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a72760f | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 | import OAI.NumberTheory.PiExponent.Ampleness.ClosedAmpleRestriction
import OAI.NumberTheory.PiExponent.Approximation.ProjectionFormula
import OAI.NumberTheory.PiExponent.Approximation.TwistPowerSection
import OAI.NumberTheory.PiExponent.LocalAlgebra.IdealModule
namespace OAI
namespace PiExponent.ClosedRestrictionTensor
noncomputable section
open AlgebraicGeometry CategoryTheory TopologicalSpace Opposite
open PiExponentSeshadri.Geometry PiExponentSeshadri.Frames PiExponentSeshadri.TensorPure
lemma tensor_map_pure_val {Z : Scheme.{0}} {M N P Q : Z.Modules}
(a : M ⟶ P) (b : N ⟶ Q) (U : Z.Opens)
(m : M.val.obj (op U)) (n : N.val.obj (op U)) :
(moduleTensorMap a b).val.app (op U) (pure M N U m n) =
pure P Q U (a.val.app (op U) m) (b.val.app (op U) n) :=
map_pure a b U m n
lemma tensor_comm_pure_val {Z : Scheme.{0}} (M N : Z.Modules) (U : Z.Opens)
(m : M.val.obj (op U)) (n : N.val.obj (op U)) :
(moduleTensorComm M N).hom.val.app (op U) (pure M N U m n) =
pure N M U n m := moduleTensorComm_pure M N U m n
lemma projection_hom_pure_val {X Y : Scheme.{0}} (f : X ⟶ Y)
(M : X.Modules) (L : Y.Modules) (U : Y.Opens)
(m : M.val.obj (op (f ⁻¹ᵁ U))) (s : L.val.obj (op U)) :
(ProjectionFormula.hom f M L).val.app (op U)
(pure ((Scheme.Modules.pushforward f).obj M) L U m s) =
pure M ((Scheme.Modules.pullback f).obj L) (f ⁻¹ᵁ U) m
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L).val.app (op U) s) :=
ProjectionFormula.hom_pure f M L U m s
lemma pushforward_hom_pure_val {X Y : Scheme.{0}} (f : X ⟶ Y)
(M N : X.Modules) (U : Y.Opens)
(m : M.val.obj (op (f ⁻¹ᵁ U))) (n : N.val.obj (op (f ⁻¹ᵁ U))) :
(PiExponentSeshadri.PushforwardTensor.hom f M N).val.app (op U)
(pure ((Scheme.Modules.pushforward f).obj M) ((Scheme.Modules.pushforward f).obj N) U m n) =
pure M N (f ⁻¹ᵁ U) m n :=
PiExponentSeshadri.PushforwardTensor.hom_pure f M N U m n
variable {X Y : Scheme.{0}} (f : X ⟶ Y) (L : LineBundle Y)
def powerProjectionIso (n : ℕ) :
(moduleTwistFunctor L n).obj ((Scheme.Modules.pushforward f).obj (O X)) ≅
(Scheme.Modules.pushforward f).obj ((L.pullback f).pow n).sheaf :=
(ProjectionFormula.twistIso f (O X) L n).symm ≪≫
(Scheme.Modules.pushforward f).mapIso (moduleTwistUnitIso (L.pullback f) n)
def powerUnit (n : ℕ) : (L.pow n).sheaf ⟶
(Scheme.Modules.pushforward f).obj ((L.pullback f).pow n).sheaf :=
(Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (L.pow n).sheaf ≫
(Scheme.Modules.pushforward f).map (PiExponentSeshadri.PullbackTensor.powIso f L n).hom
lemma powerProjectionIso_succ_pure (n : ℕ) (U : Y.Opens)
(m : ((moduleTwistFunctor L n).obj ((Scheme.Modules.pushforward f).obj (O X))).val.obj (op U))
(s : L.sheaf.val.obj (op U)) :
(powerProjectionIso f L (n+1)).hom.val.app (op U)
(pure _ L.sheaf U m s) =
pure (L.pullback f).sheaf ((L.pullback f).pow n).sheaf (f ⁻¹ᵁ U)
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf).val.app (op U) s)
((powerProjectionIso f L n).hom.val.app (op U) m) := by
change (moduleTwistUnitIso (L.pullback f) (n+1)).hom.val.app (op (f ⁻¹ᵁ U))
((ProjectionFormula.hom f ((moduleTwistFunctor (L.pullback f) n).obj (O X)) L.sheaf).val.app (op U)
((moduleTensorMap (ProjectionFormula.twistIso f (O X) L n).inv (𝟙 L.sheaf)).val.app (op U)
(pure _ L.sheaf U m s))) = _
rw [tensor_map_pure_val]
change (moduleTensorComm ((L.pullback f).pow n).sheaf (L.pullback f).sheaf).hom.val.app (op (f ⁻¹ᵁ U))
((moduleTensorMap (moduleTwistUnitIso (L.pullback f) n).hom (𝟙 (L.pullback f).sheaf)).val.app (op (f ⁻¹ᵁ U))
((ProjectionFormula.hom f ((moduleTwistFunctor (L.pullback f) n).obj (O X)) L.sheaf).val.app (op U)
(pure _ L.sheaf U ((ProjectionFormula.twistIso f (O X) L n).inv.val.app (op U) m) s))) = _
