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2c7d975 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | 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
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