''ᵁ ⊤ ≤ ⊤)).op (r • m)) = _ |
| rw [M.map_smul, Scheme.Modules.smul_restrictAppIso_inv_apply] |
| congr 1 |
| have hm := Scheme.Hom.appLE_appIso_inv f (U := ⊤) (V := ⊤) (by simp) |
| have hApp : f.appLE ⊤ ⊤ (by simp) = f.appTop := f.appLE_eq_app |
| rw [hApp] at hm |
| have he := CategoryTheory.congr_fun hm r |
| change (f.appIso ⊤).hom _ = f.appTop r |
| rw [← he] |
| exact ConcreteCategory.congr_hom (f.appIso ⊤).inv_hom_id (f.appTop r) |
|
|
| lemma sectionRestrictionLinear_surjective (M : X.Modules) (f : Y ⟶ X) |
| [IsOpenImmersion f] |
| [TopCat.Sheaf.IsFlasque ((SheafOfModules.toSheaf X.ringCatSheaf).obj M)] : |
| Function.Surjective (sectionRestrictionLinear M f) := by |
| have hs : Function.Surjective |
| (M.presheaf.map (homOfLE (le_top : f ''ᵁ ⊤ ≤ ⊤)).op) := |
| (AddCommGrpCat.epi_iff_surjective _).mp |
| (TopCat.Presheaf.IsFlasque.epi (F := ((SheafOfModules.toSheaf X.ringCatSheaf).obj M).obj) |
| (homOfLE (le_top : f ''ᵁ ⊤ ≤ ⊤)).op) |
| exact (ConcreteCategory.bijective_of_isIso (M.restrictAppIso f ⊤).inv).surjective.comp hs |
|
|
| def restrictCokernelIso {M : X.Modules} (s : O X ⟶ M) |
| (f : Y ⟶ X) [IsOpenImmersion f] : |
| (cokernel s).restrict f ≅ cokernel (restrictSection f s) := |
| PreservesCokernel.iso (Scheme.Modules.restrictFunctor f) s ≪≫ |
| cokernel.mapIso _ _ (Scheme.Modules.restrictUnitIso f) (Iso.refl _) |
| (by |
| let unitIso : (O X).restrict f ≅ O Y := Scheme.Modules.restrictUnitIso f |
| change (Scheme.Modules.restrictFunctor f).map s ≫ 𝟙 _ = |
| unitIso.hom ≫ unitIso.inv ≫ (Scheme.Modules.restrictFunctor f).map s |
| rw [Category.comp_id, unitIso.hom_inv_id_assoc]) |
|
|
| def moduleIsoTop {M N : X.Modules} (e : M ≅ N) : |
| Γ(M,⊤) ≃ₗ[Γ(X,⊤)] Γ(N,⊤) where |
| toFun := e.hom.app ⊤ |
| invFun := e.inv.app ⊤ |
| left_inv x := congrArg (fun t : M ⟶ M => t.app ⊤ x) e.hom_inv_id |
| right_inv x := congrArg (fun t : N ⟶ N => t.app ⊤ x) e.inv_hom_id |
| map_add' := by intro a b; exact map_add _ a b |
| map_smul' := by intro r m; exact e.hom.app_smul r m |
| |
| lemma LineBundle.restrict_nonzero [IsIntegral X] (L : LineBundle X) |
| (f : Y ⟶ X) [IsOpenImmersion f] [Nonempty Y] |
| (s : O X ⟶ L.sheaf) (hs : s ≠ 0) : restrictSection f s ≠ 0 := by |
| intro h |
| apply hs |
| apply L.restrict_sections_injective f |
| let unitIso : (structureSheaf X).restrict f ≅ structureSheaf Y := |
| Scheme.Modules.restrictUnitIso f |
| apply (cancel_epi unitIso.inv).mp |
| change restrictSection f s = restrictSection f 0 |
| simpa only [restrictSection_zero] using h |
|
|
| def sectionQuotientLocalMap [IsIntegral X] [IsNoetherian X] |
| (L : LineBundle X) (s : O X ⟶ L.sheaf) (hs : s ≠ 0) |
| (f : Y ⟶ X) [IsOpenImmersion f] [IsAffine Y] [IsNoetherian Y] [Nonempty Y] |
| (e : L.sheaf.restrict f ≅ O Y) : |
| Γ(cokernel s,⊤) →ₛₗ[f.appTop.hom] |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) := by |
| letI : IsIntegral Y := isIntegral_of_isOpenImmersion f |
| let E := (moduleIsoTop (restrictCokernelIso s f)).trans |
| (affine_section_quotient (L.restrict f) e (restrictSection f s) |
| (L.restrict_nonzero f s hs)).symm |
| exact E.toLinearMap.comp (sectionRestrictionLinear (cokernel s) f) |
|
|
| lemma sectionQuotientLocalMap_surjective [IsIntegral X] [IsNoetherian X] |
| (hd : topologicalKrullDim X ≤ 1) (L : LineBundle X) |
| (s : O X ⟶ L.sheaf) (hs : s ≠ 0) |
| (f : Y ⟶ X) [IsOpenImmersion f] [IsAffine Y] [IsNoetherian Y] [Nonempty Y] |
| (e : L.sheaf.restrict f ≅ O Y) : |
| Function.Surjective (sectionQuotientLocalMap L s hs f e) := by |
| let : IsIntegral Y := isIntegral_of_isOpenImmersion f |
| exact ((moduleIsoTop (restrictCokernelIso s f)).trans |
| (affine_section_quotient (L.restrict f) e (restrictSection f s) |
| (L.restrict_nonzero f s hs)).symm).surjective.comp |
| (@sectionRestrictionLinear_surjective X Y (cokernel s) f _ |
| (L.cokernel_flasque hd s hs)) |
|
|
| lemma baseScalars_comp_actual (f : Y ⟶ X) (g : X ⟶ Spec (CommRingCat.of ℂ)) : |
