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b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | 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
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