openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ClosedRestrictionTensor.lean
| 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 | |