| import OAI.Algebra.Finitistic.Selected.SelectedReferenceCohomology | |
| import OAI.Algebra.Finitistic.Selected.SelectedTensorSplitting | |
| import OAI.Algebra.Finitistic.Selected.SelectedChainAction | |
| namespace OAI | |
| noncomputable section | |
| open CategoryTheory CategoryTheory.Limits HomologicalComplex | |
| attribute [local instance] HasDerivedCategory.standard HasExt.standard | |
| namespace LittleFinitistic.SelectedRealization | |
| open SelectedFixedDiagram QuadraticEncoding SelectedNativeEvaluation | |
| variable {Y : Type} [AddCommGroup Y] [Module ℂ Y] [FiniteDimensional ℂ Y] | |
| [Module SelectionRing Y] [IsScalarTower ℂ SelectionRing Y] | |
| local instance selectedModule : Module SelectionRing (H (Y := Y)) := | |
| inferInstanceAs (Module SelectionRing (selectionStep (ModuleCat.of SelectionRing Y))) | |
| local instance selectedTower : IsScalarTower ℂ SelectionRing (H (Y := Y)) := | |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) | |
| abbrev input := (SelectedChainHomology.representation (Y := Y)).Evaluation | |
| abbrev selectedRepresentation := SourceActionEvaluation.representation (Y := H (Y := Y)) encoding | |
| abbrev reference := EvaluationVertexDiagram.module (selectedRepresentation (Y := Y)) | |
| omit [FiniteDimensional ℂ Y] in | |
| lemma selected_action (a : Fin (encoding.generators+1)) : | |
| EvaluationVertexDiagram.action (Y := H (Y := Y)) (selectedRepresentation (Y := Y)) a = | |
| HAction (Y := Y) (RelationBasisAction.arrow (SelectionEncoding.generator encoding) a) := by | |
| apply ModuleCat.hom_ext | |
| apply LinearMap.ext | |
| intro x | |
| rw [HAction_apply] | |
| refine Fin.cases ?_ (fun j => ?_) a | |
| · change x = (1 : SelectionRing) • x | |
| rw [one_smul] | |
| · rfl | |
| def coordinates (n : ℤ) (hn : n = 0 ∨ n = -3) : | |
| ∀ v : Fin 3, (SelectedTensor.source (input (Y := Y)) v).homology n ≅ H (Y := Y) := | |
| Fin.cases (SelectedChainHomology.D0Iso n hn) | |
| (Fin.cases (SelectedChainHomology.D1Iso n hn) | |
| (fun _ => SelectedChainHomology.D2Iso n hn)) | |
| lemma coordinates01 (a : Fin (encoding.generators+1)) (n : ℤ) (hn : n = 0 ∨ n = -3) : | |
| homologyMap (((BalancedTensor.tensorFunctor ℂ (B P) (input (Y := Y))).mapHomologicalComplex (.up ℤ)).map | |
| (SelectedRectification.f a)) n ≫ (coordinates (Y := Y) n hn 1).hom = | |
| (coordinates (Y := Y) n hn 0).hom ≫ EvaluationVertexDiagram.action (Y := H (Y := Y)) (selectedRepresentation (Y := Y)) a := by | |
| erw [selected_action (Y := Y)] | |
| exact SelectedChainHomology.fixed_arrow01 a n hn | |
| lemma coordinates12 (a : Fin (encoding.generators+1)) (n : ℤ) (hn : n = 0 ∨ n = -3) : | |
| homologyMap (((BalancedTensor.tensorFunctor ℂ (B P) (input (Y := Y))).mapHomologicalComplex (.up ℤ)).map | |
| (SelectedRectification.g a)) n ≫ (coordinates (Y := Y) n hn 2).hom = | |
| (coordinates (Y := Y) n hn 1).hom ≫ EvaluationVertexDiagram.action (Y := H (Y := Y)) (selectedRepresentation (Y := Y)) a := by | |
| erw [selected_action (Y := Y)] | |
| exact SelectedChainHomology.fixed_arrow12 a n hn | |
| /-- The entire cohomology B-module is the manuscript's M(H(Y)), in both degrees. -/ | |
| def cohomology (n : ℤ) (hn : n = 0 ∨ n = -3) : | |
| (SelectedTensor.complex (input (Y := Y))).homology n ≅ reference (Y := Y) := | |
| SelectedReferenceCohomology.iso _ (selectedRepresentation (Y := Y)) n (coordinates (Y := Y) n hn) | |
| (fun a => coordinates01 a n hn) (fun a => coordinates12 a n hn) | |
| lemma source_zero (n : ℤ) (hn0 : n ≠ 0) (hn3 : n ≠ -3) (v : Fin 3) : | |
| IsZero ((SelectedTensor.source (input (Y := Y)) v).homology n) := by | |
| fin_cases v | |
| · exact SelectedChainHomology.D0_zero n hn0 hn3 | |
| · exact SelectedChainHomology.D1_zero n hn0 hn3 | |
| · exact SelectedChainHomology.D2_zero n hn0 hn3 | |
| /-- The one fixed selecting bimodule realizes H on every finite-dimensional Y. | |
| This is an isomorphism in the full B-module derived category. -/ | |
| def realization : | |
| SelectedTensorSplitting.object (input (Y := Y)) ≅ | |
| (DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj (reference (Y := Y)) ⊞ | |
| ((DerivedCategory.singleFunctor (ModuleCat (B P)) 0).obj (reference (Y := Y)))⟦(3 : ℤ)⟧ := | |
| SelectedTensorSplitting.split _ _ (cohomology 0 (Or.inl rfl)) | |
| (cohomology (-3) (Or.inr rfl)) source_zero | |
| end LittleFinitistic.SelectedRealization | |
| end | |
| end OAI | |