| baseScalars (f ≫ g) = f.appTop.hom.comp (baseScalars g) := by |
| ext c |
| simp only [baseScalars, Scheme.Hom.comp_appTop, CommRingCat.hom_comp, |
| RingHom.comp_apply] |
|
|
| def sectionQuotientLocalComplex [IsIntegral X] [IsNoetherian X] |
| (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (L : LineBundle X) (s : O X ⟶ L.sheaf) (hs : s ≠ 0) |
| (f : Y ⟶ X) [IsOpenImmersion f] [IsAffine Y] [IsNoetherian Y] [Nonempty Y] |
| (e : L.sheaf.restrict f ≅ O Y) : |
| letI := Module.compHom Γ(cokernel s,⊤) (baseScalars p) |
| letI := Module.compHom |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) (baseScalars (f ≫ p)) |
| Γ(cokernel s,⊤) →ₗ[ℂ] |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) := by |
| letI := Module.compHom Γ(cokernel s,⊤) (baseScalars p) |
| letI := Module.compHom |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) (baseScalars (f ≫ p)) |
| refine { toFun := sectionQuotientLocalMap L s hs f e |
| map_add' := (sectionQuotientLocalMap L s hs f e).map_add |
| map_smul' := ?_ } |
| intro r x |
| change (sectionQuotientLocalMap L s hs f e) (baseScalars p r • x) = |
| baseScalars (f ≫ p) r • (sectionQuotientLocalMap L s hs f e) x |
| rw [baseScalars_comp_actual] |
| exact (sectionQuotientLocalMap L s hs f e).map_smulₛₗ (baseScalars p r) x |
|
|
| theorem projective_local_section_quotient_finite [IsIntegral X] [IsNoetherian X] |
| {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) |
| (f : Y ⟶ X) [IsOpenImmersion f] [IsAffine Y] [IsNoetherian Y] [Nonempty Y] |
| (e : L.sheaf.restrict f ≅ O Y) : |
| letI := Module.compHom |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) (baseScalars (f ≫ p)) |
| Module.Finite ℂ (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) := by |
| let := Module.compHom Γ(cokernel s,⊤) (baseScalars p) |
| let := Module.compHom |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) (baseScalars (f ≫ p)) |
| let := Module.compHom (cohomology (cokernel s) 0) (baseScalars p) |
| let := projective_section_cokernel_finite p hd a ha L s hs 0 |
| let : Module.Finite ℂ Γ(cokernel s,⊤) := |
| Module.Finite.of_surjective (cohomologyZeroSections p (cokernel s)).toLinearMap |
| (cohomologyZeroSections p (cokernel s)).surjective |
| exact Module.Finite.of_surjective (sectionQuotientLocalComplex p L s hs f e) |
| (sectionQuotientLocalMap_surjective hd.le L s hs f e) |
|
|
| theorem projective_local_section_length_le_degree [IsIntegral X] [IsNoetherian X] |
| {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) |
| (f : Y ⟶ X) [IsOpenImmersion f] [IsAffine Y] [IsNoetherian Y] [Nonempty Y] |
| (e : L.sheaf.restrict f ≅ O Y) : |
| letI := Module.compHom |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) (baseScalars (f ≫ p)) |
| (Module.finrank ℂ (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) : ℤ) ≤ |
| eulerCharacteristic p 1 L.sheaf - eulerCharacteristic p 1 (O X) := by |
| let := Module.compHom Γ(cokernel s,⊤) (baseScalars p) |
| let := Module.compHom |
| (Γ(Y,⊤) ⧸ Ideal.span {coefficient e (restrictSection f s)}) (baseScalars (f ≫ p)) |
| let := Module.compHom (cohomology (cokernel s) 0) (baseScalars p) |
| let := projective_section_cokernel_finite p hd a ha L s hs 0 |
| let : Module.Finite ℂ Γ(cokernel s,⊤) := |
| Module.Finite.of_surjective (cohomologyZeroSections p (cokernel s)).toLinearMap |
| (cohomologyZeroSections p (cokernel s)).surjective |
| have hb := LinearMap.finrank_le_finrank_of_surjective |
| (f := sectionQuotientLocalComplex p L s hs f e) |
| (sectionQuotientLocalMap_surjective hd.le L s hs f e) |
| rw [projective_section_euler_difference p hd a ha L s hs] |
| exact_mod_cast hb |
|
|
| end |
| end MaximalSeshadri.Geometry |
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| end |
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| end |
| end |
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| end OAI |
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