| import OAI.Algebra.AuslanderReiten.Triangular.TriangularComplete |
| import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace ArExplicit.TriangularInduction |
|
|
| open CategoryTheory |
| open CochainComplex CochainComplex.HomComplex |
|
|
| universe u |
|
|
| variable (k R F : Type u) [Field k] [Ring R] [Algebra k R] |
| variable [AddCommGroup F] [Module k F] [Module R F] [Module Rᵐᵒᵖ F] |
| variable [SMulCommClass k R F] [SMulCommClass R Rᵐᵒᵖ F] |
| variable [IsScalarTower k R F] [IsScalarTower k Rᵐᵒᵖ F] [SMulCommClass k Rᵐᵒᵖ F] |
|
|
| attribute [local instance] moduleCatBaseModule moduleCatBaseTower |
|
|
| def eFunctor : ModuleCat.{u} R ⥤ ModuleCat.{u} (Triangular R F) where |
| obj M := ModuleCat.of _ (TriangularE k R F M) |
| map f := ModuleCat.ofHom (eMap f.hom) |
| map_id M := by apply ModuleCat.hom_ext; exact eMap_id |
| map_comp f g := by apply ModuleCat.hom_ext; exact eMap_comp f.hom g.hom |
|
|
| def jFunctor : ModuleCat.{u} R ⥤ ModuleCat.{u} (Triangular R F) where |
| obj M := ModuleCat.of _ (TriangularJ k R F M) |
| map f := ModuleCat.ofHom (jMap f.hom) |
| map_id M := by apply ModuleCat.hom_ext; exact jMap_id |
| map_comp f g := by apply ModuleCat.hom_ext; exact jMap_comp f.hom g.hom |
|
|
| instance : (eFunctor k R F).Additive where |
| map_add := by |
| intro X Y f g |
| apply ModuleCat.hom_ext |
| apply eHomEquiv.injective |
| ext m |
| rfl |
|
|
| instance : (jFunctor k R F).Additive where |
| map_add := by |
| intro X Y f g |
| apply ModuleCat.hom_ext |
| apply jHomEquiv.injective |
| ext m |
| rfl |
|
|
| instance : (eFunctor k R F).Faithful where |
| map_injective := by |
| intro M N f g h |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro m |
| exact congrArg (fun a : (eFunctor k R F).obj M ⟶ (eFunctor k R F).obj N => |
| (a.hom ⟨m, 0⟩).top) h |
|
|
| instance : (jFunctor k R F).Faithful where |
| map_injective := by |
| intro M N f g h |
| apply ModuleCat.hom_ext |
| apply LinearMap.ext |
| intro m |
| exact congrArg (fun a : (jFunctor k R F).obj M ⟶ (jFunctor k R F).obj N => |
| (a.hom ⟨0, m⟩).bottom) h |
|
|
| instance : (eFunctor k R F).Full where |
| map_surjective := by |
| intro M N g |
| refine ⟨ModuleCat.ofHom (TriangularModule.topMap g.hom), ?_⟩ |
| apply ModuleCat.hom_ext |
| exact eHom_topMap g.hom |
|
|
| instance : (jFunctor k R F).Full where |
| map_surjective := by |
| intro M N g |
| refine ⟨ModuleCat.ofHom (TriangularModule.bottomMap g.hom), ?_⟩ |
| apply ModuleCat.hom_ext |
| exact jHom_bottomMap g.hom |
|
|
| theorem eToJCochain_eq_zero (P Q : CochainComplex (ModuleCat.{u} R) ℤ) (n : ℤ) |
| (x : Cochain (((eFunctor k R F).mapHomologicalComplex _).obj P) |
| (((jFunctor k R F).mapHomologicalComplex _).obj Q) n) : x = 0 := by |
| apply Cochain.ext |
| intro i j hij |
| apply ModuleCat.hom_ext |
| exact hom_E_J_eq_zero (x.v i j hij).hom |
|
|
| end ArExplicit.TriangularInduction |
|
|
| end |
|
|
| end OAI |
|
|