erw [projection_hom_pure_val f ((moduleTwistFunctor (L.pullback f) n).obj (O X)) L.sheaf U ((ProjectionFormula.twistIso f (O X) L n).inv.val.app (op U) m) s]
have ht := tensor_map_pure_val (Z := X) (moduleTwistUnitIso (L.pullback f) n).hom
(𝟙 (L.pullback f).sheaf) (f ⁻¹ᵁ U)
((ProjectionFormula.twistIso f (O X) L n).inv.val.app (op U) m)
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf).val.app (op U) s)
exact (congrArg (fun r => (moduleTensorComm ((L.pullback f).pow n).sheaf
(L.pullback f).sheaf).hom.val.app (op (f ⁻¹ᵁ U)) r) ht).trans
(tensor_comm_pure_val _ _ _ _ _)
lemma powerUnit_succ_pure (n : ℕ) (U : Y.Opens)
(s : L.sheaf.val.obj (op U)) (m : (L.pow n).sheaf.val.obj (op U)) :
(powerUnit f L (n+1)).val.app (op U) (pure L.sheaf (L.pow n).sheaf U s m) =
pure (L.pullback f).sheaf ((L.pullback f).pow n).sheaf (f ⁻¹ᵁ U)
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf).val.app (op U) s)
((powerUnit f L n).val.app (op U) m) := by
have h := PiExponentSeshadri.PullbackTensor.unit_hom f L.sheaf (L.pow n).sheaf
have hp := congrArg (fun q => q.val.app (op U) (pure L.sheaf (L.pow n).sheaf U s m)) h
change (PiExponentSeshadri.PullbackTensor.hom f L.sheaf (L.pow n).sheaf).val.app (op (f ⁻¹ᵁ U))
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app
(moduleTensor Y L.sheaf (L.pow n).sheaf)).val.app (op U) (pure L.sheaf (L.pow n).sheaf U s m)) =
(PiExponentSeshadri.PushforwardTensor.hom f _ _).val.app (op U)
((moduleTensorMap ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf)
((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (L.pow n).sheaf)).val.app (op U)
(pure L.sheaf (L.pow n).sheaf U s m)) at hp
have hm := tensor_map_pure_val (Z := Y)
((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf)
((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (L.pow n).sheaf) U s m
have hv := pushforward_hom_pure_val f ((Scheme.Modules.pullback f).obj L.sheaf)
((Scheme.Modules.pullback f).obj (L.pow n).sheaf) U
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf).val.app (op U) s)
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (L.pow n).sheaf).val.app (op U) m)
have hp' := hp.trans ((congrArg (fun r => (PiExponentSeshadri.PushforwardTensor.hom f
((Scheme.Modules.pullback f).obj L.sheaf) ((Scheme.Modules.pullback f).obj (L.pow n).sheaf)).val.app
(op U) r) hm).trans hv)
change (moduleTensorMap (𝟙 (L.pullback f).sheaf) (PiExponentSeshadri.PullbackTensor.powIso f L n).hom).app
(f ⁻¹ᵁ U)
((PiExponentSeshadri.PullbackTensor.hom f L.sheaf (L.pow n).sheaf).val.app (op (f ⁻¹ᵁ U))
(((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app
(moduleTensor Y L.sheaf (L.pow n).sheaf)).val.app (op U) (pure L.sheaf (L.pow n).sheaf U s m))) = _
exact (congrArg (fun r => (moduleTensorMap (𝟙 (L.pullback f).sheaf)
(PiExponentSeshadri.PullbackTensor.powIso f L n).hom).val.app (op (f ⁻¹ᵁ U)) r) hp').trans
(tensor_map_pure_val (Z := X) (𝟙 (L.pullback f).sheaf)
(PiExponentSeshadri.PullbackTensor.powIso f L n).hom _ _ _)
theorem twist_structureMap_power (n : ℕ) :
(moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f) ≫
(powerProjectionIso f L n).hom =
(moduleTwistUnitIso L n).hom ≫ powerUnit f L n := by
induction n with
| zero =>
change PiExponentSeshadri.IdealModule.structureMap f =
(Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (O Y) ≫
(Scheme.Modules.pushforward f).map (pullbackUnitIso f).hom
exact (pullbackUnit_adjunction f).symm
| succ n ih =>
apply PiExponentSeshadri.TensorPure.hom_ext
intro U m s
change (powerProjectionIso f L (n+1)).hom.val.app (op U)
((moduleTensorMap ((moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f))
(𝟙 L.sheaf)).val.app (op U) (pure _ L.sheaf U m s)) =
(powerUnit f L (n+1)).val.app (op U)
((moduleTensorComm (L.pow n).sheaf L.sheaf).hom.val.app (op U)
