| import OAI.Algebra.Finitistic.Rectification.LeftVertexHomology |
| import OAI.Algebra.Finitistic.Localization.EvaluationModule |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory |
| namespace LittleFinitistic.EvaluationVertexDiagram |
| open QuadraticEncoding |
| universe u |
| variable {k : Type u} [Field k] {d t : ℕ} {P : Relations k d t} |
| variable {Y : Type u} [AddCommGroup Y] [Module k Y] (X : Representation P Y) |
|
|
| abbrev module := ModuleCat.of (B P) X.Evaluation |
|
|
| /-- The vertex of the actual evaluation module is exactly the original vector space. -/ |
| def equiv (i : Fin 3) : |
| (LeftVertexHomology.vertex k (B P) (ThreeVertexAlgebra.vertex (composition P) i)).obj (module X) ≃ₗ[k] Y where |
| toFun z := X.projection i z.val |
| invFun y := ⟨X.inclusion i y, by |
| erw [VertexFunctor.mem_fixed, X.vertex_smul, X.projection_inclusion]⟩ |
| left_inv z := by |
| apply Subtype.ext |
| have hz := (VertexFunctor.mem_fixed _ _ _ _ z.val).1 z.property |
| exact (X.vertex_smul i (show X.Evaluation from z.val)).symm.trans hz |
| right_inv y := X.projection_inclusion i y |
| map_add' x y := (X.projection i).map_add x.val y.val |
| map_smul' c z := by |
| change X.projection i (algebraMap k (B P) c • z.val) = c • X.projection i z.val |
| calc |
| X.projection i (algebraMap k (B P) c • (show X.Evaluation from z.val)) = |
| X.projection i (c • (show X.Evaluation from z.val)) := |
| congrArg (X.projection i) (IsScalarTower.algebraMap_smul (B P) c (show X.Evaluation from z.val)) |
| _ = c • X.projection i z.val := (X.projection i).map_smul c (show X.Evaluation from z.val) |
|
|
| def vertexIso (i : Fin 3) := (equiv X i).toModuleIso |
|
|
| def action (a : Fin (d+1)) : ModuleCat.of k Y ⟶ ModuleCat.of k Y := |
| ModuleCat.ofHom (Fin.cases (motive := fun _ => Module.End k Y) (1 : Module.End k Y) X.generator a) |
|
|
| /-- The first arrow family acts by the given representation, including the identity arrow. -/ |
| theorem arrow01_square (a : Fin (d+1)) |
| (ha : ThreeVertexAlgebra.vertex (composition P) 1 * arrow01 P a = arrow01 P a) : |
| (LeftVertexHomology.arrow k (B P) (ThreeVertexAlgebra.vertex (composition P) 0) |
| (ThreeVertexAlgebra.vertex (composition P) 1) (arrow01 P a) ha).app (module X) ≫ |
| (vertexIso X 1).hom = (vertexIso X 0).hom ≫ action X a := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro z |
| change X.projection 1 (arrow01 P a • z.val) = |
| (Fin.cases (motive := fun _ => Module.End k Y) (1 : Module.End k Y) X.generator a) (X.projection 0 z.val) |
| exact congrArg (X.projection 1) (X.arrow01_smul a (show X.Evaluation from z.val)) |
|
|
| /-- The second arrow family has the same specified representation action. -/ |
| theorem arrow12_square (b : Fin (d+1)) |
| (hb : ThreeVertexAlgebra.vertex (composition P) 2 * arrow12 P b = arrow12 P b) : |
| (LeftVertexHomology.arrow k (B P) (ThreeVertexAlgebra.vertex (composition P) 1) |
| (ThreeVertexAlgebra.vertex (composition P) 2) (arrow12 P b) hb).app (module X) ≫ |
| (vertexIso X 2).hom = (vertexIso X 1).hom ≫ action X b := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro z |
| change X.projection 2 (arrow12 P b • z.val) = |
| (Fin.cases (motive := fun _ => Module.End k Y) (1 : Module.End k Y) X.generator b) (X.projection 1 z.val) |
| exact congrArg (X.projection 2) (X.arrow12_smul b (show X.Evaluation from z.val)) |
| end LittleFinitistic.EvaluationVertexDiagram |
|
|
| end |
|
|
| end OAI |
|
|
