| import OAI.NumberTheory.PiExponent.Approximation.LineClosedUnit |
| import OAI.NumberTheory.PiExponent.Geometry.PrincipalCartierSequence |
|
|
| namespace OAI |
|
|
|
|
| namespace PiExponentSeshadri.CartierSequence |
| noncomputable section |
| open AlgebraicGeometry CategoryTheory CategoryTheory.Limits TopologicalSpace Opposite |
| open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames |
| variable {X : Scheme.{0}} |
|
|
| lemma chart_topIso_naturality {Y : Scheme.{0}} (j : Y ⟶ X) (U : X.Opens) : |
| U.topIso.hom ≫ j.app U ≫ (j ⁻¹ᵁ U).topIso.inv = (j ∣_ U).appTop := by |
| have H := (arrowResLEAppIso j U (j ⁻¹ᵁ U) le_rfl).hom.w |
| change U.topIso.hom ≫ j.appLE U (j ⁻¹ᵁ U) le_rfl = |
| (j.resLE U (j ⁻¹ᵁ U) le_rfl).appTop ≫ (j ⁻¹ᵁ U).topIso.hom at H |
| rw [Scheme.Hom.resLE_eq_morphismRestrict, Scheme.Hom.appLE_eq_app] at H |
| apply (cancel_mono (j ⁻¹ᵁ U).topIso.hom).mp |
| simpa only [Category.assoc, Iso.inv_hom_id, Category.comp_id] using H |
|
|
| lemma restriction_principal (_p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (I : X.IdealSheafData) (U : X.affineOpens) (r : Γ(U.1.toScheme,⊤)) |
| (hI : I.ideal U = Ideal.span {U.1.topIso.hom r}) : |
| (I.subschemeι ∣_ U.1).ker.ideal ⟨⊤,isAffineOpen_top U.1.toScheme⟩ = |
| Ideal.span {r} := by |
| ext z |
| rw [Scheme.Hom.ker_apply,RingHom.mem_ker] |
| change (I.subschemeι ∣_ U.1).appTop z = 0 ↔ z ∈ Ideal.span {r} |
| have hz : (I.subschemeι ∣_ U.1).appTop z = 0 ↔ |
| U.1.topIso.hom z ∈ I.ideal U := by |
| rw [← chart_topIso_naturality] |
| simp only [CommRingCat.comp_apply] |
| rw [← I.ker_subschemeι_app U] |
| change (I.subschemeι ⁻¹ᵁ U.1).topIso.inv |
| (I.subschemeι.app U.1 (U.1.topIso.hom z)) = 0 ↔ |
| I.subschemeι.app U.1 (U.1.topIso.hom z) = 0 |
| exact map_eq_zero_iff _ (ConcreteCategory.bijective_of_isIso |
| (I.subschemeι ⁻¹ᵁ U.1).topIso.inv).injective |
| rw [hz,hI,Ideal.mem_span_singleton,Ideal.mem_span_singleton] |
| constructor |
| · rintro ⟨a,ha⟩ |
| refine ⟨U.1.topIso.inv a,?_⟩ |
| have h := congrArg U.1.topIso.inv ha |
| simpa only [map_mul,U.1.topIso.hom_inv_id_apply] using h |
| · rintro ⟨a,ha⟩ |
| refine ⟨U.1.topIso.hom a,?_⟩ |
| simpa only [map_mul] using congrArg U.1.topIso.hom ha |
|
|
| theorem exact (p : X ⟶ Spec (CommRingCat.of ℂ)) |
| (M N : LineBundle X) (φ : M.sheaf ⟶ N.sheaf) [Mono φ] |
| (I : X.IdealSheafData) |
| (heq : ∀ x : X, ∃ U : X.affineOpens, x ∈ U.1 ∧ |
| ∃ e : M.sheaf.restrict U.1.ι ≅ O U.1.toScheme, |
| ∃ d : N.sheaf.restrict U.1.ι ≅ O U.1.toScheme, |
| I.ideal U = Ideal.span {U.1.topIso.hom |
| (endValue (e.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ d.hom))}) : |
| ∃ hz : φ ≫ LineClosedUnit.map I.subschemeι N = 0, |
| (ShortComplex.mk φ (LineClosedUnit.map I.subschemeι N) hz).ShortExact := by |
| have local_exact (U : X.affineOpens) |
| (e : M.sheaf.restrict U.1.ι ≅ O U.1.toScheme) |
| (d : N.sheaf.restrict U.1.ι ≅ O U.1.toScheme) |
| (hI : I.ideal U = Ideal.span {U.1.topIso.hom |
| (endValue (e.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ d.hom))}) := |
| PrincipalSequence.exact (I.subschemeι ∣_ U.1) |
| (e.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ d.hom) |
| (restriction_principal p I U _ hI) |
| have hz : φ ≫ LineClosedUnit.map I.subschemeι N = 0 := by |
| apply PiExponent.CoherentTwist.eq_zero_of_local |
| intro x |
| obtain ⟨U,hx,e,d,hI⟩ := heq x |
| obtain ⟨hzero,hex⟩ := local_exact U e d hI |
| refine ⟨U.1,hx,?_⟩ |
| apply (cancel_epi e.inv).mp |
| apply (cancel_mono (LineClosedUnit.frame I.subschemeι N U.1 d).hom).mp |
| rw [Functor.map_comp,comp_zero,zero_comp] |
| rw [Category.assoc,Category.assoc] |
| rw [← d.hom_inv_id_assoc ((Scheme.Modules.restrictFunctor U.1.ι).map |
| (LineClosedUnit.map I.subschemeι N))] |
| simp only [Category.assoc] |
| rw [LineClosedUnit.framed_map] |
| simpa only [Category.assoc] using! hzero |
| refine ⟨hz,{ exact := ?_, epi_g := LineClosedUnit.epi _ _ }⟩ |
| apply PiExponent.CoherentTwist.exact_of_local |
| intro x |
| obtain ⟨U,hx,e,d,hI⟩ := heq x |
| obtain ⟨hzero,hex⟩ := local_exact U e d hI |
| refine ⟨U.1,hx,?_⟩ |
| let S := (ShortComplex.mk φ (LineClosedUnit.map I.subschemeι N) hz).map |
| (Scheme.Modules.restrictFunctor U.1.ι) |
| let T := ShortComplex.mk (e.inv ≫ (Scheme.Modules.restrictFunctor U.1.ι).map φ ≫ d.hom) |
| (IdealModule.structureMap (I.subschemeι ∣_ U.1)) hzero |
| let ee : S ≅ T := ShortComplex.isoMk e d (LineClosedUnit.frame I.subschemeι N U.1 d) |
| (by simp [S,T]) (by |
| change d.hom ≫ IdealModule.structureMap (I.subschemeι ∣_ U.1) = |
| (Scheme.Modules.restrictFunctor U.1.ι).map (LineClosedUnit.map I.subschemeι N) ≫ _ |
| rw [← LineClosedUnit.framed_map I.subschemeι N U.1 d] |
| exact d.hom_inv_id_assoc _) |
| exact ShortComplex.exact_of_iso ee.symm hex.exact |
|
|
| end |
| end PiExponentSeshadri.CartierSequence |
|
|
| end OAI |
|
|
