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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 | import OAI.Algebra.AuslanderReiten.Resolutions.MapResolutionExt
import OAI.Algebra.AuslanderReiten.Cocycles.CocycleFunctor
namespace OAI
noncomputable section
open CategoryTheory CategoryTheory.Abelian CategoryTheory.Limits
open CochainComplex CochainComplex.HomComplex HomologicalComplex
namespace ArExplicit.ResolutionProducts
universe u v u' v'
variable {C : Type u} [Category.{v} C] [Abelian C] [HasExt.{v} C]
variable {D : Type u'} [Category.{v'} D] [Abelian D] [HasExt.{v'} D]
variable (F : C ⥤ D) [F.Additive] [PreservesFiniteLimits F]
[PreservesFiniteColimits F] [F.PreservesProjectiveObjects]
variable {X Y : C} (P : ProjectiveResolution X) {n : ℕ}
omit [Abelian C] [Abelian D] [HasExt.{v} C] [HasExt.{v'} D] [F.Additive]
[PreservesFiniteLimits F] [PreservesFiniteColimits F] [F.PreservesProjectiveObjects] in
private theorem map_eval_cancel {U V W Z : C} {U' Z' : D}
(a : U' ⟶ F.obj U) (b : U ⟶ V) (f : V ⟶ W) (e : Z ≅ W) (t : Z' ≅ F.obj W) :
(a ≫ F.map ((b ≫ f) ≫ e.inv) ≫ (F.map e.hom ≫ t.inv)) ≫ t.hom =
(a ≫ F.map b) ≫ F.map f := by
simp only [Functor.map_comp, Category.assoc, t.inv_hom_id, Category.comp_id]
rw [← F.map_comp e.inv e.hom, e.inv_hom_id, F.map_id, Category.comp_id]
omit [HasExt.{v} C] [HasExt.{v'} D] in
theorem mappedResolutionCocycle_eq
(x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ)) :
mappedResolutionCocycle F P x =
((fullHomCocycleMap F _ _ _ x).precomp (mapResolutionCochainIso F P).hom).postcomp
((F.mapCochainComplexSingleFunctor 0).hom.app Y) := by
apply Cocycle.equivHomShift.symm.injective
rw [mappedResolutionCocycle, AddEquiv.symm_apply_apply,
Cocycle.equivHomShift_symm_postcomp, Cocycle.equivHomShift_symm_precomp, cocycle_map_hom]
exact (Category.assoc _ _ _).symm
omit [HasExt.{v} C] [HasExt.{v'} D] in
theorem mappedResolutionCocycle_v
(x : Cocycle P.cochainComplex ((CochainComplex.singleFunctor C 0).obj Y) (n : ℤ))
(i j : ℤ) (hij : i+n=j) :
(mappedResolutionCocycle F P x).val.v i j hij =
(mapResolutionCochainIso F P).hom.f i ≫ F.map (x.val.v i j hij) ≫
((F.mapCochainComplexSingleFunctor 0).hom.app Y).f j := by
rw [mappedResolutionCocycle_eq]
change (((Cochain.ofHom _).comp (x.val.map F) (zero_add (n : ℤ))).comp
(Cochain.ofHom _) (add_zero (n : ℤ))).v i j hij = _
simp only [Cochain.comp_zero_cochain_v, Cochain.zero_cochain_comp_v, Cochain.ofHom_v,
Cochain.map_v]
exact Category.assoc _ _ _
theorem map_extMk (f : P.complex.X n ⟶ Y)
(hf : P.complex.d (n+1) n ≫ f = 0) :
(P.extMk f (n+1) rfl hf).mapExactFunctor F =
(mapResolution F P).extMk (F.map f) (n+1) rfl
(by change F.map (P.complex.d (n+1) n) ≫ F.map f = 0
rw [← F.map_comp, hf, F.map_zero]) := by
unfold ProjectiveResolution.extMk
rw [map_extOfCocycle]
apply congrArg (mapResolution F P).extEquivCohomologyClass.symm
apply congrArg CohomologyClass.mk
apply Cocycle.ext
apply (Cochain.toSingleEquiv (p := -(n : ℤ)) (q := 0) (n := (n : ℤ)) (by omega)).injective
rw [Cocycle.toSingleMk_coe, Cochain.toSingleEquiv_toSingleMk]
change (mappedResolutionCocycle F P _).val.v (-(n : ℤ)) 0 (by omega) ≫
(singleObjXSelf (.up ℤ) 0 (F.obj Y)).hom = _
rw [mappedResolutionCocycle_v, Cocycle.toSingleMk_coe, Cochain.toSingleMk_v]
change ((mapResolutionCochainIso F P).hom.f (-(n : ℤ)) ≫
F.map (((P.cochainComplexXIso (-(n : ℤ)) n rfl).hom ≫ f) ≫
(singleObjXSelf (.up ℤ) 0 Y).inv) ≫
((singleMapHomologicalComplex F (.up ℤ) 0).hom.app Y).f 0) ≫
(singleObjXSelf (.up ℤ) 0 (F.obj Y)).hom = _
rw [singleMapHomologicalComplex_hom_app_self]
exact (map_eval_cancel F ((mapResolutionCochainIso F P).hom.f (-(n : ℤ)))
(P.cochainComplexXIso (-(n : ℤ)) n rfl).hom f
(singleObjXSelf (.up ℤ) 0 Y) (singleObjXSelf (.up ℤ) 0 (F.obj Y))).trans
(congrArg (fun g => g ≫ F.map f) (mapResolutionCochainIso_coordinate F P _ n rfl))
end ArExplicit.ResolutionProducts
end
end OAI
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