| import OAI.Algebra.Finitistic.Selected.SelectedUnderlyingHomology |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory CategoryTheory.Limits |
| attribute [local instance] HasDerivedCategory.standard |
| namespace LittleFinitistic.SelectedChainAction |
| open QuadraticEncoding SelectedFixedDiagram SelectedNativeEvaluation SelectedChainHomology |
| universe v |
| variable {Y : Type v} [AddCommGroup Y] [Module ℂ Y] |
| [Module SelectionRing Y] [IsScalarTower ℂ SelectionRing Y] |
|
|
| lemma HAction_one : HAction (Y := Y) 1 = 𝟙 _ := by |
| apply ModuleCat.hom_ext |
| ext x |
| rw [HAction_apply] |
| apply Subtype.ext |
| change selectionAutomorphism 1 (1 : SelectionRing) • x.val = x.val |
| rw [map_one,one_smul] |
|
|
| /-- In the underlying Y, the selected action is exactly restriction of alpha(r). -/ |
| lemma HAction_val (r : SelectionRing) (x : H (Y := Y)) : |
| ((HAction (Y := Y) r).hom x).val = alpha r • x.val := by |
| rw [HAction_apply] |
| rfl |
|
|
| variable [FiniteDimensional ℂ Y] |
|
|
| lemma distinguished01 (n : ℤ) (hn : n = 0 ∨ n = -3) : |
| HomologicalComplex.homologyMap |
| (SelectedTensorComparison.tensorMap (representation (Y := Y)) (SelectedChainDiagram.fixedChain.f 0)) n ≫ |
| (SelectedChainHomology.D1Iso n hn).hom = (SelectedChainHomology.D0Iso n hn).hom := by |
| simpa only [RelationBasisAction.arrow,Fin.cases_zero,HAction_one,Category.comp_id] using |
| (fixed_arrow01 (Y := Y) 0 n hn) |
| lemma distinguished12 (n : ℤ) (hn : n = 0 ∨ n = -3) : |
| HomologicalComplex.homologyMap |
| (SelectedTensorComparison.tensorMap (representation (Y := Y)) (SelectedChainDiagram.fixedChain.g 0)) n ≫ |
| (SelectedChainHomology.D2Iso n hn).hom = (SelectedChainHomology.D1Iso n hn).hom := by |
| simpa only [RelationBasisAction.arrow,Fin.cases_zero,HAction_one,Category.comp_id] using |
| (fixed_arrow12 (Y := Y) 0 n hn) |
|
|
| /-- Every actual first arrow, on every homology class in either selected degree. -/ |
| lemma arrow01_val (a : Fin (encoding.generators+1)) |
| (f : UnderlyingRepresentatives.complex P fixed.D0 ⟶ UnderlyingRepresentatives.complex P fixed.D1) |
| (hf : (HomotopyCategory.quotient _ _).map f = (SourceModelBridge.ambient P).map (fixed.f a)) |
| (n : ℤ) (hn : n = 0 ∨ n = -3) (x : (complex (Y := Y) fixed.D0).homology n) : |
| ((SelectedChainHomology.D1Iso n hn).hom.hom |
| ((HomologicalComplex.homologyMap (SelectedUnderlyingHomology.tensorMap (Y := Y) f) n).hom x)).val = |
| alpha (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a) • |
| ((SelectedChainHomology.D0Iso n hn).hom.hom x).val := by |
| have h := congrArg (fun q => q.hom x) (SelectedUnderlyingHomology.arrow01 (Y := Y) a f hf n hn) |
| change (SelectedChainHomology.D1Iso n hn).hom.hom |
| ((HomologicalComplex.homologyMap (SelectedUnderlyingHomology.tensorMap (Y := Y) f) n).hom x) = |
| (HAction (Y := Y) (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a)).hom |
| ((SelectedChainHomology.D0Iso n hn).hom.hom x) at h |
| exact (congrArg Subtype.val h).trans (HAction_val _ _) |
|
|
| lemma arrow12_val (a : Fin (encoding.generators+1)) |
| (g : UnderlyingRepresentatives.complex P fixed.D1 ⟶ UnderlyingRepresentatives.complex P fixed.D2) |
| (hg : (HomotopyCategory.quotient _ _).map g = (SourceModelBridge.ambient P).map (fixed.g a)) |
| (n : ℤ) (hn : n = 0 ∨ n = -3) (x : (complex (Y := Y) fixed.D1).homology n) : |
| ((SelectedChainHomology.D2Iso n hn).hom.hom |
| ((HomologicalComplex.homologyMap (SelectedUnderlyingHomology.tensorMap (Y := Y) g) n).hom x)).val = |
| alpha (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a) • |
| ((SelectedChainHomology.D1Iso n hn).hom.hom x).val := by |
| have h := congrArg (fun q => q.hom x) (SelectedUnderlyingHomology.arrow12 (Y := Y) a g hg n hn) |
| change (SelectedChainHomology.D2Iso n hn).hom.hom |
| ((HomologicalComplex.homologyMap (SelectedUnderlyingHomology.tensorMap (Y := Y) g) n).hom x) = |
| (HAction (Y := Y) (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a)).hom |
| ((SelectedChainHomology.D1Iso n hn).hom.hom x) at h |
| exact (congrArg Subtype.val h).trans (HAction_val _ _) |
| end LittleFinitistic.SelectedChainAction |
|
|
| end |
|
|
| end OAI |
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