| import OAI.Algebra.Finitistic.Rectification.VertexHomology |
| import OAI.Algebra.Finitistic.Rectification.VertexDecomposition |
|
|
| namespace OAI |
| noncomputable section |
| open CategoryTheory HomologicalComplex |
| namespace LittleFinitistic.LeftVertexHomology |
| universe u |
| variable (k B : Type u) [Field k] [Ring B] [Algebra k B] |
|
|
| lemma scalar_commute (e : B) (c : k) : e * algebraMap k B c = algebraMap k B c * e := |
| (Algebra.commutes c e).symm |
|
|
| / |
| abbrev vertex (e : B) : ModuleCat.{u} B ⥤ ModuleCat.{u} k := |
| VertexFunctor.functor (algebraMap k B) e (scalar_commute k B e) |
|
|
| / |
| def arrow (ei ej a : B) (ha : ej*a=a) : vertex k B ei ⟶ vertex k B ej where |
| app M := ModuleCat.ofHom { |
| toFun := fun x => ⟨a • (x.val : M), by |
| rw [VertexFunctor.mem_fixed] |
| rw [← mul_smul, ha]⟩ |
| map_add' := fun x y => Subtype.ext (smul_add _ _ _) |
| map_smul' := fun c x => by |
| apply Subtype.ext |
| change a • (algebraMap k B c • (x.val : M)) = |
| algebraMap k B c • (a • (x.val : M)) |
| rw [← mul_smul, ← mul_smul, scalar_commute] } |
| naturality M M' f := by |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro x |
| apply Subtype.ext |
| exact (f.hom.map_smul a x.val).symm |
| |
| instance scalarTower (M : ModuleCat.{u} B) : |
| IsScalarTower k B ((ModuleCat.restrictScalars (algebraMap k B)).obj M) := |
| IsScalarTower.of_algebraMap_smul (fun _ _ => rfl) |
| |
| / |
| def spaceEquiv (e : B) (M : ModuleCat.{u} B) : |
| (vertex k B e).obj M ≃ₗ[k] |
| IdempotentTensor.leftSpace (k := k) e ((ModuleCat.restrictScalars (algebraMap k B)).obj M) where |
| toFun x := ⟨x.val, (VertexFunctor.mem_fixed _ _ _ M x.val).1 x.property⟩ |
| invFun x := ⟨x.val, (VertexFunctor.mem_fixed _ _ _ M x.val).2 x.property⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| / |
| def homologyEquiv (e : B) (he : e*e=e) |
| (X : CochainComplex (ModuleCat.{u} B) ℤ) (n : ℤ) : |
| (((vertex k B e).mapHomologicalComplex (.up ℤ)).obj X).homology n ≃ₗ[k] |
| IdempotentTensor.leftSpace (k := k) e |
| ((ModuleCat.restrictScalars (algebraMap k B)).obj (X.homology n)) := |
| (VertexHomology.iso (algebraMap k B) e he (scalar_commute k B e) X n).toLinearEquiv.trans |
| (spaceEquiv k B e (X.homology n)) |
|
|
| / |
| theorem arrow_homology (ei ej a : B) (hi : ei*ei=ei) (hj : ej*ej=ej) (ha : ej*a=a) |
| (X : CochainComplex (ModuleCat.{u} B) ℤ) (n : ℤ) : |
| homologyMap ((NatTrans.mapHomologicalComplex (arrow k B ei ej a ha) (.up ℤ)).app X) n ≫ |
| (VertexHomology.iso (algebraMap k B) ej hj (scalar_commute k B ej) X n).hom = |
| (VertexHomology.iso (algebraMap k B) ei hi (scalar_commute k B ei) X n).hom ≫ |
| (arrow k B ei ej a ha).app (X.homology n) := |
| VertexHomology.natural (algebraMap k B) ei hi (scalar_commute k B ei) |
| ej hj (scalar_commute k B ej) _ X n |
| end LittleFinitistic.LeftVertexHomology |
|
|
| end |
|
|
| end OAI |
|
|
