| import OAI.Algebra.Finitistic.Selected.SelectedNativeEvaluation |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.SelectedNativeEvaluation |
| open QuadraticEncoding SourceImageEvaluation FiniteProjectiveHomotopy SelectedFixedDiagram |
| universe v |
| variable {Y : Type v} [AddCommGroup Y] [Module ℂ Y] |
| [Module SelectionRing Y] [IsScalarTower ℂ SelectionRing Y] |
|
|
| abbrev imageEnd (r : SelectionRing) := imageCornerEnd |
| (action (k := ℂ) (Y := Y) e).hom (action (k := ℂ) (Y := Y) (alpha r)).hom |
| abbrev moduleIso := (SelectedImageComparison.equiv (Y := Y)).toModuleIso |
|
|
| def HAction (r : SelectionRing) : End (H (Y := Y)) := |
| (moduleIso (Y := Y)).inv ≫ imageEnd (Y := Y) r ≫ (moduleIso (Y := Y)).hom |
|
|
| /-- The conjugated map is the primary selectionStep module action, on every vector. -/ |
| theorem HAction_apply (r : SelectionRing) (x : H (Y := Y)) : |
| (HAction (Y := Y) r).hom x = SelectedImageComparison.moduleAction (Y := Y) r x := by |
| have h := SelectedImageComparison.equiv_action (Y := Y) r |
| ((SelectedImageComparison.equiv (Y := Y)).symm x) |
| dsimp only [HAction,moduleIso,LinearEquiv.toModuleIso] at * |
| simp only [LinearEquiv.apply_symm_apply] at h |
| exact h |
|
|
| lemma module_square (r : SelectionRing) : |
| imageEnd (Y := Y) r ≫ (moduleIso (Y := Y)).hom = |
| (moduleIso (Y := Y)).hom ≫ HAction (Y := Y) r := by |
| simp only [HAction,Iso.hom_inv_id_assoc] |
|
|
| variable [FiniteDimensional ℂ Y] |
|
|
| def HKAction (r : SelectionRing) : End (HK (Y := Y)) := |
| (inclusion ℂ).preimage ((vectorStalk (k := ℂ)).map (HAction (Y := Y) r)) |
| def HDerivedAction (r : SelectionRing) : End (HDerived (Y := Y)) := |
| (derivedEmbedding ℂ).toEssImage.map (HKAction (Y := Y) r) |
|
|
| lemma imageIsoK_hom : (inclusion ℂ).map (imageIsoK (Y := Y)).hom = |
| (vectorStalk (k := ℂ)).map (moduleIso (Y := Y)).hom := by |
| simp only [imageIsoK,Functor.preimageIso,Functor.mapIso_hom] |
| erw [Functor.map_preimage] |
|
|
| lemma image_square (r : SelectionRing) : |
| imageCorner (k := ℂ) (Y := Y) e (alpha r) ≫ (imageIso (Y := Y)).hom = |
| (imageIso (Y := Y)).hom ≫ HDerivedAction (Y := Y) r := by |
| apply (derivedTargetInclusion (k := ℂ)).map_injective |
| rw [Functor.map_comp,Functor.map_comp] |
| dsimp only [imageCorner] |
| erw [Functor.map_preimage] |
| change DerivedCategory.Qh.map ((vectorStalk (k := ℂ)).map (imageEnd (Y := Y) r)) ≫ |
| DerivedCategory.Qh.map ((inclusion ℂ).map (imageIsoK (Y := Y)).hom) = |
| DerivedCategory.Qh.map ((inclusion ℂ).map (imageIsoK (Y := Y)).hom) ≫ |
| DerivedCategory.Qh.map ((inclusion ℂ).map (HKAction (Y := Y) r)) |
| rw [imageIsoK_hom] |
| dsimp only [HKAction] |
| erw [Functor.map_preimage] |
| erw [← Functor.map_comp,← Functor.map_comp,← Functor.map_comp,← Functor.map_comp,module_square] |
| rfl |
|
|
| abbrev targetAction (r : SelectionRing) : End (target (Y := Y)) := |
| biprod.map (HDerivedAction (Y := Y) r) ((HDerivedAction (Y := Y) r)⟦(3 : ℤ)⟧') |
| |
| lemma target_square (r : SelectionRing) : |
| SelectedDiagramEvaluation.targetAction (Y := Y) r ≫ (targetIso (Y := Y)).hom = |
| (targetIso (Y := Y)).hom ≫ targetAction (Y := Y) r := by |
| have h := image_square (Y := Y) r |
| have hs := congrArg (shiftFunctor (FiniteVectorDerived.{0,v} (k := ℂ)) (3 : ℤ)).map h |
| simp only [Functor.map_comp] at hs |
| dsimp only [SelectedDiagramEvaluation.targetAction,targetIso,targetAction,biprod.mapIso] |
| apply biprod.hom_ext |
| · simp only [Category.assoc,biprod.map_fst,biprod.map_fst_assoc,h] |
| · simp only [Category.assoc,biprod.map_snd,biprod.map_snd_assoc,Functor.mapIso_hom,hs] |
| |
| /-- The fixed V0 evaluates equivariantly to the primary H(Y) plus H(Y)[3]. -/ |
| theorem V0_action (r : SelectionRing) : |
| (SelectedDiagramEvaluation.F (Y := Y)).map (fixed.rho r) ≫ (V0Iso (Y := Y)).hom = |
| (V0Iso (Y := Y)).hom ≫ targetAction (Y := Y) r := |
| DiagramEvaluationSquares.compose _ _ _ SelectedDiagramEvaluation.V0Iso |
| SelectedDiagramEvaluation.V0Iso targetIso targetIso |
| (SelectedDiagramEvaluation.V0_action r) (target_square r) |
| |
| theorem D_arrow01 (a : Fin (encoding.generators+1)) : |
| (SelectedDiagramEvaluation.sourceEvaluation (Y := Y)).map (fixed.f a) ≫ (D1Iso (Y := Y)).hom = |
| (D0Iso (Y := Y)).hom ≫ targetAction (Y := Y) |
| (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a) := |
| DiagramEvaluationSquares.compose _ _ _ SelectedDiagramEvaluation.D0Iso |
| SelectedDiagramEvaluation.D1Iso targetIso targetIso |
| (SelectedDiagramEvaluation.D_arrow01 a) (target_square _) |
| |
| theorem D_arrow12 (a : Fin (encoding.generators+1)) : |
| (SelectedDiagramEvaluation.sourceEvaluation (Y := Y)).map (fixed.g a) ≫ (D2Iso (Y := Y)).hom = |
| (D1Iso (Y := Y)).hom ≫ targetAction (Y := Y) |
| (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a) := |
| DiagramEvaluationSquares.compose _ _ _ SelectedDiagramEvaluation.D1Iso |
| SelectedDiagramEvaluation.D2Iso targetIso targetIso |
| (SelectedDiagramEvaluation.D_arrow12 a) (target_square _) |
| end LittleFinitistic.SelectedNativeEvaluation |
| |
| end |
| |
| end OAI |
| |