((moduleTensorMap (moduleTwistUnitIso L n).hom (𝟙 L.sheaf)).val.app (op U)
(pure _ L.sheaf U m s)))
have h1 := tensor_map_pure_val (Z := Y)
((moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f)) (𝟙 L.sheaf) U m s
have h2 := powerProjectionIso_succ_pure f L n U
(((moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f)).val.app (op U) m) s
have h3 := tensor_map_pure_val (Z := Y) (moduleTwistUnitIso L n).hom (𝟙 L.sheaf) U m s
have h4 := tensor_comm_pure_val (L.pow n).sheaf L.sheaf U
((moduleTwistUnitIso L n).hom.val.app (op U) m) s
have h5 := powerUnit_succ_pure f L n U s ((moduleTwistUnitIso L n).hom.val.app (op U) m)
have hleft := (congrArg (fun r => (powerProjectionIso f L (n+1)).hom.val.app (op U) r) h1).trans h2
have hright := (congrArg (fun r => (powerUnit f L (n+1)).val.app (op U) r)
((congrArg (fun r => (moduleTensorComm (L.pow n).sheaf L.sheaf).hom.val.app (op U) r) h3).trans h4)).trans h5
exact hleft.trans ((congrArg (fun r => pure (L.pullback f).sheaf ((L.pullback f).pow n).sheaf
(f ⁻¹ᵁ U) (((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app L.sheaf).val.app (op U) s) r)
(congrArg (fun q => q.val.app (op U) m) ih)).trans hright.symm)
theorem power_restriction_eq (n : ℕ) :
(moduleTwistUnitIso L n).inv ≫
(moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f) ≫
(ProjectionFormula.twistIso f (O X) L n).inv ≫
(Scheme.Modules.pushforward f).map (moduleTwistUnitIso (L.pullback f) n).hom =
(Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (L.pow n).sheaf ≫
(Scheme.Modules.pushforward f).map (PiExponentSeshadri.PullbackTensor.powIso f L n).hom := by
change (moduleTwistUnitIso L n).inv ≫
((moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f) ≫
(powerProjectionIso f L n).hom) = powerUnit f L n
erw [twist_structureMap_power, Iso.inv_hom_id_assoc]
theorem push_pullbackSection {M : Y.Modules} (s : O Y ⟶ M) :
PiExponentSeshadri.IdealModule.structureMap f ≫
(Scheme.Modules.pushforward f).map (pullbackSection f s) =
s ≫ (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app M := by
have hu : PiExponentSeshadri.IdealModule.structureMap f =
(Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (O Y) ≫
(Scheme.Modules.pushforward f).map (pullbackUnitIso f).hom :=
(pullbackUnit_adjunction f).symm
let F := Scheme.Modules.pushforward f
let P := Scheme.Modules.pullback f
let e := pullbackUnitIso f
have h := (F.map_comp e.hom (e.inv ≫ P.map s)).symm.trans
(congrArg F.map (e.hom_inv_id_assoc (P.map s)))
calc
_ = ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (O Y) ≫
F.map e.hom) ≫ F.map (e.inv ≫ P.map s) :=
congrArg (fun q : O Y ⟶ F.obj (O X) => q ≫ F.map (pullbackSection f s)) hu
_ = (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (O Y) ≫ F.map (P.map s) :=
(Category.assoc _ _ _).trans
(congrArg (fun q => (Scheme.Modules.pullbackPushforwardAdjunction f).unit.app (O Y) ≫ q) h)
_ = _ := ((Scheme.Modules.pullbackPushforwardAdjunction f).unit.naturality s).symm
theorem section_restriction_eq (n : ℕ) (s : O Y ⟶ (L.pow n).sheaf) :
s ≫ (moduleTwistUnitIso L n).inv ≫
(moduleTwistFunctor L n).map (PiExponentSeshadri.IdealModule.structureMap f) ≫
(ProjectionFormula.twistIso f (O X) L n).inv ≫
(Scheme.Modules.pushforward f).map (moduleTwistUnitIso (L.pullback f) n).hom =
PiExponentSeshadri.IdealModule.structureMap f ≫
(Scheme.Modules.pushforward f).map (pullbackPowerSection L f n s) := by
erw [power_restriction_eq]
symm
simp only [pullbackPowerSection]
erw [(Scheme.Modules.pushforward f).map_comp (pullbackSection f s) (PiExponentSeshadri.PullbackTensor.powIso f L n).hom, ← Category.assoc, push_pullbackSection f s, Category.assoc]
end
end PiExponent.ClosedRestrictionTensor
end OAI
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